UNIVERSITY OF CALIFORNIA, LOS ANGELES
Department of Economics

Economics 143 (Cameron) - Applied Regression Analysis

Classroom Handout #18: Serially Correlated Errors and the AUTO Command


Data are from January 1984 through October 1997, on number of business failures
and manufacturing production (in constant 1987 dollars, not seasonally adjusted.

 |_read(failn.dat) yrmo fail / skiplines=1
 ...SAMPLE RANGE IS NOW SET TO:         1       166
 |_read(mfgprod.dat) yrmo mfgprod / skiplines=1
 |_genr t=time(0)

Create separate YR and MO variables from the YRMO variable in CITIBASE

 |_genr yr=(yrmo-mod(yrmo,100))/100
 |_genr mo=mod(yrmo,100)

 |_stat
 NAME        N   MEAN        ST. DEV      VARIANCE     MINIMUM      MAXIMUM
 YRMO       166   9048.6       400.25      0.16020E+06   8401.0       9710.0
 FAIL       166   5772.1       1324.2      0.17536E+07   3404.0       9143.0
 MFGPROD    166   100.19       11.296       127.60       78.989       128.30
 T          166   83.500       48.064       2310.2       1.0000       166.00
 YR         166   90.422       4.0034       16.027       84.000       97.000
 MO         166   6.4398       3.4385       11.824       1.0000       12.000

Create a set of monthly dummy variables to capture seasonality

 |_if(mo.eq.1) jan=1
 |_if(mo.eq.2) feb=1
 |_if(mo.eq.3) mar=1
 |_if(mo.eq.4) apr=1
 |_if(mo.eq.5) may=1
 |_if(mo.eq.6) jun=1
 |_if(mo.eq.7) jul=1
 |_if(mo.eq.8) aug=1
 |_if(mo.eq.9) sep=1
 |_if(mo.eq.10) oct=1
 |_if(mo.eq.11) nov=1
 |_if(mo.eq.12) dec=1

Illustrate use of EXACTDW option on OLS to check for first-order serially
correlated errors (AR(1) errors); test rejects "zero serial correlation"
 
 |_ols fail mfgprod t feb mar apr may jun jul &
 |     aug sep oct nov dec / exactdw
 
 DURBIN-WATSON STATISTIC  =   0.74405
 DURBIN-WATSON P-VALUE =    0.000000
 
  R-SQUARE =   0.6812     R-SQUARE ADJUSTED =   0.6539
 VARIANCE OF THE ESTIMATE-SIGMA**2 =  0.60688E+06
 STANDARD ERROR OF THE ESTIMATE-SIGMA =   779.02
 SUM OF SQUARED ERRORS-SSE=  0.92246E+08
 MEAN OF DEPENDENT VARIABLE =   5772.1
 LOG OF THE LIKELIHOOD FUNCTION = -1333.47
 
                      ANALYSIS OF VARIANCE - FROM MEAN
                       SS         DF             MS                 F
 REGRESSION       0.19710E+09     13.       0.15162E+08            24.983
 ERROR            0.92246E+08    152.       0.60688E+06           P-VALUE
 TOTAL            0.28935E+09    165.       0.17536E+07             0.000
 
 VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
   NAME    COEFFICIENT   ERROR     152 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
 MFGPROD   -233.70      20.94      -11.16     0.000-0.671    -1.9934    -4.0563
 T          67.705      4.780       14.16     0.000 0.754     2.4574     0.9794
 FEB        249.27      298.2      0.8359     0.405 0.068     0.0525     0.0036
 MAR        1243.0      299.6       4.149     0.000 0.319     0.2616     0.0182
 APR        607.56      299.2       2.031     0.044 0.163     0.1279     0.0089
 MAY        937.30      300.8       3.116     0.002 0.245     0.1973     0.0137
 JUN        1314.9      320.7       4.100     0.000 0.316     0.2768     0.0192
 JUL       -99.116      296.5     -0.3343     0.739-0.027    -0.0209    -0.0014
 AUG        1277.0      320.2       3.989     0.000 0.308     0.2688     0.0187
 SEP        1098.8      329.7       3.333     0.001 0.261     0.2313     0.0161
 OCT        1441.2      320.7       4.494     0.000 0.342     0.3034     0.0211
 NOV        219.87      305.5      0.7198     0.473 0.058     0.0447     0.0030
 DEC       -803.77      300.1      -2.678     0.008-0.212    -0.1636    -0.0109
 CONSTANT   22898.      1659.       13.80     0.000 0.746     0.0000     3.9669

