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FIGURE 12.7 Index of compensation (V) and index of productivity (X), United States, 1959-1998.

although there is some hint that at higher values of productivity the relationship between the two may be slightly nonlinear. Therefore, we decided to estimate a linear as well as a log-linear model, with the following results:

Yt = 29.5192 + 0.7136Xt se = (1.9423) (0.0241) t = (15.1977) (29.6066)

where d is the Durbin-Watson statistic, which will be discussed shortly. lnYt = 1.5239 + 0.6716 ln Xt se = (0.0762) (0.0175) t = (19.9945) (38.2892)

For discussion purposes, we will call (12.5.1) and (12.5.2) wages-productivity regressions.

462 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

Qualitatively, both the models give similar results. In both cases the estimated coefficients are "highly" significant, as indicated by the high t values. In the linear model, if the index of productivity goes up by a unit, on average, the index of compensation goes up by about 0.71 units. In the loglinear model, the slope coefficient being elasticity (why?), we find that if the index of productivity goes up by 1 percent, on average, the index of real compensation goes up by about 0.67 percent.

How reliable are the results given in (12.5.1) and (12.5.2) if there is autocorrelation? As stated previously, if there is autocorrelation, the estimated standard errors are biased, as a result of which the estimated t ratios are unreliable. We obviously need to find out if our data suffer from autocorrelation. In the following section we discuss several methods of detecting autocorrelation. We will illustrate these methods with the linear model (12.5.1) only, leaving the log-linear model (12.5.2) as an exercise.

Recall that the assumption of nonautocorrelation of the classical model relates to the population disturbances ut, which are not directly observable. What we have instead are their proxies, the residuals ut, which can be obtained by the usual OLS procedure. Although the ut are not the same thing as ut,17 very often a visual examination of the lis gives us some clues about the likely presence of autocorrelation in the u's. Actually, a visual examination of ut or (£¿2) can provide useful information not only about autocorrelation but also about heteroscedasticity (as we saw in the preceding chapter), model inadequacy, or specification bias, as we shall see in the next chapter. As one author notes:

The importance of producing and analyzing plots of [residuals] as a standard part of statistical analysis cannot be overemphasized. Besides occasionally providing an easy to understand summary of a complex problem, they allow the simultaneous examination of the data as an aggregate while clearly displaying the behavior of individual cases.18

There are various ways of examining the residuals. We can simply plot them against time, the time sequence plot, as we have done in Figure 12.8, which shows the residuals obtained from the wages-productivity regression (12.5.1). The values of these residuals are given in Table 12.5 along with some other data.

17Even if the disturbances ut are homoscedastic and uncorrelated, their estimators, the residuals, Ut, are heteroscedastic and autocorrelated. On this, see G. S. Maddala, Introduction to Econometrics, 2d ed., Macmillan, New York, 1992, pp. 480-481. However, it can be shown that as the sample size increases indefinitely, the residuals tend to converge to their true values, the ut's. On this see, E. Malinvaud, Statistical Methods of Econometrics, 2d ed., North-Holland Publishers, Amsterdam, 1970, p. 88.

18Stanford Weisberg, Applied Linear Regression, John Wiley & Sons, New York, 1980, p. 120. 