Dec. 9, 2003 ECON 240A-1 L. PhillipsFinalAnswer all five questions. Each question is worth 30 points.1. A survey of economics (econometrics) students was conducted about the monthly rentper person that they paid along with the information about the number of persons living together, the number of rooms in the unit, the number of blocks from the centerof campus, and the gender of the person responding, female = 1, for yes, 0 for no. The32 observations of data is appended to the exam. Figure 1.1 shows the histogram and descriptive statistics for rent per person. Figure 1.2 depicts a regression of rent per person against the number of persons living together in the unit.a. Is this cross-section or time series data? Cros-sectionb. Is the distribution more or less symmetric? Yes, but a little skewed and kurtotic.c. Is it normally distributed? Explain. No. Based on the Jarque-Bera statistic, it isskewed enough and kurtotic enough to be significantly non-normal, but only at the 1% level. It is not wildly non-normal.d. What is the central tendency of rent per person? Based on the mean about $138, based in the median about $140 per month.024681050 75 100 125 150 175 200 225 250 275 300Series: RENTPERPERSONSample 1 32Obs ervations 32Mean 138.1693Median 140.0000Max imum 285.0000Minimum 50.75000Std. Dev. 47.11474Skewness 1.017101Kurtosis 4.787240Jarque-Bera 9.776269Probability 0.007535Figure 1.1 Histogram and Statistics for Rent Per PersonDec. 9, 2003 ECON 240A-2 L. PhillipsFinale. Is this bivariate regression significant at the 5% level? Explain. No. Based on an F statistic of F1, 30 = (R2 /1-R2)(n-2/1) = (0.0959/0.9041) 30 = 3.18 with a critical value of 4.18 at the 5% level.2. Another analyst looked at the data and noticed that rent appeared to be higher for women than for men. There were 10 women in the sample and 22 men. The mean rent per person for women was $164.17 and the mean rent for men was $126.35. Thisanalyst was a student of econometrics and calculated an indicator variable for males as one minus the indicator variable for females and regressed the 32 observations for rent per person against these two indicator variables. The results are displayed in Table 2.1. Then the analyst ran another regression, dropping the male indicator variable and adding a constant term. The results are presented in Table 2.2.Table 2.1 Regression of Rent Per Person Against Gender Indicator VariablesDependent Variable: RENTPERPERSONMethod: Least SquaresSample: 1 32Included observations: 32Variable Coefficient Std. Error t-Statistic Prob. FEMALE 164.1667 14.02177 11.70799 0.0000MALE 126.3523 9.453475 13.36570 0.0000R-squared 0.142860 Mean dependent var 138.1693Adjusted R-squared 0.114289 S.D. dependent var 47.11474S.E. of regression 44.34073 Akaike info criterion 10.48215Sum squared resid 58983.00 Schwarz criterion 10.57375Log likelihood -165.7143 Durbin-Watson stat 2.522594Fig. 1.2 Regression of Monthly Rent Per Person Against Personsy = -11.066x + 165.14R2 = 0.09590501001502002503000 1 2 3 4 5 6 7Persons Living TogetherRent Per PersonDec. 9, 2003 ECON 240A-3 L. PhillipsFinalTable 2.2: Regression of Rent per Person Against the Female Indicaor Variable and a Constant TermDependent Variable: RENTPERPERSONMethod: Least SquaresSample: 1 32Included observations: 32Variable Coefficient Std. Error t-Statistic Prob. FEMALE 37.81439 16.91089 2.236097 0.0329C 126.3523 9.453475 13.36570 0.0000R-squared 0.142860 Mean dependent var 138.1693Adjusted R-squared 0.114289 S.D. dependent var 47.11474S.E. of regression 44.34073 Akaike info criterion 10.48215Sum squared resid 58983.00 Schwarz criterion 10.57375Log likelihood -165.7143 F-statistic 5.000131Durbin-Watson stat 2.522594 Prob(F-statistic) 0.032934a. Is there a significant difference in monthly rent for men and women? Explain. Yes. The t-statistic of 2.27 for the coefficient on the female variable, testing the difference in mean rents, is significant at the 5% levelb. Suppose you were not a student of econometrics, and did not know about running regressions to answer this question. Could you get a back-of-the- envelope estimate? Hint: Assume the square root of the estimate of the pooled variance estimator, sp2, is equal to the standard deviation estimate in Figure 1.1. What do you calculate for a t-statistic using this approximation? 212121/1/1(]0()[( nnsxxt =(37.82-0)/47.11[(1/10)+(1/22)]1/2 =2.10.3. Someone looking at questions 1 and 2, above, might wonder about combining the twoanalyses and regressing rent per person against both the number of persons living together and gender. In the following multivariate regression, rent per person was regressed against the number of persons living together, the number of rooms rented, the number of blocks from the center of campus, and the indicator variable female. The results follow in Table 3.1. a. Is the regression in Table 3.1 significant as a whole at the 5% level? Explain. Yes, the F-stat of 4.27 is significant at the 1% level.b. Is the indicator variable female significant at the 5 % level? No. the t-stat of 1.51 is not significant at the 5% level.c. How can the indicator variable female be significant in Table 2.2 but not significant in Table 3.1? Explain. Female was the only regressor in Table 2.2 but is one of four regressors in Table 3.1, so in the latter case, the question isDec. 9, 2003 ECON 240A-4 L. PhillipsFinalwhether gender is significant controlling for (conditional on) other factors such as rooms, persons, and blocks,Based on the regression results in Table 3.1, two variables, female and blocks were dropped, although both had t-statistics that were greater than one but not very significant. The results are reported in Table 3.2Table 3.1: Multivariat Regression for Rent Per PersonDependent Variable: RENTPERPERSONMethod: Least SquaresSample: 1 32Included observations: 32Variable Coefficient Std. Error t-Statistic Prob. ROOMS 32.54048 13.33490 2.440249 0.0215PERSONS -39.96653 12.90240 -3.097605 0.0045BLOCKS -0.607752 0.578149 -1.051202 0.3025FEMALE 24.87735 16.49201 1.508449 0.1431C 165.6059 17.72424 9.343470 0.0000R-squared 0.387446 Mean dependent var 138.1693Adjusted R-squared 0.296697 S.D. dependent var 47.11474S.E. of regression 39.51192 Akaike info criterion 10.33368Sum squared resid 42152.17 Schwarz criterion 10.56270Log likelihood -160.3389 F-statistic 4.269429Durbin-Watson stat
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