12
for t = 1, ..., N1 and It may be the case that the variance functional form contains unknown parameters that must be estimated. An estimation approach is to first estimate the model by OLS and then use the OLS estimated residuals to construct an estimate of the error variance specification. In the second step weighted least squares can be applied. This is a special case of estimated generalized least squares (EGLS) or feasible generalized least squares.
This example returns to the Griffiths, Hill and Judge data set on wheat supply in Australia that was analyzed in the section on testing for heteroskedasticity. Recall that, following OLS estimation, the Goldfeld-Quandt test showed evidence to reject the null hypothesis of equal variance in two groups of observations. That is, the variance assumption is:
var(et) =
12
for t = 1, ..., N1 and
var(et) =
22
for t = N1+1, ..., N
where
12
is the error variance in the first subset and
22
is the error variance in the second subset.
The problem is that the error variances are unknown.
The estimation approach is to first obtain estimates of the
error variance by separate OLS estimation of each sample partition.
This is shown in the SHAZAM commands
(filename: WLS2.SHA)
below. Note that, immeditately following
estimation, the error variance is available in the SHAZAM temporary
variable with the special name $SIG2.
In the second step the estimated GLS estimates are obtained
by using the weighted least squares option of SHAZAM.
SAMPLE 1 26 * Read the Griffiths, Hill and Judge (1993, p.491) wheat supply data set READ (WHEAT.txt) Q P * Generate a time index GENR T=TIME(0) * Define a variable to use as the weight variable GENR WT=0 * * STEP 1 : Separate OLS regression for the two observation subsets * First subset SAMPLE 1 13 OLS Q P T * Save the estimated error variance in the weight variable GENR WT=1/$SIG2 * Second subset SAMPLE 14 26 OLS Q P T GENR WT=1/$SIG2 * * STEP 2 : Get the weighted least squares (WLS) estimates * Griffiths, Hill and Judge (1993 p.499) - Equation (15.2.25) SAMPLE 1 26 OLS Q P T / WEIGHT=WT NONORM NOMULSIGSQ * ----------------------- comparison with OLS --------------------------- * Compare the WLS estimates with OLS - use the HETCOV option to obtain * standard errors that are adjusted for heteroskedastic errors. OLS Q P T / HETCOV STDERR=SEHET * Compare the HETCOV standard errors with the OLS standard errors. * Griffiths, Hill and Judge (1993 p.500) - Equation (15.2.29) OLS Q P T / STDERR=SEOLS * Obtain standard errors by adjusting for different error variance * in the 2 sample partitions GENR ONE=1 COPY P T ONE X GENR P=P/SQRT(WT) GENR T=T/SQRT(WT) GENR ONE=ONE/SQRT(WT) COPY P T ONE XW * Griffiths, Hill and Judge (1993 p.499) - Equation (15.2.26) MATRIX SEWLS=SQRT(DIAG(INV(X'X)*(XW'XW)*INV(X'X))) * Compare the various OLS standard errors SAMPLE 1 3 PRINT SEHET SEOLS SEWLS STOP
In this application, the error terms of the transformed model are
homoskedastic with error variance equal to 1.
To make use of this result, the OLS command for the
weighted least squares estimation specifies the options
NONORM and NOMULSIGSQ. The NONORM
option specifies that the weights are not normalized and the
NOMULSIGSQ option specifies that the covariance matrix of the
parameter estimates is estimated as the inverse of the cross-product
matrix of the transformed observations. That is, the matrix is not
multiplied by the error variance estimate.
The SHAZAM output can be viewed. The estimation results from the weighted least squares procedure (as reported in Griffiths, Hill and Judge [1993 p.499, Equation 15.2.25]) are as follows:
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 23 DF P-VALUE CORR. COEFFICIENT AT MEANS P 21.720 8.805 2.467 .022 .457 .3661 .2249 T 3.2834 .8117 4.045 .001 .645 .6003 .2406 CONSTANT 138.05 12.65 10.91 .000 .915 .0000 .5345
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The data set is from Griffiths, Hill and Judge [1993, Table 15.1, p. 491]. The 2-step GLS estimation procedure is described in Section 15.4.2, pp. 505-6 of this text.
