Inference on Trending Panel Data

Transcription

1 Inference on rending Panel Data Peter M Robinson and Carlos Velasco London School of Economics and Universidad Carlos III de Madrid November 3, 15 Abstract Semiparametric panel data modelling and statistical inference with fractional stochastic trends, nonparametrically time-trending individual effects, and general cross-sectional correlation and heteroscedasticity in innovations is developed. he fractional stochastic trends allow for a wide range of nonstationarity, indexed by a memory parameter, nesting the familiar I1 case and allowing for parametric short-memory. he individual effects can nonparametrically vary simultaneously across time and across units. he cross-sectional covariance matrix is also nonparametric. he main focus is on estimation of the time series parameters. wo methods are considered, both of which entail an only approximate differencing out of the individual effects, leaving an error which has to be taken account of in our theory. In both cases we obtain standard asymptotics, with a central limit theorem, over a wide range of possible parameter values, and unlike the nonstandard asymptotics for autoregressive parameter estimates at a unit root. For statistical inference, consistent estimation of the limiting covariance matrix of the parameter estimates requires consistent estimation of a functional of the cross-sectional covariance matrix. We examine effi ciency loss due to cross-sectional correlation in a spatial model example. A Monte Carlo study of finite-sample performance is included. Keywords: Semiparametric panel data modelling, Nonparametrically timetrending individual effects, Nonparametric cross-sectional correlation and heteroscedasticity, Spatial model, Parametric fractional dependence, Consistency, Asymptotic normality. JEL classifications: C1, C13, C3... Corresponding author. el fax address: p.m.robinson@lse.ac.uk 1

2 1. Introduction A considerable literature has developed around the theme of nonstationary panel data with individual and temporal effects. he nonstationarity can have both stochastic and deterministic origins. Unit roots in an autoregressive setting have been the main representation of stochastic trends, while individual and temporal effects are often modelled in a separable, additive way, so that temporal effects are common over the cross-section and individual ones stay constant over time, and explanatory variables, including linear and other deterministic trends, can also feature. Autoregressive models are also commonly adopted in the nonstationary time series and cointegration literature. However, the latter has also developed fractional time series modelling, which can nest I1 behaviour in a continuum of nonstationary possibilities, with the degree of nonstationarity characterized by a memory parameter, which can be estimated from the data. his approach has the advantages of flexibility and of yielding standard asymptotics, and possible local effi ciency of inference, unlike the nonstandard theory which usually emerges from an autoregressive setting. Likewise, the fixed-design nonparametric regression literature suggests a modelling of deterministic trends which is less prone to specification error than parametric functions. Robinson and Velasco 15 employed fractional models with individual effects in a panel data setting, but with no provision for other time-trending features such as deterministic trends. Another aspect of their work was the assumption of crosssectional independence and homoscedasticity, conditional on individual effects, which is increasingly seen as restrictive, and sometimes replaced by factor modelling, or spatial modelling. he present paper considers semiparametric modelling of panel data, such that dynamic parametric fractional stochastic trends are complemented by stochastic or deterministic nonparametrically time trending individual effects and allowance for cross-sectional correlation and heteroscedasticity of a nonparametric form, entailing greater generality than factor or spatial models. he number of time series observations,, is large, and the number of cross-sectional ones, N, can be large increasing with or small. he latter setting is also considered by Robinson and Velasco 15, but our introduction of nonparametric temporal variation of individual effects and relaxation of cross-sectional independence and homoscedasticity strengthens the need for an asymptotic theory based on increasing. Large- panel data are increasingly available, for example very long time series of prices of several stocks, monthly or quarterly macroeconometric series for several countries, and repeated micro-economic surveys. We consider an observable array {y it }, i = 1,..., N, t =, 1,...,. he vectors of

3 N cross-sectional observations y t = y 1t,..., y Nt, the prime denoting transposition, are assumed to be generated by the semiparametric model λ t L; θ y t α t = ε t, t =, 1,...,. 1 he parametric aspect of 1 is due to the p parameter vector θ, which is known only to lie in a given compact subset Θ of R p+1. In 1 L is the lag operator, and for any θ Θ and each t, λ t L; θ is a scalar function given by λ t L; θ = t λ j θ L j, j= where the λ j θ are given functions to be defined subsequently. he unobservable vectors ε t have elements with zero mean and are uncorrelated and homoscedastic across t. hus, 1 is an autoregressive representation for the y t α t with initial condition at t =, where the number of terms in keeps increasing with t. he assumptions on λ t L; θ that we will impose are aimed at covering fractionally integrated autoregressive moving average FARIMA sequences y t α t, with unknown memory parameter that can lie in either the stationary or nonstationary regions. Other aspects of 1 are nonparametric. We do not require uncorrelatedness or homoscedasticity across the elements of ε t, allowing it to have covariance matrix that is unrestricted apart from remaining positive definite with increasing N, thereby to reflect possible crosssectional correlation and heteroscedasticity of a nonparametric nature, though it can also be assumed to be diagonal, to reflect a lack of cross-sectional correlation but the possibility of nonparametric heteroscedasticity. he vectors α t consist of stochastic or deterministic unobservable individual effects that can time-trend in a nonparametric way, and in a manner that can vary across elements of the vector, where with N increasing the familiar incidental parameters problem arises. Our allowance for temporally varying individual effects and cross-sectional correlation and heteroscedasticity of innovations, and our relaxation of temporal independence of innovations to martingale difference structure, extend the scope of the model of Robinson and Velasco 15 to a practically significant degree, but as a consequence reduces the range of memory parameter values covered and limits the degree to which N can increase relative to. he main goal of the paper is to justify statistical inference on the unknown parameter vector θ. Detailed regularity conditions, later employed in establishing the properties of consistency and asymptotic normality, are described in the following section. In Section 3 an estimate of θ, based on initial first-differencing of 1, is shown to be consistent and asymptotically normal, and feasible statistical inference is justified. In Section 4 a more refined estimate, with some advantages, is similarly 3

