1 Panel Data Econometrics Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans University of Orléans January 2010
2 De nition A longitudinal, or panel, data set is one that follows a given sample of individuals over time, and thus provides multiple observations on each individual in the sample. (Hsiao,2003, page 2).
3 Terminology and notations: Individual or cross section unit : country, region, state, rm, consumer, individual, couple of individuals or countries (gravity models) etc. Double index : i (for cross-section unit) and t (for time) y it for i = 1,.., N and t = 1,.., T
4 A micro-panel data set is a panel for which the time dimension T is largely less important than the individual dimension N (example: the University of Michigan s Panel Study of Income Dynamics, PSID with 15,000 individuals observed since 1968): T << N A macro-panel data set is a panel for which the time dimension T is similar to the individual dimension N (example: a panel of 100 countries with quateraly data since the WW2): T ' N
5 De nition A panel is said to be balanced if we have the same time periods, t = 1,.., T, for each cross section observation. For an unbalanced panel, the time dimension, denoted T i, is speci c to each individual. Remark: While the mechanics of the unbalanced case are similar to the balanced case, a careful treatment of the unbalanced case requires a formal description of why the panel may be unbalanced, and the sample selection issues can be somewhat subtle. => issues of sample selection and attrition.
6 What are the main advantages of the panel data sets and the panel data models?
7 Panel data sets for economic research possess several major advantages over conventional cross-sectional or time-series data sets (see Hsiao 2003) Hsiao, C., (2003), Analysis of Panel Data, second edition, Cambridge Universirty Press. Wooldridge J.M., (2001), Econometric Analysis of Cross Section and Panel Data, The MIT Press.
8 What are the main advantages of the panel data sets and the panel data models? 1 Panel data usually give the researcher a large number of data points (N T ), increasing the degrees of freedom and reducing the collinearity among explanatory variables hence improving the e ciency of econometric estimates => but it is a kind of phantasm... more data points doesn t necessarily imply more information (heterogeneity bias)!! 2 More importantly, longitudinal data allow a researcher to analyze a number of important economic questions that cannot be addressed using cross-sectional or time-series data sets.
9 What are the main advantages of the panel data sets and the panel data models? De nition The oft-touted power of panel data derives from their theoretical ability to isolate the e ects of speci c actions, treatments, or more general policies.
10 Example (Ben-Porath (1973), in Hsiao (2003)) Suppose that a cross-sectional sample of married women is found to have an average yearly labor-force participation rate of 50%. 1 ) It might be interpreted as implying that each woman in a homogeneous population has a 50 percent chance of being in the labor force in any given year. 2 ) It might imply that 50 percent of the women in a heterogeneous population always work and 50 percent never work. To discriminate between these two models, we need to utilize individual labor-force histories (the time dimension) to estimate the probability of participation in di erent subintervals of the life cycle.
11 What are the main advantages of the panel data sets and the panel data models? 3. Panel data data provides a means of resolving the magnitude of econometric problems that often arises in empirical studies, namely the often heard assertion that the real reason one nds (or does not nd) certain e ects is the presence of omitted (mismeasured or unobserved) variables that are correlated with explanatory variables. Panel data allows to control for omitted (unobserved or mismeasured) variables.
12 Example Example: let us consider a simple regression model. where y it = α + β 0 x it + ρ 0 z it + ε it i = 1,.., N t = 1,.., T x it and z it are k 1 1 and k 2 1 vectors of exogeneous variables α is a constant, β and ρ are k 1 1 and k 2 1 vectors of parameters ε it is i.i.d. over i and t, with V (ε it ) = σ 2 ε Let us assume that z it variables unobservable and correlated with x it cov (x it, z it) 6= 0
13 Example (II): It is well known the least-squares regression coe cients of y it on x it are biaised. Let us assume that z i,t = z i (z values stay constant throught time for a given individual but vary accross individuals). y it = α + β 0 x it + ρ 0 z i + ε it i = 1,.., N t = 1,.., T Then, we can take the rst di erence of individual observations over time : y it y i,t 1 = β 0 (x it x i,t 1 ) + ε it ε i,t 1 Least squares regression now provides unbiased and consistent estimates of β.
14 Example (III): Let us assume that z i,t = z t (z values are common for all individuals but vary accross time: common factors). y it = α + β 0 x it + ρ 0 z t + ε it i = 1,.., N t = 1,.., T Then, we can take deviation from the mean across individuals at a given time : y it y t = β 0 (x it x t ) + ε it ε t where y t = (1/N) N i=1 y it x t = (1/N) N i=1 x it ε t = (1/N) N i=1 ε it Least squares regression now provides unbiased and consistent estimates of β.
15 What are the main advantages of the panel data sets and the panel data models? 4. Panel data involve two dimensions: a cross-sectional dimension N, and a time-series dimension T. We would expect that the computation of panel dataestimators would be more complicated than the analysis of cross-section data alone (where T = 1) or time series data alone (where N = 1). However, in certain cases the availability of panel data can actually simplify the computation and inference.
16 Example (time-series analysis of nonstationary data) Let us consider a simple AR (1) model. x t = ρx t 1 + ε t where the innovation ε t is i.i.d. 0, σ 2 ε. Under the non stationarity, assumption ρ = 1, it is well known that the asymptotic distribution of the OLS estimator bρ is given by: T (bρ 1) d 1 W (1) 2 1! R T! W (r)2 dr where W (.) denotes a standard brownian motion.
