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Page 1 Answers Chapter 11 11.1, 11.2, 11.7, 11.10 (parts a and b only). 11.1 See class notes on April 19th …we did all of these transformations in class. Specification Transformation: for var(et) 2 xt Why??? Divide the model by X1/4: independent variables: 1/X1/4 and X/X1/4 We divide by the standard deviation, which is the square root of the variance : X1/4. We can ignore the term in all of the models since it doesn’t vary over observations. 2xt Divide the model by X1/2 Independent variables: 1/X and X/X1/2 1/2 This is just like the one we did in class. The standard deviation is xt 2 xt2 Divide the model by X: independent variables: 1/x plus an intercept Here, the standard deviation is xt, so we divide by xt 2ln(xt) Divide the model by (ln(X))1/2 Here the standard deviation is Independent variables: 1/(ln(x))1/2 ln( xt ) Here is the SAS code and output from estimating the food expenditure model for each of these transformations. Make sure to look through the results and pick out the estimates of 1 and 2. options ls=78; options formdlim='*'; data food; infile 'c:\documents and settings\doylejm\my documents\classes\ec385\ue\datafiles\table3-1.dat' firstobs=2; input y x; proc reg ; white: model y = x / acov; run; data foodgls; set food; y_a = y/(x**(1/4)); x1_a = 1/(x**(1/4)); x2_a = x/(x**(1/4)); y_b = y/sqrt(x); x1_b = 1/sqrt(x); Page 2 x2_b = x/sqrt(x); y_c = y/x; x_c = 1/x; lnx = log(x); y_d = y/sqrt(lnx); x1_d = 1/sqrt(lnx); x2_d = x/sqrt(lnx); proc reg; Part_A: model y_a = x1_a x2_a / noint; Part_B: model y_b = x1_b x2_b / noint; Part_C: model y_c = x_c; Part_D: model y_d = x1_d x2_d / noint; run; Model: white Dependent Variable: y Analysis of Variance Source DF Sum of Squares Mean Square Model Error Corrected Total 1 38 39 25221 54311 79533 25221 1429.24556 Root MSE Dependent Mean Coeff Var 37.80536 130.31300 29.01120 R-Square Adj R-Sq F Value Pr > F 17.65 0.0002 0.3171 0.2991 Parameter Estimates Variable Intercept x DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 40.76756 0.12829 22.13865 0.03054 1.84 4.20 0.0734 0.0002 ****************************************************************************** The SAS System 10 The REG Procedure Model: white Dependent Variable: y Consistent Covariance of Estimates Variable Intercept x Intercept x 561.88951794 -0.886643258 -0.886643258 0.0014569032 Page 3 Below are 4 sets of regression estimates, for parts a) – d). The estimates that are highlighted in yellow are estimates of the 1 (intercept in the original model) and those highlighted blue are estimates of the 2 (slope in the original model). Model: Part_A Dependent Variable: y_a NOTE: No intercept in model. R-Square is redefined. Analysis of Variance Source DF Sum of Squares Mean Square Model Error Uncorrected Total 2 38 40 26028 1915.30947 27943 13014 50.40288 Root MSE Dependent Mean Coeff Var 7.09950 25.30918 28.05108 R-Square Adj R-Sq F Value Pr > F 258.20 <.0001 0.9315 0.9279 Parameter Estimates Variable x1_a x2_a DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 36.75257 0.13391 20.05232 0.02879 1.83 4.65 0.0747 <.0001 ****************************************************************************** Model: Part_B Dependent Variable: y_b NOTE: No intercept in model. R-Square is redefined. Analysis of Variance Source DF Sum of Squares Mean Square Model Error Uncorrected Total 2 38 40 978.84644 68.70197 1047.54841 489.42322 1.80795 Root MSE Dependent Mean Coeff Var 1.34460 4.93981 27.21965 Variable x1_b x2_b DF R-Square Adj R-Sq Parameter Estimates Parameter Standard Estimate Error 1 1 31.92438 0.14096 F Value Pr > F 270.71 <.0001 0.9344 0.9310 t Value Pr > |t| 1.77 5.22 0.0839 <.0001 17.98608 0.02700 ****************************************************************************** Model: Part_C Dependent Variable: y_c Analysis of Variance Source DF Sum of Squares Mean Square Model Error Corrected Total 1 38 39 0.00562 0.09285 0.09846 0.00562 0.00244 F Value Pr > F 2.30 0.1377 Page 4 Root MSE Dependent Mean Coeff Var 0.04943 0.19116 25.85840 R-Square Adj R-Sq 0.0571 0.0322 