Lecture 12 Panel Data
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- Merry Thornton
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1 Lecture 12 Panel Data Economics 8379 George Washington University Instructor: Prof. Ben Williams
2 Introduction This lecture will discuss some common panel data methods and problems. Random effects vs. fixed vs. alternatives IV in panel data Lagged dependent variables Binary Outcome panel data models
3 The linear panel model Basic model and assumptions: y it = β x it + η i + ν it A1 E(ν i1,..., ν it x i1,..., x it, η i ) = 0 A2 Var(ν i1,..., ν it x i1,..., x it, η i ) = σ 2 I T These assumptions can be replaced by weaker but harder to interpret assumptions.
4 Differencing and within variation Some notation first: y i = (y i1,..., y it ) x i = (x i1,..., x it ) ν i = (ν i1,..., ν it ) The basic idea you ve seen before: In matrix notation, y it = β x it + ν it and E( ν it x it ) = 0 Dy i = Dx i β + Dν i where D is the (T 1) T first difference operator.
5 Differencing and within variation The fixed effects regression is not ( n i=1 x i D Dx i ) 1 n i=1 x i D Dy i, though this first differences estimator would be consistent under assumption A1.
6 Differencing and within variation The fixed effects regression is not ( n i=1 x i D Dx i ) 1 n i=1 x i D Dy i, though this first differences estimator would be consistent under assumption A1. Because Var(Dν i x i ) = σ 2 DD, the GLS estimator is more efficient, ˆβ fe := ( n i=1 x i D (DD ) 1 Dx i ) 1 n i=1 x i D (DD ) 1 Dy i
7 Differencing and within variation But Q = D (DD ) 1 D is idempotent and equal to I T ιι /T. This is the within-group operator. The fixed effects estimator is based on within variation. The fixed effects estimator is equivalent to including entity dummies.
8 Differencing and within variation Properties of the fixed effects (or within-group) estimator: For a fixed T, ˆβ fe is unbiased and optimal 1, and as n it is consistent and asymptotically normal. Estimates of η i are unbiased but only consistent if T. If T then ˆβ fe is consistent, even if n is fixed.
9 Differencing and within variation Robust standard errors: If A2 does not hold then the usual standard error formula for OLS on the transformed data is inconsistent. If T is fixed and n is large then the clustered (on entity) standard error formula provides a HAC estimator. If T is large and n is fixed then a Newey West type std error estimator is required for consistency under serial correlation.
10 Differencing and within variation Under serial correlation in ν it, the fixed effects estimator is not optimal. Let νi = Dν i. Generally, if Var(νi x i ) = Ω(x i ) then the GLS estimator is ( n i=1 x i D Ω(x i )Dx i ) 1 n i=1 x i D Ω(x i )Dy i In the special case where Var(νi x i ) = Ω, replace Ω(x i ) with n ˆΩ = n 1 ˆν i ˆν i i=1 to get a feasible GLS estimator.
11 Random effects Pooled OLS estimator is ( n ˆβ pooled = x i x i i=1 ) n i=1 x i y i It s unbiased and consistent only under the assumption that E(η i x it ) = 0. Under assumption A2 and Var(η i x i ) = σ 2 η, Var(η i ι + ν i x i ) = σ 2 ηιι + σ 2 I T
12 Random effects The GLS estimator is then ( n ) n ˆβ GLS = x i V 1 x i x i V 1 y i i=1 where V 1 = σ 2 ( I T σ 2 ηιι /(σ 2 + T σ 2 η) ). This is the random effects estimator. When T, this becomes the fixed effects estimator. More generally, if ψ = σ 2 η/(σ 2 + T σ 2 η) goes to 0 we get fixed effects and if ψ goes to 1 we get pooled OLS. i=1
13 Random effects Feasible GLS Estimate ψ in first stage to get estimate of ˆV. Several ways to estimate ψ. This is what xtreg...,re in Stata does. An alternative is the maximum likelihood estimator that will estimate β and σ and σ 2 η simultaneously. the usual MLE assumes that η i N(0, σ 2 η) though different distributions can be used.
