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45-734 PROBABILITY AND STATISTICS II Homework Answers #5 (4th Mini AY1997-98)

    1. Here are the regressions for Clinton, Bush, and Perot: 
      ============================================================
LS // Dependent Variable is CLINT92                                   
Date: 03/27/98   Time: 14:21                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           40.71648   1.473157   27.63893   0.0000          
       AFRAM         0.417606   0.022756   18.35184   0.0000          
        HISP         0.161357   0.021758   7.416127   0.0000          
       INCOME       -0.090043   0.042894  -2.099211   0.0364          
       SOUTH        -3.292143   0.824754  -3.991666   0.0001          
     LCECON103      -14.92176   1.049951  -14.21187   0.0000          
      LCSOC103      -4.266176   0.831283  -5.132037   0.0000          
============================================================
R-squared            0.750458    Mean dependent var 43.92138          
Adjusted R-squared   0.746715    S.D. dependent var 12.17377          
S.E. of regression   6.126750    Akaike info criter 3.642378          
Sum squared resid    15014.83    Schwarz criterion  3.711326          
Log likelihood      -1311.732    F-statistic        200.4892          
Durbin-Watson stat   1.335062    Prob(F-statistic)  0.000000          
============================================================

============================================================
LS // Dependent Variable is BUSH92                                    
Date: 03/27/98   Time: 14:22                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           34.01372   1.286647   26.43594   0.0000          
       AFRAM        -0.192880   0.019875  -9.704868   0.0000          
        HISP        -0.092295   0.019003  -4.856866   0.0000          
       INCOME        0.149004   0.037463   3.977354   0.0001          
       SOUTH         7.393401   0.720336   10.26383   0.0000          
     LCECON103       12.53965   0.917021   13.67433   0.0000          
      LCSOC103       3.054497   0.726038   4.207077   0.0000          
============================================================
R-squared            0.671402    Mean dependent var 36.98034          
Adjusted R-squared   0.666473    S.D. dependent var 9.265626          
S.E. of regression   5.351069    Akaike info criter 3.371642          
Sum squared resid    11453.58    Schwarz criterion  3.440590          
Log likelihood      -1256.637    F-statistic        136.2152          
Durbin-Watson stat   1.488689    Prob(F-statistic)  0.000000          
============================================================

============================================================
LS // Dependent Variable is PEROT92                                   
Date: 03/27/98   Time: 14:23                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           24.44912   0.991061   24.66964   0.0000          
       AFRAM        -0.225134   0.015309  -14.70626   0.0000          
        HISP        -0.069345   0.014637  -4.737525   0.0000          
       INCOME       -0.054807   0.028857  -1.899291   0.0582          
       SOUTH        -3.747834   0.554850  -6.754673   0.0000          
     LCECON103       2.357762   0.706351   3.337948   0.0009          
      LCSOC103       1.109201   0.559243   1.983397   0.0480          
============================================================
R-squared            0.536450    Mean dependent var 18.50369          
Adjusted R-squared   0.529496    S.D. dependent var 6.008972          
S.E. of regression   4.121750    Akaike info criter 2.849605          
Sum squared resid    6795.529    Schwarz criterion  2.918553          
Log likelihood      -1150.403    F-statistic        77.15089          
Durbin-Watson stat   0.946190    Prob(F-statistic)  0.000000          
============================================================
      The coefficients in the Clinton and Bush regressions all have the correct signs and are statistically significant -- the largest p-value is only .0364 on the INCOME variable in the Clinton equation. Clinton did very well in districts with sizable African-American populations and in districts represented by liberal congressmen.

      Bush did very well in the South, higher income districts, and in districts represented by conservative congressmen. He did poorly in districts with sizable African-American populations but the negative effect was not as large as Clinton's positive effect.

      Perot clearly hurt Bush in 1992. Perot's voters outside the South tended to be White Conservatives from somewhat lower income districts. These voters were part of Ronald Reagan's base of support.

      Note that the size of the coefficients on the measure of economic liberalism/conservativism is much larger than the size of the coefficients on the measure of social liberalism/conservatism. Since the scale of the two measures is the same, this is a clear indication that voters place more importance on economic issues.

