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:

C_t = beta ₀ + beta ₁ W_t + beta ₂ P_t + beta ₃ A_t + e_t

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:

beta ₂ = 0.75 beta ₁ and beta ₃ = 0.625 beta 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]