## Exercises

Questions

CHAPTER TWELVE: AUTOCORRELATION 491

12.2. Given a sample of 50 observations and 4 explanatory variables, what can you say about autocorrelation if (a) d = 1.05? (b) d = 1.40? (c) d = 2.50? (d) d = 3.97?

12.3. In studying the movement in the production workers' share in the value added (i.e., labor's share), the following models were considered by Gujarati*:

Model A: Yt = 00 + 01t + u Model B: Yt = a0 + a11 + a2t2 + ut where Y = labor's share and t = time. Based on annual data for 19491964, the following results were obtained for the primary metal industry:

ModelA: Y = 0.4529 — 0.0041t R2 = 0.5284 d = 0.8252

where the figures in the parentheses are t ratios.

a. Is there serial correlation in model A? In model B?

b. What accounts for the serial correlation?

c. How would you distinguish between "pure'' autocorrelation and specification bias?

12.4. Detecting autocorrelation: von Neumann ratio test.^ Assuming that the residual ut are random drawings from normal distribution, von Neumann has shown that for large n, the ratio

S2 E(ui — ui — 1)2/(n — 1) 7 . —— = -=- Note: u = 0 in OLS

s2 Y.(u — u)2 / n called the von Neumann ratio, is approximately normally distributed with mean

and variance n - 1

a. If n is sufficiently large, how would you use the von Neumann ratio to test for autocorrelation?

b. What is the relationship between the Durbin-Watson d and the ratio?

Damodar Gujarati, "Labor's Share in Manufacturing Industries," Industrial and Labor Relations Review, vol. 23, no. 1, October 1969, pp. 65-75.

J. von Neumann, "Distribution of the Ratio of the Mean Square Successive Difference to the Variance,'' Annals of Mathematical Statistics, vol. 12, 1941, pp. 367-395.

492 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

c. The d statistic lies between 0 and 4. What are the corresponding limits for the von Neumann ratio?

d. Since the ratio depends on the assumption that the us are random drawings from normal distribution, how valid is this assumption for the OLS residuals?

e. Suppose in an application the ratio was found to be 2.88 with 100 observations. Test the hypothesis that there is no serial correlation in the data.

Note: B. I. Hart has tabulated the critical values of the von Neumann ratio for sample sizes of up to 60 observations.*

12.5. In a sequence of 17 residuals, 11 positive and 6 negative, the number of runs was 3. Is there evidence of autocorrelation? Would the answer change if there were 14 runs?

12.6. Theil-Nagar p estimate based on d statistic. Theil and Nagar have suggested that in small samples instead of estimating p as (1 — d/2), it be estimated as

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