Wzory 2

1. Macierzowa postać modelu

$$\begin{bmatrix} 1 & 0 \\ {- \beta}_{21} & 1 \\ \end{bmatrix}\begin{bmatrix} Y_{1} \\ Y_{2} \\ \end{bmatrix} = \begin{bmatrix} \gamma_{11} & 0 & \gamma_{13} \\ 0 & \gamma_{22} & 0 \\ \end{bmatrix}\begin{bmatrix} Z_{1} \\ Z_{2} \\ Z_{3} \\ \end{bmatrix} + \begin{bmatrix} \varepsilon_{1} \\ \varepsilon_{1} \\ \end{bmatrix}$$

β Γ

1. Parametry strukturalne

   1. Równanie zrewidowane:

$$Y_{2} = \beta_{21}{\hat{Y}}_{1} + \gamma_{21}Z_{1} + \gamma_{23}Z_{3} + \varepsilon_{2}$$

1. $c_{2} = \begin{bmatrix} \begin{matrix} b_{21} \\ c_{21} \\ \end{matrix} \\ c_{22} \\ \end{bmatrix}$ wektor estymatorów wektora parametrów strukturalnych $\gamma = \begin{bmatrix} \begin{matrix} \beta_{21} \\ \gamma_{21} \\ \end{matrix} \\ \gamma_{23} \\ \end{bmatrix}$

2. Wektor estymatorów:

c_j = (A_j^TA_j)^-1 A_j^Ty_j

$$A_{j} = \left\lbrack {\hat{y}}_{i}\ Z_{j} \right\rbrack$$

$$\left( {A_{j}}^{T}A_{j} \right)^{- 1} = \ \frac{1}{\left| {A_{j}}^{T}A_{j} \right|}\left( {A_{j}}^{T}A_{j} \right)^{D}$$

1. Postać teoretyczna:

$${\hat{Y}}_{2} = b_{21}{\hat{Y}}_{1} + c_{21}Z_{1} + c_{23}Z_{3}$$

1. Estymator wariancji składnika losowego: $S^{2}\left( e_{2} \right) = \frac{1}{n - k_{2}} \bullet {(y_{2}}^{T}y_{2} - \ {c_{2}}^{T}{A_{2}}^{T}y_{2})$

$${y_{2}}^{T}y_{2} = \ \left\lbrack \sum_{t = 1}^{n}{y_{t2}}^{2} \right\rbrack$$

1. Błąd standardowy składnika losowego: $S\left( e \right) = \sqrt{S^{2}(e)}$

2. Macierz ocen wariancji i kowariancji estymatorów: S(c₂,c₂) = S²(e₂)(A₂^TA₂)⁻¹

3. ${\hat{Y}}_{2} = \ \frac{b_{21}{\hat{Y}}_{1}}{\left( S\left( b_{21} \right) \right)} + \ \frac{c_{21}Z_{1}}{\left( S\left( c_{21} \right) \right)} + \frac{c_{23}Z_{3}}{\left( S\left( c_{23} \right) \right)}\text{\ \ \ }\frac{\ }{\left( S\left( e_{2} \right) \right)}$

1. Miary unormowania dopasowania modelu do danych empirycznych:

   1. $\left\{ \begin{matrix} {\Phi_{2}}^{2} = \frac{{y_{2}}^{T}y_{2} - \ {c_{2}}^{T}{A_{2}}^{T}y_{2}}{{y_{2}}^{T}y_{2} - n{{\overset{\overline{}}{y}}_{2}}^{2}}\text{\ \ \ \ \ } - \text{wsp.zbie}z\text{ny} \\ {R_{2}}^{2} = 1 - {\Phi_{2}}^{2}\ - \text{wsp.determinacji} \\ \end{matrix} \right.\ $

   2. ${\overset{\overline{}}{y}}_{2} = \ \frac{1}{n}\ \sum_{t = 1}^{n}y_{t2}$

2. Prognoza punktowa:

   1. ${\hat{y}}_{\tau 2} = b_{21}{\hat{y}}_{\tau 1} + c_{21}Z_{\tau 1} + c_{23}Z_{\tau 3}$

   2. Standardowy błąd prognozy: $S_{\tau} = S\left( e \right)\sqrt{1 + {\tilde{z}}_{\text{τ\ }}{({A_{2}}^{T}A_{2})}^{- 1}{{\tilde{z}}_{\tau}}^{T}}$

   3. ${\tilde{z}}_{\text{τ\ }} = \begin{bmatrix} {\hat{y}}_{\tau 1} & z_{\tau 1} & z_{\tau 3} \\ \end{bmatrix}$

   4. Względny błąd prognozy: $\ V_{\tau 2} = \frac{S_{\tau}}{|{\hat{y}}_{\tau}|} \bullet 100\%$