WZORY EKONOMETRIA – KOLOKWIUM II

$$S_{U} = \ \sqrt{\frac{\sum_{}^{}u_{t}^{2}}{n - k - 1}}$$

$$u_{t} = y - \hat{y}$$

D²(a) = S_u² • (X^T • X)⁻¹

$$V = \frac{S_{u}}{\overset{\overline{}}{y}} \bullet 100\%$$

$$\varphi^{2} = \frac{\sum_{}^{}u_{t}^{2}}{\sum_{}^{}\left( y - \overset{\overline{}}{y} \right)^{2}}$$

$$R^{2} = \frac{\sum_{}^{}\left( \hat{y} - \overset{\overline{}}{y} \right)^{2}}{\sum_{}^{}\left( y - \overset{\overline{}}{y} \right)^{2}}$$

φ² + R² = 1

H₀ : R = 0 

H₁ : R ≠ 0

$F = \frac{R^{2}}{1 - R^{2}} \bullet \frac{n - k - 1}{k}$ m₁ = k,      m₂ = n − k − 1

α i m₁,   m₂,   st. sw., F^* F > F^*  H₀ odrzucamy.

H₀ : α_i = 0 

H₁ : α_i ≠ 0

$$I_{i} = \frac{\left| a_{i} \right|}{S(a_{i})}$$

α i n − k − 1 st.sw., I^*

I > I^*  H₀ odrzucamy.

H₀ :  F_u = F_n

H₁ :  F_u ≠ F_n

$$W = \frac{\left\lbrack \sum_{}^{\left\lceil \frac{n}{2} \right\rceil}{a_{n - t - 1}\left( u_{n - t + 1} - u_{t} \right)} \right\rbrack^{2}}{\sum_{}^{}\left( u_{t} - \overset{\overline{}}{u} \right)^{2}}$$

n,  γ → W^*

W ≥ W^*,,  brak podst. do odrzucenia H₀

H₀ :  δ²u₁ =  δ²u₂

H₁ :  δ²u₁ <  δ²u₂

$$F = \frac{{S_{u}^{2}}_{2}}{{S_{u}^{2}}_{1}}$$

$${S_{u}^{2}}_{1} = \frac{1}{n_{1} - k - 1} \bullet \sum_{}^{}\left( u_{t} - \overset{\overline{}}{u_{1}} \right)^{2}$$

$${S_{u}^{2}}_{2} = \frac{1}{n_{2} - k - 1} \bullet \sum_{}^{}\left( u_{t} - \overset{\overline{}}{u_{2}} \right)^{2}$$

α,   m₁ = n₂ − k − 1;   m₂ =  n₁ − k − 1 ;   →  F^*

F ≤ F^* brak podst. do odrzucenia H₀

H₀ :  ρ₁ = 0

H₁ :  ρ₂  ≠ 0

$$d = \ \frac{\sum_{t = 2}^{n}\left( u_{t} - u_{t - 1} \right)^{2}}{\sum_{t = 1}^{n}u_{t}^{2}}$$

γ, n → d_l,  d_u

d > d_u brak podst. odrzucenia H₀

d < d_l H₀odrzucic

d_l ≤ d ≤ d_u test nie rozstrzyga

$$H_{0}:\ \hat{Y} = \rho\left( x_{1},\ldots,x_{n} \right)$$

$$H_{0}:\ \hat{Y} \neq \rho\left( x_{1},\ldots,x_{n} \right)$$

n₁, n₂,  S

$$\frac{\gamma}{2} \rightarrow S_{1}^{*}$$

$$1 - \frac{\gamma}{2} \rightarrow S_{2}^{*}$$

S₁^* < S < S₂^* brak podst. od odrzucenia H₀