Chapter 19 Hypothesis Testing

Note:

• Always written in terms of the population parameter ($$\beta$$) not the estimator/estimate ($$\hat{\beta}$$)

• Sometimes, different disciplines prefer to use $$\beta$$ (i.e., standardized coefficient), or $$\mathbf{b}$$ (i.e., unstandardized coefficient)

• $$\beta$$ and $$\mathbf{b}$$ are similar in interpretation; however, $$\beta$$ is scale free. Hence, you can see the relative contribution of $$\beta$$ to the dependent variable. On the other hand, $$\mathbf{b}$$ can be more easily used in policy decisions.

• $\beta_j = \mathbf{b} \frac{s_{x_j}}{s_y}$

• Assuming the null hypothesis is true, what is the (asymptotic) distribution of the estimator
• Two-sided

$H_0: \beta_j = 0 \\ H_1: \beta_j \neq 0$

then under the null, the OLS estimator has the following distribution

$A1-A3a, A5: \sqrt{n} \hat{\beta_j} \sim N(0,Avar(\sqrt{n}\hat{\beta}_j))$

• For the one-sided test, the null is a set of values, so now you choose the worst case single value that is hardest to prove and derive the distribution under the null
• One-sided

$H_0: \beta_j\ge 0 \\ H_1: \beta_j < 0$

then the hardest null value to prove is $$H_0: \beta_j=0$$. Then under this specific null, the OLS estimator has the following asymptotic distribution

$A1-A3a, A5: \sqrt{n}\hat{\beta_j} \sim N(0,Avar(\sqrt{n}\hat{\beta}_j))$