4 analysed. Section 5 employs a spatial model to illustrate relative effi ciency. A Monte Carlo study of finite-sample performance is reported in Section 6. Section 7 contains some final comments. he proofs of theorems are included in an appendix.. heoretical setting he present section presents detailed regularity conditions on the model introduced in the previous section, which will be assumed to hold in our theoretical results. he function λ t L; θ defined in is regarded as truncating the expansion λ L; θ = λ j θ L j, j= which has the structure λ L; θ = δ ψ L; ξ, where δ is a scalar, ξ is a p 1 vector, θ = δ, ξ and the functions δ and ψ L; ξ are described as follows. Defining the difference operator = 1 L, δ has the expansion δ = π j δ L j Γj δ, π j δ = Γ δγj + 1, j= for non-integer δ >, while for integer δ =, 1,..., π j δ = 1j =, 1,..., δ 1 j δ δ 1 δ j + 1 /j!, taking / = 1 and 1. to be the indicator function; ψ L; ξ is a known function of its arguments such that for complex-valued x, ψ x; ξ =, x 1 and is continuously differentiable in ξ, and in the expansion ψ L; ξ = ψ j ξ L j, j= the coeffi cients ψ j ξ satisfy ψ ξ = 1, ψj ξ + ψj ξ = O exp c ξ j, 3 where ψ j ξ = / ξ ψ ξ and c ξ is a positive-valued function of ψ. Note that λ j θ = j π j k δ ψ k ξ, j. 4 k= In general it is assumed that 3 holds for all ξ Ξ with c ψ satisfying inf Ξ c ξ = c >. 5 4

5 We impose the identifiability condition that, for all ξ ξ, ψ x; ξ ψ x; ξ on a subset of {x : x = 1} of positive Lebesgue measure. Define φ L; ξ = ψ 1 L; ξ = φ j ξ L j j= and χ L; ξ = θ log λ L; θ = log, / ξ log ψ L; ξ = χ j ξ L j, j= where the prime denotes transposition and χ j ξ = χ 1j ξ, χ j ξ, χ1j ξ = j 1, χ j ξ = j φ k ξ ψ j k ξ, k=1 with the φ k ξ given by k= φ k ξ L j = ψ L; ξ 1. hen define the p + 1 p + 1 matrix [ B ξ = χ j ξ χ π /6 ] j=1 j ξ = χ j ξ /j j=1 χ j ξ /j j=1 χ, j ξ χ j ξ j=1 and assume B ξ is non-singular. he conditions on ψ L; ξ are satisfied by the coeffi cients in stationary and invertible autoregressive moving average sequences, and the conditions on λ L; θ are satisfied by the coeffi cients in FARIMA sequences. he above setting is identical to that of Robinson and Velasco 15, but we extend their model in the following three respects. First, we relax their assumption of independence and identity of distribution of the unobservable elements ε it of the vectors ε t = ε 1t,..., ε Nt across t to martingale difference structure for levels and squares/products, in particular, almost surely, E ε it ε j,t s, s 1, j 1 =, i 1, t, 6 E ε it ε ij σ ij ε k,t s, ε l,t s, s 1, k, l 1 =, i, j 1, t, 7 where 6 implies E ε it =. We impose also the moment condition and the fourth cumulant condition sup t N i,j,k,l=1 supeε 4 it < 8 i,t cum ε it, ε jt, ε kt, ε lt = O N, 9 both of which are automatically satisfied if ε it is Gaussian. 5

6 Secondly, we relax their assumption of homoscedasticity and lack of correlation across elements of the vectors ε t, in allowing Eε t ε t = Σ N, t =, 1,...,, for an N N matrix Σ N = σ ij which is assumed to stay positive definite and have bounded elements with increasing N but is otherwise unknown, to reflect possible cross-sectional correlation and heteroscedasticity, or else is restricted to be diagonal, to reflect an assumed lack of cross-sectional correlation, but the possibility of heteroscedasticity. Since N is allowed to increase, Σ N can in either case be regarded as nonparametric. Specifically, we assume that where. denotes spectral norm, and existence of lim Σ N + Σ 1 N <, 1 N σ = lim N 1 N tr Σ N, 11 and of 1 κ = lim N N tr Σ N, 1 Of course N 1 tr Σ N Σ N while also, from the inequality tr ABB A A tr BB see e.g. Horn and Johnson 1988, p. 313, N 1 tr Σ N Σ N N 1 tr Σ N, so the first component of 1 implies σ + κ <, while the second implies σ >. Condition 1 effectively upper- and lowerbounds variances and mildly limits the extent of cross-sectional dependence. Under cross-sectional uncorrelatedness assumption 1 reduces to sup σii + σii 1 <. i Our final extension allows the α t = α 1t,..., α Nt in 1 to vary with t, being vectors of unobserved nonparametric possibly stochastic, and cross-sectionally dependent trending individual effects, such that α it = α i t/, for functions α i u which satisfy a possibly stochastic Lipschitz condition sup α i t α i t + u u as u. 13 i,t Here there is no exogeneity requirement on α i t, with no restriction on possible dependence with the ε it. However we find that if a strong exogeneity condition is imposed, and a related condition to 13 is added, we can slightly increase the range of values of δ covered and relax the restrictions on increase of N with. In particular: {α i t α i t + u, t, t + u, 1], i 1} is independent of {ε it, i 1, t } 14 6

