Heteroskedastic-Consistent Standard Errors

The least squares estimator can be used to estimate the linear model even when the errors are heteroskedastic with good results. As mentioned in the first part of this chapter, the problem with using least squares in a heteroskedastic model is that the usual estimator of precision (estimated variance-covariance matrix) is not consistent. The simplest way to tackle this problem is to use least squares to estimate the intercept and slopes and use an estimator of least squares covariance that is consistent whether errors are heteroskedastic or not. This is the so-called heteroskcedasticity robust estimator of covariance that gretl uses.

Подпись: 1 2 3 4 5 Подпись: ols food_exp const income modeltab add ols food_exp const income modeltab add modeltab show image261

In this example, the food expenditure data is used to estimate the model using least squares with both the usual and the robust sets of standard errors. Start by estimating the food expen­diture model using least squares and add the estimates to the model table the estimates (Usual). Reestimate the model using the —robust option and store the results (modeltab add).

The model table, which I edited a bit, is

OLS estimates

Dependent variable: food_exp

(Usual)

(HC3 Robust)

const

72.96*

72.96**

(38.83)

(19.91)

income

11.50**

11.50**

(2.508)

(2.078)

n

20

20

Replace sortby income with dsortby income to sort the sample by income in descending order.

R2 0.5389 0.5389

і -109.1 -109.1

Standard errors in parentheses * indicates significance at the 10 percent level ** indicates significance at the 5 percent level

Notice that the coefficient estimates are the same, but that the estimated standard errors are different. Interestingly enough, the robust standard error for the slope is actually smaller than the usual one!

A number of commands behave differently when used after a model that employs the —robust option. For instance, the omit and restrict commands will use a Wald test instead of the usual one based on the difference in sum of squared errors.

The confidence intervals can be computed manually using saved results from the regression or from the model window of a model estimated through the GUI. Estimate the model using ols from the GUI. Select Analysis > Confidence Intervals for coefficients in the model window to generate confidence intervals based on the HCCME.

When you estimate the model, check the ‘Robust standard errors’ option (see Figure 8.2) and choose the ‘Configure’ button to select one of the options for bias correction using the pull-down menu for cross-sectional data as shown earlier in Figure 8.3.

These robust standard errors are obtained from what is often referred to as the heteroskedasticity – consistent covariance matrix estimator (HCCME) that was proposed by Huber and rediscovered by White. In econometrics, the HCCME standard errors may be referred to as White’s standard errors or Huber/White standard errors. This probably accounts for the tab’s name in the dialog box.

Since least squares is inefficient in heteroskedastic models, you’d think that there might be another unbiased estimator that is more precise. And, there is. The generalized least squares (GLS) estimator is, at least in principle, easy to obtain. Essentially, with the GLS estimator of the heteroskedastic model, the different error variances are used to reweigh the data so that they are all have the same (homoskedastic) variance. If the data are equally variable, then least squares is efficient!

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