Fixed and random effects of panel data analysis

Fixed and random effectuate of dialog box info analysisPanel info (also known as longitudinal or cross-sectional conviction-series data) is a dataset in which the behavior of entities ar observed crosswise clipping. With adorn data you can include versatiles at contrary levels of analysis (i.e. students, schools, districts, states) suitable for multilevel or stratified postureing. In this document we focus on two techniques subroutine to analyze bloodshed data_DONE_Fixed effectuateRandom effectsFE explore the relationship between predictor and outcome variable quantitys within an entity (country, person, company, etc.). Each entity has its own singular 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(prenominal) country could fool some effect on trade or GDP or the business practices of a company may influence its rai lway line price).When using FE we assume that something within the respective(prenominal) may impact or bias the predictor or outcome variables and we need to control for this. This is the principle behind the assumption of the correlation between entitys misunderstanding term and predictor variables. FE remove the effect of those clock beat- immutable characteristics from the predictor variables so we can assess the predictors win effect. _DONE_An opposite important assumption of the FE bewilder is that those time-invariant characteristics atomic number 18 unique to the individual and should not be correlated with other individual characteristics. Each entity is disparate therefore the entitys error term and the constant (which captures individual characteristics) should not be correlated with the others. If the error terms argon correlated then FE is no suitable since inferences may not be correct and you need to model that relationship (probably using random-effects), th is is the main rationale for the Hausmantest (presented later on in this document).The equivalence for the wintry effects model becomesYit= 1Xit+ i+ uiteq.1Wherei(i=1.n) is the unknown intercept for each entity (nentity- particularised intercepts).Yitis the dependent variable (DV) where i= entity and t= time.Xitrepresents one independent variable (IV),1 is the co cost-efficient for that IV,uitis the error term _DONE_Random effects assume that the entitys error term is not correlated with the predictors which allows for time-invariant variables to play a role as informative variables.In random-effects you need to specify those individual characteristics that may or may not influence the predictor variables. The business 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 ideal used in the model.To nail down between repair or random effects you can run a Hausman test wher e 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 crosswise entities is zero. This is, no satisfying difference across units (i.e. no card 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.EC968Panel Data abstractSteve PudneyISER University of Essex 2007Panel data are a form of longitudinal data, involving regularly repeated observations on the same individualsIndividuals may be people, households, firms, areas, e tcRepeat observations may be different time periods orunits within clusters (e.g. workers within firms siblingswithin couple up pairs)+DONE_ whatsoever terminologyA balanced panel has the same matter of time observations (T)on each of the n individualsAn unbalanced panel has different rime of time observations(Ti) on each individualA compact panel c everywheres only consecutive time periods for eachindividual there are no gapsAttrition is the process of drop-out of individuals from the panel,leading to an unbalanced and possibly non-compact panelA short panel has a large-mouthed number of individuals provided few timeobservations on each, (e.g. BHPS has 5,500 households and 13waves)A long panel has a long run of time observations on eachindividual, permitting separate time-series analysis for each_DONE_Advantages of panel dataWith panel data We can study dynamics The sequence of events in time helps to reveal causation We can allow for time-invariant unobservable variablesBUT rendering between people usually far exceeds variation all over time for an individual a panel with T waves doesnt give T times the learningof a cross-section Variation over time may not exist or may be inflated by meter error Panel data imposes a fixed timing structure continuoustimesurvival analysis may be to a greater extent(prenominal) informativePanel Data Analysis Advantages and ChallengesCheng HsiaoMay 2006IEPR WORKING PAPER 06.49Panel data or longitudinal data typically refer to data containing time series observationsof a number of individuals. Therefore, observations in panel data involve at leasttwo dimensions a cross-sectional dimension, indicated by subscript i, and a time