Rlm_matrix

RLM (MATRIZ) {\(y}_{t}=\beta_{1}+\beta_{2}.x_{t,2}+\beta_{3}.x_{t,3}+...+\beta_{k}.x_{t,k}+\mu_{t} t=1,2,...,n [X]= \begin{bmatrix} 1 & x_{1,2} & x_{1,3} & ... & x_{1,k} \\ 1 & x_{2,2} & x_{2,3} & ... & x_{2,k} \\ . & . & . & & . \\ . & . & . & ... & . \\ 1 & x_{n,2} & x_{n,3} & ... & x_{n,k} \end{bmatrix}_{n x K} então temos: \{x}_{t} = (1,x_{t2},x_{t3},...,x_{tk})^{t} \beta=(\beta_{1},\beta_{2},\beta_{3},...,\beta_{k})^{t} \{y}_{t}=x_{t}.\beta+\mu_{t} (t=1,...,n) PARA DETERMINAR (\hat{\beta}), temos: \hat{\beta}=(X'X)^{-1}.X'.y OBS: Se (X_{n x K}) -> (X')_{k x n} => X'.X=A_{k x k} => (INV(A))_{k x k} => (INV(A)).(X') = [B]_{k x n} => [B](Y) = [C]_{k x 1} Exemplo: BD-ANDY t Vendas Precos Propog1 Propog2 1 73.2 5.69 1.3 1.69 2 71.8 6.49 2.9 8.11 ... ... ... ... ... 75 75 6.05 2.2 4.84 Dessa forma: \begin{bmatrix} y_{75 x 1} \\ X_{75 x 4} \end{bmatrix} Como \hat{\beta}=(X'.X)^{-1}.X'.y ENTÃO \{X'.X}\_{4 x 4}\{(4 x 4)}=A_{4 x 4}, X'_{4 x 75} [B]_{4 x 75} => [B](Y) = [C]= \begin{bmatrix} \beta_{1} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4} \end{bmatrix}_{4 x 1} +- ERRO PADRÃO (6) 1.) SQR=\sum{\hat{\mu}_{t}^{2}} = \sum{(x_{t}-x_{t}.\hat{\beta})^{2}} 2.) VAR(ERRO)=\frac{SQR}{(n-k)} 3.) DP(ERRO)=\sqrt{VAR(ERRO)} DP(ERRO) = ERRO PADRÃO VAR(\hat{\beta}): 1.) Dado (X'.X)^{-1}_{k x k} 2.) Matrizz VAR-COVAR: VAR(\hat{\beta}_{i})=[6^{2}](X'.X)^{-1}_{VAR(ERRO)}=[N]_{k x k} 3.) Como VAR(\hat{\beta}): é dado por (V11,V22, ..., Vkk) Diagonal Principal 4.) Dado VAR(\hat{\beta}) => ERRO(\hat{\beta}_{i})=\sqrt{VAR(\hat{\beta}_{i})}