Why OLS Works in NARDL Models: Trends, Frisch–Waugh–Lovell, and Convergence

NARDL has become a widely used tool in applied time-series econometrics. It allows economists to study whether increases and decreases in an explanatory variable have different short-run and long-run effects, while estimating the resulting dynamic equation by ordinary least squares. This combination of flexibility and ease of estimation helps explain its appeal in research on exchange rates, energy prices, trade, and other asymmetric economic relationships.

Cho (2026) fills a gap between this widespread practice and its econometric justification. He establishes the consistency and limiting distribution of the conventional one-step OLS estimator for a NARDL model with a time trend. The difficulty comes from the positive and negative partial sums used to represent asymmetry: they generate deterministic trends that make the usual normalized regressor matrix singular in the limit.

Cho uses Frisch–Waugh–Lovell transformations to separate those trends from the stochastic fluctuations and derive the behavior of the original estimator. This blog explains the estimation argument in Sections 1–4 of the paper. The main text develops the intuition and works through the principal calculations; eight linked appendices provide the complete derivations. The applied discussion also draws on the paper’s comparison of OLS and two-stage NARDL estimation.

1. The econometric foundation for a widely used estimator

The NARDL framework developed by Shin, Yu, and Greenwood-Nimmo (2014) provides a way to estimate asymmetric relationships without specifying an entirely separate dynamic model for every regime. Positive and negative changes are accumulated separately and enter a common regression. Applied researchers can then estimate different responses to increases and decreases in the same explanatory variable.

Once these partial sums have been constructed, OLS estimates the coefficients of a linear regression. To establish consistency and a limiting distribution, however, the proof must track how the regressors and their cross-products behave as the sample grows. In the model studied by Cho, several regressors share a deterministic time trend. The usual normalization makes their limiting columns proportional, so the matrix that a conventional OLS proof would invert is singular.

The stochastic fluctuations around those trends still distinguish the regressors in a finite sample. Cho derives a representation in which those fluctuations remain in the limiting calculation. His contribution establishes the asymptotic behavior of the original one-step OLS coefficients, complementing earlier work on the NARDL framework and two-stage estimation.

The method builds on the projection arguments of Park and Phillips (1988, 1989) for regressions with integrated processes. Exact FWL identities connect the original regression to regressions with nonsingular limiting matrices. The proof then reconstructs every original coefficient and shows why different estimation errors determine their convergence rates.

For applied work, this gives a theoretical justification for a familiar estimator and a basis for interpreting its precision. A coefficient in the dynamic equation, a long-run slope, and an adjustment speed are related parameters, each with its own role in the asymptotic argument. The distinction becomes visible once the regression is written in error-correction form.

Main takeaway. Cho establishes the large-sample properties of conventional one-step OLS for a NARDL model with a time trend. The proof addresses the deterministic drifts introduced by the positive and negative partial sums.

Calculations: Appendix C identifies the singularity; Appendix D derives the projection argument.

2. From economic asymmetry to a linear regression

For one explanatory variable, split each observed change into its positive and negative parts:

\[\Delta x_t^+=\max(\Delta x_t,0),\qquad \Delta x_t^-=\min(\Delta x_t,0).\]

A change of \(-3\) enters the negative component as \(-3\). Accumulate the two series separately:

\[x_t^+=\sum_{j=1}^{t}\Delta x_j^+,\qquad x_t^-=\sum_{j=1}^{t}\Delta x_j^-.\]

The original cumulative change is their sum, \(x_t-x_0=x_t^++x_t^-\). Keeping the two components separate allows increases and decreases to have different coefficients. If an energy price rises by one unit and subsequently falls by one unit, its original level returns to its starting point. The two partial sums still record a cumulative increase of one unit and a cumulative decrease of one unit. A NARDL model can therefore distinguish the effects of the two movements.

With \(k\) explanatory variables, \(x_t^+\) and \(x_t^-\) are each \(k\times1\) column vectors: each contains one signed partial sum per explanatory variable. Stacking them gives \(\ddot x_t=(x_t^{+\prime},x_t^{-\prime})’\), a \(2k\times1\) column vector. The prime denotes a transpose, which exchanges rows and columns. The double dot is a notation label. The model is nonlinear in the original changes because their sign determines which component changes. Once those components have been constructed, the regression is linear in its unknown coefficients.

The error-correction representation is

\[\begin{aligned} \Delta y_t={}&\rho_*y_{t-1}+\theta_*’\ddot x_{t-1} +\xi_*q_t+\alpha_*+\kappa_*’z_{2t}+e_t,\\ q_t:={}&t-1. \end{aligned}\]

The vector \(z_{2t}\) contains lagged changes in \(y_t\) and the current and lagged positive and negative changes in the explanatory variables. Their coefficients describe short-run dynamics. The lagged levels and the time trend describe the longer-run relationship. A star marks a population parameter. Here \(q_t\) is the time regressor; the unsubscripted \(q\) in NARDL\((p,q)\) is the explanatory-variable lag order.

Counting variables and reading dimensions. A dimension gives the number of rows followed by the number of columns. A \(2k\times1\) vector has \(2k\) rows and one column; a scalar is a single number. The following counts will be used throughout the calculations.

Variable counts used in the vectors and matrices

Symbol	What it counts
\(T\)	Observations used in the regression.
\(k\)	Original explanatory variables, giving \(2k\) positive and negative partial sums.
\(p\), \(q\)	Dependent-variable and explanatory-variable lag orders, respectively.
\(s=(p-1)+2kq\)	Short-run regressors: \(p-1\) lagged outcome changes and \(2kq\) current and lagged signed explanatory-variable changes.
\(d=2k+2\)	Regressors in the drifting block: one lagged outcome, \(2k\) partial sums, and one time trend.
\(K=d+s+1=2k+s+3\)	All regressors: the drifting block, \(s\) short-run regressors, and one constant.

Table notes. Read each row as a count of observations, variables, or regressors. For \(k=1\), \(p=2\), and \(q=1\), there are \(s=1+2=3\) short-run regressors, \(d=4\) drifting regressors, and \(K=8\) regressors altogether. Increasing the sample size \(T\) adds observations while these counts remain fixed for this specification.

In the compact equation, \(\theta_*\) and \(\ddot x_{t-1}\) are both \(2k\times1\), while \(\kappa_*\) and \(z_{2t}\) are both \(s\times1\). Transposing \(\theta_*\) makes it a \(1\times2k\) row; multiplying it by the \(2k\times1\) column \(\ddot x_{t-1}\) adds \(2k\) coefficient–regressor products and produces one scalar. The product \(\kappa_*’z_{2t}\) similarly adds \(s\) products. Every term in the equation therefore has the scalar dimension of \(\Delta y_t\). The coefficients \(\rho_*\), \(\xi_*\), and \(\alpha_*\) are scalars.

The coefficients on lagged partial sums in this dynamic equation are collected in the \(2k\times1\) vector \(\theta_*\). The equilibrium slopes form another \(2k\times1\) vector, \(\beta_*\), obtained by dividing every component by the scalar adjustment coefficient. The equilibrium trend coefficient \(\zeta_*\) is scalar:

\[\beta_*=-\frac{\theta_*}{\rho_*},\qquad \zeta_*=-\frac{\xi_*}{\rho_*},\qquad \rho_*<0.\]

For example, \(\rho_*=-0.4\) and \(\theta_*^+=0.8\) imply an equilibrium slope \(\beta_*^+=2\). The coefficient \(0.8\) appears in the equation for the next change in the outcome; the coefficient \(2\) describes the equilibrium association between the outcome and the positive partial sum.

The long-run relationship is

\[y_{t-1}=\beta_*’\ddot x_{t-1}+\zeta_*q_t+\nu_*+u_{t-1}.\]

The product \(\beta_*’\ddot x_{t-1}\) is scalar, and \(\zeta_*\), \(\nu_*\), and \(u_{t-1}\) are scalars as well. The cointegration error \(u_{t-1}\) measures the outcome’s deviation from this relationship. The intercept \(\nu_*\) centers that error at zero. Substitution into the dynamic equation gives

\[\Delta y_t=\rho_*u_{t-1}+\gamma_*+\kappa_*’z_{2t}+e_t, \qquad \gamma_*:=\alpha_*+\rho_*\nu_*.\]

If the outcome lies two units above its equilibrium relation and \(\rho_*=-0.4\), the adjustment term contributes \(-0.8\) to its next change. The other terms in the equation determine the remainder of that change. This is the economic role of the adjustment coefficient whose estimation error becomes important later.

Asymmetry concerns the equilibrium slopes as well as the short-run coefficients. In a numerical illustration with \(\beta_*^+=2\) and \(\beta_*^-=1\), a one-unit increase in the explanatory variable raises the equilibrium component by two units. A one-unit decrease lowers it by one unit because the negative partial sum changes by \(-1\). These examples describe the model’s arithmetic.

The assumptions used in the paper

Cho’s Assumption 1 maintains cointegration and stable dynamics, a nonzero mean change \(\delta_*=E(\Delta y_t)\), stationary short-run components with suitable moment and dependence conditions, and the required nonsingular covariance matrices. The dynamic innovation has conditional mean zero and constant conditional variance relative to an information set that includes contemporaneous explanatory variables. The time trend is included as a regressor, including when its true coefficient is zero.

Main takeaway. The error-correction model separates adjustment toward equilibrium from short-run dynamics. Its level coefficients \(\theta_*\) and long-run slopes \(\beta_*=-\theta_*/\rho_*\) answer different economic questions, so their estimates and convergence rates must be distinguished.

Calculations and symbol definitions: Appendix A.

3. Why partial sums generate trends

Separating a change into positive and negative components changes the means of the resulting series. Consider a simple illustration in which \(\Delta x_t\) takes the values \(-2,-1,+1,+2\), each with probability one quarter. The original mean is zero. Its positive part takes the values \(0,0,1,2\), while its negative part takes the values \(-2,-1,0,0\). Therefore,

\[ E(\Delta x_t^+)=\frac{0+0+1+2}{4}=\frac34, \qquad E(\Delta x_t^-)=-\frac34. \]

Accumulating the positive component adds an average of \(3/4\) each period. Accumulating the negative component adds an average of \(-3/4\). Thus \(E(x_t^+)=3t/4\) and \(E(x_t^-)=-3t/4\). Their deterministic parts cancel when the two series are added, which is consistent with a mean-zero change in the original variable. The NARDL regression includes them separately, so each enters with its own linear drift.

To distinguish drift from fluctuations, define the mean vector \(\mu_*:=E(\Delta\ddot x_t)\), the centered increments \(s_j=\Delta\ddot x_j-\mu_*\), and their cumulative sum \(m_{t-1}=\sum_{j=1}^{t-1}s_j\). Each is a \(2k\times1\) column vector, with one entry for each signed component. The increment vector \(s_j\) is distinct from the scalar count \(s\) in the table above. Adding the mean back to each increment gives the exact decomposition

\[ \ddot x_{t-1}=\mu_*q_t+m_{t-1}, \qquad q_t=t-1. \]

The first term describes the deterministic trend. The second records the cumulative departures from that trend. Its mean is zero, but its realized value can be substantial.

The size of those departures can be understood from one scalar component. For this illustration, let \(\eta_j\) denote one component of the vector \(s_j\), and let \(\mu\) denote the corresponding scalar mean increment. Suppose the centered increments \(\eta_j\) are independent, each with variance \(\sigma^2\). The variance of their sum equals the sum of their variances because the cross-covariances are zero:

\[ \operatorname{Var}\!\left(\sum_{j=1}^{t}\eta_j\right) =\sum_{j=1}^{t}\operatorname{Var}(\eta_j) =t\sigma^2. \]

Standard deviation is the square root of variance. Consequently,

\[ \operatorname{SD}\!\left(\sum_{j=1}^{t}\eta_j\right) =\sigma\sqrt t. \]

For unit-variance increments, 100 observations give variance 100 and standard deviation 10. With 400 observations, the variance rises to 400 and the standard deviation to 20. Quadrupling the number of increments doubles the typical fluctuation. A deterministic drift, in contrast, quadruples over the same interval.

The relative size of the stochastic fluctuation therefore decreases:

\[ \frac{\sigma\sqrt t}{|\mu|t} =\frac{\sigma}{|\mu|\sqrt t}\longrightarrow0 \qquad(\mu\ne0). \]

The fluctuations grow in absolute size while shrinking relative to the trend. The paper obtains the corresponding square-root scale under its assumptions on dependence; independence is used here to make the variance calculation transparent.

This distinction also explains the powers used to normalize the regression matrix. Near the end of a sample of length \(T\), a trend observation has order \(T\), its square has order \(T^2\), and summing over \(T\) observations gives order \(T^3\). A cumulative fluctuation has scale \(\sqrt T\), so its square has scale \(T\), and its sum of squares has order \(T^2\). The normalization that retains the trend contribution consequently reduces the fluctuation contribution by an additional factor of \(1/T\).

Main takeaway. Separately accumulating positive and negative changes creates deterministic trends even when the original change has mean zero. These trends grow linearly with time, while the centered cumulative fluctuations have a square-root scale.

The variance calculation, numerical examples, and exact sums: Appendix B.

4. Where the usual OLS argument breaks down

The difficulty becomes concrete in the preceding example. The two partial sums have the representations \(x_{t-1}^+=(3/4)q_t+m_{t-1}^+\) and \(x_{t-1}^-=-(3/4)q_t+m_{t-1}^-\). Retaining only their dominant deterministic parts would give

\[ \theta_*^+x_{t-1}^+ +\theta_*^-x_{t-1}^-+\xi_*q_t \approx \left(\frac34\theta_*^+-\frac34\theta_*^-+\xi_*\right)q_t. \]

At this leading scale, the three regressors explain the same time pattern. Only the coefficient combination in parentheses is distinguishable. Different values of the three coefficients can produce the same combination. The fluctuations \(m_{t-1}^+\) and \(m_{t-1}^-\) provide additional variation that can distinguish their separate effects in the observed sample.

The outcome also contains a deterministic drift:

\[ y_{t-1} =\delta_*q_t+\sum_{j=1}^{t-1}(\Delta y_j-\delta_*), \qquad \delta_*=E(\Delta y_t), \]

where initial values are suppressed. Collect the trending regressors and their drift coefficients:

\[ v_t= \begin{pmatrix} y_{t-1}\\ \ddot x_{t-1}\\ q_t \end{pmatrix}, \qquad a= \begin{pmatrix} \delta_*\\ \mu_*\\ 1 \end{pmatrix}. \]

Both \(v_t\) and \(a\) are \(d\times1\) column vectors, where \(d=1+2k+1=2k+2\): their displayed middle block contains \(2k\) entries. The vector \(a\) contains one drift coefficient for each regressor in \(v_t\). Its last entry is one because the explicit time trend is exactly \(1\times q_t\). Thus, \(aq_t\) is also \(d\times1\) and contains the deterministic trend in every regressor. Define the \(d\times1\) vector \(r_t\) as the departures from those trends:

\[ v_t=aq_t+r_t, \qquad r_t= \begin{pmatrix} y_{t-1}-\delta_*q_t\\ \ddot x_{t-1}-\mu_*q_t\\ 0 \end{pmatrix}. \]

The last departure is zero because the explicit trend has no random component. The other entries contain the fluctuations discussed in Section 3. Under the paper’s functional limit assumptions, their stochastic magnitude is at most \(\sqrt T\) at times proportional to sample size, whereas a nonzero deterministic drift grows like \(T\).

Multiplying the trend components. The transpose \(a’\) is a \(1\times d\) row vector. Multiplying the \(d\times1\) column \(a\) by that row produces the \(d\times d\) matrix \(aa’\): each row is paired with every column. Since \(q_t\) is a scalar,

\[ (aq_t)(aq_t)’=(aq_t)(q_ta’)=q_t^2aa’. \]

The matrices \(v_tv_t’\) and \(aa’\), their normalized sums, and the limiting matrix \(aa’/3\) all have dimension \(d\times d\). The matrix \(aa’\) contains all pairwise products of the drift coefficients. Its entry in row \(i\), column \(j\) is \(a_i a_j\), with \(i,j=1,\ldots,d\). For example, multiplying the positive partial sum’s trend \(\mu_*^+q_t\) by the explicit trend \(1\times q_t\) gives \(\mu_*^+q_t^2\).

Accounting for the fluctuations. The observed cross-product also contains the departures \(r_t\). For any two components, write \(v_{it}=a_iq_t+r_{it}\) and \(v_{jt}=a_jq_t+r_{jt}\). Expanding their product gives four terms:

\[ v_{it}v_{jt} =a_i a_jq_t^2 +a_iq_tr_{jt} +a_jq_tr_{it} +r_{it}r_{jt}. \]

The first is the product of two deterministic trends. The middle two combine a trend with a fluctuation. The last multiplies two fluctuations. Under the paper’s functional limit assumptions, \(\sum_t r_{it}^2\) has at most order \(T^2\), as explained in Appendix B, while \(\sum_t q_t^2\) has order \(T^3\).

The mixed terms can be bounded using the Cauchy–Schwarz inequality, which bounds a sum of products by the product of the two square-root sums of squares:

\[ \left|\sum_{t=1}^{T}q_tr_{it}\right| \leq \left(\sum_{t=1}^{T}q_t^2\right)^{1/2} \left(\sum_{t=1}^{T}r_{it}^2\right)^{1/2}. \]

The first factor has order \(T^{3/2}\), and the second has at most order \(T\). Their product therefore has at most order \(T^{5/2}\). After division by \(T^3\), this bound has at most order \(T^{-1/2}=1/\sqrt T\), which tends to zero. The same inequality bounds \(\left|\sum_t r_{it}r_{jt}\right|\) by a product of two terms of at most order \(T\). That product has at most order \(T^2\), so division by \(T^3\) leaves at most order \(1/T\), which also tends to zero.

Consequently, only the product of the deterministic trends survives this normalization. In matrix form,

\[ \frac1{T^3}\sum_{t=1}^{T}v_tv_t’ =aa’\left[\frac1{T^3}\sum_{t=1}^{T}q_t^2\right]+\mathcal{E}_T, \qquad \mathcal{E}_T\xrightarrow{p}0. \]

Here the \(d\times d\) matrix \(\mathcal{E}_T\) collects the normalized mixed and fluctuation products just examined. Its entries are random because they depend on the observed sample. The symbol \(\xrightarrow{p}\) means convergence in probability. For an entry \(\mathcal{E}_{ij,T}\), choose any fixed positive tolerance \(\varepsilon\). The statement is

\[ \Pr\!\left(|\mathcal{E}_{ij,T}|>\varepsilon\right)\longrightarrow0. \]

The probability refers to samples generated by the model. As sample size grows, an appreciable remaining contribution becomes less likely. For example, a tolerance of \(0.01\) means that the probability of a remainder exceeding \(0.01\) in absolute value tends to zero. The same claim holds for every smaller fixed positive tolerance.

