Part 2: Basic Econometrics [ 1/51]

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business

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www.oft.gov.uk/shared_oft/reports/Evaluating-OFTs-work/oft1416.pdf

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Econometrics

Theoretical foundations Microeconometrics and macroeconometrics Behavioral modeling

Statistical foundations: Econometric methods

Mathematical elements: the usual ‘Model’ building – the econometric model

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Estimation Platforms Model based

Kernels and smoothing methods (nonparametric)

Semiparametric analysis Parametric analysis

Moments and quantiles (semiparametric) Likelihood and M- estimators (parametric) Methodology based (?)

Classical – parametric and semiparametric Bayesian – strongly parametric

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Trends in Econometrics Small structural models vs. large scale multiple

equation models Non- and semiparametric methods vs. parametric Robust methods – GMM (paradigm shift? Nobel prize) Unit roots, cointegration and macroeconometrics Nonlinear modeling and the role of software Behavioral and structural modeling vs. “reduced

form,” “covariance analysis” Pervasiveness of an econometrics paradigm Identification and “causal” effects

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Objectives in Model Building Specification: guided by underlying theory

Modeling framework Functional forms

Estimation: coefficients, partial effects, model implications

Statistical inference: hypothesis testing Prediction: individual and aggregate Model assessment (fit, adequacy) and evaluation Model extensions

Interdependencies, multiple part models Heterogeneity Endogeneity

Exploration: Estimation and inference methods

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Regression Basics

The “MODEL” Modeling the conditional mean – Regression

Other features of interest Modeling quantiles Conditional variances or covariances Modeling probabilities for discrete choice Modeling other features of the population

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Application: Health Care UsageGerman Health Care Usage Data, 7,293 Individuals, Varying Numbers of PeriodsData downloaded from Journal of Applied Econometrics Archive. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice.  There are altogether 27,326 observations.  The number of observations ranges from 1 to 7.  (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987).  (Downloaded from the JAE Archive)Variables in the file are DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT =  health satisfaction, coded 0 (low) - 10 (high)   DOCVIS =  number of doctor visits in last three months HOSPVIS =  number of hospital visits in last calendar year PUBLIC =  insured in public health insurance = 1; otherwise = 0 ADDON =  insured by add-on insurance = 1; otherswise = 0 HHNINC =  household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC =  years of schooling AGE = age in years MARRIED = marital status

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Household Income

Kernel Density Estimator Histogram

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Regression – Income on Education

----------------------------------------------------------------------Ordinary least squares regression ............LHS=LOGINC Mean = -.92882 Standard deviation = .47948 Number of observs. = 887Model size Parameters = 2 Degrees of freedom = 885Residuals Sum of squares = 183.19359 Standard error of e = .45497Fit R-squared = .10064 Adjusted R-squared = .09962Model test F[ 1, 885] (prob) = 99.0(.0000)Diagnostic Log likelihood = -559.06527 Restricted(b=0) = -606.10609 Chi-sq [ 1] (prob) = 94.1(.0000)Info criter. LogAmemiya Prd. Crt. = -1.57279--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+-------------------------------------------------------------Constant| -1.71604*** .08057 -21.299 .0000 EDUC| .07176*** .00721 9.951 .0000 10.9707--------+-------------------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level.----------------------------------------------------------------------

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Specification and Functional Form----------------------------------------------------------------------Ordinary least squares regression ............LHS=LOGINC Mean = -.92882 Standard deviation = .47948 Number of observs. = 887Model size Parameters = 3 Degrees of freedom = 884Residuals Sum of squares = 183.00347 Standard error of e = .45499Fit R-squared = .10157 Adjusted R-squared = .09954Model test F[ 2, 884] (prob) = 50.0(.0000)Diagnostic Log likelihood = -558.60477 Restricted(b=0) = -606.10609 Chi-sq [ 2] (prob) = 95.0(.0000)Info criter. LogAmemiya Prd. Crt. = -1.57158--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+-------------------------------------------------------------Constant| -1.68303*** .08763 -19.207 .0000 EDUC| .06993*** .00746 9.375 .0000 10.9707 FEMALE| -.03065 .03199 -.958 .3379 .42277--------+-------------------------------------------------------------

