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I am running an OLS on which the dependent variable (Y) and the independent variables (X1, X2, X3, ...) are non-stationary. But the residuals are found to be stationary. Does this mean my regression is OK? Or is it spurious? I tried searching online but I got a bit confused.

Richard Hardy
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adrCoder
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    Do all of your variables have unit roots? Or what kind of nonstationarity is it? – Richard Hardy Jan 08 '20 at 10:41
  • The variables can be non-stationary provided their relation doesn't change over time (if it does and you don't model it, you will see this in the residuals, they will not be stationary). A simple example of your case where OLS is fine would be $Y_t = \beta t + u_t$ with $u_t$ stationary. If you look into "co-integration" you can find the relevant time series theory. – CloseToC Jan 08 '20 at 12:18
  • @RichardHardy yes they have unit root but the residuals are stationary. Is that ok or problematic? – adrCoder Jan 08 '20 at 13:33
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    Your point estimates are kind of OK, but probably not the standard errors, confidence intervals, p-values, etc. You would be safer to use an error correction model (ECM; see [tag:ecm]) $$\Delta y_t=\gamma_0 + \alpha \text{ect}{t-1}+\gamma_1 \Delta x{1,t}+\dots+\gamma_p \Delta x_{p,t}$$ where $\text{ect}$ is the error-correction term (a stationary linear combination of $y$ and $x$s). – Richard Hardy Jan 08 '20 at 14:14
  • @RichardHardy do you have some reference (book, paper, online article) that states and justifies why the point estimates are ok in case the variables are non-stationary but the residuals are stationary? – adrCoder Jan 23 '20 at 07:38
  • Most time series textbooks include a treatment of regression of cointegrated variables. Superconsistency is one keyword to look for (characterizing the point estimator in such a regression). – Richard Hardy Jan 23 '20 at 07:41
  • @RichardHardy thank you. Can you please recommend some good books so that I order them and read them? – adrCoder Jan 23 '20 at 12:07
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    See these references, also Lütkepohl "New Introduction to Multiple Time Series Analysis" (2005). – Richard Hardy Jan 23 '20 at 13:27
  • Many thanks Richard! – adrCoder Jan 23 '20 at 14:24

1 Answers1

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...the dependent variable (Y) and the independent variables (X1, X2, X3, ...) are non-stationary. But the residuals are found to be stationary. Does this mean my regression is OK?...

This suggests there is a cointegrating relationship among the dependent and independent variable. In this case, the OLS estimates are (super)-consistent.

For example, consider the simplest cointegration model: $$ y_t = \beta x_t + \epsilon_t $$ where $\epsilon_t \stackrel{i.i.d.}{\sim} (0,1)$ and $u_t \stackrel{i.i.d.}{\sim} (0,1)$ are two independent white noise and $$ x_t = \sum_{s = 0}^t u_s $$ is a random walk with $\{ u_t \}$ as innovations.

Both $x_t$ and $y_t$ are not non-stationary but the population error term $y_t - \beta x_t$ is stationary. There is a cointegrating relationship with cointegrating vector $(1, -\beta)$.

The OLS estimate $\hat{\beta}$ satisfies $$ T ( \hat{\beta} - \beta ) = \frac{\sum_1^T x_t \epsilon_t}{\sum_1^T x_t^2} \stackrel{d}{\rightarrow} \frac{\int_0^1 W^{(1)}_t dW^{(2)}_t}{\int_0^1 \left( W^{(1)}_t \right)^2 dt}, $$ where $W^{(1)}$ and $W^{(2)}$ are independent standard Brownian motions. Therefore $\hat{\beta} - \beta = O_p(\frac{1}{T})$, i.e. $\hat{\beta}$ is T-consistent. In contrast, in the stationary case, $\hat{\beta}$ is $\sqrt{T}$-consistent.

Here (T-)consistency holds even when $\epsilon_t$ and $u_t$ are correlated. This is in contrast to the stationary case where consistency requires $E[\epsilon_t x_t] = 0$.

Unlike the stationary case, where inference is obtained via asymptotic normal distributions, here the asymptotic distribution of $T ( \hat{\beta} - \beta )$ is non-normal. Same goes for the asymptotic distribution of the usual Wald/F-statistics (critical values can be obtained by simulation, if needed).

Notice that sum of squared residuals of this cointegration regression is $$ \sum_1^T \epsilon_t ^2 - \frac{( \sum_1^T x_t \epsilon_t)^2}{\sum_1^T x_t^2} = O_p(T) - O_p(1) = O_p(T), $$ which implies the residuals are stationary. On the other hand, if the regression is spurious, the sum of squared residuals is $$ \sum_1^T y_t ^2 - \frac{( \sum_1^T x_t y_t)^2}{\sum_1^T x_t^2} = O_p(T^2) - O_p(T^2) = O_p(T^2) $$ which implies the residuals are non-stationary. The fact that you found residuals to be stationary suggests your regression is cointegrated, rather than spurious.

Further Comments

  1. In applying unit root tests to residuals to check for non-stationarity, standard critical values cannot be used. For example, for the Augmented Dickey-Fuller test-statistic computed using residuals, the Engel-Granger critical values should be used.

  2. As already pointed out in comments, for a cointegration model there exists a corresponding error correction model (ECM). For the above simple example, the corresponding ECM is $$ \Delta y_t = \Delta x_t + \underbrace{ (y_{t-1} - \beta x_{t-1})}_\text{$\epsilon_t$}. $$ While the cointegration model describe the long-run equilibrium relationship between $x_t$ and $y_t$, the corresponding ECM describes the short-run relationship between $\Delta y_t$, $\Delta x_t$, and the deviation $\epsilon_t$ from long-run level. Notice that all variables in the ECM are stationary. Which model you estimate depends on the question of interest. To estimate the ECM, regress $\Delta y_t$ on $\Delta x_t$ and $e_t$ where $e_t$ is the residual from the cointegration regression.

Michael
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