Panel Data: Linear Models


 Rudolf Willis
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1 Panel Data: Linear Models Laura Magazzini University of Verona Laura Magazzini Panel Data: Linear Models 1 / 45
2 Introduction Outline What is Panel Data? Motivation: the omitted variable problem An example: Production function Model specification Estimation Laura Magazzini Panel Data: Linear Models 2 / 45
3 Introduction What is panel (or longitudinal) data? It is a timeseries of crosssection, where the same unit is observed over a number of periods Units can be individuals, firms, households, industries, markets, regions, countries,... Micro vs. Macropanels: different techniques are required for estimation Bank of Italy, European panel: large N & small T OECD: large N & small/medium/large T We work on micropanel (large N & small T ) Random sampling over the crosssectional dimension Micro & Macropanel: one of the most active bodies of literature in econometrics Laura Magazzini Panel Data: Linear Models 3 / 45
4 Introduction Basic model and notation We will consider the linear model y it = x itβ + v it with i = 1,..., N (sample units), t = 1,..., T (time periods) For each sample units, we have the following T equations: y i1 = x i1β + v i1 y i2 = x i2β + v i2. y it = x it β + v it Laura Magazzini Panel Data: Linear Models 4 / 45
5 Advantages of panel data Introduction Greater flexibility in the study of dynamics than CS or TS (ex.1) Repeated CS: in two points in time you observed 50% of women appear working. Onehalf of the women will be working? Or the some onehalf of women will be working over all time periods? [BenPorath (1973)] (ex.2) Production function: economies of scale (ES) versus technical change (TC). CS only provides information about ES. TS muddle the two effects. Greater precision in estimation (greater number of observations due to pooling) Heterogeneity across units: it is possible to disentangle different sources of variance of the units of interest (permanent versus transitory factors) Can solve the omitted variables bias (fixed effects) Consistent estimates can be obtained in the presence of omitted variables, if the omitted variable vary across sample units, but it is constant over time, e.g. preferences, individual ability, propensity to patent,... Laura Magazzini Panel Data: Linear Models 5 / 45
6 Introduction Example 1: Production function Max output given the value of the inputs Consider the case of agricultural production: Q = φ(l, V ) o Q: Output o L: Input that varies over time (labor) o V : Input that remains constant over time (soil quality) You can also think of a firm production function where V represents managerial capability Typically, V is known to the farmer/manager, but unknown to the econometrician Laura Magazzini Panel Data: Linear Models 6 / 45
7 Introduction Example 1: Econometric specification Let us consider a CobbDouglas production function: φ(l, V ) = AL α V β Taking logs (and adding an error term, summarizing all inputs outside the farmer s control, e.g. rainfall): q = a + αl + βv + u Parameter of interest: α, i.e. the (%) increase in Q driven by a 1 percent increase in L, holding V constant Laura Magazzini Panel Data: Linear Models 7 / 45
8 Introduction Example 1: Data availability Ideal world q = a + αl + βv + u You measure Q, L, and V on a sample of N farmers If standard hypotheses hold, the relationship can be estimated by OLS Real world V is not observable: you measure only Q and L on a sample of N farmers q = a + αl + (βv + u) = a + αl + ɛ Omitted variable bias? Laura Magazzini Panel Data: Linear Models 8 / 45
9 Introduction Example 1: Estimation by OLS? E[q l] = a + αl + (βe[v l] + E[u l]) = a + αl + E[ε l] OLS regression of q on l allows the identification of the parameter of interest α if and only if E[ε l] = 0 We assume E[u l] = 0, therefore we need the omitted variable v (1) not to affect q once l is controlled for, i.e. β = 0 or (2) uncorrelated with l: E[v l] = 0 We do not believe (1): soil quality affects harvest (managerial capabilities affect firm output) What does economic theory tell us about hypothesis (2)? Laura Magazzini Panel Data: Linear Models 9 / 45
10 Introduction Example 1: Relationship between L and V According to economic theory, a farmer/firm chooses L that maximizes the expected profit Let p l the cost of a unit of L, and p the price of the output Q π = AL α V β p Lp l Taking first derivatives and solving first order condition, the optimal L depends on V As a consequence, L is correlated with V : firms choose the optimal L on the basis of characteristics that are unobservable for the researcher but known to the farmer/firm! cov(v, l) 0 E[v l] 0 and, therefore, E[ε l] 0: OLS is inconsistent Laura Magazzini Panel Data: Linear Models 10 / 45
11 Introduction Example 1: The panel solution (1) The omitted variable bias is linked to the problem of endogeneity Instrumental Variable can be applied for estimation (need to search for external instruments) What if...? The soil quality/managerial ability V is constant over time Q and L are observed for (at least) T = 2 time periods Laura Magazzini Panel Data: Linear Models 11 / 45
12 Introduction Example 1: The panel solution (2) When t = 1: q i1 = a + αl i1 + βv i1 + u i1 When t = 2: q i2 = a + αl i2 + βv i2 + u i2 Taking the difference (we assume V constant over time v i1 = v i2 ): q i2 q } {{ i1 = α(l } i2 l i1 ) + u } {{ } i2 u } {{ i1 } q i l i u i The equation q i = α l i + u i does not depend from the unobserved variable v If u i satisfies classic assumptions, the regression of q i on l i can provide an estimate of the parameter of interest α. Laura Magazzini Panel Data: Linear Models 12 / 45
13 Introduction Example 1: The panel solution (3) Advantages: repeated observations over time on the same unit allows to use estimation methods that are robust to the presence of omitted variables in the model, if these variables are constant over time. Any transformation of the initial model that eliminates the unobservable variable v is a good starting point The linearity and additivity of the model are necessary in this context. Laura Magazzini Panel Data: Linear Models 13 / 45
14 Introduction Example 2: Return to schooling Aim: Study the variation in income associated to a change in the years of schooling The model of interest is: w i = α + ρs i + a i + ɛ i with w i indicates the income, s i is the number of years of schooling, a i represents individual ability (i = 1,..., N). Likely, individual ability affects income (cov(w, a) > 0) and is correlated with the years of schooling (cov(s, a) > 0) Unfortunately, a i is typically unobservable! Laura Magazzini Panel Data: Linear Models 14 / 45
15 Introduction Example 2: Identification and estimation Let us suppose we observe (w, s) for the same unit at two points in time Typically, s i does not vary over time, i.e. we look at the relationship between w and s when choices about s have already been done At time 1: w i1 = α + ρs i1 + a i + ɛ i1 At time 2: w i2 = α + ρs i2 + a i + ɛ i2 Taking differences (since s i1 = s i2 ): w i2 w i2 = ɛ i2 ɛ i1 The availability of repeated observations does not improve the identification of ρ Laura Magazzini Panel Data: Linear Models 15 / 45
16 The Omitted Variables Problem Motivation: The omitted variables problem Panel data can be used to obtain consistent estimators in the presence of omitted variables Let y and x = (x 1,..., x K ) be observable random variables Let c be an unobservable random variable We are interested in the partial effect of the observable explanatory variables x j in the population regression function: E[y x 1,..., x K, c] Assuming a linear model: E[y x 1,..., x K, c] = β 0 + x β + c, i.e. y = β 0 + x β + c + ɛ  Interest lies in the (K 1) vector β  c is called unobserved effect Laura Magazzini Panel Data: Linear Models 16 / 45
17 The Omitted Variables Problem What if cov(x, c) 0? y = β 0 + x β + c + ɛ 1 Find a proxy for c and estimate β using OLS 2 Find an external instrument for x and apply 2SLS 3 If we can observe the same units at different points in time (i.e. we can collect a panel data set), we can get consistent estimates of β as long as we can assume c to be constant over time Accomplished by transforming the original data ( internal instruments) Laura Magazzini Panel Data: Linear Models 17 / 45
18 The Omitted Variables Problem The panel solution to omitted variable bias (T = 2) Assume we can observe (y, x) at two different points in time: t = 1: (y 1, x 1 ) & t = 2: (y 2, x 2 ) The population regression function is: E[y t x t, c] = β 0 + x tβ + c or y t = β 0 + x tβ + c + u t where by definition E[u t x t, c] = 0 (t = 1, 2). What about E[c x t ]? If E[x tc] = 0, we can apply OLS If E[x tc] 0, pooled OLS is biased and inconsistent But we can take first difference and eliminate c: y 2 y } {{ } 1 = (x 2 x 1 ) } {{ } β + u 2 u } {{ } 1 y x u Laura Magazzini Panel Data: Linear Models 18 / 45
19 The Omitted Variables Problem Can we apply OLS for estimation? (T = 2) y = x β + u Exogeneity: E[ x u] = 0 E(x 2 u 2) + E(x 1 u 1) E(x 1 u 2) E(x 2 u 1) = 0 Stronger than E(x tu t ) = 0 (t=1,2) Strict exogeneity: cov(x t, u s ) = 0 for all t and s No restrictions on the correlation between x t and c Rank condition: ranke( x x) = K If x t contains a variable that is constant across time for every member of the population, then x contains an entry that is identically zero, and rank condition fails Laura Magazzini Panel Data: Linear Models 19 / 45
20 Linear Model Notation The basic linear panel data model (1) For a randomly drawn crosssection i, we assume (i = 1,..., N, t = 1,.., T ): y it = x itβ + c i + u it c i : individual effect or individual heterogeneity u it : idiosyncratic errors/disturbances Assume c i uncorrelated with u it Assume u it homeschedastic and serially uncorrelated We consider a balanced panel : each crosssection i is observed T times (total of N T observations) Laura Magazzini Panel Data: Linear Models 20 / 45
21 Linear Model Notation The basic linear panel data model (2) In compact form we can write: y i = x iβ + c i ι T + u i where vectors have dimension T 1 y i = (y i1,..., y it ) x i = (x i1,..., x it ) u i = (u i1,..., u it ) ι T = (1,..., 1) Different estimators are available on the basis of underlying assumptions on the correlation structure of c i Asymptotics rely on N, for fixed T Laura Magazzini Panel Data: Linear Models 21 / 45
22 Linear Model OLS estimation When pooled OLS? y it = x itβ + c i + u it = x itβ + v it v it : composite error, sum of the unobserved effect and idiosyncratic error OLS is consistent if E[x it v it] = 0: E[x it u it] = 0 E[x it c i] = 0, t = 1, 2,..., T Robust standard errors: the presence of c i induces correlation over time for the same individual OLS is not efficient Laura Magazzini Panel Data: Linear Models 22 / 45
23 Random effects structure Linear Model Random effect estimation y it = x itβ + c i + u it = x itβ + v it u it homoschedastic and serially uncorrelated: E[u i u i x i, c i ] = σ 2 ui T c i homoschedastic: E[ci 2 x i] = σc 2 As a result, the error structure has the following form: σc 2 + σu 2 σc 2... σc 2 Ω i = E[v i v i] σc 2 σc 2 + σu 2... σc 2 = σc σc 2 + σu 2 (T T ) E[vv ] = I N Ω i = Ω = σ 2 cι T ι T + σ2 ui T Laura Magazzini Panel Data: Linear Models 23 / 45
24 Linear Model GLS estimation (unfeasible) Random effect estimation ˆβ RE(GLS) = ( N i=1 X iω 1 i X i ) 1 ( N i=1 X i Ω 1 i y i The estimator can be obtained by applying OLS regression to Ω 1/2 X on Ω 1/2 y Ω 1/2 = [I N Ω i ] 1/2 = I N Ω 1/2 Ω 1/2 i = 1 [ σ IT u θ T ι T ι ] T with θ = 1 σ u i σ 2 u +T σc 2 The GLS estimator can be obtained by the OLS regression of (y it θȳ i ) on (x it θ x i ) If σc 2 = 0, θ = 0: RE = OLS (no unobs. heterogeneity; Breusch Pagan LM statistic) ) Laura Magazzini Panel Data: Linear Models 24 / 45
25 GLS estimation (feasible) Linear Model Random effect estimation In order to implement the RE procedure, we need to obtain ˆσ 2 c and ˆσ 2 u ( N ) 1 ( N ) ˆβ RE(FGLS) = X ˆΩ i 1 X i X i ˆΩ 1 y i i=1 To get ˆΩ (get ˆσ c 2 and ˆσ u), 2 Wooldridge suggests: σ2 c + σu 2 from pooled OLS residuals As σc 2 = E[v it v is ], autocorrelation in OLS residuals can be exploited to obtain an estimate of σc 2 ˆσ u 2 can be recovered by taking the difference σ c 2 + σu 2 ˆσ c 2 Alternative procedure described in Greene (Maddala and Mount, 1973) In small sample you can have ˆσ c 2 < 0! i=1 Laura Magazzini Panel Data: Linear Models 25 / 45
26 Linear Model Random effect estimation Random effect estimation y it = x itβ + c i + u it Obtained from the OLS regression of (y it θȳ i ) on (x it θ x i ) (in the more general case: OLS regression of Ω 1/2 y on Ω 1/2 X) Assumptions (stronger than OLS): (1) Strict exogeneity: E[x is u it] = 0 for each s, t = 1,..., T (2) Orthogonality between c i and each x it : E[c i x i ] = E[c i ] = 0 (3) Rank condition: rank E[X i Ω 1 X i ] = K, where Ω = E[v i v i ] Why REE? Exploit serial correlation of the error term in a GLS framework: efficient Laura Magazzini Panel Data: Linear Models 26 / 45
27 Linear Model Random effect estimation The strict exogeneity assumption y it = x itβ + c i + u it E[y it x i1, x i2,..., x it, c i ] = E[y it x it, c i ] = x it β + c i Once x it and c i are controlled for, x is has no partial effect on y it for s t {x it, t = 1,..., T } are strictly exogenous conditional on the unobserved effect c i The strict exogeneity assumption can be stated in terms of the idiosyncratic error term: E[u it x i1, x i2,..., x it, c i ] = 0 This implies that explanatory variables in each time period are uncorrelated with the idiosyncratic error in each time period: E[x is u it] = 0 for each s, t = 1,..., T Stronger than zero contemporaneous correlation: E[x it u it] = 0 Laura Magazzini Panel Data: Linear Models 27 / 45
28 Linear Model Fixed effect estimation Fixed effect framework We maintain the strict exogeneity assumption: E[u it x i, c i ] = 0 Allow c i to be arbitrarily correlated with x i FE is more robust than RE We can consistently estimate partial effects in the presence of timeconstant omitted variable, that can be related to the observables x i BUT we cannot include timeconstant factors in x i (e.g. gender, race in the analysis of individuals; foundation year for firms;...) To get estimates we transform the equation to remove c i and apply OLS Dummy variable regression Within transformation First difference Laura Magazzini Panel Data: Linear Models 28 / 45
29 Linear Model Dummy variable regression Least Squares Dummy Variables (LSDV) Fixed effect estimation y i = x i β + c i ι T + u i Collecting the terms over the N units gives: y 1 x 1 ι T y 2. = x 2. β + 0 ι T ι T y N x N c 1 c 2. c N + Or, letting d i be a dummy variable indicating unit i [ ] β y = [X d 1 d 2... d N ] + u = Xβ + Dc + u c Classical regression model with K + N parameters What if N is thousands? Laura Magazzini Panel Data: Linear Models 29 / 45 u 1 u 2. u N
30 Linear Model Fixed effect estimation Dummy variable regression Discussion The parameter of interest is β c i : nuisance parameters that only increase the computational complexity of estimation Incidental parameter problem: increasing N also increases the number of c i to be estimated Solution: use the within gruop (WG) transformation Numerically, LSDV and WG transformation lead to the same estimate for β (result of partitioned regression just algebra) Estimate of β easier to compute with WG (an important issue some years ago...) Laura Magazzini Panel Data: Linear Models 30 / 45
31 Linear Model Fixed effect estimation Within group (WG) transformation We transform the model in order to remove the term c i For individual i at time t: y it = x it β + c i + u it For individual i, the average over the T periods is: ȳ i = x i β + c i + ū i Therefore by taking deviations from group means, we get: y it ȳ i = (x it x i ) β + (u it ū i ) Under the assumption of strict exogeneity, we can apply OLS the the transformed data to get a consistent estimate of β Estimates of c i can be computed by ĉ i = ȳ i ˆβ x i (unbiased; not consistent for fixed T and N ) The F test can be applied for the joint significance of c i Laura Magazzini Panel Data: Linear Models 31 / 45
32 Fixed effect estimation Linear Model Fixed effect estimation y it = x itβ + c i + u it WG: OLS regression of y it ȳ i on x it x i (removes c i ) Assumptions: (1) Strict exogeneity: E[x is u it] = 0 for each s, t = 1,..., T T ) (2) Rank condition: rank( t=1 E[ẍ itẍit] = rank E[Ẍ i Ẍi] = K, where ẍ it = x it x i No assumption about the correlation of c i and each x it : consistent even if E[c i x i ] 0 More robust than RE, but effect of timeinvariant variables cannot be identified Efficient if u it homoschedastic and uncorrelated over time Laura Magazzini Panel Data: Linear Models 32 / 45
33 Linear Model Fixed effect estimation First difference (FD) Another way to remove the term c i from the equation is to take first differences: y it y it 1 = (x it x it 1 ) β + (u it u it 1 ) OLS can be applied for estimation if x it is uncorrelated with u it (satisfied under strict exogeneity) However it is not efficient, due to the correlation introduced among the error terms u it and u it 1 (if u it is uncorrelated over time) For example, for T = 3 y i2 = x i2β + (u i2 u i1 ) y i3 = x i3β + (u i3 u i2 ) GLS estimation could be employed to solve the problem: you get the withingroup estimator Laura Magazzini Panel Data: Linear Models 33 / 45
34 Linear Model Fixed effect estimation First difference estimation y it = x itβ + c i + u it FD: OLS regression of y it on x it (removes c i ) Assumptions: (1) E[ x it u it] = 0, that is E[x is u it] = 0 for each t = 1,..., T ; s = t 1, t, t + 1 satisfied under strict exogeneity (2) Rank condition: rank E[ X i X i] = K No assumption about the correlation of c i and each x it : consistent even if E[c i x i ] 0 More robust than RE, but effect of timeinvariant variables cannot be identified Laura Magazzini Panel Data: Linear Models 34 / 45
35 Linear Model Fixed effect estimation Nonspherical u it What if Ω i σ 2 cι T ι T + σ2 ui T? That is, u it heteroskedastic and/or correlated over time If E(c i x i ) 0, then the FE estimator is still consistent (under strict exogeneity); it is no longer efficient Robust formulas should be employed for the computation of the standard errors! ˆβ FD is efficient if u it is a random walk ( u it serially uncorrelated) If E(c i x i ) = 0 (the orthogonality condition holds), then the RE estimator remains consistent (under strict exogeneity); it is no longer efficient A more general estimator of Ω i can be obtained as: ˆΩ i = N 1 with ˆv i pooled OLS residuals (efficient in the more general case) Assume alternative specifications: parametric assumptions about the correlation structure in u it, e.g. AR(1) and perform GLS estimation Laura Magazzini Panel Data: Linear Models 35 / 45 N i=1 ˆv i ˆv i
36 WG vs. FD Which one to choose? Linear Model Which one to choose? WG: OLS regression of (y it ȳ i ) on (x it x i ) FD: OLS regression of y it on x it Both WG and FD produces unbiased and consistent estimates of the parameter of interest β, as c i is removed from the regression The estimate of β is not affected by the correlation (if any) between c i and x i Generally, if the two estimators are different, this can be interpreted as evidence against the assumption of strict exogeneity When T = 2, ˆβ WG = ˆβ FD If T 3, under homoschedasticity of u, ˆβ WG is to be preferred because efficient If uncorrelation and homoschedasticity of u is not satisfied, the choice depends on the assumptions about u it : If u it is a random walk, then u it is serially uncorrelated: ˆβFD is efficient In the more general set up, use FD or WG with robust s.e.! Laura Magazzini Panel Data: Linear Models 36 / 45
37 FE vs. RE (1) Which one to choose? Linear Model Which one to choose? Traditional approach: c i treated either as parameter to be estimated vs. random disturbance Philosophical issue Wrongheaded in microeconometrics applications Modern terminology: fixed effects estimation vs. random effects estimation The difference is in the assumptions about E[c i x i ] FE allows consistent estimation of β even in cases where c i is correlated with x i RE requires c i to be uncorrelated with x i Laura Magazzini Panel Data: Linear Models 37 / 45
38 FE vs. RE (2) Which one to choose? Linear Model Which one to choose? FE: OLS regression of (y it ȳ i ) on (x it x i ) Only within variation is considered RE: OLS regression of (y it θȳ i ) on (x it θ x i ) Both within and between variation are employed for estimation It is possible to show that ˆβ RE = Λ ˆβ B + (I K Λ) ˆβ FE with ˆβ B obtained from the OLS regression of ȳ i on x i σ θ = 1 u : if T, RE = FE you need a different framework! σ 2 u +T σ 2 c Key: correlation between c i and x it If E[c i x it ] = E[c i ] (= 0): RE is consistent and efficient, FE consistent If E[c i x it ] E[c i ]: FE consistent, but RE is not Laura Magazzini Panel Data: Linear Models 38 / 45
39 Linear Model Which one to choose? FE vs. RE The Hausman test Both FE and RE assume strict exogeneity If E[c i x] = E[c i ] (= 0) Both ˆβ FE and ˆβ RE are consistent for β: ˆβFE ˆβ RE 0 ˆβ RE is efficient: Var( ˆβ FE ) is greater than Var( ˆβ RE ) If E[c i x] E[c i ] ˆβ FE is consistent, but ˆβ RE is biased: ˆβFE ˆβ RE 0 We can apply the Hausman test ( ˆβ FE ˆβ RE ) (Var( ˆβ FE ) Var( ˆβ RE )) 1 ( ˆβ FE ˆβ RE ) χ 2 K Remark: Two maintained hypotheses (not tested!): (i) strict exogeneity; (ii) random effect structure of the covariance (under the null, RE has to be efficient: valid under spherical u it ) Laura Magazzini Panel Data: Linear Models 39 / 45
40 Linear Model Which one to choose? Between FE and RE: Correlated random effects (Mundlak, 1978; Chamberlain, 1982, 1984) RE assumes no correlation between c i and x it Richer models can be specified that relax this assumption Mundlak (1978): c i = x i π + w i with w i i.i.d. GLS estimation of the regression of y it on x it and x i produces the fixed effect estimator Chamberlain (1982, 1984): c i = x i1 π x it π T + w i Estimation of the extended model by minimum distance method produces the fixed effect estimator In nonlinear models, fixed effect models are not always estimable and richer RE models provide an alternative approach Laura Magazzini Panel Data: Linear Models 40 / 45
41 Linear Model Which one to choose? FE vs. RE A robust version of the Hausman test Starting from the Mundlak (1978) definition (linear projection): c i = x i π + w i with w i i.i.d. we can write: y it = x itβ + c i + u it = x itβ + x i π + (w i + u it ) GLS estimation produces: ˆβ GLS = ˆβ FE and ˆπ GLS = ˆβ BET ˆβ FE ( ˆβ BET : OLS estimate in the regression of ȳ i on x i ) Hausman test can be carried out by testing H 0 : π = 0 in the extended regression Robust version of the Hausman test: use a robust Wald statistic in the context of pooled OLS (strict exo is still needed, but we can relax on efficiency of RE under the null) Laura Magazzini Panel Data: Linear Models 41 / 45
42 The R 2 with panel data Goodness of fit R 2 as the square of correlation coefficient between observed and fitted values Total variability can be decomposed into within and between variability: 1 NT i,t (y it ȳ) 2 = 1 NT STATA provides three R 2 statistics: Rwithin 2 = corr 2 ((x it x i ) ˆβFE, y it ȳ i ) Rbetween 2 = corr 2 ( x i ˆβ B, ȳ i ) Roverall 2 = corr 2 (x it ˆβ OLS, y it ) i,t (y it ȳ i ) NT (ȳ i ȳ) 2 i,t Laura Magazzini Panel Data: Linear Models 42 / 45
43 Discussion Discussion Source of the examples: Wooldridge Two questions: Is the unobserved effect c i uncorrelated with x it for all t? Is the strict exogeneity assumption (conditional on c i ) reasonable? Examples: (a) Program evaluation log(wage it ) = θ t + z itγ + δ 1 prog it + c i + u it (b) Distributed Lag Model (Hausman, Hall, Griliches, 1984) patents it = θ t + z itγ + δ 0 RD it + δ 1 RD it δ 5 RD it 5 + c i + u it (c) Lagged Dependent Variable log(wage it ) = β 1 log(wage it 1 ) + c i + u it Laura Magazzini Panel Data: Linear Models 43 / 45
44 Main References Main References Baltagi BH (2001): Econometric Analysis of Panel Data, John Wiley & Sons Ltd. Chamberlein G (1984): Panel Data, in Griliches and Intriligator, (eds.) Handbook of Econometrics, Vol.2, Elsevier Science, Amsterdam Greene, WH (2003): Econometric Analysis, Prentice Hall, ch.13 Hsiao C (2003): Analysis of Panel Data, Cambridge University Press Mundlak Y (1978): On the Pooling of Time Series and Cross Section Data, Econometrica, 46(1), Verbeek M (2006): A Guide to Modern Econometrics, ch. 10 Wooldridge, JM (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press: Cambridge, ch.10 Laura Magazzini Panel Data: Linear Models 44 / 45
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