Estimating Long-run Coe Cients And Bootstrapping In Large Panels With .
Estimating long-run coefficients and bootstrapping in large panels with cross-sectional dependence 2019 Northern European Stata User Group Meeting Jan Ditzen Heriot-Watt University, Edinburgh, UK Center for Energy Economics Research and Policy (CEERP) August 30, 2019 Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 1 / 48
Introduction xtdcce2 on SSC since August 2016 Described in The Stata Journal, Vol 18, Number 3, Ditzen (2018) and in Ditzen (2019). Setting: Dynamic panel model with heterogeneous slopes and an unobserved common factor (ft ) and a heterogeneous factor loading (γi ): yi,t λi yi,t 1 βi xi,t ui,t , (1) ui,t γi0 ft ei,t βMG N N 1 X 1 X βi , λMG λi N N i 1 i 1 i 1, ., N and t 1, ., T Aim: consistent estimation of βi and βMG : I I I I Large N, T 1: Cross Section; β̂ β̂i , i N 1 , Large T: Time Series; β̂i Large N, Small T: Micro-Panel; β̂ β̂i , i Large N, Large T: Panel Time Series; β̂i and β̂MG If the common factors are left out, they become an omitted variable, leading to the omitted variable bias. xtdcce2 includes test for cross-sectional dependence (Pesaran, 2015), xtcd2, and estimation of exponent of cross-sectional dependence (Bailey et al., 2016, 2019), xtcse2. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 2 / 48
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 - Long Run Coefficients 30. August 2019 3 / 48
Dynamic Common Correlated Effects I yi,t λi yi,t 1 βi xi,t ui,t , (2) ui,t γi0 ft ei,t Individual fixed effects (αi ) 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, βi β vi , vi IID(0, Ωv ) and λi λ ςi , ςi IID(0, Ως ). ft is an unobserved common factor and γi a heterogeneous factor loading. In a static model λi 0, Pesaran (2006) shows that equation (2) can be consistently estimated by approximating the unobserved common factors with cross section averages x̄t and ȳt under strict exogeneity. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 4 / 48
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) h i show that the estimator gains consistency if the floor of pT 3 T lags of the cross-sectional averages are added. Estimated Equation: yi,t λi yi,t 1 βi xi,t pT X 0 γi,l z̄t l i,t l 0 z̄t (ȳt , x̄t ) The Mean Group Estimates are: π̂MG and the asymptotic variance is 1 N PN i 1 π̂i with π̂i (λ̂i , β̂i ) N d (π̂MG ) Var X 1 (π̂i π̂MG ) (π̂i π̂MG )0 N(N 1) i 1 Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 5 / 48
Estimation of Long Run Coefficients A more general representation of eq (1) with further lags of the dependent and independent variable in the form of an ARDL(py , px ) model is: yi,t py X λl,i yi,t l l 1 px X βl,i xi,t l ui,t . (3) l 0 where py and px is the lag length of y and x. The long run coefficient of β and the mean group coefficient are: Ppx l 0 βl,i θi , P py 1 l 1 λl,i θ̄MG N 1 X θi N (4) i 1 How to estimate θi and θ̄MG ? I I 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 - Long Run Coefficients 30. August 2019 6 / 48
CS-DL, CS-ARDL, CS-ECM CS-DL I Idea: directly estimate the long run coefficients, by adding differences of the explanatory variables and their lags. yi,t θi xi,t pX x 1 δi,l xi,t l pT X 0 γi,l z̄t l ei,t l 0 l 0 CS-ARDL and CS-ECM I Idea: first estimate short run coefficients, then calculate long run coefficients. yi,t py X λl,i yi,t l l 1 px X βl,i xi,t l l 0 pT X 0 γi,l z̄t l ei,t l 0 Ppx θ̂CS ARDL,i 1 l 0 β̂l,i P py l 1 λ̂l,i P For all estimators the mean group estimates are θ̄ˆMG N i 1 θ̂i . 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 θi , the delta method is used. Delta Method Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 7 / 48
