waldTest#

Purpose#

Performs a Wald test of joint hypothesis on model parameters.

Format#

{ waldTest, p_value } = waldTest(out[, R, q, tau, joint])#
{ waldTest, p_value } = waldTest(sigma, params[, R, q, df_residuals, varnames])
Parameters:

  • out (Struct) – Post-estimation filled output structure. Valid structure types include: olsmtOut, gmmOut, glmOut, and qfitOut.

  • sigma (Matrix) – Parameter variance-covariance estimation.

  • params (Vector) – Parameter estimates.

  • R (String Array) –

    Optional, LHS of the null hypothesis. Should be specified in terms of the model variables, with a separate row for each hypothesis. The function accepts linear combinations of the model variables. If using matrix inputs and no variable names are specified, variables labeled by default "X1", "X2", "X3", ....

    e.g to test the hypothesis "X1 - X4 = 0" jointly with the hypothesis that "2*X3 - X2 = 0". The R matrix input will be:

    // Specify R matrix
R_sa = "X1 - X4"$|"2*X3 - X2";

    To include all individual variables use “all”. Default is to test the joint hypothesis of all variables.

  • q (Vector) –

    Optional, RHS of the null hypothesis. Must be numeric vector. Default is to set RHS of all hypothesis to zero.

    e.g to test the hypothesis "X1 - X4 = 2" jointly with the hypothesis that "2*X3 - X2 = 0" The q matrix input will be:

    // Specify R matrix
R_sa = "X1 - X4"$|"2*X3 - X2";

// Specify q matrix
q = 2|0;

  • tau (Vector) – Optional, tau level corresponding to the testing hypothesis. Default is to jointly tests across all tau values. To include all tau levels use "all". Only valid for the qfitOut structure.

  • joint (Scalar) – Optional, specification to test quantileFit() hypotheses jointly across all coefficients for the qfitOut structure. Default = 1.

  • df_residuals (Scalar) – Optional, model degrees of freedom for the F-test. Default = 500.

  • varnames (String array) – Optional, variable names.

Returns:

  • waldTest (Vector) – The statistic for testing the null joint hypothesis specified by the R and q inputs.

  • p_value (Vector) – The p-value associated with the Wald statistic.

Examples#

Basic test with estimation output structure#

The default settings of the waldTest() procedure test the joint hypotheses that all variables equal zero.

// Run ols estimation
// Load data
fname = getGAUSSHome("examples/auto2.dta");

// Run OLS estimation
struct olsmtOut out;
out = olsmt(auto2, "price ~ mpg + rep78");

The OLS results are:

Valid cases:                    69      Dependent variable:               price
Missing cases:                   5      Deletion method:               Listwise
Total SS:            576796958.870      Degrees of freedom:                  63
R-squared:                   0.258      Rbar-squared:                     0.199
Residual SS:         427776355.434      Std error of est:              2605.782
F(5,63):                     4.389      Probability of F:                 0.002

                                 Standard                 Prob   Standardized  Cor with
Variable             Estimate      Error      t-value     >|t|     Estimate    Dep Var
---------------------------------------------------------------------------------------
CONSTANT                10450     2251.04     4.64229     0.000       ---         ---
mpg                  -280.261     61.5767    -4.55142     0.000   -0.564519   -0.455949
rep78: Fair           877.635     2063.28    0.425358     0.672   0.0971824  -0.0223477
rep78: Average        1425.66     1905.44    0.748204     0.457     0.24444   0.0859051
rep78: Good           1693.84     1942.67    0.871914     0.387    0.257252   -0.015317
rep78: Excellent      3131.98     2041.05      1.5345     0.130    0.396546   -0.035102

// Call waldTest
call waldTest(out);

The code above will print a test summary.

===================================
Wald test of null joint hypothesis:

CONSTANT         =  0
mpg              =  0
rep78: Fair      =  0
rep78: Average   =  0
rep78: Good      =  0
rep78: Excellent =  0
-----------------------------------
F( 6, 63 ):                 67.6332
Prob > F :                   0.0000
===================================

Example One: Testing that all variables equal zero#

The default settings of the waldTest() procedure test the joint hypotheses that all variables equal zero.

// Run ols estimation
// Load data
fname = getGAUSSHome("examples/auto2.dta");

// Run OLS estimation
struct olsmtOut out;
out = olsmt(auto2, "price ~ mpg + rep78");

// Call waldTest
call waldTest(out);

The code above will print a test summary.

===================================
Wald test of null joint hypothesis:

CONSTANT         =  0
mpg              =  0
rep78: Fair      =  0
rep78: Average   =  0
rep78: Good      =  0
rep78: Excellent =  0
-----------------------------------
F( 6, 63 ):                 67.6332
Prob > F :                   0.0000
===================================

Example Two: Testing that subset of variables equal zero#

In the first example we tested all variables. Now suppose we wish to test all variable except the constant. This is done by specifying a hypothesis matrix, R.

// Specify hypotheses
R = "mpg"$|"rep78: Fair"$|"rep78: Average"$|"rep78: Good"$|"rep78: Excellent";

// Call waldTest to test joint hypotheses that
// mpg = 0
// rep78: Fair = 0
// rep78: Average = 0
// rep78: Good = 0
// rep78: Excellent = 0
call waldTest(out, R);

Note that this is the same as the F-test reported from the OLS estimation:

===================================
Wald test of null joint hypothesis:

mpg              =  0
rep78: Fair      =  0
rep78: Average   =  0
rep78: Good      =  0
rep78: Excellent =  0
-----------------------------------
F( 5, 63 ):                  4.3893
Prob > F :                   0.0017
===================================

Example Three: Testing the equality of variables#

The true usefulness of the waldTest() procedure is the ability to more than if variables are equal to zero. For example, suppose we want to test if the coefficients for the rep78: Average and rep78: Good categories are equal. We can do this by testing the hypothesis that rep78: Average - rep78: Good = 0.

// Specify R matrix
R = "rep78: Good - rep78: Average";

// Call waldTest
call waldTest(out, R);

===================================
Wald test of null joint hypothesis:
rep78: Good - rep78: Average =  0
-----------------------------------
F( 1, 63 ):                  0.1155
Prob > F :                   0.7351
===================================

In this case, we cannot reject the null hypothesis.

See also

qFitSlopeTest()

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