## 10.1 Nested Model

\begin{aligned} y &= \beta_0 + x_1\beta_1 + x_2\beta-2 + x_3\beta_3 + \epsilon & \text{unrestricted model} \\ y &= \beta_0 + x_1\beta_1 + \epsilon & \text{restricted model} \end{aligned}

Unrestricted model is always longer than the restricted model
The restricted model is “nested” within the unrestricted model
To determine which variables should be included or exclude, we could use the same Wald Test

Adjusted $$R^2$$

• $$R^2$$ will always increase with more variables included
• Adjusted $$R^2$$ tries to correct by penalizing inclusion of unnecessary variables.

\begin{aligned} {R}^2 &= 1 - \frac{SSR/n}{SST/n} \\ {R}^2_{adj} &= 1 - \frac{SSR/(n-k)}{SST/(n-1)} \\ &= 1 - \frac{(n-1)(1-R^2)}{(n-k)} \end{aligned}

• $${R}^2_{adj}$$ increases if and only if the t-statistic on the additional variable is greater than 1 in absolute value.
• $${R}^2_{adj}$$ is valid in models where there is no heteroskedasticity
• there fore it should not be used in determining which variables should be included in the model (the t or F-tests are more appropriate)

### 10.1.1 Chow test

Should we run two different regressions for two groups?