Econometrics of Panel Data - Warsaw School of Economicsweb.sgh.waw.pl/~jmuck/EoPD/Meeting1_2017.pdf · Jakub Mućk Econometrics of Panel Data Course outline Meeting # 1 3 / 31. Topics

Econometrics of Panel Data

Jakub Mućk

Department of Quantitative Economics

Meeting # 1

Jakub Mućk Econometrics of Panel Data Meeting # 1 1 / 31

Outline

1 Course outline

2 Panel dataAdvantages of Panel DataLimitations of Panel Data

3 Pooled OLS estimator

4 Fixed effects modelThe Least Squares Dummy Variable EstimatorThe Fixed Effect (Within Group) Estimator

Jakub Mućk Econometrics of Panel Data Course outline Meeting # 1 2 / 31

Literature

Basic:1 Baltagi B. H., (2014), Econometric Analysis of Panel Data, 5th edition, Wiley.2 Wooldridge J. M., (2010), Econometric Analysis of Cross Section and Panel Data,

2nd edition, The MIT Press.Additional:1 Angrist J. D. and Pischke J.-S., (2009), Mostly Harmless Econometrics: An Em-

piricist’s Companion, Princeton University Press.2 Arellano M., (2004), Panel Data Econometrics. Advanced Texts in Econometrics,

Oxford University Press.3 Baltagi B. H. (ed.), (2015), The Oxford Handbook of Panel Data, Oxford University

Press.4 Cameron C. A. and Trivedi P. K., (2006),Microeconometrics: Methods and Applica-

tions, Cambridge University Press.5 Hsiao Ch., (2014), Analysis of Panel Data, 3rd edition, Cambridge University Press.6 Pesaran H. (ed.), (2015), Time Series and Panel Data Econometrics, Oxford Uni-

versity Press.


Topics

1 Panel data - basic definitions, characteristics, etc.2 Linear static model - common types. Fixed and random effects approaches.3 Fixed effects estimation. Random effects models.4 Hausman test. The between estimator.5 Two-way error component models.6 Heteroskedasticity, serial correlation and cross-sectional dependence in static mod-

els. GLS estimation.7 Endogeneity. Instrumental variables (IV) regression. The Hausman-Taylor estima-

tor.8 Dynamic panel data models. The FD (first differences) estimator. The Nickell’s bias.

The Anderson-Hsiao estimator.9 Estimation of dynamic models. Generalized Method of Moments (GMM). The

Arellano-Bond estimator and a system estimator.10 Heterogeneous panels. Seemingly unrelated regression. Swamy’s random coefficient

model. The Mean Group estimator. The Common Correlated Effects Mean Groupestimator.

11 Panel unit root tests. Panel cointegration tests.12 Nonstationary panels. Panel VAR.13 Limited dependent variable. The FE logit. The RE binary outcome models.14 The FE and RE Poisson models. The RE tobit model.15 Estimating average treatment effects (ATE). Difference-in-differences (DID).Jakub Mućk Econometrics of Panel Data Course outline Meeting # 1 4 / 31

Office hours: TBA.E-mail: [email protected]: http://web.sgh.waw.pl/∼jmuck/=⇒ teaching=⇒ Econometrics of Panel Data.Software: Stata (+ R).Exam rules:

I Exam: 60 %.I Classes: 40% – houseworks and activity.


mailto:[email protected]

http://web.sgh.waw.pl/~jmuck/

Outline

1 Course outline




Jakub Mućk Econometrics of Panel Data Panel data Meeting # 1 6 / 31

Panel data

Panel of data consists of a group of cross-section units (people, firms, states,countries) that are observed over the time:

Cross-section:yi wherei ∈ {1, . . . ,N}.Time series:yt wheret ∈ {1, . . . ,T}.Panel data:yit wherei ∈ {1, . . . ,N}t ∈ {1, . . . ,T}.

In general,N - the cross-sectional dimension.T - the time dimension.


Panel data

We might describe panel data using T and N:long/short describes the time dimension (T );wide/narrow describes the cross-section dimension (N );

For example: panel with relatively large N and T: long and wide panel.

In a balanced panel, each individuals(unit) has the same number of obser-vation.Unbalanced panel is a panel in which the number of time series observationsis different across units.