    2. Here are the regressions for Clinton, Dole, and Perot: 
      ============================================================
LS // Dependent Variable is CLINT96                                   
Date: 03/27/98   Time: 14:57                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           46.67272   1.421413   32.83545   0.0000          
       AFRAM         0.380007   0.022350   17.00282   0.0000          
        HISP         0.217946   0.021059   10.34908   0.0000          
       INCOME       -0.032785   0.041479  -0.790384   0.4298          
       SOUTH        -4.212347   0.775116  -5.434470   0.0000          
     LCECON104      -16.02892   0.965621  -16.59960   0.0000          
      LCSOC104      -5.472769   0.790804  -6.920517   0.0000          
============================================================
R-squared            0.791252    Mean dependent var 50.41769          
Adjusted R-squared   0.788121    S.D. dependent var 12.74948          
S.E. of regression   5.868626    Akaike info criter 3.556290          
Sum squared resid    13776.31    Schwarz criterion  3.625238          
Log likelihood      -1294.213    F-statistic        252.6980          
Durbin-Watson stat   1.378019    Prob(F-statistic)  0.000000          
============================================================

============================================================
LS // Dependent Variable is DOLE96                                    
Date: 03/27/98   Time: 14:59                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           38.47091   1.504226   25.57522   0.0000          
       AFRAM        -0.250124   0.023652  -10.57529   0.0000          
        HISP        -0.143436   0.022286  -6.436031   0.0000          
       INCOME        0.107895   0.043896   2.457973   0.0144          
       SOUTH         6.465954   0.820276   7.882661   0.0000          
     LCECON104       16.25024   1.021879   15.90231   0.0000          
      LCSOC104       5.418992   0.836877   6.475258   0.0000          
============================================================
R-squared            0.725911    Mean dependent var 39.73464          
Adjusted R-squared   0.721799    S.D. dependent var 11.77471          
S.E. of regression   6.210538    Akaike info criter 3.669545          
Sum squared resid    15428.31    Schwarz criterion  3.738492          
Log likelihood      -1317.260    F-statistic        176.5629          
Durbin-Watson stat   1.426119    Prob(F-statistic)  0.000000          
============================================================

============================================================
LS // Dependent Variable is PEROT96                                   
Date: 03/27/98   Time: 14:59                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           13.70630   0.441327   31.05702   0.0000          
       AFRAM        -0.116052   0.006939  -16.72412   0.0000          
        HISP        -0.082123   0.006539  -12.55963   0.0000          
       INCOME       -0.097065   0.012879  -7.536853   0.0000          
       SOUTH        -1.479197   0.240662  -6.146370   0.0000          
     LCECON104       0.189186   0.299811   0.631018   0.5284          
      LCSOC104       0.472191   0.245532   1.923131   0.0552          
============================================================
R-squared            0.604723    Mean dependent var 8.289926          
Adjusted R-squared   0.598794    S.D. dependent var 2.876689          
S.E. of regression   1.822119    Akaike info criter 1.217049          
Sum squared resid    1328.047    Schwarz criterion  1.285997          
Log likelihood      -818.1775    F-statistic        101.9915          
Durbin-Watson stat   1.146324    Prob(F-statistic)  0.000000          
============================================================
      The regression results for Clinton in 1996 are essentially the same as those for 1992. The only big difference is the intercept. Clinton's vote, all else held equal, was higher across the board in 1996. Interestingly, the income effect is no longer statistically significant. This could reflect Clinton's success in protraying himself as a fiscal conservative. However, on the other hand, note that the coefficients on the ideological variables have increased slightly in magnitude.

      The pattern of the coefficents for Dole in 1996 is very similar to that for Bush in 1992. However, Dole did much better than Bush in 1992 -- the intercept term for Dole is larger. Dole drew better from Conservatives of all stripes than Bush did but did more poorly with African-Americans and Hispanics.

      Perot's base of support in 1996 clearly shifted from 1992. Note that the coefficient on economic liberalism/conservatism is very small and statistically insignificant, the coefficent on social liberalism/conservatism is half what it was in 1992, the South coefficient is smaller, the income effect is more pronounced, and the coefficient on percent African-American is half what it was in 1992. In other words, Perot was appealing even more so to "downscale" voters but now they were largely non-ideological (that is, moderate) and drawn from a broader segment of the population than 1992 (that is, his support was not as concentrated demographically as it was in 1992). Perot may have hurt Dole more than Clinton but it is hard to tell from these regression results.

    3. Here are the regression results for Clinton, Dole, and Perot with the indicator variables for seat switches: 
      ============================================================
LS // Dependent Variable is CLINT96                                   
Date: 03/27/98   Time: 16:08                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           47.09082   1.381522   34.08618   0.0000          
       AFRAM         0.368268   0.021727   16.94977   0.0000          
        HISP         0.211036   0.020340   10.37540   0.0000          
       INCOME       -0.047721   0.040007  -1.192812   0.2337          
       SOUTH        -3.799335   0.748587  -5.075344   0.0000          
     LCECON104      -17.40202   0.961411  -18.10050   0.0000          
      LCSOC104      -5.134018   0.770416  -6.663957   0.0000          
 REP104*(1-REP105)   7.064975   1.326490   5.326069   0.0000          
 (1-REP104)*REP105  -4.381055   1.734261  -2.526180   0.0119          
============================================================
R-squared            0.807977    Mean dependent var 50.41769          
Adjusted R-squared   0.804118    S.D. dependent var 12.74948          
S.E. of regression   5.642741    Akaike info criter 3.482605          
Sum squared resid    12672.53    Schwarz criterion  3.571252          
Log likelihood      -1277.218    F-statistic        209.3341          
Durbin-Watson stat   1.350070    Prob(F-statistic)  0.000000          
============================================================

