A comment on ‘resolving spurious regressions and serially correlated errors’

Abstract

In order to diminish size distortions of the t test in a time series linear specification, Agiakloglou (Agiakloglou in Empir Econ, 45(3):1361–1366, 2013) proposed to (1) include the first lag of the dependent variable as a regressor or (2) estimate it using the first differences of the variables. He provided finite-sample evidence to support his proposal. In this paper, we extend the Monte Carlo experiment to different data-generating processes and calculate the asymptotic behavior of the modified specifications. We show that including the lag of the dependent variable as a regressor reduces size distortions when the variables are driftless unit roots, but this approach does not hold under the presence of long memory, nonlinearities, or structural breaks.

Appendices

Appendix 1: Proof of Theorem

1. Model 2: Given that $x_t$ and $y_t$ are unit root processes and the regression is estimated using the variables in first differences, the results in part (1) of the Theorem are straightforward. First differences of both unit root processes behave as simple iid I(0) independent processes. Therefore, classic econometric theory applies, so we know that, under the null, the t ratios converge to a standard normal random variable (rv). Note that the standard Brownian motion, $w_{x,y}(\cdot )$, stated in the first part of the Theorem, is precisely a standard Normal rv. Note too that, when the differenced variables are not iid but rather stationary ARMA processes (for instance), typical autocorrelation ensues.

2. Model 3: We present a guide as to how to obtain the asymptotic expression of the estimates and their associated t ratios using OLS where the variables $y_t$ and $x_t$ are generated as independent unit root processes, and specification 3 is estimated. Let $\hat{\varTheta }=(\hat{\alpha },\hat{\beta },\hat{\delta })'$:

$$\begin{aligned} \hat{\varTheta }&= \left( X'X\right) ^{-1}X'Y,\,\,\,\,\mathrm{Var}(\hat{\varTheta }) = \hat{\sigma }^{2}\left( X'X\right) ^{-1},\\ t_{\hat{\alpha }}&= \frac{\hat{\alpha }}{\sqrt{\hat{\sigma }^{2}_{\hat{\alpha }}}},\,\,\,\,t_{\hat{\beta }}=\frac{\hat{\beta }}{\sqrt{\hat{\sigma }^{2}_{\hat{\beta }}}},\,\,\,\,t_{\hat{\delta }}=\frac{\hat{\delta }}{\sqrt{\hat{\sigma }^{2}_{\hat{\delta }}}}, \end{aligned}$$

where,

$$\begin{aligned} X'X= \begin{bmatrix} T&\sum {x_{t}}&\sum {y_{t-1}}\\ \sum {x_{t}}&\sum {x_{t}^{2}}&\sum {x_{t}y_{t-1}}\\ \sum {y_{t-1}}&\sum {x_{t}y_{t-1}}&\sum {y_{t-1}^{2}} \end{bmatrix};&\quad X'Y= \begin{bmatrix} \sum {y_{t}}\\ \sum {x_{t}y_{t}}\\ \sum {y_{t}y_{t-1}} \end{bmatrix}; \end{aligned}$$

and,

$$\begin{aligned} \hat{\sigma }^{2}&= T^{-1}\left[ \sum {y_{t}^{2}}+\hat{\alpha }^{2}T+\hat{\beta }^{2}\sum {x_{t}^{2}}+\hat{\delta }^{2} \sum {y_{t-1}^{2}}-2\hat{\alpha }\sum {y_{t}}-2\hat{\beta }\sum {x_{t}y_{t}}\right. \\&\left. -2\hat{\delta }\sum {y_{t}y_{t-1}}+2\hat{\alpha }\hat{\beta }\sum {x_{t}}+2\hat{\alpha }\hat{\delta } \sum {t}+2\hat{\beta }\hat{\delta }\sum {x_{t}y_{t-1}}\right] . \end{aligned}$$

The estimated parameters $\hat{\alpha }$, $\hat{\beta }$, and $\hat{\delta }$ and their corresponding t ratios are functions of the following expressions (initial conditions are set as $x_{0}=y_{0}=0$). We use well-known asymptotic results (available, inter alia, in Hamilton (1994), 1994, Proposition 17.1, p.486). Let $z=y,x$ and $\xi _{zt}=\sum _{i=1}^{t}{u_{zi}}$:

$$\begin{aligned} T^{-3/2}\sum {\xi _{zt-1}}&\mathop {\rightarrow }\limits ^{D}\sigma _{z} \int _{0}^{1}{\omega _{z}(r) dr},\\ T^{-2}\sum {\xi _{zt-1}^{2}}&\mathop {\rightarrow }\limits ^{D}\sigma _{z}^{2} \int _{0}^{1}{[\omega _{z}(r)]^{2} dr},\\ T^{-1}\sum {\xi _{zt-1}u_{zt}}&\mathop {\rightarrow }\limits ^{D}\sigma _{z}^{2} \frac{1}{2} ([\omega _{z}(1)]^{2}-1),\\ T^{-2}\sum {\xi _{xt-1} \xi _{yt-1}}&\mathop {\rightarrow }\limits ^{D}\sigma _{x}\sigma _{y}\int _{0}^{1}{\omega _{x}\omega {y}(r)dr},\\ T^{-1}\sum {\xi _{yt-1}u_{xt}}&\mathop {\rightarrow }\limits ^{D}\sigma _{x}\sigma {y}\int _{0}^{1}{\omega _{x}(r)d\omega _{y}(r)}, \end{aligned}$$

where $\omega _{z}(1)$ is a standard brownian motion and $\mathop {\rightarrow }\limits ^{D}$ denotes convergence in law. The later results allow us to fill the OLS matrices and then compute the parameter estimates and the t statistic associated with each. Note that we also require sums involving the lagged dependent variable, which can be easily obtained:

$$\begin{aligned} \sum {z_{t}}&=\sum {\xi _{zt-1}}+\sum {u_{zt}},\\ \sum {y_{t-1}}&=\sum {\xi _{yt-1}},\\ \sum {z_{t}^{2}}&=\sum {\xi _{zt-1}^{2}}+\sum {u_{zt}^{2}}+2 \sum {\xi _{zt-1}u_{zt}},\\ \sum {y_{t-1}^{2}}&=\sum {\xi _{yt-1}^{2}},\\ \sum {x_{t}y_{t}}&=\sum {\xi _{xt-1} \xi _{yt-1}}+ \sum {\xi _{yt-1}u_{xt}}+\sum {\xi _{xt-1}u_{yt}}+ \sum {u_{xt}u_{yt}},\\ \sum {x_{t}y_{t-1}}&=\sum {\xi _{xt-1} \xi _{yt-1}}+\sum {\xi _{yt-1}u_{xt}},\\ \sum {y_{t-1}y_{t}}&=\sum {\xi _{yt-1}^{2}}+\sum {\xi _{yt-1}u_{yt}}. \end{aligned}$$

Replacing the sums that appear in the OLS formulae with the above asymptotic expressions and letting $T\rightarrow \infty$ yields the results stated in the Theorem. The algebra involved is cumbersome but does not present any particular complication. Codes in algebraic manipulation programs with the calculations are available upon request.

Appendix 2: Data-generating processes

See Table 4.

Table 4 Data-generating processes