Estimating long-run effects in models with cross-sectional dependence using xtdcce2 Three ways to estimate long run coefficients Jan Ditzen Heriot-Watt University, Edinburgh, UK Center for Energy Economics Research and Policy (CEERP) October 25, 2018 Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 1 / 42
Introduction xtdcce2 on SSC since August 2016 Described in The Stata Journal article in Vol 18, Number 3, Ditzen (2018). Current version 1.33 (as of 22.10.2018). Setting: Dynamic panel model with heterogeneous slopes and an unobserved common factor ( f t ) and a heterogeneous factor loading ( γ i ): y i , t = λ i y i , t − 1 + β i x i , t + u i , t , (1) u i , t = γ ′ i f t + e i , t � N � N β MG = 1 λ MG = 1 β i , λ i N N i =1 i =1 i = 1 , ..., N and t = 1 , ..., T Aim: consistent estimation of β i and β MG : ◮ Large N, T = 1: Cross Section; ˆ β = ˆ β i , ∀ i ◮ N=1 , Large T: Time Series; ˆ β i ◮ Large N, Small T: Micro-Panel; ˆ β = ˆ β i , ∀ i ◮ Large N, Large T: Panel Time Series; ˆ β i and ˆ β MG If the common factors are left out, they become an omitted variable, leading to the omitted variable bias. Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 2 / 42
Introduction Estimation of most economic models requires heterogeneous coefficients. Examples: growth models (Lee et al., 1997), development economics (McNabb and LeMay-Boucher, 2014), productivity analysis (Eberhardt et al., 2012), consumption models (Shin et al., 1999) ,... Vast econometric literature on heterogeneous coefficients models (Zellner, 1962; Pesaran and Smith, 1995; Shin et al., 1999). Theoretical literature how to account for unobserved dependencies between cross-sectional units evolved (Pesaran, 2006; Chudik and Pesaran, 2015). Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 3 / 42
Dynamic Common Correlated Effects I y i , t = λ i y i , t − 1 + β i x i , t + u i , t , (2) u i , t = γ ′ i f t + e i , t Individual fixed effects ( α i ) or deterministic time trends can be added, but are omitted in the remainder of the presentation.. The heterogeneous coefficients are randomly distributed around a common mean, β i = β + v i , v i ∼ IID (0 , Ω v ) and λ i = λ + ς i , ς i ∼ IID (0 , Ω ς ). f t is an unobserved common factor and γ i a heterogeneous factor loading. In a static model λ i = 0, Pesaran (2006) shows that equation (2) can be consistently estimated by approximating the unobserved common factors with cross section averages ¯ x t and ¯ y t under strict exogeneity. Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 4 / 42
Dynamic Common Correlated Effects II In a dynamic model, the lagged dependent variable is not strictly exogenous and therefore the estimator becomes inconsistent. Chudik and Pesaran (2015) show that the estimator gains consistency if the � � √ 3 floor of p T = T lags of the cross-sectional averages are added. Estimated Equation: p T � γ ′ y i , t = λ i y i , t − 1 + β i x i , t + z t − l + ǫ i , t i , l ¯ l =0 z t = (¯ y t , ¯ x t ) ¯ � N π MG = 1 π i = (ˆ λ i , ˆ The Mean Group Estimates are: ˆ i =1 ˆ π i with ˆ β i ) N and the asymptotic variance is N � 1 � π MG ) ′ Var (ˆ π MG ) = (ˆ π i − ˆ π MG ) (ˆ π i − ˆ N ( N − 1) i =1 Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 5 / 42
What is new? This is what xtdcce2 can do - what is new in version 1.33? A more general representation of eq (1) with further lags of the dependent and independent variable in the form of an ARDL( p y , p x ) model is: p y p x � � y i , t = λ l , i y i , t − l + β l , i x i , t − l + u i , t . (3) l =1 l =0 where p y and p x is the lag length of y and x . The long run coefficient of β and the mean group coefficient are: � p x N � l =0 β l , i ¯ θ i = 1 − � p y , θ MG = θ i (4) l =1 λ l , i i =1 xtdcce2 , version < 1 . 33, is not able to estimate the sum of coefficients and their standard errors. How to estimate θ i and ¯ θ MG ? ◮ Chudik et al. (2016) propose two methods, the cross-sectionally augmented ARDL (CS-ARDL) and the cross-sectionally augmented distributed lag (CS-DL) estimator. ◮ Using an error correction model (ECM). Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 6 / 42
