45-734 PROBABILITY AND STATISTICS II (4th Mini AY1997-98)

Note Set 11 Handouts



  1. Heteroskedasticity Examples

    To perform the White Test, click the "view" button and select "Residual Tests". Under "Residual Tests" select "White Heteroskedasticity (No Cross Terms)."
    ============================================================
    White Heteroskedasticity Test:                                        
    ============================================================
    F-statistic          2.872725    Probability        0.000038          
    Obs*R-squared        54.15035    Probability        0.000094          
    ============================================================
                                                                          
    Test Equation:                                                        
    LS // Dependent Variable is RESID^2                                   
    Date: 04/21/98   Time: 13:03                                          
    Sample: 1 336                                                         
    Included observations: 336                                            
    ============================================================
          Variable      CoefficienStd. Errort-Statistic  Prob.            
    ============================================================
             C          -5.273037   1.975978  -2.668570   0.0080          
            TAX          0.004902   0.076273   0.064269   0.9488          
           TAX^2        -0.014658   0.040564  -0.361361   0.7181          
           DRKAGE        0.562102   0.206821   2.717822   0.0069          
          DRKAGE^2      -0.014380   0.005285  -2.720967   0.0069          
           PCINC        -0.074213   0.146670  -0.505982   0.6132          
          PCINC^2        0.009711   0.019575   0.496121   0.6202          
           MILES         94.47385   36.46556   2.590769   0.0100          
          MILES^2       -3898.757   1599.262  -2.437848   0.0153          
           YNGDRV       -1.304460   0.694870  -1.877270   0.0614          
          YNGDRV^2       0.965221   0.462340   2.087686   0.0376          
            INSP         0.004712   0.009554   0.493171   0.6222          
           MORMON       -0.677308   0.233042  -2.906385   0.0039          
          MORMON^2       0.727000   0.321472   2.261470   0.0244          
            PROT        -0.260051   0.236303  -1.100500   0.2720          
           PROT^2        0.090228   0.418702   0.215494   0.8295          
            CATH        -0.312100   0.129396  -2.411972   0.0164          
           CATH^2        0.521536   0.192144   2.714302   0.0070          
           SOBAB        -0.533068   0.217292  -2.453235   0.0147          
          SOBAB^2        1.122543   0.920530   1.219453   0.2236          
            WET         -0.116754   0.847227  -0.137807   0.8905          
           WET^2         0.086463   0.522267   0.165553   0.8686          
    ============================================================
    R-squared            0.161162    Mean dependent var 0.040846          
    Adjusted R-squared   0.105061    S.D. dependent var 0.072709          
    S.E. of regression   0.068783    Akaike info criter-5.290355          
    Sum squared resid    1.485577    Schwarz criterion -5.040425          
    Log likelihood       434.0164    F-statistic        2.872725          
    Durbin-Watson stat   1.896893    Prob(F-statistic)  0.000038          
    ============================================================
    
    LS(H) LFT18T20 C TAX DRKAGE PCINC MILES YNGDRV INSP MORMON PROT CATH SOBAB WET
    ============================================================
    LS // Dependent Variable is LFT18T20                                  
    Date: 04/21/98   Time: 13:07                                          
    Sample: 1 336                                                         
    Included observations: 336                                            
    White Heteroskedasticity-Consistent Standard Errors &                 
            Covariance                                                    
    ============================================================
          Variable      CoefficienStd. Errort-Statistic  Prob.            
    ============================================================
             C           3.165057   0.293886   10.76969   0.0000          
            TAX         -0.278794   0.045943  -6.068288   0.0000          
           DRKAGE       -0.035408   0.008922  -3.968661   0.0001          
           PCINC        -0.195457   0.039210  -4.984835   0.0000          
           MILES         110.1364   8.881711   12.40036   0.0000          
           YNGDRV        1.524236   0.161228   9.453931   0.0000          
            INSP        -0.075403   0.026855  -2.807803   0.0053          
           MORMON       -0.814448   0.137447  -5.925542   0.0000          
            PROT        -0.712803   0.143324  -4.973384   0.0000          
            CATH        -0.504089   0.160777  -3.135327   0.0019          
           SOBAB        -0.707793   0.244949  -2.889549   0.0041          
            WET          0.424073   0.106305   3.989223   0.0001          
    ============================================================
    R-squared            0.570763    Mean dependent var 4.025630          
    Adjusted R-squared   0.556190    S.D. dependent var 0.308939          
    S.E. of regression   0.205812    Akaike info criter-3.126519          
    Sum squared resid    13.72424    Schwarz criterion -2.990194          
    Log likelihood       60.49186    F-statistic        39.16620          
    Durbin-Watson stat   1.156329    Prob(F-statistic)  0.000000          
    ============================================================
    
