## Y

FIGURE 13.4 In each subfigure, the solid line gives the OLS line for all the data and the broken line gives the OLS line with the outlier, denoted by an s, omitted. In (a), the outlier is near the mean value of X and has low leverage and little influence on the regression coefficients. In (b), the outlier is far away from the mean value of X and has high leverage as well as substantial influence on the regression coefficients. In (c), the outlier has high leverage but low influence on the regression coefficients because it is in line with the rest of the observations.

42Adapted from John Fox, Applied Regression Analysis, Linear Models, and Related Methods, Sage Publications, California, 1997, p. 268.

43Norman R. Draper and Harry Smith, op. cit., p. 76.

542 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

What are some of the tests that one can use to detect outliers and leverage points? There are several tests discussed in the literature, but we will not discuss them here because that will take us far afield.44 Software packages such as Shazam and Microfit have routines to detect outliers, leverage, and influential points.

In Chapter 8 we examined the question of the structural stability of a regression model involving time series data and showed how the Chow test can be used for this purpose. Specifically, you may recall that in that chapter we discussed a simple savings function (savings as a function of income) for the United States for the period 1970-1995. There we saw that the savings income relationship probably changed around 1982. Knowing the point of the structural break we were able to confirm it with the Chow test.

But what happens if we do not know the point of the structural break (or breaks)? This is where one can use recursive least squares (RELS). The basic idea behind RELS is very simple and can be explained with the savings-income regression.

where Y = savings and X = income and where the sample is for the period 1970-1995. (See the data in Table 8.9.)

Suppose we first use the data for 1970-1974 and estimate the savings function, obtaining the estimates of ¡1 and ¡2. Then we use the data for 1970-1975 and again estimate the savings function and obtain the estimates of the two parameters. Then we use the data for 1970-1976 and re-estimate the savings model. In this fashion we go on adding an additional data point on Y and X until we exhaust the entire sample. As you can imagine, each regression run will give you a new set of estimates of ¡1 and ¡2. If you plot the estimated values of these parameters against each iteration, you will see how the values of estimated parameters change. If the model under consideration is structurally stable, the changes in the estimated values of the two parameters will be small and essentially random. However, if the estimated values of the parameters change significantly, it would indicate a structural break. RELS is thus a useful routine with time series data since time is ordered chronologically. It is also a useful diagnostic tool in cross-sectional data where the data are ordered by some "size" or "scale" variable, such as

44Here are some accessible sources: Alvin C. Rencher, Linear Models in Statistics, John Wiley & Sons, New York, 2000, pp. 219-224; A. C. Atkinson, Plots, Transformations and Regression: An Introduction to Graphical Methods of Diagnostic Regression Analysis, Oxford University Press, New York, 1985, Chap. 3; Ashis Sen and Muni Srivastava, Regression Analysis: Theory, Methods, and Applications, Springer-Verlag, New York, 1990, Chap. 8; and John Fox, op. cit., Chap. 11.

### Recursive Least Squares

Yt = ßi + ß2 Xt + ut the employment or asset size of the firm. In exercise 13.30 you are asked to apply RELS to the savings data given in Table 8.9.

Software packages such as Shazam, Eviews, and Microfit now do recursive least-squares estimates routinely. RELS also generates recursive residuals on which several diagnostic tests have been based.45

We have already discussed Chow's test of structural stability in Chapter 8. Chow has shown that his test can be modified to test the predictive power of a regression model. Again, we will revert to the U.S. savings-income regression for the period 1970-1995.

Suppose we estimate the savings-income regression for the period 19701981, obtaining fai,7o-8i and fa2,70-81, which are the estimated intercept and slope coefficients based on the data for 1970-1981. Now using the actual values of income for period 1982-1995 and the intercept and slope values for the period 1970-1981, we predict the values of savings for each of 19821995 years. The logic here is that if there is no serious structural change in the parameter values, the values of savings estimated for 1982-1995 based on the parameter estimates for the earlier period, should not be very different from the actual values of savings prevailing in the latter period. Of course, if there is a vast difference between the actual and predicted values of savings for the latter period, it will cast doubts on the stability of the savings-income relation for the entire data period.

Whether the difference between the actual and estimated savings value is large or small can be tested by the F test as follows:

where n1 = number of observations in the first period (1970-1981) on which the initial regression is based, n2 = number of observations in the second or forecast period, Y u*2 = RSS when the equation estimated for all the observations (n1 + n2), and Yu2 = RSS when the equation is estimated for the first n1 observations and k is the number of parameters estimated (two in the present instance). If the errors are independent, and identically, normally distributed, the F statistic given in (13.10.1) follows the F distribution with n2 and n1 df, respectively. In exercise 13.31 you are asked to apply Chow's predictive failure test to find out if the savings-income relation has in fact changed. In passing, note the similarity between this test and the forecast x2 test discussed previously.

45For details, see Jack Johnston and John DiNardo, Econometric Methods, 4th ed., McGraw-Hill, New York, 1997, pp. 117-121.

Chow's Prediction Failure Test 