Fixed and random effects of panel data analysis

Panel data (also known as longitudinal or cross-sectional time-series data) is a dataset in which the behavior of entities are observed across time. With panel data you can include variables at different levels of analysis (i.e. students, schools, districts, states) suitable for multilevel or hierarchical modeling. In this document we focus on two techniques use to analyze panel data:_DONE_

Fixed effects

Random effects

FE explore the relationship between predictor and outcome variables within an entity (country, person, company, etc.). Each entity has its own individual characteristics that may or may not influence the predictor variables (for example being a male or female could influence the opinion toward certain issue or the political system of a particular country could have some effect on trade or GDP or the business practices of a company may influence its stock price).

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When using FE we assume that something within the individual may impact or bias the predictor or outcome variables and we need to control for this. This is the rationale behind the assumption of the correlation between entity’s error term and predictor variables. FE remove the effect of those time-invariant characteristics from the predictor variables so we can assess the predictors’ net effect. _DONE_

Another important assumption of the FE model is that those time-invariant characteristics are unique to the individual and should not be correlated with other individual characteristics. Each entity is different therefore the entity’s error term and the constant (which captures individual characteristics) should not be correlated with the others. If the error terms are correlated then FE is no suitable since inferences may not be correct and you need to model that relationship (probably using random-effects), this is the main rationale for the Hausmantest (presented later on in this document).

The equation for the fixed effects model becomes:

Yit= β1Xit+ αi+ uit[eq.1]

Where

αi(i=1….n) is the unknown intercept for each entity (nentity-specific intercepts).

Yitis the dependent variable (DV) where i= entity and t= time.

Xitrepresents one independent variable (IV),

β1 is the coefficient for that IV,

uitis the error term _DONE_

Random effects assume that the entity’s error term is not correlated with the predictors which allows for time-invariant variables to play a role as explanatory variables.

In random-effects you need to specify those individual characteristics that may or may not influence the predictor variables. The problem with this is that some variables may not be available therefore leading to omitted variable bias in the model.

RE allows to generalize the inferences beyond the sample used in the model.

To decide between fixed or random effects you can run a Hausman test where the null hypothesis is that the preferred model is random effects vs. the alternative the fixed effects (see Green, 2008, chapter 9). It basically tests whether the unique errors (ui) are correlated with the regressors, the null hypothesis is they are not.

Testing for random effects: Breusch-Pagan Lagrange multiplier (LM)The LM test helps you decide between a random effects regression and a simple OLS regression.

The null hypothesis in the LM test is that variances across entities is zero. This is, no significant difference across units (i.e. no panel effect). Here we failed to reject the null and conclude that random effects is not appropriate. This is, no evidence of significant differences across countries, therefore you can run a simple OLS regression.

EC968

Panel Data Analysis

Steve Pudney

ISER University of Essex 2007

Panel data are a form of longitudinal data, involving

regularly repeated observations on the same individuals

Individuals may be people, households, firms, areas, etc

Repeat observations may be different time periods or

units within clusters (e.g. workers within firms; siblings

within twin pairs)+DONE_

Some terminology

A balanced panel has the same number of time observations (T)

on each of the n individuals

An unbalanced panel has different numbers of time observations

(Ti) on each individual

A compact panel covers only consecutive time periods for each

individual – there are no “gaps”

Attrition is the process of drop-out of individuals from the panel,

leading to an unbalanced and possibly non-compact panel

A short panel has a large number of individuals but few time

observations on each, (e.g. BHPS has 5,500 households and 13

waves)

A long panel has a long run of time observations on each

individual, permitting separate time-series analysis for each_DONE_

Advantages of panel data

With panel data:

• We can study dynamics

• The sequence of events in time helps to reveal causation

• We can allow for time-invariant unobservable variables

BUT…

• Variation between people usually far exceeds variation

over time for an individual

⇒ a panel with T waves doesn’t give T times the information

of a cross-section

• Variation over time may not exist or may be inflated by

measurement error

• Panel data imposes a fixed timing structure; continuoustime

survival analysis may be more informative

Panel Data Analysis – Advantages and Challenges

Cheng Hsiao

May 2006

IEPR WORKING PAPER 06.49

Panel data or longitudinal data typically refer to data containing time series observations

of a number of individuals. Therefore, observations in panel data involve at least

two dimensions; a cross-sectional dimension, indicated by subscript i, and a time series

dimension, indicated by subscript t. However, panel data could have a more complicated

clustering or hierarchical structure. For instance, variable y may be the measurement of

the level of air pollution at station _ in city j of country i at time t (e.g. Antweiler (2001),

Davis (1999)). For ease of exposition, I shall confine my presentation to a balanced panel

involving N cross-sectional units, i = 1, . . .,N, over T time periods, t = 1, . . ., T._DONE_

There are at least three factors contributing to the geometric growth of panel data

studies. (i) data availability, (ii) greater capacity for modeling the complexity of human

behavior than a single cross-section or time series data, and (iii) challenging methodology.

Advantages of Panel Data

Panel data, by blending the inter-individual differences and intra-individual dynamics

have several advantages over cross-sectional or time-series data:

(i) More accurate inference of model parameters. Panel data usually contain more

degrees of freedom and more sample variability than cross-sectional data which

may be viewed as a panel with T = 1, or time series data which is a panel

with N = 1, hence improving the efficiency of econometric estimates (e.g. Hsiao,

Mountain and Ho-Illman (1995)._DONE_

(ii) Greater capacity for capturing the complexity of human behavior than a single

cross-section or time series data. These include:

(ii.a) Constructing and testing more complicated behavioral hypotheses. For instance,

consider the example of Ben-Porath (1973) that a cross-sectional

sample of married women was found to have an average yearly labor-force

participation rate of 50 percent. These could be the outcome of random

draws from a homogeneous population or could be draws from heterogeneous

populations in which 50% were from the population who always work

and 50% never work. If the sample was from the former, each woman would

be expected to spend half of her married life in the labor force and half out of

the labor force. The job turnover rate would be expected to be frequent and

3

the average job duration would be about two years. If the sample was from

the latter, there is no turnover. The current information about a woman’s

work status is a perfect predictor of her future work status. A cross-sectional

data is not able to distinguish between these two possibilities, but panel data

can because the sequential observations for a number of women contain information

about their labor participation in different subintervals of their life

cycle.

Another example is the evaluation of the effectiveness of social programs

(e.g. Heckman, Ichimura, Smith and Toda (1998), Hsiao, Shen, Wang and

Wang (2005), Rosenbaum and Rubin (1985). Evaluating the effectiveness of

certain programs using cross-sectional sample typically suffers from the fact

that those receiving treatment are different from those without. In other

words, one does not simultaneously observe what happens to an individual

when she receives the treatment or when she does not. An individual is

observed as either receiving treatment or not receiving treatment. Using

the difference between the treatment group and control group could suffer

from two sources of biases, selection bias due to differences in observable

factors between the treatment and control groups and selection bias due to

endogeneity of participation in treatment. For instance, Northern Territory

(NT) in Australia decriminalized possession of small amount of marijuana

in 1996. Evaluating the effects of decriminalization on marijuana smoking

behavior by comparing the differences between NT and other states that

were still non-decriminalized could suffer from either or both sorts of bias. If

panel data over this time period are available, it would allow the possibility

of observing the before- and affect-effects on individuals of decriminalization

as well as providing the possibility of isolating the effects of treatment from

other factors affecting the outcome.

