Exercise Sheet 5: Solutions

Transcription

1 Exercise Sheet 5: Solutions R.G. Pierse 2. Estimation of Model M1 yields the following results: Date: 10/24/02 Time: 18:06 C LPC LPF LPL LPD LNY LP D D D 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 (a) Noting the relationships between the variables: log RP C t = log(p C t /P t ) = log P C t log P t log RP F t = log(p F t /P t ) = log P F t log P t log RP L t = log(p L t /P t ) = log P L t log P t log RP D t = log(p D t /P t ) = log P D t log P t log RY t = log(y t /P t ) = log Y t log P t the model (M1) can be rewritten as log RC t = β 1 + β 2 log RP C t + β 2 log P t + β 3 log RP F t + β 3 log P t +β 4 log RP L t + β 4 log P t + β 5 log RP D t + β 5 log P t (M1) +β 6 log RY t + β 6 log P t + β 7 log P t +β 8 d1 t + β 9 d2 t + β 10 d3 t + ε t so that the sum of the coefficients on P t is β 2 + β 3 + β 4 + β 5 + β 6 + β 7. If the restriction β 7 = β 2 + β 3 + β 4 + β 5 + β 6 holds, then this is zero and the model collapses to (M2). (b) The Wald test gives: Null Hypothesis: C(2)+C(3)+C(4)+C(5)+C(6)+C(7)=0 F-statistic Probability Chi-square Probability The p-value for this test (in either form) is 0.53 so we fail to reject the null at the 5% level. Thus model (M2) is the preferred model. 2

3 3. Estimation of Model M2 yields the following results: Date: 10/24/02 Time: 18:45 C LRPC LRPF LRPL LRPD LRY D D D 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 coefficient of D1 ( 0.373) can be interpreted as an estimate of the change in the (logarithm of the) level of expenditure on clothing and footwear in the first quarter relative to the expenditure in the fourth quarter. The interpretation of D2 and D3 is similar. (b) For first order autocorrelation, we can use the Durbin-Watson statistic. The value of the statistic is which is below the critical value of d L which is (T = 150, k = 9) and so we reject the null of no autocorrelation. For fourth order autocorrelation, we use the LM test: 3

4 (c) Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability which rejects the null (both forms of the test have p-values of ). We would usually test for fourth order autocorrelation when we have quarterly data. (i) The own price elasticity is and the income elasticity is (ii) Testing for unit price elasticity: Null Hypothesis: C(2)=-1 F-statistic Probability Chi-square Probability the hypothesis of a unit price elasticity is strongly rejected (iii) Testing for unit income elasticity: Null Hypothesis: C(6)=1 F-statistic Probability Chi-square Probability we do not reject the null of a unit elasticity at the 5% level. 4

5 4. Creating a dummy variable DUM1 = 1 in 1973:1 but zero otherwise and a dummy variable DUM2 = 1 in 1973:2 but zero otherwise, we can add these dummies to the model and get the following results: Date: 10/24/02 Time: 19:14 C LRPC LRPF LRPL LRPD LRY D D D DUM DUM 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 see that dummy DUM1 (the dummy for the period before the tax change) is significant but dummy DUM2 is not. We might want to test the hypothesis that the coefficients on the dummies are equal but opposite in sign, which is the hypothesis that agents perfectly anticipated the tax change. Conversely, we could just drop DUM2. Note that all the models we have been looking at have very low Durbin- Watson statistics and so we might want to consider respecifying these models, possibly including lags of the model variables. 5