SHAZAM Estimation with Restrictions

## Estimation with Restrictions

Linear parameter restrictions can be imposed in model estimation. There are 2 ways of proceeding. One way is to substitute the restrictions into the equation to obtain a reparametrized equation. OLS applied to the new equation will yield restricted estimates.

The second way is to obtain restricted least squares estimates as the solution to a constrained least squares minimization problem. When this approach is followed, the general command format for restricted least squares is:

 ``` OLS depvar indeps / RESTRICT options RESTRICT equation1 RESTRICT equation2 . . . END ```

The `RESTRICT` option on the `OLS` command tells SHAZAM that `RESTRICT` commands follow with the specification of the linear restrictions. The `END` command is required to mark the end of the list of `RESTRICT` commands.

The `RESTRICT` commands are specified as a linear function of the variables specified in the `indeps` list on the `OLS` command. The variable names actually represent the coefficients.

Note that each restriction will add one degree of freedom.

#### Example

Parameter restrictions may be suggested by economic theory. For example, constant returns to scale may impose parameter restrictions in a production function. Another application of restricted estimation, that is developed in this example, is to obtain more precise estimates in the presence of multicollinearity.

This example, from Griffiths, Hill and Judge, uses the Klein-Goldberger data set. The study considers the relationship between aggregate consumption (C), and 3 components of income : wage income (W), nonwage-nonfarm income (P) and farm income (A) for the U.S. economy. It can be expected that components of income move together - so multicollinearity may be a problem.

The regression equation is:

Ct = 0 + 1 Wt + 2 Pt + 3 At + et

The SHAZAM commands (filename: `KLEING.SHA`) below use the `STAT` command to get the sample correlation matrix for the income variables. The equation is then estimated by `OLS`. Prior expertise suggests that reasonable parameter restrictions are: 2 = 0.75 1     and 3 = 0.625 1

The equation is estimated by restricted least squares and tests of the validity of the parameter restrictions are considered.

```SAMPLE 1 20
READ (KLEING.txt) C W P A
STAT W P A / PCOR
* Unrestricted estimation
OLS C W P A
* Restricted estimation
OLS C W P A / RESTRICT
RESTRICT P=0.75*W
RESTRICT A=0.625*W
END
* --------------------------------------------------------------------
* An equivalent way of obtaining the restricted least squares estimates
* is to make a substitution for the restrictions as follows.
GENR X = W + 0.75*P + 0.625*A
OLS C X
* Get the estimated coefficent on P
TEST 0.75*X
* Get the estimated coefficent on A
TEST 0.625*X
* --------------------------------------------------------------------
*
* Test to determine if the restrictions are accepted or rejected
* Individual Restriction Test : t-test
OLS C W P A
TEST P=0.75*W
TEST A=0.625*W
* Joint Test of the Restrictions : F-test
TEST
TEST P=0.75*W
TEST A=0.625*W
END
STOP
```

The SHAZAM output can be inspected. First look at the correlation matrix of the explanatory variables. This is reported as:

```  CORRELATION MATRIX OF VARIABLES -       20 OBSERVATIONS
W          1.0000
P          .71847       1.0000
A          .91517       .63061       1.0000
W            P            A
```

The correlation coefficient of 0.915 indicates a strong linear association between wage income (W) and farm income (A) - a sign of a multicollinearity problem. The unrestricted estimation reports the results:

```  R-SQUARE =    .9527     R-SQUARE ADJUSTED =    .9438

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      16 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
W          1.0588      .1736       6.100      .000  .836      .9226      .7683
P          .45224      .6558       .6897      .500  .170      .0542      .1106
A          .12115      1.087       .1114      .913  .028      .0151      .0088
CONSTANT   8.1328      8.921       .9116      .375  .222      .0000      .1122
```

The parameter estimate on W is 1.06 (somewhat large) - this implies that a \$1 increase in wage income leads to more than a \$1 increase in consumption expenditure. The effects of P and A do not appear to be individually significant - although a high R-square is reported.

The restricted estimation reports the results:

```  R-SQUARE =    .9517     R-SQUARE ADJUSTED =    .9490

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      18 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
W          .96639      .5133E-01   18.83      .000  .976      .8421      .7013
P          .72479      .3850E-01   18.83      .000  .976      .0868      .1773
A          .60399      .3208E-01   18.83      .000  .976      .0753      .0441
CONSTANT   5.6048      3.680       1.523      .145  .338      .0000      .0773
```

The effect of the restrictions is to lower the standard errors of each of the estimated coefficients. [SHAZAM Guide home]

#### SHAZAM output comparing unrestricted and restricted estimation

Discussion on these results is available in Griffiths, Hill and Judge [1993, Chapter 13] or Judge, Hill, Griffiths, Lutkepohl and Lee [1988, Chapter 21].

``` |_SAMPLE 1 20
|_READ (KLEING.txt) C W P A

UNIT 88 IS NOW ASSIGNED TO: KLEING.txt
4 VARIABLES AND       20 OBSERVATIONS STARTING AT OBS       1

|_STAT W P A / PCOR
NAME        N   MEAN        ST. DEV      VARIANCE     MINIMUM      MAXIMUM
W           20   52.584       16.641       276.93       33.590       80.970
P           20   17.725       2.2876       5.2333       13.390       22.120
A           20   5.2935       2.3815       5.6715       1.6700       9.3000

CORRELATION MATRIX OF VARIABLES -       20 OBSERVATIONS
W          1.0000
P          .71847       1.0000
A          .91517       .63061       1.0000
W            P            A

|_* Unrestricted estimation
|_OLS C W P A
OLS ESTIMATION
20 OBSERVATIONS     DEPENDENT VARIABLE = C
...NOTE..SAMPLE RANGE SET TO:    1,   20

R-SQUARE =    .9527     R-SQUARE ADJUSTED =    .9438
VARIANCE OF THE ESTIMATE-SIGMA**2 =   20.496
STANDARD ERROR OF THE ESTIMATE-SIGMA =   4.5272
SUM OF SQUARED ERRORS-SSE=   327.93
MEAN OF DEPENDENT VARIABLE =   72.465
LOG OF THE LIKELIHOOD FUNCTION = -56.3495

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      16 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
W          1.0588      .1736       6.100      .000  .836      .9226      .7683
P          .45224      .6558       .6897      .500  .170      .0542      .1106
A          .12115      1.087       .1114      .913  .028      .0151      .0088
CONSTANT   8.1328      8.921       .9116      .375  .222      .0000      .1122

|_* Restricted estimation
|_OLS C W P A / RESTRICT
OLS ESTIMATION
20 OBSERVATIONS     DEPENDENT VARIABLE = C
...NOTE..SAMPLE RANGE SET TO:    1,   20
|_RESTRICT P=0.75*W
|_RESTRICT A=0.625*W
|_END
F TEST ON RESTRICTIONS=   .16949     WITH    2 AND   16 DF  P-VALUE=  .84559

R-SQUARE =    .9517     R-SQUARE ADJUSTED =    .9490
VARIANCE OF THE ESTIMATE-SIGMA**2 =   18.604
STANDARD ERROR OF THE ESTIMATE-SIGMA =   4.3133
SUM OF SQUARED ERRORS-SSE=   334.88
MEAN OF DEPENDENT VARIABLE =   72.465
LOG OF THE LIKELIHOOD FUNCTION = -56.5591

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      18 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
W          .96639      .5133E-01   18.83      .000  .976      .8421      .7013
P          .72479      .3850E-01   18.83      .000  .976      .0868      .1773
A          .60399      .3208E-01   18.83      .000  .976      .0753      .0441
CONSTANT   5.6048      3.680       1.523      .145  .338      .0000      .0773

|_* --------------------------------------------------------------------
|_* An equivalent way of obtaining the restricted least squares estimates
|_* is to make a substitution for the restrictions as follows.
|_GENR X = W + 0.75*P + 0.625*A
|_OLS C X
OLS ESTIMATION
20 OBSERVATIONS     DEPENDENT VARIABLE = C
...NOTE..SAMPLE RANGE SET TO:    1,   20

