A package for model specification and estimation for linear panel time series models with cross-sectional dependence
The xtdcce2 package contains
xtdcce2- Estimating heterogeneous coefficient models using common correlated effects in a dynamic panel with a large number of observations over groups and time periods.xtcd2- Testing for weak cross-sectional dependence.xtcse2- Estimation of the exponent of cross-sectional dependence.
An introduction into the topic can be found in my slides of the 2021 Stata Economics Virtual Symposium here.
Table of Contents
- Syntax
- Description
- Options
- Econometric and Empirical Model
- Mean Group
- Common Correlated Effects
- Dynamic Common Correlated Effects
- Pooled Estimations
- Instrumental Variables
- Error Correction Models (ECM/PMG)
- Cross-Section Augmented Distributed Lag (CS-DL)
- Cross-Section Augmented ARDL(CS-ARDL)
- Regularized CCE (rCCE)
- Coefficient of Determination (R2)
- Rank Classifier
- Information Criteria
- Collinearity Issues
- Saved Values
- Postestimation Commands
- Examples
- Testing for Cross-sectional Dependence
- Estimating the exponent of Cross-sectional Dependence
- References
- About
- How to install
- Changelog
xtdcce2 _depvar_ [_indepvars_] [_varlist2_ = _varlist_iv_] [ifin] ,
crosssectional(_varlist_[,cr_lags(_numlist_) rcce[(criterion(er/gr) scale npc(integer))] rcclassifier[(er gr replications(integer) standardize(integer) randomshrinkage noshrinkage)]])
[clustercrosssectional(_varlist_, clustercr(_varlist_) [cr_lags(_numlist_)])
globalcrosssectional(_varlist_[,cr_lags(_numlist_)]) pooled(_varlist_) cr_lags(_numlist_)
NOCRosssectional ivreg2options(_string_) e_ivreg2_ ivslow noisily lr(_varlist_) lr_options(_string_)
pooledconstant reportconstant pooledvce(_string_) noconstant trend
pooledtrend jackknife recursive nocd exponent xtcse2options(_string_)
showindividual fullsample fast fast2 blockdiaguse nodimcheck useqr useinvsym noomitted mgmissing]
and for an optimized version for speed and large datasets:
xtdcce2fast _depvar_ [_indepvars_] [ifin] , crosssectional(_varlist_[,cr_lags(_numlist_) rcce[(criterion(er/gr) scale npc(integer))] rcclassifier[(er gr replications(integer) standardize(integer) randomshrinkage noshrinkage)]])
[clustercrosssectional(_varlist_, clustercr(_varlist_) [cr_lags(_numlist_)])
globalcrosssectional(_varlist_[,cr_lags(_numlist_)]) cr_lags(_string_)
NOCRosssectional lr(_varlist_) lr_options(_string_)
reportconstant noconstant cd fullsample notable cd postframe nopost ]
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.
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)
iv) The regularized CCE Estimator (rCCE, Juodis 2022), 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). The rank condition can be checked in static panels using the classifier from De Vos et al. (2024). Information criteria to select the optimal number of cross-section averages in static panels can be calculated using estat ic. It also supports instrumental variable estimations (see ivreg2).
xtdcce2fast is an optimized version for speed and large datasets. In comparison to xtdcce2 it does not perform any collinearity checks does not support pooled estimations and instrumental variable regressions. It also stores some estimation results in mata rather than e() to circumvent some restrictions on matrix dimensions in Stata.