This disappearance concerns the products after division by \(T^3\). The cumulative fluctuations themselves can keep growing, and their unnormalized cross-products can keep growing too. Their growth is slower than the denominator, so they become negligible on the scale used for this matrix limit.

Obtaining the factor \(1/3\). Because \(q_t=t-1\), its squared values are \(0^2,1^2,\ldots,(T-1)^2\). Their exact sum is derived step by step in Appendix B.3. Substituting that formula and dividing each factor by the appropriate power of \(T\) gives

\[ \begin{aligned} \frac1{T^3}\sum_{t=1}^{T}q_t^2 &=\frac{(T-1)T(2T-1)}{6T^3}\\ &=\frac16\left(\frac{T-1}{T}\right)\left(\frac{2T-1}{T}\right)\\ &=\frac16\left(1-\frac1T\right)\left(2-\frac1T\right)\\ &\longrightarrow\frac16\times1\times2=\frac13. \end{aligned} \]

The vector \(a\) contains fixed population drift coefficients, so \(aa’\) can be taken outside the sum. Combining the limit \(1/3\) with the vanishing remainder gives

\[ \frac1{T^3}\sum_{t=1}^{T}v_tv_t’ \xrightarrow{p}\frac13aa’. \]

This explains each entry of the limiting matrix: multiply the drift coefficient of its row by the drift coefficient of its column, then divide by three. The trend’s diagonal entry is \(1\times1/3=1/3\). In the scalar illustration, the entry joining the positive partial sum and the trend is \((3/4)\times1/3=1/4\).

To read the probability statement concretely, call the normalized positive-partial-sum–trend entry \(C_T\). In the example with \(\mu_*^+=3/4\),

\[ C_T:=\frac1{T^3}\sum_{t=1}^{T}x_{t-1}^+q_t \xrightarrow{p}\frac14. \]

Different samples generally produce different values of \(C_T\). Convergence to \(1/4=0.25\) means that these values become concentrated around \(0.25\). Taking a tolerance of \(0.02\) gives

\[ \Pr\!\left(|C_T-0.25|>0.02\right)\longrightarrow0. \]

Thus, the chance of a value outside \([0.23,0.27]\) tends to zero. The numbers \(0.02\), \(0.23\), and \(0.27\) illustrate the meaning of the limit; they are not estimated confidence bounds. The result also applies to any narrower fixed tolerance around \(0.25\). The deterministic trend–trend entry needs only an ordinary arrow because its value depends on \(T\) alone; entries involving observed partial sums require a probability statement.

Why this matrix has rank one. Rank counts the number of linearly independent columns. Column \(j\) of \(aa’/3\) is the vector \(a\) multiplied by the single number \(a_j/3\):

\[ \text{column }j\text{ of }\frac13aa’=\frac{a_j}{3}\,a. \]

In particular, the last column is \(a/3\), because the last entry of \(a\) is one. Every other column is its own drift coefficient times that last column. The last column is nonzero, and all the columns lie along this one direction. There is therefore exactly one independent column.

The numerical example makes the dependence visible. With one original explanatory variable, \(k=1\) and the full drifting block has \(d=4\) entries. For the following calculation, restrict attention to its two partial sums and explicit trend, whose drift coefficients are \(3/4\), \(-3/4\), and \(1\). Their \(3\times1\) drift vector multiplied by its \(1\times3\) transpose gives this \(3\times3\) block:

\[ \frac13 \begin{pmatrix}3/4\\-3/4\\1\end{pmatrix} \begin{pmatrix}3/4&-3/4&1\end{pmatrix} = \begin{pmatrix} 3/16&-3/16&1/4\\ -3/16&3/16&-1/4\\ 1/4&-1/4&1/3 \end{pmatrix}. \]

The first column equals \(3/4\) times the third; the second equals \(-3/4\) times the third. Thus these three columns contain only one independent direction. The same reasoning applies when the lagged outcome is included: its limiting column is \(\delta_*\) times the trend column.

This is the source of the singularity. At the \(T^3\) scale, the normalized matrix retains the common deterministic trend. The smaller random fluctuations that help distinguish the regressors in the observed sample disappear from this limit. Adding the constant and stationary regressors preserves the proportionality among the drifting-regressor columns throughout the full normalized matrix; Appendix C displays those additional blocks.

A singular limit can coexist with a finite-sample inverse. For example,

\[ B_T= \begin{pmatrix}1&1\\1&1+1/T\end{pmatrix}, \qquad \det(B_T)=\frac1T>0. \]

Here \(B_T\) is a \(2\times2\) illustrative matrix, and its determinant is a scalar. It is invertible for every finite positive \(T\), although its limit has identical columns. The NARDL problem likewise concerns the limiting normalized matrix; it does not require exact collinearity in the observed regression.

Let \(z_t\) denote the complete \(K\times1\) regressor vector, where \(K=2k+s+3\). Bold \(\boldsymbol\alpha_*\) is the corresponding \(K\times1\) vector of original coefficients; scalar \(\alpha_*\) denotes the intercept alone. Stacking the \(T\) observation rows \(z_t’\) forms a \(T\times K\) design matrix. Its cross-product matrix is \(\sum_t z_tz_t’\), with dimension \(K\times K\). The regressor–error sum \(\sum_t z_te_t\) is \(K\times1\), because each error \(e_t\) is scalar. Multiplying this vector by the \(K\times K\) inverse gives the \(K\times1\) coefficient-error vector in the exact OLS identity:

\[ \widehat{\boldsymbol\alpha}_T-\boldsymbol\alpha_* = \left(\sum_{t=1}^{T}z_tz_t’\right)^{-1} \sum_{t=1}^{T}z_te_t. \]

The usual asymptotic argument would normalize the two expressions on the right and replace the normalized inverse matrix by the inverse of its limit. That step fails here because the limiting matrix has no inverse. Establishing the estimator’s behavior therefore requires retaining the smaller stochastic terms before taking the limit. Cho’s transformations provide a representation in which those terms survive at their appropriate scales.

Main takeaway. The initial normalization leaves several regressor columns carrying only the same deterministic trend, making the limiting matrix singular. The observed regression can still be estimable because the smaller stochastic fluctuations distinguish its columns.

Every matrix entry, its normalizer, and a finite-sample example: Appendix C.

5. What Frisch–Waugh–Lovell accomplishes

The difficulty comes from several regressors carrying the same large trend. To see their separate contributions, Cho rewrites the columns of the regression. The fitted relationship stays the same, but the new columns distinguish the common trend from the smaller fluctuations around it. The Frisch–Waugh–Lovell argument establishes the exact correspondence between the two sets of coefficients.

Start with the lagged outcome, \(y_{t-1}\). Regress it on the partial sums, the time trend, and a constant. This gives a fitted part and a residual:

\[ y_{t-1}=\widetilde\beta_T’\ddot x_{t-1}+\widetilde\zeta_Tq_t+\widetilde\nu_T+\widetilde u_{t-1}. \]

Here \(\widetilde\beta_T\) is \(2k\times1\), while \(\widetilde\zeta_T\), \(\widetilde\nu_T\), and \(\widetilde u_{t-1}\) are scalars. The auxiliary regression uses \(2k\) partial-sum columns, one trend, and one constant, so its design matrix is \(T\times d\), with \(d=2k+2\). The fitted part records what those regressors jointly explain in the observed sample. The residual records what remains. Under the maintained cointegrating model, it estimates the deviation from the long-run relationship. It has the scale of a stationary variable, even though the variables used to construct it contain trends.

Now replace the column \(y_{t-1}\) in the dynamic regression by the column \(\widetilde u_{t-1}\). Keep the partial sums, trend, constant, and every short-run regressor. Nothing needed to reconstruct the old column has been discarded: its fitted part is made entirely from columns still present. Conversely, the displayed identity reconstructs the residual from the original columns. The two regressions can therefore produce exactly the same fitted values, with their coefficients expressed differently.

Keeping all the controls matters. The auxiliary residual is orthogonal to the partial sums, trend, and constant used to construct it. It need not be orthogonal to the short-run regressors. Those variables must remain in the dynamic regression and be estimated jointly. Although the proof uses successive regressions, this augmented representation reproduces the original one-step OLS estimator. It is distinct from adopting the separate two-step NARDL estimation procedure discussed later.

The partial sums themselves still carry large deterministic drifts. Cho therefore projects each partial-sum column on the time column \(q_t=t-1\), and subtracts the fitted line:

\[ \ddot\mu_T=\frac{\sum_tq_t\ddot x_{t-1}}{\sum_tq_t^2}, \qquad \ddot m_{t-1}=\ddot x_{t-1}-\ddot\mu_Tq_t. \]

The numerator in \(\ddot\mu_T\) is a \(2k\times1\) vector, and its denominator is the scalar sum of squared trend values. Thus \(\ddot\mu_T\) is \(2k\times1\), with one estimated drift slope per partial-sum series. The residual \(\ddot m_{t-1}\) is also \(2k\times1\). It retains the cumulative stochastic fluctuations around those fitted drifts. Removing a deterministic line does not remove the accumulated random shocks: these detrended partial sums still have integrated stochastic variation.

This particular projection uses the trend alone, so its fitted line goes through the origin. A constant is retained separately in the subsequent regressions. The resulting condition is \(\sum_tq_t\ddot m_{t-1}=0\), which says that the residual has zero sample cross-product with the trend. It does not say that its sample mean is zero. Demeaning would subtract an average; this operation subtracts a different fitted value, \(\ddot\mu_Tq_t\), at each date. Including an intercept in this projection would require different coefficient bookkeeping.

The transformed regression contains an estimated stationary equilibrium error, detrended cumulative processes, and a deterministic trend. The table summarizes their column normalizations; the calculations below explain each row.

Scales of the transformed regression columns

Regressor after transformation	Order of the sum of squared entries	Divide every entry by
Estimated cointegration residual	\(T\)	\(\sqrt T\)
Detrended partial sum	\(T^2\)	\(T\)
Time trend	\(T^3\)	\(T^{3/2}\)
Constant and stationary controls	\(T\)	\(\sqrt T\)

Table notes. Here \(T\) is the number of observations used in the regression. A “column” is the full sequence of observations for one regressor. For example, the time-trend column is \((0,1,\ldots,T-1)’\). For a generic scalar regressor \(w_t\), its sum of squared entries is \(\sum_{t=1}^{T}w_t^2\). This sum occupies that regressor’s diagonal position in the OLS cross-product matrix.

The middle column reports how this sum grows as observations are added. An entry \(T^2\), for example, says that the sum has growth order \(T^2\); it does not say that its observed value is exactly \(T^2\). Fixed scale factors and random variation remain in the normalized quantity.

Why the final column takes a square root. If each observation \(w_t\) is divided by a divisor \(\lambda_T\), the sum of squares is divided by \(\lambda_T^2\):

\[ \sum_{t=1}^{T}\left(\frac{w_t}{\lambda_T}\right)^2 =\frac{1}{\lambda_T^2}\sum_{t=1}^{T}w_t^2. \]

We choose \(\lambda_T\) so that the denominator matches the growing power in the sum of squares. A sum growing like \(T\) calls for \(\lambda_T=\sqrt T\); one growing like \(T^2\) calls for \(\lambda_T=T\); and one growing like \(T^3\) calls for \(\lambda_T=T^{3/2}\). This explains the relationship between the last two columns of the table.

Estimated cointegration residual. The first auxiliary regression subtracts the fitted long-run relationship from \(y_{t-1}\). Its residual \(\widetilde u_{t-1}\) estimates the stationary deviation from equilibrium. Under the paper’s results, the generated residual has a typical magnitude that remains on a fixed scale as \(T\) increases. Squaring preserves that fixed scale; adding \(T\) squared observations gives a sum of order \(T\). Dividing every residual by \(\sqrt T\) therefore divides that sum by \(T\).

Detrended partial sum. Subtracting a fitted deterministic line from a partial sum leaves its accumulated random fluctuations. Each component of \(\ddot m_{t-1}\) still has stochastic magnitude \(\sqrt T\) at times proportional to \(T\). Its square has magnitude \(T\), and adding the squared observations produces a sum of order \(T^2\). This gives the divisor \(T\). Detrending and forming the cointegration residual have different effects: the former removes a time pattern, while the latter combines cointegrated variables to recover their stationary equilibrium error.

The divisor must account for all the observations being added. If we divided a detrended partial-sum column by only \(\sqrt T\), its sum of squares would be divided by \(T\). Since the original sum has order \(T^2\), the result would still grow like \(T\). Dividing the column by \(T\) divides its sum of squares by \(T^2\), giving the finite stochastic scale needed in the limit calculation.

Time trend. The trend value \(q_t=t-1\) has magnitude proportional to \(T\) at a fixed fraction of the sample. Squaring gives magnitude \(T^2\), and the exact sum grows like \(T^3/3\), as shown in Section 4. The column divisor is therefore \(\sqrt{T^3}=T^{3/2}\). Its normalized sum of squares tends to \(1/3\).

Constant and stationary controls. The constant column contains \(T\) ones, whose squares add up to exactly \(T\). Dividing every one by \(\sqrt T\) makes the sum of squares exactly one. Under the maintained moment and law-of-large-numbers conditions, a stationary control also has a sum of squares growing in proportion to \(T\), giving the same divisor. Its observations can take many values; their typical magnitude remains on a fixed scale as the sample grows.

A numerical reading of the final column. With \(T=100\), divide each estimated cointegration residual by \(10\), each detrended partial-sum observation by \(100\), each trend observation by \(1000\), and each constant or stationary-control observation by \(10\). These are the theoretical column rescalings used to analyze the OLS matrix. Direct OLS can still be computed from the original model.

After scaling, the columns’ sums of squares remain on finite scales. The normalized sums involving cumulative processes generally have random limits. Lemma 2 establishes that the full transformed cross-product matrix has a nonsingular limit, allowing its inverse to enter the asymptotic argument. The original coefficient estimates’ convergence rates also depend on the error terms and the coefficient transformations developed next.

Main takeaway. The FWL transformations preserve the fitted regression while separating the equilibrium error, cumulative fluctuations, and time trend. Giving each type of column its own normalization retains the variation needed for a nonsingular limiting matrix.

Exact FWL proof: Appendix D. Detrending and coefficient adjustments: Appendix E.

6. Recovering every original coefficient

After detrending, the auxiliary long-run regression has the exact fitted-value decomposition

\[ y_{t-1}=\ddot\beta_T’\ddot m_{t-1}+\ddot\vartheta_Tq_t+\ddot\nu_T+\ddot u_{t-1}. \]

The vectors \(\ddot\beta_T\) and \(\ddot m_{t-1}\) are \(2k\times1\); the trend coefficient \(\ddot\vartheta_T\), intercept \(\ddot\nu_T\), and residual \(\ddot u_{t-1}\) are scalars. The long-run regression’s residual is unchanged: \(\ddot u_{t-1}=\widetilde u_{t-1}\). Its slope coefficients and intercept are also unchanged. Its trend coefficient absorbs the fitted trends moved out of the partial sums; the correction \(\ddot\mu_T’\ddot\beta_T\) is a scalar product of a \(1\times2k\) row and a \(2k\times1\) column:

\[ \ddot\beta_T=\widetilde\beta_T, \qquad \ddot\nu_T=\widetilde\nu_T, \qquad \widetilde\zeta_T=\ddot\vartheta_T-\ddot\mu_T’\ddot\beta_T. \]

The transformed dynamic regression is then

\[ \Delta y_t=\ddot\rho_T\ddot u_{t-1} +\ddot\eta_T’\ddot m_{t-1} +\ddot\varsigma_Tq_t+\ddot\gamma_T +\ddot\kappa_T’z_{2t}+\ddot e_t. \]

In this dynamic regression, \(\ddot\eta_T\) is a \(2k\times1\) coefficient vector and \(\ddot\kappa_T\) is \(s\times1\). The coefficients \(\ddot\rho_T\), \(\ddot\varsigma_T\), and \(\ddot\gamma_T\) are scalars, as are the residuals \(\ddot u_{t-1}\) and \(\ddot e_t\). There are still \(K\) regressor columns, giving a \(T\times K\) design matrix. The scalar coefficients \(\ddot\vartheta_T\) and \(\ddot\varsigma_T\) belong to different equations: the first describes the trend in the fitted long-run relationship; the second describes the trend in this dynamic regression.

To recover the original OLS coefficients, substitute the definitions of \(\ddot u_{t-1}\) and \(\ddot m_{t-1}\) into the last equation and collect the terms multiplying each original regressor. Proposition 4 gives:

\[\begin{aligned} \widehat\rho_T&=\ddot\rho_T,\\ \widehat\theta_T&=\ddot\eta_T-\ddot\rho_T\ddot\beta_T,\\ \widehat\xi_T&=\ddot\varsigma_T-\ddot\mu_T’\ddot\eta_T-\ddot\rho_T\widetilde\zeta_T,\\ \widehat\alpha_T&=\ddot\gamma_T-\ddot\rho_T\ddot\nu_T,\\ \widehat\kappa_T&=\ddot\kappa_T. \end{aligned}\]

Consider one coordinate \(j\) of the partial-sum vector. The estimated disequilibrium contains the term \(-\ddot\beta_{T,j}\ddot x_{t-1,j}\). Multiplying disequilibrium by \(\ddot\rho_T\) therefore contributes \(-\ddot\rho_T\ddot\beta_{T,j}\) to the coefficient on that partial sum. The transformed dynamic equation also contributes its direct coefficient \(\ddot\eta_{T,j}\). Adding both contributions gives \(\widehat\theta_{T,j}\).

For example, suppose the estimates are \(\ddot\beta_{T,j}=2\), \(\ddot\rho_T=-0.4\), and \(\ddot\eta_{T,j}=0.03\). Then

\[ \widehat\theta_{T,j}=0.03-(-0.4)(2)=0.03+0.80=0.83. \]

The contribution through estimated disequilibrium is \(0.80\); the additional partial-sum contribution is \(0.03\). The value \(2\) is an auxiliary estimate of a long-run slope. The value \(0.83\) is the reconstructed coefficient in the original dynamic equation. At the population level, the corresponding parameters satisfy \(\theta_{*,j}=-\rho_*\beta_{*,j}\).

The trend row has two corrections because both transformations moved terms into the time column. Subtracting \(\ddot\mu_T’\ddot\eta_T\) reverses the drift removed from the partial sums. Subtracting \(\ddot\rho_T\widetilde\zeta_T\) reverses the explicit trend removed when constructing disequilibrium. The intercept similarly subtracts \(\ddot\rho_T\ddot\nu_T\). The lagged-outcome and short-run coefficients require no adjustment.