Next Steps. 1 Monte Carlo simulation 2 Bootstrapping in large panels 3 Description of xtdcce2 4 Examples Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 8 / 48
Monte Carlo Simulation Aims: Assess the bias of the point estimate and standard error of the long run coefficient. Simulation follows Chudik et al. (2016). The DGP is an ARDL(2,1) model: yi,t αi λ1,i yi,t 1 λ2,i yt 2 β0,i xi.t β1,i xi,t 1 ui,t ui,t γu ft i,t The coefficients are generated as: θi IIDN(1, σθ2 ) λ1,i (1 ξλi )ηλi λ2,i ξλi ηλi β0,i ξβi ηβi , β1,i (1 ξβi ) ηβi ηλi IIDU(0, λmax ) ηβi θi / (1 λi,1 λ2,i ) , ξλi IIDU(0.2, 0.3), ξβi IIDU(0, 1) (σθ2 , λmax ) are varied between (0.2, 0.6) and (0.8, 0.8). Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients Details 30. August 2019 9 / 48
Monte Carlo Results Bias and RMSE of θ̂MG . (N,T) Bias of θ̂MG (x100) 40 50 100 CS-DL 40 -21.57 50 -19.41 100 -20.04 150 -16.99 200 -20.73 CS-ARDL 40 -2.63 50 -2.13 100 -3.53 150 -4.93 200 -2.63 150 200 RMSE of θ̂MG (x100) 40 50 100 150 200 -21.04 -19.15 -18.76 -16.41 -19.62 -19.52 -17.09 -17.40 -15.06 -18.20 -18.73 -16.64 -17.08 -14.72 -17.72 -18.26 -16.42 -16.93 -14.56 -17.37 23.50 21.12 20.39 17.35 21.04 22.48 20.19 19.02 16.64 19.80 20.10 17.51 17.25 15.05 18.24 19.04 16.84 16.81 14.62 17.70 18.46 16.52 16.61 14.46 17.31 -1.64 -186.07 -0.43 -2.29 -2.29 -1.94 -1.45 -1.21 -1.31 -1.63 -0.64 -0.75 -0.94 -0.95 -1.11 -0.48 -0.58 -0.65 -0.59 -0.61 192.31 40.85 182.04 34.46 23.47 13.65 4049.97 24.21 7.20 8.54 8.01 6.53 4.64 3.69 3.76 5.58 5.47 3.46 2.69 2.73 4.80 4.36 2.96 2.48 2.22 Monte Carlo results for θ̂MG 1/N PN i 1 2 θ̂i with pT [T 1/3 ], ρf 0 and (σθ , λmax ) (0.2, 0.6). CS-ARDL performs better in terms of bias, bias of both estimators decline with an increase in T. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 10 / 48
Monte Carlo Results Bias and RMSE of SE (θ̂MG ). (N,T) Bias of SE (θ̂MG ) (x100) 40 50 100 CS-DL 40 -53.83 50 -54.64 100 -67.21 150 -73.50 200 -76.23 CS-ARDL 40 -46.24 50 -10.73 100 -42.71 150 -35.95 200 -39.30 150 200 RMSE of SE (θ̂MG ) (x100) 40 50 100 150 200 -60.79 -60.85 -71.64 -76.87 -79.50 -71.47 -71.95 -79.56 -83.12 -85.22 -75.26 -75.80 -82.30 -85.09 -87.17 -77.61 -78.13 -83.81 -86.19 -88.23 12.06 11.40 12.91 14.17 14.77 13.54 12.63 13.75 14.81 15.40 15.85 14.87 15.26 16.01 16.51 16.68 15.66 15.79 16.39 16.88 17.19 16.13 16.07 16.60 17.09 -43.80 836.47 -53.72 -67.29 -68.12 -65.46 -66.20 -75.66 -80.78 -82.47 -71.38 -72.09 -80.31 -84.14 -85.69 -74.85 -75.85 -82.62 -85.84 -87.39 187.57 36.00 180.31 32.86 21.64 10.94 4048.46 24.47 13.31 14.47 14.57 13.72 14.53 15.56 15.98 15.84 14.91 15.41 16.21 16.60 16.59 15.67 15.85 16.53 16.93 Monte Carlo results for SE (θ̂MG ) q 1/N PN i 1 2 (θ̂i θ̂MG )2 with pT [T 1/3 ], ρf 0 and (σθ , λmax ) (0.2, 0.6). Standard errors are downward biased, increase with number of time periods. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 11 / 48
Bootstrapping in large panels Monte Carlo results show that standard errors are downward biased. Bootstrap often useful in small samples. No closed form solution for standard errors of individual long run coefficients. Delta method can fail. Bootstrap has to maintain the following properties of the DGP: I I I I Dynamic nature of the model Common factor structure Error structure across time and cross-sectional units N and T jointly to infinity Kapetanios (2008) and Westerlund et al. (2019) propose to re-sample cross-sectional units, but common factor structure changes. Gonçalves and Perron (2018) show that resampling over time is invalid in the presence of cross-sectional dependence. Praskova (2018) shows that if the common factors are known a wild bootstrap can be used. Idea: Wild Bootstrap Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 12 / 48
