1. Set up a Y, X and beta vector for each of the following regression models:

1. $Y_i=\beta_0+\beta_1X_{i1}+\beta_2X_{i2}+\beta_3X_{i1}X_{i2}+\epsilon_i$

This is a regression model with an interaction term between $X_1$ and $X_2$.

$\bold{Y=}\begin{bmatrix}Y_1\\Y_2\\.\\.\\.\\Y_n\end{bmatrix}, \bold{X}=\begin{bmatrix}1&X_{11}&X_{12}&X_{11}X_{12}\\1&X_{21}&X_{22}&X_{21}X_{22}\\.&.&.\\.&.&.\\.&.&.\\1&X_{n1}&X_{n2}&X_{n1}X_{n2}\end{bmatrix}, \bold{\beta}=\begin{bmatrix}\beta_0\\\beta_1\\\beta_2\\\beta_3\end{bmatrix}$

b. check

c. check

d. check

2. For each regression model, indicate whether it is a general LRM.

If not, state whether it can be expressed in the form of a general linear regression by a suitable transformation.

1. $Y_i=\beta_0+\beta_1X_{i1}+\beta_2logX_{i2}+\beta_3X^{-1}_{i3}+\epsilon_i$

This is a general linear regression model.

b. $Y_i=\beta_0\exp(\beta_1X_{i1})+\epsilon_i$

This is not a general linear regression model because it is not linear in parameters. It depends on exponent.

c. $Y_i=[1+\exp(\beta_0+\beta_1X_{i1}+\epsilon_i]^{-1}$

This is not a general linear regression model because it is not linear in parameters. We can take a suitable transformation, $Y'_i=log(\frac{1}{Y_i}-1).$

The model becomes: $Y'i=\beta_0+\beta_1X{i1}+\epsilon_i$, which is in the form of a general LRM.