xtdcce2

Estimating heterogeneous coefficient models using common correlated effects in a dynamic panel with a large number of observations over groups and time periods.

Table of Contents

  1. Syntax
  2. Description
  3. Options
  4. Econometric and Empirical Model
    1. Mean Group
    2. Common Correlated Effects
    3. Dynamic Common Correlated Effects
    4. Pooled Estimations
    5. Instrumental Variables
    6. Error Correction Models (ECM/PMG)
    7. Cross-Section Augmented Distributed Lag (CS-DL)
    8. Cross-Section Augmented ARDL(CS-ARDL)
  5. Saved Values
  6. Postestimation Commands
  7. Examples
    1. Mean Group
    2. Common Correlated Effects
    3. Dynamic Common Correlated Effects
    4. Pooled Estimation
    5. Instrumental Variables
    6. Error Correction Model (ECM/PMG)
    7. Cross-Section Augmented Distributed Lag (CS-DL)
    8. Cross-Section Augmented ARDL(CS-ARDL)
  8. References
  9. About
  10. How to install
  11. Changelog

1. Syntax

xtdcce2 _depvar_ [_indepvars_] [_varlist2_ = _varlist_iv_] [ifin] , crosssectional(_varlist_) [pooled(_varlist_) cr_lags(_string_) NOCRosssectional ivreg2options(_string_) e_ivreg2_ ivslow noisily lr(_varlist_) lr_options(_string_) pooledconstant reportconstant noconstant trend pooledtrend jackknife recursive nocd exponent showindividual fullsample fast blockdiaguse nodimcheck useqr useinvsym NOOMITted]

where varlist2 are endogenous variables and varlist_iv the instruments. Data has to be xtset before using xtdcce2; see tssst. varlists may contain time-series operators, see tsvarlist, or factor variables, see fvvarlist. xtdcce2 requires the moremata package.

2. Description

xtdcce2 estimates a heterogeneous coefficient model in a large panel with dependence between cross sectional units. A panel is large if the number of cross-sectional units (or groups) and the number of time periods are going to infinity.

It fits the following estimation methods:

i) The Mean Group Estimator (MG, Pesaran and Smith 1995).

ii) The Common Correlated Effects Estimator (CCE, Pesaran 2006),

iii) The Dynamic Common Correlated Effects Estimator (DCCE, Chudik and Pesaran 2015), and

For a dynamic model, several methods to estimate long run effects are possible:

a) The Pooled Mean Group Estimator (PMG, Shin et. al 1999) based on an Error Correction Model,

b) The Cross-Sectional Augmented Distributed Lag (CS-DL, Chudik et. al 2016) estimator which directly estimates the long run coefficients from a dynamic equation, and

c) The Cross-Sectional ARDL (CS-ARDL, Chudik et. al 2016) estimator using an ARDL model. For a further discussion see Ditzen (2018b).

Additionally xtdcce2 tests for cross sectional dependence (see xtcd2) and estimates the exponent of the cross sectional dependence alpha (see xtcse2). It also supports instrumental variable estimations (see ivreg2).

3. Options

Option Description
crosssectional(varlist) defines the variables which are added as cross sectional averages to the equation. Variables in crosssectional() may be included in pooled(), exogenous_vars(), endogenous_vars() and lr(). Variables in crosssectional() are partialled out, the coefficients not estimated and reported. crosssectional(all) adds all variables as cross sectional averages. No cross sectional averages are added if crosssectional(none) is used, which is equivalent to nocrosssectional. crosssectional() is a required option but can be substituted by nocrosssectional.
pooled(varlist) specifies variables which estimated coefficients are constrained to be equal across all cross sectional units. Variables may occur in indepvars. Variables in exogenous_vars(), endogenous_vars() and lr() may be pooled as well.
cr_lags(integers) sets the number of lags of the cross sectional averages. If not defined but crosssectional() contains a varlist, then only contemporaneous cross sectional averages are added but no lags. cr_lags(0) is the equivalent. The number of lags can be different for different variables, where the order is the same as defined in cr(). For example if cr(y x) and only contemporaneous cross-sectional averages of y but 2 lags of x are added, then cr_lags(0 2).
nocrosssectional suppresses adding any cross sectional averages Results will be equivalent to the Mean Group estimator.
pooledconstant restricts the constant term to be the same across all cross sectional units.
reportconstant reports the constant term. If not specified the constant is partialled out.
noconstant suppresses the constant term.
trend adds a linear unit specific trend. May not be combined with pooledtrend.
pooledtrend adds a linear common trend. May not be combined with trend.
jackknife applies the jackknife bias correction method. May not be combined with recursive.
recursive applies the recursive mean adjustment method. May not be combined with jackknife.
nocd suppresses calculation of CD test. For details about the CD test see LINK TO XTCD2.
exponent uses xtcse2 to estimate the exponent of the cross-sectional dependence of the residuals. A value above 0.5 indicates cross-sectional dependence, see xtcse2.
showindividual reports unit individual estimates in output.
fullsample uses entire sample available for calculation of cross sectional averages. Any observations which are lost due to lags will be included calculating the cross sectional averages (but are not included in the estimation itself).
fast omit calculation of unit specific standard errors.
blockdiaguse uses mata blockdiag rather than an alternative algorithm. mata blockdiag is slower, but might produce more stable results.
nodimcheck Does not check for dimension. Before estimating a model, xtdcce2 automatically checks if the time dimension within each panel is long enough to run a mean group regression. Panel units with an insufficient number are automatically dropped.