Parameter Estimates Variable Intercept x_c DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 0.15769 21.28584 0.02342 14.03797 6.73 1.52 <.0001 0.1377 ****************************************************************************** Model: Part_D Dependent Variable: y_d NOTE: No intercept in model. R-Square is redefined. Analysis of Variance Source DF Sum of Squares Mean Square Model Error Uncorrected Total 2 38 40 106685 8111.23735 114797 53343 213.45361 Root MSE Dependent Mean Coeff Var Variable x1_d x2_d 14.61005 R-Square 50.89489 Adj R-Sq 28.7063 Parameter Estimates F Value Pr > F 249.90 <.0001 0.9293 0.9256 DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 39.55015 0.12996 21.46901 0.02997 1.84 4.34 0.0733 0.0001 11.2 Code and SAS output appear below. (a) Countries with high per capita income can decide whether to spend larger amounts on education than their poorer neighbours, or to spend more of their larger income on other things. They are likely to have more discretion with respect to where public monies are spent. On the other hand, countries with low per capita income may regard a particular level of education spending as essential, meaning that they have less scope for deviating from a mean function. These differences can be captured by a model with heteroskedasticity. Remember that heteroskedasticity is more common in cross-section data. (b) The least squares estimated function is yt 01246 . 0.07317 xt (0.0485) (0.00518) R2 0.862 This function and the corresponding residuals appear in Figure 11.1. The absolute magnitude of the errors does tend to increase as x increases suggesting the existence of heteroskedasticity. Page 5 Yt 1.6 1.4 1.2 y = - 0.1246 + 0.0732x 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0 5 10 15 20 Xt Figure 11.1 Estimated Function for Education Expenditure (c) Since it is suspected that, if heteroskedasticity exists, the variance is related to xt , we begin by ordering the observations according to the magnitude of xt. Then, splitting the sample into two equal subsets of 17 observations each, and applying least squares to each subset, we obtain 12 = 0.0081608 and 22 = 0.029127 leading to a Goldfelt-Quandt statistic of GQ 0.029127 = 3.569 0.008161 The critical value from an F-distribution with (15,15) degrees of freedom and a 5% significance level is Fc = 2.40. Since 3.569 > 2.40 we reject a null hypothesis of homoskedasticity and conclude that the error variance is directly related to per capita income xt. (e) Generalized least squares estimation under the assumption var et 2 xt yields yt 0.0929 0.06932 xt (0.0289) (0.00441) (note: I have expressed these results in the model’s original form although it was estimated with no intercept and two independent variables: the reciprocal of the square root of x and x over the square root of x.) The estimated response of per capita education expenditure to per capita income has declined slightly relative to the least squares estimate. The associated 95% confidence interval is (0.0603, 0.0783). This interval is narrower than both those computed from least squares estimates. The comparison with the White-calculated interval suggests that generalized least squares is more efficient; a comparison with the conventional least squares interval is not really valid because the standard errors used to compute that interval are not valid. See below for the case were Var(et) = 2X2t. The differences of how this is carried out and how to interpret the results is important. Part B Source Least Squares results The REG Procedure Model: MODEL1 Dependent Variable: y Analysis of Variance Sum of DF Squares Model Error Corrected Total 1 32 33 3.68386 0.59063 4.27449 Mean Square 3.68386 0.01846 F Value Pr > F 199.59 <.0001 Page 6 Root MSE Dependent Mean Coeff Var Variable Intercept x DF 1 1 0.13586 0.47674 28.49753 R-Square Adj R-Sq Parameter Estimates Parameter Standard Estimate Error -0.12457 0.04852 0.07317 0.00518 0.8618 0.8575 t Value -2.57 14.13 Pr > |t| 0.0151 <.0001 ****************************************************************************** The REG