Detecting Serially Correlated Errors in an Autoregressive Model

Serial correlation, or autocorrelation, is usually fairly straightforward to test. Make sure your output has the Durbin–Watson statistic (DW), and see how it compares to the estimated border values. But this is where that gets sticky. The DW statistic just isn't useful in regressions that have a lagged dependent variable. So for autoregressive models, you'll need something else.

Suppose you had an autoregressive model that was an AR(1), so it's looking at each prior-period value. If there is no autocorrelation, what would you expect the covariance to be between these two?

Actually, that would show serial correlation—just negative serial correlation. That's still a problem.

Well, no. That would show some level of positive serial correlation.

Exactly! If there is no serial correlation, then there should be no covariance, which means a covariance of 0. That would be nice.

But suppose you end up with a covariance of 4.2, or perhaps -198.6. Don't forget that these numbers are absolutely meaningless by themselves. It depends entirely on the units being measured, and covariance is in terms of those units squared. Fortunately, so is the variance. How might you use variance to then "standardize" the covariance term into something meaningful?

Not quite. This would just give you a different numerical value still in terms of units squared. It would also be even less meaningful.

Exactly right. Take the covariance term, and divide by the variance. $$\displaystyle \rho_{\epsilon,k} = \frac{Cov(\epsilon_t,\epsilon_{t-k})}{\sigma^2_\epsilon} $$ Covariance is the average product of each value's deviation from the mean, so you can write it a couple of other ways as well, if you prefer. $$\displaystyle \rho_{\epsilon,k} = \frac{Cov(\epsilon_t,\epsilon_{t-k})}{\sigma^2_\epsilon} = \frac{E[(\epsilon_t - 0)(\epsilon_{t-k} - 0)]}{\sigma^2_\epsilon} = \frac{E(\epsilon_t \epsilon_{t-k})}{\sigma^2_\epsilon} $$

Not exactly. This would give you some value in terms of units to the fourth power. That would move you in the wrong direction.

So ideally this value will be 0. If it wanders too far away from that, then you have the problem of serial correlation. How far is "too far" depends on your chosen level of confidence and a _t_-test. The test statistic is the usual value-divided-by-standard-error ratio, and the standard error in this test is always $$\frac{1}{\sqrt{T}} $$, where _T_ is the number of time periods (or observations) that you have.

For example, suppose you're looking at an autoregressive model with 45 observations, and the autocorrelation measure is 0.1594. What is your _t_-statistic?

No. This is the autocorrelation measure divided by the square root of 45. But note that the standard error here is $$\frac{1}{\sqrt{45}} $$.

Actually, no. This is the approximate value of the standard error, but not the test statistic.

Yes! The _t_-test statistic here is $$\displaystyle t = \frac{\rho_{\epsilon,k}}{1/\sqrt{T}} = \frac{0.1594}{0.1491} = 1.07 $$. If you assume that this was a simple AR(1) model estimating just two parameters, then there are 43 degrees of freedom, leading to a critical value of around 2 for a 95% confidence level. Then this _t_-statistic is too low to reject, and you can comfortably assume that serial correlation of the error term isn't a big problem here.

But what if it were a problem? A _t_-statistic greater than 2 (in absolute value) would cause you to reject the null hypothesis of no serial correlation. Then the model is incorrectly specified. Remember that serial correlation is nothing to mess around with in time series. If it's a problem, it's a big problem, and you really should not use your estimates for any sort of inference. So in such cases, it's back to the drawing board for a respecification.

To summarize: [[summary]]

0

1

Something negative

Subtract the variance

Divide by the variance

Multiply by the variance

0.02

0.15

1.07

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