This example also explains why knowing the precision of the auxiliary long-run slope is insufficient. The reconstructed coefficient depends on the adjustment estimate and the auxiliary dynamic coefficient as well. An error in any of these estimated quantities can change the final value. The next section follows these separate errors and determines which contributions remain after scaling by the sample size.

With full sample rank, every equality above is exact. In matrix notation they read \(\widehat{\boldsymbol\alpha}_T=\mathbf T_T\ddot\tau_T\). Here \(\ddot\tau_T=(\ddot\rho_T,\ddot\eta_T’,\ddot\varsigma_T,\ddot\gamma_T,\ddot\kappa_T’)’\) stacks the transformed coefficients. Its five displayed blocks contain \(1\), \(2k\), \(1\), \(1\), and \(s\) entries, respectively, so it is a \(K\times1\) vector. The transformation matrix \(\mathbf T_T\) is \(K\times K\); multiplying it by \(\ddot\tau_T\) gives the \(K\times1\) original coefficient vector. Both the transformation matrix and the transformed estimates depend on the sample. Their joint uncertainty must therefore be followed through these identities to obtain the limiting behavior of the original OLS estimator.

Main takeaway. Every original OLS coefficient can be recovered exactly from the transformed coefficients and the auxiliary estimates. These identities allow the probability limits derived in the transformed regression to describe the estimator used in applied work.

Row-by-row reconstruction, including the trend and intercept: Appendix E.

7. From exact algebra to probability limits

The reconstruction formulas identify how the original OLS estimates depend on the auxiliary estimates. Lemmas 3 and 4 establish how their estimation errors behave as the sample grows. Their central concern is that the transformed regressors themselves contain estimated quantities: the fitted drift enters the detrended partial sums, and the estimated long-run coefficients enter the cointegration residual.

Consider first the long-run regression. Substituting \(\ddot x_{t-1}=\ddot m_{t-1}+q_t\ddot\mu_T\) into the population relationship gives

\[ y_{t-1}=\beta_*’\ddot m_{t-1} +\underbrace{(\zeta_*+\ddot\mu_T’\beta_*)}_{\vartheta_{T*}}q_t +\nu_*+u_{t-1}. \]

The coefficient required on the transformed trend is therefore \(\vartheta_{T*}=\zeta_*+\ddot\mu_T’\beta_*\). It depends on the sample because the fitted drift defines the transformed regressors. A different estimated drift changes the transformed trend coefficient by the corresponding long-run-slope-weighted amount. The underlying population slopes \(\beta_*\) and trend coefficient \(\zeta_*\) remain fixed.

Lemma 3 describes the errors around these targets. Let \(\mathcal L_1\) denote the \(2k\times1\) limiting slope-error vector, \(\mathcal L_2\) the scalar limiting transformed-trend error, and \(\mathcal L_3\) the scalar limiting intercept error. Stacking them gives a \(d\times1\) joint random vector \(\mathcal L\), with \(d=2k+2\). The display below has three blocks, and its first block contains \(2k\) entries. Both sides therefore have \(d\) entries. The symbol \(\Rightarrow\) denotes convergence in distribution:

\[ \begin{pmatrix} T(\ddot\beta_T-\beta_*)\\ T^{3/2}(\ddot\vartheta_T-\vartheta_{T*})\\ \sqrt T(\ddot\nu_T-\nu_*) \end{pmatrix} \Rightarrow \begin{pmatrix}\mathcal L_1\\\mathcal L_2\\\mathcal L_3\end{pmatrix}. \]

Each multiplier indicates the scale needed to preserve a nonzero fluctuation. The slope error is of order \(T^{-1}\); the intercept error is of order \(T^{-1/2}\). The transformed-trend error is of order \(T^{-3/2}\) around its sample-dependent target. The unscaled errors approach zero, establishing consistency, while their scaled limits describe the remaining uncertainty.

The symbols \(\mathcal L_j\) describe the limiting distributions of errors across samples generated by the model. For the long-run slope, two statements work together:

\[ \ddot\beta_T\xrightarrow{p}\beta_*, \qquad T(\ddot\beta_T-\beta_*)\Rightarrow\mathcal L_1. \]

The first says that the probability of an estimation error larger than any fixed tolerance tends to zero. The second magnifies that shrinking error by \(T\), allowing its distribution to retain a nontrivial shape. Imagine comparing many possible samples of the same length: the unscaled slope errors become more concentrated around zero as that length increases, while the errors multiplied by \(T\) approach the distribution described by \(\mathcal L_1\). This is how consistency and a random limiting distribution can hold together.

The centering in the middle row matters. The fixed limit of \(\vartheta_{T*}\) is \(\vartheta_*:=\zeta_*+\mu_*’\beta_*\), and their difference is

\[ \vartheta_{T*}-\vartheta_* =\beta_*'(\ddot\mu_T-\mu_*). \]

This difference is generally of order \(T^{-1/2}\). Thus the \(T^{3/2}\) result measures how closely the transformed regression estimates its sample-dependent trend coefficient. A comparison with the fixed \(\vartheta_*\) must also include drift-estimation uncertainty.

Returning to the original trend coefficient makes the role of slope estimation explicit. Proposition 2 gives the following exact cancellation:

\[ \widetilde\zeta_T-\zeta_* =(\ddot\vartheta_T-\vartheta_{T*}) -\ddot\mu_T'(\ddot\beta_T-\beta_*). \]

The first term has order \(T^{-3/2}\), whereas the second has order \(T^{-1}\). The original explicit-trend estimate therefore inherits the larger slope-error contribution. Multiplying by \(T\) removes the first term in the limit and gives \(T(\widetilde\zeta_T-\zeta_*)\Rightarrow-\mu_*’\mathcal L_1\). This explains why the original and transformed trend coefficients require different normalizations.

Lemma 4 then follows the long-run estimation errors into the dynamic regression. Write those errors as \(b_T=\ddot\beta_T-\beta_*\), \(v_T=\ddot\vartheta_T-\vartheta_{T*}\), and \(n_T=\ddot\nu_T-\nu_*\). The slope-error vector \(b_T\) is \(2k\times1\); the trend and intercept errors \(v_T\) and \(n_T\) are scalars. This scalar \(v_T\) is distinct from the \(d\times1\) observation vector \(v_t\) in Section 4. The product \(b_T’\ddot m_{t-1}\) is scalar, so every term in the following residual identity is scalar:

\[ u_{t-1}=\ddot u_{t-1}+b_T’\ddot m_{t-1}+v_Tq_t+n_T. \]

For example, overestimating a long-run slope subtracts too much of its regressor when constructing \(\ddot u_{t-1}\). The retained regressor must compensate for that error in the dynamic equation. Substituting the identity above into \(\Delta y_t=\rho_*u_{t-1}+\gamma_*+\kappa_*’z_{2t}+e_t\) produces

\[ \Delta y_t=\rho_*\ddot u_{t-1} +\rho_*b_T’\ddot m_{t-1} +\rho_*v_Tq_t+\gamma_*+\rho_*n_T +\kappa_*’z_{2t}+e_t. \]

The appropriate sample-dependent targets are consequently \(\rho_*b_T\) for the retained level coefficients, \(\rho_*v_T\) for the trend, and \(\gamma_*+\rho_*n_T\) for the intercept. Their first-stage errors shrink, but they remain relevant at the scales used to derive the distributions.

As a scalar illustration, suppose the slope error is \(b_T=0.02\) and \(\rho_*=-0.5\). The generated residual subtracts an extra \(0.02\) times the detrended regressor. Multiplication by the adjustment coefficient introduces an extra positive contribution of \(0.01\) times that regressor. The coefficient \(\rho_*b_T=-0.01\) on the retained level offsets it. This compensation is the reason for the centering term in Lemma 4.

Two blocks of Lemma 4 are especially useful for understanding the main result. Denote the scalar limiting adjustment-coefficient error by \(\mathcal D_1\) and the \(2k\times1\) limiting centered level-coefficient error vector by \(\mathcal D_2\). They are the first two blocks of the full \(K\times1\) dynamic-limit vector \(\mathcal D\). Its remaining blocks concern the scalar trend coefficient, scalar intercept, and \(s\times1\) short-run coefficient vector. Then

\[ \sqrt T(\ddot\rho_T-\rho_*)\Rightarrow\mathcal D_1, \qquad T\{\ddot\eta_T-\rho_*b_T\}\Rightarrow\mathcal D_2. \]

The second expression centers the estimate at the coefficient required by the generated-regressor equation. Centering directly at zero would retain the additional first-stage term. The same reasoning governs the transformed trend and intercept. The limiting quantities are jointly determined by the observations, so their dependence must be preserved when reconstructing OLS.

After this centering, the primitive dynamic errors have a mixed-normal limit: conditional on the relevant limiting regressor information, their distribution is Gaussian with a covariance that can depend on that information. Lemma 3’s static long-run errors generally have a different, nonstandard limit. The next step shows how the reconstruction removes their contribution from the leading distribution of the original estimator.

Both lemmas obtain their limits from normalized regressor cross-products and regressor–error products. The transformed regressions retain the accumulated fluctuations at their appropriate scale, so their limiting moment matrices can be inverted. Combining these inverse matrices with the corresponding score limits gives the joint coefficient distributions. The Brownian expressions in the paper specify those random limits; the centering identities above explain which estimation errors they describe.

Main takeaway. Estimating the long-run relationship also changes the regressors used in the dynamic equation. The proof follows those first-stage errors jointly with the dynamic-regression errors and keeps the appropriate centering explicit.

Lemma 3 and its centering: Appendix F. Lemma 4 and generated-regressor uncertainty: Appendix G.

8. Why the convergence rate depends on the coefficient

The equation-level coefficient combines the estimated equilibrium slope with the estimated speed of adjustment. Proposition 4 gives \(\widehat\theta_T=\ddot\eta_T-\ddot\rho_T\ddot\beta_T\), while its population value is \(\theta_*=-\rho_*\beta_*\). Expanding the product separates three sources of error:

\[ \begin{aligned} \widehat\theta_T-\theta_*={}&-\beta_*(\ddot\rho_T-\rho_*)\\ &+\{\ddot\eta_T-\rho_*(\ddot\beta_T-\beta_*)\}\\ &-(\ddot\rho_T-\rho_*)(\ddot\beta_T-\beta_*). \end{aligned} \]

Each term in this expansion is a \(2k\times1\) vector. The adjustment-speed error \(\ddot\rho_T-\rho_*\) is scalar, so it multiplies every component of the slope vector. The first term is adjustment-speed error multiplied by the true long-run slope. The second is precisely the centered error characterized by Lemma 4. The third is the product of the scalar adjustment error and the \(2k\times1\) slope error. This expansion is exact, including the product term.

When the corresponding component of \(\beta_*\) is nonzero, the first term has order \(T^{-1/2}\). The second has order \(T^{-1}\), and the product has order \(T^{-3/2}\), because \(T^{-1/2}T^{-1}=T^{-3/2}\). The following arithmetic illustrates how their scales separate as the sample grows; the actual errors also contain random multipliers.

Powers of the sample size appearing in the error expansion

Sample size	\(T^{-1/2}\)	\(T^{-1}\)	\(T^{-3/2}\)
\(T=100\)	0.1	0.01	0.001
\(T=400\)	0.05	0.0025	0.000125

Table notes. These entries are arithmetic values of three powers of \(T\), illustrating the scales of the three terms in the preceding expansion. For example, at \(T=400\), \(T^{-1/2}=1/\sqrt{400}=1/20=0.05\). Quadrupling \(T\) divides the three scales by \(2\), \(4\), and \(8\), respectively. Actual estimation errors also contain parameter loadings and random factors; the displayed numbers are neither estimated errors nor standard errors.

The adjustment error shrinks most slowly. Multiplication by \(\sqrt T\) preserves it, while the other two terms become orders \(T^{-1/2}\) and \(T^{-1}\), both approaching zero. Consequently,

\[ \sqrt T(\widehat\theta_T-\theta_*) =-\beta_*\sqrt T(\ddot\rho_T-\rho_*)+o_p(1) \Rightarrow-\beta_*\mathcal D_1. \]

The notation \(o_p(1)\) means that the omitted difference converges to zero in probability. Here the statement can be written explicitly as

\[ \sqrt T(\widehat\theta_T-\theta_*) +\beta_*\sqrt T(\ddot\rho_T-\rho_*) \xrightarrow{p}0. \]

Thus, on the \(\sqrt T\) scale used to study the estimation error, the original level-coefficient error and its leading adjustment-error contribution become arbitrarily close with probability approaching one. The omitted terms remain in the exact finite-sample identity, while their probability of changing this scaled approximation by more than a fixed tolerance tends to zero.

This rate is consistent with the economic parameterization. Holding the equilibrium slopes fixed, a change in the estimated adjustment speed rescales their coefficients in the dynamic equation. A rapidly estimated slope therefore does not make its product with a more slowly estimated adjustment coefficient equally precise.

For an illustrative long-run slope of two, an adjustment-coefficient error of \(0.01\) contributes \(-2(0.01)=-0.02\) to the level-coefficient error. A slope of one receives a contribution of \(-0.01\) from the same adjustment error. These are the loadings in the limiting formula. The remaining estimation components affect finite-sample errors but shrink more quickly in the generic case.

The same source determines the leading uncertainty in the original trend coefficient when \(\zeta_*\ne0\). The generic joint limit is

\[ \sqrt T\begin{pmatrix} \widehat\rho_T-\rho_*\\ \widehat\theta_T-\theta_*\\ \widehat\xi_T-\xi_* \end{pmatrix} \Rightarrow \begin{pmatrix}1\\-\beta_*\\-\zeta_*\end{pmatrix}\mathcal D_1. \]

The coefficient-error vector on the left is \(d\times1\): it contains one adjustment error, \(2k\) level-coefficient errors, and one trend-coefficient error. The loading vector \((1,-\beta_*’,-\zeta_*)’\) on the right has the same dimension and multiplies the single random scalar \(\mathcal D_1\). These leading errors therefore share one random component with different loadings. Their limiting linear dependence is compatible with distinct, computable coefficient estimates in a finite sample: the smaller terms in the expansion still distinguish those estimates.

The intercept also contains a surviving dynamic-regression error. Let the scalar \(\mathcal D_4\) denote the limit of \(\sqrt T\{(\ddot\gamma_T-\gamma_*)-\rho_*n_T\}\), and let the \(s\times1\) vector \(\mathcal D_5\) denote the limit of \(\sqrt T(\ddot\kappa_T-\kappa_*)\). Reconstructing the original intercept cancels its first-stage estimation contribution, giving the scalar intercept limit and the \(s\times1\) short-run coefficient limit:

\[ \sqrt T(\widehat\alpha_T-\alpha_*)\Rightarrow\mathcal D_4-\nu_*\mathcal D_1, \qquad \sqrt T(\widehat\kappa_T-\kappa_*)\Rightarrow\mathcal D_5. \]

Together, these expressions describe every original coefficient in the generic case of Theorem 1. Their joint distribution is mixed normal because the surviving components come from the centered dynamic limit. The error terms all vanish without normalization, establishing consistency.

A zero true slope changes which term dominates. The positive-slope block \(\beta_*^+\) has dimension \(k\times1\), one entry per original explanatory variable. In the theorem’s case \(\beta_*^+=0\), every entry in that block is zero and the positive-block adjustment contribution disappears. The centered term of order \(T^{-1}\) then becomes the leading error. Writing \(\mathcal D_2^+\) for the corresponding \(k\times1\) block of \(\mathcal D_2\), the theorem’s relevant case gives

\[ T(\widehat\theta_T^+-\theta_*^+)\Rightarrow\mathcal D_2^+. \]

The analogous result holds for a zero negative block. When \(\zeta_*=0\), the trend coefficient likewise requires multiplication by \(T\), with limit \(-\mu_*’\mathcal D_2\). The original \(\sqrt T\) expansion remains valid in these directions, but its limit is zero. A finer normalization reveals the remaining fluctuation. Thus the parameter values determine the informative rate even though the OLS computation itself is unchanged.

Finally, the long-run ratio has a useful cancellation of its own. Define \(h_T:=\ddot\eta_T-\rho_*b_T\). For the fixed nonzero adjustment coefficient considered here, the one-step slope estimate satisfies

\[ \widehat\beta_T^{\mathrm{ratio}}:=-\frac{\widehat\theta_T}{\widehat\rho_T}, \qquad \widehat\beta_T^{\mathrm{ratio}}-\beta_* =\frac{(\ddot\rho_T-\rho_*)b_T-h_T}{\ddot\rho_T}. \]

The product in the numerator has order \(T^{-3/2}\), and \(h_T\) has order \(T^{-1}\). The ratio therefore has rate \(T\), with scaled limit \(-\mathcal D_2/\rho_*\). This is an algebraic deduction from the preceding identities: the common adjustment-speed error cancels when the equation coefficient is divided by that adjustment estimate. It also explains why convergence rates must be compared for the same target parameter.

Division preserves these orders because \(\ddot\rho_T\) converges to the fixed nonzero \(\rho_*\). Thus the equilibrium slope and its coefficient in the dynamic equation can be consistently estimated at different rates within the same one-step regression.

Main takeaway. The original level coefficients generally inherit the \(\sqrt T\) rate of the adjustment coefficient. When a leading coefficient is zero, a finer scale can reveal the remaining variation; the long-run ratios converge at rate \(T\) because the leading adjustment error cancels.

The complete product expansion, the four cases of Theorem 1, and a long-run ratio calculation: Appendix H.

9. How applied economists should use these results

Use one-step OLS with a clear understanding of its theoretical setting

For an applied economist working with a correctly specified cointegrated NARDL model containing a time trend, Cho supplies a theoretical basis for the familiar one-step OLS estimator. The auxiliary regressions explain that estimator; they are not extra steps that must be run to reproduce its point estimates. They are also distinct from the separate bias-correcting two-stage NARDL procedure.

Report the parameter that answers the economic question

The coefficient \(\theta_*^+\) multiplies the lagged positive partial sum in the dynamic equation. The equilibrium response is \(\beta_*^+=-\theta_*^+/\rho_*\). Reporting these objects under distinct labels helps readers interpret both their economic meaning and their statistical behavior. The adjustment coefficient \(\rho_*\) describes the response to disequilibrium, conditional on the short-run dynamics.

In the one-step regression, the corresponding estimated equilibrium slope is \(-\widehat\theta_T^+/\widehat\rho_T\). The cancellation of the common adjustment-speed error is explained in Section 8 and derived in Appendix H. The primitive static long-run estimate \(\ddot\beta_T\) and the one-step ratio are different finite-sample estimators, even when both have rate \(T\).

