Respuesta :
Answer:
See explanation below.
Step-by-step explanation:
If we assume a linear model with two variables and one intercept the model is given by:
[tex] y_i = \beta_0 + \beta_1 x_{1j} +\beta_2 x_{2j}+ e_i [/tex]
The extension of this to a multiple regression modelwith p predictors is:
[tex] y_i = \beta_0 + \sum_{j=1}^p \beta_j x_{ij} +e_i [/tex]
We assume that we have n individuals [tex] i \in [1,...,n][/tex]
And the distribution for the errors is [tex]e_i \sim N(0,\sigma^2)[/tex]
and we can write this model with a design matrix X like this:
[tex] y = Xb + e[/tex]
[tex] y= (y_1,....,y_n)' \in R^n [/tex] a nx1 response vector
[tex] X = [1_n , x_1,....,x_p] \in R^{nx(p+1)}[/tex] represent the design matrix nx(p+1)
Where [tex]1_n[/tex] is a nx1 vector of ones, and [tex]x_j =(x_{1j}, ...,x_{nj})\in R^{p+1}[/tex] is a (p+1)x1 vector of coeffcients
And [tex]e=(e_1, ...,e_n) \in R^n[/tex] is a nx1 error vector
[tex]e \sim N(0_n , \sigma^2 I_n)[/tex]
[tex] y|X \sim N (Xb, \sigma^2 I_n)[/tex]
Using ordinary least squares we need to minimize the following quantity:
[tex] min_{b \in R^{p+1}} ||y-Xb||^2 [/tex]
And for this case if we find the best estimator for [tex]b[/tex] we got:
[tex] \hat b= (X'X)^{-1} X' y[/tex]
And the fitted values can be written as:
[tex]\hat y = X \hat b[/tex]
[tex]\hat y = X(X'X)^{-1}X' y= Hy[/tex]
Where [tex] H= X(X'X)^{-1} X'[/tex]
In order to see if any coefficnet is significant we can conduct the following hyppthesis:
Null hypothesis: [tex]b_j = b_j^*[/tex]
Alternative hypothesis: [tex] b_j \neq b_j^*[/tex]
For some j in {0,1,....,p}
We need to use the following statistic:
[tex] Z =\frac{\hat b_j -b_j^*}{\sigma_{b_j}}[/tex]
Where [tex] Z \sim N(0, 1)[/tex]
And [tex]\sigma_{b_o} , \sigma_{b_j}[/tex] are square roots of the diagonals of the diagomals of [tex] V(\hat b) = \sigma^2 (X'X)^{-1}[/tex]