7 and sup E α i t α i t + u = O u as u. 15 i,t Note that neither 13 nor 15 implies the other, though they are closely related. o illustrate these various conditions, suppose α i t has the formal separable infinite series representation α i t = β i + β ij g ij t, i 1, 16 j=1 where β i is completely unrestricted as it can be fully differenced out, the functions g ij t are nonstochastic, and the β ij can be random variables. hen since α i t α i t + u a suffi cient condition for 13 is sup i j=1 while the latter and sup E β ij <, i j=1 i β ij j=1 j=1 1/ g ij t g ij t + u β ij 1, sup g ij t g ij t + u = O u, j=1 { βij, i, j 1 } is independent of {ε it, i 1, t } are collectively suffi cient for 14 and 15. An example of 16 are polynomials in t, with degree and possibly-stochastic coeffi cients allowed to vary with i. In any case the trending individual effects are nonparametric, as in fixed-design nonparametric regression see eg Cai 7, though here they can be stochastic, and also the fact that there are N of them, where N like will be regarded as increasing in our asymptotic theory, lends a further nonparametric aspect. We can compare our approach with the familiar one see e.g. Hsiao 14 in which our α it is replaced by the addition of a separable individual effect and time effect, namely β i + γ t. hen first differencing, which is employed in the current paper, eliminates the individual effect β i completely, but not the time effect γ t. If, however we take γ t = γt/, for a Lipschitz continuous function γu, we obtain γ t γ t 1 1, which becomes small as diverges. We obtain the same error bound by differencing our α it given condition 13, but clearly they afford more generality than the additive β i + γt/, since our time trends can vary over individuals. It is thus possible that differenced estimation of θ that ignores the α it is more or less robust to their presence, and the extent of this will be examined below. he more specialised structure β i + γt/ was also employed by Robinson 1, but there the focus was on estimating the nonparametric function γu. Note that the dependence on of the α t in 1 implies that the y t form a triangular array but our notation suppresses reference to this fact. 7

8 3. First difference estimates We consider in the current section one of the estimates of θ proposed by Robinson and Velasco 15 in their more specialised model, with our goal being to achieve the same property of consistency as if α t were absent from 1 or constant over t and the elements of ε t were cross-sectionally uncorrelated and homoscedastic, and asymptotic normality with the same convergence rate as before but with a different asymptotic variance matrix, and also to justify feasible large sample inference. From 1 and 13 or 15 it follows that y t = v t + α it 17 = v t + O p 1, as, 18 where the final term on the right of 18 in fact represents a vector with elements that are O p 1 uniformly in i and t, and v t = λ 1 t L; θ α it, t =,...,. 19 For any θ Θ, we attempt to fully whiten the data by forming the N matrix Z θ = z 1 θ..., z θ, where z t θ = λ t L; θ 1 y t, t = 1,...,, with θ 1 = δ 1, ξ. From 7 of Robinson and Velasco 15 and 17 z t θ = λ t L; θ v t τ t θ ε + λ t L; θ 1 α it 1 where τ t θ = λ t 1; θ. We have λ t L; θ v t = ε t but our estimates will be based on as if the two error terms in 1 were absent, though our theoretical justification will take account of them. Define the set Θ = D Ξ, where Ξ is a compact subset of R p and D = [δ, δ], where δ > max, δ 1, δ <, and we assume that δ D. he feasibly bias-corrected difference estimate of Robinson and Velasco 15 is θd = arg min θ Θ LD θ 1 b D arg min θ Θ LD θ, where L D θ = 1 N tr Z θ Z θ, b D θ = B 1 ξ S τ τ θ S τχ θ, 8

9 S τ τ θ = τ t θ τ t θ, S τχ θ = where π t δ = / δ π t δ and τ t θ χ t ξ, τ t θ = θ τ t θ = [ t k= π k δ t k j= ψ j ξ t k= π k δ t k j= ψ j ξ ], 3 he basic objective function is of conditional sum of squares type, but corrupted by the second and third components on the right hand side of 1, whose presence will be accounted for in the theoretical development. Finally define heorem 3.1 As µ = κ /σ 4. θd p θ. 4 If also 14 and 15 are not imposed but 1 < δ 1 and N 1 δ log or δ > 1 and N 1, or if 14 and 15 are imposed with 1 < δ 4 1 and N 1 4δ log 4, as, N 1 θd θ d N, µ B 1 ξ. 5 he consistency 4 requires no further restriction on δ, beyond δ D, and no restriction on the rate of increase of N with. However, comparing with Robinson and Velasco s 15 heorem 5. for their much more special model, we restrict δ and N in order to achieve asymptotic normality 5, in particular requiring δ to take nonstationary values when the α t are not exogenous; this is due to bias produced by temporal variation in individual effects. In order to base statistical inference on heorem 3.1 we estimate B ξ by B ξd, with ξ D denoting the final p elements of θ D, and estimate µ by µ D = Ntr Σ N θd /tr ΣN θd, 6 where Σ N θ = 1 Z θ Z θ. 7 An analogous unrestricted estimate of the cross-sectional covariance matrix was considered by Robinson 1 in the context of a panel model with no autocorrelation. If instead we maintain cross-sectional uncorrelatedness we take Σ N θ = 1 diag Z θ Z θ. 8 9

10 heorem 3. Under the conditions of heorem 3.1, as, B ξd p B ξ, 9 µ D p µ, 3 with µ D given by 6 where Σ N θd is defined either by 7 or, if σ ij = for all i j, 8, N 1 1/ B ξd / µ D θd θ converges in distribution to a vector of independent standard normal random variables. 4. Pseudo maximum likelihood estimate based on first differences Our next estimate is the difference pseudo maximum likelihood estimate PMLE of Robinson and Velasco 15. Define the matrix, Ω θ = ω st θ, ω st θ = 1 s = t+τ s θ τ t θ, so that Ω θ is proportional to the exact covariance matrix of the vector ε i1 τ t θ ε i,..., ε i τ θ ε i, cf 1. Unlike in the difference estimate of the previous section we thus allow for the initial value effect in our estimation, though as there we attempt to incorporate cross-sectional correlation or heteroscedasticity only in studentization, not in the point estimation of θ, and we have to contend in the theory with the O p 1 error of differencing the α it in 1. We estimate θ by θp = arg min θ Θ LP θ, where Θ is as defined in the previous section and L P θ = Ω θ 1 σ θ, 31 in which. σ θ = 1 N tr Z θ Ω 1 θ Z θ. heorem 4.1 As, θp p θ. 3 If also 14 and 15 are not imposed but 1<δ 1 and N 1 δ log or δ > 1 and N 1, or if 14 and 15 hold with 3 8 < δ 1 and N 3 8δ log 4 + N 1 log, as, N 1 θp θ d N, µ B 1 ξ. 33 1