Compare OLS and two-stage NARDL on the same basis

Cho’s comparison gives a conditional practical recommendation. With a sufficiently large sample, direct OLS can be attractive because it avoids the preliminary long-run covariance estimation used for bias correction in two-stage NARDL. With a smaller sample, two-stage NARDL can be preferable when that correction successfully removes the relevant bias.

Table notes. The first two rows describe conditional choices: the small-sample advantage depends on effective bias correction. The final row concerns parameter definitions. Here \(\rho_*\) is the adjustment coefficient, \(\beta_*\) is the long-run slope vector, and \(\theta_*=-\rho_*\beta_*\) contains the coefficients in the dynamic equation. A meaningful rate comparison follows the same parameter across estimators.

The paper supplies no universal sample-size threshold. Its recommendation depends on finite-sample bias and the effectiveness of estimating the correction. Consistency establishes convergence as the sample grows; it does not make the estimator unbiased in a given finite sample.

Interpret the trend and the matrix problem precisely

The theorem includes an explicit time-trend regressor, even when its true coefficient is zero. That coefficient describes a deterministic time effect conditional on the partial sums and other included regressors. The partial sums themselves already carry drift, so the observed trend in the outcome reflects their contributions as well as the explicit time coefficient.

An asymptotically singular normalized matrix also differs from exact sample collinearity. The former is the theoretical problem resolved here. If software cannot estimate a coefficient because columns are exactly dependent in the observed sample, that is a separate rank problem.

Finally, the distinction between estimation and testing remains useful: the maintained \(\rho_*<0\) theory does not provide the null distribution for testing \(\rho_*=0\). The estimation results developed in this note should be used within that cointegrated setting.

Main takeaway. Report the adjustment coefficient, dynamic coefficients, and long-run ratios according to the economic question. The OLS versus two-stage choice concerns finite-sample bias: the paper supports a conditional comparison, with no universal sample-size cutoff.

Parameter definitions: Appendix A. Same-target comparisons and the long-run ratio: Appendix H.

10. Conclusion

Cho establishes the consistency and limiting distribution of the original OLS estimator in a NARDL model whose normalized regressor matrix is asymptotically singular. The singularity comes from the deterministic trends generated by the positive and negative partial sums. Those trends make several columns proportional under the initial normalization.

The FWL transformations separate the common trend from the stochastic fluctuations and replace the lagged dependent variable with its estimated cointegration residual. Each component can then be normalized at its own scale. The resulting nonsingular matrix allows the transformed estimators’ joint limit to be derived, and the exact coefficient identities recover the distribution of the original OLS estimates.

The reconstruction explains why the original level coefficients generally converge at rate \(\sqrt T\): they contain the error in estimating the adjustment speed. The long-run slopes converge faster. When a population coefficient makes the leading adjustment-error term vanish, the remaining terms determine a different rate.

For applied economists, the results justify the familiar one-step OLS procedure within the cointegrating model studied by Cho. They also clarify how to report the estimates: distinguish the adjustment coefficient, the coefficients in the dynamic equation, and the long-run ratios. In choosing between OLS and two-stage NARDL, the practical issue is finite-sample bias. Direct OLS is convenient in sufficiently large samples; two-stage estimation can be preferable in smaller samples when its bias correction is effective.

Main takeaway. The theoretical contribution is to recover the behavior of the original OLS estimator despite a singular initial matrix limit. Preserving the cumulative stochastic variation before taking that limit makes the argument possible.

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References and further reading

Cho, J. S. (2026). “Estimation and Inference of Nonlinear Autoregressive Distributed Lag Models with Time Trend by Ordinary Least Squares.” Econometric Theory, First View. Publisher and DOI · Author’s page · Author’s manuscript. The estimation argument developed here follows Lemma 1, Propositions 1–4, Lemmas 2–4, and Theorem 1.

Shin, Y., Yu, B., and Greenwood-Nimmo, M. (2014). “Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework.” In Festschrift in Honor of Peter Schmidt: Econometric Methods and Applications, pp. 281–314. Springer. Publisher and DOI.

Park, J. Y., and Phillips, P. C. B. (1988, 1989). “Statistical Inference in Regression with Integrated Processes,” Parts 1 and 2. Econometric Theory, 4, 468–498, and 5, 95–131. Part 1 · Part 2. These papers develop the integrated-regression projection approach on which Cho builds.

Related EconMacro reading: AR Filtering or Lag Controls? What the Frisch–Waugh–Lovell Theorem Really Says.

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Appendix A. From the levels equation to the error-correction model

Return to the main text: Section 2: the error-correction model · Contents.

The error-correction representation follows from exact substitutions in the levels equation. This appendix shows those substitutions before introducing the compact vector notation. Throughout, a star denotes a fixed population parameter. The shorthand \(q_t=t-1\) denotes the time-trend regressor; the lag order \(q\), which has no time subscript, is a different object.

A.1. A worked example with two lags of the dependent variable

Take one explanatory variable, two lags of \(y_t\), and one lag of each partial sum. The levels equation is

\[ \begin{aligned} y_t={}&\alpha_{0*}+\xi_*t +\varphi_{1*}y_{t-1}+\varphi_{2*}y_{t-2}\\ &+\theta_{0*}^+x_t^++\theta_{1*}^+x_{t-1}^+ +\theta_{0*}^-x_t^-+\theta_{1*}^-x_{t-1}^-+e_t. \end{aligned} \]

Subtract \(y_{t-1}\) from both sides. On the left this produces \(\Delta y_t=y_t-y_{t-1}\). On the right, the coefficient on \(y_{t-1}\) becomes \(\varphi_{1*}-1\). Next use the identities

\[ y_{t-2}=y_{t-1}-\Delta y_{t-1}, \qquad x_t^\pm=x_{t-1}^\pm+\Delta x_t^\pm, \qquad t=q_t+1. \]

The lagged dependent-variable terms become

\[ \begin{aligned} (\varphi_{1*}-1)y_{t-1}+\varphi_{2*}y_{t-2} &=(\varphi_{1*}-1)y_{t-1} +\varphi_{2*}(y_{t-1}-\Delta y_{t-1})\\ &=(\varphi_{1*}+\varphi_{2*}-1)y_{t-1} -\varphi_{2*}\Delta y_{t-1}. \end{aligned} \]

For each sign, the explanatory-variable terms become

\[ \begin{aligned} \theta_{0*}^{\pm}x_t^\pm+\theta_{1*}^{\pm}x_{t-1}^\pm &=\theta_{0*}^{\pm}(x_{t-1}^\pm+\Delta x_t^\pm) +\theta_{1*}^{\pm}x_{t-1}^\pm\\ &=(\theta_{0*}^{\pm}+\theta_{1*}^{\pm})x_{t-1}^\pm +\theta_{0*}^{\pm}\Delta x_t^\pm. \end{aligned} \]

Finally, \(\alpha_{0*}+\xi_*t=(\alpha_{0*}+\xi_*)+\xi_*q_t\). Collecting the terms gives

\[ \begin{aligned} \Delta y_t={}&\rho_*y_{t-1} +\theta_*^+x_{t-1}^++\theta_*^-x_{t-1}^- +\xi_*q_t+\alpha_*\\ &+\phi_{1*}\Delta y_{t-1} +\pi_{0*}^+\Delta x_t^+ +\pi_{0*}^-\Delta x_t^-+e_t, \end{aligned} \] \[ \begin{gathered} \rho_*=\varphi_{1*}+\varphi_{2*}-1, \qquad \phi_{1*}=-\varphi_{2*},\\ \theta_*^\pm=\theta_{0*}^\pm+\theta_{1*}^\pm, \qquad \pi_{0*}^\pm=\theta_{0*}^\pm, \qquad \alpha_*=\alpha_{0*}+\xi_*. \end{gathered} \]

The disturbance remains \(e_t\). The operation subtracts the observed variable \(y_{t-1}\); it does not subtract the entire model dated \(t-1\). Consequently, it does not produce \(\Delta e_t\).

A.2. The same calculation for arbitrary lag orders

With \(p\) autoregressive lags and explanatory-variable lag order \(q\geq1\), the general coefficient definitions are

\[ \begin{aligned} \rho_*&=\sum_{i=1}^{p}\varphi_{i*}-1, &\phi_{j*}&=-\sum_{i=j+1}^{p}\varphi_{i*}, \quad j=1,\ldots,p-1,\\ \theta_*^\pm&=\sum_{i=0}^{q}\theta_{i*}^\pm, &\pi_{0*}^\pm&=\theta_{0*}^\pm,\\ \pi_{j*}^\pm&=-\sum_{i=j+1}^{q}\theta_{i*}^\pm, &&j=1,\ldots,q-1. \end{aligned} \]

For \(k\) explanatory variables, positive and negative changes are defined component by component. Stack their partial sums as

\[ \ddot x_t= \begin{pmatrix}x_t^+\\x_t^-\end{pmatrix}, \qquad \theta_*= \begin{pmatrix}\theta_*^+\\\theta_*^-\end{pmatrix}. \]

Each of \(x_t^+\), \(x_t^-\), \(\theta_*^+\), and \(\theta_*^-\) is a \(k\times1\) column vector. Stacking a positive and a negative block gives \(\ddot x_t\) and \(\theta_*\), each of dimension \(2k\times1\). The double dot is a notation label, not a time derivative.

Let \(z_{2t}\) collect the \(p-1\) lagged changes in \(y_t\) and the \(2kq\) current and lagged signed changes in the explanatory variables. It has \(s=(p-1)+2kq=p+2kq-1\) entries, so both \(z_{2t}\) and its coefficient vector \(\kappa_*\) are \(s\times1\). The compact equation is

\[ \Delta y_t=\rho_*y_{t-1}+\theta_*’\ddot x_{t-1} +\xi_*q_t+\alpha_*+\kappa_*’z_{2t}+e_t. \]

The whole regressor vector is \(z_t=(y_{t-1},\ddot x_{t-1}’,q_t,1,z_{2t}’)’\). The corresponding coefficient vector is \(\boldsymbol\alpha_*=(\rho_*,\theta_*’,\xi_*,\alpha_*,\kappa_*’)’\). Bold \(\boldsymbol\alpha_*\) denotes this entire vector; scalar \(\alpha_*\) denotes its intercept entry.

For the products in this scalar equation, a transpose changes a column into a row. Thus \(\theta_*’\) is \(1\times2k\), and multiplying it by \(\ddot x_{t-1}\), which is \(2k\times1\), gives a \(1\times1\) scalar. Similarly, \(\kappa_*’z_{2t}\) multiplies a \(1\times s\) row by an \(s\times1\) column. Every term added to \(\Delta y_t\) is therefore scalar.

The full regressor vector and the full coefficient vector each have \(1+2k+1+1+s=2k+s+3\) entries: one lagged outcome, \(2k\) partial sums, one trend, one constant, and \(s\) short-run terms. We write \(K=2k+s+3\) for this total. Both \(z_t\) and \(\boldsymbol\alpha_*\) are \(K\times1\), and their scalar product is \(z_t’\boldsymbol\alpha_*\).

A.3. Identify the equilibrium relationship

Define the long-run slopes and trend coefficient by

\[ \beta_*=-\frac{\theta_*}{\rho_*}, \qquad \zeta_*=-\frac{\xi_*}{\rho_*}, \qquad u_{t-1}=y_{t-1}-\beta_*’\ddot x_{t-1}-\zeta_*q_t-\nu_*. \]

Dividing the \(2k\times1\) vector \(\theta_*\) by the scalar \(\rho_*\) leaves \(\beta_*\) with dimension \(2k\times1\). The product \(\beta_*’\ddot x_{t-1}\) is scalar. The quantities \(\zeta_*\), \(\nu_*\), and \(u_{t-1}\) are also scalars, so the equation defining the equilibrium error is dimensionally consistent.

The constant \(\nu_*\) centers the cointegration error so that \(E[u_t]=0\). Multiplying its defining equation by \(\rho_*\) gives

\[ \rho_*u_{t-1} =\rho_*y_{t-1}+\theta_*’\ddot x_{t-1} +\xi_*q_t-\rho_*\nu_*. \]

It follows that the dynamic equation can be written as

\[ \Delta y_t=\rho_*u_{t-1}+\gamma_*+\kappa_*’z_{2t}+e_t, \qquad \gamma_*=\alpha_*+\rho_*\nu_*. \]

This calculation explains the intercept adjustment as well as the signs in the long-run ratios. The coefficient \(\rho_*\) governs adjustment to disequilibrium; the vector \(\beta_*\) describes the equilibrium slopes.

For example, a positive \(u_{t-1}\) means that the observed outcome lies above the level prescribed by the long-run relationship. With \(\rho_*<0\), the contribution \(\rho_*u_{t-1}\) to the next change is negative. The adjustment term therefore moves the outcome back toward equilibrium, holding the other terms fixed. Its size depends on both the distance from equilibrium and the adjustment coefficient.

A.4. A guide to the notation used in the calculations

Table notes. Each row distinguishes symbols or notation serving different purposes. For example, \(\ddot\beta_T\) is an auxiliary estimate of the population vector \(\beta_*\): the double dot identifies the estimation representation, and the subscript \(T\) records its dependence on the observed sample. Transposing a vector changes its orientation, while these notation labels leave its dimensions unchanged.

For example, an estimation error of order \(T^{-1/2}\) becomes bounded in probability after multiplication by \(\sqrt T\). A theorem establishing a nondegenerate distribution for that scaled error gives more information than the order statement alone: it identifies the fluctuation that survives.

Dimensions and multiplication. A dimension is written as rows \(\times\) columns. Two matrices can be multiplied when the number of columns of the first equals the number of rows of the second. The result retains the first matrix’s number of rows and the second matrix’s number of columns. For example, \((K\times1)(1\times K)\) produces a \(K\times K\) matrix, while \((1\times K)(K\times1)\) produces a scalar.

Dimensions of the original regression objects

Object	Dimensions	What the rows and columns represent
\(\ddot x_{t-1}\), \(\theta_*\), \(\beta_*\), \(\mu_*\)	\(2k\times1\)	One entry for each positive or negative component.
\(z_{2t}\), \(\kappa_*\)	\(s\times1\)	One entry for each short-run regressor or its coefficient.
\(z_t\), \(\boldsymbol\alpha_*\)	\(K\times1\)	One entry for every regressor or coefficient in the full equation.
\(Z\), with row \(t\) equal to \(z_t’\)	\(T\times K\)	\(T\) observations in rows and \(K\) regressors in columns.
\(Z’Z=\sum_tz_tz_t’\)	\(K\times K\)	One row and one column for every regressor; each entry sums a pair of regressor products.

Table notes. Read dimensions as rows \(\times\) columns. With \(T=100\) and \(K=8\), \(Z\) is \(100\times8\), its transpose is \(8\times100\), and \(Z’Z\) is \(8\times8\). Thus its rows and columns identify regressors; the observations have already been summed in each entry.

The sample size \(T\) determines how many rows the data matrix has. The number of regressors \(K\) determines the size of its cross-product matrix. Adding observations increases \(T\), while these matrix dimensions in regressor space remain fixed when the model specification is fixed.

Main takeaway. The error-correction representation follows from exact substitutions in the levels equation. Each inner product is scalar, and the complete regressor vector has \(K=2k+s+3\) entries.

Return to the main text: Section 2: the error-correction model · Contents

Appendix B. Why partial sums drift, and where the powers of T come from

Return to the main text: Section 3: drift and stochastic fluctuations · Section 5: column normalizations · Contents.

B.1. Separating increases and decreases changes their means

For a scalar change \(v=\Delta x_t\), define \(v^+=\max(v,0)\) and \(v^-=\min(v,0)\). The negative part retains its sign. Two useful identities are

\[ v^+=\frac{v+|v|}{2}, \qquad v^-=\frac{v-|v|}{2}. \]

Taking expectations gives

\[ \mu_*^+=\frac{E[v]+E|v|}{2}, \qquad \mu_*^-=\frac{E[v]-E|v|}{2}. \]

If \(E[v]=0\) and changes are nondegenerate, then \(\mu_*^+=E|v|/2>0\) and \(\mu_*^-=-E|v|/2<0\). Zero mean for the original change therefore coexists with positive and negative means for its two components.

For a simple calculation, let a change take the values \(-2,-1,+1,+2\), each with probability one quarter. The original mean is \((-2-1+1+2)/4=0\). The positive part takes the values \(0,0,1,2\), giving mean \((0+0+1+2)/4=3/4\). The negative part takes the values \(-2,-1,0,0\), giving mean \((-2-1+0+0)/4=-3/4\). The component means cancel when added, although each signed partial sum has its own nonzero drift.

Define centered increments \(s_t^\pm=\Delta x_t^\pm-\mu_*^\pm\). Summing the identity \(\Delta x_t^\pm=\mu_*^\pm+s_t^\pm\) gives

\[ x_t^\pm=\mu_*^\pm t+m_t^\pm, \qquad m_t^\pm=\sum_{j=1}^{t}s_j^\pm. \]

In vector form, \(\ddot x_t=\mu_*t+m_t\). Each of \(\ddot x_t\), \(\mu_*\), and \(m_t\) is \(2k\times1\); multiplying the drift vector \(\mu_*\) by the scalar time index \(t\) preserves that dimension. The deterministic component grows linearly with time; the centered cumulative component fluctuates around it. Appendix B.2 isolates one scalar component to explain its variance.

B.2. Why a centered cumulative process has a square-root scale

To isolate the arithmetic, consider independent increments \(\eta_j\) with zero mean and variance \(\sigma^2\), and let \(m_t=\sum_{j=1}^{t}\eta_j\). Its mean is zero. Its variance is

\[ \begin{aligned} \operatorname{Var}(m_t) &=\sum_{j=1}^{t}\operatorname{Var}(\eta_j) +2\sum_{i<j}\operatorname{Cov}(\eta_i,\eta_j)\\ &=t\sigma^2. \end{aligned} \]

The covariance terms vanish under independence. Standard deviation is the square root of variance, so

\[ \operatorname{SD}(m_t)=\sigma\sqrt t. \]

Variances add across independent increments. This produces a standard deviation proportional to \(\sqrt t\), even though there are \(t\) increments. For \(\sigma=1\), the standard deviation is \(10\) after \(100\) increments and \(20\) after \(400\).

With nonzero drift \(\mu\), the relative stochastic scale is

\[ \frac{\sigma\sqrt t}{|\mu|t} =\frac{\sigma}{|\mu|\sqrt t}\longrightarrow0. \]

The cumulative fluctuations continue to grow in absolute size, while becoming smaller relative to the deterministic drift. Under suitable short-memory dependence, an analogous calculation uses the long-run variance of the increments. The paper’s assumptions provide the functional limit theory needed for its processes.

