7 7th Turorial

7.1 Recap

Previously we have been discussing multivariate linear regression that have the formula $\textbf{Y} = \textbf{X}\beta + \epsilon$. Now assume we have two linear model $M_1$ and $M_2$ where $M_2$ is a simplification of $M_1$ such as ($M_1: y_i = \beta_0 + \beta_1~x_{1_{i}} + \beta_2~x_{2_{i}}+ \beta_3~x_{3_{i}}$ and $M_2: y_i = \beta_0 + \beta_1~x_{1_{i}}$) A question arises here is which model is preferred and that is equivalent to test

$$H_0 = \beta_2 = \beta_3 =0 ~~~ vs ~~~ H_1 = \beta_2 \neq 0 , \beta_3 \neq0$$

The decision rule is to reject $H_0$ if $$F = \frac{(D_2-D_1)/q}{D_1/(n-p)} > F_{q,n−p,\alpha}$$

where $n$ is the number of observations, $p$ is the number of parameters in $M_1$ and $q$ is the number of parameters fixed to reduce $M_1$ to $M_2$. For the example above, $p = 4$ and $q = 2$, and $D_1$ and $D_2$ are the SSR of $M_1$ and $M_2$ respectively. Equivalently, we could use the notation $SSE(...)$. For instance $SSE(X_1,X_2,X_3)$ denotes the sum of squared error for a multiple linear regression that includes $X_1$, $X_2$ and $X_3$ to draw the model.

7.2 Exercises

Exercise 7.1 Download the csv file for the dataset here

1. Estimate the liner model for the given data and interpret its coefficients.
2. Discuss the efficiency of the model by two different approaches.
3. Write the ANOVA table that factorize the sum square regression $X_1$ and $X_2$ given $X_1$.
4. Use partial F to test whether you can remove $X_2$ from model.
5. Calculate $R^2$ , $r^2_{Y,2.1}$ , $r_{Y,1.2}$ and $r^2_{Y,2}$
6. Estimate the corresponding standard model and discuss its coefficient.