xtdcce2 checks for collinearity in three different ways. It checks if matrix of the cross-sectional averages is of full rank. After partialling out the cross-sectional averages, it checks if the entire model across all cross-sectional units exhibits multicollinearity. The final check is on a cross-sectional level. The outcome of the checks influence which method is used to invert matrices. If a check fails xtdcce2 posts a warning message. The default is cholinv and invsym if a matrix is of rank-deficient. For a further discussion see collinearity issues) .

The following options are available to alter the behaviour of xtdcce2 with respect to matrices of not full rank:

Option Description
useqr calculates the generalized inverse via QR decomposition. This was the default for rank-deficient matrices for xtdcce2 pre version 1.35.
useinvsym calculates the generalized invers via mata invsym.
showomitted displays a cross-sectional unit - variable breakdown of omitted coefficients.
noomitted no omitted variable checks on the entire model.

xtdcce2 supports IV regressions using ivreg2. The IV specific options are:

Option Description
ivreg2options(string) passes further options to ivreg2, see ivreg2, options
e_ivreg2 posts all available results from ivreg2 in e() with prefix ivreg2, see ivreg2, macros.
noisily displays output of ivreg2.
ivslow For the calculation of standard errors for pooled coefficients an auxiliary regressions is performed. In case of an IV regression, xtdcce2 runs a simple IV regression for the auxiliary regressions. This is faster. If option is used ivslow, then xtdcce2 calls ivreg2 for the auxiliary regression. This is advisable as soon as ivreg2 specific options are used.

xtdcce2 is able to estimate long run coefficients. Three models are supported:

The pooled mean group models (Shin et. al 1999), similar to xtpmg (see xtdcce2, ecm), the CS-DL (see xtdcce2, CSDL) and CS-ARDL method (see xtdcce2, ardl) as developed in Chudik et. al 2016. No options for the CS-DL model are necessary.

Options Description
lr(varlist) specifies the variables to be included in the long-run cointegration vector. The first variable(s) is/are the error-correction speed of adjustment term. The default is to use the pmg model. In this case each estimated coefficient is divided by the negative of the long-run cointegration vector (the first variable). If the option ardl is used, then the long run coefficients are estimated as the sum over the coefficients relating to a variable, divided by the sum of the coefficients of the dependent variable.
lr_options(string) options for the long run coefficients. Options are:
ardl estimates the CS-ARDL estimator. For further details see xtdcce2, ardl.
nodivide coefficients are not divided by the error correction speed of adjustment vector. Equation (7) is estimated, see xtdcce2, ecm.
xtpmgnames coefficient names in e(b_p_mg) (or e(b_full)) and e(V_p_mg) (or e(V_full)) match the name convention from xtpmg.

4. Econometric and Empirical Model

Econometric Model

Assume the following dynamic panel data model with heterogeneous coefficients:

(1) y(i,t) = b0(i) + b1(i)*y(i,t-1) + x(i,t)*b2(i) + x(i,t-1)*b3(i) + u(i,t) u(i,t) = g(i)*f(t) + e(i,t)

where f(t) is an unobserved common factor loading, g(i) a heterogeneous factor loading, x(i,t) is a (1 x K) vector and b2(i) and b3(i) the coefficient vectors. The error e(i,t) is iid and the heterogeneous coefficients b1(i), b2(i) and b3(i) are randomly distributed around a common mean. It is assumed that x(i,t) is strictly exogenous. In the case of a static panel model (b1(i) = 0) Pesaran (2006) shows that mean of the coefficients 0, b2 and b3 (for example for b2(mg) = 1/N sum(b2(i))) can be consistently estimated by adding cross sectional means of the dependent and all independent variables. The cross sectional means approximate the unobserved factors. In a dynamic panel data model (b1(i) <> 0) pT lags of the cross sectional means are added to achieve consistency (Chudik and Pesaran 2015). The mean group estimates for b1, b2 and b3 are consistently estimated as long as N,T and pT go to infinity. This implies that the number of cross sectional units and time periods is assumed to grow with the same rate.