B.3. From an observation’s size to a column’s normalization

With \(T\) observations, the time index \(t\) runs from \(1\) to \(T\). The trend regressor \(q_t=t-1\) therefore takes the values

\[ q_1=0,\quad q_2=1,\quad q_3=2,\quad\ldots,\quad q_T=T-1. \]

There are \(T\) entries in this column, including the initial zero. We need two sums. Write \(S_1\) for the sum of the entries and \(S_2\) for the sum of their squares:

\[ \begin{aligned} S_1&=\sum_{t=1}^{T}q_t=0+1+2+\cdots+(T-1),\\ S_2&=\sum_{t=1}^{T}q_t^2=0^2+1^2+2^2+\cdots+(T-1)^2. \end{aligned} \]

These sums enter the OLS regressor cross-product matrix. Multiplying the constant column by the trend column gives \(S_1\), because every entry of the constant column equals one. Multiplying the trend column by itself gives \(S_2\). Thus, \(S_2\) is the diagonal entry associated with the trend, while \(S_1\) appears in the two off-diagonal positions linking the trend and the constant.

Calculating the sum of the trend values. Write \(S_1\) twice, with the second line in reverse order:

\[ \begin{aligned} S_1&=0+1+\cdots+(T-2)+(T-1),\\ S_1&=(T-1)+(T-2)+\cdots+1+0. \end{aligned} \]

Add the two lines term by term. The first pair is \(0+(T-1)=T-1\), the second is \(1+(T-2)=T-1\), and every remaining pair also equals \(T-1\). There are \(T\) pairs, so

\[ 2S_1=T(T-1), \qquad \boxed{S_1=\frac{T(T-1)}2.} \]

Calculating the sum of squared trend values. Squaring each entry before adding produces a different sum. To derive its formula, start with the expansion of a cube:

\[ (j+1)^3=j^3+3j^2+3j+1, \qquad (j+1)^3-j^3=3j^2+3j+1. \]

Here \(j\) runs through the trend values \(0,1,\ldots,T-1\). Add this identity over those \(T\) values. On the left, the consecutive cube differences give

\[ \begin{aligned} \sum_{j=0}^{T-1}\bigl[(j+1)^3-j^3\bigr] &=(1^3-0^3)+(2^3-1^3)+\cdots+\bigl[T^3-(T-1)^3\bigr]\\ &=T^3. \end{aligned} \]

The \(+1^3\) in the first bracket cancels the \(-1^3\) in the second; the same cancellation occurs for every intermediate cube. Only \(T^3-0^3=T^3\) remains. On the right, adding \(3j^2\) gives \(3S_2\), adding \(3j\) gives \(3S_1\), and adding \(T\) copies of one gives \(T\). Hence

\[ T^3=3S_2+3S_1+T. \]

Subtract \(3S_1+T\), divide by three, and substitute the formula already obtained for \(S_1\):

\[ \begin{aligned} S_2 &=\frac{T^3-3S_1-T}{3}\\ &=\frac{T^3-\frac32T(T-1)-T}{3}\\ &=\frac{2T^3-3T(T-1)-2T}{6}\\ &=\frac{2T^3-3T^2+T}{6}\\ &=\frac{T(2T^2-3T+1)}6. \end{aligned} \]

The quadratic in the last line factors as \(2T^2-3T+1=(T-1)(2T-1)\). This gives the second formula:

\[ \boxed{S_2=\sum_{t=1}^{T}q_t^2=\frac{(T-1)T(2T-1)}6.} \]

For a numerical check, take \(T=4\). The trend column is \((0,1,2,3)’\), and direct addition gives

\[ S_1=0+1+2+3=6, \qquad S_2=0^2+1^2+2^2+3^2=14. \]

The formulas give exactly the same numbers:

\[ \frac{4(4-1)}2=6, \qquad \frac{(4-1)\times4\times(2\times4-1)}6=14. \]

Why the sum of squares grows like \(T^3\). Expanding the exact expression separates the three powers of sample size:

\[ S_2=\frac{T^3}{3}-\frac{T^2}{2}+\frac{T}{6}. \]

To compare their importance as \(T\) grows, divide every term by \(T^3\):

\[ \frac{S_2}{T^3} =\frac13-\frac1{2T}+\frac1{6T^2} \longrightarrow\frac13. \]

The second and third terms tend to zero. Thus, saying that \(S_2\) is approximately \(T^3/3\) means that the ratio \(S_2/(T^3/3)\) approaches one. The growing power is \(T^3\); the factor \(1/3\) is the constant that remains after normalization.

Why the column is divided by \(T^{3/2}\). The sum of squares is built by multiplying each column entry by itself. If every entry \(q_t\) is divided by a number \(\lambda_T\), its square is divided by \(\lambda_T^2\):

\[ \sum_{t=1}^{T}\left(\frac{q_t}{\lambda_T}\right)^2 =\frac{S_2}{\lambda_T^2}. \]

Since \(S_2\) grows like \(T^3\), choose \(\lambda_T^2=T^3\). Taking the positive square root gives \(\lambda_T=\sqrt{T^3}=T^{3/2}=T\sqrt T\). Consequently,

\[ \sum_{t=1}^{T}\left(\frac{q_t}{T^{3/2}}\right)^2 =\frac{1}{T^3}\sum_{t=1}^{T}q_t^2 \longrightarrow\frac13. \]

This is what the column normalization accomplishes: the trend’s diagonal entry in the normalized OLS matrix approaches a finite, positive number. The factor \(T^{3/2}\) scales the column, and its square \(T^3\) scales the corresponding sum of squares.

Applying the same reasoning to the other columns. In the example from Appendix B.2, the centered cumulative process \(m_{t-1}\) has mean zero and variance \(\sigma^2(t-1)\). Its expected square therefore equals that variance. Adding across observations gives

\[ E\left[\sum_{t=1}^{T}m_{t-1}^{\,2}\right] =\sigma^2\sum_{t=1}^{T}(t-1) =\frac{\sigma^2T(T-1)}2. \]

The formula for \(S_1\) now explains the last equality: the expected squared cumulative fluctuations add up to \(\sigma^2\) times \(0+1+\cdots+(T-1)\). This expected sum grows like \(T^2\). Under the functional limit assumptions discussed above, the random sum of squares has that scale, giving column normalization \(\sqrt{T^2}=T\).

For the constant column, each entry and each squared entry equals one. There are \(T\) observations, so its sum of squares is exactly \(T\). Dividing the column by \(\sqrt T\) therefore gives

\[ \sum_{t=1}^{T}\left(\frac{1}{\sqrt T}\right)^2 =\sum_{t=1}^{T}\frac1T =1. \]

How to read the table. Each row follows one type of regressor from the magnitude of an individual observation to the sum of squared observations, then to the divisor applied to its entire column. The entries give growth orders as the sample size \(T\) increases. Constants such as \(\sigma\) and \(1/3\) are omitted from these orders; their values remain in the formulas above.

The observation column compares values at the same relative position in expanding samples. For example, halfway through a sample of \(100\) observations, \(t=50\); halfway through a sample of \(400\), \(t=200\). In both cases \(t=T/2\). This explains the earlier phrase “sample times proportional to \(T\)”: the time index grows with the sample size. An early observation such as \(q_2=1\) stays fixed when later observations are added.

First row: the linear trend. At a time proportional to \(T\), a trend value has magnitude proportional to \(T\). Squaring gives magnitude \(T^2\); summing across the sample gives order \(T^3\). The exact calculation above establishes the factor \(1/3\). The sequence of powers is

\[ T\quad\longrightarrow\quad T^2\quad\longrightarrow\quad T\times T^2=T^3\quad\longrightarrow\quad\sqrt{T^3}=T^{3/2}. \]

These four steps are the magnitude of one observation, its square, the sum of squares, and the divisor for the column. The last step takes a square root because dividing each observation by a divisor divides its square by the square of that divisor.

Second row: centered cumulative fluctuations. These are the accumulated random changes left after removing their mean drift. In the independent-increment example, their standard deviation grows as \(\sqrt T\) at times proportional to \(T\). This describes a typical stochastic magnitude: an individual realization can lie above or below that scale. Squaring gives magnitude \(T\), and the sum of squares has order \(T^2\). Thus,

\[ \sqrt T\quad\longrightarrow\quad T\quad\longrightarrow\quad T\times T=T^2\quad\longrightarrow\quad\sqrt{T^2}=T. \]

Dividing the cumulative-process column by \(T\) keeps its sum of squares on a finite scale. Under the functional limit theory, this normalized sum generally converges in distribution to a random quantity.

Third row: a stationary regressor or the constant. The entry \(1\) means that the typical magnitude of an observation remains on a fixed scale as the sample grows. A stationary regressor can take many values; the entry describes its growth order. With a finite, positive second moment and the relevant law of large numbers, its average squared value approaches a finite, positive constant. Its sum of squares therefore grows in proportion to \(T\). For the constant column, every observation is exactly one. The powers are

\[ 1\quad\longrightarrow\quad1\quad\longrightarrow\quad T\times1=T\quad\longrightarrow\quad\sqrt T. \]

For example, when \(T=100\), the divisors in the final column are \(100^{3/2}=1000\) for the trend, \(100\) for the centered cumulative process, and \(\sqrt{100}=10\) for a stationary regressor or the constant. Each divisor applies to every observation in its column. These normalizations are used to study the OLS matrix as the sample grows; the convergence rates of the original coefficient estimates also depend on the error terms and the coefficient transformations.

A cross-product uses the product of the two column scales. Thus a trend–constant entry is divided by \(T^{3/2}\sqrt T=T^2\); a trend–trend entry is divided by \(T^3\); and a constant–constant entry is divided by \(T\).

For example, multiply the normalized constant by the normalized trend at each observation and then add. The formula for \(S_1\) gives

\[ \begin{aligned} \sum_{t=1}^{T}\frac{1}{\sqrt T}\frac{q_t}{T^{3/2}} &=\frac{1}{T^2}\sum_{t=1}^{T}q_t\\ &=\frac{T(T-1)}{2T^2} =\frac12-\frac1{2T} \longrightarrow\frac12. \end{aligned} \]

The normalized constant’s diagonal entry is therefore \(1\), the trend’s diagonal entry tends to \(1/3\), and their cross-product tends to \(1/2\). These are the entries calculated again within the larger OLS matrix in Appendix C.

B.4. Signed error products require their own calculation

A sum of regressor–error products contains positive and negative contributions. Its size cannot be obtained by simply adding the absolute sizes of its terms. For independent mean-zero errors with variance \(\sigma_e^2\), a deterministic trend gives

\[ \operatorname{Var}\left(\sum_{t=1}^{T}q_te_t\right) =\sigma_e^2\sum_{t=1}^{T}q_t^2, \]

so this signed sum has standard-deviation scale \(T^{3/2}\), whereas the trend’s sum of squares has scale \(T^3\). Similarly, with independent errors conditional on a cumulative regressor path, the conditional variance of \(\sum_t m_{t-1}e_t\) is \(\sigma_e^2\sum_t m_{t-1}^2\), giving scale \(T\).

These examples explain the arithmetic behind the normalized scores. The full multivariate result additionally accounts for cross-products, dependence, and generated regressors. A column normalization alone does not establish a coefficient’s convergence rate around a fixed population parameter.

B.5. The dependent variable also has a drift

Cho defines \(\delta_*=E[\Delta y_t]\), and then centers the change as \(d_t=\Delta y_t-\delta_*\). Summing gives

\[ y_t=y_0+\delta_*t+\sum_{j=1}^{t}d_j. \]

The equilibrium relationship also implies \(\delta_*=\beta_*’\mu_*+\zeta_*\). Taking expectations in the dynamic equation provides another expression:

\[ \delta_*= \frac{\displaystyle \gamma_*+ \sum_{j=0}^{q-1} \left(\pi_{j*}^{+\prime}\mu_*^+ +\pi_{j*}^{-\prime}\mu_*^-\right)} {\displaystyle 1-\sum_{j=1}^{p-1}\phi_{j*}}. \]

The denominator is the paper’s \(\varrho_*=1-\sum_j\phi_{j*}\). It is distinct from the adjustment coefficient \(\rho_*\). This identity follows by collecting the lagged mean changes, each equal to \(\delta_*\), on the left side of the expectation equation.

In this expectation calculation, \(\pi_{j*}^+\) and \(\mu_*^+\) are \(k\times1\) vectors, as are their negative counterparts. Each product \(\pi_{j*}^{+\prime}\mu_*^+\) or \(\pi_{j*}^{-\prime}\mu_*^-\) multiplies a \(1\times k\) row by a \(k\times1\) column and gives a scalar. The numerator, denominator, and resulting mean change \(\delta_*\) are therefore all scalars.

More explicitly, the expectation equation contains \(\delta_*\) on the left and \(\sum_j\phi_{j*}\delta_*\) on the right. Subtract the latter from both sides to obtain \((1-\sum_j\phi_{j*})\delta_*\). The cointegration-error contribution disappears because its mean is zero. The signed changes contribute their respective means, which explains the remaining terms in the numerator.

Main takeaway. Trend, cumulative-fluctuation, and stationary columns have sums of squares of orders \(T^3\), \(T^2\), and \(T\), respectively. Dividing each column by the square root of its sum-of-squares order explains the normalizers \(T^{3/2}\), \(T\), and \(\sqrt T\).

Return to the main text: Section 3: drift and stochastic fluctuations · Section 5: column normalizations · Contents

Appendix C. Locate the singularity inside the OLS matrix

Return to the main text: Section 4: the singular limiting matrix · Contents.

C.1. Normalize each row and column explicitly

Collect the drifting regressors in \(g_t=(y_{t-1},\ddot x_{t-1}’,q_t)’\). This is the vector called \(v_t\) in Section 4. It contains \(d=1+2k+1=2k+2\) entries and has dimension \(d\times1\). The drift vector \(a=(\delta_*,\mu_*’,1)’\) also has dimension \(d\times1\), so its product with the scalar \(q_t\) has the right dimension to describe the leading component of \(g_t\).

Write \(z_t=(g_t’,1,z_{2t}’)’\). Its three blocks have \(d\), \(1\), and \(s\) rows, totaling \(K=d+1+s=2k+s+3\). Define

\[ D=\operatorname{diag} \left(T^{3/2}I_{2k+2},\sqrt T,\sqrt T I_s\right), \qquad A_T=D^{-1}\left(\sum_{t=1}^{T}z_tz_t’\right)D^{-1}. \]

The left multiplication divides each row by its regressor’s scale; the right multiplication divides each column by its scale. An entry involving two trending regressors is consequently divided by \(T^3\).

Here \(I_{2k+2}=I_d\) is a \(d\times d\) identity matrix, and \(I_s\) is an \(s\times s\) identity matrix. An identity matrix has ones on its diagonal and zeros elsewhere. The notation \(\operatorname{diag}\) places the displayed square blocks along one diagonal, with zero blocks between them. Thus \(D\), \(D^{-1}\), \(\sum_tz_tz_t’\), and \(A_T\) are all \(K\times K\). The normalization changes their entries while preserving their dimensions.

Let \(h=E[z_{2t}]\), an \(s\times1\) vector of short-run regressor means, and \(M_{22}=E[z_{2t}z_{2t}’]\), an \(s\times s\) matrix of second moments. In this ordering, Lemma 1 gives the full \(K\times K\) limit

\[ A_T\xrightarrow{p}M= \begin{pmatrix} aa’/3 & a/2 & ah’/2\\ a’/2 & 1 & h’\\ ha’/2 & h & M_{22} \end{pmatrix}. \]

The three row and column groups are the drifting regressors, the constant, and the stationary short-run regressors. The stationary variables can have nonzero means; this is why \(h\) appears.

The matrix statement \(A_T\xrightarrow{p}M\) applies entry by entry. The entries of \(A_T\) vary with the sample; the corresponding entries of \(M\) are fixed numbers determined by population drift coefficients and stationary moments. With the number of regressors fixed, the probability that any entry differs from its target by more than a chosen tolerance tends to zero:

\[ \Pr\!\left(\max_{i,j}|(A_T)_{ij}-M_{ij}|>\varepsilon\right) \longrightarrow0 \qquad\text{for every }\varepsilon>0. \]

The maximum simply checks the largest entrywise discrepancy. The limiting matrix can be singular even though all these approximations become accurate. Its drifting columns approach proportional columns, so the limiting matrix loses the independent variation needed for inversion. The inverse of a finite-sample matrix therefore cannot be obtained here by inverting \(M\).

The displayed array has three block rows and three block columns. Its full dimensions are \(K\times K\), because most displayed entries are themselves matrices. Their dimensions are

Dimensions of the nine blocks of the limiting matrix

Row group	Drifting columns: \(d\)	Constant column: \(1\)	Short-run columns: \(s\)
Drifting rows: \(d\)	\(aa’/3:\ d\times d\)	\(a/2:\ d\times1\)	\(ah’/2:\ d\times s\)
Constant row: \(1\)	\(a’/2:\ 1\times d\)	\(1:\ 1\times1\)	\(h’:\ 1\times s\)
Short-run rows: \(s\)	\(ha’/2:\ s\times d\)	\(h:\ s\times1\)	\(M_{22}:\ s\times s\)

Table notes. Select a row group and a column group to locate a block. Here \(d=2k+2\), \(s\) counts short-run regressors, \(a\) contains the \(d\) drift coefficients, \(h=E[z_{2t}]\), and \(M_{22}=E[z_{2t}z_{2t}’]\). The displayed three-by-three arrangement partitions a larger \(K\times K\) matrix.

For example, \(ah’\) multiplies a \(d\times1\) column by a \(1\times s\) row and produces a \(d\times s\) block. Every block in a given block row has the same number of rows, and every block in a given block column has the same number of columns. The three row counts add to \(d+1+s=K\).

C.2. Calculate representative entries

The explicit trend’s diagonal entry is its normalized sum of squares:

\[ \frac1{T^3}\sum_{t=1}^{T}(t-1)^2 =\frac{(T-1)(2T-1)}{6T^2} \longrightarrow\frac13. \]

The constant’s diagonal entry is exactly

\[ \frac1T\sum_{t=1}^{T}1^2=1. \]

The entry joining the trend and constant is

\[ \frac1{T^2}\sum_{t=1}^{T}(t-1) =\frac{T-1}{2T}\longrightarrow\frac12. \]

For \(k=1\), the regressor order is \(y_{t-1},x_{t-1}^+,x_{t-1}^-,q_t,1,z_{2t}’\). The trend entry is therefore \(A_{44,T}\), and the constant entry is \(A_{55,T}\). These are diagonal entries of the normalized cross-product matrix \(A_T\).