Panel data

We might describe panel data using T and N:long/short describes the time dimension (T );wide/narrow describes the cross-section dimension (N );

For example: panel with relatively large N and T: long and wide panel.

In a balanced panel, each individuals(unit) has the same number of obser-vation.Unbalanced panel is a panel in which the number of time series observationsis different across units.


Examples of panel data

Micro dataPSID – the Panel Study of Income Dynamics collected by the Institute forSocial Research at the University of Michigan.NLS – the National Longitudinal Surveys collected by the Bureau of LaborStatistics.CPS – Current Population Survey conducted by the Bureau of Census (monthlyfrequency).

Macro dataPenn World Table,WDI Indicators.

Sectoral dataKLEMS database.


Advantages of Panel Data (Baltagi, 2014)

Controlling for individual heterogeneity.Panel data offer more informative data, more variability, less collinearityamong the dependent variables, more degrees of freedom and more efficiencyin estimation.Identification and measurement of effects that are simply not detectable inpure cross-section or pure time-series data.Testing more complicated behavioral models than purely cross-section ortime-series data.Reduction in biases resulting from aggregation over firms or individuals.Overcome the problem of nonstandard distributions typical of unit roots tests=⇒ macro panels.


Limitations of Panel Data

Design and data collection problems:I coverage;I nonresponse;I frequency of interviewing;

Distortions of measurement errorsSelectivity problems:

I self-selectivity;I nonresponse;I attrition;

Short T .Cross-sectional dependence.


Two examples (Arellano, 2003)

Classical exampleAgricultural Cobb-Douglas production function. Consider the following model:

yit = βxit + uit + ηi (1)

I yit – the log output.I xit – the log of a variable output;I ηi – an farm-specific input that is constant over time, e.g., soil quality.I uit – a stochastic input that is outside framer’s control, e.g., rainfalls.I β - the technological parameter.

An example in which panel data does not workReturns to education. Consider the following model:

yit = α+ βxit + uit (2)

I yit – the log wage;I xit – years of the full-time education;I β – returns to education.

In addition:uit = ηi + εit (3)

where ηi stands for the unobserved individual ability.Problem: xit lacks of time variation.


Two examples (Arellano, 2003)

Classical exampleAgricultural Cobb-Douglas production function. Consider the following model:

yit = βxit + uit + ηi (1)

I yit – the log output.I xit – the log of a variable output;I ηi – an farm-specific input that is constant over time, e.g., soil quality.I uit – a stochastic input that is outside framer’s control, e.g., rainfalls.I β - the technological parameter.

An example in which panel data does not workReturns to education. Consider the following model:

yit = α+ βxit + uit (2)

I yit – the log wage;I xit – years of the full-time education;I β – returns to education.

In addition:uit = ηi + εit (3)

where ηi stands for the unobserved individual ability.Problem: xit lacks of time variation.


Outline

1 Course outline




Jakub Mućk Econometrics of Panel Data Pooled OLS estimator Meeting # 1 13 / 31

Pooled OLS estimator

Pooled model is one where the data on different units are pooled togetherwith no assumption on individual differences:

yit = α+ β1x1it + . . .+ βkxkit + uit (4)

whereI yit – the dependent variable;I xkit – the k − th explanatory variable;I uit – the error/disturbance term;I α – the intercept;I β1, . . ., βk – the structural parameters;

Note that the coefficients α, β1, . . ., βk are the same for all unit (don’t havei or t subscript).



Pooled model is one where the data on different units are pooled togetherwith no assumption on individual differences:

yit = α+ β1x1it + . . .+ βkxkit + uit (4)

whereI yit – the dependent variable;I xkit – the k − th explanatory variable;I uit – the error/disturbance term;I α – the intercept;I β1, . . ., βk – the structural parameters;

Note that the coefficients α, β1, . . ., βk are the same for all unit (don’t havei or t subscript).



In vector form:y = Xβ + u (5)

where y is NT × 1, β is (K + 1)× 1, X is NT ×K and u is NT × 1 and:

X =

1 x11 . . . xk11 x12 . . . xk2...

.... . .