11 he cost of allowing temporal variation in individual effects is greater than with the difference estimates, especially as the corresponding result heorem 4.4 of Robinson and Velasco 15 imposed no restrictions on δ and N. Now define µ P = Ntr Σ N θp /tr ΣN θp 34 and denote by ξ P the final p elements of θ P. heorem 4. Under the conditions of heorem 4.1, as, B ξp p B ξ, µ P p µ, with µ D given by 34 where Σ N θp is defined either using 7 or, if σ ij = for all i j, 8, N 1 1/ B ξp / µ P θp θ converges in distribution to a vector of independent standard normal random variables. 5. Ineffi ciency of estimation In the limiting covariance matrix µ B 1 ξ in heorems 3.1 and 4.1, the factor µ = 1 when the ε it are homoscedastic and uncorrelated across i, but in general µ 1 so heteroscedasticity and/or cross-sectional correlation inflates the variance matrix in the limiting distribution by a scalar factor, relative to the outcome of Robinson and Velasco 15. Note also that µ D 1 whether it is based on either of the estimates 7 or 8 of Σ N. he potential ineffi ciency of our estimates, or equivalently the degree of invalidity of the inference rules which assume homoscedasticity and lack of correlation across i, can be examined by considering a specific model for ε t. he spatial moving average model is defined by ε t = I N + ρw η t, 35 where I N is the N N identity matrix, W is an N N user-chosen spatial weight matrix with zero diagonal elements, taken here to be symmetric and normalised such that W = 1, and correspondingly the scalar ρ satisfies ρ < 1, while the elements of η t are mutually uncorrelated with common variance ς. hus tr Σ N = ς tr I + ρw = ς N + ρ tr W, tr Σ N = ς 4 tr I + ρw 4 = ς 4 N + 6ρ tr W + 4ρ 3 tr W 3 + ρ 4 tr W 4, 11

12 and so Ntr Σ N tr Σ N = N N + 6ρ tr W + 4ρ 3 tr W 3 + ρ 4 tr W 4 N + ρ tr W = 1 + 4ρ Ntr W + 4ρ 3 Ntr W 3 + ρ 4 Ntr W 4 tr W N + ρ tr W 1 + 4ρ N tr W + ρtr W 3 N + ρ tr W. 36 his lower bound is 1 when there is no spatial correlation, ρ =, but in general 36 exceeds 1, noting that tr W + ρtr W 3 > even when 1 < ρ < since W 1 implies tr W 3 tr W, and 36 increases in ρ. A simple W, proposed by Case 1991, is W = I r B s, B s = s s 1 s I s, 37 where rs = N and 1 s is the s 1 vector of 1 s, representing r districts each containing s farms, so farms are neighbours if and only if they lie in the same district and neighbours are equally weighted. Since we have B s = s 1 s 1 s 1 s + I s, B 3 s = s 1 3 s 3s s 1 s I s, tr W = r s 1 s s + s = N s 1 1, tr W 3 = r s 1 3 s s 3s + 3 s = N s s 1, and the lower bound 36 becomes 1 + 4ρ s ρ s s ρ s 1 1 = 1 + 4ρ s 1 + ρ s s 1 + ρ 38 A lower bound for µ is obtained by letting N, and if s this tends to 1, but if s stays fixed the bound exceeds 1, and again increases with ρ. 6. Monte Carlo simulations In this section we report the results of a simulation study of the properties of our estimates in finite samples in the presence of cross-sectional dependence and trends. We extend a similar set-up of Robinson and Velasco 15, where only constant fixed effects and independent and identically distributed ε it, across both i and t, were employed. 1

13 We generate the ε t as the spatial moving average 35, where η t is N, I N and ρ =.5 or.9, while W is generated as in 37, setting r = 4 when N = 8 or r = 5 N > 8. We consider both a pure fractional model, so p = 1, θ = δ and ψ L; ξ 1, and a model with FARIMA1,δ, dynamics, so p = and θ = δ, ξ with ψ L; ξ = 1 ξl. We consider different choices of N, and θ. In particular we employ three basic values of N, namely 1,, 4 with two combinations of N and for each, to account for relatively short = 1, and moderate time series = 1, 5, 5, and also N = 96 for = 1. he range of values of N thus varies from 8 through from the smallest to the largest sample size. he values of δ include a stationary one δ =.3, which our theorems predict will be the most problematic from the point of view of bias, a moderately non-stationary one δ =.6, a value close to the unit root δ =.9, and a more nonstationary one δ = 1.. For FARIMA models we consider two autoregressive parameter values, ξ =.5,.8. Optimizations were carried out using the Matlab function fmincon with D = [.1, 1.5] and Ξ = [.95,.95], and the results are based on 1, independent replications. For all combinations of sample sizes and parameter values we report scaled empirical bias of both the feasibly bias-corrected difference estimate θ D and the PMLE θp, root-mean square error MSE and empirical size of the corresponding t or Wald test based on estimates of µ and B for the asymptotic variance as studied in heorems 3. and 4. "corrected" tests to account for cross sectional dependence, while we also report the empirical size for tests based on a pooled estimate of the variance of innovations, which would be only valid in case of uncorrelated homoscedastic innovations "uncorrected" tests. he results in ables 1-8 concern the pure fractional case, θ = δ. We first consider the case without trends, where constant fixed effects are exactly removed by first differencing. able 1 provides a bias comparison of the estimates of δ. In general bias is not affected by cross-sectional correlation compared with the results in Robinson and Velasco 15, who did not allow for such, and typically reduces with increase of δ and as expected. Bias of the difference estimate δ D for δ =.3 can be of an order of magnitude larger than in the other cases for the smallest for a given N, despite bias-correction having a large beneficial effect results without bias correction are not reported here. he PMLE δ P does much better in this diffi cult setup, but the difference for larger δ is much smaller. MSE results in able confirm the consistency of estimates, no clear superiority of any of the two estimates apart from the bias effect, and increase in the variance of estimates with cross-sectional correlation that is, with increasing ρ. he empirical size of the properly studentized t-test is sensitive to the 13

