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t(Inft), all in percent form, for the U.S. economy for the period 1954-1981.

a. Regress RRt on inflation.

b. Regress RRt on OGt+1 and Inft c. Comment on the two regression results in view of Eugene Fama's observation that "the negative simple correlation between real stock returns and inflation is spurious because it is the result of two structural relationships: a positive relation between current real stock returns and expected output growth [measured by OG t+1], and a negative relationship between expected output growth and current inflation."

d. Would you expect autocorrelation in either of the regressions in a and b? Why or why not? If you do, take the appropriate corrective action and present the revised results.

13.36. The Durbin h statistic: Consider the following model of wage determination:

where Y = wages = index of real compensation per hour X = productivity = index of output per hour.

a. Using the data in Table 12.4, estimate the above model and interpret your results.

b. Since the model contains lagged regressand as a regressor, the Durbin-Watson d is not appropriate to find out if there is serial correlation in the data. For such models, called autoregressive models, Durbin has developed the so-called h statistic to test for first-order autocorrelation, which is defined as*:

where n = sample size, var(03) = variance of the coefficient of the lagged Yt—1, and p = estimate of the first-order serial correlation.

For large sample size (technically, asymptotic), Durbin has shown that, under the null hypothesis that p = 0, that is, the h statistic follows the standard normal distribution. From the properties of the normal distribution we know that the probability of |h| > 1.96 is about 5 percent. Therefore, if in an application |h| > 1.96, we can reject the null hypothesis that p = 0, that is, there is evidence of first-order autocorrelation in the autoregressive model given above.

To apply the test, we proceed as follows: First, estimate the above model by OLS (don't worry about any estimation problems at this stage). Second, note var(03) in this model as well as the routinely

J. Durbin, "Testing for Serial Correlation in Least-squares Regression When Some of the Regressors Are Lagged Dependent Variables," Econometrica, vol. 38, pp. 410-421.

504 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

computed d statistic. Third, using the d value, obtain p & (1 — d/2). It is interesting to note that although we cannot use the d value to test for serial correlation in this model, we can use it to obtain an estimate of p. Fourth, now compute the h statistic. Fifth, if the sample size is reasonably large and if the computed \h\ exceeds 1.96, we can conclude that there is evidence of first-order autocorrelation. Of course, you can use any level of significance you want.

Apply the h test to the autoregressive wage determination model given earlier and draw appropriate conclusions and compare your results with those given in regression (12.5.1).

12.37. Dummy variables and autocorrelation. Refer to the savings-income regression discussed in Chapter 9. Using the data given in Table 9.2, and assuming an AR(1) scheme, reestimate the savings-income regression, taking into account autocorrelation. Pay close attention to the transformation of the dummy variable. Compare your results with those presented in Chapter 9.

12.38. Using the wages-productivity data given in Table 12.4, estimate model

(12.9.8) and compare your results with those given in regression

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