The results from estimation commands display only two-sided tests for the coefficients. How can I perform a one-sided test?

Estimation commands provide a t test or z test for the null hypothesis that a coefficient is equal to zero. The test command can perform Wald tests for simple and composite linear hypotheses on the parameters, but these Wald tests are also limited to tests of equality.

One-sided t tests

To perform one-sided tests, you can first perform the corresponding two-sided Wald test. Then you can use the results to calculate the test statistic and p-value for the one-sided test. Let’s say that you perform the following regression:

. sysuse auto, clear
(1978 Automobile Data)

. regress price mpg weight 

      Source |       SS       df       MS              Number of obs =      74
-------------+------------------------------           F(  2,    71) =   14.74
       Model |   186321280     2  93160639.9           Prob > F      =  0.0000
    Residual |   448744116    71  6320339.67           R-squared     =  0.2934
-------------+------------------------------           Adj R-squared =  0.2735
       Total |   635065396    73  8699525.97           Root MSE      =    2514

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -49.51222   86.15604    -0.57   0.567    -221.3025     122.278
      weight |   1.746559   .6413538     2.72   0.008      .467736    3.025382
       _cons |   1946.069    3597.05     0.54   0.590    -5226.245    9118.382
------------------------------------------------------------------------------

If you wish to test that the coefficient on weight, β_weight, is negative (or positive), you can begin by performing the Wald test for the null hypothesis that this coefficient is equal to zero.

. test _b[weight]=0
 ( 1)  weight = 0

       F(  1,    71) =    7.42
            Prob > F =    0.0081

The Wald test given here is an F test with 1 numerator degree of freedom and 71 denominator degrees of freedom. The Student’s t distribution is directly related to the F distribution in that the square of the Student’s t distribution with d degrees of freedom is equivalent to the F distribution with 1 numerator degree of freedom and d denominator degrees of freedom.

As long as the F test has 1 numerator degree of freedom, the square root of the F statistic is the absolute value of the t statistic for the one-sided test. To determine whether this t statistic is positive or negative, you need to determine whether the fitted coefficient is positive or negative. To do this, you can use the sign() function.

. local sign_wgt = sign(_b[weight])

Then, using the ttail() function along with the returned results from the test command, you can calculate the p-values for the one-sided tests in the following manner:

. display "Ho: coef <= 0  p-value = " ttail(r(df_r),`sign_wgt'*sqrt(r(F)))
Ho: coef <= 0  p-value = .00406491

. display "Ho: coef >= 0  p-value = " 1-ttail(r(df_r),`sign_wgt'*sqrt(r(F)))
Ho: coef >= 0  p-value = .99593509	

In the special case where you are interested in testing whether a coefficient is greater than, less than, or equal to zero, you can calculate the p-values directly from the regression output. When the estimated coefficient is positive, as for weight, you can do so as follows:

When the estimated coefficient is negative, as for mpg, the same code can be used:

. test _b[mpg]=0

 ( 1)  mpg = 0

       F(  1,    71) =    0.33
            Prob > F =    0.5673

. local sign_mpg = sign(_b[mpg])  

. display "Ho: coef <= 0  p-value = " ttail(r(df_r),`sign_mpg'*sqrt(r(F)))
Ho: coef <= 0  p-value = .71633814

. display "Ho: coef >= 0  p-value = " 1-ttail(r(df_r),`sign_mpg'*sqrt(r(F)))
Ho: coef >= 0  p-value = .28366186

However, to calculate the p-values from the regression output directly, you use the following formulas:

On the other hand, if you want to perform a test such as H₀: β_weight <= 1, you cannot calculate the p-value directly from the regression results. Here you would have to perform the Wald test first.

  . test _b[weight]=1

   ( 1)  weight = 1

         F(  1,    71) =    1.35
              Prob > F =    0.2483

  . local sign_wgt2 = sign(_b[weight]-1)

  . display "Ho: coef <= 1  p-value = " ttail(r(df_r),`sign_wgt2'*sqrt(r(F)))
  Ho: coef <= 1  p-value = .12415299

One-sided z tests

In the output for certain estimation commands, you will find that z statistics are reported instead of t statistics. In these cases, when you use the test command, you will get a chi-squared test instead of an F test. The relationship between the standard normal distribution and the chi-squared distribution is similar to the relationship between the Student’s t distribution and the F distribution. In fact, the square root of the chi-squared distribution with 1 degree of freedom is the standard normal distribution. Therefore, one-sided z tests can be performed similarly to one-sided t tests. For example,

. use http://www.stata-press.com/data/r12/union.dta, clear
(NLS Women 14-24 in 1968)

. xtset id
. xtlogit union age grade, nolog


Random-effects logistic regression              Number of obs      =     26200
Group variable: idcode                          Number of groups   =      4434

Random effects u_i ~ Gaussian                   Obs per group: min =         1
                                                               avg =       5.9
                                                               max =        12

                                                Wald chi2(2)       =     69.07
Log likelihood  = -10623.006                    Prob > chi2        =    0.0000

------------------------------------------------------------------------------
       union |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0149252   .0036634     4.07   0.000      .007745    .0221055
       grade |   .1130089   .0177088     6.38   0.000     .0783002    .1477175
       _cons |  -4.313313   .2426307   -17.78   0.000    -4.788861   -3.837766
-------------+----------------------------------------------------------------
    /lnsig2u |   1.800793    .046732                        1.7092    1.892386
-------------+----------------------------------------------------------------
     sigma_u |   2.460578   .0574939                      2.350434    2.575884
         rho |   .6479283   .0106604                      .6267624    .6685287
------------------------------------------------------------------------------
Likelihood-ratio test of rho=0: chibar2(01) =  6343.94 Prob >= chibar2 = 0.000

. test grade


 ( 1)  [union]grade = 0

           chi2(  1) =   40.72
         Prob > chi2 =    0.0000

Here the test command returns r(chi2), which can be used along with the normal() function to calculate the appropriate p-values.

. local sign_grade = sign(_b[grade])

. display "H_0: coef<=0  p-value = " 1-normal(`sign_grade'*sqrt(r(chi2)))
H_0: coef<=0  p-value = 8.768e-11

. display "H_0: coef>=0  p-value = " normal(`sign_grade'*sqrt(r(chi2)))
H_0: coef>=0  p-value = 1

Finally, if you want to perform a test of inequality for two of your coefficients, such as H₀: β_age >= β_grade, you would first perform the following Wald test:

.  test age-grade = 0

 ( 1)  [union]age - [union]grade = 0

           chi2(  1) =   27.44
         Prob > chi2 =    0.0000

Then calculate the appropriate p-value:

. local sign_ag = sign(_b[age]-_b[grade])

. display "H_0: age coef >= grade coef. p-value = " normal(`sign_ag'*sqrt(r(chi2)))
H_0: age coef >= grade coef. p-value = 8.112e-08

Again, this approach (performing a Wald test and using the results to calculate the p-value for a one-sided test) is appropriate only when the Wald F statistic has 1 degree of freedom in the numerator or the Wald chi-squared statistic has 1 degree of freedom. The distributional relationships discussed above are not valid if these degrees of freedom are larger than 1.