How to Perform a Wald Test in Python

A Wald test can be used to test if one or more parameters in a model are equal to certain values.

This test is often used to determine if one or more predictor variables in a regression model are equal to zero.

We use the following null and alternative hypotheses for this test:

• H₀: Some set of predictor variables are all equal to zero.
• H_A: Not all predictor variables in the set are equal to zero.

If we fail to reject the null hypothesis, then we can drop the specified set of predictor variables from the model because they don’t offer a statistically significant improvement in the fit of the model.

The following example shows how to perform a Wald test in Python

Example: Wald Test in Python

For this example, we’ll the famous mtcars dataset to fit the following multiple linear regression model:

mpg  = β₀ + β₁disp + β₂carb + β₃hp + β₄cyl

The following code shows how to fit this regression model and view the model summary:

import statsmodels.formula.api as smf
import pandas as pd
import io

#define dataset as string
mtcars_data="""model,mpg,cyl,disp,hp,drat,wt,qsec,vs,am,gear,carb
Mazda RX4,21,6,160,110,3.9,2.62,16.46,0,1,4,4
Mazda RX4 Wag,21,6,160,110,3.9,2.875,17.02,0,1,4,4
Datsun 710,22.8,4,108,93,3.85,2.32,18.61,1,1,4,1
Hornet 4 Drive,21.4,6,258,110,3.08,3.215,19.44,1,0,3,1
Hornet Sportabout,18.7,8,360,175,3.15,3.44,17.02,0,0,3,2
Valiant,18.1,6,225,105,2.76,3.46,20.22,1,0,3,1
Duster 360,14.3,8,360,245,3.21,3.57,15.84,0,0,3,4
Merc 240D,24.4,4,146.7,62,3.69,3.19,20,1,0,4,2
Merc 230,22.8,4,140.8,95,3.92,3.15,22.9,1,0,4,2
Merc 280,19.2,6,167.6,123,3.92,3.44,18.3,1,0,4,4
Merc 280C,17.8,6,167.6,123,3.92,3.44,18.9,1,0,4,4
Merc 450SE,16.4,8,275.8,180,3.07,4.07,17.4,0,0,3,3
Merc 450SL,17.3,8,275.8,180,3.07,3.73,17.6,0,0,3,3
Merc 450SLC,15.2,8,275.8,180,3.07,3.78,18,0,0,3,3
Cadillac Fleetwood,10.4,8,472,205,2.93,5.25,17.98,0,0,3,4
Lincoln Continental,10.4,8,460,215,3,5.424,17.82,0,0,3,4
Chrysler Imperial,14.7,8,440,230,3.23,5.345,17.42,0,0,3,4
Fiat 128,32.4,4,78.7,66,4.08,2.2,19.47,1,1,4,1
Honda Civic,30.4,4,75.7,52,4.93,1.615,18.52,1,1,4,2
Toyota Corolla,33.9,4,71.1,65,4.22,1.835,19.9,1,1,4,1
Toyota Corona,21.5,4,120.1,97,3.7,2.465,20.01,1,0,3,1
Dodge Challenger,15.5,8,318,150,2.76,3.52,16.87,0,0,3,2
AMC Javelin,15.2,8,304,150,3.15,3.435,17.3,0,0,3,2
Camaro Z28,13.3,8,350,245,3.73,3.84,15.41,0,0,3,4
Pontiac Firebird,19.2,8,400,175,3.08,3.845,17.05,0,0,3,2
Fiat X1-9,27.3,4,79,66,4.08,1.935,18.9,1,1,4,1
Porsche 914-2,26,4,120.3,91,4.43,2.14,16.7,0,1,5,2
Lotus Europa,30.4,4,95.1,113,3.77,1.513,16.9,1,1,5,2
Ford Pantera L,15.8,8,351,264,4.22,3.17,14.5,0,1,5,4
Ferrari Dino,19.7,6,145,175,3.62,2.77,15.5,0,1,5,6
Maserati Bora,15,8,301,335,3.54,3.57,14.6,0,1,5,8
Volvo 142E,21.4,4,121,109,4.11,2.78,18.6,1,1,4,2"""

#convert string to DataFrame
df = pd.read_csv(io.StringIO(mtcars_data), sep=",")

#fit multiple linear regression model
results = smf.ols('mpg ~ disp + carb + hp + cyl', df).fit()

#view regression model summary
results.summary()

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept34.0216 2.523	13.482	0.000	28.844	39.199
disp	-0.0269	 0.011	-2.379	0.025	-0.050	-0.004
carb	-0.9269	 0.579	-1.601	0.121	-2.115	0.261
hp	0.0093	 0.021	0.452	0.655	-0.033	0.052
cyl	-1.0485	 0.784	-1.338	0.192	-2.657	0.560

Next, we can use the wald_test() function from statsmodels to test if the regression coefficients for the predictor variables “hp” and “cyl” are both equal to zero.

The following code shows how to use this function in practice:

#perform Wald Test to determine if 'hp' and 'cyl' coefficients are both zero
print(results.wald_test('(hp = 0, cyl = 0)'))

F test: F=array([[0.91125429]]), p=0.41403001184235005, df_denom=27, df_num=2

From the output we can see that the p-value of the test is 0.414.

Since this p-value is not less than .05, we fail to reject the null hypothesis of the Wald test.

This means we can assume the regression coefficients for the predictor variables “hp” and “cyl” are both equal to zero.

We can drop these terms from the model since they don’t statistically significantly improve the overall fit of the model.

Additional Resources

The following tutorials explain how to perform other common operations in Python:

How to Perform Simple Linear Regression
How to Perform Polynomial Regression in Python
How to Calculate VIF in Python