14 parameter and sample size values, though they converge with increasing N to the nominal value. In general, performance tends to deteriorate with the larger ρ and smaller δ, and PMLE-based tests do better than difference ones. ests not using consistent estimates of µ under cross-sectional correlation are systematically very oversized in all cases, cf. able 4. We repeat the experiment in ables 5-8 for ρ =.8 but with a linear trend specified as α i u = β i u cf 16, where the β i are generated independently from the N, γ distribution with γ = 1, 3 and our estimates are of course invariant to fixing a temporally constant component of the individual effects at zero. he larger value of γ can generate relatively large trends that can dominate the behaviour of the time series as shown by the bias and MSE results in ables 5 and 6, respectively. However, for the smaller γ, first differencing seems to account properly for the heterogenous deterministic component, though bias is substantially increased, as is MSE, for the small values of N and. In this case PMLE-based t-tests perform in a similar way as in the absence of trend for not too small and N and δ.6. In ables 9-14 we consider the results for the FARIMA model, again considering for only the value of ρ,.9. he estimation of δ is substantially affected by the autoregressive short run dynamic component, and the bias in able 9 can change sign with the value of δ and of, for ξ =.5, the results improving in most cases with increasing δ, while bias is always positive for the largest ξ,.8, so persistence is incorporated in the estimation of δ in finite samples. In general, the PMLE dominates bias-corrected difference estimates again. here is an overall increase in variability in able 1 compared to able, since both parameter estimates are highly correlated. Estimation results for ξ in ables 11-1 are parallel to the ones for δ : large bias for small N and δ, negative bias in all cases for ξ =.8, but no clear pattern for ξ =.5, and MSE decreasing with N, and δ. he feasible asymptotic inference results reported in ables 1 and 14 confirm previous ideas on the need for consistent estimation of µ to account for cross-sectional correlation, though now oversizing is more severe for the larger values of across the whole range of values of δ and ξ. 7. Final Comments In a semiparametric panel data model with fractional dynamics we have established desirable and useful asymptotic properties of estimates of time series parameters that are robust to nonparametric, time-varying individual effects and to cross-sectional correlation and heteroscedasticity, at the cost of restrictions on the range of possible 14

15 values of the memory parameter and the rate of increase of N with. Some further issues that might be considered are as follows. 1. As an alternative to our nonparametric estimation of the factor µ that inflates the limiting covariance matrix we could invest in a parametric model for the covariance matrix of ε t, such as a factor model see eg Ergemen and Velasco 14 or a spatial model cf. the discussion in Section 5. With a correct specification improved finitesample properties are likely to result, though a misspecified parameterization would lead to inconsistent estimation of µ and thus invalidate inferences based on heorems 3. and 4... Point estimates of θ that use either our nonparametric estimates of Σ N, or parametric ones such as just described, to correct for cross-sectional correlation/heteroscedasticity to the extent of being asymptotically effi cient in a Gaussian context, can be constructed. heir investigation would be worthwhile but likely entail a considerable amount of further work and possibly some further restrictions on δ and. 3. Whereas we impose the same dynamics over the cross-section, Hassler, Demetrescu and arcolea 11 developed tests in a panel with a fractional structure which is allowed to vary across units, and with allowance for cross-sectional dependence, but without allowing for individual effects and keeping N fixed as. In our context where N can increase it would be possible to keep the number of time series parameters fixed by assuming they are constant within finitely-many known cross-sectional subsets, over which the parameters can vary. 4. hough the nonstochastic conditional covariance matrix of innovations assumed in 7 is common in the time series literature, the implications of allowing for conditional heteroscedasticity could be explored. 5. Observable explanatory variables might be allowed to enter, in either a parametric or nonparametric way. In the former case, if they are linearly involved our differencing will leave only their first differences and initial value, but with nonlinear or nonparametric modelling we will get a difference of the functions, which structure, in the nonparametric case, needs to be exploited via additive nonparametric regression methodology in order to minimize a curse of dimensionality. Appendix Proof of heorem

16 o prove 4 we first prove consistency of arg min θ Θ L D θ. he proof of this extends that of heorem 3.3 of Robinson and Velasco 15 hereafter RV which uses their Lemma, which clearly still holds in our setting, and their Proposition 1, which needs extending because we have relaxed their iid assumption on ε it. First consider RV s A θ, and their U θ, which is defined as there but with σ defined in 11. A θ A θ U θ is 1 N N j ν jθ ε it σ ii + N j=1 t= N tr Σ N σ σ N j + 1ν jθ j=1 j 1 ν jθ σ j=1 j= +1 j 1 t ν j θν k θε i,t j ε i,t k j=1 k= ν jθ 39 there is a typo in the last term on p.449 of RV. As in RV j N sup ν Θ jθ ε N j it σ ii sup ν Θ jθ ε it σ ii, j=1 t= which is uniformly o p N much as in RV using also the proof of heorem 1 of Hualde and Robinson 11 6 and 8. he second term in 39 is uniformly o p 1 in much the same way, and the remaining terms are uniformly o1 from 11 and 5 of RV. Since from 1 L D θ = λ t L; θ v t τ t θ ε + λ t 1 L; θ 1 α t it remains to show that the term in α t contributes negligibly. First, sup λ t 1 L; θ 1 t 1 α t sup α t sup λ j θ 1 Θ t Θ j= N t 1 δ + 1 j=1 t= N 1 δ + 1 = o p N 4 as for δ >. Next using the Cauchy inequality, sup λ Θ t 1 L; θ 1 α t λt L; θ v t 1/ sup λ t L; θ v t sup λ t 1 L; θ 1 Θ Θ 1/ α t N 1/ o p N 1/ = o p N, 41 16

