The Spatial Economics and Econometrics Centre (SEEC) D ISCUSSION PAPER S ERIES No 8 / July, 2016

xtdcce: Estimating Dynamic Common Correlated Effects in Stata Jan Ditzen

Please see the SEEC website (http://seec.hw.ac.uk/) for more information.

xtdcce: Estimating Dynamic Common Correlated Effects in Stata Jan Ditzen∗ Spatial Economics and Econometrics Centre (SEEC) Heriot-Watt University, Edinburgh, UK July 7, 2016

Abstract This article introduces a new Stata command, xtdcce, to estimate a dynamic common correlated effects model with heterogeneous coefficients. The estimation procedure mainly follows Chudik and Pesaran (2015b), in addition the common correlated effects estimator (Pesaran, 2006), as well as the mean group (Pesaran and Smith, 1995) and the pooled mean group estimator (Shin et al., 1999) are supported. Coefficients are allowed to be heterogeneous or homogeneous. In addition instrumental variable regressions and unbalanced panels are supported. The Cross Sectional Dependence Test (CD Test) is automatically calculated and presented in the estimation output. Small sample time series bias can be corrected by jackknife correction or recursive mean adjustment. Examples for empirical applications of all estimation methods mentioned above are given. Keywords: xtdcce, parameter heterogeneity, dynamic panels, cross section dependence, common correlated effects, pooled mean-group estimator, mean-group estimator, instrumental variables, ivreg2 ∗

Correspondence: Jan Ditzen, PhD Student, Spatial Economics and Econometrics Centre (SEEC), School of Management & Languages, Heriot-Watt University, Scotland, United Kingdom, EH14 4AS; Email: [email protected], Web: www.jan.ditzen.net

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1

Introduction

Estimating panels with heterogeneous coefficients in a large N and T setting became standard in the last years, thanks to seminal work in theoretical econometrics (Pesaran and Smith, 1995; Shin et al., 1999) and availability of data and computer power. Allowing for heterogeneous slopes allows the researcher to identify effects on a much more detailed thus local level. At the same time the theoretical literature on how to account for unobserved dependence between cross sectional units evolved (Pesaran, 2006; Chudik and Pesaran, 2015b). Cross sectional means are added to account for the unobserved dependence between countries. This paper introduces a new Stata program which combines these two strands of the literature. xtdcce allows for (pooled) mean group estimations in a dynamic panel with dependence between countries. It controls for dependence by adding cross sectional means and lags, as proposed by Pesaran (2006) and Chudik and Pesaran (2015b).1 Furthermore it tests for cross sectional dependence in the error terms and allows for instrumental variable estimation as well. Additionally xtdcce allows to correct for small sample time series bias by using the jackknife correction method or the recursive mean adjustment as proposed by Chudik and Pesaran (2015b). xtdcce differs in several ways from the existing estimation procedures for common correlated effects in a heterogeneous panel. In comparison to xtmg it allows the consistent estimation of a dynamic panel by adding lags of the cross sectional means. Moreover coefficients may be constrained to be homogeneous across all units. Additionally unbalanced panels are supported. Compared to xtpmg, xtdcce avoids maximum likelihood estimations, offering the possibility to estimate models including endogenous independent variables. Hence, the main novelties within the setting of xtpmg and xtmg are the inclusion of a test for cross sectional dependence, small T bias correction methods and the support for instrumental variable (IV) regressions. IV regressions benefit from the ivreg2 package. Possible applications for an IV estimation are endogenous spatial lags, which are instrumented by exogenous measures such as distance, other variables or higher order spatial lags. Furthermore adding cross sectional means implies to account for unobserved heterogeneity across units. The xtdcce packages includes xtcd2, which tests for cross sectional de1

Chudik and Pesaran (2015a) give a comprehensive overview over the literature on (dynamic) common correlated effects, while Chudik and Pesaran (2015b) focuses on dynamic common correlated effects. In the following common correlated effects cites Pesaran (2006), while dynamic cites Chudik and Pesaran (2015b), even though both are found in Chudik and Pesaran (2015a).

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pendence (henceforth CD test) as proposed by Pesaran (2015) and Chudik and Pesaran (2015a). Two other programs, xtcd (by Markus Eberhardt) and xtcsd by De Hoyos and Sarafidis (2006), made the CD test already available in Stata. The novelties of xtcd2 is the support of unbalanced panels, the possibility to test any variable for cross sectional dependence and the option to plot the cross correlations as a histogram. The remainder of the paper is structured as the following: the next two sections give a brief introduction into dynamic common correlated effects and testing for cross sectional dependence. Then the syntax, options and saved values of xtdcce and xtcd2 are explained. The paper closes with examples for an empirical application and a comparison of regression results obtained by xtdcce with results from estimation procedure already available in Stata.

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Common Correlated Effects Estimators

Assume the following equation with heterogeneous coefficients (Pesaran, 2006): yi,t = αi + βi xi,t + ui,t

(1)

ui,t = γi0 ft + ei,t where ft is an unobserved common factor and γi a heterogeneous factor loading.2 The heterogeneous coefficients are randomly distributed around a common mean, such that βi = β + vi , vi ∼ IID(0, Ωv ) (Pesaran and Smith, 1995). Pesaran (2006) shows that equation 1 can be consistently estimated by approximating the unobserved common factors with cross section means x¯t under strict exogeneity of xi,t . This estimator is commonly know as the Common Correlated Effects estimator (henceforth CCE). Furthermore it was proved to be consistent under a variety of further assumptions on the error term Chudik et al. (2011); Kapetanios et al. (2011). In empirical applications the estimator was used for example in Eberhardt et al. (2012), Bond and Eberhardt (2013) or McNabb and LeMay-Boucher (2014). The CCE estimator was made available in Stata by Markus Eberhard’s xtmg command (Eberhardt, 2012). The CCE estimator is, however, consistent only in non-dynamic panels (Chudik and Pesaran, 2015b; Everaert and De Groote, 2015). In a dynamic panel as: yi,t = αi + λi yi,t−1 + βi xi,t + ui,t , 2

Unlike Pesaran (2006) the intercept is kept and not partialled out. See discussion in Section 4.5.

