Assignment on Serial Correlation

Assignment on Serial Correlation Chapter 9

Suggested Solutions:

Answer 9-2 a. Reject H0 of ‘no positive serial correlation (d < dL = 1.03) b. Cannot reject H0 of no positive serial correlation (d > dU = 1.25) c. Inconclusive (dL = 0.98 < d < 1.73 = dU) d. Inconclusive (4 – dU = 2.37 < d < 4 – dL = 2.87) e. Cannot reject H0 of no positive serial correlation (d > dU = 1.57) f. Reject H0 of no serial correlation (d < dL = 1.04) g. Inconclusive (dL = 0.9 < d < 1.99 = dU)

Answer 9-4 a. B*0 = B0 (1 – ρ)  to get B0, divide by B*0 by (1 – ρ) b. To account for the fact that the equation was estimated with GLS c. B0 = 23.5/( 1 - 0.8) = 117.5 d. The constant term is different for different econometric equations. This doesn’t concern us because it acts as a ‘garbage collector’ for the equation it is in. As a result, we should not analyze the estimated values of the constant term.

Answer 9-8 Parts b. and c

Technically speaking, GLS can be applied but theoretically the procedure will not make sense because this time error term is now hypothesized to be a function of next time’s error term. In theory, this will only occur when the decision makers accurately predict and adjust to future random events before they occur, which would be in the world of rational expectations and perfect information.

Part a

Assume the data has been collected for four years: 1990, 1991, 1992 and 1993. During these years, et = ρet – 1. Now, suppose the data were entered in reverse chronological order: 1993, 1992, 1991 and 1990. The serial correlation would still exist; only et = ρet+1. The estimate of ρ is the same in both equations as the error terms are still correlated and DW can detect that. However, the serial correlation would not be detected correctly for the first term as 1993 error term would be a function of 1994 error term (not in the data set).

Answer 9-10 a. F = 13.99 – 0.7RP + 0.85P, with standard errors of 1.84 and 0.25 respectively and t-statistics of t = -0.38 and 3.41 respectively. R2 = 0.288, N = 25 and DW = 1.248 b. The relative price coefficient is now insignificant, and the dummy variable is now significant in the unanticipated direction (that is, the Pope’s decision significantly increased fish consumption). In addition the DW is inconclusive in testing for serial correlation, but the DW of 1.247 is quite close to dL of 1.21 at 5% level of significance. Thus, the omitted variable has not only caused bias, it also has moved the DW d just about into the positive serial correlation range. c. This exercise is a good example of why it makes sense to reach for specification errors before adjusting for serial correlation.

Answer 9-12 a. Table 1:

Coefficient βlnY βPB βPP βD Hypothesized + - + - sign Calculated T- 6.6 - 2.6 2.7 - 3.7 score T-c = 1.714 at All variables are significantly different from zero in the expected 5% direction

b. With a 5%, one sided test and N = 28, K = 4, the critical values are dL = 1.1 and dU = 1.75. Since, d = 0.94 < dL , we conclude positive serial correlation.

c. The probable serial correlation suggests GLS

d. We prefer GLS equation because we’ve rid the equation of much of the serial correlation while retaining estimated coefficients that make economic sense. Note that the improved fit is not an evidence of better equation!

Answer 9-14 a. Equation 9.26 Coefficient Β1 Β2 Β3 Hypothesized + + + sign Calculated T- 0.76 14.98 1.8 score T-c = 1.721 at Insig Sig sig 5%

Equation 9.27 Coefficient Β1 Β2 Hypothesized + + sign Calculated T- 1.44 28.09 score T-c = 1.721 at Insig Sig 5% b. All three specification criteria imply that SP is a relevant variable. Adjusted R2 increases as SP is added, SP’s coefficient is significantly different from zero, and the estimated coefficient of LnUSD changes when SP is added. In addition while, authors expected the sign would be negative (an idea supported by the fact that the authors obtaine a negative sign for SP coefficient for the subset of the sample from 1960 to 1976) and that equation 9.26 therefore has a significant unexpected sign by an omitted variable. However, the authors could explain the positive hypothesized sign by stating that “Soviet” leadership became more competitive after 1977, leading the USSR to increase defense spending as SP increased. No matter which sign you expect, SP cannot be considered to be irrelevant. c. For both equations, DW is far below the critical values for a 5% one sided test, so we can reject the null hypothesis of no positive serial correlation. This result makes us worry that SP’s coefficient may be inflated, making it more likely to be an irrelevant variable. d. Such a small improvement in DW statistic is no evidence whatsoever that the serial correlation is impure. e. Just as we were beginning to suspect, running GLS makes SP insignificant making it even more likely to be an irrelevant variable.