| Option | Description |
|---|---|
| crosssectional(varlist, [cr_lags(numlist)]) | 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. If cr(..., cr_lags()) is used, then the global option cr_lags() (see below) is ignored. |
| rcce[(criterion(er/gr) scale npc(integer))] | implements the regularized CCE estimator from Juodis (2022). criterion() sets the er or gr criterion from Ahn and Horenstein (2023). scale scales cross-section averages, see Juodis (2022). npc(real) specifies number of eigenvectors without estimating it. Cannot be combined with criterion. |
| rcclassifier[er gr replications(integer) standardize(integer) randomshrinkage noshrinkage] | performs the rank condition classifier, see De Vos et al. (2024). RC = 1 implies the rank condition holds and CCE is consistent. |
| globalcrosssectional(varlistcr1 [,cr_lags(numlist)]) | define global cross-section averages. global cross-section averages are cross-section averages based on observeations which are excluded using if statements. If cr(..., cr_lags()) is used, then the global option cr_lags() (see below) is ignored. |
| clusterosssectional(varlistcr1 [,cr_lags(numlist)] clustercr(varlist)) | are clustered or local cross-section averages. That is, the cross-section averages are the same for each realisation of the variables defined in clustercr(). For example, we have data observations regions of multiple countries, defined by variable country Now we want to add cross-section averages for each country. We can define those by using the option clustercr(varlist , clustercr(country)). If cr(..., cr_lags()) is used, then the global option cr_lags() (see below) is ignored. |
| 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. |
| pooledvce(type) | specifies the variance estimator for pooled regression. The default is the non-parametric variance estimator from Pesaran (2006). type can be nw for Newey West heteroscedasticity autocorrelation robust standard errors (Pesaran 2006) or wpn for fixed T adjusted standard errors from Westerlund et. al (2019). ols for standard OLS standard errors. |
| 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. |
| mgmissing | if it is not possible to estimate individual coefficient for a cross-section because of missing data or perfect collinearity, individual coefficient is excluded for MG estimation. Coefficient will still be displayed as zero in e(bi). |
| 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. |
| fast2 | use xtdcce2fast instead of xtdcce2. |
| 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. |
| notable | do not display output (only xtdcce2fast). |
| cd | calculate CD test statistic, see xtcd2 (only xtdcce2fast). |
| postframe | save predicted values to frame. Speeds up predict (only xtdcce2fast). |
| nopost | do not save/post predicted values (only xtdcce2fast). |
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 inverse 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. |
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 the averages of the coefficients b0, 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))'
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.
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
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
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
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
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
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
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
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
The CCE approach can involve a large number of cross-section averages which is larger than the number of factors and can lead to a non-trivial bias for the pooled and mean group estimator, see Karabiyik et. al. (2017). Juodis (2022) propose a solution for linear static panels which uses singular value decomposition to remove redundant singular values in the cross-section averages. The so-called rCCE method involves the following steps:
1. Calculate cross-sectional averages.
2. Estimate number of common factors using the ER or GR criterion from Ahn and Horenstein (2013).
3. Replace the cross-sectional averages with eigenvectors from the cross-section averages.
The eigenvectors are the eigenvectors of the largest eigenvalues and the number is obtained in step 2.
The method requires bootstrapped standard errors, see bootstrapping.
See Example
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].
A key condition for a consistent estimation in the presence of common factors and strong cross-sectional dependence is the so called Rank Condition in Pesaran (2016) and Chudik and Pesaran (2015). The rank condition implies that the rank of the average factor loadings is is smaller than the number of common factors. Karabiyik et al. (2017) show that if rank condition fails, the CCE estimator is inconsistent. In an empirical setting this implies that 1) the unknown number of unobserved common factors has to be equal or larger than the rank of the unobserved average factor loadings; 2) cross-section averages with a zero loading can pose a problem if the number of cross-section averages is small.
DeVos et al. (2024) propose an classifier which indicates if the rank conditions holds:
RC = 1 − I (g <m)
where m is number of factors in the data and g is the rank of the matrix of cross-sectional averages of the data. m is estimated using the ER or GR criterion on the cross/product of the observed data. g is the rank of the average factor loadings and estimated from the cross-section averages.If RC = 1, then the rank condition holds.
The Rank Condition Classifier is calculated on request using the option cr(varlist,rccl).
Notes: The estimation of the rank of the factor loadings requires a bootstrap. Consistency depends on fixed T, however CCE requires large T. The solution is to bound the dimension of the loadings with shrinkage. Most importantly for the practitioner, the classifier is only valid for static panels!
See Example
The selection of the optimal set of cross-section averages is non-trivial and difficult to establish ex-anti. The inclusion of too many cross-section averages can cause inefficiency. To guide the selection of the optimal set of cross-section averages, Margaritella and Westerlund (2023) propose 4 information criteria:
IC1 = ln(S(FM)^2) + m (N+T)/(NT) ln(NT/(N+T))
IC2 = ln(S(FM)^2) + m (N+T)/(NT) ln(c(NT))
PC1 = S(FM)^2 + m S(FMb)^2 (N+T)/(NT) ln(NT/(N+T))
PC2 = S(FM)^2 + m S(FMb)^2 (N+T)/(NT) ln(c(NT))
where m is the number of cross-section averages of set M and Mb denotes the set of cross-section averages with the largest number of cross-section averages. S(FM) is the standard error of regression with the set of cross-section averages corresponding to set m, S(FMb) is the standard error of regression with the largest set of cross-section averages. c(NT) is an additional penalty term.