seriesdimension, indicated by subscript t. However, panel data could have a more complicatedclustering or hierarchical structure. For instance, variable y may be the measurement ofthe level of air pollution at station _ in city j of country i at time t (e.g. Antweiler (2001),Davis (1999)). For ea se of exposition, I shall confine my presentation to a balanced panelinvolving 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 datastudies. (i) data availability, (ii) greater capacity for modeling the complexity of humanbehavior than a single cross-section or time series data, and (iii) challenging methodology.Advantages of Panel DataPanel data, by blending the inter-individual differences and intra-individual dynamicshave several advantages over cross-sectional or time-series data(i) More accurate inference of model parameters. Panel data usually contain moredegrees of granting immunity and more sample variability than cross-sectional data whichmay be viewed as a panel with T = 1, or time series data which is a panelwith N = 1, hence improving the efficiency of econometric estimates (e.g. Hsiao,Mountain and Ho-Illman (1995)._DONE_(ii) Greater capacity for capturing t he complexity of human behavior than a singlecross-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-sectionalsample of married women was found to have an add up yearly labor-forceparticipation rate of 50 percent. These could be the outcome of randomdraws from a homogeneous population or could be draws from heterogeneouspopulations in which 50% were from the population who always workand 50% never work. If the sample was from the former, each woman wouldbe expected to spend half of her married flavour in the labor force and half out ofthe labor force. The job turnover rate would be expected to be frequent and3the average job duration would be about two years. If the sample was fromthe latter, there is no turnover. The current information about a womanswork status is a perfect predictor of her future work status. A cross-sectionaldata is not able to di stinguish between these two possibilities, but panel datacan because the sequential observations for a number of women contain informationabout their labor participation in different subintervals of their lifecycle.Another example is the evaluation of the effectiveness of social programs(e.g. Heckman, Ichimura, Smith and Toda (1998), Hsiao, Shen, Wang andWang (2005), Rosenbaum and Rubin (1985). Evaluating the effectiveness ofcertain programs using cross-sectional sample typically suffers from the factthat those receiving intervention are different from those without. In otherwords, one does not simultaneously observe what happens to an individualwhen she receives the treatment or when she does not. An individual isobserved as either receiving treatment or not receiving treatment. exploitationthe difference between the treatment group and control group could sufferfrom two sources of biases, selection bias callable to differences in observablefactors between the treatment and con trol groups and selection bias due toendogeneity of participation in treatment. For instance, Northern Territory(NT) in Australia decriminalized possession of small amount of marijuanain 1996. Evaluating the effects of decriminalization on marijuana smokingbehavior by comparing the differences between NT and other states thatwere still non-decriminalized could suffer from either or some(prenominal) sorts of bias. Ifpanel data over this time period are available, it would allow the possibilityof observing the before- and affect-effects on individuals of decriminalizationas well as providing the possibility of isolate the effects of treatment fromother factors affecting the outcome.4(ii.b) Controlling the impact of omitted variables. It is frequently argued that thereal reason one finds (or does not find) certain effects is due to ignoring theeffects of certain variables in ones model specification which are correlatedwith the included explanatory variables. Panel data contain inform ationon both the intertemporal dynamics and the individuality of the entitiesmay allow one to control the effects of missing or unobserved variables. Forinstance, MaCurdys (1981) life-cycle labor supply model under sure thingimplies that because the logarithm of a workers hours worked is a linearfunction of the logarithm of her wage rate and the logarithm of workersmarginal utility of initial wealth, leaving out the logarithm of the workersmarginal utility of initial wealth from the regression of hours worked on wagerate because it is unobserved can lead to seriously biased inference on thewage elasticity on hours worked since initial wealth is likely to be correlatedwith wage rate. However, since a workers marginal utility of initial wealth remain constant over time, if time series observations of an individual areavailable, one can bind