There are lots of apparently individually statistically significant coefficients
when the standard errors are calculated using the default OLS formulas (which
are not valid if there is autocorrelation in the errors).
 
 |_auto fail mfgprod t feb mar apr may jun jul &
 |     aug sep oct nov dec
 
 DEPENDENT VARIABLE =  FAIL
 ..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
 
 LEAST SQUARES ESTIMATION            166 OBSERVATIONS
 BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100
 
     ITERATION          RHO               LOG L.F.            SSE
         1             0.00000        -1333.47             0.92246E+08
         2             0.63181        -1289.89             0.54398E+08
         3             0.69439        -1289.02             0.53783E+08
         4             0.72625        -1288.77             0.53591E+08
         5             0.74688        -1288.66             0.53502E+08
         6             0.76166        -1288.61             0.53452E+08
         7             0.77279        -1288.58             0.53423E+08
         8             0.78138        -1288.57             0.53405E+08
         9             0.78809        -1288.57             0.53393E+08
        10             0.79337        -1288.57             0.53386E+08
        11             0.79754        -1288.57             0.53381E+08
        12             0.80081        -1288.57             0.53379E+08
        13             0.80339        -1288.58             0.53377E+08
        14             0.80542        -1288.58             0.53376E+08
        15             0.80700        -1288.58             0.53375E+08
        16             0.80825        -1288.58             0.53374E+08
        17             0.80922        -1288.59             0.53374E+08
 
  LOG L.F. =   -1288.59       AT RHO =     0.80922
 
                     ASYMPTOTIC  ASYMPTOTIC  ASYMPTOTIC
           ESTIMATE    VARIANCE    ST.ERROR     T-RATIO
 RHO        0.80922     0.00208     0.04560    17.74625
 
  R-SQUARE =   0.8155     R-SQUARE ADJUSTED =   0.7998
 VARIANCE OF THE ESTIMATE-SIGMA**2 =  0.35115E+06
 STANDARD ERROR OF THE ESTIMATE-SIGMA =   592.58
 SUM OF SQUARED ERRORS-SSE=  0.53374E+08
 MEAN OF DEPENDENT VARIABLE =   5772.1
 LOG OF THE LIKELIHOOD FUNCTION = -1288.59
 
                      ANALYSIS OF VARIANCE - FROM MEAN
                       SS         DF             MS
 REGRESSION       0.23597E+09     13.       0.18152E+08
 ERROR            0.53374E+08    152.       0.35115E+06
 TOTAL            0.28935E+09    165.       0.17536E+07
 
 VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
   NAME    COEFFICIENT   ERROR     152 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
 MFGPROD   -20.689      52.54     -0.3938     0.694-0.032    -0.1765    -0.3591
 T          21.127      12.74       1.658     0.099 0.133     0.7668     0.3056
 FEB       -232.77      201.3      -1.156     0.249-0.093    -0.0490    -0.0034
 MAR        676.51      256.4       2.638     0.009 0.209     0.1424     0.0099
 APR        64.903      281.0      0.2310     0.818 0.019     0.0137     0.0009
 MAY        307.31      307.9      0.9980     0.320 0.081     0.0647     0.0045
 JUN        15.490      418.8      0.3699E-01 0.971 0.003     0.0033     0.0002
 JUL       -452.88      296.5      -1.527     0.129-0.123    -0.0953    -0.0066
 AUG       -10.246      415.8     -0.2464E-01 0.980-0.002    -0.0022    -0.0001
 SEP       -419.54      452.2     -0.9278     0.355-0.075    -0.0883    -0.0061
 OCT        139.95      398.3      0.3514     0.726 0.028     0.0295     0.0020
 NOV       -358.41      271.0      -1.322     0.188-0.107    -0.0729    -0.0049
 DEC       -811.59      170.3      -4.765     0.000-0.360    -0.1652    -0.0110
 CONSTANT   6169.2      4141.       1.490     0.138 0.120     0.0000     1.0688

Not much left that matters when standard errors are computed correctly


Updated: 4:24 PM 11/23/98; Prepared by: Trudy Ann Cameron; Site Index