Problem Set 4 Answer Key

Economics 31 Menzie D. Chinn Fall 4 Social Sciences 7418 University of Wisconsin-Madison Problem Set 4 Answer Key This problem set is due in lecture on Wednesday, December 1st. No late problem sets will be accepted. Be sure to show your work (that is, do not use a spreadsheet or statistical program to generate your answers), and to write your name, ID number, as well as the name of your Teaching Assistant, on your problem set. Numbered problems refer to the textbook problems. 8.16 8. 8.3 8.5 8.56 8.6 8.7 8.8 8.8 9. 9. 9.8 9.36 9.44 9.6 9.7

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Problem X.1 During the late 199s, there was a belief that Europe lagged behind the United States in productivity growth, especially over the 1995q1-q4 period (shaded area in the graph below)..6.5 US (4 qtr growth rate).4.3..1. -.1 -. Eurozone (4 qtr growth rate) 8 84 86 88 9 9 94 96 98 D(LPRODUS9,,4) D(LPRODEU9,,4) Figure 1: US and Eurozone 4 quarter growth rates. Here are some statistics below on quarter-on-quarter annualized growth rates and differences in growth rates between the two economies. 5 4 3 1 -....4.6.8 Series: D(LPRODUS9)*4 Sample 1995:1 :4 Observations 4 Mean.6989 Median.3134 Maximum.8838 Minimum -.1684 Std. Dev..5717 Skewness.57541 Kurtosis 3.1793 Jarque-Bera 1.33777 Probability.5146 Figure : Histogram for United States qoq annualized GDP growth 9

7 6 5 4 3 1 -.4 -....4.6.8 Series: D(LPRODEU9)*4 Sample 1995:1 :4 Observations 4 Mean.161 Median.8859 Maximum.8951 Minimum -.3664 Std. Dev..5384 Skewness.84397 Kurtosis 4.86637 Jarque-Bera 6.331998 Probability.417 Figure 3: Histogram for Eurozone qoq annualized GDP growth 5 4 Series: DIF Sample 1995:1 :4 Observations 4 3 1 -.6 -.4 -....4.6 Mean.14377 Median.13389 Maximum.61469 Minimum -.551 Std. Dev..3564 Skewness -.357949 Kurtosis.34387 Jarque-Bera.94445 Probability.63739 Figure 4: Histogram for DIF = (d(lprodus9)-d(lprodeu9))*4 X.1.a. Conduct a test for difference in means between the US and Eurozone, assuming independent sampling, and setting the probability of Type I error equal to.5%. The hypothesis test that corresponds to this text is: H dyus = dy H : dy dy Eu A US Eu The general formula for this test, assuming independent random sampling, is given by: ( X1 X s ) D ( n1 1) s1 + ( n 1) s t = where s p = n + n s p 1 1 + n n 1 1 or in this case where 1 refers to the US and refers to Eu, and D =, t = ( XUS X Eu ) (. 7. 16). 144 = = = 138. 1 1 1 1. 143 s p + (. 136) + n n 4 4 US Eu 1

The critical value for tα / = t. 15 for 46 (=4+4-) degrees of freedom cannot be read off the t-table directly, for two reasons. First, there are no 46 df entries, just 4 and 6 df entries. Use 4 df; then there is no entry for t.15 (remember this is a two-tailed test). One way to proceed is to realize that even using a lower probability of Type I, error, say.5, means the critical t value is.1, and 1.38 <.1, one fails to reject the null hypothesis in favor of the alternative. The other approach is to be more careful, and to linearly interpolate between the two values;.5 (=.15-.1) is 1/6 of the distance between.5 and.1, so one could say that.1 + (5/6) (.43-.1) =.356 is the critical t value. X.1.b. Conduct a test for difference in means between the US and Eurozone assuming paired sampling, and once again setting the probability of Type I error equal to.5%. The hypothesis test is now: H : dy$ = where dy$ = dy dy H dy A: $ x t = s D D US. 144 = 1. / n. 36 / 3 Eu Now it becomes obvious that being careful is important. Using the t.5 critical value for 3 df (=4-1), one would reject the null. However, linearly interpolating, one finds that the t.15 critical value is approximately.69+(5/6) (.5-.69) =.48. Notice, if one performed the one tail test corresponding to the hypothesis test: H : dy$ = H dy A: $ > then one could use the entry in the t-tables, where the critical value is.69 <.48, and one would reject the null hypothesis in favor of the alternative. X.1.c. Which testing procedure is more appropriate? Why? The paired differences test is more appropriate because the data are sampled in a systematically related manner. Problem X. Consider the following data pertaining to annualized quarter-on-quarter growth rates of real US GDP (in 1996 chained dollars) over the last 5 years. 11