IC1 and IC2 are calculated based only on the current (m-set) of cross-section averages, while PC1 and PC2 are in relation to the largest possible set of cross-section averages (Mb-set). Example: 3 possible cross-section averages, y, x1 and x2. Then the Mb-set would be y, x1 and x2. The m-sets can be 1) y, 2) x1, 3) x3, 4) y+x1, 5) y+x2, 6) x1+x2. Hence a total of 7 different combinations of CSA and ICs can be calculated.
IC1 and IC2 are automatically calculated when running xtdcce2. PC1 and PC2 can be calculated using estat ic, see estat ic.
Notes: The IC and PC are only valid for static panel models. The IC and PC are intended to identify the optimal set of cross-section averages. DO NOT use the criteria to select the number of lags in a dynamic model.
See Example
(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:
- In the cross-sectional averages (z(i,s)) stacked in Z are collinear.
- The cross-sectional averages and the explanatory variables are collinear.
- 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.
- 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:
-
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.
-
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.
-
X'X is collinear for all i. xtdcce2 uses
_rmcollto 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). -
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.
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 |
| e(IC1) | Information Criteria 1 to select CSA from Margaritella and Westerlund (2023). |
| e(IC2) | Information Criteria 2 to select CSA from Margaritella and Westerlund (2023). |
| 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 |
predict and estat can be used after xtdcce2.
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 |
|---|
predict coeff, coeff| 1.predict coeff, coeffpredict partial, partial| 2.gen xb2 = coeff_x * xgen 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)).
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 |
| dropzero | does not display coefficients with zeros in bar or rcap graphs. |
The name of the combined graph is saved in r(graph_name).
xtdcce2 can bootstrap confidence intervals and standard errors. It supports two types of bootstraps: the wild bootstrap and the cross-section bootstrap. The syntax is:
estat bootstrap , [options]
| Options | Description |
|---|---|
| reps(integer) | Number of repetitions. Default 100. |
| seed(string) | Set seed, see seed. |
| wild | Use wild bootstrap rather than cross-section bootstrap. |
| cfresdiduals | Use residuals including common factors for wild bootstrap. |
| percentile | Bootstrap confidence intervals. |
| showindividual | show unit specific results. |
estat bootstrap implements two types of bootstraps, the wild bootstrap and the cross-section bootstrap. The cross-section bootstrap is the default.
The cross-section bootstrap draws with replacement from the cross-sectional dimension. That is it draws randomly cross-sectional units with their entire time series. It then estimates the model using xtdcce2. The cross-section bootstrap has been proposed in Westerlund et. al. (2019) or Goncalves and Perron (2014).
The wild bootstrap is a slower from of the wild bootstrap implemented in boottest (Roodman et. al. 2019). It reweighs the residuals with Rademacher weights from the initial regression, recalculates the dependent variable and then runs xtdcce2.
The default is to bootstrap standard errors and then use the bootstrapped standard errors to calculate the confidence intervals. Option percentile directly bootstraps confidence intervals.
estat ic calculates the information criteria (IC) to select the optimal number of cross-section averages from Margaritella and Westerlund (2023). A total of 4 IC are implemented. IC1 and IC2 are information criteria on the current set of cross-section averages as set by the option crosssectional() of the xtdcce2. PC1 and PC2 are the panel criteria and compare a set of cross-section averages to the largest possible set. The largest possible set is either defined by all variables in crosssectional() or by (model1) when using the model() option of estat ic.
See Examples for details on syntax.
estat ic , [options]
| Options | Description |
|---|---|
| sequential | Calculate IC for all combinations of cross-section averages defined in cr(). |
| model(models) | Compare different models, where models are defined as model((model1) (model2) ... (modelK)). |
| single | Calculate IC for only 1st model. |
| noprogress | Omit Progress bar. |
The crtieria are only valud for static panels and to select the optimal set of cross-section averages. Do not use the information criteria in dynamic panels or to select the optimal number of lags of the dependent, independent variab ags of cross-section averages.
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%).
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
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.
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)
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. If the option replace is used, then the newvar is replaced if it exists.
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
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.
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)
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).