4

(ii.b) Controlling the impact of omitted variables. It is frequently argued that the

real reason one finds (or does not find) certain effects is due to ignoring the

effects of certain variables in one’s model specification which are correlated

with the included explanatory variables. Panel data contain information

on both the intertemporal dynamics and the individuality of the entities

may allow one to control the effects of missing or unobserved variables. For

instance, MaCurdy’s (1981) life-cycle labor supply model under certainty

implies that because the logarithm of a worker’s hours worked is a linear

function of the logarithm of her wage rate and the logarithm of worker’s

marginal utility of initial wealth, leaving out the logarithm of the worker’s

marginal utility of initial wealth from the regression of hours worked on wage

rate because it is unobserved can lead to seriously biased inference on the

wage elasticity on hours worked since initial wealth is likely to be correlated

with wage rate. However, since a worker’s marginal utility of initial wealth

stays constant over time, if time series observations of an individual are

available, one can take the difference of a worker’s labor supply equation

over time to eliminate the effect of marginal utility of initial wealth on hours

worked. The rate of change of an individual’s hours worked now depends

only on the rate of change of her wage rate. It no longer depends on her

marginal utility of initial wealth._DONE_

(ii.c) Uncovering dynamic relationships.

“Economic behavior is inherently dynamic so that most econometrically interesting

relationship are explicitly or implicitly dynamic”. (Nerlove (2002)).

However, the estimation of time-adjustment pattern using time series data

often has to rely on arbitrary prior restrictions such as Koyck or Almon distributed

lag models because time series observations of current and lagged

variables are likely to be highly collinear (e.g. Griliches (1967)). With panel

5

data, we can rely on the inter-individual differences to reduce the collinearity

between current and lag variables to estimate unrestricted time-adjustment

patterns (e.g. Pakes and Griliches (1984))._DONE_

(ii.d) Generating more accurate predictions for individual outcomes by pooling

the data rather than generating predictions of individual outcomes using

the data on the individual in question. If individual behaviors are similar

conditional on certain variables, panel data provide the possibility of learning

an individual’s behavior by observing the behavior of others. Thus, it is

possible to obtain a more accurate description of an individual’s behavior by

supplementing observations of the individual in question with data on other

individuals (e.g. Hsiao, Appelbe and Dineen (1993), Hsiao, Chan, Mountain

and Tsui (1989)).

(ii.e) Providing micro foundations for aggregate data analysis.

Aggregate data analysis often invokes the “representative agent” assumption.

However, if micro units are heterogeneous, not only can the time series properties

of aggregate data be very different from those of disaggregate data

(e.g., Granger (1990); Lewbel (1992); Pesaran (2003)), but policy evaluation

based on aggregate data may be grossly misleading. Furthermore, the

prediction of aggregate outcomes using aggregate data can be less accurate

than the prediction based on micro-equations (e.g., Hsiao, Shen and Fujiki

(2005)). Panel data containing time series observations for a number of individuals

is ideal for investigating the “homogeneity” versus “heterogeneity”

issue.

(iii) Simplifying computation and statistical inference.

Panel data involve at least two dimensions, a cross-sectional dimension and a

time series dimension. Under normal circumstances one would expect that the

6

computation of panel data estimator or inference would be more complicated than

cross-sectional or time series data. However, in certain cases, the availability of

panel data actually simplifies computation and inference. For instance:

(iii.a) Analysis of nonstationary time series.

When time series data are not stationary, the large sample approximation

of the distributions of the least-squares or maximum likelihood estimators

are no longer normally distributed, (e.g. Anderson (1959), Dickey and Fuller

(1979,81), Phillips and Durlauf (1986)). But if panel data are available,

and observations among cross-sectional units are independent, then one can

invoke the central limit theorem across cross-sectional units to show that the

limiting distributions of many estimators remain asymptotically normal (e.g.

Binder, Hsiao and Pesaran (2005), Levin, Lin and Chu (2002), Im, Pesaran

and Shin (2004), Phillips and Moon (1999)).

(iii.b) Measurement errors.

Measurement errors can lead to under-identification of an econometric model

(e.g. Aigner, Hsiao, Kapteyn and Wansbeek (1985)). The availability of

multiple observations for a given individual or at a given time may allow a

researcher to make different transformations to induce different and deducible

changes in the estimators, hence to identify an otherwise unidentified model

(e.g. Biorn (1992), Griliches and Hausman (1986), Wansbeek and Koning

(1989)).

