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FIGURE 12.12 Hypothetical regression residuals plotted against time.

12.13. Based on the Durbin-Watson d statistic, how would you distinguish "pure" autocorrelation from specification bias?

12.14. Suppose in the model

Yt = Pi + Pi Xt + ut the us are in fact serially independent. What would happen in this situation if, assuming that ut = p ut-i + et, we use the generalized difference regression

Yt - p Yt-i = Pi(i - p) + Pi Xt - pPi Xt_i + et Discuss in particular the properties of the disturbance term et.

496 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

12.15. In a study of the determination of prices of final output at factor cost in the United Kingdom, the following results were obtained on the basis of annual data for the period 1951-1969:

PFt = 2.033 + 0.273Wt — 0.521Xt + 0.256Mt + 0.028Mt—1 + 0.121PFt—1 se = (0.992) (0.127) (0.099) (0.024) (0.039) (0.119)

where PF = prices of final output at factor cost, W = wages and salaries per employee, X = gross domestic product per person employed, M = import prices, Mt—1 = import prices lagged 1 year, and PFt—1 = prices of final output at factor cost in the previous year.*

"Since for 18 observations and 5 explanatory variables, the 5 percent lower and upper d values are 0.71 and 2.06, the estimated d value of 2.54 indicates that there is no positive autocorrelation." Comment.

12.16. Give circumstances under which each of the following methods of estimating the first-order coefficient of autocorrelation p may be appropriate:

a. First-difference regression b. Moving average regression c. Theil-Nagar transform d. Cochrane and Orcutt iterative procedure e. Hildreth-Lu scanning procedure f. Durbin two-step procedure

12.17. Consider the model:

Ut = P1 Ut-1 + P2 Ut-2 + St that is, the error term follows an AR(2) scheme, and where et is a white noise error term. Outline the steps you would take to estimate the model taking into account the second-order autoregression. 12.18. Including the correction factor C, the formula for j?GLS given in (12.3.1) is

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