Wild Bootstrap Steps: 1 2 3 PpT 0 γi,l z̄t l i,t Estimate Model, eg: yi,t λi yi,t 1 βi xi,t l 0 Remove residual: ỹi,t yi,t ˆi,t Following Roodman et al. (2018) generate weights 1 with p 0.5 (b) ki,t 1 with p 0.5 (b) 4 5 (b) and calculate yi,t ỹi,t ki,t ˆi,t Estimate model and save coefficients. Repeat 3 - 4 B times and calculate standard errors or percentile confidence interval. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 13 / 48
xtdcce2 General Syntax Syntax: indepvars varlist2 varlist iv if crosssectional(varlist cr) , nocrosssectional pooled(varlist p ) xtdcce2 depvar cr lags(#) ivreg2options(string ) e ivreg2 ivslow lr(varlist lr ) lr options(string ) pooledconstant noconstant reportconstant trend pooledtrend jackknife recursive exponent xtcse2options(string) nocd fullsample showindividual fast blockdiaguse nodimcheck useinvsym useqr noomitted showomitted , Stored in e() , For Bootstrap: More Details bootstrap xtdcce2 Bias Correction , reps(intger) seed(string) cfresiduals percentile showindividual Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 14 / 48
xtdcce2 General Syntax yi,t αi py X λl,i yi,t l l 1 pȳ X l 0 γy ,i,l ȳt l px X βl,i xi,t l l 0 px̄ X γx,i,l x̄t l ei,t l 0 crosssectional(varlist) specifies cross sectional means, i.e. variables in z̄t . These variables are partialled out. cr lags(#) defines number of lags (pT ) 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(pȳ px̄ ). lr(varlist lr ) and lr options(string ) define the long run coefficients and options. For an ARDL (2,2) model it would be: lr(L(1/2).y L(0/2).x) lr options(ardl) Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 15 / 48
xtdcce2 CS-DL Example Chudik et al. (2013) estimate the long run effect of public debt on output growth with the following equation: yi,t ci θ 0i xi,t pX x 1 β i,l xi,t l γy ,i ȳt l 0 3 X γ x,i,l x̄i,t l ei,t l 0 where yi,t is the log of real GDP, xi,t ( di,t , πi,t )0 , di,t is log of debt to GDP ratio and π is the inflation rate. The results from Chudik et al. (2013, Table 18) with 1 lag of the explanatory variables (px 1) in the form of an ARDL(1,1,1) and three lags of the cross sectional averages are estimated with: xtdcce2 d.y dp d.gd d.(dp d.gd) , cr(d.y dp d.gd) cr lags(0 3 3) fullsample Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 16 / 48
xtdcce2 CS-DL Example . xtdcce221 d.y dp d.gd d.(dp d.gd) /// , cr(d.y dp d.gd) cr lags(0 3 3) fullsample (Dynamic) Common Correlated Effects Estimator - Mean Group Panel Variable (i): ccode Number of obs Time Variable (t): year Number of groups Degrees of freedom per group: Obs per group (T) without cross-sectional averages 35.025 with cross-sectional averages 26.025 Number of F(560, 1041) cross-sectional lags 0 to 3 Prob F variables in mean group regression 160 R-squared variables partialled out 400 R-squared (MG) Root MSE CD Statistic p-value D.y Coef. Std. Err. 1601 40 40 0.90 0.93 0.67 0.40 0.03 1.11 0.2667 z P z [95% Conf. Interval] -3.47 -6.05 0.13 0.46 0.001 0.000 0.897 0.646 -.1391961 -.1145398 -.0757413 -.022261 Mean Group: dp D.gd D.dp D2.gd -.0889339 -.0865123 .0053284 .0068065 .0256445 .0143 .0413629 .0148306 -.0386717 -.0584849 .0863981 .035874 Mean Group Variables: dp D.gd D.dp D2.gd Cross Sectional Averaged Variables: D.y(0) dp(3) D.gd(3) Heterogenous constant partialled out. The long run coefficients are θ̂π,MG 0.0889 and θ̂ d,MG 0.0865. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 17 / 48