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 then calculates the 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
The regularized CCE approach is only possible for static models. To estimate a static model of growth on human, physical captial and population growth, we can use:
xtdcce2 log_rgdpo log_hc log_ck log_ngd , cr(log_rgdpo log_hc log_ck log_ngd, rcce)
xtdcce2 selects the first and second eigenvector of the cross-section averages and adds it as a variable. The selection criterion is the ER criterion from Ahn and Horenstein (2013). To use the GR criterion instead, the option criterion(gr) is used:
xtdcce2 log_rgdpo log_hc log_ck log_ngd , cr(log_rgdpo log_hc log_ck log_ngd, rcce(criterion(gr)))
Three regularized cross-section averages are added. Instead of specifying the criteria to estimate the number of eigenvectors of the rcce approach, we can hard set it using the option npc():
xtdcce2 log_rgdpo log_hc log_ck log_ngd , cr(log_rgdpo log_hc log_ck log_ngd, rcce(npc(3)))
To bootstrap standard errors with a fixed seed:
estat bootstrap, seed(123)
To run a wild bootstrap and bootstrap confidence intervals, the options wild and percentile are added:
estat bootstrap, seed(123) wild percentile
The rank condition is key for a consistent estimation using the CCE estimator. Rank condition implies that - loosely speaking - the estimated rank of the matrix of cross-sectional averages of the data has to be larger or equal to the rank of the factors. To calculate the classifier from DeVos et al. (2024), option rcclassifier is added to the static model from the first Example:
xtdcce2 d.log_rgdpo log_hc log_ck log_ngd , cr(_all, rccl) reportc
The estimated rank of the matrix of cross-sectional averages of the data is 3 and the rank of the factors is 1, thus the rank condition holds. Using the ER criterion to estimate the number of common factors in the cross-section averages and use fold-over matrix based on random normal values to shrink dimension:
xtdcce2 d.log_rgdpo log_hc log_ck log_ngd , cr(_all, rccl(er random))
The estimated rank of the averages factor loadings reduces to 2, but is still arger than the number of factors. Rank condition holds.
The Information Criteria from Margaritella and Westerlund (2023) can be used to identify the optimal set of cross-section averages in static panels. We return to the model from the first Example:
xtdcce2 d.log_rgdpo log_hc log_ck log_ngd , cr(_all) reportc
Obtain IC1 and IC2 for the current set of cross-section averages defined in cr(_all):
estat ic
Next, calculate IC for all possible combination of cross-section and indicate the lowest ones:
estat ic, seqential
IC for three different sets of cross-section averages indicated by model((model1) (model2) ... (modelK)), where model1 is the reference model with the largest set of cross-section averages:
estat ic, model( (d.log_rgdpo log_hc log_ck log_ngd) (log_hc log_ck log_ngd) (log_hc log_ck ) )
Get IC for a sinlge model of cross-section averages using option single. Compare to output when option single not used, then all combinations are tried.
estat ic, model(log_hc log_ck ) single
estat ic, model(log_hc log_ck log_ngd)
The xtdcce2 package includes xtcd2 which tests for weak cross-sectional dependence. The syntax is:
xtcd2 [varlist] [if] [,pesaran cdw pea cdstar rho
pca(integer) reps(integer) seed(integer)
kdensity name(string) heatplot[(absolute options_heatplot)]
contour[(absolute options_contour) noadjust] ]
varlist is the name of residuals or variables to be tested for weak cross sectional dependence. varlist may contain time-series operators, see tsvarlist. varlist is optional if the command is performed after an estimation (postestimation).
xtcd2 tests residuals or a variables for weak cross sectional dependence in a panel data model. It implements the tests by Pesaran (2015, 2021), the weighted CD test (CDw) by Juodis & Reese (2021) including the power enhancement (Fang et. al., 2015). It also implements the CD* from Pesaran & Xie (2021). As a default all four test statistics are calculated and presented next to each other. p-values are displayed in parenthesis.
Cross sectional dependence in the error term occurs if dependence between cross sectional units in a regression is not accounted for. The dependence between units violates the basic OLS assumption of an independent and identically distributed error term. In the worst case cross sectional dependence in the error term can lead to omitted variable bias or endogeneity and therefore to inconsistent estimates. Cross sectional dependence can be measured as the correlation between units. For example the correlation of the errors of unit i and unit j can be calculated. Obviously, if the correlation is large, cross sectional dependence is present.
