Appunti Econometrics

COEFF. CORREL - R²

* _y = 5_x / 5_x·5y → UNIT FREE

1 ≤ k ≤ 1

Correlation ⇒ No Causation

measure no of stand. deviat. thaty changes by when X changes by 1 SD.

* R² = square

* qtadato coeff. correlazione

_c = √R²

0 ≤ R² ≤ 1

→ measure of goodness of fit → semper positiva

• percentage of variance of the depend. variab. Y explained by the
• measures the fraction of variable Y explained by x
• Low R²

  small part of x explain only

  an improvement of the FIT of the model.

R̅² = adjusted R²

• add a regressor has 2 opposite effect in R²
• can be negative

IF R² increases

• not means that an added variate is stat.sign. (t) test
• high R²

• not mean that the regressor are true cause of Y and have
• appropriate set of regressors (not imply no omitt. var. bias)
• low R̅²
• not imply necesario. OMISS. VAR. BIAS

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 depend. 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)

XXXXX ⇒

• Low R² ⇒ factors omitted in the variation of X explain only small part of the variation in Y

R² increases if a regressor is added, even if this NOT MEAN an improvement of the FIT of the model.

• R̅² = adjusted R²
• R̅² ≤ R²
• R̅² = 1 - (n - 1 / n - k - 1) · (1 - R²)

• add a Regressor has 2 opposite effects in R² can be NEGATIVE

IF R² increases ⇒ NOT MEANS that an ADDED VARIABLE IS STAT. SIGN. (t Test)

• high R² ⇒ NOT mean that regressor are true cause of Y and have an appropriate set of regressor

(NOT IMPLY NO OMITT. VAR. BIAS)

• low R̅² ⇒ NOT IMPLY Necesoz. OMITT. VAR. BIAS

LINEAR REGRESSION (1R)

= ₀ + ₁

Expected Value of when = 0̂₀

0 = Expected Value of when = 0(slope coefficient) ➝ it is the effect on when is increased by 1

₁ = /

MULTI REGRESSION LIN. ( MLR)

measures ∈ good prediction of

if ⇔➝ if ⇔

Hyp. Testing

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

= (̂₁ - ) / (̂₁)

Confid. Intervalis the interval that has (95%) probability of containing the true value of ₁

̂₁ ±

• Hyp. Test. 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)