The standard OLS estimation output from SHAZAM reports a t-ratio
for testing the null hypothesis that the true regression coefficient
is zero. When the regression equation contains more than 1 explanatory
variable it may be of interest to test the null hypothesis
that all slope coefficients are jointly equal to zero.
This is called a test of the overall significance of the regression line.
The F-test statistic for this test is computed with the
In practice, the economist is likely to be interested in other types of hypotheses that may involve linear (or nonlinear) combinations of the regression coefficients.
Testing a single linear combination of coefficients
Test statistics are computed with the
The SHAZAM output reports a t-test statistic and a p-value for a 2-sided test. The null hypothesis can be rejected if the p-value is less than a selected level of significance (say, 0.05).
One-tailed tests can also be considered. For example, consider testing hypotheses about some unknown parameter . Suppose the null and alternative hypotheses are:
H0: < c and H1: > c
where c is some scalar constant.
The test statistic for the one-tailed test is computed in the same way as for a two-tailed test. However, the null hypothesis will be rejected only if the value of the test statistic is excessively large (giving support to the alternative hypothesis). Suppose that p is the p-value reported for the two-tailed test. The p-value for the inequality hypotheses stated above can be computed as follows:
Testing more than one linear combination of coefficients
A test statistic for a joint test that involves two or more functions of the coefficients can be obtained in SHAZAM with the general command format:
The tests involved in the hypothesis are enclosed between a header
that is a blank
Typically, an assumption in hypothesis testing is that the residuals are normally distributed. This assumption is then used to determine the distribution of the test statistic.
This example uses the Theil textile data set to illustrate hypothesis testing in SHAZAM. The textile demand equation is specified in log-log form so that the parameter estimates have interpretations as income elasticities and price elasticities. A number of hypotheses about consumer behaviour can be tested. For example, a negative price elasticity is expected. A price elasticity that is less than 1 in absolute value implies that demand is price inelastic.
The command file (filename:
Note that the indentation used for the
The SHAZAM output can be viewed.
ANALYSIS OF VARIANCE - FROM MEAN SS DF MS F REGRESSION .51733 2. .25867 266.018 ERROR .13613E-01 14. .97236E-03 P-VALUE TOTAL .53094 16. .33184E-01 .000
A test of the null hypothesis that all slope coefficients are zero
reports a F-test statistic of
Possibly more interesting tests about consumer behaviour are
given with the
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 14 DF P-VALUE CORR. COEFFICIENT AT MEANS LY 1.1432 .1560 7.328 .000 .891 .3216 1.1432 LP -.82884 .3611E-01 -22.95 .000 -.987 -1.0074 -.8288 CONSTANT 3.1636 .7048 4.489 .001 .768 .0000 3.1636
The income elasticity is the estimated coefficient on
|_TEST LY=1 TEST VALUE = .14316 STD. ERROR OF TEST VALUE .15600 T STATISTIC = .91766674 WITH 14 D.F. P-VALUE= .37433
The next output shows the computation of a test statistic for the null
hypothesis that the price elasticity is equal to
|_TEST LP=-1 TEST VALUE = .17116 STD. ERROR OF TEST VALUE .36111E-01 T STATISTIC = 4.7398530 WITH 14 D.F. P-VALUE= .00032