R-SQUARE =    .9517     R-SQUARE ADJUSTED =    .9490
VARIANCE OF THE ESTIMATE-SIGMA**2 =   18.604
STANDARD ERROR OF THE ESTIMATE-SIGMA =   4.3133
SUM OF SQUARED ERRORS-SSE=   334.88
MEAN OF DEPENDENT VARIABLE =   72.465
LOG OF THE LIKELIHOOD FUNCTION = -56.5591

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      18 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
X          .96639      .5133E-01   18.83      .000  .976      .9755      .9227
CONSTANT   5.6048      3.680       1.523      .145  .338      .0000      .0773

|_* Get the estimated coefficent on P
|_TEST 0.75*X
TEST VALUE =   .72479     STD. ERROR OF TEST VALUE   .38496E-01
T STATISTIC =   18.827780     WITH   18 D.F.    P-VALUE=  .00000
F STATISTIC =   354.48529     WITH    1 AND   18 D.F.  P-VALUE=  .00000
WALD CHI-SQUARE STATISTIC =   354.48529     WITH    1 D.F.  P-VALUE=  .00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY =  .00282

|_* Get the estimated coefficent on A
|_TEST 0.625*X
TEST VALUE =   .60399     STD. ERROR OF TEST VALUE   .32080E-01
T STATISTIC =   18.827780     WITH   18 D.F.    P-VALUE=  .00000
F STATISTIC =   354.48529     WITH    1 AND   18 D.F.  P-VALUE=  .00000
WALD CHI-SQUARE STATISTIC =   354.48529     WITH    1 D.F.  P-VALUE=  .00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY =  .00282
|_* --------------------------------------------------------------------

|_*
|_* Test to determine if the restrictions are accepted or rejected
|_* Individual Restriction Test : t-test
|_OLS C W P A
OLS ESTIMATION
20 OBSERVATIONS     DEPENDENT VARIABLE = C
...NOTE..SAMPLE RANGE SET TO:    1,   20

R-SQUARE =    .9527     R-SQUARE ADJUSTED =    .9438
VARIANCE OF THE ESTIMATE-SIGMA**2 =   20.496
STANDARD ERROR OF THE ESTIMATE-SIGMA =   4.5272
SUM OF SQUARED ERRORS-SSE=   327.93
MEAN OF DEPENDENT VARIABLE =   72.465
LOG OF THE LIKELIHOOD FUNCTION = -56.3495

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      16 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
W          1.0588      .1736       6.100      .000  .836      .9226      .7683
P          .45224      .6558       .6897      .500  .170      .0542      .1106
A          .12115      1.087       .1114      .913  .028      .0151      .0088
CONSTANT   8.1328      8.921       .9116      .375  .222      .0000      .1122

|_TEST P=0.75*W
TEST VALUE =  -.34184     STD. ERROR OF TEST VALUE   .72396
T STATISTIC =  -.47218600     WITH   16 D.F.    P-VALUE=  .64317
F STATISTIC =   .22295962     WITH    1 AND   16 D.F.  P-VALUE=  .64317
WALD CHI-SQUARE STATISTIC =   .22295962     WITH    1 D.F.  P-VALUE=  .63679
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000

|_TEST A=0.625*W
TEST VALUE =  -.54059     STD. ERROR OF TEST VALUE   1.1812
T STATISTIC =  -.45764552     WITH   16 D.F.    P-VALUE=  .65336
F STATISTIC =   .20943942     WITH    1 AND   16 D.F.  P-VALUE=  .65336
WALD CHI-SQUARE STATISTIC =   .20943942     WITH    1 D.F.  P-VALUE=  .64721
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000

|_* Joint Test of the Restrictions : F-test
|_TEST
|_TEST P=0.75*W
|_TEST A=0.625*W
|_END
F STATISTIC =   .16949471     WITH    2 AND   16 D.F.  P-VALUE=  .84559
WALD CHI-SQUARE STATISTIC =   .33898942     WITH    2 D.F.  P-VALUE=  .84409
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_STOP
``` [SHAZAM Guide home]