(iii.c) Dynamic Tobit models. When a variable is truncated or censored, the actual

realized value is unobserved. If an outcome variable depends on previous

realized value and the previous realized value are unobserved, one has to

take integration over the truncated range to obtain the likelihood of observables.

In a dynamic framework with multiple missing values, the multiple

7

integration is computationally unfeasible. With panel data, the problem can

be simplified by only focusing on the subsample in which previous realized

values are observed (e.g. Arellano, Bover, and Labeager (1999)).

The advantages of random effects (RE) specification are: (a) The number of parameters

stay constant when sample size increases. (b) It allows the derivation of efficient

10

estimators that make use of both within and between (group) variation. (c) It allows the

estimation of the impact of time-invariant variables. The disadvantage is that one has

to specify a conditional density of αi given x

˜

_

i = (x

˜ it, . . ., x

˜iT ), f(αi | x

˜ i), while αi are

unobservable. A common assumption is that f(αi | x

Ëœi) is identical to the marginal density

f(αi). However, if the effects are correlated with x

Ëœit or if there is a fundamental difference

among individual units, i.e., conditional on x

Ëœit, yit cannot be viewed as a random draw

from a common distribution, common RE model is misspecified and the resulting estimator

is biased.

The advantages of fixed effects (FE) specification are that it can allow the individualand/

or time specific effects to be correlated with explanatory variables x

˜ it. Neither does

it require an investigator to model their correlation patterns. The disadvantages of the FE

specification are: (a’) The number of unknown parameters increases with the number of

sample observations. In the case when T (or N for λt) is finite, it introduces the classical

incidental parameter problem (e.g. Neyman and Scott (1948)). (b’) The FE estimator

does not allow the estimation of the coefficients that are time-invariant.

In order words, the advantages of RE specification are the disadvantages of FE specification

and the disadvantages of RE specification are the advantages of FE specification.

To choose between the two specifications, Hausman (1978) notes that if the FE estimator

(or GMM), ˆθ_DONE_

˜FE, is consistent whether αi is fixed or random and the commonly used RE

estimator (or GLS), ˆθ

˜RE, is consistent and efficient only when αi is indeed uncorrelated

with x

˜it and is inconsistent if αi is correlated with x

Ëœit.

The advantage of RE specification is that there is no incidental parameter problem.

The problem is that f(αi | x

˜ i) is in general unknown. If a wrong f(αi | x

Ëœi) is postulated,

maximizing the wrong likelihood function will not yield consistent estimator of β

˜

.

Moreover, the derivation of the marginal likelihood through multiple integration may be

computationally infeasible. The advantage of FE specification is that there is no need to

specify f(αi | x

˜ i). The likelihood function will be the product of individual likelihood (e.g.

(4.28)) if the errors are i.i.d. The disadvantage is that it introduces incidental parameters.

Longitudinal (Panel and Time Series Cross-Section) Data

Nathaniel Beck

Department of Politics

NYU

New York, NY 10012

nathaniel.beck@nyu.edu

http://www.nyu.edu/gsas/dept/politics/faculty/beck/beck home.html

Jan. 2004

What is longitudinal data?

Observed over time as well as over space.

Pure cross-section data has many limitations (Kramer, 1983). Problem is that only have

one historical context.

(Single) time series allows for multiple historical context, but for only one spatial location.

Longitudinal data – repeated observations on units observed over time

Subset of hierarchical data – observations that are correlated because there is some tie

to same unit.

E.g. in educational studies, where we observe student i in school u. Presumably there

is some tie between the observations in the same school.

In such data, observe yj,u where u indicates a unit and j indicates the j’th observation

drawn from that unit. Thus no relationship between yj,u and yj,u0 even though they have

the same first subscript. In true longitudinal data, t represents comparable time.

