36—Chapter 19. Additional Regression Tools

Dependent Variable: 100*D(LOG(POJ)) Method: Least Squares

Date: 04/14/09 Time: 14:27 Sample: 1950:01 2000:12 Included observations: 612

HAC standard errors & covariance (Bartlett kernel, User bandwidth = 8.0000)

Note in particular that the top of the equation output shows the use of HAC covariance estimates along with relevant information about the settings used to compute the long-run covariance matrix.

Weighted Least Squares

Suppose that you have heteroskedasticity of known form, where the conditional error variances are given by j2t . The presence of heteroskedasticity does not alter the bias or consistency properties of ordinary least squares estimates, but OLS is no longer efficient and conventional estimates of the coefficient standard errors are not valid.

If the variances j2t are known up to a positive scale factor, you may use weighted least squares (WLS) to obtain efficient estimates that support valid inference. Specifically, if

38—Chapter 19. Additional Regression Tools

Alternately, if your series INVARWGT contains the values proportional to the inverse of the standard deviation of the residuals you should choose Inverse std. dev.

Next, you should enter an expression for your weight series in the Weight series edit field.

Lastly, you should choose a scaling method for the weights. There are three choices: Average, None, and (in some cases) EViews default. If you select Average, EViews will, prior to use, scale the weights prior so that

the wi sum to T . The EViews default specification scales the weights so the square roots of the wi sum to T . (The latter square root scaling, which offers backward compatibility to EViews 6 and earlier, was originally introduced in an effort to make the weighted residuals

wt (yt xt¢ˆ ) comparable to the unweighted residuals.) Note that the EViews default b

–

method is only available if you select Inverse std. dev. as weighting Type.

Unless there is good reason to do so, we recommend that you employ Inverse std. dev. weights with EViews default scaling, even if it means you must transform your weight series. The other weight types and scaling methods were introduced in EViews 7, so equations estimated using the alternate settings may not be read by prior versions of EViews.

We emphasize the fact that bWLS and ˆ WLS are almost always invariant to the scaling of

S

weights. One important exception to this invariance occurs in the special case where some of the weight series values are non-positive since observations with non-positive weights will be excluded from the analysis unless you have selected EViews default scaling, in which case only observations with zero weights are excluded.

As an illustration, we consider a simple example taken from Gujarati (2003, Example 11.7, p. 416) which examines the relationship between compensation (Y) and index for employment size (X) for nine nondurable manufacturing industries. The data, which are in the workfile “Gujarati_wls.WF1”, also contain a series SIGMA believed to be proportional to the standard deviation of each error.

To estimate WLS for this specification, open an equation dialog and enter

y c x

as the equation specification.

Click on the Options tab, and fill out the Weights section as depicted here. We select Inverse std. dev. as our Type, and specify “1/SIGMA” for our Weight series. Lastly, we select

EViews default as our Scaling method.

Click on OK to estimate the specified equation. The results are given by:

Weighted Least Squares—39

Dependent Variable: Y

Method: Least Squares

Date: 06/17/09 Time: 10:01

Sample: 1 9

Included observations: 9

Weighting series: 1/SIGMA

Weight type: Inverse standard deviation (EViews default scaling)

Weighted Statistics

The top portion of the output displays the estimation settings which show both the specified weighting series and the type of weighting employed in estimation. The middle section shows the estimated coefficient values and corresponding standard errors, t-statistics and probabilities.

The bottom portion of the output displays two sets of statistics. The Weighted Statistics show statistics corresponding to the actual estimated equation. For purposes of discussion, there are two types of summary statistics: those that are (generally) invariant to the scaling of the weights, and those that vary with the weight scale.

The “R-squared”, “Adjusted R-squared”, “F-statistic” and “Prob(F-stat)”, and the “DurbinWatson stat”, are all invariant to your choice of scale. Notice that these are all fit measures or test statistics which involve ratios of terms that remove the scaling.

One additional invariant statistic of note is the “Weighted mean dep.” which is the weighted mean of the dependent variable, computed as: