1、Contents12 THE PANEL DATA MODEL 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.1 Individual Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1.2 Fixed Effects and Random Effects . . . . . . . . . . . . . . . . . . . . . .
2、 512.1.3 Some Algebraic Results Related to the G Matrix . . . . . . . . . . . . . . 512.1.4 Partition of the Sum of Squares . . . . . . . . . . . . . . . . . . . . . . . 612.1.5 The Within and Between Estimators . . . . . . . . . . . . . . . . . . . . . 912.2 The Fixed-Effects Model . . . . . . . .
3、. . . . . . . . . . . . . . . . . . . . . . . 912.2.1 The Identification Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.2.2 The Least Square Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 1112.2.3 Properties of the Least Square Estimators . . . . . . . . . . . . . . .
4、 . . . 1312.2.4 The Between Estimator in the Fixed-Effects Model . . . . . . . . . . . . . 1412.2.5 The Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . 1612.3 The Random-Effects Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1712.3.1 The Generalized Least Squ
5、are Estimation . . . . . . . . . . . . . . . . . . 1812.3.2 Properties of the GLS Estimators . . . . . . . . . . . . . . . . . . . . . . 1912.3.3 Variance Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.3.4 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . .
6、 . . . 2212.3.5 Time-Invariant Regressors . . . . . . . . . . . . . . . . . . . . . . . . . . 2412.3.6 The Correlation between Regressors and the Effect . . . . . . . . . . . . . 2712.4 Some Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Chapter 12THE PANEL DATA MO
7、DEL12.1 IntroductionIf data on each individual in an economy, such as a household, a firm, a region, a country, etc., canbe collected repeatedly over time, and there are many such individuals, then we have a sample thatis called panel data, or longitudinal data. At each point of time the many indivi
8、duals consists of across-section data set, while the repeated observations on each individual over time represent timeseries. Panel data are not simply a collection of several cross-section data sets over time, since inthe latter case we do not necessarily have repeated time series data on the same
9、individuals.Because of the “two-dimensional” characteristic of the panel data, we need to change thesubscript notation for various variables. For example, the notation for the dependent variable yichanges to yit, where the first subscript i indicates a particular individual in a cross-section whilet
10、he second subscript t indicates the point of the time when individual i is observed. Given thedouble subscripts of each variable, the linear regression model for panel data is expressed as1yit D oCx0it C“it; (12.1)i D 1;:;n, and t D 1;:;T . The double subscripts also reminds us of the sample size of
11、 thepanel data, as a product of the size n of the cross-section and the number T of time periods, isgenerally quite large. However, the usefulness of panel data is more than its large sample size.The very nature of repeated observations on the same individual offers the opportunity to inferbehaviora
12、l changes of an individual over time while controlling the differences across individuals.To see this, consider the standard regression equation for the estimation of income elasticity of1We separate the constant term from all other k 1 non-constant explanatory variables in the .k 1/-dimensionalvect
13、or xit . So 0 is the intercept and is the vector of k 1 slopes. The reason for such separate treatments willbecome clear shortly.2CHAPTER 12. THE PANEL DATA MODEL 3health care:ln ci D 0C 1 ln mi C“iwhere ci and mi are the ith observations on health care expenditure and income, respectively,while the
14、 coefficient 1 is the income elasticity to be estimated. The estimation can be conductedusing cross-sectional micro data in which case i represents an individual consumer or household.Time series data can also be employed, in which case i represents a particular time and the ciand mi usually are agg
15、regate data (for the entire country, for instance) at time i.2 No matterwhich type of data are used, the above regression equation is obviously too simple since there aremany factors other than income that can influence health care consumption decision. To controlthose other factors, we may want to
16、include as many explanatory variables as possible into theregression equation. This consideration is particularly important in the cross sectional case wherethe data are collected at the micro level. Numerous personal traits like age, gender, marital status,etc., may all exert substantial influence
17、on the health care consumption decision. Moreover, anumber of individual characteristics, such as current health condition, attitude toward exercises,etc., are usually not covered by cross sectional survey data but can greatly affect the health careconsumption decision. As a result, no matter how we