CS-DL If λ i lies within the unit circle, the general ARDL model in (3) can be re-written as a level equation: y i , t = θ i x i , t + δ i ( L )∆ x i , t + ˜ u i , t (5) and L is the lag operator. Idea: directly estimate the long run coefficients, by adding differences of the explanatory variables and their lags. Details Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 7 / 42
CS-DL Lags of the cross-sectional averages are added to account for cross-sectional dependence. Together with the lags, equation (5) can be written as: p x − 1 p ¯ p ¯ � � � y x y i , t = θ i x i , t + δ i , l ∆ x i , t − l + γ y , i , l ¯ y i , t − l + γ x , i , l ¯ x i , t − l + e i , t l =0 l =0 l =0 where p ¯ y and p ¯ x is the number of lags of the cross-sectional averages. The mean group estimates are then � N ˆ ¯ ˆ θ MG = θ i i =1 The variance/covariance matrix for the mean group coefficients is the same as for the ”normal” (D)CCE estimator. Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 8 / 42
CS-ARDL Idea: Estimate the short run coefficients first and then calculate the long run coefficients. Equation (3) is extended by cross-sectional averages p y p x p T � � � γ ′ y i , t = λ l , i y i , t − l + β l , i x i , t − l + i , l ¯ z t − l + e i , t . l =1 l =0 l =0 with ¯ z t − l = (¯ y i , t − l , ¯ x i , t − l ) and the long run coefficients and the mean group estimates are � p x l =0 ˆ � N β l , i ˆ ˆ ¯ ˆ θ CS − ARDL , i = 1 − � p y , θ MG = θ i l =1 ˆ λ l , i i =1 The variance/covariance matrix for the mean group coefficients is the same as for the ”normal” (D)CCE estimator. For the calculation of the variance/covariance matrix of the individual long run coefficients θ i , the delta method is used. Delta Method Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 9 / 42
Error Correction Model Equation (3) can be transformed into an ECM 1 : ∆ y i , t = φ i [ y i , t − 1 − θ i x i , t ] p y − 1 p x p T � � � − λ l , i ∆ l y i , t − 1 − β l , i ∆ l x i , t + γ i , l ¯ z i , t + u i , t l =1 l =1 l =0 where ∆ l = t − t − l , for example ∆ 3 x i , t = x i , t − x i , t − 3 and � � � p x p y N � l =0 ˆ � β l , i and ˆ ˆ ˆ ˆ ¯ ˆ φ i = − 1 − θ i = θ MG = λ l , i , θ i ˆ φ i l =1 i =1 For the calculation of the variance/covariance matrix of the individual long run coefficients θ i , the delta method is used. Delta Method 1 This function was already available in xtdcce2 < 1 . 33. Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 10 / 42
xtdcce2 General Syntax Syntax: � � � � � � xtdcce2 depvar indepvars if varlist2 = varlist iv � crosssectional(varlist cr) , nocrosssectional pooled( varlist p ) cr lags( # ) ivreg2options( string ) e ivreg2 ivslow lr( varlist lr ) lr options( string ) pooledconstant noconstant reportconstant trend pooledtrend jackknife recursive noomitted nocd fullsample � showindividual fast More Details , Stored in e() , Bias Correction Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 11 / 42
xtdcce2 General Syntax p y p x � � y i , t = α i + λ l , i y i , t − l + β l , i x i , t − l l =1 l =0 p ¯ p ¯ � � y x + γ y , i , l ¯ y t − l + γ x , i , l ¯ x t − l + e i , t l =0 l =0 crosssectional ( varlist ) specifies cross sectional means, i.e. variables in ¯ z t . These variables are partialled out. cr lags (#) defines number of lags ( p T ) of the cross sectional averages. The number of lags can be variable specific. The same order as in cr () applies, hence if cr(y x) , then cr lags( p ¯ y p ¯ x ) . pooled ( varlist ) constraints coefficients to be homogeneous ( β i = β, ∀ i ∈ N ). reportonstant reports constant and pooledconstant pools it. Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 12 / 42
xtdcce2 CS-DL Assume an ARDL(1,2) and p T = ( p ¯ y , p ¯ x ) = (0 , 2) such as: y i , t = λ i y i , t − 1 + β 0 , i x i , t + β 1 , i x i , t − 1 + β 2 , i x i , t − 2 2 � + γ y , i ¯ y t + γ x , i , l ¯ x t − l + e i , t l =0 To estimate the model directly using the CS-DL estimator the following auxiliary regression is needed 2 � y i , t = θ i x i , t + δ 0 , i ∆ x i , t + δ 1 , i ∆ x i , t − 1 + γ y , i ¯ y t + γ x , i , l ¯ x t − l + ǫ i , t (6) l =0 To estimate it in xtdcce2 the command line would be: xtdcce2 y x d.x d2.x , cr(y x) cr lags(0 2) No specific commands for the long run estimation are required. Jan Ditzen (Heriot-Watt University) xtdcce2 25. October 2018 13 / 42
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