21. Panel: Linear A. Colin Cameron Pravin K. Trivedi Copyright 2006

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1 21. Panel: Linear A. Colin Cameron Pravin K. Trivedi Copyright 2006 These slides were prepared in They cover material similar to Chapter 21 of our subsequent book Microeconometrics: Methods and Applications, Cambridge University Press, 2005.

2 Table of Contents D Introduction D2 Overview of Basic Panel Data Estimators D Example: Patents and R&D De Fixed Effects Model DD Random Effects Model DS Constant Coef cients Model D. Modelling Issues DH Extensions Db Nonlinear Models 2

3 INTRODUCTION Data on the same cross-section (of rms or individuals or countries) that is observed over several years is called panel data or longitudinal or cohort data. In microeconometrics most often a short panel where? $4and A f (even A '2). This has different theory to long panel where? xed and A $4used in time series applications such as cross-region studies 3

4 For panel data The error is correlated over. Usual reported standard errors are then too small and statistics too large. Richer models permit regression coef cients to vary across individuals. In particular, the intercept. Muchalgebra is used. This is to obtain analytical formula so large matrices need not be inverted. 4

5 PATENTS - R&D EXAMPLE Annual data on 346 U.S. rms for ten years Source: Hall, Griliches and Hausman (1986). Patents = number of patents applied for during the year that were eventually granted. R&D = real R&D spending during the year in 1972 $ Might expect that doubling R&D doubles patents *?EPatents ' k n q *?ER&D n 0 c where k is the rm-speci c component. When Patents equals f we set ln(patents)' *?EfD. 5

6 OVERALL: + ' k n 3 n 6 4 Ln(Patents) Ln(R&D) Overall regression: Ln(Patents) on LOG(R&D) 1.OLS on all 346 rms in all 10 years. 6

7 BETWEEN: 7+ ' k n 7 3 n Ln(Patents) Ln(R&D) Between: Ln(PATENTS) on LOG(R&D) 2.OLS on 10-year averages for 346 rms. 7

8 WITHIN (or Fixed Effects) + ' k n 3 n 0, 7+ ' k n 7 3 n70, + 7+ 'E 7 3 ne0 70 8

9 WITHIN (cont.) 4 Ln(Patents) Ln(R&D) Within (fixed effects) regression: Ln(Patents) on LOG(R&D 3.OLS of diff from means for 346 rms and 10 years. 9

10 FIRST-DIFFERENCED + ' k n 3 n 0, + c ' k n 3 c n 0 c, + + c 'E 3 c 3 ne0 0 c 10

11 . FIRST-DIFFERENCED (cont.) 4 2 DLPAT DLOGR First differences regression: dln(patents) on dln(r&d) 4.OLS of rst differences for 346 rms and 10 years. 11

12 RANDOM EFFECTS GLS in the model + ' 3 nek n 0 k Efcj 2 k, 0 Efcj 2 0 same as OLS in the model + w7+ 'E w> ne w7 3 ne0 w70 w ' j 0 *EAj 2 k n j 2 0 *2 12

13 CC CC-rob Betwn Within 1st Diff RE-GLS k (.021) (.052) (.054) (.037) (.043) (.052) q (.009) (.021) (.054) (.029) (.013) (.019) R-sq ESS TSS j 2 k j ?

14 BASICLINEARMODELS A general linear model for panel data is + ' k n 3 n 0 c ' cc?c ' cc Ac where + is a scalar dependent variable is a & vector of independent variables 0 is a scalar disturbance term. Usually indexes individual (or rm or country) in a cross-section and indexes time. 14

15 The distinguishing feature is: The intercept k and slope coef cients may differ across individuals and/or across time. Such variation in coef cients reàects: individual-speci c and time-speci c effects. 15

16 But the preceding model is too general and is not estimable. Further restrictions are placed on the extent to which k and vary with and, and on the behavior of the error 0. 16

17 INDIVIDUAL SPECIFIC EFFECTS The one-way individual-speci c effect model lets just the intercept k vary across individuals. + ' k n 3 n 0 The two-way individual-speci c effect model also allows for time-speci ceffects + ' k n B n 3 n 0 For short panels we work with one-way model as there will be just a few B and these can be included in as time-speci c indicator variables. 17

18 For this model the two standard models are The xed effects model. This treats k as a parameter to be estimated. Problem is that there are? of these and? $4 Therandom effects model. This treats k as a random variable i.i.d. with mean f and variance j 2 k. 18

19 CONSTANT COEFFICIENTS MODEL Specialization where k ' > is the same for all individuals + ' > n 3 n 0 If 0 is i.i.d. then OLS is consistent and ef cient. But for panel data this model is too restrictive. Ifdgpis xedeffects then OLS is inconsistent. If dgp is random effects then OLS is still consistent, though is inef cient and the usual reported OLS standard errors will be incorrect. 19

20 FIXED EFFECTS MODEL The xed effects model is + ' k n 3 n 0 The individual-speci c effects k are xed coef cients to be estimated from the data. There are? of them. It is hoped that controlling for individual effects in this way leaves an error 0 that is i.i.d. Efcj 2. This is an obvious simple model Individual-speci c effects due to different intercepts. But keep slope coef cients the same over individuals. 20

21 OLS of + on only gives inconsistent estimates of. The usual consistent estimator is most commonly called the within groups estimator. There are several ways to obtain this estimator 1. OLS on a model with dummy variables for each of the? xed effects. 2. OLS after a differencing transformation that eliminates k :OLSofE+ 7+ on Ef GLS after this same differencing transformation. We go through these in detail. Takes time. 21

22 FIXED EFFECTS: OLS ESTIMATOR OLS estimation is ef cient if the error is i.i.d.. Implement by de ning a set of? indicator variables _ c ' + if ' c ' c c? f if i 9' Then + ' _ c k n n _?c k? n 3 n 0 c and OLS gives estimates of k c c k? and q. For this reason the xed effects model is also called the dummy variable model. 22

23 Conceptually this is the easiest way to proceed. But in many applications it is not practical as the number of regressors,? n A n &, may be too large to permit estimation by a package OLS program. Fortunately, there is an alternative simpler way to compute the estimator. 23

24 FIXED EFFECTS: DIFFERENCING ESTIMATOR For xed effects + ' k n 3 n 0, 7+ ' k n 7 3 n70 averaging over time, E+ 7+ 'E 7 3 ne0 70 Thus OLS of E+ 7+ on E 7 provides consistent estimates of. This is called the within-group estimator. It is ef cient and can be shown to equal the dummy variable estimator. 24

25 FIXED EFFECTS: ESTIMATOR DISTRIBUTION For 0 i.i.d. the within estimator has 9DUd e )( o'j 2 5 7?[ ' 6 A[ E 7 E ' We can also estimate the intercepts e c)( ' e )( c ' cc? This estimator of k is consistent only if? $4. So for short panel this is inconsistent. 25

26 FIXED EFFECTS: MECHANICS Here we present OLS for the xed effects model and show that for this yields the within estimator. The model is + ' k n 3 n 0 For individual stack all A observations Or A 6 : 8 ' : 8 k n A 6 : 8 n A 6 : 8 c 26

27 ) ' ik n j n % c Here i 'E c c c 3 is a A vector of ones and j is a A & matrix. 27

28 Now stack over all? individuals Or Here ) G )? 6 : 8 ' if f f... f ff i 6 5 : k G k? 6 : 8 n j. j? 6 : 8 n % & ) 'dew? i jo n %c % G %? 6 : 8 c EW? i is an?a? block-diagonal matrix and j is the?a & matrix of nonconstant regressors. Since 9DUdo 'j 2 W?A we apply OLS. 28

29 This yields % e)( e )( & ' % EW? i 3 EW? i EW? i 3 j j 3 EW? i j 3 j & % EW? i 3 ) j 3 ) & We want to compare this to differencing estimator. 29

30 At this stage it is useful to use the following properties of the A vector i 'E c c c 3 1.i 3 i ' A. 2.EW? i 3 EW? i 'W? i 3 i 'A W? 3.ForA6matrix ~ : i 3 ~ ' A7~ 3, where 7~ 'E *A S A ' ~ is a 6 vector. 4.ForA 6 matrix ~ 'd~ 3 ~3?o 3 : EW? i 3 ~ 'di 3 ~ i 3 ~? o'a7~, where 7~ 'd7~ 3 7~ 3?o is a? 6 matrix. 30

31 Then % e)( e )( & ' % EW? i 3 EW? i EW? i 3 j j 3 EW? i j 3 j & % EW? i 3 ) j 3 ) & ' % A W? A 7j A 7j 3 j 3 j & % 7) j 3 ) & Here the sample means 7j 'd7j 3 7j 3?o 3, 7j 'E *A S A ' j, 7) 'd7+ 7+? o 3 and 7+ 'E *A S A ' +. 31

32 Using the formula for partitioned inverse, further algebra leads to the solution Or % e)( e )( & ' % 7) 7j e & )( dj 3 j 7j 3 7jo Ej 3 ) 7j 3 7) e )( ' 5?[ 7 ' 6 A[ Ej 7j Ej 7j 3 8? [ ' ' A[ Ej 7j E+ 7+ ' 32

33 It follows that the xed effects estimator for the model can be obtained by 1.Get e )( by OLS of the differenced data E+ 7+ on Ej 7j 2. Calculate e c)( '7+ 7j 3 e )(. 33

34 FIXED EFFECTS: FURTHER MECHANICS We return to the differencing estimator presented earlier. De ne the A A differencing matrix " ' W A A ii3 Then for A 6 matrix ~ "~ ' ~ i7~ ' A[ E~ 7~ ' 34

35 Useful properties of " are 1. "i ' f as "i ' W A i E *A ii 3 i ' i E *A ia ' i i ' f 2. "" 3 ' " as "" 3 ' W A E *A ii 3 E *A ii 3 n E *A ii 3 E *A ii 3 ' W A E *A ii 3 ' " 3."~ ' ~ i7~ for A 6 matrix ~ and 6 vector 7~ 'E *A S A ' ~ as "~ ' W A ~ E *A ii 3 ~ ' ~ E *A ia 7~ ' ~ i7~. 4. E"~ 3 E"~ 'E~ i7~ 3 E~ i7~ ' S A ' E~ 7~ E~ 7~ 3 35

36 Premultiply the stacked equation ) ' ik n j n %, ") ' "ik n "j n "%, ") ' "j n "% as "i ' f, E) 7+ 'Ej 7j ne% 7% Here 9DUd"% o'"9dud% o" 3 ' j 2 "" 3 ' j 2 " " has generalized inverse " where " 3 " " ' ". 36

37 GLS estimator using independence over is 5 e ' 7 5 ' 7 5 ' 7 6?[ j 3 "3 " "j 8 ' 6?[ j 3 "j 8 '?[ j 3 "3 " "+ '?[ j 3 "+ ' 6?[ Ej 7jEj 7j 3 8? [ ' ' Ej 7jE

38 Thus this is the same estimator as 1. That obtained by more simple OLS estimation of the transformed equation. 2.The xedeffects estimator. 38

39 RANDOM EFFECTS MODEL The random effects model is where now + ' > n k n 3 n 0 c individual-speci ceffectsk are i.i.d. Efcj 2 k. 0 is i.i.d. Efcj 2 0. a non-random intercept > is added so the random effects can be normalized to have zero mean. 39

40 This is a random coef cient model, withonlythe intercept coef cient being random. This can be re-expressed as + ' > n 3 n c where the error has two components ' k n 0 For this reason it is also called the random components model. 40

41 RANDOM EFFECTS ESTIMATION OLS of + on intercept and is consistent, inef cient with usual standard erors are wrong. OLS of E+ 7+ on E 7 is consistent, inef cient with usual standard erors are wrong. The ef cient estimator is GLS MLE if additionally assume errors are normal. 41

42 RANDOM EFFECTS GLS GLS allowing for correlated errors is ef cient. Start with + ' > n 3 n k n 0 For individual stack observations over time + ' i> n j n ik n % ' h j % where i is a A vector of ones. The variance of the A vector error ' ik n % > & n c 42

43 is or l'(deik n % Eik n % 3 o'(d% % 3 on(dk2 oii3 l'j 2 0W A n j 2 kii 3 43

44 GLS using 0 independent of 0, 9',gives % & e> 5( e 5( where h j 3 'di ' 5 7 j o?[ ' 6 hj 3 l j h 8 [ The matrix l is of dimension A A. ' hj 3 l + c Forlongpanelsthismaybeoftoolargedimensionto invert. So use analytical expression for l. 44

45 RANDOM EFFECTS GLS... Use l ' j 2 0dEW A E *A ii 3 ne *A ii 3 oc where ' j 2 0*Ej 2 0 n Aj 2 k After some algebra can be expressed for the slope coef cients as e 5( '{ e % new & { e )( Here e )( is the within groups estimator. And e % is the between groups estimator. 45

46 5 e % ' 7 6?[ E 7j 7jE 7j 7j 3 8 ' The & & weighting matrix is? [ ' E 7j 7jE7+ 7+c {' 5?[ 7 j 3 "j n A ' 6?[ E 7j 7jE 7j 7j 3 8 '?[ E 7j 7jE 7j 7j 3 ' 46

47 RANDOM EFFECTS GLS... The estimator of the intercept is e -. '7+ 7j e 5( For more details see, for example, Hsiao (1986, p.36) or Greene (1993, p.472). To implement these estimators additionally requires estimates of the variance components j 2 0 and j 2 k. 47

48 From the within groups (or xed effects) regression we obtain ej 2 0 ' E+ 7+ 'Ej 7j 3 ne0 70 c?ea & [ [ EE+ 7+ Ej 7j 3 5( e 2 From the between groups regression we obtain ej 2 k '? E& n 7+ ' > n 7j 3 n k n70 [ E7+ e> 7j 3e % 2 A ej2 0 48

49 More ef cient estimators of the variance components j 2 0 and j 2 > are possible, see Amemiya (1985), but will not necessarily increase the ef ciency of e 5(. The above regressions for the within groups and between groups estimators make it clear that consistency requires? $4. 49

50 RANDOM EFFECTS MLE Now assume the errors are normally distributed. Then maximize the log-likelihood function with respect to, >, j 2 0 and j 2 >. Then for given j 2 0 and j 2 > the MLE for and > is the same as the GLS estimator. But the MLE gives different estimators for the variance components j 2 0 and j 2 > than those given above, and hence different and >. 50

51 RANDOM EFFECTS: CORRELATION Random effects model lets &RYd c r o'j 2 k for all 9' r. Thus correlation between errors for the same individual in different time periods is the same regardless of how far apart the time periods are. This is reasonable in some applications models, such as different members ( ) inthesamefamily() or different measurements ( ) of the same quantity ( ). But it may be unreasonable when denotes time. Then generalize to allow 0 to be serially correlated. 51

52 FIXED VS. RANDOM EFFECTS There is a dif cult literature on the conceptual differences between xed effects and random effects. Fixed effects is a conditional analysis. We look at the effect of on + controlling for the individual effect k. Prediction is possible for individuals in this particular sample, but not for other individuals with different and unknown individual effects. The random effects analysis is a marginal analysis. 52

53 The individual effects are integrated out as i.i.d. random variables. Model estimates can be applied outside the sample. If analysis is for a random sample of countries then do random effects. If intrinsically interested in the particular countries in the sample then do xed effects. 53

54 FIXED VS. RANDOM EFFECTS... There are also practical differences between the two. The xed effects model cannot estimate a coef cient of any time-invariant regressor, such as sex, since the individual intercepts can take a different value for each individual. Similarly, problems arise for a model with xed time effects, such as a nationwide unemployment rate. A model with random effects does not suffer from this problem. 54

55 The random effects model requires that the random effects (part of the error term) be uncorrelated with the regressors. In many situations this will not be the case. E.g. in a model of returns to schooling a random individual effect may capture unobserved ability that is correlated with the years of schooling regressor variables. Mundlak (1978) showed that if the individual effects are determined by k ' 7j 3 k n then the random effects estimator is biased, while the GLS estimator is the 55

56 xed effects estimator. The random effects model with regressors correlated with the random effects is discussed further below. A related result is that the OLS regression of + on j is inconsistent if the xed effects model is the d.g.p. This inconsistency would disappear, however, if the xed effects were uncorrelated with the regressors. This is precisely the assumption made in the random effects model. 56

57 IMPLEMENTATION Models easily estimated using a canned program. The main practical issue is presenting data in the order expected by the program. Can rst stack all years for a given individual and then stack all observations. Or can rst stack all individuals for a given year and then stack all years. For actual data with small A and? largetherecanbe quite a difference between xed and random effects estimates. An example is given in Hsiao (1986, p.41). 57

58 APPLICATION: PATENTS DATA From Hall, Griliches, and Hausman (1986), Patents and R&D: Is There a Lag?, IER 27: rms by 5 years PAT = number of patents applied for during the year that were eventually granted. LOGR = Logarithm of R&D spending during the year (in 1972 dollars). AVELOGR = Average of LOGR for current and past 5 years. 58

59 Variable Coeff St. Errors u7 FE RE u7 FE RE T.uC- 2eD D b2 fs 2D 2 (t.-2 f ef (t.- f. fb ef (t.-e eb e eh ef (t.-d S eb DH ef 7AA.D b 2-2 es b. e 59

60 Similar results for year effects. Big difference for effect of AVELOGR. Note: PAT has average of 35 and standard deviation 71. This is a potential problem. R-squared varies as expected. 60

61 RANDOM COEFFICIENTS MODELS Random effects model permits only the intercept term to be randomly distributed across individuals. Can additionally allow the slope coef cients to vary. For example ) ' k n 3 n 0 c where both k is i.i.d. Ekc j 2 k and q is i.i.d. EqcP q. For details see Greene (1993, pp ). These models are more widely used in statistics than in econometrics. 61

62 ENDOGENOUS VARIABLES A weakness of the random effects model is that it leads to inconsistent parameter estimates if the random effects are correlated with the regressors, as they often will be. Obvious solution is to instead use the xed effects model, but then we could not identify the effects of time-invariant regressors. Hausman and Taylor (1981) proposed estimation of the random effects model by instrumental variables to 62

63 overcome this problem. 63

64 They consider + ' 3 n 3 n k n 0 The error 0 is i.i.d. Efcj 2 0 uncorrelated with the regressors. The random effects k are i.i.d. Efcj 2 k and potentially correlated with some of the regressors. The time-invariant regressors are separately included, since if all regressors were time-varying we could estimate all the parameters by xed effects. A constant term is included in. 64

65 ENDOGENOUS VARIABLES... In matrix form the model is ) ' j n ` n n % First pre-multiply the system by l *2 where l ' 9DUd n %o to obtain l *2 ) 'l *2 j nl *2` n c where the error term is an i.i.d. error. This transformed system is estimatedbyiv. 65

66 The instruments E" ~ c j c ` are " ~ ' W ~E~ 3 ~ ~ 3 ~ 'EW? i is the?a? vector of ones and zeroes j,and` are the components of f and ` that are uncorrelated with k. Thechoiceoff and ` as instruments is obvious. Theeffectofusing" ~ as instruments is to use the time-varying regressors that are uncorrelated with twice as means and as deviations as means. Other choices of instruments leading to more ef cient 66

67 IV estimators are given in Amemiya and MaCurdy (1986) and Breusch et al. (1989). A similar approach may be taken if instead the regressors are correlated with the error 0. We would again premultiply the system by l *2 before performing instrumental variables estimation. The dif cult part, as usual, is selecting the instruments. 67

68 ENDOGENEITY TESTS... Can test for whether the random effects component of the error is correlated with the regressors. This is a Hausman test of M f G (d mj c ~ o'f against G (d mj c ~ o 9' f. One possibility is to look at the difference between e 5( consistent and ef cient under M f but inconsistent under e )( consistent but inef cient under M f and consistent under 68

69 AUTOCORRELATED ERRORS Analysis to date assumes 0 are i.i.d. Now allow 0 to be serially correlated, but continue to assume that there are no lagged dependent variables. The estimators above will remain consistent, but be inef cient and have incorrect reported standard errors as in the pure time series case. 69

70 For xed effect models, see MaCurdy (1982). This paper gives a Box-Jenkins type analysis for identi cation and estimation of ARMA processes for 0 for short panels. We note that in this case it is not necessary to assume an ARMA process for 0 or even stationarity, since for? $4and A xed we can always consistently estimate &RYd c r o by E *? S e e r. But nonetheless we may be interested in knowing the ARMA process for the errors. 70

71 For random effects model, see Baltagi and Li (1991). This paper considers only AR(1) errors, but can be easily generalized to the AR(p) case. Methods for MA and ARMA errors have been developed more recently. A summary is given in Baltagi (1995). 71

72 HETEROSKEDASTIC AND AUTOCORRELATED ERRORS Consider the constant coef cients model + ' > n 3 n with no individual efects. Methods exist for errors that are both serially correlated and cross-sectionally heteroskedastic (of unspeci ed form). Consider method of Kmenta. 72

73 Heteroskedasticity of unspeci ed form (d 2 o'j2 c where errors uncorrelated for different individuals. First-order autoregressive disturbances with individualspeci c AR(1) coef cient ' 4 n 0 c where 0 is i.i.d. Efcj 2 0. It follows that &RYd c r o'4 r j 2. OLS estimates are consistent, but inef cient. 73

74 Kmenta s procedure is 1. Estimate the original model by OLS 2. From OLS residuals form e4 ' S e e * S e e 3. Estimate by OLS the Prais-Winsten transformed model + e4 + c 'E e4 k ne e4 c 3 n 0 with modi cation for the case ' E e4 2 *2 + 'E e4 2 *2 k ne e4 2 *2 j n 0 4. From step 3. residuals form ej 2 'EA & S e2 *E e4 2 74

75 5. Estimate original model by GLS using ej 2 and e4 to estimate the covariance matrix of the errors. This can be performed by OLS estimation of the model in step 3 where all variables (including the constant) are divided by ej 2. For more details see Kmenta (1986) and also Greene (1993). This procedure requires both A $ 4(for the AR(1) coef cients) and? $4(for heteroskedasticity). Biggest weakness is assumption that coef cients do not vary across individuals and/or time. 75

76 DYNAMIC MODELS Suppose regressors include lagged dependent variables + c. Then analysis ecomes considerably more complicated. References include Anderson and Hsiao (1981) and Nickell (1981). 76