====================================================
Wald Test:                                                    
Equation: Untitled                                            
====================================================
Null HypothesisC(8)=-C(9)                                     
====================================================
F-statistic     1.515547    Probability     0.219022          
Chi-square      1.515547    Probability     0.218295          
====================================================

============================================================
LS // Dependent Variable is DOLE96                                    
Date: 03/27/98   Time: 16:11                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           38.00511   1.464857   25.94458   0.0000          
       AFRAM        -0.237542   0.023038  -10.31105   0.0000          
        HISP        -0.136022   0.021567  -6.306931   0.0000          
       INCOME        0.123671   0.042421   2.915344   0.0038          
       SOUTH         6.039659   0.793742   7.609093   0.0000          
     LCECON104       17.68563   1.019404   17.34898   0.0000          
      LCSOC104       5.049261   0.816888   6.181092   0.0000          
 REP104*(1-REP105)  -7.203941   1.406505  -5.121874   0.0000          
 (1-REP104)*REP105   4.807518   1.838874   2.614382   0.0093          
============================================================
R-squared            0.746888    Mean dependent var 39.73464          
Adjusted R-squared   0.741801    S.D. dependent var 11.77471          
S.E. of regression   5.983118    Akaike info criter 3.599749          
Sum squared resid    14247.49    Schwarz criterion  3.688396          
Log likelihood      -1301.057    F-statistic        146.8036          
Durbin-Watson stat   1.411552    Prob(F-statistic)  0.000000          
============================================================

====================================================
Wald Test:                                                    
Equation: Untitled                                            
====================================================
Null HypothesisC(8)=-C(9)                                     
====================================================
F-statistic     1.074688    Probability     0.300518          
Chi-square      1.074688    Probability     0.299889          
====================================================

============================================================
LS // Dependent Variable is PEROT96                                   
Date: 03/27/98   Time: 16:13                                          
Sample: 1 407                                                         
Included observations: 407                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           13.76346   0.446793   30.80498   0.0000          
       AFRAM        -0.116878   0.007027  -16.63351   0.0000          
        HISP        -0.082622   0.006578  -12.56016   0.0000          
       INCOME       -0.097749   0.012939  -7.554774   0.0000          
       SOUTH        -1.475876   0.242098  -6.096195   0.0000          
     LCECON104       0.149666   0.310927   0.481356   0.6305          
      LCSOC104       0.506470   0.249158   2.032728   0.0427          
 REP104*(1-REP105)  -0.081821   0.428996  -0.190726   0.8488          
 (1-REP104)*REP105  -0.483862   0.560871  -0.862696   0.3888          
============================================================
R-squared            0.605498    Mean dependent var 8.289926          
Adjusted R-squared   0.597568    S.D. dependent var 2.876689          
S.E. of regression   1.824900    Akaike info criter 1.224915          
Sum squared resid    1325.443    Schwarz criterion  1.313562          
Log likelihood      -817.7782    F-statistic        76.35835          
Durbin-Watson stat   1.143984    Prob(F-statistic)  0.000000          
============================================================

====================================================
Wald Test:                                                    
Equation: Untitled                                            
====================================================
Null HypothesisC(8)=-C(9)                                     
====================================================
F-statistic     0.643692    Probability     0.422856          
Chi-square      0.643692    Probability     0.422377          
====================================================
      The magnitude of the coefficients on the party-switch indicator variables in the Clinton and Dole regressions is about the same. In those districts that switched from Republican to Democrat in the 1996 elections, Clinton picked up 7 percentage points and Dole lost 7 percentage points; and in those districts that switched from Democrat to Republican in 1996, Clinton lost a bit over 4 percentage points and Dole picked up a bit under 5 percentage points. This makes sense and is consistent with a story of voter punishment of the House Republicans for shutting down the government. However, the Wald tests for Clinton and Dole show that we cannot reject the null hypothesis that the effect is the same. Even so, the effect appears to be present but it is not very large in magnitude. Finally, note that in the Perot regresssion these coefficients are both negative and statistically insignificant! This does not support a story about Perot appealling to the disaffected!