In an empirical setting this can be interpreted as N/T being constant. A dataset with one dimension being large in comparison to the other would lead to inconsistent estimates, even if both dimension are large in numbers. For example a financial dataset on stock markets returns on a monthly basis over 30 years (T=360) of 10,000 firms would not be sufficient. While individually both dimension can be interpreted as large, they do not grow with the same rate and the ratio would not be constant. Therefore an estimator relying on fixed T asymptotics and large N would be appropriate. On the other hand a dataset with lets say N = 30 and T = 34 would qualify as appropriate, if N and T grow with the same rate.

The variance of the mean group coefficient b1(mg) is estimated as:

var(b(mg)) = 1/N sum(i=1,N) (b1(i) - b1(mg))^2

or if the vector pi(mg) = (b0(mg),b1(mg)) as:

var(pi(mg)) = 1/N sum(i=1,N) (pi(i) - pi(mg))(p(i)-pi(mg))'

Empirical Model

The empirical model of equation (1) without the lag of variable x is:

(2) y(i,t) = b0(i) + b1(i)*y(i,t-1) + x(i,t)*b2(i) + sum[d(i)*z(i,s)] + e(i,t),

where z(i,s) is a (1 x K+1) vector including the cross sectional means at time s and the sum is over s=t…t-pT. xtdcce2 supports several different specifications of equation (2).

xtdcce2 partials out the cross sectional means internally. For consistency of the cross sectional specific estimates, the matrix z = (z(1,1),…,z(N,T)) has to be of full column rank. This condition is checked for each cross section. xtdcce2 will return a warning if z is not full column rank. It will, however, continue estimating the cross sectional specific coefficients and then calculate the mean group estimates. The mean group estimates will be consistent. For further reading see, Chudik, Pesaran (2015, Journal of Econometrics), Assumption 6 and page 398.

4.1 Mean Group

If no cross sectional averages are added (d(i) = 0), then the estimator is the Mean Group Estimator as proposed by Pesaran and Smith (1995).

The estimated equation is:

(3) y(i,t) = b0(i) + b1(i)*y(i,t-1) + x(i,t)*b2(i) + e(i,t).

Equation (3) can be estimated by using the nocross option of xtdcce2. The model can be either sta#tic (b(1) = 0) or dynamic (b(1) <> 0).

See example

4.2 Common Correlated Effects

The model in equation (3) does not account for unobserved common factors between units. To do so, cross sectional averages are added in the fashion of Pesaran (2006):

(4) y(i,t) = b0(i) + x(i,t)*b2(i) + d(i)*z(i,t) + e(i,t).

Equation (4) is the default equation of xtdcce2. Including the dependent and independent variables in crosssectional() and setting cr_lags(0) leads to the same result. crosssectional() defines the variables to be included in z(i,t). Important to notice is, that b1(i) is set to zero.

See example

4.3 Dynamic Common Correlated Effects

If a lag of the dependent variable is added, endogeneity occurs and adding solely contemporaneous cross sectional averages is not sufficient any longer to achieve consistency.

Chudik and Pesaran (2015) show that consistency is gained if pT lags of the cross sectional averages are added:

(5) y(i,t) = b0(i) + b1(i)*y(i,t-1) + x(i,t)*b2(i) + sum [d(i)*z(i,s)] + e(i,t).

where s = t,…,t-pT. Equation (5) is estimated if the option cr_lags() contains a positive number.

See example

4.4 Pooled Estimations

Equations (3) - (5) can be constrained that the parameters are the same across units. Hence the equations become:

(3-p) y(i,t) = b0 + b1*y(i,t-1) + x(i,t)*b2 + e(i,t),
(4-p) y(i,t) = b0 + x(i,t)*b2 + d(i)*z(i,t) + e(i,t),
(5-p) y(i,t) = b0 + b1*y(i,t-1) + x(i,t)*b2 + sum [d(i)*z(i,s)] + e(i,t).

Variables with pooled (homogenous) coefficients are specified using the pooled(varlist) option. The constant is pooled by using the option pooledconstant. In case of a pooled estimation, the standard errors are obtained from a mean group regression. This regression is performed in the background. See Pesaran (2006).

See example

4.5 Instrumental Variables

xtdcce2 supports estimations of instrumental variables by using the ivreg2 package. Endogenous variables (to be instrumented) are defined in varlist_2 and their instruments are defined in varlist__iv.