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where the idiosyncratic errors ui,t are cross sectionally weakly dependent and E(λi ) = λ. The lagged dependent variable is no longer strictly exogenous and therefore the estimator becomes inconsistent. √ Chudik and Pesaran (2015b) show that the estimator gains consistency if 3 T lags of the cross section means are added. The equation to be estimated is then yi,t = αi + λi yi,t−1 + βi xi,t +

pT X

0 δi,l z¯t−l + ei,t ,

(2)

l=0

where z¯t = (¯ yt−1 , x¯t ) and pT is the number of lags. λi and βi are stacked into πi = (λi , βi ). The mean group estimates are then π ˆM G =

N 1 X π ˆi . N i=1

π ˆi and π ˆM G are consistently estimated with convergence rate

√

N if

(N, T, pT ) ⇒ ∞ and under full rank of the factor loadings (Chudik and Pesaran, 2015a,b). The asymptotic variance can be consistently estimated by ˆπ = V ˆar(ˆ πM G ) = N −1 Σ

N X 1 (ˆ πi − π ˆM G ) (ˆ πi − π ˆ M G )0 . N (N − 1) i=1

The mean group estimates have the following asymptotic distribution (Chudik and Pesaran, 2015b): √ d N (ˆ πM G − π) → N (0, ΣM G ). The pooled mean group estimator (Shin et al., 1999) can be seen as an intermediate between a pure pooled estimation (homogeneous coefficients) and a mean group estimation (heterogeneous coefficients). The assumption of the pooled mean group estimator (PMG) is, that regressors have a homogeneous long run and a heterogeneous short run effect on the dependent variable. Equation 1 is transformed into an error correction model, such that ∆yi,t = φi (yi,t−1 − θ1,i xi,t ) + δ0,i + δ1,i ∆xi,t + i,t . φi is the error-correction speed of adjustment parameter and expected to be negative. θ are the long run coefficients and assumed to be homogeneous, while δ capture short term dynamics and are heterogeneous across units. Shin et al. (1999) propose to estimate the long run coefficients by maximum 4

likelihood and the short run coefficients by OLS. The estimator is consistent as long as the disturbances are independently distributed across all individuals and time periods with a zero mean and a variance strictly larger than zero. The mean group estimate and the variance of the short run coefficients are: N N  2 X 1 1 X δˆi , V ˆar(δˆM G ) = δˆM G = δˆi − δˆM G N i=1 N (N − 1) i=1 The mean group and the pooled mean group estimator, in the static and the dynamic version rely on large N and T. The literature on small sample time series bias corrections in dynamic heterogeneous panels is somewhat scare and the reason why Chudik and Pesaran (2015b) focus on ”‘half-panel”’ jackknife and recursive mean adjustment bias correction methods. Both do not require any knowledge of the error factor structure and can be applied to the mean group estimates.3 The mean group estimate of the jackknife bias-corrected CCE estimator is π ˜M G = 2ˆ πM G −

 1 a b π ˆM G + π ˆM G , 2

T a where π ˆM G is the mean group estimate of the first half (t = 1, ..., 2 ) of the T b panel and π ˆM G of the second half (t = 2 + 1, ..., T ) of the panel. The recursive mean adjustment removes the partial mean from the all variables, meaning: t−1 1 X ω ˜ it = ωit − ωis , t − 1 s=1

where ωit = (yit , xit ) or any other variable except the constant. In line with Chudik and Pesaran (2015b) the partial mean is lagged by one period to prevent it from being influenced by contemporaneous observations.

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Testing for cross sectional dependence

If the cross sectional means are not included in the equation or do not account for all dependence between units, the error term will contain cross sectional dependence. Therefore it is not iid any more and OLS becomes inconsistent (Everaert and De Groote, 2015). Pesaran (2015) and Chudik and Pesaran (2015a) develop a procedure to test for cross sectional dependence. Under the 3

For a further discussion see Chudik and Pesaran (2015b) or Everaert and De Vos (2016).

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null hypothesis, the error terms are weakly cross sectional dependent. More formally, if the error term for unit i in period t is ui,t , then the hypothesis is H0 : E(ui,t uj,t ) = 0, ∀ t and i 6= j. and the test statistic is s

CD =





N −1 X N X 2T  ρˆij  N (N − 1) i=1 j=i+1

PT

t=1

ρˆij = ρˆji = P T

ˆ2it t=1 u

uˆi,t uˆjt

1/2 P

T t=1

uˆ2jt

1/2

where ρˆij is the correlation coefficient. In the case of an unbalanced panel, the correlation coefficient is calculated for the common sample ρˆij = ρˆji = 



P

where

t∈Ti ∩Tj

¯ˆi = u

P

P

t∈Ti ∩Tj

¯ˆi uˆit − u

t∈Ti ∩Tj

Tij



¯ˆi uˆit − u



2 (1/2) P

uˆit

¯ˆj uˆjt − u

t∈Ti ∩Tj





¯ˆj uˆjt − u

2 (1/2)

, Tij = # (Ti ∩ Tj )

and the CD test statistic becomes then CD =

s





N −1 X N q X 2  Tij ρˆij  N (N − 1) i=1 j=i+1

Under the null the CD test statistic is asymptotically CD ∼ N (0, 1) distributed. For a more in depth discussion see Pesaran (2015); Chudik and Pesaran (2015a).

