Appunti Econometrics

Coeff. correl

R²* _y = 5_x / 5_x·5y → UNIT FREE 1 ≤ k ≤ 1

Correlation ⇒ No causation

Measure number of standard deviations that y changes by when X changes by 1 SD.

* R² = square quadato coeff. correlazione_c = √R² 0 ≤ R² ≤ 1 → Measure of goodness of fit → Sempre positiva.

Percentage of variance of the dependent variable Y explained by the model. Measures the fraction of variable Y explained by x.

Low R² means that only a small part of x explains an improvement of the fit of the model.

R̅² = adjusted R²

Adding a regressor has two opposite effects in R² and can be negative.

If R² increases, it does not mean that an added variable is statistically significant (t test).

High R² does not mean that the regressors are the true cause of Y and have an appropriate set of regressors (this does not imply no omitted variable bias).

Low R̅² does not imply necessarily omitted variable bias.

Linear regression

COEFF. CORREL - R² μ_x = 5x 5x.5y UNIT FREE -1 ≤ r ≤ +1 CORRELATION ⇒ NO CAUSATION Measure no of STAND. DEVIAT. that y changes by when X changes by 1 SD.

R² = square quadrato coeff. correlazione χ_t= √R² 0 ≤ R² ≤ 1 Measure of goodness of fit ⇒ Sempre positiva

Percentage of variance of the dependent variable y explained by the model (by variation of the variables, x₁, x_k, ...). Measures the fraction of variable Y explained by X_i (Regressors).

Low R² ⇒ Factors omitted in the variation of X explain only a small part of the variation in Y.

R² increases if a regressor is added, even if this does not mean an improvement of the fit of the model.

R̅² = adjusted R² R̅² ≤ R²

R̅² = 1 - (n - 1 / n - k - 1) · (1 - R²) Adding a regressor has two opposite effects in R² and can be negative.

If R² increases ⇒ it does not mean that an added variable is statistically significant (t Test).

High R² ⇒ Does not mean that regressors are the true cause of Y and have an appropriate set of regressors (does not imply no omitted variable bias).

Low R̅² ⇒ Does not imply necessary omitted variable bias.

Linear regression details

LINEAR REGRESSION (1R) = ₀ + ₁Expected Value of when = 0̂₀0 = Expected Value of when = 0 (slope coefficient) ➝ It is the effect on when is increased by 1₁ = /

Multi regression lin. (MLR)

Measures ∈ good prediction of if ⇔➝ if ⇔

Hypothesis testing

Signif. Test 1₀: ̂₁ = 0 NO SIGN.₁: ̂₁ ≠ 0 2sided (SIG.) = (̂₁ - ) / (̂₁)

Confidence interval is the interval that has 95% probability of containing the true value of ₁̂₁ ±

Hypothesis testing I

p-value NON LINEAR REGRESSION (heteroskedasticity) _ΔX

1. Effects on y of a change in X depends on _ΔX (INITIAL POINT X₀)

Δy = ƒ(cx₀ + Δx) - ƒ(cx₀)

Polynomial quadratic

Y = β₀ + β₁X + β₂X²

H₀: β₂ = 0

H₁: β₂ ≠ 0

ΔY = {f(cx₀ + Δx)} - {f(cx₀)} = β₁(x₀ + x) + β₂(cx₀ + x)² - β₁X - β₂X²

^{Linear Model} t = β₂ / se(β₂)