See example

4.6 Error Correction Models (ECM/PMG)

As an intermediate between the mean group and a pooled estimation, Shin et. al (1999) differentiate between homogenous long run and heterogeneous short run effects. Therefore the model includes mean group as well as pooled coefficients. Equation (1) (without the lag of the explanatory variable x and for a better readability without the cross sectional averages) is transformed into an ARDL model:

(6)y(i,t) = phi(i)*(y(i,t-1) - w0(i) - x(i,t)*w2(i)) + g1(i)*[y(i,t)-y(i,t-1)] + [x(i,t) - x(i,t-1)] * g2(i) + e(i,t),

where phi(i) is the cointegration vector, w(i) captures the long run effects and g1(i) and g2(i) the short run effects. Shin et. al estimate the long run coefficients by ML and the short run coefficients by OLS.

xtdcce2 estimates a slightly different version by OLS:

(7)	y(i,t) = o0(i) + phi(i)*y(i,t-1) + x(i,t)*o2(i) + g1(i)*[y(i,t)-y(i,t-1)] + [x(i,t) - x(i,t-1)] * g2(i) + e(i,t),

where w2(i) = - o2(i) / phi(i) and w0(i) = - o0(i)/phi(i).

Equation (7) is estimated by including the levels of y and x as long run variables using the lr(varlist) and pooled(varlist) options and adding the first differences as independent variables.

xtdcce2 estimates equation (7) but automatically calculates estimates for w(i) = (w0(i),…,wk(i)). The advantage estimating equation (7) by OLS is that it is possible to use IV regressions and add cross sectional averages to account for dependencies between units. The variance/covariance matrix is calculated using the delta method, for a further discussion, see Ditzen (2018).

See Example

4.7 Cross-Section Augmented Distributed Lag (CS-DL)

Chudik et. al (2016) show that the long run effect of variable x on variable y in equation (1) can be directly estimated.

Therefore they fit the following model, based on equation (1):

(8)  y(i,t) = w0(i) + x(i,t) * w2(i)  + delta(i) * (x(i,t) - x(i,t-1)) + sum [d(i)*z(i,s)] + e(i,t)

where w2(i) is the long effect and sum [d(i) z(i,s)] the cross-sectional averages with an appropriate number of lags. To account for the lags of the dependent variable, the corresponding number of first differences are added.

If the model is an ARDL(1,1), then only the first difference of the explanatory variable is added. In the case of an ARDL(1,2) model, the first and the second difference are added. The advantage of the CS-DL approach is, that no short run coefficients need to be estimated.

A general ARDL(py,px) model is estimated by:

(8) y(i,t) = w0(i) + x(i,t) * w2(i)  + sum(l=1,px) delta(i,l) * (x(i,t-l) - x(i,t-l-1))  + sum [d(i)*z(i,s)] + e(i,t)

The mean group coefficients are calculated as the unweighted averages of all cross-sectional specific coefficient estimates. The variance/covariance matrix is estimated as in the case of a Mean Group Estimation.

See Example

4.8 Cross-Section Augmented ARDL (CS-ARDL)

As an alternative approach the long run coefficients can be estimated by first estimating the short run coefficients and then the long run coefficients.

For a general ARDL(py,px) model including cross-sectional averages such as:

(9) y(i,t) = b0(i) + sum(l=1,py) b1(i,l) y(i,t-l) + sum(l=0,px) b2(i,l) x(i,t-l) +  sum [d(i)*z(i,s)] + e(i,t),

the long run coefficients for the independent variables are calculated as:

(10) w2(i) = sum(l=0,px) b2(i,l) / ( 1 - sum(l=1,py) b1(i,l))

and for the dependent variable as:

(11) w1(i) = 1 - sum(l=1,py) b1(i,l).

This is the CS-ARDL estimator in Chudik et. al (2016).

The variables belonging to w(1,i) need to be enclosed in parenthesis, or tsvarlist need to be used. For example coding lr(y x L.x) is equivalent to lr(y (x lx)), where lx is a variable containing the first lag of x (lx = L.x).

The disadvantage of this approach is, that py and px need to be known. The variance/covariance matrix is calculated using the delta method, see Ditzen (2018b).

See Example

4.9 Coefficient of Determination (R2)

xtdcce2 calculates up to three different coefficients of determination (R2). It calculates the standard un-adjusted R2 and the adjusted R2 as common in the literature. If all coefficients are either pooled or heterogeneous, xtdcce2 calculates an adjusted R2 following Holly et. al (2010); Eq. 3.14 and 3.15. The R2 and adjusted R2 are calculated even if the pooled or mean group adjusted R2 is calculated. However the pooled or mean group adjusted R2 is displayed instead of the adjusted R2 if calculated.