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The xtdcce command

4.1

Syntax

xtdcce depvar



indepvars



if



, pooled(varlist) crosssectional(varlist)

nocrosssectional cr lags(#) exogenous vars(varlist)

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endogenous vars(varlist) ivreg2options(string) lr(varlist) lr options(string) pooledconstant reportconstant trend pooledtrend residuals(string) jackknife recursive nocd cluster(string) noomit full lists noisily  post full

Data has to be [TS] tsset before using xtdcce. varlist may contain timeseries operators, see [TS] tsvarlist. xtdcce requires the moremata package by Jann (2005).

4.2

Options

pooled(varlist) specifies homogenous coefficients. For these variables the estimated coefficients are constrained to be equal across all units (βi = β ∀ i). Variable may occur in indepvars. Variables in exogenous vars(), endogenous vars() and lr() may be pooled as well. crosssectional(varlist) defines the variables which are included in zt and added as cross sectional averages (¯ zt−l ) to the equation. Variables in crosssectional() may be included in pooled(), exogenous vars(), endogenous vars() and lr(). Default option is to include all variables from depvar, indepvars and endogenous vars() in zt . Variables in crosssectional() are partialled out, the coefficients not estimated and reported. cr lags(#) specifies the number of lags of the cross sectional averages. If not defined but crosssectional() contains varlist, then only contemporaneous cross sectional averages are added, but no lags. cr lags(0) is equivalent to omitting it. nocrosssectional prevents adding cross sectional averages. Results will be equivalent to the Pesaran and Smith (1995) Mean Group estimator, or if lr(varlist) specified to the Shin et al. (1999) Pooled Mean Group estimator. xtdcce supports instrumental variable regression using ivreg2 by Baum et al. (2003, 2007). Endogenous and exogenous variables are set by: endogenous vars(varlist) specifies the endogenous and exogenous vars(varlist) the exogenous variables. See for a further description ivreg2. ivreg2options passes further options on to ivreg2. See ivreg2, options for more information. fulliv posts all available results from ivreg2 in e() with prefix ivreg2 . noisily shows the output of wrapped ivreg2 regression command. lr(varlist): Variables to be included in the long-run cointegration vector. The first variable is the error-correcting speed of adjustment term. 7

lr options(string) Options for the long run coefficients. Options may be: nodivide, coefficients are not divided by the error correction speed of adjustment vector (i.e. estimate equation 4). xtpmgnames, coefficients names in e(b p mg) and e(V p mg) match the name convention from xtpmg. noconstant suppress constant term. pooledconstant restricts the constant to be the same across all groups (β0,i = β0 , ∀i). reportconstant reports the constant. If not specified the constant is treated as a part of the cross sectional averages. trend adds a linear unit specific trend. May not be combined with pooledtrend. pooledtrend a linear common trend is added. May not be combined with trend. jackknife applies the jackknife bias correction for small sample time series bias. May not be combined with recursive. recursive applies recursive mean adjustment method to correct for small sample time series bias. May not be combined with jackknife. residuals(varname) saves residuals as new variable. nocd suppresses calculation of CD test statistic. cluster(varname) clustered standard errors, where varname is the cluster identifier. nomit suppress checks for collinearity. full reports unit individual estimates in output. lists shows all variables names and lags of cross section means. post full requests that the individual estimates, rather than the mean group estimates are saved in e(b) and e(V). Mean group estimates are then saved in e(b p mg) and e(V p mg).

4.3

Saved results

xtdcce saves the following in e():

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Scalars e(N) e(T) e(N partial) e(N pooled) e(rss) e(ll) e(df m) e(r2) e(cd) Scalars e(minT) e(avgT) Macros e(tvar) e(depvar) e(omitted) e(pooled) e(cmd full)

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 (unbalanced panel) minimum time average time

e(N g) e(K) 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(maxT)

maximum time

name of time variable name of dependent variable name of omitted variables name of pooled variables command line including options

e(idvar) e(indepvar) e(lr) e(cmd)

name of unit variable name of independent variables long run variables command line

e(instd)

instrumented (endogenous) variables

e(V)

variance–covariance matrix (mean group or individual) variance–covariance matrix (mean group and pooled) variance–covariance matrix (individual and pooled)

Macros (iv-specific) e(insts) instruments (exogenous) variables Matrices e(b) coefficient vector (mean group or individual) e(b p mg) coefficient vector (mean group and pooled) coefficient vector e(b full) (individual and pooled) Functions e(sample) marks estimation sample

4.4

e(V p mg) e(V full)

xtcd2

Included in the xtdcce package is the xtcd2 command, which tests for weak cross sectional dependence. The command supports balanced as well as unbalanced panels. For a discussion of the teststatistic see section 3. 4.4.1 xtcd2

Syntax 

varname(max=1), noestimation rho histogram name(string)



varname is the name of the residuals or variables and is optional in case the command is performed after an estimation (postestimation). 4.4.2

Options

If noestimation is specified, then xtcd2 is not run as a postestimation command and does not require e(sample) to be set. This option allows to test any variable. If not set, then xtcd2 uses either the variable specified in

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varname or predicts the residuals using predict, residuals. In both cases the sample is restricted to e(sample). histogram plots a histogram of the cross correlations. The number of observations, the mean, percentiles, minimum and maximum of the cross correlations are reported. If name(string) is set, then the histogram is saved and not drawn. rho saves the matrix with the cross correlations in r(rho). 4.4.3

Saved results

Scalars r(CD) r(p) Matrices r(rho)