    1. If b1 = 1, then we can divide both sides of the top model by SQFT to get the bottom model.

    2. Note that the estimated coefficient, b1, for LOG(SQFT), in the first model is very close to 1.0. 
      
============================================================
LS // Dependent Variable is LOG(PRICE)                                
Date: 03/28/98   Time: 14:26                                          
Sample: 1 574                                                         
Included observations: 574                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           5.192543   0.257102   20.19645   0.0000          
     LOG(SQFT)       0.997349   0.041457   24.05732   0.0000          
        BED          0.049292   0.021497   2.292995   0.0222          
        BATH         0.007593   0.025845   0.293801   0.7690          
       STORY        -0.009927   0.002515  -3.946731   0.0001          
       CDIST         4.59E-06   3.53E-06   1.298839   0.1945          
        PARK         0.141336   0.025300   5.586488   0.0000          
        MDUM        -0.249986   0.079382  -3.149150   0.0017          
        BDUM        -0.242321   0.077275  -3.135843   0.0018          
        CDUM        -0.304950   0.078438  -3.887802   0.0001          
============================================================
R-squared            0.836759    Mean dependent var 11.69985          
Adjusted R-squared   0.834154    S.D. dependent var 0.574891          
S.E. of regression   0.234120    Akaike info criter-2.886577          
Sum squared resid    30.91400    Schwarz criterion -2.810747          
Log likelihood       23.97687    F-statistic        321.2241          
Durbin-Watson stat   1.976597    Prob(F-statistic)  0.000000          
============================================================

============================================================
LS // Dependent Variable is LOG(PRICE/SQFT)                           
Date: 03/28/98   Time: 14:28                                          
Sample: 1 574                                                         
Included observations: 574                                            
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           5.176905   0.079368   65.22628   0.0000          
        BED          0.048406   0.016428   2.946602   0.0033          
        BATH         0.006803   0.022677   0.299990   0.7643          
       STORY        -0.009910   0.002498  -3.967230   0.0001          
       CDIST         4.57E-06   3.52E-06   1.299165   0.1944          
        PARK         0.141187   0.025171   5.609190   0.0000          
        MDUM        -0.249641   0.079128  -3.154881   0.0017          
        BDUM        -0.242077   0.077113  -3.139273   0.0018          
        CDUM        -0.304846   0.078352  -3.890743   0.0001          
============================================================
R-squared            0.157690    Mean dependent var 4.978085          
Adjusted R-squared   0.145764    S.D. dependent var 0.253085          
S.E. of regression   0.233913    Akaike info criter-2.890054          
Sum squared resid    30.91423    Schwarz criterion -2.821807          
Log likelihood       23.97479    F-statistic        13.22181          
Durbin-Watson stat   1.976815    Prob(F-statistic)  0.000000          
============================================================
      The R2 for the first model is the proportion of the variance in price per dwelling explained by the model. The R2 for the second model is the proportion of the variance in price per square foot explained by the model. The models are essentially the same. The size of the dwelling seems to explain much of the price of a dwelling. Indeed, the correlation between LOG(PRICE) and LOG(SQFT) is .902! Neither model is better than the other. We can say that it is harder to explain price per square foot than price per dwelling (note that the s.d. of the dependent variable in the second model is less than one-half the size of the s.d. of the first model). This example shows why being overly concerned with R2 can be misleading. By simply rewriting the second model in the form of the first model, the R2 looks better.

    3. To solve the problem, plug the values given in the problem into the second equation (all except STORY):

      5.176905 + 0.048406*3 + 0.006803*2 + 0.00000457*2640 + 0.141187*1 -0.249641*1 - 0.242077*0 -0.304846*0 = 5.23934

      Hence:

      exp(å (bi*valuei))= exp(5.23934) = 188.5456.

      To find the price per square foot for a building, we multiply
      188.5456 by exp(-.009910*#number of stories). And to find the total revenue, we multiply by the number of stories. We find 21 to be the profit maximizing number of floors. That is:

      GENR REV=188.5456*EXP(-.00991*FLOORS)*FLOORS
      GENR COST=40*FLOORS + 2*FLOORS*FLOORS
      GENR PROFIT=REV - COST
      =================================
  obs     FLOORS       PROFIT              
=================================
   1      1.000000     144.6863            
   2      2.000000     281.6908            
   3      3.000000     411.0679            
   4      4.000000     532.8714            
   5      5.000000     647.1542            
   6      6.000000     753.9688            
   7      7.000000     853.3668            
   8      8.000000     945.3991            
   9      9.000000     1030.116            
  10      10.00000     1107.567            
  11      11.00000     1177.802            
  12      12.00000     1240.868            
  13      13.00000     1296.813            
  14      14.00000     1345.685            
  15      15.00000     1387.529            
  16      16.00000     1422.392            
  17      17.00000     1450.318            
  18      18.00000     1471.353            
  19      19.00000     1485.541            
  20      20.00000     1492.924            
  21      21.00000     1493.546            
  22      22.00000     1487.448            
  23      23.00000     1474.673            
  24      24.00000     1455.262            
  25      25.00000     1429.256            
  26      26.00000     1396.693            
  27      27.00000     1357.615            
  28      28.00000     1312.059            
  29      29.00000     1260.065            
  30      30.00000     1201.670