xtdcce2 CS-DL Example . bootstrap xtdcce2 , reps(500) (running on xtdcce2 sample) Wild-Bootstrap replications ( 500 ) using residuals 1 2 3 4 5 . . . . . . . . . . Observed Coef. Bootstrap Std. Err. -.0889339 -.0865123 .0053284 .0068065 .1532996 .0600872 .2093117 .094243 z 50 100 150 200 250 300 350 400 450 500 P z Normal-based [95% Conf. Interval] 0.562 0.150 0.980 0.942 -.3893956 -.204281 -.404915 -.1779065 Mean Group: dp D.gd D.dp D2.gd -0.58 -1.44 0.03 0.07 .2115279 .0312563 .4155719 .1915195 The long run coefficients are not significant any longer. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 18 / 48
xtdcce2 CS-ARDL Assume an ARDL(1,2) and pT (pȳ , px̄ ) (2, 2) such as: yi,t λi yi,t 1 β0,i xi,t β1,i xi,t 1 β2,i xi,t 2 2 X γy ,i,l ȳt l 0 2 X γx,i,l x̄t l ei,t l 0 The model is directly estimated and then the long run coefficients are calculated as: θ̂CS ARDL,i β̂0,i β̂1,i β̂2,i 1 λ̂i Using xtdcce2 the command line is: xtdcce2 y , lr(L.y x L.x L2.x) lr options(ardl) cr(y x) cr lags(2) lr() defines the long run variables. xtdcce2 automatically detects the variables and their lags if time series operators are used. Alternatively variables can be enclosed in parenthesis, for example lr(L.y (x lx l2x)), with lx L.x and l2x L2.x. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 19 / 48
xtdcce2 CS-ARDL Example - ARDL(1,1,1) from Chudik et al. (2013, Table 17). . xtdcce221 d.y , lr(L.d.y L.dp dp L.d.gd d.gd) /// lr options(ardl) cr(d.y dp d.gd) cr lags(3) /// fullsample (Dynamic) Common Correlated Effects Estimator - (CS-ARDL) Panel Variable (i): ccode Number of obs Time Variable (t): year Number of groups Degrees of freedom per group: Obs per group (T) without cross-sectional averages 33.975 with cross-sectional averages 21.975 Number of F(720, 879) cross-sectional lags 3 Prob F variables in mean group regression 200 R-squared variables partialled out 520 R-squared (MG) Root MSE CD Statistic p-value D.y Coef. Std. Err. 1599 40 40 0.79 1.00 0.61 0.44 0.03 0.57 0.5690 z P z [95% Conf. Interval] Short Run Est. Mean Group: LD.y dp D.gd L.dp LD.gd .0475614 -.1036029 -.0745686 -.0199465 -.0132482 .0393514 .0402888 .0122305 .0462873 .0156115 1.21 -2.57 -6.10 -0.43 -0.85 0.227 0.010 0.000 0.667 0.396 -.0295659 -.1825675 -.0985398 -.1106679 -.0438463 .1246888 -.0246383 -.0505973 .0707749 .0173498 -.1639748 -.0873991 -.9524386 .0378594 .0164432 .0393514 -4.33 -5.32 -24.20 0.000 0.000 0.000 -.2381778 -.1196271 -1.029566 -.0897718 -.0551711 -.8753112 Long Run Est. Mean Group: lr dp lr gd lr y Mean Group Variables: Cross Sectional Averaged Variables: D.y dp D.gd Long Run Variables: lr dp lr gd lr y Cointegration variable(s): lr y Heterogenous constant partialled out. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 20 / 48
xtdcce2 CS-ARDL Example - ARDL(1,1,1) from Chudik et al. (2013, Table 17), bootstrapped. . bootstrap xtdcce2 , reps(500) percentile (running on xtdcce2 sample) Wild-Bootstrap replications ( 500 ) using residuals 1 2 3 4 5 . . . . . . . . . . Observed Coef. Observed Std. Err. .0475614 -.1036029 -.0745686 -.0199465 -.0132482 .0393514 .0402888 .0122305 .0462873 .0156115 -.1639748 -.0873991 -.9524386 .0378594 .0164432 .0393514 50 100 150 200 250 300 350 400 450 500 P z percentile t [95% Conf. Interval] 1.21 -2.57 -6.10 -0.43 -0.85 0.227 0.010 0.000 0.667 0.396 .7006659 -.2056815 -.0594676 -.0205935 -.0282485 1.241012 -.1088788 -.0451482 .0986097 .0010115 -4.33 -5.32 -24.20 0.000 0.000 0.000 -.2314161 -.0905856 -.2993341 -.142728 -.0492529 .241012 z Short Run Est. Mean Group: LD.y dp D.gd L.dp LD.gd Long Run Est. Mean Group: lr dp lr gd lr y Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 21 / 48
