The Consequences of Stochastic Trends for Regression (Time Series)

Dependent variables, explanatory variables and the disturbances can all contain stochastic trends.

Such stochastic trends can undermine the consistency and the asymptotic normality of all the estimators.

When our usual estimators and test procedures  are undermined, we need alternative methodsof estimation and inference.

To know when to use such alternative methods, we need formal tests to expose stochastic trends.

This discussion identifies the various cases which stochastic trends make the Law of Large Numbers (LLN) or Central Limit Theorem (CLT) inapplicable and undermine our usual procedure.

Lets we focus on data generating process (DGP) with one explanator and an unknown intercept. 

                ${{Y}_{t}}={{\beta }_{0}}+{{\beta }_{1}}{{X}_{t}}+{{u}_{t}}$                                       (1)

which the disturbances are contemporaneously uncorrelated with the explanators.

The explanators $X$  may or may not contain a stochastic trend, and the disturbances may or may not contain a stochastic trend.

Notice that, if the disturbances contain a stochastic trend and $X$   does not, $Y$   must contain a stochastic trend.

We assume that all nontrending variables that appear in regression are measure as deviations from their means, and therefore have a mean of zero.

There are four cases must be considered:

Case 1

When neither $X$  or $u$    contains a stochastic trend, ${{\hat{\beta }}_{1}}$ is consisten.

This is a standard result and the t-statistics based on t-distribution, and for multiple regression, F-statistics based on F-distribution and are asymptotically valid.

Case 2

When $X$ contains a stochastic trend, but $u$ does not contain a stochastic trend, ${{\hat{\beta }}_{1}}$ converges in probability to ${{\beta }_{1}}$ . 

Unfortunately, the asymptotic distribution of ${{\hat{\beta }}_{1}}$ is not generally the normal distribution, so our usual t-statistics and F-statistics are generally not valid.

Then ${{\hat{\beta }}_{1}}$ is super consistent , converging at faster that root $n$ rate.

Case 3

When $X$ contains no stochastic trend, but $u$ does contain a stochastic trend (an unlikely case to encounter in practice), the numerator of

 $\frac{\frac{1}{T}\sum{{{x}_{t}}{{u}_{t}}}}{\frac{1}{T}\sum{x_{t}^{2}}}$

has a term with an unbound variance, $u$ , and the denominator converges to a nonzero constant (under our usual large-sample assumptions).

The LLN and CLT both fail in this case.

Instead of converging to ${{\beta }_{1}}$ , the ${{\hat{\beta }}_{1}}$ converge to a random variable.

OLS is not consistent. Our usual tests are invalid.

Case 4

When both $X$ and $u$ contain stochastic trends, there are two terms with unbounded variances in both the numerator and the denominator of 

$\frac{\frac{1}{T}\sum{{{x}_{t}}{{u}_{t}}}}{\frac{1}{T}\sum{x_{t}^{2}}}$

The LLN and CLT both fail in this case.

Instead of converging to ${{\beta }_{1}}$ ,the ${{\hat{\beta }}_{1}}$ converge to a random variable.

OLS is not consistent. Our usual tests are invalid.

We call the regression of a stochastically trending variable on another, unrelated, stochastically 

trending variable a spurious regression because in this case, if ${{\beta }_{1}}=0$, ordinary t-statistic will far to too often reject a true null of ${{\beta }_{1}}=0$.

In multiple regression model, if all the parameter involved in a test fall into Case 1, the usual t-statistic and F-statistics can be apply.

If one or more parameter does not fall into Case 1, usual t-statistic and F-statistics are not valid where the usual t-distribution and F-distribution provide incorrect critical values.

If one or more parameter fall into Case 3 or 4, conventional test involving those parameters are inconsistent. Econometrician have used  Monte Carlo methods to obtain valid critical value for the t-statistic and F-statistics.