Econometric Hypothesis Testing and Regression Model Specification
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Econometric Notation: Residuals and Estimates
Residuals are denoted by e (Sample Regression Function, SRF) and the true error term by u (Population Regression Function, PRF).
- Coefficients: Estimated coefficients (e.g., $\hat{\beta}_0$ and $\hat{\beta}_1$) in the SRF are denoted with hats (caps). True coefficients ($\beta_0, \beta_1$) in the PRF are without hats.
- Dependent Variable: The predicted value $\hat{Y}_t$ (the value derived from the regression equation without the residual) is denoted with a hat. The actual observed value $Y_t$ (which includes the residual or error term) is without a hat.
Fundamentals of Hypothesis Testing
Hypothesis testing makes inferences about the validity of specific economic (or other) theories from a sample of the population for which the theories are supposed to be true.
Four Basic Steps of Hypothesis Testing (Using the t-Test)
- Set up the null and alternative hypotheses.
- Choose a level of significance ($\alpha$) and, therefore, a critical t-value.
- Run the regression and obtain an estimated (calculated) t-value.
- Apply the decision rule by comparing the calculated t-value with the critical t-value in order to reject or not reject the null hypothesis.
Defining Null and Alternative Hypotheses
The null hypothesis ($H_0$) states the range of values that the regression coefficient is expected to take on if the researcher’s theory is not correct. The alternative hypothesis ($H_A$) is a statement of the range of values that the regression coefficient is expected to take if the researcher’s theory is correct.
Types of Errors in Hypothesis Testing
The two kinds of errors we can make in hypothesis testing are:
- Type I Error: We reject a null hypothesis that is true (False Positive).
- Type II Error: We do not reject a null hypothesis that is false (False Negative).
Applying the t-Test for Individual Coefficients
The t-test tests hypotheses about individual coefficients from regression equations. In many regression applications, the hypothesized value of the coefficient ($β_{H_0}$) is zero.
The Decision Rule for Rejecting $H_0$
Once you have calculated a t-value and chosen a critical t-value, you reject the null hypothesis if:
- The calculated t-value is greater in absolute value than the critical t-value, AND
- The t-value has the sign implied by the alternative hypothesis ($H_A$).
Important Caution: While the t-test is straightforward, care must be taken to avoid confusing statistical significance with theoretical validity or empirical importance.
Addressing Omitted Variable Bias (OVB)
Understanding Omission Effects
The omission of a relevant variable from a regression equation will cause bias in the estimates of the remaining coefficients, specifically to the extent that the omitted variable is correlated with the included variables.
The bias expected from leaving a variable out of an equation equals the coefficient of the excluded variable multiplied by a function of the simple correlation coefficient between the excluded variable and the included variable in question.
Consequences of Including Irrelevant Variables
Including a variable in an equation when it is actually irrelevant does not cause bias. However, this practice will usually:
- Increase the variances of the included variables’ estimated coefficients.
- Lower their t-values.
- Lower $R^2$.
Criteria for Variable Inclusion in Regression Models
Four useful criteria for the inclusion of a variable in an equation are:
- Theory
- t-Test (Statistical Significance)
- Bias (Assessing potential OVB if omitted)
- [The original list skipped 'c' but provided 'd' as 'bias'. We list the three provided criteria.]
Crucial Principle: Theory, not statistical fit alone, should be the most important criterion for the inclusion of a variable in a regression equation. Relying solely on statistical fit risks producing incorrect and/or disbelieved results.