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Interesting Partial Effects----------------------------------------------------------------------Ordinary least squares regression ............LHS=LOGINC Mean = -.92882 Standard deviation = .47948 Number of observs. = 887Model size Parameters = 5 Degrees of freedom = 882Residuals Sum of squares = 171.87964 Standard error of e = .44145Fit R-squared = .15618 Adjusted R-squared = .15235Model test F[ 4, 882] (prob) = 40.8(.0000)Diagnostic Log likelihood = -530.79258 Restricted(b=0) = -606.10609 Chi-sq [ 4] (prob) = 150.6(.0000)Info criter. LogAmemiya Prd. Crt. = -1.62978--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+-------------------------------------------------------------Constant| -5.26676*** .56499 -9.322 .0000 EDUC| .06469*** .00730 8.860 .0000 10.9707 FEMALE| -.03683 .03134 -1.175 .2399 .42277 AGE| .15567*** .02297 6.777 .0000 50.4780 AGE2| -.00161*** .00023 -7.014 .0000 2620.79--------+-------------------------------------------------------------

2

[ | ]2 Age Age

E IncomeAge

Age

x

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Function: Log Income | Age Partial Effect wrt Age

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A Statistical Relationship A relationship of interest:

Number of hospital visits: H = 0,1,2,… Covariates: x1=Age, x2=Sex, x3=Income,

x4=Health Causality and covariation

Theoretical implications of ‘causation’ Comovement and association Intervention of omitted or ‘latent’ variables Temporal relationship – movement of the

“causal variable” precedes the effect.

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(Endogeneity)

A relationship of interest: Number of hospital visits: H = 0,1,2,… Covariates: x1=Age, x2=Sex, x3=Income,

x4=Health Should Health be ‘Endogenous’ in this model?

What do we mean by ‘Endogenous’ What is an appropriate econometric method

of accommodating endogeneity?

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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

Conditional probabilities: P(y|x) What is the sense in which “y varies with x?”

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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

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Application: Doctor Visits German individual health care data: N=27,236 Model for number of visits to the doctor:

Poisson regression (fit by maximum likelihood) E[V|Income]=exp(1.412 - .0745 income) OLS Linear regression: g*(Income)=3.917 - .208

incomeHistogram for Number of Doctor Visits

Freq

uen

cy

DOCVIS

0

2788

5576

8364

11152

0 4 8 12162024283236404448525660646872768084889296100104108112116120

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Conditional Mean and Linear Projection

Notice the problem with the linear projection. Negative predictions.

Projection and Conditional Mean Functions

INCOME

.66

1.57

2.48

3.40

4.31

-.26

5 10 15 200

CONDMEAN PROJECTN

Fu

nctio

n

This area is outside the range of the data Most of

the data are in here

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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 Linear projection equals the conditional mean

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Partial Effects

What did the model tell us? Covariation and partial effects: How does

the y “vary” with the x? Marginal Effects: Effect on what?????

For continuous variables

For dummy variables

Elasticities: ε(x)=δ(x) x / E[y|x]

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

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

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Average Partial Effects When δ(x) ≠β, APE = Ex[δ(x)]= Approximation: Is δ(E[x]) = Ex[δ(x)]? (no) Empirically: Estimated APE = Empirical approximation: Est.APE = For the doctor visits model

δ(x)= β exp(α+βx)=-.0745exp(1.412-.0745income)

Sample APE = -.2373 Approximation = -.2354 Slope of the linear projection = -.2083 (!)

x (x)f(x)dx

Ni=1 i

ˆ(1/ N) (x )

(x)

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APE and PE at the Mean

2

2x

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 ( )

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 = -.2373

Approximation = -.2354

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The Linear Regression Model

y = X+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]=0

Regression? If E[y|X] = X then X is the projection of y

on X

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Estimation of the Parameters Least squares, LAD, other estimators – we

will focus on least squares

Classical vs. Bayesian estimation of Properties Statistical inference: Hypothesis tests Prediction (not this course)

( )

s /N or /(N-K)

-1

2

b= X'X X'y

e'e e'e

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Properties of Least Squares

Finite sample properties: Unbiased, etc. No longer interested in these.

Asymptotic properties Consistent? Under what assumptions? Efficient?

Contemporary work: Often not important Efficiency within a class: GMM

Asymptotically normal: How is this established?