4.5

value of the CD statistic p-value cross correlations matrix, if requested

The Constant in xtdcce

xtdcce can treat the constant (αi ) in several ways.4 In Pesaran (2006) and Chudik and Pesaran (2015a) the constant is part of the matrix which includes the cross sectional averages and is partialled out. In an empirical setting it can be of use to obtain estimates of the constant. Therefore xtdcce estimates and reports the constant if the option reportconstant is used. Otherwise it is partialled out or removed from the model and not reported. Additionally xtdcce allows the constant to be the same across all units by specifying the option pooledconstant.5 As a final option, the constant can be completely removed from the model by using the noconstant option. The constant is removed from the model if all parameters including the constant are constrained to be homogeneous, the cross sectional means include all variables and the dataset is strongly balanced. Loosely speaking, by adding the time averages of the dependent variable and all independent variables together with a homogeneous constant, the data is demeaned and the constant is rendered to be zero. Thus xtdcce automatically removes the constant from the model to improve the estimation. If the option reportconstant is used, then the constant is still estimated and reported in the output. 4

In order to stick to Stata notation, the intercept is called the constant. If pooledconstant is used but not reportconstant, the constant is internally calculated but not displayed. 5

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5

Empirical Examples

In the following three empirical examples are carried out to demonstrate the use of the xtdcce command. As a first exercise an augmented Solow model with dynamic common correlated effects is estimated. This section includes IV regressions and introduces the novelties of xtdcce. Afterwards mean group and common correlated effects estimations and finally pooled mean group estimations are shown. These last two parts compare xtdcce to the existing Stata commands xtmg and xtpmg.

5.1

Dynamic Common Correlated Effects and testing for cross sectional dependence

As a first exercise the augmented Solow Model in style of Mankiw et al. (1992), Islam (1995) and Lee et al. (1997) is estimated by regressing the difference in log GDP per capita on lagged GDP per capita, human and physical capital and the population growth rate. The latest version of the Penn World Tables (Feenstra et al., 2015) are used. The dataset is restricted to the years from 1960 to 2007, meaning 48 years are available initially. After tsset, the estimation is run, with the dependent and all four independent variables as cross sectional means, set by crosssectional.6 The number of cross section means, specified by cr lags() is set to 3. . xtdcce d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd , /* > */ cr(d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd) /* > */ reportc cr_lags(3) res(residuals) Dynamic Common Correlated Effects - Mean Group Panel Variable (i): id Time Variable (t): year

D.log_rgdpo Mean Group Estimates: L.log_rgdpo log_hc log_ck log_ngd _cons

Number of obs = Number of groups = Obs per group (T) = F( 465, 1767)= Prob > F = R-squared = Adj. R-squared = Root MSE = CD Statistic = p-value = Coef. -.636661 -1.3089 .220947 .041286 -1.87339

Std. Err. .030397 .396571 .051488 .105526 1.83475

z -20.94 -3.30 4.29 0.39 -1.02

P>|z| 0.000 0.001 0.000 0.696 0.307

4092 93 44 0.81 1.00 0.52 0.52 0.06 1.37 0.1716

[95% Conf. Interval] -.6962375 -2.086161 .1200325 -.1655399 -5.469432

-.5770841 -.5316326 .3218616 .2481125 1.722646

6 For a better readability, the commands are abbreviated in the outputs, but not in the text.

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Mean Group Variables: L.log_rgdpo log_hc log_ck log_ngd _cons Cross Sectional Averaged Variables: D.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd Degrees of freedom per country: in mean group estimation = 39 with cross-sectional averages = 24 Number of cross sectional lags = 3 variables in mean group regression = 2325 variables partialled out = 1860

On the lower right of the upper panel, the output shows a CD test statistic of 1.37, or a p-value of 0.17, so the hypothesis of weak cross sectional dependence can not be rejected. Below the coefficient estimates, xtdcce displays the names of the 4 mean group variables, the constant and 5 cross sectional averages. As the regression without any pooled variables is essentially a regression run on each country separately, the degree of freedom for each country is shown as well, which equals to the number of time periods (T=44) minus the number of variables (5). The degree of freedom for each country with cross-sectional averages indicates the potential degree of freedom of a regression which would include the cross sectional averages. It equals the number of time periods (44) minus the number of variables (K = 5), minus the number of cross section averages times the number of lags plus one for the contemporaneous averages (5 ∗ (3 + 1) = 20). At the end of the table, the number of lags of the cross sectional means is displayed, together with the number of variables in the mean group regression and the number of variables partialled out, which equals to the number of cross sectional means. As the cross sectional means are purely treated as controls and have no interpretation, no information is lost by partialling out. Therefore the means are regressed on each of the explanatory variables of interest and then the residuals collected. The residuals are then used as the new explanatory and dependent variables. The partialling out is performed in Mata. The variables to be partialled out (the cross sectional means and if requested the heterogeneous intercept) are stacked in a block diagonal matrix, with zeros on the off diagonals. For a large number of units the matrix becomes sparse and calculating and inverting the cross product becomes computational intensive, hence time consuming. To improve speed, the partialling out is done sequentially unit by unit, which is possible as long as the coefficients on the cross sectional means, δi,l , are heterogeneous.7 Within this process, the program 7

The precision lies in a negligible order of magnitude and is offsetted by the improvement in speed. A simulation supporting these results are available upon request from the author. The standard solver for the calculation of the inverse of the cross product of the factor loadings is cholsolve. cholsolve can not solve positive definite or singular matrices. In this case qrsolve is used. Thanks to Mark Schaffer for providing the code for this routine.