...1 x1N . . . xkN

and y =

y1y2...

yN

and β =

αβ1...βK

where xij is the vector of observations of the jth independent variable for uniti over the time and yi is the vector of observation of the explained variablefor unit i.Then:

βPOOLED = (X ′X)−1 X ′y =

αPOOLED

βPOOLED1

...βPOOLED

K

(6)



Assumptions (for linear pooled model):

E(u) = 0 (7)E(uu′) = σ2

uI (8)rank(X) = K + 1 < NT (9)E(u|X) = 0 (10)

(10): X is nonstochastic and is not correlated with u.(8): the error term (u) is not autocorrelated and homoscedastic.(10) =⇒ strictly exogeneity of independent variables.

Gauss-Markov TheoremIf (7)-(10)and are satisfied then βPOOLED is BLUE (the best linear unbiasedestimator).


Empirical example

Grunfeld’s (1956) investment model:

Iit = α+ β1Kit + β2Fit + uit (11)

whereIit – the gross investment for firm i in year t;Kit – the real value of the capital stocked owned by firm i;Fit – is the real value of the firm (shares outstanding).

The dataset consists of 10 large US manufacturing firms over 20 years(1935–54).


Empirical example

Pooled OLSα −42.714

(9.512)[0.000]

β1 0.231(0.025)[0.000]

β2 0.116(0.006)[0.000]

Note: The expressions in round and squared brackets stand for standard errors andprobability values corresponding to the null about parameter’s insignificance, respectively.

Iit = α+ β1Kit + β2Fit + uit


Robust/ Clustered Standard Errors

The general assumption in pooled regression on the error terms are verystrong or even unrealistic.The lack of correlation between errors corresponding to the sameindividuals.Let’s relax the above assumption:

cov(ui,t , ui,s) 6= 0 (12)

Then we have problem of both autocorrelation and heteroskedasticity.The OLS estimator is still consistent but the standard errors are incorrect.We might use the clustered/robust standard errors. Here, the time seriesfor each individual are clusters.


Standard errors

If the assumptions (7)-(10) are satisfied then the variance-covariance matrixof β is equal to:

Var(β) = σ2 (X ′X)−1 = D (13)

where σ2 is the variance of the error term.The standard errors stand for the diagonal elements:

s.e.(βi) =√

Dii (14)

One step back in (13):

Var(β) = (X ′X)−1 X ′ΣX (X ′X)−1 (15)

(15) simplifies to (13) when Var(u) = σ2I .But when we don’t know Σ?White’s (1980) heteroscedasticity-consistent estimator of the variance-covariancematrix:

Σ = diag(u2

1 , u22 , . . . , u2

N), (16)

where u is the vector of residuals.Jakub Mućk Econometrics of Panel Data Pooled OLS estimator Meeting # 1 20 / 31

Empirical example

Pooled OLS Pooled OLSrobust se

α −42.714 −42.714(9.512) (11.575)[0.000] (0.000)

β1 0.231 0.231(0.025) (0.049)[0.000] [0.000]

β2 0.116 0.231(0.006) (0.007)[0.000] [0.000]




Outline

1 Course outline




Jakub Mućk Econometrics of Panel Data Fixed effects model Meeting # 1 22 / 31

One-way Error Component Model

In the model:

yit = α+ X ′itβ + uit i ∈ {1, . . . ,N}, t ∈ {1, . . . ,T} (17)

it is assumed that all units are homogeneous. Why?One-way error component model:

ui,t = µi + εi,t (18)

where:I µi – the unobservable individual-specific effect;I εi,t – the remainder disturbance.


Fixed effects model

We can relax assumption that all individuals have the same coefficients

yit = αi + β1x1it + . . .+ βkxkit + uit (19)

Note that an i subscript is added to only intercept αi but the slope coefficients,β1, . . ., βk are constant for all individuals.An individual intercept (αi) are include to control for individual-specificand time-invariant characteristics. That intercepts are called fixedeffects.Fixed effects capture the individual heterogeneity.The estimation:i) The least squares dummy variable estimatorii) The fixed effects estimator


The Least Squares Dummy Variable Estimator

The natural way to estimate fixed effect for all individuals is to include anindicator variable. For example, for the first unit:

D1i ={

1 i = 10 otherwise (20)

The number of dummy variables equals N . It’s not feasible to use theleast square dummy variable estimator when N is largeWe might rewrite the fixed regression as follows:

yit =N∑

j=1αiDji + β1x1it + . . .+ βkxkit + ui,t (21)

Why α is missing?



