Wald test

The Wald test is a statistical test named after the Hungarian statistician Abraham Wald. It is used to test the significance of individual coefficients in a statistical model, particularly in the context of maximum likelihood estimation. The Wald test is commonly applied in the fields of econometrics, biostatistics, and other areas where regression analysis is used.

Overview

The Wald test evaluates the null hypothesis that a particular parameter in a statistical model is equal to a specified value, usually zero, which implies that the parameter has no effect on the dependent variable. The test is based on the estimated value of the parameter and its estimated standard error.

Mathematical Formulation

Suppose we have a statistical model with a parameter \( \theta \) that we wish to test. The Wald test statistic is given by:

\[ W = \frac{(\hat{\theta} - \theta_0)^2}{\text{Var}(\hat{\theta})} \]

where:

• \( \hat{\theta} \) is the estimated value of the parameter,
• \( \theta_0 \) is the value of the parameter under the null hypothesis,
• \( \text{Var}(\hat{\theta}) \) is the estimated variance of \( \hat{\theta} \).

The Wald test statistic \( W \) follows a chi-squared distribution with one degree of freedom under the null hypothesis.

Applications

The Wald test is widely used in the context of generalized linear models (GLMs), where it is employed to test the significance of individual regression coefficients. It is also used in logistic regression, probit regression, and other types of regression models.

Advantages and Limitations

Advantages

• The Wald test is straightforward to compute, as it only requires the parameter estimate and its standard error.
• It is applicable to a wide range of models, including those with non-normal error distributions.

Limitations

• The Wald test can be unreliable if the sample size is small or if the parameter estimate is near the boundary of the parameter space.
• It may not perform well if the model is misspecified or if there is multicollinearity among the predictors.

Also see

• Likelihood-ratio test
• Score test
• Chi-squared test
• Regression analysis
• Maximum likelihood estimation

References

• Wald, A. (1943). "Tests of statistical hypotheses concerning several parameters when the number of observations is large." Transactions of the American Mathematical Society, 54(3), 426-482.
• Agresti, A. (2015). "Foundations of Linear and Generalized Linear Models." Wiley.