Exercise Sheet 6: Solutions

Transcription

1 Exercise Sheet 6: Solutions R.G. Pierse 1. (a) Regression yields: Dependent Variable: LC Date: 10/29/02 Time: 18:37 Sample(adjusted): Included observations: 36 after adjusting endpoints C LC(-1) LY LY(-1) INF INF(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

2 (i) The estimated short run income elasticity is The estimated long run income elasticity is β 3 + β 4 1 β = = Using the EViews Wald coefficient restrictions test facility, we can test the hypothesis that this long-run elasticity is zero: Wald Test on Long Run Elasticity Wald Test: Equation: Untitled Null Hypothesis: (C(3)+C(4))/(1-C(2)) F-statistic Probability Chi-square Probability (ii) The test for serial correlation yields: Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability On the basis of the Breusch-Godfrey serial correlation test, we would not reject the null of no autocorrelation. 2

3 (iii) Wald Test of hypothesis: β 2 + β 3 + β 4 = 1 Wald Test: Equation: Untitled Null Hypothesis: C(2)+C(3)+C(4)=1 F-statistic Probability Chi-square Probability We do not reject H 0 at the 5% significance level. (iv) Subtracting C t 1 from both sides of we get LC t = β 1 + β 2 LC t 1 + β 3 LY t + β 4 LY t 1 +β 5 INF t + β 6 INF t 1 + ε t LC t = β 1 + β 2 LC t 1 LC t 1 + β 3 LY t + β 4 LY t 1 +β 5 INF t + β 6 INF t 1 + ε t, and, adding and subtracting β 2 LY t 1 and β 3 LY t 1, we obtain LC t = β 1 + β 2 (LC t 1 LY t 1 ) LC t 1 + β 3 LY t +(β 2 + β 3 + β 4 )LY t 1 + β 5 INF t + β 6 INF t 1 + ε t. If the restriction β 2 + β 3 + β 4 = 1 holds then we can rewrite this as or β 3 + β 4 1 β 2 = 1 LC t = β 1 + (β 2 1)(LC t 1 LY t 1 ) + β 3 LY t +β 5 INF t + β 6 INF t 1 + ε t which is the same as the restricted model LC t = γ 1 + γ 2 LY t + γ 3 (LY t 1 LC t 1 ) +γ 4 INF t + γ 5 INF t 1 + ε t. with the parameter correspondences: γ 1 = β 1, γ 2 = β 3, γ 3 = 1 β 2, and γ 4 = β 5 and γ 5 = β 6. 3

4 2. Unrestricted Distributed Lag Dependent Variable: Y Date: 10/29/02 Time: 19:28 Sample(adjusted): 1953: :12 Included observations: 53 after adjusting endpoints C X X(-1) X(-2) X(-3) X(-4) X(-5) X(-6) X(-7) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) (a) The estimated impact (short-run) multiplier is The estimated long-run multiplier is = Using the EViews Wald coefficient restrictions test facility, we can test the hypothesis that this long-run multiplier is zero: 4

5 Wald Test on Long Run Multiplier Wald Test: Equation: Untitled Null Hypothesis: C(2)+C(3)+C(4)+C(5)+C(6)+C(7)+C(8)+C(9)=0 F-statistic Probability Chi-square Probability (b) Form the Almon variables: Z0 = X + X( 1) + X( 2) + X( 3) + X( 4) +X( 5) + X( 6) + X( 7) Z1 = X( 1) + 2 X( 2) + 3 X( 3) + 4 X( 4) +5 X( 5) + 6 X( 6) + 7 X( 7) Z2 = X( 1) + 4 X( 2) + 9 X( 3) + 16 X( 4) +25 X( 5) + 36 X( 6) + 49 X( 7) Z3 = X( 1) + 8 X( 2) + 27 X( 3) + 64 X( 4) +125 X( 5) X( 6) X( 7) Z4 = X( 1) + 16 X( 2) + 81 X( 3) X( 4) +625 X( 5) X( 6) X( 7) 5

6 Fourth Degree Almon Lag Polynomial Dependent Variable: Y Date: 10/29/02 Time: 19:45 Sample(adjusted): 1953: :12 Included observations: 53 after adjusting endpoints C Z Z Z Z Z4 7.05E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) We can see from the p-value that the coefficient on the fourth order term, Z4, is not significant so we drop it and estimate the third degree polynomial Almon lag. 6

7 Third Degree Almon Lag Polynomial Dependent Variable: Y Date: 10/29/02 Time: 19:49 Sample(adjusted): 1953: :12 Included observations: 53 after adjusting endpoints C Z Z Z Z R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) We can see from the p-value that the coefficient on the third order term, Z3, is not significant so we drop it and estimate the second degree polynomial Almon lag. 7

8 Second Degree Almon Lag Polynomial Dependent Variable: Y Date: 10/29/02 Time: 19:56 Sample(adjusted): 1953: :12 Included observations: 53 after adjusting endpoints C Z Z Z R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) This time the p-value of the coefficient on the highest order term, Z2, shows that it is significant and so we stop searching and choose a second degree polynomial. (c) One way to obtain an estimate of the long-run (equilibrium) multiplier is to note that in equilibrium so that X t = X t 1 = X t 1 = = X t 7 Y t = c + 7 α 0 X t + ( ) α 1 X t +( ) α 2 X t. The long-run multiplier is therefore given by 7 α 0 + ( ) α 1 + ( ) α 2 =

9 A Wald test for the hypothesis that the long-run multiplier is zero is given by: Wald Test on Long Run Multiplier Wald Test: Equation: Untitled Null Hypothesis: 7*C(2)+7*4*C(3)+( )*C(4)=0 F-statistic Probability Chi-square Probability