xtdcce2 CS-ARDL Example - ARDL(3,3,3) from Chudik et al. (2013, Table 17). . xtdcce221 d.y , cr lags(3) fullsample /// lr(L(1/3).(d.y) (L(0/3).dp) (L(0/3).d.gd) ) /// lr options(ardl) cr(d.y dp d.gd) (Dynamic) Common Correlated Effects Estimator - (CS-ARDL) Panel Variable (i): ccode Number of obs Time Variable (t): year Number of groups Degrees of freedom per group: Obs per group (T) without cross-sectional averages 27.05 with cross-sectional averages 15.05 Number of F(960, 602) cross-sectional lags 3 Prob F variables in mean group regression 440 R-squared variables partialled out 520 R-squared (MG) Root MSE CD Statistic p-value D.y Coef. Std. Err. 1562 40 39 0.96 0.71 0.39 0.51 0.02 -0.51 0.6108 z P z [95% Conf. Interval] Short Run Est. Mean Group: LD.y L2D.y L3D.y dp D.gd L.dp L2.dp L3.dp LD.gd L2D.gd L3D.gd .0123776 -.1395721 -.0829106 -.0707066 -.0853072 -.0312738 .098219 -.0424672 -.0270313 -.0114101 .0283559 .0349374 .0948493 .1072972 .0503045 .0137595 .0513445 .101743 .0581718 .0204755 .012726 .0177672 0.35 -1.47 -0.77 -1.41 -6.20 -0.61 0.97 -0.73 -1.32 -0.90 1.60 0.723 0.141 0.440 0.160 0.000 0.542 0.334 0.465 0.187 0.370 0.110 -.0560984 -.3254733 -.2932092 -.1693015 -.1122754 -.1319071 -.1011937 -.1564818 -.0671624 -.0363525 -.0064671 .0808536 .046329 .1273881 .0278883 -.0583391 .0693595 .2976317 .0715474 .0130999 .0135324 .0631789 -.0795232 -.1198351 -1.210105 .0587003 .0402246 .2006012 -1.35 -2.98 -6.03 0.176 0.003 0.000 -.1945738 -.1986738 -1.603276 .0355274 -.0409964 -.8169339 Long Run Est. Mean Group: lr dp lr gd lr y Mean Group Variables: Cross Sectional Averaged Variables: D.y dp D.gd Long Run Variables: lr dp lr gd lr y Cointegration variable(s): lr y Heterogenous constant partialled out. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 22 / 48
xtdcce2 CS-ARDL Example - ARDL(3,3,3) from Chudik et al. (2013, Table 17), bootstrapped. . bootstrap xtdcce2 , reps(500) (running on xtdcce2 sample) Wild-Bootstrap replications ( 500 ) using residuals 1 2 3 4 5 . . . . . . . . . . Observed Coef. Bootstrap Std. Err. .0123776 -.1395721 -.0829106 -.0707066 -.0853072 -.0312738 .098219 -.0424672 -.0270313 -.0114101 .0283559 .0523665 .0619368 .0383575 .1551891 .0408892 .2973936 .3611995 .2644735 .07608 .1131884 .0859342 -.0795232 -.1198351 -1.210105 .0790329 .0324949 .1392513 50 100 150 200 250 300 350 400 450 500 P z Normal-based [95% Conf. Interval] 0.24 -2.25 -2.16 -0.46 -2.09 -0.11 0.27 -0.16 -0.36 -0.10 0.33 0.813 0.024 0.031 0.649 0.037 0.916 0.786 0.872 0.722 0.920 0.741 -.0902588 -.2609661 -.1580899 -.3748717 -.1654486 -.6141546 -.6097191 -.5608257 -.1761453 -.2332553 -.1400721 .115014 -.0181782 -.0077312 .2334585 -.0051658 .551607 .8061571 .4758914 .1220827 .2104352 .1967839 -1.01 -3.69 -8.69 0.314 0.000 0.000 -.2344249 -.183524 -1.483033 .0753785 -.0561463 -.9371776 z Short Run Est. Mean Group: LD.y L2D.y L3D.y dp D.gd L.dp L2.dp L3.dp LD.gd L2D.gd L3D.gd Long Run Est. Mean Group: lr dp lr gd lr y Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 23 / 48