In the case of a pure homogenous model, the adjusted R2 is calculated as:

R2(CCEP) = 1 - s(p)^2 / s^2

where s(p)^2 is the error variance estimator from the pooled regressions and s^2 the overall error variance estimator. They are defined as

s(p)^2 = sum(i=1,N) e(i)'e(i) / [N ( T - k - 2) - k],
s^2 = 1/(N (T -1)) sum(i=1,N) sum(t=1,T) (y(i,t) - ybar(i) )^2.

k is the number of regressors, e(i) is a vector of residuals and ybar(i) is the cross sectional specific mean of the dependent variable.

For mean group regressions the adjusted R2 is the mean of the cross-sectional individual R2 weighted by the overall error variance:

R2(CCEMG) = 1 - s(mg)^2 / s^2
s(mg)^2 = 1/N sum(i=1,N) e(i)'e(i) / [T - 2k - 2].

4.10 Collinearity Issues

(Multi-)Collinearity in a regression models means that two or more explanatory variables are linearly dependent. The individual effect of a collinear explanatory variable on the dependent variable cannot be differentiated from the effect of another collinear explanatory variable. This implies it is impossible to estimate the individual coefficient of the collinear explanatory variables. If the explanatory variables are stacked into matrix X, one or more variables (columns) in x are collinear, then X’X is rank deficient. Therefore it cannot be inverted and the OLS estimate of beta = inverse(X’X)X’Y does not exist.

In a model in which cross-sectional dependence in which dependence is approximated by cross-sectional averages, collinearity can easily occur. The empirical model (2) can exhibit collinearity in four ways:

  1. In the cross-sectional averages (z(i,s)) stacked in Z are collinear.
  2. The cross-sectional averages and the explanatory variables are collinear.
  3. In the global set of model of explanatory variables (the constant, y(i,t-1), x(i,t), x(i,t-1) stacked in X) are collinear for all i.
  4. In a cross-sectional unit specific model of explanatory variables (the constant, y(i,t-1), x(i,t), x(i,t-1) stacked in X(i)) are collinear for some i.

xtdcce2 checks all types of collinearity and according to the prevalent type decides how to continue and invert (X’X). It uses as a default cholinv. If a matrix is rank deficient it uses invsym, where variables (columns) are removed from the right. If X = (X1 X2 X3 X4) and X1 and X4 are collinear, then X4 will be removed. This is done by invsym, specifying the order in which columns are dropped. Older versions of xtdcce2 used qrinv for rank deficient matrices. However results can be unstable and no order of which columns to be dropped can be specified. The use of qrinv for rank deficient matrices can be enforced with the option useqr.

xtdcce2 takes the following steps if:

  1. Z’Z is not of full rank Before partialling out xtdcce2 checks of Z’Z is of full rank. In case Z’Z is rank deficient, then xtdcce2 will return a warning. Cross-section unit specific estimates are not consistent, however the mean group estimates are. See Chudik, Pesaran (2015, Journal of Econometrics), Assumption 6 and page 398.

  2. The cross-sectional averages and the explanatory variables are collinear. In this case regressors from the right are dropped, this means the cross-sectional averages are dropped. This case corresponds to the first because the cross-sectional averages are regressors for the partialling out.

  3. X’X is collinear for all i. xtdcce2 uses _rmcoll to remove any variables which are collinear on the global level. A message with the list of omitted variables will be posted. A local of omitted variables is posted in e(omitted_var) and the number in e(K_omitted).

  4. X(i)’X(i) is collinear for some i. xtdcce2 automatically drops variables (columns) from the right for those cross-sectional units with collinear variables (columns). An error message appears. More details can be obtained using the option showomitted by showing a matrix with a detailed break down on a cross-section - variable level. The matrix is stored in e(omitted_var_i) as well.

Results obtained with xtdcce2 can differ from those obtained with reg or xtmg. The reasons are that xtdcce2, partialles out the cross-sectional averages and enforces the use of doubles, both is not done in xtmg. In addition it use as a default a different alogorithm to invert matrices.

5. Saved Values

xtdcce2 stores the following in e():

