Control Variable Significance in Regression Model: When to Include z1

To decide whether to include the control variables z1 in the model or not, you can use a variety of techniques to assess the necessity and impact of these variables. Here are a few methods you can consider:

1. Theoretical justification: Consider the theoretical framework and existing literature related to your study. If there are strong theoretical reasons to believe that z1 has a direct effect on y or interacts with x1 and x2, then it is important to include z1 in the model.

2. Statistical significance: Fit the first model with z1 included and examine the statistical significance of the coefficients associated with z1 (γ1 in this case). If γ1 is statistically significant (usually determined by a p-value below a predetermined threshold, such as 0.05), it suggests that z1 has a significant effect on y and should be included in the model.

3. Model fit comparison: Compare the overall fit of the two models using a goodness-of-fit measure, such as the R-squared value or the adjusted R-squared value. If the inclusion of z1 significantly improves the fit of the model (i.e., the R-squared value increases substantially), it suggests that z1 is an important control variable and should be included.

4. Residual analysis: Examine the residuals of both models. If the inclusion of z1 reduces the heteroscedasticity (unequal variance of residuals) or autocorrelation in the model, it suggests that z1 is an important control variable and should be included.

5. Robustness checks: Perform robustness checks by running alternative specifications or models. For example, you can run the model with and without z1 using different estimation techniques (e.g., OLS, fixed effects, or instrumental variable regression) to see if the results are consistent across different approaches.

Ultimately, the decision to include or exclude control variables z1 depends on the specific context, theoretical framework, data availability, and statistical analysis. It is important to carefully examine and consider the results from these techniques to make an informed decision.