1998 Flex-Time and Flex-Mode Prob-Stat II Homework Answers #6 Page

45-734 PROBABILITY AND STATISTICS II Homework Answers #6 (4^th Mini AY1997-98)

The Correlogram for Y1 is shown below. It is somewhat ambiguous but it does indicate at least two and possibly three MA terms. I would try three MA terms first.

                      Correlogram of Y1
==============================================================
Date: 03/24/98   Time: 21:28                                            
Sample: 1 100                                                           
Included observations: 96                                               
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
  ******| .     |  ******| .     |  1-0.708-0.708 49.601 0.000          
      . |****   |      . | .     |  2 0.474-0.054 72.078 0.000          
      **| .     |      . | .     |  3-0.316-0.001 82.177 0.000          
      . |*.     |      **| .     |  4 0.108-0.207 83.360 0.000          
      . |*.     |      . |*.     |  5 0.104 0.190 84.479 0.000          
==============================================================

LS Y1 C MA(1) MA(2) MA(3)

============================================================
LS // Dependent Variable is Y1                                        
Date: 03/24/98   Time: 21:31                                          
Sample(adjusted): 5 100                                               
Included observations: 96 after adjusting endpoints                   
Convergence achieved after 23 iterations                              
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C          -0.024567   0.046734  -0.525667   0.6004          
       MA(1)        -0.445189   0.083942  -5.303556   0.0000          
       MA(2)         0.405915   0.086802   4.676344   0.0000          
       MA(3)        -0.580366   0.088593  -6.550918   0.0000          
============================================================
R-squared            0.497198    Mean dependent var-0.029085          
Adjusted R-squared   0.480802    S.D. dependent var 1.601392          
S.E. of regression   1.153889    Akaike info criter 0.327050          
Sum squared resid    122.4943    Schwarz criterion  0.433897          
Log likelihood      -147.9165    F-statistic        30.32489          
Durbin-Watson stat   2.338553    Prob(F-statistic)  0.000000          
============================================================
Inverted MA Roots          .8  -.19 -.82  -.19+.82i                   
============================================================

This looks very good. The Correlogram of the residuals is shown below.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 21:34                                            
Sample: 5 100                                                           
Included observations: 96                                               
Q-statistic probabilities adjusted for 3 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      .*| .     |      .*| .     |  1-0.174-0.174 3.0023                
      . | .     |      . | .     |  2 0.035 0.005 3.1259                
      . | .     |      . | .     |  3 0.049 0.058 3.3724                
      . |**     |      . |**     |  4 0.214 0.240 8.0398 0.005          
      . |**     |      . |**     |  5 0.204 0.310 12.350 0.002          
==============================================================

The p-values at lag 4 and 5 indicate that we reject the null hypothesis of white noise. Consequently, another MA term might improve the model.

LS Y1 C MA(1) MA(2) MA(3) MA(4)

============================================================
LS // Dependent Variable is Y1                                        
Date: 03/24/98   Time: 21:39                                          
Sample(adjusted): 5 100                                               
Included observations: 96 after adjusting endpoints                   
Convergence achieved after 16 iterations                              
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C          -0.023937   0.079127  -0.302518   0.7629          
       MA(1)        -0.793839   0.075327  -10.53859   0.0000          
       MA(2)         0.704617   0.078089   9.023306   0.0000          
       MA(3)        -0.774051   0.048615  -15.92190   0.0000          
       MA(4)         0.628794   0.076743   8.193488   0.0000          
============================================================
R-squared            0.602554    Mean dependent var-0.029085          
Adjusted R-squared   0.585083    S.D. dependent var 1.601392          
S.E. of regression   1.031521    Akaike info criter 0.112747          
Sum squared resid    96.82724    Schwarz criterion  0.246307          
Log likelihood      -136.6299    F-statistic        34.49042          
Durbin-Watson stat   1.895787    Prob(F-statistic)  0.000000          
============================================================
Inverted MA Roots      .71 -.   .71+.47i  -.31 -.88  -.31+.88i        
============================================================

The correlogram of the residuals confirms that this seems to be the right model. The Q-Stat with 1 (5-0-4) degree of freedom at lag 5 has a p-value of .286.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 21:43                                            
Sample: 5 100                                                           
Included observations: 96                                               
Q-statistic probabilities adjusted for 4 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . | .     |      . | .     |  1 0.052 0.052 0.2628                
      . | .     |      . | .     |  2-0.045-0.048 0.4673                
      . |*.     |      . |*.     |  3 0.081 0.086 1.1311                
      . | .     |      . | .     |  4-0.001-0.012 1.1311                
      . | .     |      . | .     |  5-0.008 0.001 1.1383 0.286          
==============================================================

Below is the correlogram for Y2. Given the three sizable spikes in the PACFs, I am inclined to try three AR terms.