17 since the first term converges uniformly to a bounded function after standardization by N 1, while proceeding in a similar way, sup λ Θ t 1 L; θ 1 α t τ t θ ε 1/ 1/ sup τ t θ ε i sup λ t 1 L; θ 1 α t Θ Θ = o p N 1/ o p N 1/ = o p N. 4 hus consistency of arg min θ Θ L D θ is established, and thence straightforwardly 4, using smoothness properties of b D. he latter are also used, along with 4, in proving 5, as in RV s heorem 5., which incidentally required weaker conditions on δ and N than ours, so we do not repeat the details. But we have to extend RV s heorem 4.3 on arg min θ Θ L D θ bd θ to our setting, and this requires first extending the CL for scores in Proposition of RV. We can write their w as w = 1 N 1 N θ A i θ = 1 N 1 t 1 χ t j θ ε jε t. j= hus Ew w = 1 N = 1 N t 1 j= s 1 χ t j θ ε jε t ε sε k χ s k θ s=1 k= t 1 s 1 χ t j θ ε jσ N ε k χ t k θ = tr Σ N N j= k= t 1 χ t j θ χ t j θ. j= hen given 1 and much as in RV, w d N, κ B ξ. Apart from the extra terms discussed below the score and Hessian are handled much as in RV, where the latter has probability limit σ B ξ, with σ is as in 11. We consider first the extra terms in the score when 14 and 15 are not imposed. We have N 1 θ LD θ = N 1 ft ε τ εt t + λ t 1 L α t τ t ε + αt λ t 1 L, 17

18 where τ t = τ t θ, τ t = τ t θ, τ t θ = τ tθ θ, λ t 1 L = λ t 1 L; θ 1, λ t 1 L = λ t 1 L; θ 1 λ t 1 L; θ 1 = λ t 1 L; θ 1 t 1 θ, f t = ε t χ t j θ. he additional terms contributing to the asymptotic bias of N 1 are not covered in RV are thus N 1 + N 1 j=, θ LD θ that α t λ t 1 L εt τ t ε + ft ε N 1 τ t αt λ t 1 L α t λ t 1 L αt λ t 1 L. 43 Denoting λ j = λ j and thus N 1 θ 1 sup α it i,t, we have λ j = θ λ j θ 1 α t λ t 1 L εt τ t ε N 1 N 1 N 1 N N λ εt j τ t ε t 1 j= t 1 δ + 1 log t δ + log N 1 = O λ log j = O j δ log t, j 1 δ + 1 log which is o p 1 as N 1 δ log + N 1 log, with δ > 1. he second term in 43 can be bounded similarly, given E ft ε τ t = O N 1/. he third is O p N N 1 t 1 δ + 1 log t N δ + log N 1 which is o p 1 since N 1 4δ log + N 1. with δ > δ + 3/ log, Now impose 14 and 15. Let K denote a generic finite constant. From 14 the first term in 43 has zero mean because E [ε it ] = and, using 1, variance 18

19 bounded by K 3 Σ N t 1 δ + log t log t + K Σ N t 1 δ + log t t δ log t 3 = O 3 3 δ + log log + O = O δ + 1 4δ log 3 = o δ + log log since δ > 1, so is negligible, because 4 [ E λ t 1 L α it λ t 1 L α it ] E λ t 1 L α it t 1 = O E α it j δ log j j= = O t 1 δ + 1 log t. he second term in 43 can be bounded similarly, and the last term is N O p N 1 t 1 δ + log t log t N δ + log log N 1 1 δ + 3/ log log, which is o p 1 since N 1 4δ log + N 1 with δ > 1 4 Proof of heorem 3. Given heorem 3.1 and continuity of B ξ it suffi ces to prove 3. We give the proof only for 7 because that for 8 is simpler. From 11, µ D µ differs by o 1 from Ntr Σ N θd /tr ΣN θd Ntr ΣN /tr Σ N, which is Ntr Σ Ntr N Σ N θd Σ N /tr Σ N θd tr ΣN θd tr Σ N / 19 tr ΣN θd tr Σ N

20 so the result follows on showing that tr ΣN θd Σ N tr Σ N θd Σ N noting that 44 implies that tr We have Σ N θd Σ N = Now Σ N θ = 1 = o p N, 44 = o p N, 45 ΣN θd /N has a positive, finite probability limit. ΣN θd Σ N θ + ΣN θ Σ N. ε t + s t ε t + s t = A + R, where A = 1 t ε ε t, R = 1 s t ε t + ε t s t + s t s t, with s t = λ t 1 L; θ 1 α t τ t θ ε. hus tr ΣN θ Σ N = tr A Σ N + R. Since E ε it σ ii ε jt σ jj = σ ij + cum ε it, ε it, ε jt, ε jt, we have 1 Etr A Σ N = E εt tr Σ N = 1 E N N ε it σ ii ε jt σ jj using 1 and 9, since N. Also j=1 = 1 N σ ij + cum ε it, ε it, ε jt, ε jt i,j=1 = tr Σ 1 N N + cum ε it, ε it, ε jt, ε jt i,j=1 N = O = o N 1 tr R = tr s t ε t + ε t s t + s t s t,

21 where 1 s t λ t θ 1 = o p N, as, ε + λ t 1 L; θ 1 α t as in 4-4. Also 1 tr s t ε t 1 s t 1/ 1 1/ ε t = o p N from the above and the fact that the second factor in brackets is O p N. hus we have shown that tr ΣN θ Σ N = o p N. Next, from heorem 3.1 and for θ such that θ θ θd θ, tr ΣN θd Σ N θ = 1 zt θd zt θ = z t θ z t θ θd θ θ 1 z N 3 t θ z t θ 1/ θ.. 46 From 1, z t θ = ε t τ t θ ε + λ t 1 L; θ 1 α t = ε t + O p 1, and thus z t θ N. Next z t θ = λ t L; θ v t + τ t θε + θ θ λ t 1 L; θ 1 α t. As in RV the derivative with respect to δ dominates, and we have λ t L; θ v t θ K t N 1/ log j j δ δ 1 j=1 ε i,t j N 1/ t δ δ + 1 log t, while τ t θ = O log t t δ, λ t 1 L; θ 1 α t N 1/ t log j j δ j=1 N 1/ t. 1