With \(k=1\), the drifting block has \(d=4\) rows and columns, and the complete matrix is \((s+5)\times(s+5)\). For the worked lag choice \(p=2\), \(q=1\), there are \(s=3\) short-run regressors, so the complete matrix is \(8\times8\). Its \((4,4)\) and \((5,5)\) positions identify the trend and constant regardless of the sample size.

Two drifting regressors with leading coefficients \(a_i\) and \(a_j\) produce the limit \(a_i a_j/3\). A drifting regressor and the constant produce \(a_i/2\). The weighted sample means of the stationary regressors give the trend–short-run block \(h’/2\), and hence the general block \(ah’/2\).

C.3. Show why the stochastic variation disappears at this scale

For an illustrative scalar drifting regressor \(v_{t-1}=\mu q_t+m_{t-1}\), expand its square:

\[ \frac1{T^3}\sum_t v_{t-1}^2 = \mu^2\frac{\sum_t q_t^2}{T^3} +2\mu\frac{\sum_t q_tm_{t-1}}{T^3} +\frac{\sum_t m_{t-1}^2}{T^3}. \]

The last sum has order \(T^2/T^3=T^{-1}\). For the middle sum, the Cauchy–Schwarz inequality gives

\[ \left|\sum_t q_tm_{t-1}\right| \leq \left(\sum_t q_t^2\right)^{1/2} \left(\sum_t m_{t-1}^2\right)^{1/2} =O_p(T^{3/2}T)=O_p(T^{5/2}). \]

After division by \(T^3\), that term is \(O_p(T^{-1/2})\). Both stochastic contributions vanish, leaving \(\mu^2/3\). The stochastic component remains in the observations; this normalization suppresses its contribution to the limit.

In probability language, each of the last two normalized terms becomes smaller than any fixed positive tolerance with probability approaching one. Their disappearance is relative to the \(T^3\) normalization. At the finer \(T^2\) scale, the cumulative fluctuations’ sum of squares retains a random limit. The FWL argument uses that finer scale after removing the deterministic trend, which preserves the information lost in the first limiting matrix.

C.4. Establish dependence across the whole matrix

The time-trend position within \(g_t\) is \(2k+2\), and its entry in \(a\) equals one. Every other drifting-regressor column of \(M\) is its corresponding \(a_j\) times that time-trend column. This statement holds in all three row blocks:

\[ \begin{pmatrix} aa_j/3\\ a_j/2\\ ha_j/2 \end{pmatrix} = a_j \begin{pmatrix} a/3\\ 1/2\\ h/2 \end{pmatrix}. \]

Thus the full limiting matrix is singular. This explicit column calculation shows how the dependence extends through the constant and short-run rows, so its location in the complete OLS matrix is visible.

Each displayed stacked column has \(d+1+s=K\) entries: a \(d\times1\) upper block, a scalar middle block, and an \(s\times1\) lower block. The dependence therefore concerns full columns of \(M\). The drifting \(d\times d\) block has rank one; the complete matrix can have additional independent columns from the constant and short-run regressors, while remaining singular because its drifting columns are dependent.

C.5. A finite inverse can coexist with a singular limit

A \(2\times2\) matrix illustrates the distinction. Its dimension is fixed, while its entries depend on the sample-size index \(T\):

\[ B_T= \begin{pmatrix}1&1\\1&1+1/T\end{pmatrix}, \qquad \det(B_T)=\frac1T>0. \]

For every finite positive \(T\), the inverse exists:

\[ B_T^{-1}= \begin{pmatrix}T+1&-T\\-T&T\end{pmatrix}. \]

Yet \(B_T\) converges to a matrix with two identical columns. Its diagonal entries stay near one while its determinant tends to zero. The original regression can likewise be estimable in finite samples even though its normalized matrix limit is singular.

The normalized OLS error identity is

\[ D(\widehat{\boldsymbol\alpha}_T-\boldsymbol\alpha_*) = A_T^{-1} \left(D^{-1}\sum_{t=1}^{T}z_te_t\right). \]

The coefficient-error vector has dimension \(K\times1\). Since \(e_t\) is scalar, each \(z_te_t\), and hence their sum, also has dimension \(K\times1\). The matrices \(D\), \(D^{-1}\), and \(A_T^{-1}\) are \(K\times K\), so both sides of the error identity are \(K\times1\) vectors. Each matrix multiplication combines all coefficient directions without changing the number of coefficients.

Since \(M\) has no inverse, the usual step \(A_T^{-1}\to M^{-1}\) is unavailable. This identifies the obstacle in the proof. The transformations in the next appendices retain the stochastic variation at scales that permit a nonsingular limiting matrix.

Main takeaway. The drifting-regressor block has a rank-one limit, and the corresponding full columns remain dependent after the constant and short-run controls are included. A finite-sample inverse can therefore exist even though the normalized matrix has a singular limit.

Return to the main text: Section 4: the singular limiting matrix · Contents

Appendix D. Derive the first Frisch–Waugh–Lovell transformation

Return to the main text: Section 5: the FWL transformation · Contents.

D.1. Construct an exact fitted-value and residual decomposition

Regress \(y_{t-1}\) on \(\ddot x_{t-1},q_t,1\), using the same estimation sample as the dynamic regression. With \(r_{t-1}=(\ddot x_{t-1}’,q_t,1)’\) and \(\widetilde\upsilon_T=(\widetilde\beta_T’,\widetilde\zeta_T,\widetilde\nu_T)’\), define

\[ \widetilde u_{t-1} =y_{t-1}-r_{t-1}’\widetilde\upsilon_T. \]

Here \(r_{t-1}\) denotes the auxiliary regressor vector; Section 4 used \(r_t\) for departures from deterministic trends. The auxiliary regressor \(r_{t-1}\) and its coefficient vector \(\widetilde\upsilon_T\) each have \(2k+2\) rows and one column: \(2k\) partial-sum entries, a trend entry, and an intercept entry. Their product \(r_{t-1}’\widetilde\upsilon_T\) multiplies a \(1\times(2k+2)\) row by a \((2k+2)\times1\) column, giving the scalar fitted value for one observation. Stacking all observations would give an auxiliary design matrix of dimension \(T\times(2k+2)\).

The definition gives the exact sample identity

\[ y_{t-1} =\widetilde\beta_T’\ddot x_{t-1} +\widetilde\zeta_Tq_t+\widetilde\nu_T +\widetilde u_{t-1}. \]

OLS implies \(\sum_t r_{t-1}\widetilde u_{t-1}=0\): the residual is orthogonal to each regressor used in this projection. The short-run variables \(z_{2t}\) were not part of that projection, so \(\sum_tz_{2t}\widetilde u_{t-1}\) generally need not be zero.

The first sum is a \((2k+2)\times1\) vector, and its zero is a zero vector of that dimension: one orthogonality condition for each auxiliary regressor. The second sum has dimension \(s\times1\). The estimated residual \(\widetilde u_{t-1}\) itself is scalar at each date.

D.2. Retain the other columns in the dynamic regression

Now regress \(\Delta y_t\) on \(\widetilde u_{t-1},\ddot x_{t-1},q_t,1,z_{2t}\). Its fitted component is

\[ \widetilde\rho_T\widetilde u_{t-1} +\widetilde\eta_T’\ddot x_{t-1} +\widetilde\psi_Tq_t+\widetilde\gamma_T +\widetilde\kappa_T’z_{2t}. \]

Substitute the residual’s definition into the first term:

\[ \begin{aligned} \widetilde\rho_T\widetilde u_{t-1} ={}&\widetilde\rho_Ty_{t-1} -\widetilde\rho_T\widetilde\beta_T’\ddot x_{t-1}\\ &-\widetilde\rho_T\widetilde\zeta_Tq_t -\widetilde\rho_T\widetilde\nu_T. \end{aligned} \]

Collecting coefficients of the original columns gives

\[ \begin{aligned} \widehat\rho_T&=\widetilde\rho_T,\\ \widehat\theta_T&=\widetilde\eta_T-\widetilde\rho_T\widetilde\beta_T,\\ \widehat\xi_T&=\widetilde\psi_T-\widetilde\rho_T\widetilde\zeta_T,\\ \widehat\alpha_T&=\widetilde\gamma_T-\widetilde\rho_T\widetilde\nu_T,\\ \widehat\kappa_T&=\widetilde\kappa_T. \end{aligned} \]

Each minus sign comes from subtracting a fitted component when constructing the residual. For an arithmetic illustration, if a component of \(\widetilde\beta_T\) equals \(2\), \(\widetilde\rho_T=-0.4\), and the corresponding \(\widetilde\eta_T=0.03\), then the reconstructed coefficient is \(0.03-(-0.4)(2)=0.83\). These numbers illustrate the identity; they are not empirical estimates.

D.3. Verify equality of the OLS estimators using matrices

Let \(Z\) have rows \(z_t’\), so \(Z\) is \(T\times K\), and let \(Y\) stack the observations of \(\Delta y_t\), so \(Y\) is \(T\times1\). Here \(K=2k+s+3\) is the number of regressors. Define the \(K\times K\) coefficient-transformation matrix

\[ R_T= \begin{pmatrix} 1&0&0\\ -\widetilde\upsilon_T&I_{2k+2}&0\\ 0&0&I_s \end{pmatrix}. \]

The three block rows and columns of \(R_T\) have sizes \(1\), \(2k+2\), and \(s\). The first block corresponds to the lagged outcome, the middle block to the partial sums, trend, and constant, and the last block to the short-run regressors. In particular, \(-\widetilde\upsilon_T\) is a \((2k+2)\times1\) block. The zeros in the first row are \(1\times(2k+2)\) and \(1\times s\) blocks; all other zero blocks likewise take the row count and column count required by their positions.

The product \(ZR_T\) has dimensions \((T\times K)(K\times K)=T\times K\). Its first column is the long-run residual, and all other columns are unchanged. Hence the transformed design is \(\widetilde Z=ZR_T\). Its coefficient vector \(\widetilde\omega_T=(\widetilde\rho_T,\widetilde\eta_T’,\widetilde\psi_T,\widetilde\gamma_T,\widetilde\kappa_T’)’\) is \(K\times1\). The matrix \(R_T\) is triangular with ones on its diagonal, so it is invertible and has determinant one. Under full sample rank,

\[ \begin{aligned} \widetilde\omega_T &=(\widetilde Z’\widetilde Z)^{-1}\widetilde Z’Y\\ &=(R_T’Z’ZR_T)^{-1}R_T’Z’Y\\ &=R_T^{-1}(Z’Z)^{-1}Z’Y =R_T^{-1}\widehat{\boldsymbol\alpha}_T. \end{aligned} \]

In the first line, \(\widetilde Z’\widetilde Z\) is \(K\times K\), while \(\widetilde Z’Y\) is \(K\times1\). Multiplying the inverse cross-product matrix by this vector therefore gives the \(K\times1\) coefficient vector. In the second line, each factor in \(R_T’Z’ZR_T\) has compatible dimensions, and the resulting matrix is again \(K\times K\).

Multiplying by \(R_T\) produces Proposition 1:

\[ \boxed{\widehat{\boldsymbol\alpha}_T=R_T\widetilde\omega_T.} \]

The fitted values also coincide: \(\widetilde Z\widetilde\omega_T=ZR_TR_T^{-1}\widehat{\boldsymbol\alpha}_T=Z\widehat{\boldsymbol\alpha}_T\). Their dynamic regression residuals are therefore identical.

Both fitted-value vectors have dimension \(T\times1\): a \(T\times K\) design matrix multiplies a \(K\times1\) coefficient vector to produce one fitted value for every observation. The transformation changes the coordinate system for the coefficients while retaining the same number of regressors and fitted observations.

The population coefficients on the extra levels and trend would be zero if the true cointegration error were used. With an estimated residual, those columns are retained to preserve the exact sample equivalence. The representation reconstructs every original coefficient and supplies the algebra needed for the asymptotic analysis.

Appendices A–D unpack equations (1)–(7), Lemma 1, and Proposition 1 in Cho’s paper. The small numerical and matrix examples are explanatory constructions.

Main takeaway. Replacing the lagged outcome by its auxiliary residual, while retaining the other regressors, is an invertible change of columns. The fitted values stay the same, and the original coefficients follow from an exact matrix identity.

Return to the main text: Section 5: the FWL transformation · Contents

Appendix E. Propositions 2–4: remove the fitted trends, then recover every OLS coefficient

Return to the main text: Section 5: detrending · Section 6: recovering the original coefficients · Contents.

E.1. A change of regressors that preserves the regression

Let \(q_t=t-1\), a shorthand for the paper’s lagged time trend, and let \(\ddot x_{t-1}\) stack the \(2k\) positive and negative partial sums. Double dots are notation labels, not derivatives; on \(\ddot x_{t-1}\), they identify the stacked partial sums. Cho first projects each partial sum on \(q_t\), without an intercept:

\[ \ddot\mu_T=\frac{\sum_{t=1}^{T}q_t\ddot x_{t-1}}{\sum_{t=1}^{T}q_t^2}, \qquad \ddot m_{t-1}=\ddot x_{t-1}-q_t\ddot\mu_T. \]

These definitions give two exact sample identities:

\[ \ddot x_{t-1}=\ddot m_{t-1}+q_t\ddot\mu_T, \qquad \sum_{t=1}^{T}q_t\ddot m_{t-1}=0. \]

Dimensions. At one date, \(\ddot x_{t-1}\), \(\ddot m_{t-1}\), and \(\ddot\mu_T\) are \(2k\times1\) column vectors. The numerator in the drift estimate is a sum of \(2k\times1\) vectors, while its denominator is a scalar; the resulting estimate therefore has one entry for each partial sum. The zero in the orthogonality identity above is a \(2k\times1\) zero vector, giving one orthogonality condition per partial sum.

The first reconstructs the original partial sums. The second is the OLS orthogonality condition: the residuals have zero sample cross-product with the fitted trend. Their sample means generally remain nonzero, because this projection has no intercept. The subsequent regressions therefore retain a constant.

The original auxiliary long-run regression uses \((\ddot x_{t-1}’,q_t,1)’\), with coefficient vector \(\widetilde\upsilon_T=(\widetilde\beta_T’,\widetilde\zeta_T,\widetilde\nu_T)’\). Replacing the partial sums by their detrended counterparts gives a regression on \((\ddot m_{t-1}’,q_t,1)’\), with coefficients \(\ddot\upsilon_T=(\ddot\beta_T’,\ddot\vartheta_T,\ddot\nu_T)’\). Substitution yields Proposition 2:

\[ \widetilde\beta_T=\ddot\beta_T, \qquad \widetilde\zeta_T=\ddot\vartheta_T-\ddot\mu_T’\ddot\beta_T, \qquad \widetilde\nu_T=\ddot\nu_T. \]

The transformed trend coefficient absorbs the trends previously carried inside the partial sums. Recovering the original explicit trend requires subtracting their fitted contribution. The intercept stays unchanged because the decomposition of \(\ddot x_{t-1}\) adds only a multiple of \(q_t\).

The two auxiliary coefficient vectors \(\widetilde\upsilon_T\) and \(\ddot\upsilon_T\) each have \(2k+2\) entries: \(2k\) partial-sum slopes, one trend coefficient, and one intercept. Their regression matrices have \(T\) rows and \(2k+2\) columns. In the trend correction, \(\ddot\mu_T’\) is \(1\times2k\) and \(\ddot\beta_T\) is \(2k\times1\), so their product is the scalar needed to adjust a single trend coefficient. For \(k=1\), the transformation below is a \(4\times4\) matrix multiplying a \(4\times1\) coefficient vector.

For one explanatory variable, the matrix form makes every adjustment visible:

\[ \begin{pmatrix}\widetilde\beta_T^+\\\widetilde\beta_T^-\\\widetilde\zeta_T\\\widetilde\nu_T\end{pmatrix} = \underbrace{\begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ -\ddot\mu_T^+&-\ddot\mu_T^-&1&0\\ 0&0&0&1 \end{pmatrix}}_{\mathbf P_T} \begin{pmatrix}\ddot\beta_T^+\\\ddot\beta_T^-\\\ddot\vartheta_T\\\ddot\nu_T\end{pmatrix}. \]

For example, take \(\ddot\mu_T=(0.4,-0.4)’\), \(\ddot\beta_T=(2,1)’\), and \(\ddot\vartheta_T=0.7\). The fitted trend contributed by the partial sums is \(0.4(2)-0.4(1)=0.4\), giving \(\widetilde\zeta_T=0.7-0.4=0.3\).

Both regressor sets span the same sample column space. With full column rank, their fitted values are identical, so the estimated cointegration residual is also identical: \(\widetilde u_{t-1}=\ddot u_{t-1}\). Detrending changes the coordinates used to describe the fitted relationship.

This equivalence can be checked in both directions. Starting with the original regression, replace each partial sum by its detrended component plus fitted trend. Starting with the transformed regression, add the fitted trend back into each partial sum. Every fitted observation is reproduced. The coefficient on a detrended partial sum remains the original slope because the coefficient multiplying that component has not changed; the trend coefficient accounts for the contribution moved into the time regressor.

E.2. Apply the same transformation to the dynamic regression

Proposition 3 repeats this operation with \(\Delta y_t\) as the dependent variable. The fully augmented transformed regression is

\[ \Delta y_t=\ddot\rho_T\ddot u_{t-1} +\ddot\eta_T’\ddot m_{t-1}+\ddot\varsigma_Tq_t +\ddot\gamma_T+\ddot\kappa_T’z_{2t}+\ddot e_t. \]

Here, \(z_{2t}\) contains the \(s=p+2kq-1\) short-run regressors. Distinguish \(\ddot\vartheta_T\), the trend coefficient in the auxiliary long-run regression, from \(\ddot\varsigma_T\), the trend coefficient in this dynamic regression. They belong to different equations.

For dimension counting, write \(K:=1+2k+1+1+s=2k+s+3\) for the total number of coefficients in the dynamic regression. The five blocks correspond to the adjustment coefficient, the \(2k\) detrended partial sums, the trend, the intercept, and the \(s\) short-run regressors. Thus, \(\ddot\eta_T\) and \(\ddot m_{t-1}\) are \(2k\times1\), while \(\ddot\kappa_T\) and \(z_{2t}\) are \(s\times1\). Both inner products in the equation are scalars: \((1\times2k)(2k\times1)=1\times1\) and \((1\times s)(s\times1)=1\times1\).