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checks if the factor loadings are full rank.8 If the check fails, xtdcce aborts with error code 506, see section 6. While for the computation of the cross sectional means, all available data is used, the partialling out is restricted to the observations used in the regression. For the example above this means the following: 1 period is lost to create L.log rgdpo and further 3 periods are lost due to the creation of the lags of the cross sectional means. The cross sectional means of the variables with exception of the lag level of the dependent variable are computed using the full available sample size. The average for L.log rgdpo is based on the years 1961 - 2007 as the value for 1960 is missing. The regression and the partialling out is then applied on the restricted sample. In total 4 periods are lost, so the time span for the regression are the years 1964 - 2007, making the time dimension T = 44. The regression results are not in favour of the Solow Model as the coefficient on human capital is negative and the coefficient on physical capital positive but not significant. For a more detailed discussion of the Solow Model in growth empirics see Mankiw et al. (1992); Islam (1995); Lee et al. (1997) or Durlauf et al. (2005); Jones (2015); and with a focus on slope heterogeneity see Islam (1998) and Lee et al. (1998). As the residuals are saved as a new variable called residuals, the test on cross sectional dependence can be done by hand to confirm the result from above. . xtcd2 residuals Chudik and Pesaran (2015) test for cross sectional dependence Postestimation. H0: errors are weakly cross sectional dependent. CD = 1.3669791 p_value = .17163187

Using the option noestimation leads to the same result, as long as the observations which are omitted in the estimation are missings in the variable residuals. The advantage of noestimation is, that it allows testing variables for cross sectional dependence. For example testing the independent variable for cross sectional dependence reads: . xtcd2 log_d_rgdpo , noest Chudik and Pesaran (2015) test for cross sectional dependence H0: errors are weakly cross sectional dependent. CD = 72.243906 p_value = 0

In a next step all 5 coefficients are constrained to be the same across 8

The condition of a full rank is checked on the unit specific matrices containing the cross sectional means (¯ zi = (y¯i , x¯i ), where y¯i and x¯i are T × 1 and T × K matrices containing the cross sectional means). This is possible as the matrix over all units is block diagonal and the rank of it is equal to the sum of the rank of the blocks.

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countries, βi,k = βk , ∀i = 1, .., N, k = 0, ..., 4, by specifying the pooled() option, pooling the constant by using the pooledconstant option and forcing xtdcce to display the constant using reportconstant.9 . xtdcce d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd , /* > */ p(L.log_rgdpo log_hc log_ck log_ngd) reportc /* > */ cr(d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd) cr_lags(3) pooledc Dynamic Common Correlated Effects - Pooled Mean Group Panel Variable (i): id Time Variable (t): year

D.log_rgdpo Pooled Variables: L.log_rgdpo log_hc log_ck log_ngd _cons

Coef. -.291652 .061267 .147058 .003515 -1.4e-15

Std. Err. .014082 .096776 .013429 .022033 .469827

Number of obs = Number of groups = Obs per group (T) = F( 5, 2227)= Prob > F = R-squared = Adj. R-squared = Root MSE = CD Statistic = p-value = z

-20.71 0.63 10.95 0.16 -0.00

P>|z| 0.000 0.527 0.000 0.873 1.000

4092 93 44 0.23 0.95 0.16 0.16 0.07 -0.92 0.3567

[95% Conf. Interval] -.3192522 -.1284114 .1207373 -.0396682 -.9208433

-.2640528 .2509449 .1733795 .0466986 .9208433

Pooled Variables: L.log_rgdpo log_hc log_ck log_ngd _cons Cross Sectional Averaged Variables: D.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd Degrees of freedom per country: in mean group estimation = 39 with cross-sectional averages = 24 Number of cross sectional lags = 3 variables in mean group regression = 1865 variables partialled out = 1860

The estimate for the constant is close to zero. This result is expected, as outlined in the section above, because all coefficients are pooled, the panel is balanced and all variables are added as cross sectional means. As a final exercise, let’s assume that investments into physical capital are endogenous and are instrumented by its lagged values. The endogenous variable log sk is specified in endogenous vars() and the exogenous variable L.log sk in exogenous vars(). . xtdcce d.log_rgdpo L.log_rgdpo log_hc log_ngd , /* > */ cr(d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd) reportc /* > */ endo(log_ck) exo(L.log_ck) cr_lags(3) ivreg2options(nocollin noid) Dynamic Common Correlated Effects - Mean Group IV 9

Pesaran (2006) discusses the mean group and the pooled version of the CCE estimator. A pooled version in the dynamic setting is not mentioned by Chudik and Pesaran (2015b). However Everaert and De Groote (2015) show that the unrestricted CCEP estimator is consistent as long as N, T ⇒ ∞.

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Panel Variable (i): id Time Variable (t): year

D.log_rgdpo Mean Group Estimates: log_ck L.log_rgdpo log_hc log_ngd _cons

Number of obs = Number of groups = Obs per group (T) = F( 2325, 1767)= Prob > F = R-squared = Adj. R-squared = Root MSE = CD Statistic = p-value = Coef. -.085406 -.557558 -1.04877 .127136 -1.51382

Std. Err. .073381 .034126 .403293 .167222 1.92312

z -1.16 -16.34 -2.60 0.76 -0.79

P>|z| 0.244 0.000 0.009 0.447 0.431

4092 93 44 3.55 0.00 0.48 -0.21 0.04 1.16 0.2474

[95% Conf. Interval] -.2292295 -.6244428 -1.839213 -.2006135 -5.283056

.0584181 -.4906724 -.2583344 .4548857 2.255424

Mean Group Variables: L.log_rgdpo log_hc log_ngd _cons Cross Sectional Averaged Variables: D.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd Endogenous Variables: log_ck Exogenous Variables: L.log_ck Degrees of freedom per country: in mean group estimation = 39 with cross-sectional averages = 24 Number of cross sectional lags = 3 variables in mean group regression = 2325 variables partialled out = 1860

5.2

Pooled Mean Group

In the following xtdcce is compared to results from xtpmg, by Blackburne and Frank (2007). xtpmg implements the Pooled Mean Group estimator by Shin et al. (1999) into Stata. The two programs differs in two ways. First of all xtpmg estimates the following equation ∆ci,t = φi (ci,t−1 − θ1,i yi,t − θ2,i πi,t ) + δ0,i + δ1,i ∆yi,t + δ2,i ∆πi,t + i,t ,

