## Gls

Given this formula and (12.3.1), find the expression for the correction factor C.

12.19. Show that estimating (12.9.5) is equivalent to estimating the GLS discussed in Section 12.3, excluding the first observation on Y and X.

*Source: Prices and Earnings in 1951-1969: An Econometric Assessment, Department of Employment, Her Majesty's Stationery Office, 1971, Table C, p. 37, Eq. 63.

CHAPTER TWELVE: AUTOCORRELATION 497

12.20. For regression (12.9.9), the estimated residuals have the following signs, which for ease of exposition are bracketed.

(++++)(—)(+++++++)(—)(++++)(——)(+)(——)(+)(——)(++)(—)

On the basis of the runs test, do you reject the null hypothesis that there is no autocorrelation in the residuals? "12.21. Testing for higher-order serial correlation. Suppose we have time series data on a quarterly basis. In regression models involving quarterly data, instead of using the AR(1) scheme given in (12.2.1), it may be more appropriate to assume an AR(4) scheme as follows:

Ut = P4 U-4 + St that is, to assume that the current disturbance term is correlated with that of the same quarter in the previous year rather than that of the preceding quarter.

To test the hypothesis that p4 = 0, Wallis1 suggests the following modified Durbin-Watson d test:

The testing procedure follows the usual d test routine discussed in the text.

Wallis has prepared d4 tables, which may be found in his original article.

Suppose now we have monthly data. Could the Durbin-Watson test be generalized to take into account such data? If so, write down the appropriate d12 formula.

12.22. Suppose you estimate the following regression:

Aln output t = fa1 + fa2 Aln Lt + fa3 Aln Kt + ut where Y is output, L is labor input, and K is capital input, and A is the first-difference operator. How would you interpret fa1 in this model? Could it be regarded as an estimate of technological change? Justify your answer.

12.23. As noted in the text, Maddala has suggested that if the Durbin-Watson d is smaller than R2, one may run the regression in the first-difference form. What is the logic behind this suggestion?

### Optional.

^Kenneth Wallis, "Testing for Fourth Order Autocorrelation in Quarterly Regression Equations," Econometrica, vol. 40, 1972, pp. 617-636. Tables of ¿4 can also be found in J. Johnston, op. cit., 3d ed., p. 558.

498 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

12.24. Refer to Eq. (12.4.1). Assume r = 0 but p = 0. What is the effect on E(o2) if (a) 0 < p < 1 and (b) —1 < p < 0? When will the bias in o2 be reasonably small?

12.25. The residuals from the wages-productivity regression given in (12.5.1) were regressed on lagged residuals going back six periods [i.e., AR(6)], yielding the following results:

Dependent Variable: RES1 Method: Least Squares Sample(adjusted): 1965-1998

Included Observations: 34 after adjusting endpoints 