Conclusion xtdcce2. introduced a routine to estimate a panel model with heterogeneous slopes and dependence across cross-sectional untis by using the dynamic common correlated effects estimator. supports estimation of long run coefficients using three different models, using the I I I CS-DL estimator - direct estimation of the long run coefficients CS-ARDL estimator - calculation of long run coefficients out of short run coefficients an ECM approach is available on SSC (current version 2.01). standard errors and confidence intervals can be bootstrapped. includes estimation of cross-sectional exponent. Further developments: I I I Two-step ECM. Speed improvements and fitting it for ”big” data. Compare bootstrapped standard errors and delta method standard errors. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 24 / 48
The Delta Method back Allows calculation of an approximate probability distribution for a matrix function a(β) based on a random vector with a known variance. Assume βi p β and n(βi β) d N(0, σ) and first derivate of a(β): A(β) a(β) β 0 then the distribution of the function a() is n [a(βi ) a(β)] d N 0, A(β)ΣA(β)0 . Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 25 / 48
The Delta Method I back Assume an ARDL(2,1) model with the following long run coefficients: yi,t αi λ1,i yi,t 1 λ2,i yi,t 2 β0,i xi,t β1,i xi,t 1 ei,t φi (1 λ1,i λ2,i ) β0,i β1,i θ1,i 1 λ1,i λ2,i Stack the short run coefficients into πi (λ1,i , λ2,i , β0,i , β1,i ) The vector function a(πi ) maps the short run coefficients into a vector of the short run and long run coefficients: a(πi ) (λ1,i , λ2,i , β0,i , β1,i , φi , θ1,i ), where φi 1 λ1,i λ2,i and β β1,i θ1,i 1 λ0,i1,i λ . 2,i Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 26 / 48
The Delta Method II back The covariance matrix is: Var (λ1,i ) Cov (λ1,i , λ2,i ) Cov (λ1,i , β0,i ) Cov (λ1,i , β1,i ) . . Σi . . Var (β1,i ) The first derivative of a(πi ) is: Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 27 / 48
The Delta Method III back λ1,i λ1,i λ2,i λ1,i β0,i λ 1,i A(πi ) β1,i λ1,i φ i λ1,i θ1,i λ1,i Jan Ditzen (Heriot-Watt University) λ1,i λ2,1 λ1,i β0,i λ2,i λ2,i λ2,i β0,i β0,i λ2,i β0,i β0,i β1,i λ2,i β1,i β0,i φi λ2,i φi β0,i θ1,i λ2,i θ1,i β0,i xtdcce2 - Long Run Coefficients λ1,i β1,i λ2,i β1,i β0,i β1,i β1,i β1,i φi β1,i θ1,i β1,i 30. August 2019 28 / 48
The Delta Method IV back with φi φi 1 λ1,i λ2,i θ1,i θ1,i 1 β0,i β1,i 1 λ1,i λ2,i θ1,i θ1,i β0,i β1,i λ1,i λ2,i (1 λ1,i λ2,i )2 Then A(πi ) becomes: Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 29 / 48
The Delta Method V back A(πi ) 1 0 0 0 1 0 1 0 0 1 β0,i β1,i β0,i β1,i 2 (1 λ1,i λ2,i ) 2 (1 λ1,i λ2,i ) 0 0 1 0 0 0 0 0 1 0 1 1 λ1,i λ2,i 1 1 λ1,i λ2,i Then the covariance matrix including the long run coefficients is Σlri A(πi )Σi A(πi )0 Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 30 / 48
Monte Carlo Setup back As in Chudik et al. (2016) the data generating processes are the following:1 yi,t αi λ1,i yi,t 1 λ2,i yt 2 β0,i xi.t β1,i xi,t 1 ui,t ui,t γi0 ft i,t xi,t cxi αxi yi,t 1 γxi ft vxi,t yi,t is the dependent variable and xi,t the only independent variable. For a matter of ease, it is assumed that only one explanatory variable exists. The common factors are calculated as below: ft ρf ft 1 ςft , ςft IIDN(0, 1 ρ2f ) 2 vxi,t ρxi vxi,t 1 ςxi,t , ςxi,t IIDN(0, σvxi ) ρxi IIDU(0, 0.95) ρf 0 if serially uncorrelated factors, or if correlated ρf 0.6 2 q 2 2 2 σvxi σvi β0i 1 [E (ρxi )] 1 This paper focuses on the baseline cases with heterogenous slopes and stationary factors. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 31 / 48