(3)

while xtdcce internally estimates (leaving out any cross sectional means): ∆ci,t = φi ci,t−1 + γ1,i yi,t + γ2,i πi,t + δ0,i + δ1,i ∆yi,t + δ2,i ∆πi,t + i,t ,

(4)

Secondly xtpmg calculates the long run coefficients using maximum likelihood. xtdcce treats the long run coefficients, defined in lr(), as further covariates and estimates equation 4 entirely by OLS. To calculate the long run coefficients, the coefficients are divided by the negative of the long run cointegration vector to match equation 3, θ1,i = −γ1,i /φi . The variances are calculated using the Delta method. Equation 4 and the coefficients γ1,i , ..., γK,i can be estimated by using lr options(nodivide). 15

As in Blackburne and Frank (2007) the jasa2 dataset is used to explain consumption with inflation and income after 1962. esttab produces the following output: . use jasa2, clear . tsset id year panel variable: id (unbalanced) time variable: year, 1960 to 1993 delta: 1 unit . qui xtpmg d.c d.pi d.y if year>=1962, lr(l.c pi y) ec(ec) replace pmg . eststo xtpmg: . . eststo xtdcce1: qui xtdcce d.c d.pi d.y if year >= 1962 , /* > */ lr(l.c pi y) p(l.c pi y) nocross lr_options(xtpmgnames) reportc . eststo xtdcce2: qui xtdcce d.c d.pi d.y if year >= 1962 , /* > */ lr(l.c pi y) p(l.c pi y) nocross lr_options(nodivide xtpmgnames) reportc . eststo xtdcce3: qui xtdcce d.c d.pi d.y if year >= 1962 , /* > */ lr(l.c pi y) p(l.c pi y) cr(d.c d.pi d.y) cr_lags(0) /* > */ lr_options(xtpmgnames) reportc . esttab xtpmg xtdcce1 xtdcce2 xtdcce3 /* > */ , se mlabels(xtpmg "xtdcce (mg)" "xtdcce (mg)" "xtdcce (cce)") s(N cd cdp) (1) xtpmg ec pi y SR ec D.pi D.y _cons N cd cdp

(2) xtdcce (mg)

(3) xtdcce (mg)

(4) xtdcce (cce)

-0.466*** (0.0567) 0.904*** (0.00868)

-0.194** (0.0701) 0.903*** (0.0163)

-0.0327** (0.0121) 0.152*** (0.0144)

-0.276*** (0.0698) 0.940*** (0.0170)

-0.200*** (0.0322) -0.0183 (0.0278) 0.327*** (0.0574) 0.154*** (0.0217)

-0.168*** (0.0151) -0.0548 (0.0299) 0.380*** (0.0350) 0.137*** (0.00765)

-0.168*** (0.0151) -0.0548 (0.0299) 0.380*** (0.0350) 0.137*** (0.00765)

-0.184*** (0.0172) 0.0237 (0.0317) 0.384*** (0.0431) 0.0643*** (0.00585)

767 4.101 0.0000410

767 4.101 0.0000410

767

767 0.671 0.502

Standard errors in parentheses * p<0.05, ** p<0.01, *** p<0.001

Column (1) shows the results using xtpmg, columns (2) - (4) using the xtdcce estimator. Column (1) matches the results from Blackburne and Frank (2007) p. 203. Column (2) displays the results using xtdcce. The signs of the results are the same and the magnitudes, especially for the short run coefficients, are very similar. In column (3) the option nodivide is used, producing estimates for equation 4.10 10

An alternative to obtain long run coefficients in a dynamic panel using the between

16

As the last row indicates, using no cross sectional means leads to a rejection of the null of weak cross section dependence. Therefore in column (4) cross sectional means are added. The short run coefficients can be restricted to be equal across all units by including them in the pooled() option. At the same time the long run coefficients can be allowed to vary as well. To test under which constraints the model is consistent, the Hausman test can be performed: . > . > . >

eststo mg: qui xtdcce d.c d.pi d.y if year >= 1962 , /* */ lr(l.c pi y) nocross reportc eststo pmg: qui xtdcce d.c d.pi d.y if year >= 1962 , /* */ lr(l.c pi y) p(l.c pi y) nocross reportc eststo pooled: qui xtdcce d.c d.pi d.y if year >= 1962 , /* */ lr(l.c pi y) p(l.c pi y d.pi d.y) nocross reportc

. hausman mg pooled, sigmamore Coefficients (b) (B) mg pooled pi D1. y D1. c L1. pi y

(b-B) Difference

sqrt(diag(V_b-V_B)) S.E.

-.0253642

-.0280826

.0027184

.0305406

.2337588

.3811944

-.1474357

.0533472

-.3063473 -.3529095 .9181344

-.1794146 -.266343 .9120574

-.1269326 -.0865666 .0060771

.0328851 .1230082 .0287949

b = consistent under Ho and Ha; obtained from xtdcce B = inconsistent under Ha, efficient under Ho; obtained from xtdcce Test: Ho: difference in coefficients not systematic chi2(5) = (b-B)´[(V_b-V_B)^(-1)](b-B) = 18.13 Prob>chi2 = 0.0028 . hausman pmg pooled, sigmamore Coefficients (b) (B) (b-B) sqrt(diag(V_b-V_B)) pmg pooled Difference S.E. c L1. pi y pi D1. y D1.

Test:

-.1683577 -.1941238 .9025766

-.1794146 -.266343 .9120574

.0110569 .0722191 -.0094807

.0050032 .0316818 .007498

-.0548234

-.0280826

-.0267408

.0264268

.3802491

.3811944

-.0009453

.0279448

b = consistent under Ho and Ha; obtained from xtdcce B = inconsistent under Ha, efficient under Ho; obtained from xtdcce Ho: difference in coefficients not systematic chi2(5) = (b-B)´[(V_b-V_B)^(-1)](b-B) = 2.66 Prob>chi2 = 0.7515 (V_b-V_B is not positive definite)

estimator is outlined in Ditzen and Gundlach (2015).