    LS(WT=WEIGHT) LFT18T20 C TAX DRKAGE PCINC MILES YNGDRV INSP MORMON PROT CATH SOBAB WET
    ============================================================
    LS // Dependent Variable is LFT18T20                                  
    Date: 04/21/98   Time: 13:10                                          
    Weighting series: WEIGHT                                              
    Sample: 1 336                                                         
    Included observations: 336                                            
    ============================================================
          Variable      CoefficienStd. Errort-Statistic  Prob.            
    ============================================================
             C           3.222418   0.337739   9.541154   0.0000          
            TAX         -0.274002   0.067616  -4.052302   0.0001          
           DRKAGE       -0.035095   0.009525  -3.684584   0.0003          
           PCINC        -0.196498   0.035992  -5.459494   0.0000          
           MILES         106.9708   9.091373   11.76618   0.0000          
           YNGDRV        1.486685   0.151822   9.792312   0.0000          
            INSP        -0.076806   0.026530  -2.895037   0.0040          
           MORMON       -0.804665   0.148464  -5.419917   0.0000          
            PROT        -0.687807   0.146028  -4.710117   0.0000          
            CATH        -0.523134   0.129111  -4.051817   0.0001          
           SOBAB        -0.701809   0.247495  -2.835654   0.0049          
            WET          0.422053   0.184489   2.287683   0.0228          
    ============================================================
    Weighted Statistics                                                   
    ============================================================
    R-squared            0.166752    Mean dependent var 4.018315          
    Adjusted R-squared   0.138463    S.D. dependent var 0.221887          
    S.E. of regression   0.205953    Akaike info criter-3.125154          
    Sum squared resid    13.74299    Schwarz criterion -2.988829          
    Log likelihood       60.26252    F-statistic        5.894548          
    Durbin-Watson stat   1.167913    Prob(F-statistic)  0.000000          
    ============================================================
    Unweighted Statistics                                                 
    ============================================================
    R-squared            0.570406    Mean dependent var 4.025630          
    Adjusted R-squared   0.555821    S.D. dependent var 0.308939          
    S.E. of regression   0.205898    Sum squared resid  13.73565          
    Durbin-Watson stat   1.151230                                         
    ============================================================
    
  2. Auto Correlation Examples

    1. The first example is first-order autocorrelation with parameter .5. The true values for the intercept and slope are both equal to 2. The first output is for a simple two variable regression. Note the value for the Durban-Watson statistic and remember that the estimators are unbiased. The ordinary regression works well here because of low error and a large sample size.
      ============================================================
      LS // Dependent Variable is YYY                                       
      Date: 04/18/98   Time: 14:38                                          
      Sample(adjusted): 2 500                                               
      Included observations: 499 after adjusting endpoints                  
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.039942   0.055456   36.78497   0.0000          
               X           2.019524   0.053867   37.49101   0.0000          
      ============================================================
      R-squared            0.738775    Mean dependent var 2.100864          
      Adjusted R-squared   0.738250    S.D. dependent var 2.420290          
      S.E. of regression   1.238258    Akaike info criter 0.431411          
      Sum squared resid    762.0415    Schwarz criterion  0.448295          
      Log likelihood      -813.6873    F-statistic        1405.576          
      Durbin-Watson stat   0.850402    Prob(F-statistic)  0.000000          
      ============================================================
      
      Given the value of the Durban-Watson statistic, the first step is to try estimating the first-order autocorrelation model. To do this in EVIEWS, use the command:

      LS YYY C X AR(1)