.1.5. -.5 -.1 8 8 84 86 88 9 9 94 96 98 4 D(LGDP96C)*4 Figure 5: US qoq annualized GDP growth rate, 1979q1-4q3 X..a. Suppose one wanted to test the hypothesis that the variability of US GDP growth was greater than.5%. Write out the hypothesis test, given the following information. 1 1 Series: D(LGDP96C)*4 Sample 1979:3 4:4 Observations 11 8 6 4 -.5..5 Mean.9616 Median.3474 Maximum.8918 Minimum -.8157 Std. Dev..9318 Skewness -.971839 Kurtosis 5.5774 Jarque-Bera 37.33745 Probability. Figure 6: Histogram for United States qoq annualized GDP growth, 1979q3-4q3. H : σ = ( 5. ) H : σ > (. 5) X..b. Conduct the hypothesis test, using the data provided. What is your conclusion, setting α =.5? We calculate the χ statistic: ( n 1) s χ = = 137. 36 σ The critical value for 1 df (=11-1) is 14.3; hence, we reject the null hypothesis that the standard deviation of the annualized growth rate is.5% in favor of the alternative that it is greater than.5%. 1

X..c. What assumptions do you need to impose in order to arrive at your conclusion? It is necessary to assume that the growth rate is normally distributed. X.3. There is a widespread view that the US macroeconomy has been more stable since 1984. X.3.a. Write out a hypothesis test that the economy has been more stable in the post-1984 era. Let sample 1 correspond to the pre-1984 era, and sample correspond to the post-1984 era. Then H :σ = σ 1 H 1 :σ > σ s X.3.b. Conduct the appropriate test, using the statistics provided below, and setting Type I error equal to 1%. 16 Series: D(LGDP96C)*4 Sample 1953:1 1984:4 Observations 17 1 8 4 -.1 -.5 -..5.1.15 13 Mean.3815 Median.3167 Maximum.15467 Minimum -.117 Std. Dev..4575 Skewness -.37195 Kurtosis 3.4857 Jarque-Bera 3.19589 Probability.31 Figure 7: Histogram for United States qoq annualized GDP growth, 1953q1-1984q4 1 8 6 4 -....4.6 Series: D(LGDP96C)*4 Sample 1985:1 4:4 Observations 79 Mean.3981 Median.3474 Maximum.71446 Minimum -.335 Std. Dev..1 Skewness -.46968 Kurtosis 3.4974 Jarque-Bera 3.197991 Probability.99 Figure 8: Histogram for United States qoq annualized GDP growth, 1985q1-4q3

Calculate the F-statistic: s1 (. 451) F = = 4985. s (. ) The critical F-statistic for α=.1 and 16 (=17-1) numerator degrees of freedom and 78 (=79-1) denominator degrees of freedom is between approximately 1.53 (1 denominator df) and 1.73 (6 denominator df). Using linear interpolation, one finds that the critical value is approximately 1.67. Since the F-statistic is far in excess of this value, we can reject the null hypothesis that the variances are the same in the two samples in favor of the alternative that the variance in the early subsample is greater than that in the later. 7.11.4 14