Appunti schematici Econometria

GOLDFELD

• based
• organ
• we est
• we c

WHITE

• OLS e
• auxil
• we t
• we

AR in ge

• mu

DYNAMIC M

• AUTOREGR

TIME SERI

• It includes

FDL (fi

Yt: d0 + d0

AR

ACF PQL 0

PCF 0.0 0

Linear Bivariate Model

y = β₁ + β₂x₂ + μ

Linear Model Hypothesis

E(y | x) = β₁ + βx

var(y | x) = σ²

cov(yi ,yi) = 0

yi ~ N(β₁+β₂x₂, σ²)

1. μ to β (E(μ | x) = 0)
2. homoscedasticity, var(μ | x) = σ²
3. corr(μj, μi) = 0
4. strict exogeneity, corr(μi, xi) = 0
5. μ ~ N(0, σ²) n.d.

to derive the OLS estimators for β₁, β₂ we solve:

min β₁, β₂ Σ(yᵢ - β₁ - β₂x)²

• ∂Σε²/∂β₁ - 2Σ(y - β₁ - β₂x) = 0
• ∂Σε²/∂β₂ - 2xΣ(y - β₁ - βx)

Σy = Nβ₁ + b₂Σx

Σxyᵢ = b₁Σx + b₂Σx²

b₁ = ȳ - b₂x

we use this here

bi = ȳt + b₂x̄

b₂[Σxᵢ² - ȳnΣx] Σxy - ȳnΣx̄Σy

bi = ȳt + b₂x

b₂ = Σxy / Σx²

Σxy = ȳtΣx̄ + b₂x̄Σxₓ + b₂Σx̄²

TSS Decomposition

-${\Sigma y_{i}^{2}, r^{2}Σy_{i} + Σe²}$

TSS ESS RSS → 1, 1 - RSS/TSS

Non Linear Relationships

• Squared trasf. a quadratic function! if change in x percent change in E(y) is 2b(x / y)
• log. trasf. we then consider the exp(log y) a + exp(bx) e:bᴮx
• log.log trasf. elastacity β log y = a + (log x / β)
• cubic (reciprocal)

the elastacity

• measures the error of fit = ^√Σεᵢ²/₍n-2₎ = ^√1/n-2 Σεᵢ²
• spread or distribution of μ
• average size of OLS residuals

restricted regression

A: A₁ regression model with all covariates, where ESS: Ŷ'Ŷ - ê'ê = TSS-RSS

A₂ regression model with only the constant, where ESS: 0; RSS: TSS: Ŷ'Ŷ

We use the test F = [(e'e-e_r'e_r)/q] / [e_r'e_r/(n-k-r)] = (q, n-k-r)

• restricted RSS non restricted RSS
• n: n° rows in matrix R
• q: n° elements in null hypothesis
• k: diff. between n. coeff. considered in the restr.-non restr.

FITTING THE RESTRICTED REGRESSION

Given the Ŷ: β₁ + β₂x₂ + β₃x₃ +u and the H₀: β₂ + β₃ = 1, we have the restricted model

Ŷ - β₁ + β₂x₂ + (1 - β₂)x₃ + u , OLS obtained sub b₀

Minimization problem: φ: (y - xb_r)' (y - x If we have autocorrelation instead, not diagonal, we have : E (μ_tμ_t+S)

 -> In homosch, Ω are symmetric with respect to and independent from

  [es = cov(u_t, u_t+s) / √(voi(ub) voi(uts))] (autocorrelatious) es: %/√%

The variance matrix is:

[ ^{y₁ 0 0} _{0 y2 0 0 0 yn-2} ] ²ρ_s[ ^{i ρ ...} _{ρ¹ ...}] ²

OLS are: unbiased ✔ consistent ✔ inefficient ❌ incorrect std. errors ❌

(correct v. matrix isn’t δ’δ ^-1 (x’x)

but: δ ^’x^’x’^x’ Ω x’x’)

• We can find a P matrix => Ω^-1ρρ, as BNL (x’x’Ωx)x’Ω^-1ŷ

(energy_ML / (y - xb0LS) Ω^-1(y-xb0ml)

(x’px’px) x’px’y ⟹ [(px)’(px) (px)’(py)]

we obtain y