the difference of a workers labor supply equationover time to eliminate the effect of marginal utility of initial wealth on hoursworked. The ra te of vary of an individuals hours worked now dependsonly on the rate of change of her wage rate. It no longer depends on hermarginal utility of initial wealth._DONE_(ii.c) Uncovering dynamic relationships.Economic behavior is inherently dynamic so that virtually econometrically interestingrelationship are explicitly or implicitly dynamic. (Nerlove (2002)).However, the estimation of time-adjustment pattern using time series dataoften has to rely on arbitrary prior restrictions such as Koyck or Almon distributedlag models because time series observations of current and laggedvariables are likely to be highly collinear (e.g. Griliches (1967)). With panel5data, we can rely on the inter-individual differences to reduce the collinearitybetween current and lag variables to estimate unrestricted time-adjustmentpatterns (e.g. Pakes and Griliches (1984))._DONE_(ii.d) Generating more accurate predictions for individual outcomes by poolingthe data rather than generating predictions of indivi dual outcomes usingthe data on the individual in question. If individual behaviors are similarconditional on certain variables, panel data provide the possibility of learningan individuals behavior by observing the behavior of others. Thus, it ispossible to obtain a more accurate description of an individuals behavior bysupplementing observations of the individual in question with data on otherindividuals (e.g. Hsiao, Appelbe and Dineen (1993), Hsiao, Chan, Mountainand Tsui (1989)).(ii.e) Providing micro foundations for aggregate data analysis.Aggregate data analysis often invokes the case agent assumption.However, if micro units are heterogeneous, not only can the time series propertiesof aggregate data be very different from those of disaggregate data(e.g., husbandman (1990) Lewbel (1992) Pesaran (2003)), but policy evaluationbased on aggregate data may be grossly misleading. Furthermore, theprediction of aggregate outcomes using aggregate data can be less accuratethan the predi ction based on micro-equations (e.g., Hsiao, Shen and Fujiki(2005)). Panel data containing time series observations for a number of individualsis ideal for canvas the homogeneity versus heterogeneityissue.(iii) Simplifying computation and statistical inference.Panel data involve at least two dimensions, a cross-sectional dimension and atime series dimension. Under normal circumstances one would expect that the6computation of panel data estimator or inference would be more complicated thancross-sectional or time series data. However, in certain cases, the availability ofpanel 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 approximationof the distributions of the least-squares or maximum likelihood estimatorsare no longer usually distributed, (e.g. Anderson (1959), Dickey and Fuller(1979,81), Phillips and Durlauf (1986)). But if panel data are available,and obser vations among cross-sectional units are independent, then one caninvoke the central limit theorem across cross-sectional units to show that thelimiting distributions of many estimators remain asymptotically normal (e.g.Binder, Hsiao and Pesaran (2005), Levin, Lin and Chu (2002), Im, Pesaranand 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 ofmultiple observations for a given individual or at a given time may allow aresearcher to mystify different mutations to induce different and deduciblechanges 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 actualrealized value is unobserved. If an outcome variable depends on previousrealized value and the previous realized value are unobserved, one has totake consolidation over the truncated range to obtain the likelihood of observables.In a dynamic framework with multiple missing values, the multiple7integration is computationally unfeasible. With panel data, the problem canbe simplified by only focusing on the subsample in which previous realizedvalues are observed (e.g. Arellano, Bover, and Labeager (1999)).The advantages of random effects (RE) specification are (a) The number of parametersstay constant when sample size increases. (b) It allows the derivation of efficient10estimators that make use of both within and between (group) variation. (c) It allows theestimation of the impact of time-invariant variables. The disadvantage is that one hasto specify a conditional density of i given x_i = (x it, . . ., xiT ), f(i x i), while i areunobservable. A common assumption is that f(i xi) is identical to the marginal densityf(i). However, if the effects are correlated with xit or if there is a fundamental differenceamong individual units, i.e., conditional on xit, yit cannot be viewed as a random drawfrom a common distribution, common RE model is misspecified and the resulting estimatoris 