The MacKinnon, White, and Davidson (MWD) test, also known as the Heteroscedasticity and Autocorrelation Consistent (HAC) test, is a statistical test used in econometrics to assess the presence of heteroscedasticity and autocorrelation in regression models. It is an extension of the White test that allows for robust inference when these issues are present. Here’s a step-by-step approach to conducting the MWD test:

Step 1: Formulate the null and alternative hypotheses.

Define the null and alternative hypotheses for the MWD test. 

The null hypothesis assumes no heteroscedasticity and autocorrelation, while the alternative hypothesis assumes their presence.

H0: No heteroscedasticity and autocorrelation

Ha: Heteroscedasticity and/or autocorrelation exist

Step 2: Estimate the regression model

Estimate the regression model of interest using ordinary least squares (OLS) or any other appropriate estimation method.

Obtain the coefficient estimates and residuals from the regression.

Step 3: Calculate the residual squared values

Compute the squared residuals by squaring the residuals obtained in Step 2.

These squared residuals represent the heteroscedasticity component.

Step 4: Determine the lag selection criteria

Decide on the appropriate lag selection criteria for capturing autocorrelation.

Common criteria include;

Akaike Information Criterion (AIC), Schwarz Bayesian Criterion (SBC), or the Hannan-Quinn Criterion (HQC).

Step 5: Estimate the variance-covariance matrix 

Estimate the variance-covariance matrix of the residuals using the lag selection criteria determined in Step 4. 

This matrix captures the autocorrelation component.

Step 6: Compute the robust standard errors 

Calculate the robust standard errors using the estimated variance-covariance matrix from Step 5. 

These standard errors account for both heteroscedasticity and autocorrelation.

Step 7: Perform the MWD test

Calculate the MWD test statistic, which is based on the ratio of the coefficient estimate to its corresponding robust standard error. The formula for the MWD test statistic is:

MWD test statistic = Coefficient estimate / Robust standard error

Step 8: Determine the critical value

Determine the critical value for the MWD test statistic based on the desired level of significance (e.g., 5% or 1%). 

The critical values can be obtained from statistical tables or software.

Step 9: Compare the test statistic with the critical value

Compare the calculated MWD test statistic from Step 7 with the critical value from Step 8. 

If the test statistic exceeds the critical value, you reject the null hypothesis and conclude that there is evidence of heteroscedasticity and/or autocorrelation. 

If the test statistic is below the critical value, you fail to reject the null hypothesis.