Monte Carlo Setup back I Fixed Effects The cross-section specific fixed effects are generated as: cyi IIDN(1, 1) cxi cyi ςcx i , ςcx i IIDN(0, 1). Dependence between xi,t , gi,t and cyi is introduced by adding cyi to the equations for cxi and cgi . Coefficients First the long run coefficient θ is drawn and then the short run coefficients are backed out. θi IIDN(1, σθ2 ) λ1,i (1 ξλi )ηλi , λ2,i ξλi ηλi β0,i ξβi ηβi , β1,i (1 ξβi ) ηβi ηλi IIDU(0, λmax ), ηβi θi / (1 λi,1 λ2,i ) ξλi IIDU(0.2, 0.3), ξβi IIDU(0, 1) Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 32 / 48
Monte Carlo Setup back II Factor Loadings γi γ ηiγ , γxi γx ηiγx , 2 σγ2 σγx 0.22 p γ bγ , p γ x bx , ηiγ IIDN(0, σγ2 ) 2 ηiγx IIDN(0, σγx ) 1 σγ2 m 2 2 bx σ2 m(m 1) m 1 γx bγ where m is the number of unobserved factors. In comparison to Chudik and Pesaran (2015) it is restricted to 1. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 33 / 48
Monte Carlo Setup back III Error Term The errors are generated such that heteroskedasticity, autocorrelation and weakly cross-sectional dependence is allowed. i,t ρ i i,t 1 ζi,t ζt (ζ1,t , ζ2,t , ., ζN,t ) αCSD S t e t ζt (1 αCSD S ) 1 e t 1 e t IIDN(0, σi2 1 ρ2 i ), 2 ρ i IIDU(0, 0.8) 0 1 0 0 . 1 0 1 0 2 2 . 1 . 0 0 2 S 0 0 . . . . . . 1 2 . 1 . 0 2 0 0 . 0 1 Jan Ditzen (Heriot-Watt University) with σi2 χ2 (2) 0 0 . . 0 1 2 0 xtdcce2 - Long Run Coefficients 30. August 2019 34 / 48
xtdcce2 pmg-Options lr(varlist) defines the variables in the long run relationship. xtdcce2 estimates internally yi,t φi yi,t 1 γi xi,t 1 βi xi,t pT X γi,l z̄i,t ui,t (5) l 0 while xtpmg (with common factors) is based on: yi,t φi [yi,t 1 θi xi,t 1 ] βi xi,t pT X γi,l z̄i,t ui,t l 0 where θi φγii . θi is calculated and the variances calculated using the Delta method. lr option(string ) I I nodivide, coefficients are not divided by the error correction speed of adjustment vector (i.e. estimate (5)). xtpmgnames, coefficients names in e(b p mg) and e(V p mg) match the name convention from xtpmg. Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 35 / 48
xtdcce2 Test for cross sectional dependence xtdcce2 package includes the xtcd2 command, which tests for cross sectional dependence (Pesaran, 2015). Under the null hypothesis, the error terms are weakly cross sectional dependent. H0 : E (ui,t uj,t ) 0, t and i 6 j. s N 1 N X X 2T CD ρ̂ij N (N 1) i 1 j i 1 PT t 1 ûi,t ûjt ρ̂ij ρ̂ji 1/2 P 1/2 . PT T 2 2 û û t 1 it t 1 jt Under the null the CD test statistic is asymptotically CD N(0, 1). Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coefficients 30. August 2019 36 / 48
Saved values Scalars e(N) e(T) e(N partial) e(N pooled) e(rss) e(ll) e(df m) e(r2) e(cd) e(cr lags) Scalars e(Tmin) e(Tbar) Macros e(tvar) e(depvar) e(omitted) e(pooled) e(cmdline) e(insts) e(alpha) Matrices e(b) e(bi) Functions e(sample) back number of observations number of time periods number of variables partialled out number of pooled variables residual sum of squares log-likelihood (only IV) model degrees of freedom R -squared CD test statistic number of lags of cross sectional averages (unbalanced panel) minimum time average time e(N g) e(K mg) e(N omitted) number of groups number of regressors number of omitted variables e(mss) e(F) e(rmse) e(df r) e(r2 a) e(cdp) model sum of square F statistic root mean squared error residual degree of freedom R -squared adjusted p-value of CD test statistic e(Tmax) maximum time name of time variable name of dependent variable name of omitted variables name of pooled variables command line including options instruments (exogenous) variables estimated of exponent of cross-section dependence e(idvar) e(indepvar) e(lr) e(cmd) e(version) e(instd) e(alphaSE) name of unit variable name of independent variables long run variables command line xtdcce2 version, if xtdcce2, version used instrumented (endogenous) variables estimated standard error of exponent of cross-section dependence coefficient vector (mean group or individual) coefficient vector (individual and pooled) e