|_SAMPLE 1 26
|_* Read the Griffiths, Hill and Judge (1993, p.491) wheat supply data set
|_READ (WHEAT.txt) Q P
UNIT 88 IS NOW ASSIGNED TO: WHEAT.txt
2 VARIABLES AND 26 OBSERVATIONS STARTING AT OBS 1
|_* Generate a time index
|_GENR T=TIME(0)
|_* Define a variable to use as the weight variable
|_GENR WT=0
|_*
|_* STEP 1 : Separate OLS regression for the two observation subsets
|_* First subset
|_SAMPLE 1 13
|_OLS Q P T
OLS ESTIMATION
13 OBSERVATIONS DEPENDENT VARIABLE = Q
...NOTE..SAMPLE RANGE SET TO: 1, 13
R-SQUARE = .6639 R-SQUARE ADJUSTED = .5967
VARIANCE OF THE ESTIMATE-SIGMA**2 = 641.64
STANDARD ERROR OF THE ESTIMATE-SIGMA = 25.331
SUM OF SQUARED ERRORS-SSE= 6416.4
MEAN OF DEPENDENT VARIABLE = 203.65
LOG OF THE LIKELIHOOD FUNCTION = -58.7570
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 10 DF P-VALUE CORR. COEFFICIENT AT MEANS
P 19.148 32.42 .5906 .568 .184 .1733 .1683
T 6.8853 3.005 2.291 .045 .587 .6723 .2367
CONSTANT 121.17 44.20 2.741 .021 .655 .0000 .5950
|_* Save the estimated error variance in the weight variable
|_GENR WT=1/$SIG2
..NOTE..CURRENT VALUE OF $SIG2= 641.64
|_* Second subset
|_SAMPLE 14 26
|_OLS Q P T
OLS ESTIMATION
13 OBSERVATIONS DEPENDENT VARIABLE = Q
...NOTE..SAMPLE RANGE SET TO: 14, 26
R-SQUARE = .9010 R-SQUARE ADJUSTED = .8812
VARIANCE OF THE ESTIMATE-SIGMA**2 = 57.759
STANDARD ERROR OF THE ESTIMATE-SIGMA = 7.5999
SUM OF SQUARED ERRORS-SSE= 577.59
MEAN OF DEPENDENT VARIABLE = 263.20
LOG OF THE LIKELIHOOD FUNCTION = -43.1066
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 10 DF P-VALUE CORR. COEFFICIENT AT MEANS
P 22.063 9.281 2.377 .039 .601 .4175 .2308
T 3.2574 .9944 3.276 .008 .719 .5753 .2475
CONSTANT 137.29 14.68 9.354 .000 .947 .0000 .5216
|_GENR WT=1/$SIG2
..NOTE..CURRENT VALUE OF $SIG2= 57.759
|_*
|_* STEP 2 : Get the weighted least squares (WLS) estimates
|_* Griffiths, Hill and Judge (1993 p.499) - Equation (15.2.25)
|_SAMPLE 1 26
|_OLS Q P T / WEIGHT=WT NONORM NOMULSIGSQ
OLS ESTIMATION
26 OBSERVATIONS DEPENDENT VARIABLE = Q
...NOTE..SAMPLE RANGE SET TO: 1, 26
SUM OF LOG(SQRT(ABS(WEIGHT))) = -68.382
R-SQUARE = .8799 R-SQUARE ADJUSTED = .8695
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.0271
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.0135
SUM OF SQUARED ERRORS-SSE= 23.624
MEAN OF DEPENDENT VARIABLE = 258.28
LOG OF THE LIKELIHOOD FUNCTION = -104.029
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 23 DF P-VALUE CORR. COEFFICIENT AT MEANS
P 21.720 8.805 2.467 .022 .457 .3661 .2249
T 3.2834 .8117 4.045 .001 .645 .6003 .2406
CONSTANT 138.05 12.65 10.91 .000 .915 .0000 .5345
|_* ----------------------- Optional ------------------------------
|_* Compare the WLS estimates with OLS - use the HETCOV option to obtain
|_* standard errors that are adjusted for heteroskedastic errors.
|_OLS Q P T / HETCOV STDERR=SEHET
OLS ESTIMATION
26 OBSERVATIONS DEPENDENT VARIABLE = Q
...NOTE..SAMPLE RANGE SET TO: 1, 26
USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX
R-SQUARE = .8089 R-SQUARE ADJUSTED = .7923
VARIANCE OF THE ESTIMATE-SIGMA**2 = 398.68
STANDARD ERROR OF THE ESTIMATE-SIGMA = 19.967
SUM OF SQUARED ERRORS-SSE= 9169.5
MEAN OF DEPENDENT VARIABLE = 233.42
LOG OF THE LIKELIHOOD FUNCTION = -113.145
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 23 DF P-VALUE CORR. COEFFICIENT AT MEANS
P 19.541 19.50 1.002 .327 .205 .2777 .1902
T 3.6391 1.606 2.265 .033 .427 .6353 .2105
CONSTANT 139.90 24.58 5.691 .000 .765 .0000 .5993
|_* Compare the HETCOV standard errors with the OLS standard errors.
|_* Griffiths, Hill and Judge (1993 p.500) - Equation (15.2.29)
|_OLS Q P T / STDERR=SEOLS
OLS ESTIMATION
26 OBSERVATIONS DEPENDENT VARIABLE = Q
...NOTE..SAMPLE RANGE SET TO: 1, 26
R-SQUARE = .8089 R-SQUARE ADJUSTED = .7923
VARIANCE OF THE ESTIMATE-SIGMA**2 = 398.68
STANDARD ERROR OF THE ESTIMATE-SIGMA = 19.967
SUM OF SQUARED ERRORS-SSE= 9169.5
MEAN OF DEPENDENT VARIABLE = 233.42
LOG OF THE LIKELIHOOD FUNCTION = -113.145
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 23 DF P-VALUE CORR. COEFFICIENT AT MEANS
P 19.541 17.42 1.122 .273 .228 .2777 .1902
T 3.6391 1.418 2.567 .017 .472 .6353 .2105
CONSTANT 139.90 23.22 6.026 .000 .782 .0000 .5993
|_* Obtain standard errors by adjusting for different error variance
|_* in the 2 sample partitions
|_GENR ONE=1
|_COPY P T ONE X
|_GENR P=P/SQRT(WT)
|_GENR T=T/SQRT(WT)
|_GENR ONE=ONE/SQRT(WT)
|_COPY P T ONE XW
|_* Griffiths, Hill and Judge (1993 p.500) - Equation (15.2.26)
|_MATRIX SEWLS=SQRT(DIAG(INV(X'X)*(XW'XW)*INV(X'X)))
|_* Compare the various OLS standard errors
|_SAMPLE 1 3
|_PRINT SEHET SEOLS SEWLS
SEHET SEOLS SEWLS
19.50298 17.41501 15.85248
1.606431 1.417651 1.321047
24.58077 23.21761 21.55064
|_STOP
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