Panel Data Econometrics: Conditional Mean, Projection, and Regression, Slides of Econometrics and Mathematical Economics

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Econometric Analysis of Panel Data

Econometric Analysis of Panel Data

2. Statistical Models

Models

 Conditional mean function: E[y | x ]

 Other conditional characteristics – what is ‘the

model?’

 Conditional variance function: Var[y | x ]  Conditional quantiles, e.g., median [y | x ]  Other conditional moments

Using the Model

 Understanding the relationship:

 Estimation of quantities of interest such as elasticities

 Prediction of the outcome of interest

 Control of the path of the outcome of interest

Projection and Regression

 Linear projection is not the regression, and is

not the Taylor series.

Example:

α β ≤ ≤

f(y|x)=[1/λ(x)]exp[-y/λ(x)] λ(x)=exp( + x)=E[y|x] x~U[0,1]; f(x)=1, 0 x 1

0 1

0

1

ˆg(x|x=E[x])=δ +δ (x-1/2)

exp( / 2)

exp( / 2)

δ = α + β

δ = β α + β

The Linear Projection

1 x x (^0) x x 1 1 0 0

g*(x)=E[y]+ Covx,y Var[x] E[x]=1/2 Var[x]=1/ E[y]=E E[y|x]=E [exp(α+βx)]= exp(α+βx)1dx Cov[x,y]=Cov[x,E[y|x]]=E [xE[y|x]]-E[x]E E[y|x] = x exp(α+βx)1dx 1 exp(α+βx)1dx 2 Us

−

∫

∫ ∫ 2

0 1

ing exp( x)dx=[exp( x)]/ and xexp( x)dx={[exp( x)]/ }( x 1), γ =exp( )/ -1], γ [exp( )/ ]{exp( )(1/2 - 1/ )+1/2+1/ }

β β β β β β β − α β β = α β β β β

∫ ∫

Omitted details of the proof are left for the reader. Docsity.com

(FYI)

Calc ; alpha=1;beta=2 $ Samp ; 1-200$ Crea ; x=trn(0,.005)$ Crea ; ey_x = exp(alpha + betax) $ Calc ; gamma1 =exp(alpha)/beta ( exp(beta) (.5-1/beta) +.5+1/beta )12 $ Calc ; gamma0 =exp(alpha)/beta * (exp(beta)-1) $ Calc ; delta1 = betaexp(alpha+beta/2) $ Calc ; delta0 = exp(alpha+beta/2) $ Crea ; projectn=gamma0+gamma1(x-.5)$ Crea ; taylor =delta0 +delta1*(x-.5)$ Plot ; lhs=x;endpoints=0,1;rhs=ey_x,projectn,taylor;fill$

What About the Linear Projection?

 What we do when we linearly regress a variable

on a set of variables

 Assuming there exists a conditional mean

 There usually exists a linear projection. Requires finite variance of y.  Approximation to the conditional mean

 If the conditional mean is linear

 Taylor series equals the conditional mean  Linear projection equals the conditional mean

Conditional Mean and Projection

Notice the problem with the linear approach. Negative predictions.

Doctor Visits: Conditional Mean and Linear Projection

INCOME

.

1.

2.

3.

4.

-.26 0 4 8 12 16 20 CONDMEAN PROJECTN

DocVisit

Most of the data are in here

This area is outside the range of the data

Partial Effects

 What did the model tell us?

 Covariation and partial effects

 Marginal Effects: Effect on what?????

 For continuous variables

 For dummy variables

 Elasticities: ε(x)=δ(x) * x / E[y|x]

∂ E[y|x]/ x=∂ δ(x), usually not coefficients

E[y|x,d=1]-E[y|x,d=0]

APE and PE at the Mean

2 2 x

δ(x)= E[y|x]/ x, =E[x] δ(x) δ( )+δ ( )(x- )+(1/2)δ ( )(x- ) + E[δ(x)]=APE δ( ) + (1/2)δ ( )

∂ ∂ μ ≈ μ ′^ μ μ ′′ μ μ ε ≈ μ ′′ μ σ

Implication: Computing the APE by averaging over observations (and counting on the LLN and the Slutsky theorem) vs. computing partial effects at the means of the data. In the earlier example: Sample APE = -. Approximation = -.

The Linear Model

 y = X β +ε , N observations, K columns in X ,

including a column of ones.

 Standard assumptions about X  Standard assumptions about ε|X  E[ε|X]=0, E[ε]=0 and Cov[ε,x]=

 Regression?

 If E[ y | X ] = X β  Approximation: Then this is an LP, not a Taylor series.

Endogeneity

 Definition: E[ε| x ]≠

 Why not?

 Omitted variables  Unobserved heterogeneity (equivalent to omitted variables)  Measurement error on the RHS (equivalent to omitted variables)  Simultaneity (?)

Structure and Regression

 Simultaneity?

 y=xβ+ε, x=δy+u. Cov[x, ε]≠

 xβ is not the regression?  What is the regression?  Reduced form: Assume ε and u are uncorrelated.  y = [β/(1- βδ)]u + [1/(1- βδ)]ε  x= [1/(1- βδ)]u + [δ /(1- βδ)]ε  Cov[x,y]/Var[x] =λ

 The regression is y = λx + v, where E[v|x]=

2 2 2 2 2 2 2 2 2

[ ] /[ ] (1 )(1 / ) where w= /[ ]

u u w w u u

ε ε ε

= βσ + δσ σ + δ σ = β + − δ σ σ + δ σ