Appunti Econometrics

COEFF. CORREL - R²

• UNIT FREE
• Correlation -> NO CAUSATION
• Measure no. of STAND. DEVIAT. that y changes by when X changes by 1 SD.

R²

• Square
• 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, x₂, ...)
• Measures the fraction of variable Y_i explained by X_i (Regressors)
• Low R² -> the variation of x explain only small part of the variation in y

Other important factors influence y

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

_{(1 - R²)}

R̅² = adjusted R²

• R̅² ² < R²
• Add 1 Regressor have 2 opposite effects in R²

_{(t TEST)}

• IF R̅² increases -> NOT MEANS that an ADDED VARIAB is STAT. SIGN.
• High R̅² -> NOT mean that regressor are true cause of Y and have an appropriate set of regressors
• Low R̅² -> NOT IMPLY Necessor. OMITT. VAR. BIAS

LINEAR REGRESSION (1R)

Ŷ_i = β̂₀ + β̂₁X_i

• β̂₀ (INTERCEPT)
  • Expected Value Ŷ when X = 0
  • E (Y / x = 0)
• β̂₁ (dummy variable)
  • difference to model
  • Slope Coefficient: its effect on Y when X is increased by 1
  • β̂₁ = ΔŶ / ΔX
  • β̂₁ = Β̂₁ ≥ 1

R²

• Measures of FIT (Coefficient of determination)
• 0 ≤ R² ≤ 1
• Measures the fraction of the variance Yi that is explained by Xi
• X_i good predictions _Yi

Assumptions

• I error term = U LSA: E(μ / x) = 0
• (If ≠ 0 = Bias)
• IV LSA: HOMOSKEDASTIC
• E(μ² / x) = constant

(Summary)

Hyp. Testing

Significance Test 1

• H₀: β̂₁ = 0 NOSIGN.
• H₁: β̂₁ ≠ 0 2 sided (SIG)
• H₁: β̂₁ ≥ 0 1 sided

• t = β̂₁ / SE (β̂₁)

• Decision: Reject H₀
• (β̂₁ ≠ SIGNIF)

|t| > t_α P < α

0 ∉ C.I.

MULTI REGRESSION LIN. (1 HLR)

Ŷ_i = β̂₀ + β̂₁X₁ + β̂₂X₂ + ...

K regressors X_i

β̂₀ NO SIGNIF.

holding all other things constant

R̅² < R²

• No PERFECT MULTICOLLINEARITY (dummy variable trap)

Hyp. Testing

• Joint HYD
• H₀: β̂₂ = 0
• H₁: at least 1 different by zero

F-Test

Confidence Interval

Interval that has (95%) probability of containing the true value of β̂₁

β̂₁ ± t_α SE(β̂₁)

p-value

• α
• C.I
• 1 sided
• 2 sided

• 1% 99% 2.58 1.28
• 5% 95% 1.645 1.96
• 10% 90% 1.29 1.64

Y = β₀ + β₁X

se c’è BIAS Z

de non ho incluso nel modello

Y = β₀ + β₁X + (λ + β₂ Z)

= β₀ + β₁X + μ

Z è OMITT.BIAS

CORR(z, x) ≠ 0

⟷ E (z|x) ≠ 0

Adesso voglio calcolare.

CORR(μ, x) = cov(β₂z, x) = β₂ con(z, x)

≠ 0 ≠ 0

E(μ/x) ≠ 0

Z è ENDOGENA

NOTA:

CORR(μ, x) = E(μ/x) = 0

Z è ESOGENA

X NO CORR con μ

General IV model Conditions for valid Instruments:

1. Instrument exogeneity
   • corr(_zm,u)=0
2. Instrument Relevance:
   • general: _{W hat, W2, ..., Wm, 1} are not perfectly multicollinear: the second stage regression could be run using the predicted values from the population first stage regression.

IV Regression Assumptions

1. E(u / W₂, ..., W_r, Z₁, ..., _Zm)=0
2. (Y, X₁, ..., X_k, W₁, ..., W_r, Z₁, ..., _Zm) are I.I.D.
3. X’s, W’s, Z’s and Y have nonzero, finite 4th moments
4. Instruments (Z₁, ..., _Zm) are valid.

TSLS is normally distributed.

I PANEL DATA (TUTTI I MODELLI)

I

STRICT EXOGENEITY:

E(μi / xi1, xi2, ...) = 0

Kapranas

tutti i repressn x' _i sono considerati

Esogeni ⇒ NO ENDOGENI

NO CORRELATI CON IL TERMINE DI ERRORE μi

Modello

1

FIXED EFFECT = ogni individuo/entità ha una intercetta diversa (≠ αi) diversa nel modello di popolazione

2

E(αi / xi) ≠ 0

(NO ORTOGONALI)

xit and xit CAN BE CORRELATED

Assumptions

1. Linearity

   dependent variab y is linearly related to the coefficients

2. Zero Condit. Mean

   E(ε_i | x_i) = 0

   No relationship between the error term ε_i and the indep. variab. x_i.

   estimate biased

3. No Perfect Multicollinearity

   high degree of collinearity between 2 explanatory variables

   (DUMMY VAR. TRAP)

4. Homoskedasticity

   var (ε_i | x_i) = const

   var (y_i | x_i) = const

   OLS LOSS EFFICIENCY (NO LONGER BLUE)

5. No Serial Correl (Residuals Indip)

   residuals must be statistically independent and uncorrelated from each other

   cov (ε_i, ε_j) = 0

   introduce controls for seasonality and time indicating a set of N-1 Dummy Variable