18、 expand the above regression model, taheremay always be some important variables left out from the estimation.Lets now consider the case of panel data where we have intertemporal observations on each ofthe many individuals.ln cit D 0C 1 ln mit C“itIt it not hard to see that while health care consump
19、tion cit and income mit change from person toperson (over i) as well as from time to time (over t), most of those personal characteristics, eitherobserved or unobserved, differ among individuals but remain the same over time; that is, they aretime-invariant and notation for these variables do not ne
20、ed the t subscript. Hence, it seems possibleto control all those time-invariant inter-personal (i.e., cross-sectional) differences by employingthe information only from intertemporal changes in each individuals behavior. In other words,by comparing individuals behavior changes over time only, we mig
21、ht be able to eliminate theeffects of all those observed and unobserved inter-personal differences when estimating the incomeelasticity of health care. This possibility offered by the panel structure is the main advantage ofthe panel data. Certainly, in order to improve the estimation efficiency, we
22、 may also considersome information from the inter-personal behavioral differences and include it in a systematic butcontrolled manner. The goal of the present study is to try to fully utilize the structure of the paneldata and then develop consistent and efficient statistical inference.2The interpre
23、tations of the estimated income elasticities may be different between the cross-section and time seriesdata.CHAPTER 12. THE PANEL DATA MODEL 412.1.1 Individual EffectsWe can represent the effects of all those time-invariant variables by the term i and consider it asan additive component of the distu
24、rbance in (12.1) so that we have“it D i Cuit: (12.2)Here, i represents a systematic difference in yit across individuals and is generally referred to asthe individual effect. We can rewrite (12.1) asyit D oCx0it C i Cuit; i D1;:;nI t D1;:;T: (12.3)This expression provides the basic framework for the
25、 analysis of the panel data.If we collect all T observations on each individual as follows:yiT 1D26666664yi1yi2:yiT37777775; XiT .K 1/D266666664x0i1x0i2:x0iT377777775; and uiT 1D26666664ui1ui2:uiT37777775then the model can be written asyi D1T oCXi C1T i Cui; i D1;:;n; (12.4)where 1T is a T 1 vector
26、of ones. We can further stack these vectors and matrices as follows:ynT 1D26666664y1y2:yn37777775; XnT .K 1/D26666664X1X2:Xn37777775; and unT 1D26666664u1u2:un37777775;then we have an expression of the model for all observations:yD1nT oCX CG Cu; (12.5)whereGnT nDIn 1T D2666666641T 0 00 1T 0: : : :0
27、0 1T377777775CHAPTER 12. THE PANEL DATA MODEL 5and IT is the T T identity matrix.A careful inspection of the G matrix points out the fact that adding individual effects essen-tially is to expand the set of explanatory variables with n dummy variables that characterize thedifferences across the n ind
28、ividuals. Note that we use the same notation i of the individual effectsas the coefficients of these dummy variables which measure the magnitudes of individual effects.12.1.2 Fixed Effects and Random EffectsThere are mainly two approaches to the analysis of the individual effect i . One is to separa
29、tei from the disturbance and treat it as a fixed parameter to be estimated, as implied by the abovedummy variables analysis. In this approach i is referred to as a fixed effect and the correspondingpanel data model is called the fixed-effects model. The other approach is to consider i as a partof th
30、e disturbance and, therefore, a random variable. In this second approach i is referred to as arandom effect and we have a random-effects model.Generally speaking, if the sampled individuals are assumed to be drawn randomly from a gen-eral population and we try to make inference about this underlying
31、 population, then the random-effects model model should be used. However, if each of the sampled individuals is assumed to bedrawn from a individual-specific population and inference is made about these different individualpopulation, then the fixed-effects model is more appropriate.12.1.3 Some Alge
32、braic Results Related to the G MatrixBefore we proceed with the statistical analysis of the panel-data models, lets first derive a numberof useful algebraic results associated with the matrix G. Consider the pair of projection (idempo-tent) matrices based on the G matrix:PG DG.G0G/ 1G0DIn 1T.10T 1T/
33、 110T DIn 1T 1T 10T;MG DInT PG DIn hIT 1T.10T 1T/ 110TiDIn IT 1T 1T 10T:It is important to note that, given an arbitrary n J matrix C, for all nT J matrices of the formC 1T which is said to be time-invariant matrices, we havePG.C 1T/DC 1T and MG.C 1T/D0: (12.6)In particular,PG 1nT DPG.1n 1T/D1nT and