Generalized Least Squares

An alternative is GLS. If is known (up to a scale factor), GLS is fully efficient and yields

consistent estimates of the standard errors. The GLS estimates of _ are given by

(X0−1X)

−1X0−1Y (14)

with estimated covariance matrix

(X0−1X)

−1

. (15)

(Usually we simplify by finding some “trick” to just do a simple transform on the observations

to make the resulting variance-covariance matrix of the errors satisfy the Gauss-Markov

assumptions. Thus, the common Cochrane-Orcutt transformation to eliminate serial

correlation of the errors is almost GLS, as is weighted regression to eliminate

heteroskedasticity.)

The problem is that is never known in practice (even up to a scale factor). Thus an

estimate of , ˆ, is used in Equations 14 and 15. This procedure, FGLS, provides consistent

estimates of _ if ˆ is estimated by residuals computed from consistent estimates of _; OLS

provides such consistent estimates. We denote the FGLS estimates of _ by ˜_.

In finite samples FGLS underestimates sampling variability (for normal errors). The basic

insight used by Freedman and Peters is that X0−1X is a (weakly) concave function of .

FGLS uses an estimate of , ˆ, in place of the true . As a consequence, the expectation of

the FGLS variance, over possible realizations of ˆ, will be less than the variance, computed

with the . This holds even if ˆ is a consistent estimator of . The greater the variance of

ˆ, the greater the downward bias.

This problem is not severe if there are only a small number of parameters in the

variance-covariance matrix to be estimated (as in Cochrane-Orcutt) but is severe if there are

a lot of parameters relative to the amount of data.

Beck – TSCS – Winter 2004 – Class 1 8

ASIDE: Maximum likelihood would get this right, since we would estimate all parameters and

take those into account. But with a large number of parameters in the error process, we

would just see that ML is impossible. That would have been good.

PANEL DATA ANALYSIS USING SAS

ABU HASSAN SHAARI MOHD NOR

Faculty of Economics and Business

Universiti Kebangsaan Malaysia

ahassan@pkrisc.cc.ukm.my

FAUZIAH MAAROF

Faculty of Science

Universiti Putra Malaysia

fauziah@putra.edu.my 2007

Advantages of panel data

According to Baltagi (2001) there are several advantages of using panel data as compared to

running the models using separate time series and cross section data. They are as follows:

Large number of data points

2)Increase degrees of freedom & reduce collinearity

3) Improve efficiency of estimates and

4) Broaden the scope of inference

The Econometrics of Panel Data

Michel Mouchart 1

Institut de statistique

Université catholique de Louvain (B)

3rd March 2004

1 text book

Statistical modelling : benefits and limita-

tions of panel data

1.5.1 Some characteristic features of P.D.

Object of this subsection : features to bear in mind when modelling P.D.

• Size : often

N (] of individual(s)) is large

Ti (size of individual time series) is small

thus:N >> Ti BUT this is not always the case

] of variables is large (often: multi-purpose survey)

•• Sampling : often

individuals are selected randomly

Time is not

rotating panels

split panels _ : individuals are partly renewed at each period

• • • non independent data

among data relative to a same individual: because of unobservable

characteristics of each individual

among individuals : because of unobservable characteristics common

to several individuals

between time periods : because of dynamic behaviour

CHAPTER 1. INTRODUCTION 10

1.5.2 Some benefits from using P.D.

a) Controlling for individual heterogeneity

Example : state cigarette demand (Baltagi and Levin 1992)

• Unit : 46 american states

• Time period : 1963-1988

• endogenous variable : cigarette demand

• explanatory variables : lagged endogenous, price, income

• consider other explanatory variables :

Zi : time invariant

religion (± stable over time)

education

etc.

Wt state invariant

TV and radio advertising (national campaign)

Problem : many of these variables are not available

This is HETEROGENEITY (also known as “frailty”)

(remember !) omitted variable ) bias (unless very specific hypotheses)

Solutions with P.D.