Scalars Description
e(N) number of observations
e(N_g) number of groups (cross sectional units)
e(T) number of time periods
e(K_mg) number of regressors (excluding variables partialled out)
e(N_partial) number of partialled out variables
e(N_omitted) number of omitted variables
e(N_pooled) number of pooled (homogenous) coefficients
e(mss) model sum of squares
e(rss) residual sum of squares
e(F) F statistic
e(rmse) root mean squared error
e(df_m) model degrees of freedom
e(df_r) residual degree of freedom
e(r2) R-squared
e(r2_a) R-squared adjusted
e(cd) CD test statistic
e(cdp) p-value of CD test statistic
e(Tmin) minimum time (only unbalanced panels)
e(Tbar) average time (only unbalanced panels)
e(Tmax) maximum time (only unbalanced panels)
e(cr_lags) number of lags of cross sectional averages
Macros Description
e(tvar) name of time variable
e(idvar) name of unit variable
e(depvar) name of dependent variable
e(indepvar) name of independent variables
e(omitted) omitted variables
e(lr) variables in long run cointegration vector
e(pooled) pooled (homogenous) coefficients
e(cmd) command line
e(cmdline) command line including options
e(insts) instruments (exogenous) variables (only IV)
e(istd) instrumented (endogenous) variables (only IV)
e(version) xtdcce2 version, if stata xtdcce2, version used.
Matrices Description
e(b) coefficient vector
e(V) variance-covariance matrix
e(bi) coefficient vector of individual and pooled coefficients
e(Vi) variance-covariance matrix of individual and pooled coefficients
e(alpha) estimates of the exponent of cross-sectional dependence
e(alpha) estimates of the standard error exponent of cross-sectional dependence

Estimated long run coefficients of the ARDL model are marked with the prefix lr.

Functions Description
e(sample) marks estimation sample

6. Postestimation Commands

predict and estat can be used after xtdcce2.

6.1 Predict

The syntax for predict is:

predict [type] _newvar_ _ifin_ [ options ]
Options Description
xb calculate linear prediction on partialled out variables
xb2 calculate linear prediction on non partialled out variables
stdp calculate standard error of the prediction
residuals calculate residuals (e(i,t))
cfresiduals calculate residuals including the common factors (u(i,t))
coefficients a variable with the estimated cross section specific values for all coefficients is created. The name of the new variable is newvar_varname.
se as coefficient, but with standard error instead.
partial create new variables with the partialled out values.
replace replace the variable if existing.

Option xb2 is equivalent to calculate the coefficients and then multiply the explanatory variables with it, while xb first partialles out the cross sectional averages and then multiplies the coefficients.

The following Table summarizes the differences with the command line xtdcce2 y x , nocross:

xb xb2
1. predict coeff, coeff 1. predict coeff, coeff
2. predict partial, partial 2. gen xb2 = coeff_x * x
3. gen xb = coeff_x * partial_x  

xtdcce2 is able to calculate both residuals from equation (1). predict _newvar_ , eesiduals calculates e(i,t). That is, the residuals of the regression with the cross sectional averages partialled out.

predict _newvar , cfresiduals calculates u(i,t) = g(i)f(g) + e(i,t). That is, the error including the cross-sectional averages. Internally, the fitted values are calculated and then subtracted from the dependent variable. Therefore it is important to note, that if a constant is used, the constant needs to be reported using the xtdcce2 option reportconstant. Otherwise the u(i,t) includes the constant as well (u(i,t) = b0(i) + g(i)f(g) + e(i,t)).

6.2 estat

estat can be used to create a box, bar or range plot. The syntax is:

estat _graphtype_ [_varlist_] {ifin} [,combine(_string_) individual(_string_)} nomg cleargraph]
graphtype Description
box box plot
bar bar plot
rcap range plot
Options Description
individual(string) passes options for individual graphs (only bar and rcap)
combine(_string) passes options for combined graphs
nomg mean group point estimate and confidence interval are not included in bar and range plot graphs
cleargraph clears the option of the graph command and is best used in combination with the combine() and individual() options

The name of the combined graph is saved in r(graph_name).

7 Examples

An example dataset of the Penn World Tables 8 is available for download here. The dataset contains yearly observations from 1960 until 2007 and is already tsset.

To estimate a growth equation the following variables are used: log_rgdpo (real GDP), log_hc (human capital), log_ck (physical capital) and log_ngd (population growth + break even investments of 5%).

7.1 Mean Group Estimation

To estimate equation (3), the option nocrosssectional is used. In order to obtain estimates for the constant, the option reportconstant is enabled.

xtdcce2 d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd , nocross reportc

Omitting reportconstant leads to the same result, however the constant is partialled out:

xtdcce2 d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd , nocross

7.2 Common Correlated Effects

Common Correlated effects (static) models can be estimated in several ways. The first possibility is without any cross sectional averages related options:

xtdcce2 d.log_rgdpo log_hc log_ck log_ngd , cr(_all) reportc

Note, that as this is a static model, the lagged dependent variable does not occur and only contemporaneous cross sectional averages are used. Defining all independent and dependent variables in crosssectional(varlist) leads to the same result:

xtdcce2 d.log_rgdpo log_hc log_ck log_ngd , reportc cr(log_rgdpo log_hc log_ck log_ngd)

The default for the number of cross sectional lags is zero, implying only contemporaneous cross sectional averages are used. Finally the number of lags can be specified as well using the cr_lags option.

xtdcce2 d.log_rgdpo log_hc log_ck log_ngd , reportc cr(log_rgdpo log_hc log_ck log_ngd) cr_lags(0)

All three command lines are equivalent and lead to the same estimation results.