                      Correlogram of Y2
==============================================================
Date: 03/24/98   Time: 21:54                                            
Sample: 1 100                                                           
Included observations: 100                                              
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |****   |      . |****   |  1 0.482 0.482 23.922 0.000          
     ***| .     |  ******| .     |  2-0.348-0.756 36.523 0.000          
  ******| .     |     ***| .     |  3-0.779-0.367 100.41 0.000          
    ****| .     |      . | .     |  4-0.465 0.000 123.40 0.000          
      . |**     |      . | .     |  5 0.212 0.007 128.24 0.000          
==============================================================

LS Y2 C AR(1) AR(2) AR(3)

============================================================
LS // Dependent Variable is Y2                                        
Date: 03/24/98   Time: 21:56                                          
Sample(adjusted): 4 100                                               
Included observations: 97 after adjusting endpoints                   
Convergence achieved after 3 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           0.021432   0.091892   0.233232   0.8161          
       AR(1)         0.578643   0.095971   6.029343   0.0000          
       AR(2)        -0.461242   0.103059  -4.475528   0.0000          
       AR(3)        -0.362344   0.096140  -3.768934   0.0003          
============================================================
R-squared            0.727193    Mean dependent var 0.044761          
Adjusted R-squared   0.718392    S.D. dependent var 2.122938          
S.E. of regression   1.126573    Akaike info criter 0.278724          
Sum squared resid    118.0326    Schwarz criterion  0.384897          
Log likelihood      -147.1551    F-statistic        82.63329          
Durbin-Watson stat   2.019926    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots      .50+.7   .50 -.79      -.41                    
============================================================

All three AR terms are statistically significant and the correlogram of the residuals shows no further correlated structure in the residuals. The Q-statistic at lag 5 has 2 (5-3-0) degrees of freedom and a p-value of .548. Hence we do not reject the null hypothesis that the residuals are white noise.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 21:57                                            
Sample: 4 100                                                           
Included observations: 97                                               
Q-statistic probabilities adjusted for 3 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . | .     |      . | .     |  1-0.010-0.010 0.0104                
      . | .     |      . | .     |  2-0.028-0.028 0.0877                
      . | .     |      . | .     |  3 0.064 0.064 0.5102                
      . |*.     |      . |*.     |  4 0.071 0.072 1.0289 0.310          
      . | .     |      . | .     |  5-0.041-0.036 1.2044 0.548          
==============================================================

Given the very large spike at the first PACF, I am inclined to start with a simple AR(1) model.

                      Correlogram of Y3
==============================================================
Date: 03/24/98   Time: 22:00                                            
Sample: 1 100                                                           
Included observations: 100                                              
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |*******|      . |*******|  1 0.880 0.880 79.727 0.000          
      . |*****  |     ***| .     |  2 0.694-0.352 129.89 0.000          
      . |****   |      . | .     |  3 0.528 0.059 159.19 0.000          
      . |***    |      .*| .     |  4 0.380-0.091 174.51 0.000          
      . |**     |      . | .     |  5 0.263 0.034 181.91 0.000          
==============================================================

LS Y3 C AR(1)

============================================================
LS // Dependent Variable is Y3                                        
Date: 03/24/98   Time: 22:02                                          
Sample(adjusted): 2 100                                               
Included observations: 99 after adjusting endpoints                   
Convergence achieved after 3 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           1.961704   1.768729   1.109104   0.2701          
       AR(1)         0.923736   0.044381   20.81372   0.0000          
============================================================
R-squared            0.817054    Mean dependent var 0.940025          
Adjusted R-squared   0.815168    S.D. dependent var 2.907858          
S.E. of regression   1.250149    Akaike info criter 0.466521          
Sum squared resid    151.5987    Schwarz criterion  0.518948          
Log likelihood      -161.5677    F-statistic        433.2110          
Durbin-Watson stat   1.126531    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots          .92                                        
============================================================

The AR(1) term is statistically significant but the correlogram of the residuals show that we still have a spike in the first ACF and PACF indicating that we need an MA(1) term.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 22:03                                            
Sample: 2 100                                                           
Included observations: 99                                               
Q-statistic probabilities adjusted for 1 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |***    |      . |***    |  1 0.424 0.424 18.362                
      . | .     |      **| .     |  2-0.022-0.247 18.414 0.000          
      . | .     |      . |*.     |  3-0.010 0.139 18.424 0.000          
      . | .     |      .*| .     |  4-0.035-0.127 18.556 0.000          
      .*| .     |      . | .     |  5-0.086-0.014 19.350 0.001          
==============================================================

