SHAZAM Confidence Intervals

Confidence Intervals

The CONFID command computes confidence intervals using the estimated coefficients and standard errors from the previous estimation. With OLS estimation, the general format of commands is:

OLS depvar indeps
CONFID indeps / options

where indeps is a list of the explanatory variables and options is a list of desired options. On the CONFID command, the variable names actually represent the coefficients for which interval estimates are required. The variable names listed on the CONFID command must be entered as explanatory variables on the estimation command. The variable name CONSTANT can also be specified to obtain an interval estimate for the intercept coefficient. A useful option is:

TCRIT=   Specifies the t-distribution critical values for calculating confidence intervals. If this option is not specified then SHAZAM computes 90% and 95% confidence intervals.

Suppose the regression equation has N observations and K coefficients. For a parameter estimate b with estimated standard error se(b) the 100(1-alpha)% confidence interval estimate is:

      [b - t se(b) , b + t se(b)]

where t is the alpha/2 critical value from a t-distribution with N-K degrees of freedom.

SHAZAM computes 90% and 95% confidence intervals using the critical values of the t-distribution that are tabulated in econometrics textbooks. Alternatively, the user can specify critical values with the TCRIT= option on the CONFID command.

Example

This example uses the Theil textile data set. The textile demand equation is specified in log-log form. The commands (filename: CONFID.SHA) below obtain 90% and 95% interval estimates for the coefficients. 

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 
CONFID LY LP CONSTANT
STOP

The CONFID command specifies the variable name CONSTANT. SHAZAM will then compute an interval estimate for the intercept coefficient as well as for the slope coefficients.

The SHAZAM output can be viewed. The estimated coefficient on the variable LP has an interpretation as a price elasticity. The estimation results show that the point estimate for the price elasticity is: -.83. (For presentation purposes the numerical results are rounded to 2 decimal places). The 90% interval estimate is: 

         [-.89, -.77]

The 95% interval estimate for the price elasticity is: 

         [-.91, -.75]

The next example computes 99% interval estimates. The DISTRIB command is used to obtain an appropriate critical value for the interval estimates. The general command format for obtaining critical values of the t-distribution is:

DISTRIB prob / INVERSE TYPE=T DF=df CRITICAL=crit

where prob is a variable that contains tail area probabilities and df is the degrees of freedom. The CRITICAL= option can be used to save the critical values in the variable specified. This is shown in the commands (filename: CONFID1.SHA) below. 

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)
* Get the critical value to use for interval estimates
SAMPLE 1 1
GEN1 ALPHA=.01
GEN1 A2=ALPHA/2
DISTRIB A2 / INVERSE TYPE=T DF=14 CRITICAL=Z
* Compute point estimates and interval estimates
SAMPLE 1 17
OLS LC LY LP / LOGLOG 
CONFID LY LP CONSTANT / TCRIT=Z 
STOP

The SHAZAM output can be viewed. The DISTRIB command uses a numerical algorithm to compute critical values. The critical value is computed as 2.9774. When rounded to 3 decimal places, this is identical to the value given in statistical tables in textbooks.

The 99% interval estimate for the price elasticity is: 

         [-.94, -.72]
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SHAZAM output - 90% and 95% interval estimates

 |_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
 
  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
 
 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

 |_CONFID LY LP CONSTANT
 USING 95% AND 90% CONFIDENCE INTERVALS
 
 CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH  14 D.F.
      - T CRITICAL VALUES =   2.145 AND   1.761
 NAME   LOWER 2.5%   LOWER 5%   COEFFICIENT   UPPER 5%   UPPER 2.5%   STD. ERROR
 LY         .8085       .8684       1.1432       1.418       1.478        .156
 LP        -.9063      -.8924      -.82884      -.7652      -.7514        .036
 CONSTANT   1.652       1.922       3.1636       4.405       4.675        .705
 |_STOP
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SHAZAM output - 99% interval estimates for regression coefficients

 |_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)

 |_* Get the critical value to use for interval estimates
 |_SAMPLE 1 1
 |_GEN1 ALPHA=.01
 |_GEN1 A2=ALPHA/2
 |_DISTRIB A2 / INVERSE TYPE=T DF=14 CRITICAL=Z
 T DISTRIBUTION DF=   14.000
 VARIANCE=   1.1667       H=   1.0000
 
               PROBABILITY CRITICAL VALUE   PDF
   A2
  ROW     1     .50000E-02  2.9774      .98931E-02

 |_* Compute point estimates and interval estimates
 |_SAMPLE 1 17
 |_OLS LC LY LP / LOGLOG
 
  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
 
 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

 |_CONFID LY LP CONSTANT / TCRIT=Z
 
 CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH  14 D.F.
      - T CRITICAL VALUE =   2.977
 NAME        LOWER     COEFFICIENT     UPPER       STD. ERROR
  LY         .67868       1.1432       1.6076       .15600
  LP        -.93636      -.82884      -.72132       .36111E-01
  CONSTANT   1.0651       3.1636       5.2620       .70480
 |_STOP
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