7.3 Dynamic Common Correlated Effects

The lagged dependent variable is added to the model again. To estimate the mean group coefficients consistently, the number of lags is set to 3:

xtdcce2 d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd , reportc cr(log_rgdpo  log_hc log_ck log_ngd) cr_lags(3)

Using predict

predict, _[options]_ can be used to predict the linear prediction, the residuals, coefficients and the partialled out variables. To predict the residuals, options residuals is used:

predict residuals, residuals

The residuals do not contain the partialled out factors, that is they are e(i,t) in equation (1) and (2). To estimate u(i,t), the error term containing the common factors, option cfresiduals is used:

predict uit, cfresiduals

In a similar fashion, the linear prediction (option xb, the default) and the standard error of the prediction can be obtained. The unit specific estimates for each variable and the standard error can be obtained using options coefficients and se.

For example, obtain the coefficients for log_hc from the regression above and calculate the mean, which should be the same as the mean group estimate:

predict coeff, coefficients

sum coeff_log_hc.

The partialled out variables can be obtained using

predict partial, partial

Then a regression on the variables would lead to the same results as above.{break} If the option replace is used, then the newvar is replaced if it exists.

7.4 Pooled Estimations

All coefficients can be pooled by including them in pooled(varlist). The constant is pooled by using the pooledconstant option:

xtdcce2 d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd , reportc cr(log_rgdpo  log_hc log_ck log_ngd) pooled(L.log_rgdpo  log_hc log_ck log_ngd) cr_lags(3) pooledconstant

7.5 Instrumental Variables

Endogenous variables can be instrumented by using options endogenous_vars(varlist) and exogenous_vars(varlist). Internally ivreg2 estimates the individual coefficients. Using the lagged level of physical capital as an instrument for the contemporaneous level, leads to:

xtdcce2 d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd  (log_ck = L.log_ck), reportc cr(log_rgdpo  log_hc log_ck log_ngd) cr_lags(3) ivreg2options(nocollin noid)

Further ivreg2 options can be passed through using ivreg2options. Stored values in e() from ivreg2options can be posted using the option fulliv.

7.6 Error Correction Models (ECM/PMG)

Variables of the long run cointegration vector are defined in lr(varlist), where the first variable is the error correction speed of adjustment term. To ensure homogeneity of the long run effects, the corresponding variables have to be included in the pooled(varlist) option. Following the example from Blackburne and Frank (2007) with the jasa2 dataset (the dataset is available at here from Pesaran’s webpage:

xtdcce2 d.c d.pi d.y if year >= 1962 , lr(L.c pi y) p(L.c pi y) nocross

xtdcce2 internally estimates equation (7) and then recalculates the long run coefficients, such that estimation results for equation (8) are obtained. Equation (7) can be estimated adding nodivide to lr_options().

A second option is xtpmgnames in order to match the naming convention from xtpmg.

xtdcce2 d.c d.pi d.y if year >= 1962 , lr(L.c pi y) p(L.c pi y) nocross lr_options(nodivide)
xtdcce2 d.c d.pi d.y if year >= 1962 , lr(L.c pi y) p(L.c pi y) nocross lr_options(xtpmgnames)

7.7 Cross-Section Augmented Distributed Lag (CS-DL)

Chudik et. al (2013) estimate the long run effects of public debt on output growth (the data is available here on Kamiar Mohaddes’ personal webpage.

In the dataset, the dependent variable is d.y and the independent variables are the inflation rate (dp) and debt to GDP ratio (d.gd).

For an ARDL(1,1,1) only the first difference of dp and d.gd are added as further covariates. Only a contemporaneous lag of the cross-sectional averages (i.e. cr_lags(0) of the dependent variable and 3 lags of the independent variables are added. The lag structure is implemented by defining a numlist rather than a number in cr_lags(). For the example here cr_lags(0 3 3) is used, where the first number refers to the first variable defined in cr(), the second to the second etc.

To replicate the results in Table 18, the following command line is used:

xtdcce2 d.y dp d.gd d.(dp d.gd), cr(d.y dp d.gd) cr_lags(0 3 3) fullsample

For an ARDL(1,3,3) model the first and second lag are of the first differences are added by putting L(0/2) in front of the d.(dp d.gd):

xtdcce2 d.y dp d.gd L(0/2).d.(dp d.gd), cr(d.y dp d.gd) cr_lags(0 3 3) fullsample

Note, the fullsample option is used to reproduce the results in Chudik et. al (2013).

