Why autocorrelation affects OLS coefficient standard errors?

Question

It seems that OLS residuals autocorrelation is not always an issue, depending on the problem at hand. But why residuals autocorrelation would affect the coefficient standard errors? From the Wikipedia article on autocorrelation:

While it does not bias the OLS coefficient estimates, the standard errors tend to be underestimated (and the t-scores overestimated) when the autocorrelations of the errors at low lags are positive.

Answer 1

Suppose your OLS regression is well specified and contains all the right explanatory variables, but you have an unspecified correlation structure of the residuals: $$ y_t = x_t' \beta + \epsilon_t, \mathbb{V}[\mathbf{\epsilon}]=\Omega $$ The OLS estimates are $$ \hat\beta = (X'X)^{-1} X'Y = \beta + (X'X)^{-1} X'\mathbf{\epsilon} $$ and their variance is $$ \mathbb{V}[\hat\beta] = \mathbb{E} [ (X'X)^{-1} X'\mathbf{\epsilon}\mathbf{\epsilon}'X (X'X)^{-1} ] $$ Typically, at this stage, we'd have to assume something like existence of the probability limit of $\frac1T (X'X) \to \Sigma$, so that $$ T \mathbb{V}[\hat\beta] \to \Sigma^{-1} {\rm plim} \bigl[ \frac1T X'\mathbf{\epsilon}\mathbf{\epsilon}'X \bigr] \Sigma^{-1} = \Sigma^{-1} {\rm plim} \bigl[ \frac1T X'\Omega X \bigr] \Sigma^{-1} $$ This expression is different from what the naive OLS standard errors produce, and so in general the OLS standard errors are wrong.

Of course, if $X$ can be considered fixed, then there is no need for asymptotic approximations, and $X$ can be carried through the expectations, so that $$ \mathbb{V}[\hat\beta] = (X'X)^{-1} X'\Omega X (X'X)^{-1} $$ to the same effect.

Answer 2

Auto correlation (Exogeneity) : If the observation of residual in t1 is dependent on the observation of residual in t0, then it basically violates the underlying assumption of OLS which states "Error term is independently distributed and not correlated". This can bias the estimation of Beta coefficient of OLS term.

When such trend is observed in the residuals, the residuals of regression model picks up the influence of those variables that are affecting the dependent variables that aren't part of the regression equation.

The persistence in excluded variables is most of the times a cause for autocorrelation. This is more prevalent in Time series data.

This can be mitigated by using some of the transformations techniques like,

1. Paris Winston Transformation of Data
2. Differencing the dependent variables
3. Differencing all variables

One simpler approach would be,

1. estimating the linear model using OLS.
2. Compute the residuals.
3. Regress the residuals of all Independent variables and lagged variables.
4. Use t-test, if the coefficient of lagged residual is significant, we can reject the null of independent errors.

Test to detect Auto correlation: **

1. Durbin - Watson
2. Bruesch - Godfrey

This assumption/issue is violated/exempted if the explanatory variables are stochastic, for instance when they are measured with error, or are endogenous.