BUSS 230 Homework Assignment 2 Solution, Spring 2012

1 BUSS 230: Managerial Economics Fall 2011-2012 Regression Assignment ANSWER KEY Sections 1 to 6 Question 1 a. 2  is the output elasticity of labor. It measures the percentage change in output due to a percentage change in labor. 3  is the output elasticity of capital. It measures the percentage change in output due a to a percentage change in capital. b. Regression Analysis R² 0.688 Adjusted R² 0.668 R 0.830 Std. Error 0.217 33 observations 2 predictor variables LNQ is the dependent variable ANOVA table Source SS df MS F p-value Regression 3.1123 2 1.5561 33.12 2.55E-08 Residual 1.4094 30 0.0470 Total 4.5217 32 Regression output confidence interval variables coefficients std. error t (df=30) p- value 95% lower 95% upper intercept -0.1287 0.5461 -0.236 .8153 -1.2440 0.9867 LNL 0.5590 0.8164 0.685 .4988 -1.1084 2.2264 2 LNK 0.4877 0.7039 0.693 .4937 -0.9498 1.9252 c. We can use p-values to test the significance of the coefficients 2  and 3  at the 5% level. For 2  : p-value = 0.49>0.05=α. So we do not reject the null that 2  is different from zero. Conclusion: 2  is insignificant For 3  : p-value = 0.49>0.05=α. So we do not reject the null that 3  is different from zero. Conclusion: 3  is insignificant. d. For the joint significance of the variables in this regression, we need to use the p-value of the F-test. p-value of F-test = 2.55×10-8< 0.05. So we reject the null that the variables in this regression are jointly insignificant. Conclusion: the variables in this regression are jointly significant. e. The R2 of this regression is 68% and is reasonably high. This regression exhibits good fir. Note that R2 can be interpreted as 68% of the variation in log quantity being due to variation in log labor and log capital. f. This production function exhibits increasing returns to scale. This is due to the fact that 2ˆ + 3ˆ =0.5590+0.4877>1. g. The variables are individually insignificant but jointly highly significant. This is not very intuitive and might suggest the presence of multicollinearity. h. We can compute the correlation between the two independent variables ln(K) and ln(L). The correlation coefficient is 0.98 and this indicates the presence of multicollinearity. Question 2 a. Due to the law of demand, 12  , 22 and 32 are expected to be negative. The signs of 13  , 23 and 33  depend on whether we expect the good to be an inferior or normal good. Meat is expected to be a normal good so 13  is expected to be positive. Fruits and vegetables are also expected to be normal goods so 23 is expected to be positive. It can be argued that cereals and bakery products are either a normal or inferior good. Therefore, 33  can be either positive or negative. 3 b. 12  , 22 and 32 are, respectively, the price elasticities of demand for meat, fruits and vegetables and cereals and bakery products. 13  , 23 and 33  are, respectively, the income elasticity of demand for meat, fruits and vegetables and cereals and bakery products. c. Regression Analysis R² 0.623 Adjusted R² 0.595 R 0.789 Std. Error 0.394 30 observations 2 predictor variables LNq1 is the dependent variable ANOVA table Source SS df MS F p-value Regression 6.9437 2 3.4719 22.33 1.89E-06 Residual 4.1977 27 0.1555 Total 11.1415 29 Regression output confidence interval variables coefficients std. error t (df=27) p-value 95% lower 95% upper intercept 1.0174 1.3541 0.751 .4590 -1.7611 3.7958 LNy 1.4339 0.2288 6.267 1.05E- 06 0.9644 1.9033 LNp1 -0.5670 0.2149 -2.639 .0136 -1.0079 -0.1261 Regression Analysis R² 0.541 Adjusted R² 0.507 R 0.736 Std. Error 0.451 30 observations 2 predictor variables LNq2 is the dependent variable ANOVA table Source SS df MS F p-value Regression 6.4654 2 3.2327 15.92 2.71E-05 Residual 5.4830 27 0.2031 4 Total 11.9484 29 Regression output confidence interval variables coefficients std. error t (df=27) p- value 95% lower 95% upper intercept 2.4628 1.4529 1.695 .1016 -0.5183 5.4439 LNp2 -0.6482 0.1875 -3.456 .0018 -1.0330 -0.2634 LNy 1.1435 0.2612 4.378 .0002 0.6075 1.6794 Regression Analysis R² 0.915 Adjusted R² 0.909 R 0.956 Std. Error 0.187 30 observations 2 predictor variables LNq3 is the dependent variable ANOVA table Source SS df MS F p-value Regression 10.1149 2 5.0575 145.07 3.60E-15 Residual 0.9413 27 0.0349 Total 11.0562 29 Regression output confidence interval variables coefficients std. error t (df=27) p-value 95% lower 95% upper intercept 4.8696 0.5467 8.908 1.60E- 09 3.7479 5.9913 LNp3 -0.9639 0.0653 -14.769 1.87E- 14 -1.0978 -0.8300 LNy 0.8713 0.1082 8.050 1.19E- 08 0.6492 1.0934 d. For 12  , p-value = 0.0136<0.05=α, therefore we reject the null that 12  is insignificant (i.e. it is significant) For 13  , p-value=1.05×10-6<0.05=α, therefore we reject the null that 13  is insignificant (i.e. it is significant) For 22 , p-value=0.0018<0.05=α, therefore we reject the null that 22 is insignificant (i.e. it is significant) For 23 , p-value = 0.0002 <0.05=α, therefore we reject the null that 23 is insignificant (i.e. it is significant) 5 For 32 , p-value = 1.87×10-14<0.05=α, therefore we reject the null that 32 