Procedure Model: MODEL1 Dependent Variable: y This is part D, White standard Error for b2 would the the square root of 0.0000363146 = 0.006. This is larger than the 0.00518 value reported y least squares. Consistent Covariance of Estimates Variable Intercept x Intercept 0.0015372135 -0.000211654 x -0.000211654 0.0000363146 ****************************************************************************** This regression gets you the numerator for the GQ-statistic The REG Procedure Model: MODEL1 Dependent Variable: y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 Error 15 Corrected Total 16 Root MSE Dependent Mean Coeff Var Variable Intercept x DF 1 1 0.42220 0.42220 14.50 0.0017 0.43690 0.02913 0.85910 0.17067 R-Square 0.4914 0.78115 Adj R-Sq 0.4575 21.84803 Parameter Estimates Parameter Standard Estimate Error t Value Pr > |t| -0.14087 0.24569 -0.57 0.5749 0.07516 0.01974 3.81 0.0017 This regression gets you the denominator for the GQ-statistic The REG Procedure Model: MODEL1 Dependent Variable: y Analysis of Variance Sum of Mean Source DF Squares Square F Value Model 1 0.14225 0.14225 17.43 Error 15 0.12241 0.00816 Corrected Total 16 0.26466 Pr > F 0.0008 Root MSE Dependent Mean Coeff Var Variable Intercept x DF 1 1 0.09034 R-Square 0.5375 0.17232 Adj R-Sq 0.5066 52.42382 Parameter Estimates Parameter Standard Estimate Error t Value Pr > |t| -0.03807 0.05495 -0.69 0.4990 0.05047 0.01209 4.17 0.0008 ****************************************************************************** These are the critical values The SAS System Obs fc tc 17 Page 7 1 2.40345 2.03693 This regression corrects for heteroskedasticity of the form var(et) = 2Xt The REG Procedure Model: MODEL1 Dependent Variable: ystar NOTE: No intercept in model. R-Square is redefined. Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Uncorrected Total 2 32 34 0.96083 0.06341 1.02423 0.48041 0.00198 242.45 <.0001 Root MSE Dependent Mean Coeff Var 0.04451 R-Square 0.9381 0.15116 Adj R-Sq 0.9342 29.44875 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| x1star 1 -0.09292 0.02890 -3.21 0.0030 x2star 1 0.06932 0.00441 15.71 <.0001 We predict that if GDP per capita increases by $1.00, pubic expenditures on education per capital will increase by $0.069 ****************************************************************************** This regression corrects for heteroskedasticity of the form var(et) = 2X2t The REG Procedure Model: MODEL1 Dependent Variable: ystar Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 0.00349 0.00349 12.69 0.0012 Error 32 0.00880 0.00027504 Corrected Total 33 0.01229 Root MSE 0.01658 R-Square 0.2840 Dependent Mean 0.05153 Adj R-Sq 0.2616 Coeff Var 32.18259 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.06443 0.00460 13.99 <.0001 xstar 1 -0.06739 0.01892 -3.56 0.0012 We predict that if GDP per capita increases by $1.00, pubic expenditures on education per capital will increase by $0.064, because the intercept in this transformed model is actually the slope coefficient the original model. Here is the code that generated the results: data pubexp; infile 'A:pubexp.dat' firstobs=2; input ee gdp pop; y = ee/pop; x = gdp/pop; proc sort; by descending x; proc reg; model y = x / acov; output out=pubout p=yhat r=ehat; symbol1 i=join c=green v=circle; proc gplot; plot y*x = '*' yhat*x = 1 /overlay; plot ehat*x; * create data set; Page 8 data one; set pubexp; if _n_ <= 17; proc reg; model y = x; data two; set pubexp; if _n_ >= 18; proc reg; model y = x; * critcal values for tests; data; fc = finv(.95,15,15); tc = tinv(.975,32); proc print; *PART E GLS via weighted least squares;; data two; set pubexp; ystar = y/sqrt(x); x1star = 1/sqrt(x); x2star = x/sqrt(x); ** this code corrects for hetero assuming var(et) = sigmasq*xt; proc reg ; model ystar = x1star x2star / noint; run; data three; set pubexp; ystar = y/x; xstar = 1/x; ** this code corrects for hetero assuming var(et) = sigmasq*xtsquare; proc reg ; model ystar = xstar; run; 11.7 (a) The least squares estimates of equation (11.7.5) are y t = 2.243 + 0.164 xt + 1.145 nt (2.669) (0.035) (0.414) R2 = 0.45 These results suggest that an increase in income of $1000 will increase food expenditure by $164; an additional person in the household will increase food expenditure by $1,145. Both the estimated slope coefficients are significantly different from zero. (b) See Figures 11.2 and 11.3. Overall, the residuals tend to increase in absolute value as x increases and as n increases. Thus, the plots suggest the existence of heteroskedasticity that is dependent on both xt and nt. Page 9 10 5 RESID 0 -5 -10 -15 20 40 60 80 100 X Figure 11.2 Residuals Plotted Against Income. 