Robust standard error (HC3) smaller than OLS standard error

Question

I performed a linear regression of the temperature monthly time series to get a temperature trend. I considered temperature variable as a dependent variable (y) and time (e.g., month from a certain year) as an explanatory variable (x).

When I plotted residuals, I found significant autocorrelations. This means that the standard error of the regression coefficient is likely underestimated.

When I applied the Cochrane-Orcutt procedure for the analysis, it mostly removed autocorrelations. The Cochrane-Orcutt procedure gives about 2-3 times larger standard error than the OLS method, but the method does not work for all cases (I am obtaining the temperature trends for many locations). Sometimes the regression slope is unphysical when there are AR(1), AR(2), and AR(3) together. In theory, I think the Cochrane-Orcutt procedure would give a similar regression slope while the standard error would change more significantly. I am wondering if it is logical not to use the results from the Cochrane-Orcutt procedure if the regression coefficient from the Cochrane-Orcutt procedure is significantly different from the Ordinary Least Squares (OLS) method.

As an alternative, I considered the robust standard error (or Heteroskedasticity-consistent standard errors) while keeping the regression coefficients from the OLS method. I considered HC0 (McKinnon and White, 1985). I found that the calculated HC0 robust error was often smaller than the standard error based on the OLS method. I used time as the x variable, meaning that larger weights were used in obtaining the robust standard error when (xi - E(x)) is larger (e.g., x=1, x=2, x=n-1, x=n-1). Sometimes residuals at those data points are smaller than the mean residuals, resulting in the robust standard error being smaller than the OLS standard error. I am wondering if this is the right way to use the robust error (the larger weight given to the data points is far from the midpoint)? When I applied HC3, it gave a slightly larger standard error than HC0, but it was still often smaller than the OLS standard error.

Therefore, the robust standard error is not larger than the OLS standard error, which is the opposite to what I've expected. I am wondering if it is still worth applying HC3 method to handle autocorrelations of residuals. Is it normal?

Answer 1

With time series data, use OLS with a corrected variance-covariance matrix (such as Newey-West). Other more sophisticated models exist, but this is an excellent first pass.

Het-Robust SEs only allow the variance of the epsilons to vary; they don't allow for autocorrelation, which plagues time series data.

Also, here's one simulated example where the robust errors are smaller than OLS.