LS Y3 C AR(1) MA(1)

============================================================
LS // Dependent Variable is Y3                                        
Date: 03/24/98   Time: 22:04                                          
Sample(adjusted): 2 100                                               
Included observations: 99 after adjusting endpoints                   
Convergence achieved after 6 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           1.435282   1.143186   1.255510   0.2123          
       AR(1)         0.844586   0.061323   13.77276   0.0000          
       MA(1)         0.622440   0.083629   7.442859   0.0000          
============================================================
R-squared            0.867037    Mean dependent var 0.940025          
Adjusted R-squared   0.864266    S.D. dependent var 2.907858          
S.E. of regression   1.071314    Akaike info criter 0.167606          
Sum squared resid    110.1805    Schwarz criterion  0.246246          
Log likelihood      -145.7714    F-statistic        313.0015          
Durbin-Watson stat   1.928668    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots          .84                                        
Inverted MA Roots         -.62                                        
============================================================

The correlogram of the residuals shows that we have white noise. The Q-statistic at 5 lags has 3 (5-1-1) degrees of freedom with a p-value of .899. Hence we do not reject the null hypothesis that the residuals are white noise.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 22:05                                            
Sample: 2 100                                                           
Included observations: 99                                               
Q-statistic probabilities adjusted for 2 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . | .     |      . | .     |  1 0.025 0.025 0.0643                
      . | .     |      . | .     |  2 0.046 0.045 0.2830                
      . | .     |      . | .     |  3 0.032 0.030 0.3901 0.532          
      . | .     |      . | .     |  4 0.022 0.019 0.4418 0.802          
      . | .     |      . | .     |  5-0.037-0.041 0.5870 0.899          
==============================================================

The correlogram of Y4 is ambiguous but it looks like there is at least one AR term in the error structure. Several different starting places could be chosen here. I will start with AR(1) and AR(2).

                      Correlogram of Y4
==============================================================
Date: 03/24/98   Time: 22:07                                            
Sample: 1 100                                                           
Included observations: 100                                              
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |*******|      . |*******|  1 0.847 0.847 73.868 0.000          
      . |*****  |     ***| .     |  2 0.597-0.423 111.00 0.000          
      . |***    |      .*| .     |  3 0.347-0.062 123.68 0.000          
      . |**     |      . |**     |  4 0.197 0.208 127.80 0.000          
      . |*.     |      **| .     |  5 0.066-0.296 128.27 0.000          
==============================================================

LS Y4 C AR(1) AR(2)

============================================================
LS // Dependent Variable is Y4                                        
Date: 03/24/98   Time: 22:08                                          
Sample(adjusted): 3 100                                               
Included observations: 98 after adjusting endpoints                   
Convergence achieved after 3 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C          -0.591764   0.523659  -1.130057   0.2613          
       AR(1)         1.218762   0.092320   13.20143   0.0000          
       AR(2)        -0.432515   0.092587  -4.671425   0.0000          
============================================================
R-squared            0.774750    Mean dependent var-0.595948          
Adjusted R-squared   0.770008    S.D. dependent var 2.309977          
S.E. of regression   1.107807    Akaike info criter 0.234899          
Sum squared resid    116.5875    Schwarz criterion  0.314031          
Log likelihood      -147.5660    F-statistic        163.3768          
Durbin-Watson stat   2.044549    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots      .61 -.   .61+.25i                              
============================================================

The correlogram of the residuals shows that we need some further terms. The spike at lag 3 could be either an MA(3) or a AR(3) term. Lets try the AR(3) term first.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 22:09                                            
Sample: 3 100                                                           
Included observations: 98                                               
Q-statistic probabilities adjusted for 2 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . | .     |      . | .     |  1-0.048-0.048 0.2349                
      . |*.     |      . |*.     |  2 0.121 0.119 1.7362                
      **| .     |      **| .     |  3-0.252-0.245 8.3038 0.004          
      . |*.     |      . |*.     |  4 0.115 0.093 9.6766 0.008          
      . |*.     |      . |**     |  5 0.194 0.275 13.644 0.003          
==============================================================

LS Y4 C AR(1) AR(2) AR(3)