Replacing \(\ddot m_{t-1}\) by \(\ddot x_{t-1}-q_t\ddot\mu_T\) gives

\[ \widetilde\rho_T=\ddot\rho_T,\quad \widetilde\eta_T=\ddot\eta_T,\quad \widetilde\psi_T=\ddot\varsigma_T-\ddot\mu_T’\ddot\eta_T,\quad \widetilde\gamma_T=\ddot\gamma_T,\quad \widetilde\kappa_T=\ddot\kappa_T. \]

Only the explicit trend coefficient changes. The correction now uses \(\ddot\eta_T\), the coefficients from the dynamic regression. Writing these identities as \(\widetilde\omega_T=\mathbf Q_T\ddot\tau_T\), the matrix \(\mathbf Q_T\) is an identity matrix with the additional entries \(-\ddot\mu_T’\) in the trend row and the \(\ddot\eta_T\) columns.

\[ \mathbf Q_T=\begin{pmatrix}I_{1+2k}&0\\\mathbf Q_T^{21}&I_{s+2}\end{pmatrix}, \qquad \mathbf Q_T^{21}=\begin{pmatrix}0&-\ddot\mu_T’\\0_{(s+1)\times1}&0_{(s+1)\times2k}\end{pmatrix}. \]

The first block contains the adjustment coefficient and the partial-sum coefficients. The lower block contains the trend, constant, and short-run coefficients. Its leading zero means that the trend correction contains no adjustment-coefficient term at this step: the cointegration residual itself remains unchanged.

The matrix \(\mathbf Q_T\) is \(K\times K\). Its upper block has \(1+2k\) rows and its lower block has \(s+2\) rows. Consequently, \(\mathbf Q_T^{21}\) is \((s+2)\times(1+2k)\), and the upper-right zero block is \((1+2k)\times(s+2)\). Within \(\mathbf Q_T^{21}\), the first row contains one scalar zero followed by the \(1\times2k\) row \(-\ddot\mu_T’\). The remaining \(s+1\) rows are zeros. This locates the correction in the trend equation while leaving the intercept and short-run coefficients unchanged.

E.3. Reconstruct the original one-step estimator

Proposition 1 already gives \(\widehat{\boldsymbol\alpha}_T=\mathbf R_T\widetilde\omega_T\), using the identity defining the estimated cointegration residual. Combining the transformations gives Proposition 4:

\[ \mathbf R_T=\operatorname{diag}\left[ \begin{pmatrix}1&0_{1\times(2k+2)}\\-\widetilde\upsilon_T&I_{2k+2}\end{pmatrix},I_s\right]. \]

The negative signs come directly from \(\widetilde u_{t-1}=y_{t-1}-\widetilde\beta_T’\ddot x_{t-1}-\widetilde\zeta_Tq_t-\widetilde\nu_T\). Multiplying this identity by the estimated adjustment coefficient distributes its contribution across the lagged dependent variable, partial sums, trend, and constant.

The vector \(\widetilde\upsilon_T\) is \((2k+2)\times1\). The first diagonal block of \(\mathbf R_T\) therefore has \(2k+3\) rows and columns: one for the adjustment coefficient and \(2k+2\) for the auxiliary coefficients. Appending the \(s\times s\) identity block gives a \(K\times K\) matrix. Thus \(\mathbf R_T\), \(\mathbf Q_T\), and their product \(\mathbf T_T\) are all \(K\times K\); each maps one \(K\times1\) coefficient vector into another.

\[ \widehat{\boldsymbol\alpha}_T =\mathbf R_T\mathbf Q_T\ddot\tau_T =\mathbf T_T\ddot\tau_T. \]

The transformations act from right to left. The bold matrix \(\mathbf T_T\) depends on the sample; it is distinct from the scalar sample size \(T\). For \(k=1\), the full matrix has size \((s+5)\times(s+5)\). Its leading \(5\times5\) block maps the five coefficients shown below; the omitted columns for the short-run coefficients are zeros in these five rows:

\[ \begin{pmatrix}\widehat\rho_T\\\widehat\theta_T^+\\\widehat\theta_T^-\\\widehat\xi_T\\\widehat\alpha_T\end{pmatrix} = \begin{pmatrix} 1&0&0&0&0\\ -\ddot\beta_T^+&1&0&0&0\\ -\ddot\beta_T^-&0&1&0&0\\ -\widetilde\zeta_T&-\ddot\mu_T^+&-\ddot\mu_T^-&1&0\\ -\ddot\nu_T&0&0&0&1 \end{pmatrix} \begin{pmatrix}\ddot\rho_T\\\ddot\eta_T^+\\\ddot\eta_T^-\\\ddot\varsigma_T\\\ddot\gamma_T\end{pmatrix}. \]

The remaining block is an identity matrix for the short-run coefficients. Thus every original coefficient is recovered:

\[ \begin{aligned} \widehat\rho_T&=\ddot\rho_T,\qquad \widehat\theta_T=\ddot\eta_T-\ddot\rho_T\ddot\beta_T,\\ \widehat\xi_T&=\ddot\varsigma_T-\ddot\rho_T\widetilde\zeta_T -\ddot\mu_T’\ddot\eta_T,\\ \widehat\alpha_T&=\ddot\gamma_T-\ddot\rho_T\ddot\nu_T, \qquad\widehat\kappa_T=\ddot\kappa_T. \end{aligned} \]

Using Proposition 2, the trend formula also becomes \(\widehat\xi_T=\ddot\varsigma_T-\ddot\rho_T\ddot\vartheta_T-\ddot\mu_T’\widehat\theta_T\). These are exact finite-sample identities wherever the required regressions have full rank. They retain all regressors and their jointly estimated coefficients. Their purpose is to expose the components whose probability limits must now be studied.

Each transformation matrix is triangular with ones on its diagonal, so its determinant is one. This establishes a one-to-one correspondence between coefficient vectors. The auxiliary residual is orthogonal to the regressors in the long-run regression, but it can remain correlated with the short-run regressors. Keeping those regressors in the dynamic equation preserves the complete original estimation problem.

Main takeaway. Detrending transfers the fitted drift contributions into the explicit trend coefficient. Combining this transformation with the first FWL identity recovers every original coefficient, including the intercept and short-run coefficients.

Source: Cho, Section 3, Propositions 2–4. Return to the main text: Section 5: detrending · Section 6: recovering the original coefficients · Contents.

Appendix F. Lemmas 2–3: nonsingular limits and the long-run estimator

Return to the main text: Section 5: the transformed matrix · Section 7: probability limits · Contents.

F.1. Preserve the stochastic fluctuations at their own scale

Removing a fitted trend leaves integrated stochastic variation. Indeed, if \(\ddot x_{t-1}=\mu_*q_t+m_{t-1}\), then

\[ \ddot m_{t-1}=m_{t-1}-q_t(\ddot\mu_T-\mu_*). \]

The drift-estimation error is \(O_p(T^{-1/2})\). Multiplying it by a time index of order \(T\) gives a component of order \(\sqrt T\), the same scale as the centered integrated process. The detrended partial sums consequently have sums of squares of order \(T^2\). Their column normalizer is \(T\); the trend and constant retain normalizers \(T^{3/2}\) and \(\sqrt T\).

The auxiliary regressor vector \(\ddot r_{t-1}\) below stacks a \(2k\times1\) vector and two scalars, so it is \((2k+2)\times1\). Its normalization matrix \(\ddot D_1\) is \((2k+2)\times(2k+2)\). In \(TI_{2k}\), the \(2k\times2k\) identity applies the same divisor \(T\) separately to each detrended partial sum.

\[ \ddot r_{t-1}=(\ddot m_{t-1}’,q_t,1)’,\qquad \ddot D_1=\operatorname{diag}(TI_{2k},T^{3/2},T^{1/2}). \]

The functional central limit theorem expresses scaled cumulative innovations as Brownian paths. Here \(\mathcal B_s(r)\) is the limiting path of the centered partial sums at sample fraction \(r\in[0,1]\); its covariance follows the underlying process. It need not be standard Brownian motion. The fitted-trend subtraction produces

\[ \overline{\mathcal B}_s(r) =\mathcal B_s(r)-3r\int_0^1v\mathcal B_s(v)\,dv. \]

The factor three is the inverse of \(\int_0^1r^2dr=1/3\), the limiting squared length of the trend. The projection ensures \(\int_0^1r\overline{\mathcal B}_s(r)dr=0\), matching the exact sample orthogonality.

At each sample fraction \(r\), both \(\mathcal B_s(r)\) and \(\overline{\mathcal B}_s(r)\) are \(2k\times1\) vectors. Their subscript labels the centered-increment process \(s_j\). The integral \(\int_0^1v\mathcal B_s(v)\,dv\) is also \(2k\times1\), because integration is performed separately for each component. Multiplication by the scalar \(3r\) preserves that size.

To make the matrix transparent, write its entries directly as the limiting Gram matrix of these functions:

\[ \mathcal M_{11}= \begin{pmatrix} \int_0^1\overline{\mathcal B}_s\overline{\mathcal B}_s’\,dr &0&\int_0^1\overline{\mathcal B}_s\,dr\\ 0’&1/3&1/2\\ \int_0^1\overline{\mathcal B}_s’\,dr&1/2&1 \end{pmatrix}. \]

This direct representation uses the detrended path just defined. Under the paper’s nondegeneracy assumptions, the stochastic functions preserve independent directions alongside the constant and trend, and Lemma 2 establishes nonsingularity almost surely. The full transformed dynamic regression likewise has a nonsingular limiting moment matrix \(\mathcal M\). Its inverse can therefore enter the subsequent limit calculations.

Reading the blocks of \(\mathcal M_{11}\). This is a \((2k+2)\times(2k+2)\) matrix, partitioned into block sizes \(2k,1,1\). Its upper-left block is \(2k\times2k\): the product of a \(2k\times1\) Brownian vector and its \(1\times2k\) transpose. The upper-middle zero and the upper-right integral are \(2k\times1\) columns. Their transposes occupy the corresponding \(1\times2k\) row blocks. The bottom-right four entries are scalars for the trend and constant. The full dynamic-regression matrix \(\mathcal M\) has the larger size \(K\times K\), where \(K=2k+s+3\) counts the dynamic-regression coefficients.

A quadratic form clarifies what nonsingularity requires. For a vector \(b\) and scalars \(c,\ell\), consider the corresponding quadratic form of this Gram matrix.

Here \(b\) is \(2k\times1\), while \(c\) and \(\ell\) are scalars. Stacking them gives \((b’,c,\ell)’\), a \((2k+2)\times1\) vector. The product \(b’\overline{\mathcal B}_s(r)\) is a scalar, so it can be added to \(cr+\ell\) and squared in the integral.

\[ \int_0^1\left[b’\overline{\mathcal B}_s(r)+cr+\ell\right]^2dr. \]

It could vanish only if that combination of functions were zero throughout the interval. Under the maintained assumptions, a nonzero combination of the stochastic paths cannot be reproduced exactly by a constant and a linear time trend. This identifies the information retained by the new normalization. In the earlier normalization, the fluctuations disappeared and the trending columns collapsed onto a common deterministic direction.

The zero cross-block with the trend has a narrower meaning: it records the fitted projection. The cross-block with the constant generally survives because detrending through the origin does not demean the partial sums. Thus the matrix retains both the stochastic variation and its association with the constant. Orthogonality to one regressor does not imply orthogonality to all the other regressors.

F.2. Why the trend coefficient has a sample-dependent target

Expressing the true long-run relationship in the transformed regressors gives

\[ y_{t-1}=\beta_*’\ddot m_{t-1}+\vartheta_{T*}q_t+\nu_*+u_{t-1}, \qquad \vartheta_{T*}=\zeta_*+\ddot\mu_T’\beta_*. \]

The target changes with the fitted drift because that drift defines the transformed regressors. Its fixed probability limit is \(\vartheta_*:=\zeta_*+\mu_*’\beta_*\). Define \(\overline\upsilon_{T*}=(\beta_*’,\vartheta_{T*},\nu_*)’\). The OLS error has the familiar matrix form

\[ \begin{aligned} \ddot D_1(\ddot\upsilon_T-\overline\upsilon_{T*}) &= \left[\ddot D_1^{-1}\sum\ddot r_{t-1}\ddot r_{t-1}’\ddot D_1^{-1}\right]^{-1} \left[\ddot D_1^{-1}\sum\ddot r_{t-1}u_{t-1}\right]\\ &\Rightarrow\mathcal L:=\mathcal M_{11}^{-1}\mathcal S. \end{aligned} \]

The Brownian paths summarize how the integrated regressors vary over the entire sample. Their randomness remains in the moment matrix as well as in the score. Both objects therefore enter the coefficient distribution jointly; replacing the random moment matrix by its expectation would change the limit.

The dimension calculation in the OLS formula is the same on both sides. The normalized cross-product matrix and \(\mathcal M_{11}^{-1}\) are \((2k+2)\times(2k+2)\); the normalized score and its limit \(\mathcal S\) are \((2k+2)\times1\). Their product is therefore the \((2k+2)\times1\) coefficient-error limit \(\mathcal L\). Its first block \(\mathcal L_1\) has \(2k\) entries; \(\mathcal L_2\) and \(\mathcal L_3\) are scalars.

Lemma 3(i) partitions \(\mathcal L=(\mathcal L_1′,\mathcal L_2,\mathcal L_3)’\). Reading its normalization component by component gives

\[ \begin{aligned} T(\ddot\beta_T-\beta_*)&\Rightarrow\mathcal L_1,\\ T^{3/2}(\ddot\vartheta_T-\vartheta_{T*})&\Rightarrow\mathcal L_2,\\ \sqrt T(\ddot\nu_T-\nu_*)&\Rightarrow\mathcal L_3. \end{aligned} \]

The second line centers at \(\vartheta_{T*}\). Centering instead at \(\vartheta_*\) introduces \(\beta_*'(\ddot\mu_T-\mu_*)\), generally of order \(T^{-1/2}\). Keeping the target explicit prevents the \(T^{3/2}\) normalization from being misinterpreted.

For the slope, the statement \(T(\ddot\beta_T-\beta_*)\Rightarrow\mathcal L_1\) means that multiplication by \(T\) preserves a nontrivial limiting fluctuation. This implies that the unscaled error is of order \(T^{-1}\). The intercept requires only \(\sqrt T\), corresponding to an error of order \(T^{-1/2}\). These rates are read from the normalization of the coefficient errors after the limiting matrix and score have been established.

The arrow \(\Rightarrow\) describes convergence in distribution: the probabilities attached to the scaled error approach those specified by its limiting random vector. For example, for one slope component, the probability that \(T(\ddot\beta_{j,T}-\beta_{j,*})\) falls in a given interval approaches the corresponding probability under \(\mathcal L_{1,j}\), provided the interval endpoints carry no probability mass in that limiting distribution.

Multiplication by \(T\) magnifies the shrinking slope error. Without that multiplication, the error converges to zero in probability; with it, the uncertainty remains visible in a stable limiting distribution. Brownian paths describe the randomness retained from the integrated regressors over the sample. This is why the transformed moment matrix and the coefficient-error limit remain random even though the original normalized matrix in Appendix C converged to a fixed matrix.

F.3. Read the stochastic-integral vector

Let \(\mathcal B_u(r)\) denote the Brownian limit of the normalized cumulative cointegration errors, \(T^{-1/2}\sum_{t=1}^{\lfloor Tr\rfloor}u_t\). It is considered jointly with \(\mathcal B_s(r)\); the two processes need not be independent. With every integral below taken over \([0,1]\), the three score limits are

\[ \begin{aligned} \mathcal S_1&=\int\mathcal B_s\,d\mathcal B_u -3\left(\int r\mathcal B_s\,dr\right)\left(\int r\,d\mathcal B_u\right) +\varrho_{m*},\\ \mathcal S_2&=\int r\,d\mathcal B_u,\qquad \mathcal S_3=\int d\mathcal B_u. \end{aligned} \]

An integral with \(dr\) accumulates a random path over time. An integral with \(d\mathcal B_u\) accumulates weighted Brownian increments. In particular, \(\int d\mathcal B_u=\mathcal B_u(1)\). The first component combines the centered partial sums with the cointegration error; its subtraction accounts for the fitted trend.

In this score vector, \(\mathcal B_u\) is scalar. Hence \(\mathcal S_1\) is \(2k\times1\), whereas \(\mathcal S_2\) and \(\mathcal S_3\) are scalars. The correction \(\varrho_{m*}\) below is \(2k\times1\), since each product \(s_ju_t\) multiplies a \(2k\times1\) increment vector by a scalar error. Together the three blocks form \(\mathcal S=(\mathcal S_1′,\mathcal S_2,\mathcal S_3)’\), with \(2k+2\) entries.

The additional term is

\[ \varrho_{m*}:=\lim_{T\to\infty}\frac1T \sum_{t=1}^{T}\sum_{j=1}^{t-1}E[s_ju_t], \qquad s_j=\Delta\ddot x_j-\mu_*. \]

It records dependence between past centered increments and the cointegration error. This contributes to the asymptotic bias component of the normalized limit. Consistency remains compatible with that contribution because the unscaled slope error shrinks at rate \(T^{-1}\). The primitive limit is generally nonstandard; mixed normality of the final one-step estimator is established later.

F.4. Recover the original trend coefficient

Combining the definition of \(\vartheta_{T*}\) with Proposition 2 gives an exact cancellation:

\[ \widetilde\zeta_T-\zeta_* =(\ddot\vartheta_T-\vartheta_{T*}) -\ddot\mu_T'(\ddot\beta_T-\beta_*). \]

After multiplication by \(T\), the first term is \(O_p(T^{-1/2})\), while the second retains the slope-estimation limit. Therefore

\[ T(\widetilde\zeta_T-\zeta_*)\Rightarrow-\mu_*’\mathcal L_1, \qquad \ddot D_\dagger:=\operatorname{diag}(TI_{2k},T,T^{1/2}). \]

This explains Lemma 3(ii): the original slope and explicit-trend estimates share \(\mathcal L_1\), while the intercept retains \(\mathcal L_3\). Restoring the original partial sums reallocates their estimated trend contribution to the explicit trend coefficient.

The replacement normalizer \(\ddot D_\dagger\) is again \((2k+2)\times(2k+2)\); only the trend’s multiplier changes. The limit \(-\mu_*’\mathcal L_1\) is scalar because it multiplies a \(1\times2k\) row by a \(2k\times1\) column.

Main takeaway. The transformed moment matrix retains the random variation of the detrended partial sums and has a nonsingular limit. This gives the auxiliary coefficient distributions; the trend coefficient must be centered at the target implied by the fitted drift.

Source: Cho, Lemmas 2–3. The Gram matrix above is written directly from the stated detrended Brownian path. Return to the main text: Section 5: the transformed matrix · Section 7: probability limits · Contents.

Appendix G. Lemma 4: account for the estimated cointegration residual

Return to the main text: Section 7: generated regressors and Lemma 4 · Contents.