• dummies (specific to i and/or to t)

WITHOUT “killing” the data

•• differences w.r.t. to i-averages

i.e. : yit 7! (yit − ¯yi.)_DONE_

CHAPTER 1. INTRODUCTION 11

b) more information data sets

• larger sample size due to pooling _ individual

time

dimension

In the balanced case: NT observations

In the unbalanced case: P1_i_N Ti observations

•• more variability

! less collinearity (as is often the case in time series)

often : variation between units is much larger than variation within

units_DONE_

c) better to study the dynamics of adjustment

• distinguish

repeated cross-sections : different individuals in different periods

panel data : SAME individuals in different periods

•• cross-section : photograph at one period

repeated cross-sections : different photographs at different periods

only panel data to model HOW individuals ajust over time . This is

crucial for:

policy evaluation

life-cycle models

intergenerational models_DONE_

CHAPTER 1. INTRODUCTION 12

d) Identification of parameters that would not be identified with pure

cross-sections or pure time-series:

example 1 : does union membership increase wage ?

P.D. allows to model BOTH union membership and individual

characteristics for the individuals who enter the union during

the sample period.

example 2 : identifying the turn-over in the female participation

to the labour market.

Notice: the female, or any other segment !

i.e. P.D. allows for more sophisticated behavioural models

e) • estimation of aggregation bias

•• often : more precise measurements at the micro level

Comparing the Fixed Effect and the Ran-

dom Effect Models

2.4.1 Comparing the hypotheses of the two Models

The RE model and the FE model may be viewed within a hierarchical specification

of a unique encompassing model. From this point of view, the two

models are not fundamentally different, they rather correspond to different

levels of analysis within a unique hierarchical framework. More specifically,

from a Bayesian point of view, where all the variables (latent or manifest)

and parameters are jointly endowed with a (unique) probability measure, one

CHAPTER 2. ONE-WAY COMPONENT REGRESSION MODEL 37

may consider the complete specification of the law of (y, μ, _ | Z, Zμ) as

follows:

(y | μ, _, Z, Zμ) _ N( Z_ _ + Zμμ, _2

” I(NT)) (2.64)

(μ | _, Z, Zμ) _ N(0, _2

μ I(N)) (2.65)

(_ | Z, Zμ) _ Q (2.66)

where Q is an arbitrary prior probability on _ = (_, _2

” , _2

μ). Parenthetically,

note that this complete specification assumes:

y ??_2

μ | μ, _, _2

” , Z, Zμ μ??(_, Z, Zμ) | _2

μ

The above specification implies:

(y | _, Z, Zμ) _ N( Z_ _ , _2

μ Zμ Z0μ + _2

” I(NT)) (2.67)

Thus the FE model, i.e. (2.64), considers the distribution of (y | μ, _, Z, Zμ)

as the sampling distribution and the distributions of (μ | _, Z, Zμ) and

(_ | Z, Zμ) as prior specification. The RE model, i.e. (2.67), considers the

distribution of (y | _, Z, Zμ) as the sampling distribution and the distribution

of (_ | Z, Zμ) as prior specification. Said differently, in the RE model,

μ is treated as a latent (i.e. not obervable) variable whereas in the FE model

μ is treated as an incidental parameter. Moreover, the RE model is obtained

from the FE model through a marginalization with respect to μ.

These remarks make clear that the FE model and the RE model should

be expected to display different sampling properties. Also, the inference on

μ is an estimation problem in the FE model whereas it is a prediction problem

in the RE model: the difference between these two problems regards the

difference in the relevant sampling properties, i.e. w.r.t. the distribution of

(y | μ, _, Z, Zμ) or of (y | _, Z, Zμ), and eventually of the relevant risk

functions, i.e. the sampling expectation of a loss due to an error between an

estimated value and a (fixed) parameter or between a predicted value and

the realization of a (latent) random variable.

This fact does however not imply that both levels might be used indifferently.

Indeed, from a sampling point of view:

(i) the dimensions of the parameter spaces are drastically different. In

the FE model, when N , the number of individuals, increases, the μi ‘s being

CHAPTER 2. ONE-WAY COMPONENT REGRESSION MODEL 38

incidental parameters also increases in number: each new individual introduces

a new parameter.