      The Durban-Watson is now near 2 indicating that our correction has done the trick. Also note that the standard errors are now smaller.
      ============================================================
      LS // Dependent Variable is YYY                                       
      Date: 04/18/98   Time: 14:47                                          
      Sample(adjusted): 3 500                                               
      Included observations: 498 after adjusting endpoints                  
      Convergence achieved after 4 iterations                               
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.029702   0.107408   18.89720   0.0000          
               X           1.976182   0.037777   52.31180   0.0000          
             AR(1)         0.576651   0.036850   15.64846   0.0000          
      ============================================================
      R-squared            0.823824    Mean dependent var 2.090885          
      Adjusted R-squared   0.823113    S.D. dependent var 2.412425          
      S.E. of regression   1.014617    Akaike info criter 0.035029          
      Sum squared resid    509.5768    Schwarz criterion  0.060394          
      Log likelihood      -712.3535    F-statistic        1157.349          
      Durbin-Watson stat   2.029978    Prob(F-statistic)  0.000000          
      ============================================================
      Inverted AR Roots          .58                                        
      ============================================================
      
      As a further check, we can run higher order AR models. For example, to add a second-order autocorrelation issue the EVIEWS command:

      LS YYY C X AR(1) AR(2)

      ============================================================
      LS // Dependent Variable is YYY                                       
      Date: 04/18/98   Time: 14:51                                          
      Sample(adjusted): 4 500                                               
      Included observations: 497 after adjusting endpoints                  
      Convergence achieved after 4 iterations                               
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.026031   0.111789   18.12370   0.0000          
               X           1.981206   0.038435   51.54703   0.0000          
             AR(1)         0.552113   0.045357   12.17258   0.0000          
             AR(2)         0.040334   0.045379   0.888813   0.3745          
      ============================================================
      R-squared            0.824106    Mean dependent var 2.093776          
      Adjusted R-squared   0.823036    S.D. dependent var 2.413992          
      S.E. of regression   1.015496    Akaike info criter 0.038769          
      Sum squared resid    508.3972    Schwarz criterion  0.072641          
      Log likelihood      -710.8467    F-statistic        769.9438          
      Durbin-Watson stat   1.974323    Prob(F-statistic)  0.000000          
      ============================================================
      Inverted AR Roots          .6      -.07                               
      ============================================================
      
      To add a second-order and third-order autocorrelation terms issue the EVIEWS command:

      LS YYY C X AR(1) AR(2) AR(3)

      ============================================================
      LS // Dependent Variable is YYY                                       
      Date: 04/18/98   Time: 14:54                                          
      Sample(adjusted): 5 500                                               
      Included observations: 496 after adjusting endpoints                  
      Convergence achieved after 4 iterations                               
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.033835   0.110996   18.32347   0.0000          
               X           1.981932   0.038471   51.51745   0.0000          
             AR(1)         0.554496   0.045383   12.21827   0.0000          
             AR(2)         0.050379   0.052013   0.968582   0.3332          
             AR(3)        -0.015515   0.045462  -0.341276   0.7330          
      ============================================================
      R-squared            0.824542    Mean dependent var 2.099213          
      Adjusted R-squared   0.823113    S.D. dependent var 2.413381          
      S.E. of regression   1.015019    Akaike info criter 0.039843          
      Sum squared resid    505.8590    Schwarz criterion  0.082248          
      Log likelihood      -708.6746    F-statistic        576.8487          
      Durbin-Watson stat   1.979999    Prob(F-statistic)  0.000000          
      ============================================================
      Inverted AR Roots          .6       .14       -.18                    
      ============================================================
      
    2. This second example is a two variable model with first and second-order autocorrelation. The intercept and slope are both equal to 2.0 and the autocorrelation parameters are .5 and .3 respectively.