17

The result of the Hausman test is similar to the one obtained in (Blackburne and Frank, 2007, Section 4.3 and 4.4). The first Hausman test implies that the pooled model is preferred over the mean group model. The second Hausman test compares the mean group and the pooled model. The conclusion is in line with Blackburne and Frank (2007), that the pooled mean group model is preferred.

5.3

Mean Group and Common Correlated Effects

xtdcce is able to compute the mean group and common correlated effects estimator by Pesaran and Smith (1995) and Pesaran (2006), introduced by Eberhardt (2012) xtmg command. Following Eberhardt (2012) using the dataset manu stata9.dta, xtmg leads to the following mean group results:11 . use manu_stata9.dta . xtset nwbcode year panel variable: time variable: delta: . . . . . > >

nwbcode (strongly balanced) year, 1970 to 2002 1 unit

eststo xtmg06: qui xtmg ly lk, cce trend eststo xtdcce06: qui xtdcce ly lk , cr(ly lk) cr_lags(0) trend reportc eststo xtmg95: qui xtmg ly lk, trend eststo xtdcce95: qui xtdcce ly lk , cr(ly lk) trend nocross reportc estout xtmg95 xtdcce95 xtmg06 xtdcce06 , c(b(star fmt(4)) se(fmt(4) par)) /* */ mlabels("xtmg - mg" xtdcce "xtmg - cce" xtdcce) s(N cd cdp , fmt(0 3 3 )) /* */ drop(*_ly *_lk) rename(__000007_t trend) collabels(,none) xtmg - mg

lk trend _cons N cd cdp

xtdcce

xtmg - cce

xtdcce

0.1789* (0.0805) 0.0174*** (0.0030) 7.6528*** (0.8546)

0.1789* (0.0805) 0.0174*** (0.0030) 7.6354*** (0.8531)

0.3125*** (0.0849) 0.0108** (0.0035) 4.7860*** (1.3227)

0.3125*** (0.0849) 0.0108** (0.0035) 4.7752*** (1.3202)

1194

1194 6.686 0.000

1194

1194 -0.201 0.841

The first two columns show regressions using the mean group estimator without any cross section means. Column (1) and (3) match the estimation results from Table 1, p. 67 in Eberhardt (2012). Column (3) and (4) are based on the common correlated effects estimator, which includes a contemporaneous cross sectional means. In column (2) the CD teststatistic rejects the hypothesis of weak cross sectional dependence. Including cross sectional averages improves the statistic such that the hypothesis is can not be rejected 11

The dataset manu stata9.dta is taken from Eberhardt and Teal (2014) and is available at https://sites.google.com/site/medevecon/.

18

any longer. Estimation results produced by xtmg and xtdcce differ slightly, as seen here by the constant. The difference between both is, that xtdcce converts and creates internally all variables into doubles, to allow for best precision.12

6

Error Messages

xtdcce produces the following error codes:

7

r(109)

ivreg2 not installed

r(184)

r(506)

Rank condition on cross section means not satisfied.

r(2001)

options noconstant and pooledconstant, trend and trendconstant or jackknife and recursive are combined. More variables than observations.

Conclusion

The Stata user written program xtdcce introduces new routines to estimate a heterogeneous panel model using dynamic common correlated effects. It combines estimation procedures proposed in Pesaran and Smith (1995) and (Shin et al., 1999) with those in Pesaran (2006) and Chudik and Pesaran (2015b). It allows coefficients to be pooled or estimated as mean groups. Furthermore it supports unbalanced panels, estimation of instrumental variables, small sample time series bias corrections and tests for cross sectional dependence, using the included xtcd2 routine. An empirical example estimating a growth regression is given.

8

Acknowledgments

I am grateful to Arnab Bhattacharjee and Mark Schaffer for many valuable comments and suggestions. To Achim Ahrens for his contributions to the xtcd2 command and to Kyle McNabb for testing the program. The code and especially the output greatly benefited from Markus Eberhardt’s xtmg command. Any remaining errors are my own. Jan Ditzen, PhD Student, Spatial Economics and Econometrics Centre (SEEC), School of Management & Languages, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, United Kingdom; Email: [email protected], Web: www.jan.ditzen.net 12

Another difference to xtmg is that xtdcce supports time-series operators.

19

References Baum, C. F., M. E. Schaffer, and S. Stillman. 2003. Instrumental variables and GMM: Estimation and testing. Stata Journal 1(3): 1–31. . 2007. Enhanced routines for instrumental variables/generalized method of moments estimation and testing. Stata Journal 7(4): 465–506. Blackburne, E. F., and M. W. Frank. 2007. Estimation of nonstationary heterogeneous panels. Stata Journal 7(2): 197–208. Bond, S. R., and M. Eberhardt. 2013. Accounting for unobserved heterogeneity in panel time series models. Chudik, A., and M. H. Pesaran. 2015a. Large Panel Data Models with CrossSectional Dependence: A Survey. In The Oxford Handbook Of Panel Data, ed. B. H. Baltagi, chap. 1, 2–45. Oxford University Press. . 2015b. Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors. Journal of Econometrics 188(2): 393–420. Chudik, A., M. H. Pesaran, and E. Tosetti. 2011. Weak and strong crosssection dependence and estimation of large panels. The Econometrics Journal 14(1): C45–C90. De Hoyos, R. E., and V. Sarafidis. 2006. Testing for cross-sectional dependence in panel-data models. The Stata Journal 6(4): 482–496. Ditzen, J., and E. Gundlach. 2015. A Monte Carlo study of the BE estimator for growth regressions. Empirical Economics forthcoming. Durlauf, S. N., P. A. Johnson, and J. R. W. Temple. 2005. Growth econometrics. In Handbook of economic growth, ed. P. Aghion and S. N. Durlauf, volume 1, ed., chap. Chapter 8, 555–677. Eberhardt, M. 2012. Estimating panel time series models with heterogeneous slopes. Stata Journal 12(1): 61–71. Eberhardt, M., C. Helmers, and H. Strauss. 2012. Do Spillovers Matter When Estimating Private Returns to R&D? Review of Economics and Statistics 95(May): 120207095627009. Eberhardt, M., and F. Teal. 2014. The magnitude of the task ahead: Productivity analysis with heterogeneous technology . 20

