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

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