In the matrix form:y = Xβ + u (22)

where y is NT × 1, β is (K + N ) × 1, X is NT × (K + N ) and u is NT × 1.

X =

1 0 . . . 0 x11 . . . xk10 1 . . . 0 x12 . . . xk2...

.... . .

......

. . ....

0 0 . . . 1 x1N . . . xkN

and y =

y1y2...

yN

and β =

α1...αNβ1...βK

where xij is the vector of observations of the jth independent variable for unit i overthe time and yi is the vector of observation of the explained variable for unit i.Then:

βLSDV =(X ′X

)−1 X ′y = [αLSDV1 . . . αLSDV

N βLSDV1 . . . βLSDV

K ]′ (23)



We can test estimates of intercept to verify whether the fixed effects aredifferent among units:

H0 : α1 = α2 = . . . = αN (24)

To test (24) we estimate: i) unrestricted model (the least squares dummyvariable estimator) and ii) restricted model (pooled regression). Then wecalculate sum of squared errors for both models: SSEU and SSER.

F = (SSER − SSEU )/(N − 1)SSEU/(NT −K ) (25)

if null is true then F ∼ F(N−1,NT−K).


The Fixed Effect (Within Group) Estimator

Let’s start with simple fixed effects specification for individual i:

yit = αi + β1x1it + . . .+ βkxkit + uit t = 1, . . . ,T (26)

Average the observation across time and using the assumption on time-invariantparameters we get:

yi = αi + β1x1i + . . .+ βk xki + ui (27)

where yi = 1T∑T

t=1 yit , x1i = 1T∑T

t=1 x1it , xki = 1T∑T

t=1 xkit and ui = 1T∑T

t=1 ui

Now we substract (27) from (26)

(yit − yi) = (αi − αi)︸︷︷︸=0

+ β1(x1it − x1i) + . . .+ βk(xitk − xki) + (uit − ui) (28)

Using notation: yit = (yit − yi), x1it = (x1it − x1i), xkit = (xkit − xki) uit = (uit − ui),we get

yit = β1x1t + . . .+ βk xkt + ui,t (29)Note that we don’t estimate directly fixed effects.


The Fixed Effect (Within Group) EstimatorThe Fixed Effect Estimator or the within (group) estimator in vector form:

y = Xβ + u (30)where y is NT × 1, β is K × 1, X is NT × K and u is NT × 1 and:

X =

x11 . . . xk1x12 . . . xk2...

. . ....

x1N . . . xkN

and y =

y1y2...

yN

and β =

β1...βK

Then:

βFE =(X ′X

)−1 X ′y = [βFE1 . . . βFE

K ]′ (31)The standard errors should be adjusted by using correction factor:√

NT − KNT − N − K (32)

where K is number of estimated parameters without constant.The fixed effects estimates can be recovered by using the fact that OLS fittedregression passes the point of the means.

αFEi = yi − βFE

1 x1i − βFE2 x2i − . . .− βFE

k xki (33)


Empirical example

Pooled OLS FEα −42.714 −58.744

(9.512) (12.454)[0.000] [0.000]

β1 0.231 0.310(0.025) (0.017)[0.000] [0.000]

β2 0.116 0.110(0.006) (0.012)[0.000] [0.000]




Empirical example

Pooled OLS Pooled OLS FE FErobust se robust se

α −42.714 −42.714 −58.744 −58.744(9.512) (11.575) (12.454) (27.603)[0.000] [0.000] [0.000] [0.062]

β1 0.231 0.231 0.310 0.310(0.025) (0.049) (0.017) (0.053)[0.000] [0.000] [0.000] [0.000]

β2 0.116 0.231 0.110 0.110(0.006) (0.007) (0.012) (0.015)[0.000] [0.000] [0.000] [0.000]




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Econometrics of Panel Data - Warsaw School of Economicsweb.sgh.waw.pl/~jmuck/EoPD/Meeting1_2017.pdf · Jakub Mućk Econometrics of Panel Data Course outline Meeting # 1 3 / 31. Topics