The price elasticity is
Individual tests on the income and price elasticities have been
considered. Now consider a joint test of the null hypothesis that the
income elasticity is
|_TEST |_ TEST LY=1 |_ TEST LP=-1 |_END F STATISTIC = 13.275308 WITH 2 AND 14 D.F. P-VALUE= .00058
By consulting printed statistical tables, the 1% critical value from the
F-distribution with (2,14) degrees of freedom is
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|_SAMPLE 1 17 |_READ (THEIL.txt) YEAR CONSUME INCOME PRICE UNIT 88 IS NOW ASSIGNED TO: THEIL.txt 4 VARIABLES AND 17 OBSERVATIONS STARTING AT OBS 1 |_* Transform the data to logarithms |_GENR LC=LOG(CONSUME) |_GENR LY=LOG(INCOME) |_GENR LP=LOG(PRICE) |_* Estimate the log-log model |_OLS LC LY LP / LOGLOG ANOVA OLS ESTIMATION 17 OBSERVATIONS DEPENDENT VARIABLE = LC ...NOTE..SAMPLE RANGE SET TO: 1, 17 R-SQUARE = .9744 R-SQUARE ADJUSTED = .9707 VARIANCE OF THE ESTIMATE-SIGMA**2 = .97236E-03 STANDARD ERROR OF THE ESTIMATE-SIGMA = .31183E-01 SUM OF SQUARED ERRORS-SSE= .13613E-01 MEAN OF DEPENDENT VARIABLE = 4.8864 LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -46.5862 MODEL SELECTION TESTS - SEE JUDGE ET AL. (1985,P.242) AKAIKE (1969) FINAL PREDICTION ERROR - FPE = .11440E-02 (FPE IS ALSO KNOWN AS AMEMIYA PREDICTION CRITERION - PC) AKAIKE (1973) INFORMATION CRITERION - LOG AIC = -6.7770 SCHWARZ (1978) CRITERION - LOG SC = -6.6300 MODEL SELECTION TESTS - SEE RAMANATHAN (1992,P.167) CRAVEN-WAHBA (1979) GENERALIZED CROSS VALIDATION - GCV = .11807E-02 HANNAN AND QUINN (1979) CRITERION = .11565E-02 RICE (1984) CRITERION = .12376E-02 SHIBATA (1981) CRITERION = .10834E-02 SCHWARZ (1978) CRITERION - SC = .13202E-02 AKAIKE (1974) INFORMATION CRITERION - AIC = .11397E-02 ANALYSIS OF VARIANCE - FROM MEAN SS DF MS F REGRESSION .51733 2. .25867 266.018 ERROR .13613E-01 14. .97236E-03 P-VALUE TOTAL .53094 16. .33184E-01 .000 ANALYSIS OF VARIANCE - FROM ZERO SS DF MS F REGRESSION 406.42 3. 135.47 139325.591 ERROR .13613E-01 14. .97236E-03 P-VALUE TOTAL 406.44 17. 23.908 .000 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 14 DF P-VALUE CORR. COEFFICIENT AT MEANS LY 1.1432 .1560 7.328 .000 .891 .3216 1.1432 LP -.82884 .3611E-01 -22.95 .000 -.987 -1.0074 -.8288 CONSTANT 3.1636 .7048 4.489 .001 .768 .0000 3.1636 |_* Hypothesis testing |_TEST LY=1 TEST VALUE = .14316 STD. ERROR OF TEST VALUE .15600 T STATISTIC = .91766674 WITH 14 D.F. P-VALUE= .37433 F STATISTIC = .84211225 WITH 1 AND 14 D.F. P-VALUE= .37433 WALD CHI-SQUARE STATISTIC = .84211225 WITH 1 D.F. P-VALUE= .35879 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000 |_TEST LP=-1 TEST VALUE = .17116 STD. ERROR OF TEST VALUE .36111E-01 T STATISTIC = 4.7398530 WITH 14 D.F. P-VALUE= .00032 F STATISTIC = 22.466206 WITH 1 AND 14 D.F. P-VALUE= .00032 WALD CHI-SQUARE STATISTIC = 22.466206 WITH 1 D.F. P-VALUE= .00000 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = .04451 |_* |_* A joint test |_TEST |_ TEST LY=1 |_ TEST LP=-1 |_END F STATISTIC = 13.275308 WITH 2 AND 14 D.F. P-VALUE= .00058 WALD CHI-SQUARE STATISTIC = 26.550616 WITH 2 D.F. P-VALUE= .00000 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = .07533 |_* |_* Now duplicate the F-test that is reported with the ANOVA option |_TEST |_ TEST LY=0 |_ TEST LP=0 |_END F STATISTIC = 266.01794 WITH 2 AND 14 D.F. P-VALUE= .00000 WALD CHI-SQUARE STATISTIC = 532.03587 WITH 2 D.F. P-VALUE= .00000 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = .00376 |_STOP
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