G.1. The generated regressor carries first-stage estimation error

The dynamic regression uses \(\ddot u_{t-1}\), whereas the population error-correction equation contains \(u_{t-1}\). Their difference is determined exactly by the auxiliary long-run estimates. Put

\[ b_T=\ddot\beta_T-\beta_*,\qquad v_T=\ddot\vartheta_T-\vartheta_{T*},\qquad n_T=\ddot\nu_T-\nu_*. \]

Subtracting the fitted long-run relation from the true relation gives

\[ u_{t-1}=\ddot u_{t-1}+b_T’\ddot m_{t-1}+v_Tq_t+n_T. \]

Consequently, the equation \(\Delta y_t=\rho_*u_{t-1}+\gamma_*+\kappa_*’z_{2t}+e_t\) can be expressed in the generated regressors as

\[ \Delta y_t=\rho_*\ddot u_{t-1} +\rho_*b_T’\ddot m_{t-1}+\rho_*v_Tq_t +\gamma_*+\rho_*n_T+\kappa_*’z_{2t}+e_t. \]

The retained levels and trend have sample-dependent coefficients. Their fixed population targets are zero, but their coefficients in this exact reparameterization reflect first-stage estimation error. This is why their contribution must be followed through the asymptotic calculation.

For a concrete scalar illustration, suppose the estimated long-run slope exceeds its true value by \(0.02\), and \(\rho_*=-0.5\). The fitted cointegration residual subtracts too much of that regressor. Restoring the true dynamic equation requires a coefficient \(\rho_*b_T=-0.01\) on its retained detrended level. This coefficient compensates exactly for the error already embedded in the generated residual.

Collecting these coefficients gives the sample-dependent target \(\tau_{T*}\) and its fixed population counterpart \(\tau_*\):

\[ \tau_{T*}= \begin{pmatrix}\rho_*\\\rho_*b_T\\\rho_*v_T\\\gamma_*+\rho_*n_T\\\kappa_*\end{pmatrix}, \qquad \tau_*= \begin{pmatrix}\rho_*\\0_{2k}\\0\\\gamma_*\\\kappa_*\end{pmatrix}. \]

Dimensionally, \(b_T\) is \(2k\times1\), while \(v_T\) and \(n_T\) are scalars. The product \(b_T’\ddot m_{t-1}\) is therefore scalar. Both target vectors displayed above have block lengths \(1,2k,1,1,s\), giving \(K=2k+s+3\) entries. The notation \(0_{2k}\) denotes a \(2k\times1\) column of zeros.

G.2. Part (i): center at the coefficient vector of the generated-regressor equation

Stack the transformed regressors as \(\ddot z_t=(\ddot u_{t-1},\ddot m_{t-1}’,q_t,1,z_{2t}’)’\). Their column normalizer is

\[ \ddot D=\operatorname{diag}(T^{1/2},TI_{2k},T^{3/2},T^{1/2},T^{1/2}I_s). \]

The first block corresponds to the cointegration residual, the next to integrated fluctuations, and the third to the deterministic trend. The constant and short-run regressors complete the stationary blocks.

The regressor vector \(\ddot z_t\), coefficient vector \(\ddot\tau_T\), and target vector \(\tau_{T*}\) are each \(K\times1\). Stacking the rows \(\ddot z_t’\) over all dates gives a \(T\times K\) regression matrix. Both \(\ddot D\) and the summed cross-product \(\sum_t\ddot z_t\ddot z_t’\) are \(K\times K\). The score \(\sum_t\ddot z_te_t\) is \(K\times1\). Thus the inverse matrix times the score has the required \(K\times1\) size. Normalizing the exact OLS identity gives

\[ \begin{aligned} \ddot D(\ddot\tau_T-\tau_{T*}) &=\left[\ddot D^{-1}\sum\ddot z_t\ddot z_t’\ddot D^{-1}\right]^{-1} \left[\ddot D^{-1}\sum\ddot z_te_t\right]\\ &\Rightarrow\mathcal D:=\mathcal M^{-1}\mathcal J. \end{aligned} \]

The inverse is now legitimate because Lemma 2 establishes a nonsingular limit \(\mathcal M\). The random vector \(\mathcal D\), distinct from the normalization matrix \(\ddot D\), has five blocks:

\[ \mathcal D=(\mathcal D_1,\mathcal D_2′,\mathcal D_3,\mathcal D_4,\mathcal D_5′)’, \quad \mathcal J= \begin{pmatrix} \int d\mathcal B_{ue}\\ \int\overline{\mathcal B}_s\,d\mathcal B_e\\ \int r\,d\mathcal B_e\\ \int d\mathcal B_e\\ \int d\mathcal B_{ze} \end{pmatrix}. \]

Each score block pairs its regressor with the innovation \(e_t\). Here \(\mathcal B_e(r)\) is the Brownian limit of \(T^{-1/2}\sum_{t=1}^{\lfloor Tr\rfloor}e_t\). The processes \(\mathcal B_{ue}\) and \(\mathcal B_{ze}\) arise from normalized cumulative stationary products \(u_{t-1}e_t\) and \(z_{2t}e_t\). The subscript \(ue\) labels a process; it does not mean a product of two Brownian motions.

The matrix \(\mathcal M\) is \(K\times K\), while \(\mathcal J\) and \(\mathcal D=\mathcal M^{-1}\mathcal J\) are \(K\times1\). Their five block sizes are \(1,2k,1,1,s\). Accordingly, \(\mathcal D_1,\mathcal D_3,\mathcal D_4\) are scalars, \(\mathcal D_2\) is \(2k\times1\), and \(\mathcal D_5\) is \(s\times1\). The processes \(\mathcal B_e\) and \(\mathcal B_{ue}\) are scalar; \(\mathcal B_{ze}\) is \(s\times1\).

The second entry is the regression-limit expression

\[ \int\overline{\mathcal B}_s\,d\mathcal B_e :=\int\mathcal B_s\,d\mathcal B_e -3\left(\int r\mathcal B_s\,dr\right)\left(\int r\,d\mathcal B_e\right). \]

This expanded definition retains the effect of estimating the trend from the entire sample. Lemma 4(i) establishes mixed normality: conditional on the relevant limiting regressor information, the distribution is Gaussian, with a covariance matrix that can depend on that information.

G.3. Part (ii): change from the random target to the fixed target

The change of centering follows an elementary addition:

\[ \ddot\tau_T-\tau_* =(\ddot\tau_T-\tau_{T*})+(\tau_{T*}-\tau_*). \]

Lemma 3 supplies the scaled limit of the second term. Joint convergence therefore gives

\[ \ddot D(\ddot\tau_T-\tau_*) \Rightarrow\mathcal D+\rho_*[0,\mathcal L’,0′]’. \]

The limits \(\mathcal D\) and \(\mathcal L\) originate from the same observations and must be treated jointly. Their dependence matters for the cancellations in the final OLS estimator. Centering at \(\tau_*\) generally adds the nonstandard first-stage component, so the mixed-normal conclusion of part (i) cannot simply be transferred to this differently centered vector.

The added vector \([0,\mathcal L’,0′]’\) has the same length as \(\mathcal D\): its first zero is scalar, its middle block contains the \(2k+2\) entries of \(\mathcal L\), and its last zero block has \(s\) entries. The total is \(1+(2k+2)+s=K\). It inserts the first-stage slope, trend, and intercept limits in their corresponding dynamic-regression positions.

The distinction can also be read coefficient by coefficient. The adjustment coefficient and short-run coefficients have the same targets in \(\tau_{T*}\) and \(\tau_*\). Their scaled limits remain \(\mathcal D_1\) and \(\mathcal D_5\). The intervening blocks acquire \(\rho_*\mathcal L_1\), \(\rho_*\mathcal L_2\), and \(\rho_*\mathcal L_3\). These additions arise because the estimated residual carries slope, trend, and intercept estimation error simultaneously.

Returning to the scalar illustration, suppose the estimated coefficient on the retained level is \(-0.006\). Its deviation from the random target \(-0.01\) is \(0.004\); its deviation from the fixed target zero is \(-0.006\). The centering identity reads \(-0.006=0.004+(-0.01)\). The first-stage adjustment is added with its own sign. The same accounting applies to every affected coordinate of the normalized vector.

Main takeaway. The estimated cointegration residual carries first-stage estimation error into the dynamic equation. Tracking that error in the coefficient target gives the correct joint limit and prepares the cancellations used to reconstruct OLS.

Source: Cho, Section 4.2 and Lemma 4, with the sign discrepancy identified explicitly above. Return to the main text: Section 7: generated regressors and Lemma 4 · Contents.

Appendix H. From the primitive limits to the original OLS estimator

Return to the main text: Section 8: coefficient convergence rates · Section 9: implications for applied work · Contents.

H.1. Equation (15) is an exact product expansion

Proposition 4 gives \(\widehat{\boldsymbol\alpha}_T=\mathbf T_T\ddot\tau_T\). Its population counterpart is \(\boldsymbol\alpha_* =\mathbf T\tau_*\), where \(\mathbf T\) is the probability limit of \(\mathbf T_T\). Adding and subtracting the corresponding products gives

\[ \begin{aligned} \widehat{\boldsymbol\alpha}_T-\boldsymbol\alpha_* ={}&(\mathbf T_T-\mathbf T)\tau_*\\ &+\mathbf T(\ddot\tau_T-\tau_*)\\ &+(\mathbf T_T-\mathbf T)(\ddot\tau_T-\tau_*). \end{aligned} \]

The first term is error in the transformation, the second is error in the primitive estimator, and the third is their product. The matrices and coefficient vectors are estimated jointly. Their orders can nevertheless be multiplied without assuming independence: globally, the last term is \(O_p(T^{-1})\), so multiplying it by \(\sqrt T\) makes it vanish in probability.

With \(K=2k+s+3\) coefficients in the full model, both transformation matrices \(\mathbf T_T\) and \(\mathbf T\) are \(K\times K\); every coefficient vector in this identity is \(K\times1\). Each term therefore multiplies a \(K\times K\) matrix by a \(K\times1\) vector, yielding a \(K\times1\) vector that can be added to the other two.

H.2. Identify the leading error coefficient by coefficient

Put \(a_T=\ddot\rho_T-\rho_*\), retaining \(b_T\) and \(n_T\) from Appendix G. The identity \(\widehat\theta_T=\ddot\eta_T-\ddot\rho_T\ddot\beta_T\), together with \(\theta_*=-\rho_*\beta_*\), gives the exact expansion

\[ \widehat\theta_T-\theta_* =-\beta_*a_T +[\ddot\eta_T-\rho_*b_T]-a_Tb_T. \]

The three terms have orders \(T^{-1/2}\), \(T^{-1}\), and \(T^{-3/2}\) when the corresponding component of \(\beta_*\) is nonzero. Multiplication by \(\sqrt T\) leaves

\[ \sqrt T(\widehat\theta_T-\theta_*) =-\beta_*\sqrt T a_T+o_p(1) \Rightarrow-\beta_*\mathcal D_1. \]

The dominant uncertainty comes from estimating the adjustment coefficient. The long-run slope estimate converges faster, and its remaining contribution is combined with \(\ddot\eta_T\) in the bracket governed by Lemma 4(i). Retaining the product term above keeps the expansion exact before the asymptotic simplification.

The slope expansion is a vector identity: \(\widehat\theta_T\), \(\theta_*\), \(\beta_*\), \(\ddot\eta_T\), and \(b_T\) are all \(2k\times1\). The adjustment error \(a_T\) and limit \(\mathcal D_1\) are scalars, so multiplication by them preserves this vector size. The intercept and trend expansions that follow are scalar identities; the short-run coefficient identity is \(s\times1\).

The powers can be compared numerically without assigning values to the stochastic coefficients. At \(T=100\), the magnitudes \(T^{-1/2}\), \(T^{-1}\), and \(T^{-3/2}\) are \(0.1\), \(0.01\), and \(0.001\). At \(T=400\), they are \(0.05\), \(0.0025\), and \(0.000125\). The second and third scales shrink more quickly. Multiplying by \(\sqrt T\) preserves the first scale while the others approach zero.

The intercept requires two leading components:

\[ \widehat\alpha_T-\alpha_* =-\nu_*a_T +[(\ddot\gamma_T-\gamma_*)-\rho_*n_T]-a_Tn_T. \]

The bracket has order \(T^{-1/2}\) and scaled limit \(\mathcal D_4\). It subtracts the first-stage intercept error embedded in the generated-regressor coefficient. The final product is \(O_p(T^{-1})\). Similarly,

\[ \widehat\xi_T-\xi_*=-\zeta_*a_T+O_p(T^{-1}), \qquad \widehat\kappa_T-\kappa_*=\ddot\kappa_T-\kappa_*. \]

H.3. Equation (16) collects the surviving terms

The first column of \(\mathbf T\) collects the coefficients multiplying the adjustment error:

\[ c_*=(1,-\beta_*’,-\zeta_*,-\nu_*,0′)’, \quad d_T= \begin{pmatrix} 0\\0_{2k}\\0\\ (\ddot\gamma_T-\gamma_*)-\rho_*(\ddot\nu_T-\nu_*)\\ \ddot\kappa_T-\kappa_* \end{pmatrix}. \]

Stacking the component expansions gives equation (16). When every relevant component of \(\beta_*^+\), \(\beta_*^-\), and \(\zeta_*\) is nonzero, this yields the generic normalization in Theorem 1:

\[ \begin{aligned} \sqrt T(\widehat{\boldsymbol\alpha}_T-\boldsymbol\alpha_*) &=c_*\sqrt T a_T+\sqrt T d_T+o_p(1)\\ &\Rightarrow c_*\mathcal D_1+ \begin{pmatrix}0\\0_{2k}\\0\\\mathcal D_4\\\mathcal D_5\end{pmatrix}. \end{aligned} \]

The first-stage terms have canceled from this final limit. The remaining vector is determined by \(\mathcal D\) and is mixed normal. Several coefficients share the same dominant random component \(\mathcal D_1\), which explains the dependence of their limiting estimation errors.

Both \(c_*\) and \(d_T\) have \(K\) entries, partitioned as \(1,2k,1,1,s\). The trailing zero in \(c_*\) has \(s\) entries. In the displayed limit, \(c_*\mathcal D_1\) is a \(K\times1\) vector and the added column also has \(K\) entries: one zero, \(2k\) zeros, one zero, the scalar \(\mathcal D_4\), and the \(s\times1\) block \(\mathcal D_5\). This verifies that each original coefficient receives exactly one corresponding limit entry.

The common component has a direct interpretation. Holding the long-run slopes fixed, changing the estimated adjustment speed changes every equation-level coefficient through \(\theta_*=-\rho_*\beta_*\). Their leading estimation errors therefore move together, with loadings given by the long-run coefficients. The smaller terms still distinguish the separate estimates within an observed sample and become decisive when a leading loading vanishes.

H.4. A zero leading coefficient changes the informative rate

If a component of \(\beta_*\) equals zero, its contribution \(-\beta_*a_T\) disappears. The \(T^{-1}\) bracket then determines that coefficient’s first nonzero fluctuation. The same logic applies to the trend coefficient when \(\zeta_*=0\). The four cases stated in Theorem 1 are summarized below; parameters not set to zero satisfy the theorem’s stated nonzero conditions.

Table notes. Apply each multiplier to the estimation error, meaning the estimate minus its population target. For example, the second row uses \(T(\widehat\theta_T^+-\theta_*^+)\) when the entire positive long-run slope block \(\beta_*^+\) is zero. A multiplier \(T\) corresponds to errors on the \(T^{-1}\) scale; \(\sqrt T\) corresponds to the \(T^{-1/2}\) scale. These rows describe the theorem’s specified parameter cases.

For \(k>1\), a zero \(\beta_*^+\) or \(\beta_*^-\) in these listed cases means the corresponding whole block is zero. The faster limits are \(\mathcal D_2^+\), \(\mathcal D_2^-\), and \(-\mu_*’\mathcal D_2\), respectively. The \(\sqrt T\) expansion still holds in those directions, but yields zero; the finer normalization reveals their remaining variation.

The positive and negative slope blocks \(\widehat\theta_T^+\) and \(\widehat\theta_T^-\) are each \(k\times1\). Their limit blocks \(\mathcal D_2^+\) and \(\mathcal D_2^-\) also have \(k\) entries and together form the \(2k\times1\) vector \(\mathcal D_2\). The trend limit \(-\mu_*’\mathcal D_2\) is scalar. The long-run ratio below is a \(2k\times1\) vector divided by a scalar adjustment coefficient, so its limit has \(2k\) entries.

The mechanism can be represented by an error of the form \(a/\sqrt T+b/T\). When the leading coefficient \(a\) is nonzero, the first term dominates, and the \(\sqrt T\) multiplier preserves it. If \(a=0\), the error becomes \(b/T\), and multiplication by \(T\) is needed to preserve a nonzero limit. For the estimator, the error terms also contain random factors, while a zero population loading removes the first contribution.

The zero-parameter cases therefore describe changes in which source of estimation uncertainty is dominant. They do not require changing the computational OLS procedure. The coefficient identity remains the same; its expansion reveals that a term previously of smaller order now provides the leading information. This is why the theorem states separate parameter cases instead of assigning one informative normalization to every possible coefficient configuration.

Comparisons with two-step NARDL should keep the target parameter fixed. A two-step estimator of \(\beta_*\) converges at rate \(T\), while its estimator of \(\rho_*\) has rate \(\sqrt T\). Their product estimates \(\theta_*=-\rho_*\beta_*\) at the generic \(\sqrt T\) rate, matching the corresponding one-step coefficient.

A further algebraic deduction concerns the one-step long-run ratio. For fixed \(\rho_*\ne0\), define \(\widehat\beta_T^{\mathrm{ratio}}=-\widehat\theta_T/\widehat\rho_T\). The exact reconstruction gives

\[ \begin{aligned} \widehat\beta_T^{\mathrm{ratio}}-\beta_* &=\frac{a_Tb_T}{\ddot\rho_T} -\frac{\ddot\eta_T-\rho_*b_T}{\ddot\rho_T},\\ T(\widehat\beta_T^{\mathrm{ratio}}-\beta_*) &\Rightarrow-\frac{\mathcal D_2}{\rho_*}. \end{aligned} \]

This deduction from the identities and lemmas distinguishes the superconsistent long-run ratio from the equation coefficient \(\widehat\theta_T\). The common adjustment error cancels in the ratio. Throughout these results, the maintained cointegrating framework has \(\rho_*<0\).

Main takeaway. The exact product expansion identifies which estimation error governs each original coefficient. It explains the common adjustment-error component, the faster rates when that component vanishes, and its cancellation in the long-run ratios.

Source: Cho, Section 4.3, equations (15)–(17), and Theorem 1. The ratio result is an explicit deduction from the preceding identities. Return to the main text: Section 8: coefficient convergence rates · Section 9: implications for applied work · Contents.