14 Random effects vs fixed effects The primary difference between the two is that random effects assumes η i is uncorrelated with x it. The idea of fixed (non-random) versus random effects is not the real distinction. Mundlak (1978) showed that the fixed effects estimator is equivalent to a random effects type (GLS) estimator of the model where η i = a x i + ω i where ω i is independent of x i. Not true in nonlinear models!
15 Measurement error Motivating example Bover and Watson (2000) consider a simplified version of the model from Arellano (2003) Conditional money demand equation: y it denotes cash holdings (real money balances) of firm i in year t x it denotes sales η i = log(a i ) where a i denotes a firm s financial sophistication
16 Measurement error Suppose x it = x it + ε it and the true regressor values, x it are unobserved. Fixed effects can exacerbate measurement error bias:
17 Measurement error Suppose x it = x it + ε it and the true regressor values, x it are unobserved. Fixed effects can exacerbate measurement error bias: The measurement ( error ) bias in the FE estimator when T = 2 is β λ where λ = Var( ε it )/Var( x it ) If ε it and x it are both iid then this attentuation bias is identical to the cross-sectional bias. If ε it is iid but x it is positively serially correlated then the bias is larger than in the cross-section.
18 Measurement error When T > 2, ε it is iid and x it is positively serially correlated Griliches and Hausman (1986) show that the bias of the fixed effects estimator lies between the bias of pooled OLS and that of OLS in first-differences.
19 Measurement error When T > 2, ε it is iid and x it is positively serially correlated Griliches and Hausman (1986) show that the bias of the fixed effects estimator lies between the bias of pooled OLS and that of OLS in first-differences. Panel IV can be a solution to the measurement error problem when ε it is not serially correlated and x it is. If η i is independent (random effects/pooled OLS model) then E( x is (y it β x it )) = 0 for s t
20 Measurement error If η i is correlated with x it, one solution is to take first differences and use the moment conditions E( x is ( y it β x it )) = 0 for s = 1,..., t 2, t + 1,..., T This requires T 3. Also, the rank condition should be considered carefully. What if x it is white noise? What is x it is a random walk? What if x it = α i + ξ it? With larger T, there is a tradeoff between allowing serial correlation in ε it and needing serial correlation in x it.
21 Measurement error Table from Bover and Watson (2000):
22 Measurement error The relationship among the pooled OLS, FE, and first difference estimators is consistent with measurement error in sales. Column (4) is GMM on first differences using other time periods as instruments. The Sargan test here is also marginally suggestive of measurement error. Columns (5) and (6) seem to correct for measurement error and are consistent with the expectation that pooled OLS should be downward biased.
23 Panel IV Start with the RE/pooled model where y it = β x it + u it. Moment conditions: E(z i u i) = 0 where z i is a T r matrix of instruments. Given r r weighting matrix W N, the panel GMM estimator is ( 1 n n n n ˆβ PGMM = ( x i z i)w N ( z i i)) x ( x i z i)w N ( z i y i) i=1 i=1 i=1 i=1
24 Panel IV The weighting matrix doesn t matter if r = dim(β) so that the model is just identified. The one step GMM estimator (which is just 2SLS) uses W N = ( n i=1 z i z i) 1 As we ve seen, this is optimal under homoskedasticity. The two step GMM estimator calculates a robust estimate of the asymptotic variance of n 1 n i=1 z i u i. This is Ŝ and then W N = Ŝ 1.
25 Panel IV Moment conditions. Pooled OLS is equivalent to taking z i = x i. Note that the moment conditions are then of the form: E( T x it u it ) = 0 t=1
26 Panel IV Moment conditions. Pooled OLS is equivalent to taking z i = x i. Note that the moment conditions are then of the form: E( T x it u it ) = 0 t=1 The moment conditions E(z it u it ) = 0 can be enforced by defining Z i as the Tdim(z it ) T block diagonal matrix with z i1,..., z it on the diagonal.
27 Panel IV Moment conditions. Pooled OLS is equivalent to taking z i = x i. Note that the moment conditions are then of the form: E( T x it u it ) = 0 t=1 The moment conditions E(z it u it ) = 0 can be enforced by defining Z i as the Tdim(z it ) T block diagonal matrix with z i1,..., z it on the diagonal. You get even more moment conditions under weak exogeneity (T (T 1)r) or strong exogeneity (T 2 r). Careful of the weak/many instruments finite sample bias!
28 Panel IV with fixed effects Suppose u it = η i + ν it. The GMM estimators take the same form as before except applied to y it = β x it + ν it where y it, x it, and ν it represent some differencing transformation.
29 Panel IV with fixed effects Suppose u it = η i + ν it. The GMM estimators take the same form as before except applied to y it = β x it + ν it where y it, x it, and ν it represent some differencing transformation. The available moment conditions depend on what type of differencing is used. Random vs. fixed The relevant distinction now is whether we want to assume that E(Z i u i) = 0 or E(Z i ν i ) = 0; the former requires any fixed effect η i to be uncorrelated with Z i, though not necessarily with X i.
30 Panel IV with fixed effects Suppose u it = η i + ν it. The GMM estimators take the same form as before except applied to y it = β x it + ν it where y it, x it, and ν it represent some differencing transformation. The available moment conditions depend on what type of differencing is used. Random vs. fixed The relevant distinction now is whether we want to assume that E(Z i u i) = 0 or E(Z i ν i ) = 0; the former requires any fixed effect η i to be uncorrelated with Z i, though not necessarily with X i. Under the random effects assumption it is again possible to take advantage of the error component structure to improve efficiency.
31 Strong vs weak exogeneity Assumption A1 for the fixed effects estimator was that E(ν it x i, η i ) = 0 This is strong exogeneity. This implies no feedback x it cannot be informed by ν is for s < t so it rules out lagged dependent variables x it = y i(t 1) We will focus on the lagged dependent variable problem and consider two basic solutions: figure out which past values can be used as instruments fixed effects is ok for large T
32 AR model without fixed effects Consider as a simple example the autoregressive model: y it = αy i(t 1) + βx it + ν it What happens if we do fixed effects? With T = 2, y i2 y i1 = (α 1)y i1 + β(x i2 x i1 ) + βx i1 + ν i2 Regressing y i2 y i1 on x i2 x i1 produces a bias that depends on the sign of α 1, of Cov(y i1, x i2 x i1 ), of β and of Cov(x i1, x i2 x i1 ).
33 AR model without fixed effects Consider as a simple example the autoregressive model: y it = αy i(t 1) + βx it + ν it What happens if we do fixed effects? With T = 2, y i2 y i1 = (α 1)y i1 + β(x i2 x i1 ) + βx i1 + ν i2 Regressing y i2 y i1 on x i2 x i1 produces a bias that depends on the sign of α 1, of Cov(y i1, x i2 x i1 ), of β and of Cov(x i1, x i2 x i1 ). In some cases, the effect estimate from the AR model without fixed effects and the fixed effects estimate (without lagged y it ) bound the true parameter but this is not always true.
34 AR model with fixed effects Consider as a simple example the autoregressive model: y it = αy i(t 1) + η i + ν it B1 E(ν it y t 1 i, η i ) = 0 B2 E(νit 2 y t 1 i, η i ) = σ 2 B3 (mean stationarity) E(y i0 η i ) = η i /(1 α) B4 (covariance stationarity) Var(y i0 η i ) = σ 2 /(1 α 2 ) The fixed effects estimator has a bias that is equal to (1 + α)/2 when T = 2 approximately (1 + α)/t for large T This is called the Nickell bias due to pioneering work of Nickell (1981).
35 AR model with fixed effects Without assumptions B3 and B4 the bias is more complicated. E.g., if T = 2 and σ 2 η/var(ν i1 ) is large then the bias is very small. What if T is large but the same order of magnitude as n? Formally, if n/t c > 0 then nt (ˆαfe α) N( c(1 + α), (1 α 2 )/(nt )) For moderate values of T, a bias-corrected estimator: ˆα fe,bc = ˆα fe ˆα fe T
36 IV solution Anderson and Hsiao (1981, 1982) suggested using an IV estimator that uses y i(t 2) or y i(t 2) as an instrument for y it when T 3 or T 4.
37 IV solution Anderson and Hsiao (1981, 1982) suggested using an IV estimator that uses y i(t 2) or y i(t 2) as an instrument for y it when T 3 or T 4. There are potentially many more moment conditions under assumption B1: E(y t 1 i ( y it α y i(t 1) )) = 0, t = 2,..., T
38 IV solution Holtz-Eakin, Newey, and Rosen (1988) and Arellano and Bond (1991) suggest implementing a GMM estimator that uses all (T 1)T /2 moment conditions. The Arellano Bond estimator uses a one-step optimal weighting matrix that accounts for serial correlation due to differencing, n ˆV = z i DD z i i=1 There is a bias however when n T that is proportional to 1/n.
39 IV solution Advice: When T is larger than n, use FE. When n is larger than T, use Arellano-Bond. When n is similar in magnitude to T, use bias-correction or limited number of instruments/moments.
40 With lagged dependent variables and other regressors Suppose y it = αy i(t 1) + β x it + η i + ν it. Use additional lags of x it as instruments for y i(t 1) this helps when ν it is serially correlated. tradeoff between exogeneity assumptions on x it and serial correlation in ν it, especially for small T Again, fixed effects will be consistent, and sometimes preferred, for large T
41 Extensions Arellano and Bover (1995) and Blundell and Bond (1998) suggest also using differences to instrument for levels. Use additional moments for time invariant regressors. (Hausman and Taylor, 1981) Using covariance structure to improve efficiency.
42 Static binary choice panel model Static model: T Pr(y i x i, η i ) = F(β x it + η i ) t=1 The log likelihood function is l(β, {η i }) = n T log(f (β x it + η i )) i=1 t=1 The log of the integrated likelihood function is ( n T ) l(β) = log F(β x it + η i )f η x (η i x i )dη i i=1 t=1
43 Static model Random effects models are based on the integrated likelihood. Random effects probit/logit assume that f η x = f η. Similar assumption to RE in linear models. Similarly, this is more efficient than a pooled probit/logit estimator but potentially more efficient by accounting for cross-equation dependence. The Mundlak/Chamberlain approach: η i = a x i + ω i, or η i is some other function of x i. This is implemented using the integrated likelihood with f η x (η i x i ) = f ω(η i a x i ) This reduces to T t=1 F(β x it + a x i + ω i )f ω(ω i )dω i Cannot identify β k if x itk is time invariant.
44 Static model Fixed effects models are based on the full likelihood, l(β, {η i }) Treat the η i as separate parameters. This introduces the incidental parameter problem (Neyman and Scott, 1948). The fixed effects estimator is biased for a fixed T, but is consistent as T. If T and n are of similar magnitude, or T is smaller, then FE doesn t work. When T and n are of similar magnitude bias corrections have been suggested (see work of Fernandez-Val and others)
45 Static model Conditional logit: In the logit model, when T = 2, exp(x i1 Pr(y i1 = 0, y i2 = 1 y i1 +y i2 = 1, x i ) = β) exp(x i1 β) + exp(x i2 β) This conditional likelihood estimator is implemented in Stata via clogit Not logit with i.caseid!! For larger T, condition on T t=1 y it. This approach works for dynamic logit and multinomial logit models as well.
46 Dynamic model A dynamic model: T Pr(y i x i, η i ) = Pr(y it y i,t 1, x it, η i ) t=1 Dynamic logit, fixed effects Random effects The initial conditions problem. Predetermined x it is difficult to model.
47 Linear probability model In practice a linear probability model is often used That is, y it = β x it + η i + ν it, despite the fact that y it is binary. This allows for fixed effects, various types of endogeneity, Arellano and Bond GMM estimator, etc.
48 Acemoglu et al. line Results 4.1 Baseline Results Our main linear regression model takes the form yct = βdct + The effect of democracy on income. j=1 inear regression model takes the form baseline model: p y ct = βd ct + γ j y ct j + α c + δ t + ε ct, j=1 p γjyct j + αc + δt + εct, (1) where yct is the log of GDP per capita in country c at time t, and Dct is our dichotomous measure of democracy in country c at time t. The αc s denote a full set of country fixed effects, which will absorb the impact of any time-invariant country characteristics, and the δt s denote a full set of year effects. The error term εct includes all other time-varying unobservable shocks to GDP per capita. The specification includes p lags of log GDP per capita on the right-hand side to control for the dynamics of GDP as discussed in the Introduction. We impose the following assumption: the log of GDP main per exogeneity capita in country c at time t, and D ct is our dichoto Assumption 1 (sequential exogeneity): E(εct yct 1,..., yct0, Dct,..., Dct0, αc, δt) = 0. assumption cy in country c at time t. The This is the α c standard s denote assumption awhen full dealing set with of dynamic country panel models. fixed It implieseffec that democracy and past GDP are orthogonal to contemporaneous and future error terms, and that the impact of any time-invariant error term country εct is serially uncorrelated. characteristics, and the δ t s denot Assumption 1 effectively requires sufficiently many lags of GDP to be included in equation (1). The error term ε ct includes both to eliminate all residual other serial time-varying correlation in the error term unobservable of this equation and to remove shock the pre-democratization dip in GDP. specification includes p lags We alsoof assume log (andgdp test below) that per GDPcapita and democracy on are stationary the right-hand processes (conditional s on country and year fixed effects). Under Assumption 1 and stationarity, equation (1) can be estimated using the standard within estimator. 7 Columns 1-4 of Table 2 report the results of this
49 Introduction Random/FE/GLS for the dynamics IVof GDP as discussed Laggedin Ythe Introduction. Binary outcome models Application. Acemoglu et al. (2008, 2015) P We impose the following assumption: Acemoglu et al. Assumption 1 (sequential exogeneity): E(ε ct y ct 1,..., y ct0, D ct,..., D ct0, α c, δ t) = 0. This is the standard assumption when dealing with dynamic panel models. It implies that democracy and past GDP are orthogonal to contemporaneous and future error terms, and that the error term ε ct is serially uncorrelated. Assumption 1 effectively requires sufficiently many lags of GDP to be included in equation (1) both to eliminate residual serial correlation in the error term of this equation and to remove the pre-democratization dip in GDP. We also assume (and test below) that GDP and democracy are stationary processes (conditional on country and year fixed effects). Under Assumption 1 and stationarity, equation (1) can be estimated using the standard within estimator. 7 Columns 1-4 of Table 2 report the results of this estimation controlling for different numbers of lags on our baseline sample of 175 countries between the years of 1960 and Throughout, the reported coefficient of democracy is multiplied by 100 to ease its interpretation, and we report standard errors robust against heteroskedasticity. The first column of the table controls for a single lag of GDP per capita on the right-hand side. In a pattern common with all of the results we present in this paper, there is a sizable amount of persistence in GDP, with a coefficient on lagged (log) GDP of (standard error = 0.006). 7 For future reference, we note that this involves the following within transformation, ( ) ( ) ( ) yct 1 ycs = β Dct 1 p Dcs + γj yct j 1 ycs j + δt + εct 1 εcs, Tc Tc Tc Tc s s j=1 s s with Tc being the number of times a country appears in the estimation sample. The within estimator has an asymptotic bias of order 1/T when Dct and yct j are sequentially exogenous but not strictly exogenous, and when both series are cointegrated (this is the case if both are stationary as assumed). Thus, for long panels, as the one we use, the within estimator provides a natural starting point.
50 Introduction Random/FE/GLS coefficients for the dynamics and IV just of GDP report asthe discussed Lagged p-valuein Y for thea Introduction. joint Binary test outcome of significance, models Application. which suggests Acemoglu they do et not al. (2008, 2015) P jointly Weaffect impose current the following GDP dynamics. assumption: The overall degree of persistence and the long-run effect of Acemoglu et al. democracy on GDP per capita are very similar to the estimates in column 3. Assumption 1 (sequential exogeneity): E(ε ct y ct 1,..., y ct0, D ct,..., D ct0, α c, δ t) = 0. The within group estimates of the dynamic panel model in columns 1-4 have an asymptotic biasthis of order is the 1/T standard, which is assumption known, after when Nickell dealing (1981), with as dynamic the Nickell panelbias. models. This It bias implies results that from the democracy failure of and strict pastexogeneity GDP are orthogonal in modelstowith contemporaneous lagged dependent and future variables erroron terms, the right-hand that the side as error in our termequation ε ct is serially (1) (Nickell uncorrelated. 1981, Alvarez and Arellano 2003). Because T is fairly large in our panel Assumption (on average, 1 effectively each country requires is observed sufficiently 38.8many times), lagsthis of GDP biasto should be included be small in equation in our setting, (1) motivating both to eliminate our useresidual of the model serial correlation in columnsin 1-4the aserror the baseline. term of 10 this equation and to remove the pre-democratization The rest of Table dip 2 reports in GDP. various GMM estimators that deal with the Nickel bias, and produce consistent We alsoestimates assume (and of the testdynamic below) that panel GDPmodel and democracy for finite T are. stationary Sequentialprocesses exogeneity (conditional implies the following on country moment and year conditions fixed effects). Under Assumption 1 and stationarity, equation (1) can be estimated using the standard within estimator. 7 Columns 1-4 of Table 2 report the results of this estimation controlling fore[(ε different ct εnumbers ct 1)(y cs, of Dlags cs+1) on ] = our 0 baseline for all ssample t 2. of 175 countries between the years of 1960 and Throughout, the reported coefficient of democracy is multiplied by 100 Arellano and Bond (1991) develop a GMM estimator based on these moments. In columns 5-8, we to ease its interpretation, and we report standard errors robust against heteroskedasticity. report estimates from the same four models reported in columns 1-4 using this GMM procedure. The first column of the table controls for a single lag of GDP per capita on the right-hand side. Consistent with our expectations that the within estimator has at most a small bias, the GMM In a pattern common with all of the results we present in this paper, there is a sizable amount estimates are very similar to our preferred specification in column 3. The only notable difference of persistence in GDP, with a coefficient on lagged (log) GDP of (standard error = 0.006). is that GMM estimates imply slightly smaller persistence for the GDP process, leading to smaller 7 For future reference, we note that this involves the following within transformation, long-run effects. For example, ( column 7, ) which corresponds ( to the GMM ) estimates ( analogous ) to our yct 1 ycs = β Dct 1 p Dcs + γj yct j 1 ycs j + δt + εct 1 εcs, preferred specification Tc in column Tc 3, shows a long-run impact Tc of 16.45% (standard Tc error=8.436%) s s j=1 s s on with GDP Tc being per capita the number following of times a permanent a country appears transition in the to estimation democracy. sample. The within estimator has an asymptotic bias of order 1/T when Dct and yct j In addition, we also report the p-values are of asequentially test for serial exogenous correlation but not in strictly the residuals exogenous, of and equation when both series are cointegrated (this is the case if both are stationary as assumed). Thus, for long panels, as the one we (1) use, which, the within as estimator requiredprovides by Assumption a natural starting 1, tests point. whether there is AR2 correlation in the differenced residuals. The p-values for this test indicate that this assumption is not rejected when we include
51 Introduction Random/FE/GLS coefficients for the dynamics and IV just of GDP report asthe discussed Lagged p-valuein Y for thea Introduction. joint Binary test outcome of significance, models Application. which suggests Acemoglu they do et not al. (2008, 2015) P jointly Weaffect impose current the following GDP dynamics. assumption: The overall degree of persistence and the long-run effect of Acemoglu et al. democracy on GDP per capita are very similar to the estimates in column 3. Assumption 1 (sequential exogeneity): E(ε ct y ct 1,..., y ct0, D ct,..., D ct0, α c, δ t) = 0. The within group estimates of the dynamic panel model in columns 1-4 have an asymptotic biasthis of order is the 1/T standard, which is assumption known, after when Nickell dealing (1981), with as dynamic the Nickell panelbias. models. This It bias implies results that from the democracy failure of and strict pastexogeneity GDP are orthogonal in modelstowith contemporaneous lagged dependent and future variables erroron terms, the right-hand that the side as error in our termequation ε ct is serially (1) (Nickell uncorrelated. 1981, Alvarez and Arellano 2003). Because T is fairly large in our panel Assumption (on average, 1 effectively each country requires is observed sufficiently 38.8many times), lagsthis of GDP biasto should be included be small in equation in our setting, (1) motivating both to eliminate our useresidual of the model serial correlation in columnsin 1-4the aserror the baseline. term of 10 this equation and to remove the pre-democratization The rest of Table dip 2 reports in GDP. various GMM estimators that deal with the Nickel bias, and produce consistent We alsoestimates assume (and of the testdynamic below) that panel GDPmodel and democracy for finite T are. stationary Sequentialprocesses exogeneity (conditional implies the following on country moment and year conditions fixed effects). Under Assumption 1 and stationarity, equation (1) can be estimated using the standard within estimator. 7 Columns 1-4 of Table 2 report the results of this estimation controlling fore[(ε different ct εnumbers ct 1)(y cs, of Dlags cs+1) on ] = our 0 baseline for all ssample t 2. of 175 countries between the years of 1960 and Throughout, the reported coefficient of democracy is multiplied by 100 Arellano and Bond (1991) develop a GMM estimator based on these moments. In columns 5-8, we to ease its interpretation, and we report standard errors robust against heteroskedasticity. report estimates from the same four models reported in columns 1-4 using this GMM procedure. The first column of the table controls for a single lag of GDP per capita on the right-hand side. instruments Consistent with ourbias expectations so they that also within estimator use ahas weak-instrument at most a small bias, the GMM In a pattern common with all of the results we present in this paper, there is a sizable amount estimates are very similar to our preferred specification in column 3. The only notable difference of persistence in GDP, with a coefficient on lagged (log) GDP of (standard error = 0.006). is that GMM estimates imply slightly smaller persistence for the GDP process, leading to smaller Kuersteiner 7 For future reference, (2003) we note that this involves the following within transformation, long-run effects. For example, ( column 7, ) which corresponds ( to the GMM ) estimates ( analogous ) to our yct 1 ycs = β Dct 1 p Dcs + γj yct j 1 ycs j + δt + εct 1 εcs, preferred specification Tc in column Tc 3, shows a long-run impact Tc of 16.45% (standard Tc error=8.436%) s s j=1 s s on with GDP Tc being per capita the number following of times a permanent a country appears transition in the to estimation democracy. sample. The within estimator has an asymptotic bias of order 1/T when Dct and yct j In addition, we also report the p-values are of asequentially test for serial exogenous correlation but not in strictly the residuals exogenous, of and equation when both series are cointegrated (this is the case if both are stationary as assumed). Thus, for long panels, as the one we (1) use, which, the within as estimator requiredprovides by Assumption a natural starting 1, tests point. whether there is AR2 correlation in the differenced The Arellano-Bond approach is susceptible to many/weak robust method for panel data due to Hahn, Hausman, and residuals. The p-values for this test indicate that this assumption is not rejected when we include
52 Acemoglu et al. TABLE 2: EFFECT OF DEMOCRACY ON (LOG) GDP PER CAPITA. Within estimates Arellano and Bond estimates HHK estimates (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 42 Democracy (0.294) (0.248) (0.226) (0.245) (0.477) (0.417) (0.374) (0.378) (0.467) (0.357) (0.346) (0.321) log GDP first lag (0.006) (0.038) (0.038) (0.039) (0.009) (0.041) (0.041) (0.038) (0.009) (0.046) (0.047) (0.038) log GDP second lag (0.037) (0.046) (0.043) (0.038) (0.045) (0.042) (0.043) (0.055) (0.043) log GDP third lag (0.028) (0.028) (0.028) (0.027) (0.026) (0.024) log GDP fourth lag (0.017) (0.034) (0.020) (0.033) (0.017) (0.028) Long-run effect of democracy (13.998) (8.595) (7.215) (7.740) (10.609) (9.152) (8.436) (7.829) (7.790) (6.269) (9.031) (9.597) Effect of democracy after 25 years (5.649) (5.550) (5.297) (5.455) (7.281) (7.371) (7.128) (6.653) (6.086) (5.311) (6.814) (6.815) Persistence of GDP process (0.006) (0.005) (0.005) (0.007) (0.009) (0.009) (0.009) (0.009) (0.009) (0.009) (0.008) (0.008) Unit root test adjusted t stat p value (rejects unit root) [0.000] [0.000] [0.000] [0.000] AR2 test p-value Observations 6,790 6,642 6,336 5,688 6,615 6,467 6,161 5,513 6,615 6,467 6,161 5,513 Countries in sample Notes: This table presents estimates of the effect of democracy on log GDP per capita. The reported coefficient on democracy is multiplied by 100. Columns 1-4 present results using the within estimator. Columns 5-8 present results using Arellano and Bond s GMM estimator. The AR2 row reports the p-value for a test of serial correlation in the residuals. Columns 9-12 present results using the HHK estimator. In all specifications we control for a full set of country and year fixed effects. Columns 4, 8 and 12 include 8 lags of GDP per capita as controls, but we only report the p-value of a test for joint significance of lags 5 to 8. Standard errors robust against heteroskedasticity and serial correlation at the country level are reported in parentheses.
53 Basic panel commands Stata has a suite of panel commands such as xtreg But you can use cross-sectional tools to do most of this, sometimes with some difficulty. I won t go over the basics but some commands you should be familiar with: xtset reshape egen xtreg xtsum, xttab,xttrans
54 Panel regression in Stata xtreg can be used to estimate pooled OLS, FE, and RE The first differences estimator can be estimated using reg D.(yvar xvar1 xvar2), options Standard errors that are robust to heterosk. and serial correlation vce(cluster id) or vce(robust) for xtreg Both of these can handle unbalanced panels as well. Other FGLS estimators can be implemented manually or, in some cases, using xtreg, pa, xtgls or xtgee Comparing FE and RE use hausman to conduct the specification test
55 Panel IV estimation in Stata ivregress and ivreg2 can be used for most panel IV regressions if the data is appropriately transformed and the instrument vector is appropriately defined. This is useful sometimes because it gives more transparent control over what moment conditions are used. The xtivreg command, however, can be easier to work with.
56 Panel IV estimation in Stata xtivreg: options fe, fd,re allow different transformations of data this command uses what Cameron and Trivedi call the summation moment conditions it differences the instruments as well
57 Panel IV estimation in Stata lagged dependent variables: one solution is to work with the first differences model, using ivregress or xtivreg xtabond can be used to implement versions of the Arellano Bond (1991) estimator that incorporate other instruments
58 Panel IV estimation in Stata xtabond: this only works if you want to include a lagged dependent variable you can flexibly specify how many lags to include you can flexibly specify which regressors are exogenous, endogenous, or predetermined we will see some examples of the syntax in a few minutes
59 Other panel IV commands in Stata xtabond2 is a user-written code that has a few additional features the command xtdpdsys command implements Arellano and Bover (1995) and Blundell and Bond (1998) xthtaylor implements an alternative panel IV estimator (Hausman-Taylor)
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