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 doesit require an investigator to model their correlation patterns. The disadvantages of the FEspecification are (a) The number of unknown parameters increases with the number ofsample observations. In the case when T (or N for t) is finite, it introduces the classical sequent parameter problem (e.g. Neyman and Scott (1948)). (b) The FE estimatordoes not allow the estimation of the coefficients that are time-invariant.In order words, the advantages of RE specification are the disadvantages of FE specificationand the disadvantages of RE specification are the advantages of FE specification.To aim between the two specifications, Hausman (1978) notes that if the FE estimator(or GMM), _DONE_FE, is consistent whether i is fixed or random and the commonly used REestimator (or GLS), RE, is consistent and efficient only when i is indeed uncorrelatedwith xit and is inconsistent if i is correlated with xit.The advantage of RE specification is that there is no incidental parameter problem.The problem is that f(i x i) is in general unknown. If a wrong f(i xi) is postulated,maximizing the wrong likelihood function go out not yield consistent estimator of .Moreover, the derivation of the marginal likelihood through multiple integration may becomputationally infeasible. The advantage of FE specification is that there is no need tospecify f(i 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) DataNathaniel BeckDepartment of Po liticsNYUNew York, NY 10012emailprotectedhttp//www.nyu.edu/gsas/dept/politics/faculty/beck/beck home.htmlJan. 2004What is longitudinal data?Observed over time as well as over space.Pure cross-section data has many limitations (Kramer, 1983). Problem is that only haveone historical context.(Single) time series allows for multiple historical context, but for only one spatial location.Longitudinal data repeated observations on units observed over timeSubset of hierarchical data observations that are correlated because there is some tieto same unit.E.g. in educational studies, where we observe student i in school u. presumably thereis some tie between the observations in the same school.In such data, observe yj,u where u indicates a unit and j indicates the jth observation worn-out from that unit. Thus no relationship between yj,u and yj,u0 even though they havethe same first subscript. In professedly longitudinal data, t represents comparable time. extrapolate Least SquaresAn altern ative is GLS. If is known (up to a scale factor), GLS is fully efficient and yieldsconsistent estimates of the standard errors. The GLS estimates of _ are given by(X01X)1X01Y (14)with estimated covariance matrix(X01X)1. (15)(Usually we change by finding some trick to just do a simple transform on the observationsto make the resulting variance-covariance matrix of the errors satisfy the Gauss-Markovassumptions. Thus, the common Cochrane-Orcutt transformation to eliminate serialcorrelation of the errors is almost GLS, as is weighted regression to eliminateheteroskedasticity.)The problem is that is never known in practice (even up to a scale factor). Thus anestimate of , , is used in Equations 14 and 15. This procedure, FGLS, provides consistentestimates of _ if is estimated by residuals computed from consistent estimates of _ OLSprovides such consistent estimates. We denote the FGLS estimates of _ by _.In finite samples FGLS underestimates sampling variability (for normal errors). T he basicinsight used by Freedman and Peters is that X01X is a (weakly) concave function of .FGLS uses an estimate of , , in place of the true . As a consequence, the expectation ofthe FGLS variance, over possible realizations of , will be less than the variance, computedwith the . This holds even if is a consistent estimator of . The greater the variance of, the greater the downward bias.This problem is not severe if there are only a small number of parameters in thevariance-covariance matrix to be estimated (as in Cochrane-Orcutt) but is severe if there area lot of parameters relative to the amount of data.Beck TSCS Winter 2004 Class 1 8ASIDE Maximum likelihood would commence this right, since we would estimate all parameters andtake those into account. But with a large number of parameters in the error process, wewould just see that ML is impossible. That would have been good.PANEL selective information ANALYSIS USING SASABU HASSAN SHAARI MOHD NORFaculty of Economics and Busi nessUniversiti Kebangsaan MalaysiaemailprotectedFAUZIAH MAAROFFaculty of ScienceUniversiti Putra Malaysiaemailprotected 2007Advantages of panel dataAccording to Baltagi (2001) there are several advantages of using panel data as compared torunning the models using separate time series and cross section data. They are as followsLarge number of data lodges2)Increase degrees of freedom reduce collinearity3) Improve