17 Example (time-series analysis of nonstationary data) Hence, the behavior of the usual test statistics will often have to be inferred through computer simulations. But if panel data are available, and observations among cross-sectional units are independent, then one can invoke the central limit theorem across cross-sectional units to show that the limiting distributions of many estimators remain asymptotically normal and the Waldtype test statistics are asymptotically chi-square distributed (e.g., Levin and Lin (1993); Im, Pesaran, Shin (1999), Phillips and Moon (1999, 2000); Quah (1994) etc.).
18 Example (time-series analysis of nonstationary data) Let us consider the panel data model x i,t = ρx i,t 1 + ε i,t where the innovation ε i,t is i.i.d. 0, σ 2 ε over i and t, then: T p N (bρ 1) d! N (0, 2) N,T!
20 There are three main issues in utilizing panel data: 1 Heterogeneity bias => Chapter 1 2 Dynamic panel data models (Nickel, 1981, bias) => Chapter 2 3 Selectivity bias (not speci c to panel data models)
21 The heterogeneity bias. When important factors peculiar to a given individual are left out, the typical assumption that economic variable y is generated by a parametric probability distribution function P (Y jθ)), where θ is an m-dimensional real vector, identical for all individuals at all times, may not be a realistic one.
22 The heterogeneity bias. Ignoring the individual or time-speci c e ects that exist among cross-sectional or time-series units but are not captured by the included explanatory variables can lead to parameter heterogeneity in the model speci cation.
23 Example Let us consider the example of a simple production function (Cobb Douglas) with two factors (labor and capital). We have N countries and T periods. Let us denote: - y it the log of the GDP for country i at time t. - n it the log of the labor employment for country i at time t. - y it the log of the capital stock for country i at time t. with ε i,t i.i.d. 0, σ 2 ε, 8 i, 8 t. y i,t = α i + β i k i,t + γ i n i,t + ε i,t
24 Example In this speci cation, the elasticities α i and β i are speci c to each country y i,t = α i + β i k i,t + γ i n i,t + ε i,t with ε i,t i.i.d. 0, σ 2 ε, 8 i, 8 t. But, several alternative speci cations can be considered. First, we can assume that the production function is the same for all countries: in this case we have an homogeneous speci cation: y i,t = α + βk i,t + γn i,t + ε i,t α i = α β i = β γ i = γ
25 Example However, an homogeneous speci cation of the production function for macro aggregated data is meaningless. We can introduce an heterogeneity of the Total Factor Productivity: more precisely, we can assume that the mean of TFP (given by E (α i + ε i,t ) = α i ) is di erent accross countries (due to institutional organisational factors, etc.). Then, we can use a speci cation with individual e ects, α i and common slope parameters (elasticities β and γ). y i,t = α i + βk i,t + γn i,t + ε i,t β i = β γ i = γ
26 Example Finally, we can assume that the labor and/or capital elasticities are di erent accross countries.in this case, we will have an heterogeneous speci cation of the panel data model (heterogeneous panel). y i,t = α i + β i k i,t + γ i n i,t + ε i,t
27 Example In this case, there are two solutions: 1 ) The rst solution consists in using N times series models to produce some group-mean estimates of the elasticities. 2 ) The second solution consists in using a model with random (slope) parameters => random coe cient model. In this case, we assume that parameters β i and γ i and randomly distributed, but follows the same distribution: β i i.i.i β, σ 2 β γ i i.i.i γ, σ 2 γ
28 The heterogeneity bias. Fact Ignoring such heterogeneity (in slope and/or constant) could lead to inconsistent or meaningless estimates of interesting parameters.
29 The heterogeneity bias. Let us consider a simple linear with individual e ects and only one explicative variable x i (common slope) as a DGP. y it = α i + βx it + ε it Let us assume that all NT observations fx it, y it g are used to estimate the homogeneous model. y it = α + βx it + ε it
30 The heterogeneity bias. Source: Hsiao (2003) Broken ellipses= point scatter for an individual over time Broken straight lines = individual regressions. Solid lines = least-squares regression using all NT observations
31 The heterogeneity bias. All of these gures depict situations in which biases (on bβ) arise in pooled least-squares estimates because of heterogeneous intercepts. Obviously, in these cases, pooled regression ignoring heterogeneous intercepts should never be used. Moreover, the direction of the bias of the pooled slope estimates cannot be identi ed a priori; it can go either way.
32 The heterogeneity bias. Another example: the true DGP is heterogeneous y it = α i + β i x it + ε it and we use all NT observations fx it, y it g to estimate the homogeneous model. y it = α + βx it + ε it
33 In this case it is straightforward that pooling of all NT observations, assuming identical parameters for all cross-sectional units, would lead to nonsensical results because itwould represent an average of coe cients that di er greatly across individuals (the phantasm of the NT observations..)
34 In this case, pooling gives rise to the false inference that the pooled relation is curvilinear. Fact In both cases, the classic paradigm of the representative agent simply does not hold, and pooling the data under homogeneity assumption makes no sense.
36 The lecture is organized as follows: 1 General Introduction on Panel Data Econometrics 2 Chapter 1. Heterogeneity and the linear model: from the unobserved e ect model to heterogeneous panel data models 3 Chapter 2. Dynamic panel data models.
37 References Articles (chapter 2) Baltagi, B.H. et Kao, C. (2000), Nonstationary panels, cointegration in panels and dynamic panels : a survey, in Advances in Econometrics, 15, edited by B. Baltagi et C. Kao, pp. 7-51, Elsevier Science. Hurlin, C. et Mignon, V. (2005), Une synthèse des tests de racine unitaire sur données de panel, Economie et Prévision, , pp