============================================================
LS // Dependent Variable is Y4                                        
Date: 03/24/98   Time: 22:11                                          
Sample(adjusted): 4 100                                               
Included observations: 97 after adjusting endpoints                   
Convergence achieved after 3 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C          -0.707052   0.478346  -1.478118   0.1428          
       AR(1)         1.170528   0.100622   11.63292   0.0000          
       AR(2)        -0.322918   0.152465  -2.117982   0.0368          
       AR(3)        -0.078392   0.100694  -0.778520   0.4382          
============================================================
R-squared            0.783610    Mean dependent var-0.628997          
Adjusted R-squared   0.776629    S.D. dependent var 2.298569          
S.E. of regression   1.086352    Akaike info criter 0.206014          
Sum squared resid    109.7550    Schwarz criterion  0.312188          
Log likelihood      -143.6287    F-statistic        112.2596          
Durbin-Watson stat   2.023847    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots      .66+.2   .66 -.29      -.15                    
============================================================

The AR(3) term is not the right guess. Consequently, I now try the MA(3) term:

LS Y4 C AR(1) AR(2) MA(3)

============================================================
LS // Dependent Variable is Y4                                        
Date: 03/24/98   Time: 22:13                                          
Sample(adjusted): 3 100                                               
Included observations: 98 after adjusting endpoints                   
Convergence achieved after 9 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C          -0.808939   0.368806  -2.193402   0.0307          
       AR(1)         1.364322   0.091908   14.84443   0.0000          
       AR(2)        -0.451403   0.093193  -4.843747   0.0000          
       MA(3)        -0.701170   0.093237  -7.520296   0.0000          
============================================================
R-squared            0.828102    Mean dependent var-0.595948          
Adjusted R-squared   0.822616    S.D. dependent var 2.309977          
S.E. of regression   0.972893    Akaike info criter-0.015003          
Sum squared resid    88.97293    Schwarz criterion  0.090506          
Log likelihood      -134.3208    F-statistic        150.9453          
Durbin-Watson stat   2.087724    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots          .8       .56                               
Inverted MA Roots          .8  -.44 -.77  -.44+.77i                   
============================================================

This looks very good and the correlogram of the residuals shows that we now have white noise. The Q-statistic at 5 lags has 2 (5-2-1) degrees of freedom and the p-value is .175. Hence we do not reject the null hypothesis that the residuals are white noise.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 22:14                                            
Sample: 3 100                                                           
Included observations: 98                                               
Q-statistic probabilities adjusted for 3 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      .*| .     |      .*| .     |  1-0.069-0.069 0.4854                
      . | .     |      . | .     |  2 0.057 0.053 0.8177                
      . |*.     |      . |*.     |  3 0.088 0.096 1.6192                
      . | .     |      . |*.     |  4 0.055 0.066 1.9397 0.164          
      . |*.     |      . |*.     |  5 0.121 0.122 3.4817 0.175          
==============================================================

Below is the correlogram of DRWAGE -- the first difference (y_t - y_t-1) of the real wage series. Given that there is only one sizable spike and it is at the first lag, an AR(1) model seems appropriate.

                    Correlogram of DRWAGE
==============================================================
Date: 03/24/98   Time: 22:18                                            
Sample: 1900 1970                                                       
Included observations: 70                                               
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |***    |      . |***    |  1 0.392 0.392 11.200 0.001          
      . |*.     |      . | .     |  2 0.105-0.057 12.016 0.002          
      . | .     |      .*| .     |  3-0.030-0.060 12.082 0.007          
      .*| .     |      .*| .     |  4-0.173-0.159 14.366 0.006          
      .*| .     |      . | .     |  5-0.147-0.021 16.046 0.007          
==============================================================

LS DRWAGE C AR(1)

============================================================
LS // Dependent Variable is DRWAGE                                    
Date: 03/24/98   Time: 22:23                                          
Sample(adjusted): 1902 1970                                           
Included observations: 69 after adjusting endpoints                   
Convergence achieved after 3 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           0.703185   0.247827   2.837402   0.0060          
       AR(1)         0.398629   0.113199   3.521492   0.0008          
============================================================
R-squared            0.156181    Mean dependent var 0.719420          
Adjusted R-squared   0.143587    S.D. dependent var 1.337106          
S.E. of regression   1.237393    Akaike info criter 0.454570          
Sum squared resid    102.5864    Schwarz criterion  0.519327          
Log likelihood      -111.5894    F-statistic        12.40091          
Durbin-Watson stat   1.931600    Prob(F-statistic)  0.000777          
============================================================
Inverted AR Roots          .40                                        
============================================================