22 hus from heorem 3.1 and with large enough and any ɛ > z t θ θ = O p N 1/ t ɛ + t ɛ δ N 1/ t ɛ, uniformly in t, and so z t θ θ N 1+ɛ. It follows that 46 is N 1+ɛ O p = O N 1/ 3/ p N 1/ ɛ 1/ = o p N, so tr ΣN θd Σ N θ = o p N. he proof of 44 is completed. Next, to establish 45 we have tr Σ N θd Σ tr N = ΣN θd Σ N ΣN θd + Σ N 1/ 1/ tr ΣN θd Σ N tr ΣN θd + Σ N, where tr ΣN θd Σ N = tr ΣN θd Σ N θ + Σ N θ Σ N tr ΣN θd Σ N θ + tr ΣN θ Σ N. Now tr ΣN θ Σ N = tr A Σ N + R tr A Σ N + tr R, where Etr A Σ N 1 = E tr ε t ε t Σ N = 1 E εt 4 tr Σ N + tr 1 E ε t 4 = 1 = O s=1 3tr N Σ N + N, i,j,k,l=1 s,,s t cum ε it, ε jt, ε kt, ε lt εs ε sε t ε t Σ N

23 from 1 and 9. hus tr A Σ N = o p N. From the proof of 44 it is readily seen that tr R = o pn, to complete the proof of 45. Proof of heorem 4.1 he proof of 3 extends that of RV s heorem 4.4, inspection of which indicates that it suffi ces to consider the additional term in L D θ σ θ beyond ones of the type considered in the proof of heorem 3.1, namely τ t θ λ t 1 L; θ 1 α it, 47 S ττ θ which comes from {z i θ τ θ} /S ττ θ. We can bound 47 by sup α it τ t θ t 1 j= λ j θ 1 sup i,t D,Ξ S ττ θ t δ t 1 δ + 1 sup D 1 δ δ + 1 sup D 1 δ δ + 1 so its contribution to L D θ σ θ is O p δ + 1 = o p 1 since δ >. he proof of 3 then follows straightforwardly from those of RV s heorem 3.4 and our heorem 3.1. o prove 33 we consider first the case where 14 and 15 are not imposed. We have first to bound the extra terms in σ θ depending on α it in the normalized score based on L D, namely N 1 S ττ { λ t 1 L α it + εit τ t ε i λ t 1 L α it } N We have N λ t 1 L α it N τ t λ t 1 L α it + τ t εit τ t ε i λ t 1 L α it.48 t 1 δ + 1 N 3 δ + N 1 δ + 1, 3

24 so its contribution to σ θ is O p δ +. Next S ττ N ε it τ t ε i λ t 1 L α it 1 N t 1 δ N δ +, which is O p N 1 δ + 1, so its contribution to σ θ is O p δ + 1. Finally 1 N τ t λ t 1 L α it S 1 ττ N t δ t 1 δ + 1 S 1 ττ N 1 δ + log S 1 ττ N 1 δ + log N 1 δ + N 1 δ + N log, so its contribution to σ θ is O p δ + 1 4δ + 3 log, as is that of S ττ N τ t εit τ t ε i λ t 1 L α it. Overall the contribution of the extra terms in σ θ to N 1/ / θ L P θ is O p N 1/ δ + 1 4δ + 1 N 1 δ + 3 8δ + 1 1/ = o p 1, since N 1 δ + N 1 and δ > 1, and they do not contribute to the asymptotic bias. With the notation [a t ] 1 meaning the 1 column vector with tth element a t, with respect to the contribution of θ j σ θ to / θ L P θ we have to consider the extra terms in z i θ = [z it θ ] 1 depending on λ t 1 L α it, N = N + N N N [ λ t 1 L α it ] [ λ t 1 L α it ] N 1 Ω 1 1 Ω 1 [ λ t 1 L α it ] 1 Ω 1 θ Ω j θ Ω 1 θ z i θ θ Ω j θ Ω 1 θ [ λ t 1 L α it ] 1 and those in ż i θ = [ż it θ ] 1 depending on λ t 1 L α it, N N [ λ t 1 L α it ] 1 Ω 1 θ ż i θ + N 4 θ Ω j θ Ω 1 θ [ ε it τ t ε i ] 1, 49 N [ ] λ t 1 L α it 1 Ω 1 θ z i θ. 5

25 Since Ω 1 θ = I τ τ, Ω j Sττ θ = τ j, τ + τ τ j,,where τ j, = τ j,1,..., τ j, and τ j,k = O k δ log k as k, the first extra term is N +O p N 3 t [ λ t 1 L α it ] 3 S 1 ττ S ττ 1 Ω 1 θ Ω j θ Ω 1 θ [ ] λ t 1 L α it 1 t 1 δ + 1 t δ t 1 δ + 1 t δ log t r t 1 δ + 1 t δ 3 1 δ + log log 1 4δ log + 3 log 3, while using E τ t ε it E τ t ε it t 1/ = [ σ S ττ 1 ] 1/ = O 1 δ + log 1, and similarly E τ t ε it = O 1 δ log + log, E τ t τ t ε i = O Sτ τ = O 1 δ log + log, the second one is N [ λ N t +O p t 1 L α it ] 1 Ω 1 θ Ω j θ Ω 1 θ [ ε it τ t ε i ] 1 t 1 δ + 1 t δ 1 δ log + log S 1 ττ S ττ t 1 δ + 1 t δ 1 δ log + log 1 δ + log 1 δ log + log 1 4δ log + log 3. heir joint contribution to N 1/ / θ L P θ is thus O p N 1/ 1 4δ log + log 3 t N 3 8δ log + 3 log 4 1/, 5

26 which is o p 1 under N 1 δ log + N 1 and δ > 1. he extra terms due to the new element in ż it θ = f it τ t ε i + λ t 1 L α it, where f it is the ith row of f t, are N N N N Sττ + ż it θ τ t λ t 1 L α it z it θ + ż it θ λ t 1 L α it λ t 1 L α it τ t z it θ τ t λ t 1 L α it τ t. 51 Using similar methods as before, the first term is O p t 1 δ + 1 log t = O p δ + 1 log. Next, since E z it θ τ t = O Sττ = O 1 δ + log, E ż it θ τ t = O Sτ τ = O S ττ log, the other terms are 1 O p t 1 δ + 1 t δ log t S Sττ ττ + Sτ τ t 1 δ + 1 t δ 1 1 δ + log log + 1 δ + log log δ + log log. hus the contribution to the standardized score of these two terms is O p N 1 δ + log log N 1 4δ + 3 log log 1, which is o p 1 since N 1 δ log + N 1 for δ > 1. proof when 14 and 15 are not imposed. his completes the Now imposing 14 and 15, we have first to reanalyze two terms in σ θ depending on α it in the normalized score based on L D, given in 48. Now N ε it τ t ε i λ t 1 L α it 6

