Hypothesis Testing

Hypothesis Testing

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 ANOVA option on the OLS command.

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 TEST command that immediately follows the estimation command. With OLS estimation, the general format of commands for testing a single hypothesis is:

OLS depvar indeps / options
TEST equation

The equation is specified as a function of the variables in the indeps list on the OLS command. Note: The variable names actually represent the coefficients involved in the hypothesis test. If a hypothesis test involving the intercept coefficient is required then the name CONSTANT can be used to represent the intercept.

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 beta. Suppose the null and alternative hypotheses are:

            H0: beta < c       and     H1: beta > 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:

If the test statistic is positive the p-value is p/2.
If the test statistic is negative the p-value is 1-p/2.

  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:

OLS depvar indeps / options
TEST
TEST equation1
TEST equation2
. . .
END

The tests involved in the hypothesis are enclosed between a header that is a blank TEST command and an END command.

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.

Example

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: TEST.SHA) below transforms the data to logarithms and estimates the demand equation by OLS. A series of hypothesis tests are then considered. 

SAMPLE 1 17
READ (THEIL.txt) YEAR CONSUME INCOME PRICE
* 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
* Hypothesis testing
TEST LY=1
TEST LP=-1
*
* A joint test
TEST
  TEST LY=1
  TEST LP=-1
END
*
* Now duplicate the F-test that is reported with the ANOVA option
TEST
  TEST LY=0
  TEST LP=0
END
STOP

Note that the indentation used for the TEST commands is optional and is intended to improve the readability of the command file. Tab marks should not be used for indentation - the space bar should be used for this.

The SHAZAM output can be viewed. The ANOVA option on the OLS command produces the following output. 

                      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 266. The p-value is reported as .000 (this actually means less than .0005) and so there is strong evidence to reject the null hypothesis and conclude that the estimated relationship is a significant one. Note that a critical value for the test is obtained from a F-distribution with (2,14) degrees of freedom.

Possibly more interesting tests about consumer behaviour are given with the TEST commands that follow the model estimation. The model estimation reports the following: 

 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 LY and this is reported as 1.1432. The next output shows the computation of a test statistic for the null hypothesis that the income elasticity is equal to one. 

 |_TEST LY=1
 TEST VALUE =   .14316     STD. ERROR OF TEST VALUE   .15600
 T STATISTIC =   .91766674     WITH   14 D.F.    P-VALUE=  .37433

The TEST VALUE reported in the above output is obtained as 1.1432 - 1 = .1432. (In discussing the output some rounding of results is introduced). Note that the standard error of this test value is identical to the standard error for the coefficient on LY that is listed on the OLS estimation output. The t-statistic is computed as .1432 / .15600 = 0.918 . For a test of the null hypothesis against the two-sided alternative that the income elasticity is not equal to 1 the computed p-value is .37. This suggests that there is no evidence to reject the null hypothesis. For a one-sided test of the null hypothesis that the income elasticity is less than or equal to 1 against the alternative that the income elasticity is greater than 1 the p-value is 0.37433/2 = 0.187. Again, the null hypothesis is not rejected.

The next output shows the computation of a test statistic for the null hypothesis that the price elasticity is equal to -1. 

 |_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 -.82884 and the TEST VALUE on the above output is computed as
-.82884 - (-1) = .17116. The t-statistic is computed by dividing the test value by the standard error. The associated p-value gives strong evidence to reject the null hypothesis.

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 1 and the price elasticity is -1. The output below shows the computed F-statistic for this test. 

 |_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 6.51. The test statistic clearly exceeds this. So the null hypothesis is rejected. The p-value reported on the SHAZAM output gives this conclusion immediately.

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SHAZAM output 

 |_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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