Robust estimation: To be considered later

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Least Squares Summary

2 1

2 1

2 1

ˆ plim =

N( ) N{0, [plim( / N)] }

N{ ,( / N)[plim( / N)] }

Estimated Asy.Var[ ]=s ( )

b, b

b X'X

b X'X

b X'X

d

a

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Hypothesis Testing Nested vs. nonnested tests

y=b1x+e vs. y=b1x+b2z+e: Nested y=bx+e vs. y=cz+u: Not nested y=bx+e vs. logy=clogx: Not nested y=bx+e; e ~ Normal vs. e ~ t[.]: Not nested Fixed vs. random effects: Not nested Logit vs. probit: Not nested x is endogenous: Maybe nested. We’ll see …

Parametric restrictions Linear: R-q = 0, R is JxK, J < K, full row rank General: r(,q) = 0, r = a vector of J functions, R (,q) = r(,q)/’. Use r(,q)=0 for linear and nonlinear cases

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Example: Panel Data on Spanish Dairy Farms

Units Mean Std. Dev. Minimum Maximum

OutputMilk

Milk production (liters) 131,107 92,584 14,410 727,281

InputCows

# of milking cows 22.12 11.27 4.5 82.3

InputLabor

# man-equivalent units 1.67 0.55 1.0 4.0

InputLand

Hectares of land devoted to pasture and crops.

12.99 6.17 2.0 45.1

InputFeed

Total amount of feedstuffs fed to dairy cows (Kg)

57,941 47,981 3,924.14 376,732

N = 247 farms, T = 6 years (1993-1998)

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Application y = log output x = Cobb douglas production: x = 1,x1,x2,x3,x4

= constant and logs of 4 inputs (5 terms) z = Translog terms, x1

2, x22, etc. and all cross

products, x1x2, x1x3, x1x4, x2x3, etc. (10 terms)

w = (x,z) (all 15 terms) Null hypothesis is Cobb Douglas, alternative is

translog = Cobb-Douglas plus second order terms.

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Translog Regression Model

H0:z=0

x

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Wald Tests

r(b,q)= close to zero? Wald distance function: r(b,q)’{Var[r(b,q)]}-1 r(b,q) 2[J] Use the delta method to estimate

Var[r(b,q)] Est.Asy.Var[b]=s2(X’X)-1

Est.Asy.Var[r(b,q)]= R(b,q){s2(X’X)-1}R’(b,q) The standard F test is a Wald test; JF =

2[J].

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1Wald= { }- 0 Var[ - 0] - 0z z zb b b

Closeto 0?

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Likelihood Ratio Test The normality assumption Does it work ‘approximately?’ For any regression model yi = h(xi,)+εi where

εi ~N[0,2], (linear or nonlinear), at the linear (or nonlinear) least squares estimator, however, computed, with or without restrictions,

2 2ˆlogL( , /N) (N/2)[1+log2 +log ]ˆ ˆˆ ˆ This forms the basis for likelihood ratio tests.

22

2

ˆ ˆ2[log ( ) log ( )]

ˆNlog [ ]

ˆ

unrestricted restricted

drestricted

unrestricted

L L

J

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Score or LM Test: General Maximum Likelihood (ML) Estimation A hypothesis test

H0: Restrictions on parameters are true

H1: Restrictions on parameters are not true

Basis for the test: b0 = parameter estimate under H0 (i.e., restricted), b1 = unrestricted

Derivative results: For the likelihood function under H1, logL1/ | =b1 = 0 (exactly, by definition)

logL1/ | =b0 ≠ 0. Is it close? If so, the restrictions look

reasonable

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Why is it the Lagrange Multiplier Test?Maximize logL( ) subject to restrictions ( )=

such as = or = when = = ( : and = .

Use LM approach:

Maximize wrt ( ) L* = logL( ) ( ).

ˆ* /FOC:

* /

loRL

L

XZ

Z

r 0

Rβ q 0 0 ,R 0 I) q 0

r

-

ˆ( )

ˆ ˆ ˆ ˆIf = 0, the constraints are not binding and logL( ) / so .

ˆ ˆIf 0, the constraints are binding and logL( ) / .

Direct test

ˆ ˆg ˆ ˆ( )L( )

: T

//R R

R

R R R U

R

R R

R

0

0r

0

0

r

0

0

est H : = 0

ˆ ˆEquivalent test: Test H : logL( ) / 0.LMR R

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Computing the LM Statistic

N 2i=1 i|0i 2

(1|0),i i|0 i|0 i i 02i0

0iN N(1|0) i=1 (1|0),i i=1 i|02 2 2

0 0i0 0 0

iN N 2(1|0) i=1 (1|0),i (1|0),i i=1 i|04

0

e1ˆ e , e =y , s

Ns

1 1 1ˆ ˆ= = e =

s s s

1ˆ ˆ ˆ es

xg xb

z

X'e 0xg g

Z'e Z'ez

xH g g

z

i

i i

2 i2 4 00 0 i i|0

i

10 0

'

Note this is the middle matrix in the White estimator.