Everaert, G., and T. De Groote. 2015. Common Correlated Effects Estimation of Dynamic Panels with Cross-Sectional Dependence. Econometric Reviews 4938(1981): 1–31. Everaert, G., and I. De Vos. 2016. Bias-corrected Common Correlated Effects Pooled estimation in homogeneous dynamic panels. University of Ghent Working Paper . Feenstra, R. C., R. Inklaar, and M. Timmer. 2015. The Next Generation of the Penn World Table. American Economic Review . URL www.ggdc.net/pwt. Hayashi, F. 2000. Econometrics. Islam, N. 1995. Growth Empirics: A Panel Data Approach. The Quarterly Journal of Economics 110(4): 1127 – 1170. . 1998. Growth Empirics: A Panel Data Approach - A reply. Quarterly journal of economics 113(1): 325–329. Jann, B. 2005. moremata: Stata module (Mata) to provide various functions. URL Available from http://ideas.repec.org/c/boc/bocode/s455001.html. Jones, C. I. 2015. The Facts of Economic Growth. NBER Working Paper (w21142). Kapetanios, G., M. H. Pesaran, and T. Yamagata. 2011. Panels with nonstationary multifactor error structures. Journal of Econometrics 160(2): 326–348. Lee, K., M. H. Pesaran, and R. Smith. 1997. Growth and Convergence in a Multi-Country Empirical Stochastic Solow Model. Journal of Applied Economics 12(4): 357–392. Lee, K., M. H. Pesaran, and R. P. Smith. 1998. Growth empirics: a panel data approach - A Comment. The Quarterly Journal of Economics 113(1): 319–323. Mankiw, N. G., D. Romer, and D. N. Weil. 1992. A Contribution to the Empirics of Economic Growth. The Quarterly Journal of Economics 107(2): 407–437. McNabb, K., and P. LeMay-Boucher. 2014. Tax Structures, Economic Growth and Development. ICTD Working Paper 22. 21

Pesaran, M. 2006. Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74(4): 967–1012. Pesaran, M. H. 2015. Testing Weak Cross-Sectional Dependence in Large Panels. Econometric Reviews 34(6-10): 1089–1117. Pesaran, M. H., and R. Smith. 1995. Econometrics Estimating long-run relationships from dynamic heterogeneous panels. Journal of Econometrics 68: 79–113. Shin, Y., M. H. Pesaran, and R. P. Smith. 1999. Pooled Mean Group Estimation of Dynamic Heterogeneous Panels. Journal of the American Statistical Association 94(446): 621 –634.

9 9.1

Technical Appendix Delta Method

For the calculation of the long run coefficients the estimates of γˆk,i obtained by OLS from equation 4 are divided by estimates of the long run cointegration vector φˆi . To calculate the variance covariance matrix the delta method is used. The delta method allows the calculation of an approximate probability distribution for a matrix function a(β) based on a random vector with a known variance (see for example √ Hayashi, 2000, p. 93). Suppose that for the random vector βi →p β and n(βi − β) →d N (0, σ). Denote the first derivates of a(β) as ∂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 ) . For the calculation of the long run coefficients and using the notation from equation 4, assume that βi = (φi , γ1,i , γ2,i , δ1,i , δ2,i )0 The variance covariance matrix is: 

Σ=

      

V (φi ) Cov(φi γ1,i ) Cov(φi γ2,i ) Cov(φi δ1,i ) Cov(φi δ2,i ) Cov(φi γ1,i ) V (γ1,i ) Cov(γ1,i γ2,i ) ... . . . . . . . Cov(φi δ2,i ) ... ... ... V (δ2,1 ) 22

       

The function a() maps the long run coefficients and leaves the short run coefficients: a(βi ) = (φi , −γ1,i /φi , −γ2,i /φi , δ1,i , δ2,i )0 = (φi , θ1,i , θ2,i , δ1,i , δ2,i )0 The first derivative of a() is then: 

Ai (β) =

         



=

       

∂φi ∂φi ∂φi ∂γ1,i ∂φi ∂γ2,i ∂φi ∂δ1,i ∂φi ∂δ2,i

∂θ1,i ∂φi ∂θ1,i ∂γ1,i ∂θ1,i ∂γ2,i ∂θ1,i ∂δ1,i ∂θ1,i ∂δ2,i

∂θ2,i ∂φi ∂θ2,i ∂γ1,i ∂θ2,i ∂γ2,i ∂θ2,i ∂δ1,i ∂θ2,i ∂δ2,i

1 − γφ1,i2 − γφ2,i2 i i 0 − φ1i 0 0 0 − φ1i 0 0 0 0 0 0

∂δ1,i ∂φi ∂δ1,i ∂γ1,i ∂δ1,i ∂γ2,i ∂δ1,i ∂δ1,i ∂δ1,i ∂δ2,i

0 0 0 1 0

0 0 0 0 1

∂δ2,i ∂φi ∂δ2,i ∂γ1,i ∂δ2,i ∂γ2,i ∂δ2,i ∂δ1,i ∂δ2,i ∂δ2,i

        

          

All components of the variance covariance matrix are then known and it can be calculated as: Σa = A(β)ΣA(β)0

23