10 5 RESID 0 -5 -10 -15 0 2 4 6 8 N Figure 11.3 Residuals Plotted Against Number of Persons (c) (i) To perform the first Goldfeld-Quandt test we order the observations according to decreasing values of xt. Then, we find the least squares regression of yt 1 2 xt 3 nt et for both the first and second halves of the observations, to obtain estimates 12 and 22 , respectively. We find that 12 = 31.129 and 22 = 5.8819. Although we are not hypothesizing constant error variances within each subsample, to perform the Goldfeld-Quandt test we proceed as if H0 and H1 are given by H0: 12 22 and H1: 22 12 . The test statistic value is: GQ ˆ 12 31.129 5.2923 ˆ 22 5.8819 The 5% critical value for (16, 16) degrees of freedom is Fc = 2.33. Thus, because GQ = 5.2923 > Fc = 2.33, we reject H0 and conclude that heteroskedasticity exists, and is dependent on xt. (ii) When we order the observations with respect to nt , there is not a unique ordering because nt takes on repeated integer values. There are 8 observations where nt = 3. One of these values must be included in the first 19 observations, the other 7 in the last 19 observations. There are 8 ways of doing this. The results from SAS, EViews and SHAZAM are as follows. Page 10 GQ 12 22 28.233 2.88 9.799 (SAS) These values are greater than 2.33, and so we reject a null hypothesis of homoskedasticity and conclude that the error variances are dependent on nt. These test outcomes are consistent with the evidence provided by the residual plots in part (b). (d) The alternative variance estimators yield the following standard errors: Standard Errors Coefficients White Least Squares 2 3 0.0287 0.4360 0.0354 0.4140 The results from White's variance estimator suggest the usual least squares results would underestimate the reliability of estimation for 2 and overestimate the reliability of estimation for 3. (e) To find generalized least squares estimates when 2t 2 ht 2 exp0.055xt 012 . nt we begin by calculating ht for each observation. Then we apply least squares to the transformed model. yt 1 x n e 1 2 t 3 t t ht ht ht ht ht The resulting estimates, with those from least squares, and the White standard errors are in the table below. The two estimates for 2 are similar, but the GLS estimate for the response of food expenditure to an additional household member is noticeably higher. The standard errors suggest that 1 and 3 have been more precisely estimated by GLS, but not 2. However, we do need to keep in mind that standard errors are square roots of estimated variances. It is possible for an improvement in precision to take place even when it is not reflected by the standard errors. Variable constant xt nt GLS LS (White) 1.682 2.243 (1.760) (2.270) 0.160 0.165 (0.032) (0.029) 1.364 1.145 (0.285) (0.436) Page 11 11.10 (a) The graphs for plotting the residuals against income and age show that the absolute values of the residuals increase as income increases but they appear to be constant as age increases. This indicates that the error variance depends on income. (b) Since the residual plot shows that the error variance may increase when income increases, and this is a reasonable outcome since greater income implies greater flexibility in travel, we set the null and alternative hypotheses as H0 : 12 22 against H1 : 12 22 . The test statistic is GQ ˆ 12 (2.9471 107 ) (100 4) 2.8124 ˆ 22 (1.0479 107 ) (100 4) The 5% critical value for (96, 96) degrees of freedom is Fc 1.35 . Thus, we reject H 0 and conclude that the error variance depends on income.