The s and s disappear from the product. Let e .

Then LM= ( ) , which is simple to compute.0 0

x

z

xw

z

i'W W W W i

The derivation on page 64 of Wooldridge’s text is needlessly complex, and the second form of LM is actually incorrect because the first derivatives are not ‘heteroscedasticity robust.’

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Application of the Score Test

Linear Model: Y = X+Zδ+ε Test H0: δ=0 Restricted estimator is [b’,0’]’

Namelist ; X = a list… ; Z = a list … ; W = X,Z $ Regress ; Lhs = y ; Rhs = X ; Res = e $Matrix ; list ; LM = e’ W * <W’[e^2]W> * W’ e $

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Restricted regression and derivatives for the LM Test

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Tests for Omitted Variables

? Cobb - Douglas ModelNamelist ; X = One,x1,x2,x3,x4 $

? Translog second order terms, squares and cross products of logs

Namelist ; Z = x11,x22,x33,x44,x12,x13,x14,x23,x24,x34 $? Restricted regression. Short. Has only the log terms

Regress ; Lhs = yit ; Rhs = X ; Res = e $Calc ; LoglR = LogL ; RsqR = Rsqrd $

? LM statistic using basic matrix algebraNamelist ; W = X,Z $Matrix ; List ; LM = e'W * <W’[e^2]W> * W'e $

? LR statistic uses the full, long regression with all quadratic terms

Regress ; Lhs = yit ; Rhs = W $Calc ; LoglU = LogL ; RsqU = Rsqrd ; List ; LR = 2*(Logl -

LoglR) $? Wald Statistic is just J*F for the translog terms

Calc ; List ; JF=col(Z)*((RsqU-RsqR)/col(Z)/((1-RsqU)/(n-kreg)) )$

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Regression Specifications

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Model Selection Regression models: Fit measure = R2

Nested models: log likelihood, GMM criterion function (distance function)

Nonnested models, nonlinear models: Classical

Akaike information criterion= – (logL – 2K)/N Bayes (Schwartz) information criterion = –(logL-

K(logN))/N Bayesian: Bayes factor = Posterior odds/Prior

odds (For noninformative priors, BF=ratio of posteriors)

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Remaining to Consider for the Linear Regression Model

Failures of standard assumptions Heteroscedasticity Autocorrelation and Spatial

Correlation Robust estimation Omitted variables Measurement error

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Appendix: Computing the LM

Statistic

Part 2: Basic Econometrics [ 48/51]

LM Test

0 0

0

0 0

N1| =b i=1 (1| =b ),i

1| =b

1| =b 1| =b

ˆˆ ˆ

ˆwhere each row of is a vector of derivatives.

ˆUse a Wald statistic to assess if is close to .

By the information matrix equality,

ˆVar[ ] E[ ] whe

g g G'i

G

g 0

g H

0 0 0

N1| =b i=1 (1| =b ),i (1| =b ),i

re is the Hessian.

Estimate this with the sum of squares as usual:

ˆ ˆˆ ˆ ˆ

H

H g g G'G

Part 2: Basic Econometrics [ 49/51]

LM Test (Cont.)

0 0 01| = 1| = 1| =

2 2

ˆˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆN N

NR

R = the R in the '

-1

b b b

-1

-1

2

Wald statistic = g H g

= i G G'G G'i

= i G G'G G'i /

=

uncentered' regression of

a column of ones on the vector of first derivatives.

(N is equal to the total sum of squares.)

This is a general result. We did not assume any

specific model in the derivation. (We did assume

independent observations.)

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Representing Covariation

Conditional mean function: E[y | x] = g(x)

Linear approximation to the conditional mean function: Linear Taylor series

The linear projection (linear regression?)

k

0 K 0 0k=1 k k k

K 00 k=1 k k

g( ) =g( )+Σ [g | = ](x -x )

= +Σ (x -x )

x x x x

0 1 k k

0

-1

g*(x)= (x -E[x ])

E[y]

Var[ ]} {Cov[ ,y]}x x

Kk k

Part 2: Basic Econometrics [ 51/51]

Projection and Regression The linear projection is not the regression,

and is not the Taylor series. Example:

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

Part 2: Basic Econometrics [ 52/51]

For the Example: α=1, β=2

Linear Projection

X

4.18

8.35

12.53

16.70

20.88

.00.20 .40 .60 .80 1.00.00

EY_X PROJECTN TAYLOR

Varia

ble

Linear Projection

Conditional Mean

Taylor Series