7.8 Cross-Section Augmented ARDL (CS-ARDL)

Chudik et. al (2013) estimate besides the CS-DL model a CS-ARDL model. To estimate this model all variables are treated as long run coefficients and thus added to varlist in lr(varlist). xtdcce2 first estimates the short run coefficients and the calculates then long run coefficients, following Equation 10. The option lr_options(ardl) is used to invoke the estimation of the long run coefficients. Variables with the same base (i.e. forming the same long run coefficient) need to be either enclosed in parenthesis or tsvarlist operators need to be used. In Table 17 an ARDL(1,1,1) model is estimated with three lags of the cross-sectional averages:

xtdcce2 d.y , lr(L.d.y dp L.dp d.gd L.d.gd) lr_options(ardl) cr(d.y dp d.gd) cr_lags(3) fullsample

xtdcce2 calculates the long run effects identifying the variables by their base. For example it recognizes that dp and L.dp relate to the same variable. If the lag of dp is called ldp, then the variables need to be enclosed in parenthesis.

Estimating the same model but as an ARDL(3,3,3) and with enclosed parenthesis reads:

xtdcce2 d.y , lr((L(1/3).d.y) (L(0/3).dp) (L(0/3).d.gd) ) lr_options(ardl) cr(d.y dp d.gd) cr_lags(3) fullsample

which is equivalent to coding without parenthesis:

xtdcce2 d.y , lr(L(1/3).d.y L(0/3).dp L(0/3).d.gd) lr_options(ardl) cr(d.y dp d.gd) cr_lags(3) fullsample

8. References

Baum, C. F., M. E. Schaffer, and S. Stillman 2007. Enhanced routines for instrumental variables/generalized method of moments estimation and testing. Stata Journal 7(4): 465-506

Chudik, A., K. Mohaddes, M. H. Pesaran, and M. Raissi. 2013. Debt, Inflation and Growth: Robust Estimation of Long-Run Effects in Dynamic Panel Data Model. Chudik, A., and M. H. Pesaran. 2015.

Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors. Journal of Econometrics 188(2): 393-420.

Chudik, A., K. Mohaddes, M. H. Pesaran, and M. Raissi. 2016. Long-Run Effects in Large Heterogeneous Panel Data Models with Cross-Sectionally Correlated Errors Essays in Honor of Aman Ullah. 85-135.

Ditzen, J. 2018. Estimating Dynamic Common Correlated Effcts in Stata. The Stata Journal, 18:3, 585 - 617.

Ditzen, J. 2018b. Estimating long run effects in models with cross-sectional dependence using xtdcce2.

Blackburne, E. F., and M. W. Frank. 2007. Estimation of nonstationary heterogeneous panels. Stata Journal 7(2): 197-208.

Eberhardt, M. 2012. Estimating panel time series models with heterogeneous slopes. Stata Journal 12(1): 61-71.

Feenstra, R. C., R. Inklaar, and M. Timmer. 2015. The Next Generation of the Penn World Table. American Economic Review. www.ggdc.net/pwt

Jann, B. 2005. moremata: Stata module (Mata) to provide various functions. Available from http://ideas.repec.org/c/boc/bocode/s455001.html.

Pesaran, M. 2006. Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74(4): 967-1012.

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. About

Author

Jan Ditzen (Heriot-Watt University)

Email: j.ditzen@hw.ac.uk

Web: www.jan.ditzen.net

Acknowledgments

I am grateful to Achim Ahrens, Arnab Bhattacharjee, David M. Drukker, Markus Eberhardt, Tullio Gregori, Erich Gundlach and Mark Schaffer, to the participants of the 2016 and 2018 Stata Users Group meeting in London and Zuerich, and two anonymous referees of The Stata Journal for many valuable comments and suggestions.

The routine to check for positive definite or singular matrices was provided by Mark Schaffer, Heriot-Watt University, Edinburgh, UK.

xtdcce2 was formally called xtdcce.

Citation

Please cite as follows:

Ditzen, J. 2018. xtdcce2: Estimating dynamic common correlated effects in Stata. The Stata Journal, 18:3, 585 - 617.

10. Installation

The latest versions can be obtained via

net install xtdcce2 , from("https://janditzen.github.io/xtdcce2/")

or including beta versions

net from https://janditzen.github.io/xtdcce2/

and a full history of xtdcce2, pre version 1.34 from

net from http://www.ditzen.net/Stata/xtdcce2_beta

xtdcce2 is available on SSC as well:

ssc install xtdcce2

11. Change log

This version: 2.0 (was 1.35) - 13. July 2019

Version 1.33 to Version 1.34

Version 1.32 to Version 1.33

Version 1.31 to Version 1.32

Version 1.2 to Version 1.31