efficiency of estimates and4) Broaden the scope of inferenceThe Econometrics of Panel DataMichel Mouchart 1Institut de statistiqueUniversit catholique de Louvain (B)3rd March 20041 text bookstatistical modelling benefits and limita-tions of panel data1.5.1 Some characteristic features of P.D.Object of this subsection features to bear in mind when modelling P.D. Size oftenN ( of individual(s)) is largeTi (size of individual time series) is smallthusN Ti BUT this is not always the case of variables is large (often multi-purpose survey) Sampling oftenindividuals are selec ted randomlyTime is notrotating panelssplit panels _ individuals are partly renewed at each period non independent dataamong data relative to a same individual because of unobservablecharacteristics of each individualamong individuals because of unobservable characteristics commonto several individualsbetween time periods because of dynamic behaviourCHAPTER 1. INTRODUCTION 101.5.2 Some benefits from using P.D.a) Controlling for individual heterogeneityExample 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 invariantreligion ( stable over time)educationetc.Wt state invariantTV and radio advertising (national campaign)Problem many of these variables are not availableThis 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-averagesi.e. yit 7 (yit yi.)_DONE_CHAPTER 1. INTRODUCTION 11b) more information data sets larger sample size due to pooling _ individualtimedimensionIn the balanced case NT observationsIn 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 withinunits_DONE_c) better to study the dynamics of adjustment distinguishrepeated cross-sections different individuals in different periodspanel data SAME individuals in different periods cross-section photograph at one periodrepeated cross-sections different photographs at different periodsonly panel data to model HOW individuals ajust over time . This iscrucial forpolicy evaluationlife-cycle modelsintergenerational models_DONE_CHAPTER 1. INTRODUCTION 12d) Identification of parameters that would not be identified with purecross-se ctions or pure time-seriesexample 1 does union membership increase wage ?P.D. allows to model BOTH union membership and individualcharacteristics for the individuals who enter the union duringthe sample period.example 2 identifying the turn-over in the female participationto the labour market.Notice the female, or any other segment i.e. P.D. allows for more sophisticated behavioural modelse) estimation of aggregation bias often more precise measurements at the micro levelComparing the Fixed fix and the Ran-dom Effect Models2.4.1 Comparing the hypotheses of the two ModelsThe RE model and the FE model may be viewed within a hierarchical specificationof a unique encompassing model. From this point of view, the twomodels are not fundamentally different, they rather correspond to differentlevels 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 (u nique) probability measure, oneCHAPTER 2. ONE-WAY COMPONENT REGRESSION present 37may consider the complete specification of the law of (y, , _ Z, Z) asfollows(y , _, Z, Z) _ N( Z_ _ + Z, _2 I(NT)) (2.64)( _, Z, Z) _ N(0, _2 I(N)) (2.65)(_ Z, Z) _ Q (2.66)where Q is an arbitrary prior probability on _ = (_, _2 , _2). Parenthetically,note that this complete specification assumesy _2 , _, _2 , Z, Z (_, Z, Z) _2The above specification implies(y _, Z, Z) _ N( Z_ _ , _2 Z Z0 + _2 I(NT)) (2.67)Thus the FE model, i.e. (2.64), considers the distribution of (y , _, Z, Z)as the sampling distribution and the distributions of ( _, Z, Z) and(_ Z, Z) as prior specification. The RE model, i.e. (2.67), considers thedistribution of (y _, Z, Z) as the sampling distribution and the distributionof (_ Z, Z) as prior specification. Said differently, in the RE model, is treated as a latent (i.e. not obervable) variable whereas in the FE model is treated as an incidental parameter. Moreover, th e RE model is obtainedfrom the FE model through a marginalization with respect to .These remarks make clear that the FE model and the RE model shouldbe expected to display different sampling properties. Also, the inference on is an estimation problem in the FE model whereas it is a prediction problemin the RE model the difference between these two problems regards thedifference in the relevant sampling properties, i.e. w.r.t. the distribution of(y , _, Z, Z) or of (y _, Z, Z), and at last of the relevant riskfunctions, i.e. the sampling expectation of a loss due to an error between anestimated value and a (fixed) parameter or between a predicted value andthe 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. Inthe FE model, when N , the number of individuals, increases, the i s beingCHAPTER 2. ONE-WAY COMPO NENT REGRESSION MODEL 38incidental parameters also increases in number each new individual introducesa new parameter.