1 for the (n + l)th observation 0 for all others

We have discussed this procedure in Chapter 8.

Now we just run a multiple regression with all the (« + 1) observations and this extra variable D. The estimate of the coefficient of this variable D is the prediction error and its standard error is ad„+f. This result has been proved in Chapter 8. Thus we get from the regression program and d„+i a or t„+i the /-ratio, which is e„+y/d„+, a. But for the recursive residuals we need e„+/J„+,. Thus all we have to do is multiply the /-ratio we get for the coefficient of D by . This gives us the recursive residual.

The calculation of recursive residuals by the dummy variable method is similar to that for predicted residuals except that the regressions for predicted residuals involve all the observations in the sample and for recursive residuals the observation set is sequential and the dummy variable is also sequential.

Illustrative Example

In Table 12.1 we present the OLS residual, the forward recursive, and the backward recursive residuals for the production function (4.24). The sum of squares of the recursive residuals should be equal to the sum of squares of the OLS residuals, but there are small discrepancies which are due to rounding errors. The recursive residuals are useful for:

1. Tests for heteroskedasticity described in Chapter 5.

2. Tests for autocorrelation described in Chapter 6.

3. Tests for stabihty described in Chapter 4.

For stability analysis. Brown, Durbin, and Evans suggest computing the cumulative sums (CUSUM) and cumulative sums of squares (CUSUMSQ) of the recursive residuals and comparing them with some percentage points that they have tabulated. A discussion of this is beyond the scope of this book." Instead,

"Phillips and Harvey, "A Simple Test."

"Those interested in this detail can refer to the paper by Brown, Durbin, and Evans, "Techniques for Testing."

3. Their sum of squares is equal to RSS, the residual sum of squares from the least squares regression.

The third property is useful in checking the accuracy of the calculations.

There are algorithms for calculating the regression coefficients p, and also the prediction variance dfa, in a recursive fashion.* There is, however, an alternative method that we can use for the recursive residuals as we have done for the predicted residuals described eadier. Note that recursive residuals are similar to predicted residuals except that the predictions are sequential.

Suppose that we use n observations and want to get the prediction error and the variance of the prediction error for the (n + l)th observation. Then all we have to do is to create a dummy variable D which is defined as

Table 12.1 Different Residuals for the Production Function (4.25) (Multipled by 100)

12.5 DFFITS and Bounded Influence Estimation

In Section 3.8 we discussed briefly the problem of outliers. The usual approach to outliers based on least squares residuals is as follows:

1. Look at the OLS residuals.

2. Delete the observations with large residuals.

3. Reestimate the equation.

Two major problems with this approach are that the OLS residuals (as we showed earlier) do not all have the same variance and furthermore the OLS residuals do not give us any idea of how important this particular observation is for the overall results. The idea behind the studentized residual is to allow for these differences in the variances and to look at the prediction error resulting from the deletion of this observation. Using a plus or minus 2a rule of thumb, the studentized residuals shown in Table 12.2 suggest that the observations for the years 1945 and 1947 are outliers. With the OLS residuals in Table 12.1 we might have included 1944 and 1948 as well.

One other measure that is used to detect outliers is to see the change in the fitted value of that results from dropping a particular observation. Let y,,, be the fitted value of if the rth observation is dropped. The quantity (y, - y,,,) divided by the scaling factor h„S„ where Sj is the estimator of a from a regression with the ith observation omitted is called DFFITS,.

It has been shown that*

DFFITS, = I j- ) ,

where w, is the rth studentized residual and h,, is the quantity that figures repeatedly in equations (12.1)-(12.3). (It is also known as the rth diagonal term of the "hat matrix.")

There are many computer programs available to compute studentized residuals and DFFITS. For instance, the SAS regression program gives these statis-

"D. A. Belsley, E. Kuh, and R. E. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity (New York: Wiley, 1980).

we will apply some simple Mests. If a test of the hypothesis that the mean of the recursive residual is zero gives us a significant r-statistic, this is an indication of instability of the coefficients. In Table 12.1 we present these f-ratios. With the forward recursive residuals we do not reject the hypothesis of zero mean. With the backward recursive residuals, we do reject at the 5% level, the hypothesis of zero mean. Thus the conclusions from the recursive residuals are similar to the conclusions we arrived at eariier in Chapter 4 from the predictive tests for stabihty.