      Again, start with the simple linear regression. Note that the Durban-Watson is quite small so that the next thing we should try is estimating simple first-order autocorrelation.
      ============================================================
      LS // Dependent Variable is YYYY                                      
      Date: 04/18/98   Time: 14:57                                          
      Sample: 1 500                                                         
      Included observations: 500                                            
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.127447   0.074982   28.37268   0.0000          
               X           2.089195   0.072906   28.65600   0.0000          
      ============================================================
      R-squared            0.622489    Mean dependent var 2.189925          
      Adjusted R-squared   0.621731    S.D. dependent var 2.724954          
      S.E. of regression   1.675945    Akaike info criter 1.036746          
      Sum squared resid    1398.778    Schwarz criterion  1.053604          
      Log likelihood      -966.6557    F-statistic        821.1663          
      Durbin-Watson stat   0.463438    Prob(F-statistic)  0.000000          
      ============================================================
      
      First, try estimating first-order autocorrelation.
      ============================================================
      LS // Dependent Variable is YYYY                                      
      Date: 04/18/98   Time: 14:59                                          
      Sample(adjusted): 2 500                                               
      Included observations: 499 after adjusting endpoints                  
      Convergence achieved after 4 iterations                               
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.091612   0.214367   9.757170   0.0000          
               X           1.973435   0.036009   54.80381   0.0000          
             AR(1)         0.777316   0.028582   27.19551   0.0000          
      ============================================================
      R-squared            0.847857    Mean dependent var 2.187698          
      Adjusted R-squared   0.847244    S.D. dependent var 2.727233          
      S.E. of regression   1.065914    Akaike info criter 0.133659          
      Sum squared resid    563.5415    Schwarz criterion  0.158985          
      Log likelihood      -738.3982    F-statistic        1382.045          
      Durbin-Watson stat   2.448294    Prob(F-statistic)  0.000000          
      ============================================================
      Inverted AR Roots          .78                                        
      ============================================================
      
      Note that the Durban-Watson is somewhat high. This is suspicious so the next step is to try estimating the model with second-order autocorrelation.
      ============================================================
      LS // Dependent Variable is YYYY                                      
      Date: 04/18/98   Time: 15:01                                          
      Sample(adjusted): 3 500                                               
      Included observations: 498 after adjusting endpoints                  
      Convergence achieved after 5 iterations                               
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.062221   0.296098   6.964657   0.0000          
               X           2.002662   0.037604   53.25624   0.0000          
             AR(1)         0.537732   0.043203   12.44650   0.0000          
             AR(2)         0.308084   0.043190   7.133140   0.0000          
      ============================================================
      R-squared            0.860985    Mean dependent var 2.177893          
      Adjusted R-squared   0.860141    S.D. dependent var 2.721157          
      S.E. of regression   1.017651    Akaike info criter 0.042993          
      Sum squared resid    511.5928    Schwarz criterion  0.076813          
      Log likelihood      -713.3367    F-statistic        1019.860          
      Durbin-Watson stat   1.951773    Prob(F-statistic)  0.000000          
      ============================================================
      Inverted AR Roots          .8      -.35                               
      ============================================================
      
      The model with first and second-order autocorrelation looks good. The Durban-Watson statistic is near 2. To be cautious, check for a third-order effect.
      ============================================================
      LS // Dependent Variable is YYYY                                      
      Date: 04/18/98   Time: 15:02                                          
      Sample(adjusted): 4 500                                               
      Included observations: 497 after adjusting endpoints                  
      Convergence achieved after 5 iterations                               
      ============================================================
            Variable      CoefficienStd. Errort-Statistic  Prob.            
      ============================================================
               C           2.060181   0.279652   7.366943   0.0000          
               X           2.007865   0.037373   53.72512   0.0000          
             AR(1)         0.553574   0.045326   12.21324   0.0000          
             AR(2)         0.337364   0.049643   6.795838   0.0000          
             AR(3)        -0.054397   0.045379  -1.198729   0.2312          
      ============================================================
      R-squared            0.861482    Mean dependent var 2.180052          
      Adjusted R-squared   0.860356    S.D. dependent var 2.723471          
      S.E. of regression   1.017734    Akaike info criter 0.045167          
      Sum squared resid    509.6051    Schwarz criterion  0.087507          
      Log likelihood      -711.4364    F-statistic        764.9705          
      Durbin-Watson stat   1.981769    Prob(F-statistic)  0.000000          
      ============================================================
      Inverted AR Roots          .8       .14       -.45                    
      ============================================================