Estimating long-run coe cients and bootstrapping in large panels with cross-sectional dependence 2019 Northern European Stata User Group Meeting Jan Ditzen Heriot-Watt University, Edinburgh, UK Center for Energy Economics Research and Policy (CEERP) August 30, 2019 Jan Ditzen (Heriot-Watt University) xtdcce2 - Long Run Coe cients 30. August .
solution to a possibly under-determined system of linear equations with known coe cients. This hypothesis testing problem arises naturally in a number of settings, including random coe cient, treatment e ect, and discrete choice models, as well as a class of linear programming problems. As a rst contribution, we obtain a novel
The degree of a polynomial is the largest power of xwith a non-zero coe cient, i.e. deg Xd i 0 a ix i! d if a d6 0 : If f(x) Pd i 0 a ixiof degree d, we say that fis monic if a d 1. The leading coe cient of f(x) is the coe cient of xd for d deg(f) and the constant coe cient is the coe cient of x0. For example, take R R.
Electrical Construction Estimating Introduction to Electrical Construction Estimating Estimating activites will use the North State Electric estimating procedures. Estimating and the Estimator Estimating is the science and the art by which a person or organization determines in advance of t
1.2 Rappels sur les suites de matrices Une matrice p 2pr eelle est donn ee par p coe cients et elle peut donc se voir comme un el ement de Rp2. On peut munir l’espace des matrices d’une norme qui d e nisse la topologie usuelle de Rp2 (par exemple le supdes valeurs absolues des coe cients, ou n’importe quelle autre norme). On a ainsi une
2 White-Colebrook (7.5) The coe cients Cand M are denoted as the Chezy, respectively Manning coe cients. In case of a rough bed, the roughness length z 0 can be calculated using the following relation: z 0 0:11 u k b 30 ˇ k b 30 (7.6) Here, k b is the roughness height from Nikuradse, the kinematic viscosity and u the friction velocity.
2n rst-order ordinary di erential equations (ODE’s). A solution (with 2n arbitrary coe cients) can be obtained by using the dsolve command without any boundary conditions. The symbolic solution looks di erent from the one in [2]. By simplifying the expression and rearrange the arbitrary coe cients, it is not di cult to see that the two are .
mechanics. No matter how complicated the function fis, the solution has the form on the right hand side since the integrand is covariant under the three dimensional euclidian rotation group. If the symmetry is not assumed then one ought to write 3 3 9 coe cients instead of the two coe cients Aand Babove.
Counselling and therapy theoretical approaches may be viewed as possess-ing four main dimensions if they are to be stated adequately. In this context behaviour incorporates both observable behaviour and internal behaviour or thinking. The dimensions are: 1 a statement of the basic concepts or assumptions underlying the theory; 2 an explanation of the acquisition of helpful and unhelpful .