34、 MG 1nT D0: (12.7)We are also interested in the relationship between MG and the idempotent matrixM1DInT P1DInT 1nT.10nT 1nT/ 110nT: (12.8)CHAPTER 12. THE PANEL DATA MODEL 6Specifically, it is straightforward to prove the following results3MG M1DMG DM1MG M1 and PG M1DM1 MG DM1PG M1: (12.9)Both MG M1
35、and PG M1 are symmetrical and idempotent.Finally, we note that, for an nT J matrix of the form C 1T with an arbitrary n J matrixC, we haveMG M1.C 1T/D0 and PG M1.C 1T/DM1.C 1T/: (12.10)12.1.4 Partition of the Sum of SquaresGiven group sample averages of the non-constant explanatory variables xit for
36、 the n individualsNx0i D 1TTXtD1x0it D 1T 10T Xi; i D1;:;n;where Xi is the T .k 1/ matrix containing the T observations for the ith individual, and theoverall sample averageNx0D 1nTnXiD1TXtD1x0it D 1nT 10nT X;where X is the nT .k 1/ matrix containing all nT observations, lets consider the deviationf
37、ormM1XDX 1nT 1nT 10nT XDX 1nTNx0D266666664X1 1TNx0X2 1TNx0:Xn 1TNx0377777775: (12.11)Using the earlier result (12.9) on MG M1 and PG M1, we havePG M1XDIn 1T 1T 10T266666664X1 1TNx0X2 1TNx0:Xn 1TNx0377777775D2666666641TNx01 1TNx01TNx02 1TNx0:1TNx0n 1TNx0377777775; (12.12)3For the idempotent matrix P1
38、 D 1nT.10nT 1nT/110nT , it is useful to note that PG P1 D P1PG D P1 and MG P1 DP1MG D0.CHAPTER 12. THE PANEL DATA MODEL 7andMG M1XDM1X PG M1XD266666664X1 1TNx0X2 1TNx0:Xn 1TNx03777777752666666641TNx01 1TNx01TNx02 1TNx0:1TNx0n 1TNx0377777775D266666664X1 1TNx01X2 1TNx02:Xn 1TNx0n377777775: (12.13)It i
39、s interesting to note that the term PG M1X has a time-invariant structure since all T rows in eachof the n blocks are identical. That is, it can be written as of the form C 1T . More specifically,PG M1XD266666664Nx01 Nx0Nx02 Nx0:Nx0n Nx03777777751T: (12.14)Also, from the result (12.10) on MG M1, we
40、note that if the matrix X has a time-invariant structure(i.e., it is equal to C 1T for some C), then MG M1XD0. Similarly, if some columns of X havea time-invariant structure, then the corresponding columns of MG M1X will be zero so that thematrix MG M1X does not have full column rank.The partitionM1
41、XDMG M1XCPG M1X (12.15)can be expressed in terms of n blocks for individuals as follows266666664X1 1TNx0X2 1TNx0:Xn 1TNx0377777775D266666664X1 1TNx01X2 1TNx02:Xn 1TNx0n377777775C2666666641TNx01 1TNx01TNx02 1TNx0:1TNx0n 1TNx0377777775; (12.16)CHAPTER 12. THE PANEL DATA MODEL 8which is also expressibl
42、e in terms of nT rows as follows26666666666666666666664x011 Nx0:x01T Nx0. . . . . . .x021 Nx0:x02T Nx0. . . . . . .x031 Nx0:x0nT Nx037777777777777777777775D26666666666666666666664x11 Nx01:x1T Nx01. . . . . . . .x21 Nx02:x2T Nx02. . . . . . . .x31 Nx03:xnT Nx0n37777777777777777777775C2666666666666666
43、6666664Nx01 Nx0:Nx01 Nx0. . . . . .Nx02 Nx0:Nx02 Nx0. . . . . .Nx03 Nx0:Nx0n Nx037777777777777777777775: (12.17)We now consider sums of squares of xit using the above results:Txx DX0M1XDX0M1 M1XDnXiD1TXtD1.xit Nx/.xit Nx/0; (12.18)Wxx DX0M1MG M1XDX0M1MG MG M1XDnXiD1TXtD1.xit Nxi/.xit Nxi/0; (12.19)B
44、xx DX0M1PG M1XDX0M1PG PG M1XDTnXiD1.Nxi Nx/.Nxi Nx/0: (12.20)We have the following partitionTxx DWxx CBxx: (12.21)While the term Txx is called the total sum of squares, Wxx and Bxx can be referred to as thewithin and between sums of squares, respectivly. This is because Wxx measures the intertempora
45、lvariations in xit “within” the group of observations for each individual, while Bxx measures thevariations “between” individual group averages and the overall average.If, and only if, no explanatory variables are time-invariant so that no columns of X have time-invariant structure, then M1X, PG M1X
46、, and MG M1X will all have full column rank. In such a case,Wxx, Bxx, and Txx are all nonsingular .k 1/ .k 1/ matrices.Finally, we can define the three.k 1/ 1 vectors txy, wxy, bxy and the three scalars txy, wxy,bxy in a similar fashion. For example,wxy DX0M1MG M1yDnXiD1TXtD1.xit Nxi/.yit Nyi/CHAPTE
47、R 12. THE PANEL DATA MODEL 9andwyy Dy0M1MG M1yDnXiD1TXtD1.yit Nyi/2:With these items, we have the equalities:txy DwxyCbxy and tyy DwyyCbyy12.1.5 The Within and Between EstimatorsGiven the total sums of squares, we recall that the OLS estimator of the regression coefficient isbDT 1xx txy; (12.22)and
48、the corresponding sum of squared errors isSSED.y Xb/0M1.y Xb/Dtyy t0xyT 1xx txy: (12.23)It is possible to apply the same design to the other two sums of squares and define the withinestimatorO wDW 1xx wxy (12.24)and its sum of squared errorsSSEwD.y XO w/0M1MG M1.y XO w/Dwyy w0xyW 1xx wxy; (12.25)as
49、well as the between estimatorO b DB 1xx bxy (12.26)and the corresponding sum of squared errorsSSEb D.y XO b/0M1PG M1.y XO b/Dbyy b0xyB 1xx bxy: (12.27)12.2 The Fixed-Effects ModelAs mentioned earlier, if the panel data modelyit D oCx0it C i Cuit; i D1;:;nI t D1;:;T; .12:3/CHAPTER 12. THE PANEL DATA MODEL 10is assumed to be a fixed-effects model, then the n individual effects i are treated as fixed param-eters to be estimated. More specifically, i become regression coefficients of n dummy variables,which character