27 has zero mean and variance bounded by K N Σ N t 1 δ t 1 δ + 1 t δ = O N 3 δ +, so it is O p N 1/ 1 δ + 1 Next, N S ττ, and its contribution to σ θ is O p N 1/ 1 δ + 3. has zero mean and variance bounded by τ t εit τ t ε i λ t 1 L α it KN t 1 δ + 1 KN + t 1 δ + 1 t δ Sττ S ττ = O N 3 δ log + 1+δ 1 {δ <1/} + log = O N 1 δ log + 1 log, so its contribution to σ θ is O p δ log + 1 log. Overall, the contribution of the extra terms in σ θ to N 1/ / θ L P θ is N 1 4δ O 1/ p + O p δ + 1 N +O p 3 8δ + log + 1 4δ log 4 1/ +O p N 1 4δ log + 1 log 1/, which is o p 1, since N 1 4δ log 4 + N 1 log and δ > 1, and they do 4 not to contribute to the asymptotic bias. Regarding the contribution of θ j σ θ to / θ L P θ, we need to reconsider the term in 49 N N [ λ t 1 L α it ] 1 Ω 1 θ Ω j θ Ω 1 θ [ ε it τ t ε i ] 1, which has zero mean and variance O N 1 4 S ττ t 1 δ + 1 t δ t = O N δ + log 1 δ + log log = O N 1 8δ log + 4δ log log 6, 7

28 so this term is O p N 1 8δ log + 4δ log log 6 1/. hen the joint contribution of all terms depending on the differenced trend to N 1/ / θ L P θ is thus O p N 3 8δ log log 8 1/ +O p 3 8δ log + 1 4δ log log 6 1/, which is o p 1 since N 3 8δ log 4 + N 1 log and δ > 3. 8 the term in 51 under 14 and 15: N fit τ t ε i λ t 1 L α it N Now consider has zero mean and has variance O N 1 4 t 1 δ + 1 log + t 1 δ + 1 t δ log = O N δ + log + δ + log log = O N 1 1 δ + 3 log + δ + log log, and a similar result holds for the term depending on ε it τ t ε i λ t 1 L α it, so the contribution to the standardized score of this first term is N O p 1 4δ + log 1/ +O p δ + log 1/ + O 1 4δ + 3 log log 1/, which is o p 1 since N 1 4δ log 4 + N 1 log for δ > 3, while the second 8 term is shown o p 1 as before. Proof of heorem 4. his straightforwardly extends the proofs of heorem 3. and heorem 4.1, and is thus omitted. 8

29 References Cai, Z., 7. rending time-varying coeffi cient time series models with serially correlated errors. Journal of Econometrics 136, Case, A.C., Spatial patterns in household demand. Econometrica 59, Ergemen, Y.E. and C. Velasco, 14. Estimation of fractionally integrated panels with fixed effects and cross-section dependence. Preprint UC3M. Hassler, U., M. Demetrescu and A.L. arcolea, 11. Asymptotic normal tests for integration in panels with cross-dependent units. Advances in Statistical Analysis 95, Horn, R.A. and C.R. Johnson, Matrix analysis. Cambridge University Press. Hsiao, C., 14. Analysis of panel data, hird Edition. Press. Cambridge University Hualde, J. and P. M. Robinson, 11. Gaussian pseudo-maximum likelihood estimation of fractional time series models. Annals of Statistics 39, Robinson, P. M., 1. Nonparametric trending regression with cross-sectional dependence. Journal of Econometrics 169, Robinson, P. M. and C. Velasco, 15. Effi cient inference on fractionally integrated panel data models with fixed effects. Journal of Econometrics 185,

30 able 1. Empirical bias 1. Iδ ρ =.5 ρ =.9 δd δp δd δp δ : N able. Empirical Root-MSE 1. Iδ ρ =.5 ρ =.9 δd δp δd δp δ : N

31 able 3. Empirical Size % Corrected 5% t-test. Iδ ρ =.5 ρ =.9 δd δp δd δp δ : N able 4. Empirical Size % Uncorrected 5% t-test. Iδ ρ =.5 ρ =.9 δd δp δd δp δ : N

32 able 5. Empirical bias 1. Iδ, ρ =.9, linear trend β i t/, β i IIN, γ. γ = 1 γ = 3 δd δp δd δp δ : N able 6. Empirical Root-MSE 1. Iδ, ρ =.9, linear trend β i t/, β i IIN, γ. γ = 1 γ = 3 δd δp δd δp δ : N

33 able 7. Empirical Size % Corrected 5% t-test. Iδ, ρ =.9, linear trend β i t/, β i IIN, γ. γ = 1 γ = 3 δd δp δd δp δ : N able 8. Empirical Size % Uncorrected 5% t-test. Iδ, ρ =.9, linear trend β i t/, β i IIN, γ. γ = 1 γ = 3 δd δp δd δp δ : N

34 able 9. Empirical bias δ 1. FARIMAξ, δ, ρ =.9. ξ =.5 ξ =.8 δd δp δd δp δ : N able 1. Empirical Root-MSE δ 1. FARIMAξ, δ, ρ =.9. ξ =.5 ξ =.8 δd δp δd δp δ : N

35 able 11. Empirical bias ξ 1. ARFIξ, δ, ρ =.9. ξ =.5 ξ =.8 P ξd ξδ ξd ξp δ : N able 1. Empirical Root-MSE ξ 1. FARIMAξ, δ, ρ =.9. ξ =.5 ξ =.8 ξd ξp ξd ξp δ : N