The AR(1) term is statistically significant and the correlogram of the residuals shown below shows no further correlated structure in the error. The Q-statistic at lag 5 has 4 (5-1-0) degrees of freedom with a p-value of .637. Hence we do not reject the null hypothesis that the residuals are white noise.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 22:24                                            
Sample: 1902 1970                                                       
Included observations: 69                                               
Q-statistic probabilities adjusted for 1 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . | .     |      . | .     |  1 0.025 0.025 0.0435                
      . | .     |      . | .     |  2-0.020-0.021 0.0726 0.788          
      . | .     |      . | .     |  3-0.013-0.012 0.0850 0.958          
      .*| .     |      .*| .     |  4-0.154-0.154 1.8633 0.601          
      .*| .     |      .*| .     |  5-0.094-0.090 2.5444 0.637          
==============================================================

I also tried an MA(1) model but the fit of it to the wage series was below the AR(1) model. Namely:

============================================================
LS // Dependent Variable is DRWAGE                                    
Date: 03/24/98   Time: 22:25                                          
Sample(adjusted): 1901 1970                                           
Included observations: 70 after adjusting endpoints                   
Convergence achieved after 5 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           0.717603   0.203548   3.525471   0.0008          
       MA(1)         0.383531   0.112143   3.420016   0.0011          
============================================================
R-squared            0.148007    Mean dependent var 0.722857          
Adjusted R-squared   0.135478    S.D. dependent var 1.327693          
S.E. of regression   1.234485    Akaike info criter 0.449463          
Sum squared resid    103.6288    Schwarz criterion  0.513705          
Log likelihood      -113.0569    F-statistic        11.81287          
Durbin-Watson stat   1.913881    Prob(F-statistic)  0.001008          
============================================================
Inverted MA Roots         -.38                                        
============================================================

Below is the correlogram for the first difference in logs; that is: ln(y_t) - ln(y_t-1). Given the large spike at the first lag and the way the ACFs trail off, this looks like an AR(1).

                    Correlogram of DLMONEY
==============================================================
Date: 03/24/98   Time: 22:29                                            
Sample: 1 81                                                            
Included observations: 80                                               
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |*****  |      . |*****  |  1 0.622 0.622 32.152 0.000          
      . |**     |      .*| .     |  2 0.302-0.139 39.830 0.000          
      . |*.     |      . | .     |  3 0.135 0.011 41.374 0.000          
      . | .     |      .*| .     |  4-0.005-0.105 41.376 0.000          
      .*| .     |      . | .     |  5-0.067-0.003 41.765 0.000          
==============================================================

LS DLMONEY C AR(1)

============================================================
LS // Dependent Variable is DLMONEY                                   
Date: 03/24/98   Time: 22:30                                          
Sample(adjusted): 3 81                                                
Included observations: 79 after adjusting endpoints                   
Convergence achieved after 3 iterations                               
============================================================
      Variable      CoefficienStd. Errort-Statistic  Prob.            
============================================================
         C           0.057614   0.014682   3.924036   0.0002          
       AR(1)         0.622290   0.089040   6.988878   0.0000          
============================================================
R-squared            0.388133    Mean dependent var 0.058072          
Adjusted R-squared   0.380187    S.D. dependent var 0.062605          
S.E. of regression   0.049288    Akaike info criter-5.995169          
Sum squared resid    0.187055    Schwarz criterion -5.935183          
Log likelihood       126.7130    F-statistic        48.84441          
Durbin-Watson stat   1.809760    Prob(F-statistic)  0.000000          
============================================================
Inverted AR Roots          .62                                        
============================================================

The AR(1) term is statistically significant and the correlogram of the residuals shows no further correlation in the error structure. The Q-statistic at lag 5 has 4 (5-1-0) degrees of freedom with a p-value of .571. Hence we do not reject the null hypothesis that the residuals are white noise.

                   Correlogram of Residuals
==============================================================
Date: 03/24/98   Time: 22:31                                            
Sample: 3 81                                                            
Included observations: 79                                               
Q-statistic probabilities adjusted for 1 ARMA term(s)                   
==============================================================
 Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob           
==============================================================
      . |*.     |      . |*.     |  1 0.092 0.092 0.6869                
      .*| .     |      .*| .     |  2-0.090-0.099 1.3562 0.244          
      . | .     |      . | .     |  3 0.019 0.038 1.3878 0.500          
      .*| .     |      .*| .     |  4-0.082-0.098 1.9551 0.582          
      .*| .     |      .*| .     |  5-0.106-0.084 2.9214 0.571          
==============================================================

45-734 PROBABILITY AND STATISTICS II Homework Answers #6 (4th Mini AY1997-98)

45-734 PROBABILITY AND STATISTICS II Homework Answers #6 (4^th Mini AY1997-98)