is insignificant (i.e. it is significant) For 33  , p-value = 1.19×10-8<0.05=α, therefore we reject the null that 33  is insignificant (i.e. it is significant) e. All three regressions have a high R2 ranging from 54% to 91%. The best “fit” corresponds to the third demand equation (that of cereals and bakery with an R2 of around 91%) Question 3 a. The portfolio manager is postulating that S&P500 prices follow a linear (or secular) trend model with seasonal variation. b. See excel output. Regression Analysis R² 0.820 Adjusted R² 0.814 R 0.906 Std. Error 198.928 123 observations 4 predictor variables P is the dependent variable ANOVA table Source SS df MS F p-value Regression 21,330,410.8066 4 5,332,602.7017 134.76 4.98E-43 Residual 4,669,552.1591 118 39,572.4759 Total 25,999,962.9657 122 Regression output confidence interval variables coefficients std. error t (df=118) p-value 95% lower 95% upper intercept -26.4871 47.9633 -0.552 .5818 - 121.4675 68.4933 t 11.7296 0.5053 23.214 8.24E- 46 10.7290 12.7302 D1 -5.0832 50.9497 -0.100 .9207 - 105.9774 95.8110 D2 -2.9312 50.9472 -0.058 .9542 - 103.8205 97.9581 D3 -20.1559 50.9497 -0.396 .6931 - 121.0502 80.7383 6 c. The parameter estimate for b is positive. We need to test for the significance of b at the 5% to establish whether a time trend exists. p-value=8.24×10-48 <0.05=α, therefore we reject the null that b is insignificant (i.e. it is significant). Therefore, b is positive and significant and there is evidence of a time trend in S&P500 prices. d. p-value of c1 = 0.9207 > 0.05, we do not reject the null that c1 is insignificant. p-value of c2 = 0.9542>0.05, we do not reject the null that c2 is insignificant. p-value of c3 = 0.6937>0.05, we do not reject the null that c2 is insignificant. Given that all three seasonal dummy variables are insignificant, we conclude that there is no evidence of seasonality in S&P500 prices. e. Forecast of S&P500 for 2010Q4: P(2010Q4) = -26.4871+11.72×124=1428.80 f. 3 quarter MA forecast is: 1113.78, while 5 quarter moving average forecast is: 1102.704 g. The adjusted closing price on 31 December 2010 is 1,257.64. The best forecasting method is the 3 quarter moving average as it is the closest to the actual value that materializes in December 2010. Question 4 a. 7 b. The correlation between E-commerce retail sales and business inventories is 0.879. This is a high correlation coefficient indicating a strong positive linear relationship between the variables. c. Yes, a seasonal pattern is expected in both variables. Sales are expected to increase during the holiday season (In Christmas, the 4th quarter of the year) while business are expected to hold higher inventories in the quarter preceding Christmas (3rd quarter) in anticipation for the increase in sales. d. and e. Denote business inventories by and E-commerce retail sales by Xt and inventories by Yt. The following 2 equations can be estimated to check for a time trend and seasonal pattern in sales: Xt = a+ bt + c1D1+ c2D2+ c3D3+et Yt = a+ bt + c1D1+ c2D2+ c3D3+et Where: t = 1, 2,…, 44. D1 = 1 if t is quarter 1 D2 = 1 if t is quarter 2 D3 = 1 if t is quarter 3 Estimating the 2 equations yields: 8 Regression Analysis R² 0.691 Adjusted R² 0.659 R 0.831 Std. Error 77666.022 44 observations 4 predictor variables BUSSINESSINVENTORIES is the dependent variable ANOVA table Source SS df MS F p-value Regression 525,777,004,336.9070 4 131,444,251,084.2270 21.79 1.65E-09 Residual 235,248,427,936.9800 39 6,032,010,972.7431 Total 761,025,432,273.8860 43 Regression output confidence interval variables coefficients std. error t (df=39) p- value 95% lower 95% intercept 1,090,454.5017 30,433.8979 35.830 1.85E- 31 1,028,896.1334 1,152,012 time 8,617.3960 925.6456 9.310 1.86E- 11 6,745.1011 10,489 Q1 -2,206.5778 33,129.8368 -0.067 .9472 -69,217.9973 64,804 Q2 -3,599.6102 33,168.6079 -0.109 .9141 -70,689.4515 63,490 Q3 -3,732.0972 33,233.1257 -0.112 .9112 -70,952.4382 63,488 Regression Analysis R² 0.965 Adjusted R² 0.962 R 0.982 Std. Error 2324.924 44 observations 4 predictor variables E- COMMERCERETAILSALES is the dependent variable ANOVA table Source SS df MS F p-value Regression 5,848,914,539.6000 4 1,462,228,634.9000 270.52 7.16E-28 Residual 210,805,548.9455 39 5,405,270.4858 9 Total 6,059,720,088.5455 43 Regression output confidence interval variables coefficients std. error t (df=39) p-value 95% lower 95% upper intercept 5,613.6409 911.0354 6.162 3.09E- 07 3,770.8980 7,456.3838 time 905.3591 27.7091 32.674 6.01E- 30 849.3122 961.4060 Q1 -4,404.1773 991.7380 -4.441 .0001 - 6,410.1567 - 2,398.1979 Q2 -4,781.9909 992.8986 -4.816 2.23E- 05 - 6,790.3179 - 2,773.6639 Q3 -4,995.4409 994.8299 -5.021 1.17E- 05 - 7,007.6744 - 2,983.2075 Which shows that business inventories have a positive time trend and no clear seasonal pattern (time trend is significant while seasonal dummy variables are not) whereas sales exhibit both a positive and significant time trend and a significant seasonal pattern (both the time trend and seasonal dummies are significant as evidenced by the very small p-vales on the time trend and seasonal dummy variables).