Pesaran (2015) develops a test for weak cross sectional dependence based on this principle. Weak cross sectional dependence means that the correlation between units at each point in time converges to zero as the number of cross section goes to infinity. Under strong dependence the correlation converges to a constant. The null hypothesis of the test is, that the error term (or variable) is weakly cross sectional dependent. This means that correlation between an observation of unit i in time t and unit j in time t is zero. The hypothesis is:
H0: errors are weakly cross sectional dependent.
Pesaran (2015) derives a test statistic, which sums the correlation coefficients of the different units. The test statistic for a balanced panel is:
CD = [2*T / (N*(N-1))]^(1/2) * sum(i=1,N-1) sum(j=i+1,N) rho(ij),
and for an unbalanced panel (see Chudik, Pesaran, 2015):
CD = [2 / (N*(N-1))]^(1/2) * sum(i=1,N-1) sum(j=i+1,N) [T(ij)^(1/2) * rho(ij)],
where rho(ij) is the correlation coefficient of unit i and j and T(ij) the number of common observations between i and j. Under the null hypothesis the statistic is asymptotically
CD ~ N(0,1)
distributed.
xtcd2 further implements three alternatives to test for weak cross-sectional dependence. It includes the weighted CD (CDw) test proposed by Juodis and Reese (2021). Juodis and Reese (2021) show that the CD test diverges if the time dimension grows and the test is applied to residuals after a CCE or FE regression. The CDw test weights each observation by cross-section specific Rademacher weights. The pair wise correlations are calculated as:
rho(ij) = sum(t=1,T) w(i)eps(i,t)eps(j,t)w(j)
where w(i) and w(j) are the Rademacher weights which take on the values 1 or -1 with equal probability. To reduce the dependence on the random Rademacher weights, the draw can be repeated using the reps() option.
A second alternative proposed by Juodis and Reese (2021) is the Power Enhancement Approach (PEA) by Fan et. al. (2015). The power of the CD test is improved by calculating the CD test as:
CD = [2*T / (N*(N-1))]^(1/2) * sum(i=1,N-1) sum(j=i+1,N) rho(ij) + sum(i=2,N)sum(j=1,N-1}|rho(ij)|*(|rho(ij)>2 log(N)^(1/2)T^(-1)
Fan et. al. (2015) show that the PEA works if the number of cross-sectional units (N) is very large. Therefore it is advisable to use the PEA method only for such datasets.
As forth test xtcd2 implements the bias corrected CD* test from Pesaran & Xie (2021). The bias corrected test statistic is based on the following:
CD* = (CD + (T/2*Theta)^(1/2))/(1-Theta)
where Theta is the bias correction and a function of the estimated factor loadings. The factor loadings are estimated using the first p principal components as factors. Option pca() specifies the number of principal components. Default is 4. In case of unbalanced panels an Expected Maximisation algorithm taken from xtnumfac is used.
xtcd2 calculates the CD test statistic for given variables, or if run after an estimation command which supports predict and e(sample). In the latter case xtcd2 calculates the error term using predict, residuals and then applies the CD test from above. xtcd2 supports balanced as well as unbalanced panels. Furthermore by specifying the kdensity option, a kernel density plot with the distribution of the cross correlations is drawn.
If xtcd2 is used after xtreg, then the residuals are calculated using predict, e rather than predict, res. That is the residuals including the fixed- or random-error component, see xtreg postestimation. In all other cases predict, residuals is used to calculate the residuals. xtcd2 can draw heatplots and contour plots of the cross-correlations. To draw heatplots Ben Jann's heatplot is required. Contour plots are drawn using Stata's twoway contour.
| Options | Description |
|---|---|
| pesaran | calculates the original CD test by Pesaran (2015), see Description of Pesaran (2015). |
| cdw | calculates the weighted CD test following Juodis and Reese (2022), see Description of Juodis and Reese (2022). Results vary if seed not specified. |
| pea | uses the Power Enhancement Approach (PEA) by Fan et. al. (2015), see Description of Fan et. al. (2015). This method is designed for large panel panel datasets. |
| cdstar | calculates the bias corrected CD test following Pesaran & Xie (2021), see Description of Pesaran & Xie (2021). |
| reps(integer) | number of repetitions for the weighted CD test. Implies option cdw. Default is 30. |
| pca(integer) | number of Principal Components when using the bias corrected CD test. Requires option cdstar. Default is 4. |
| rho | saves the matrix with the cross correlations in r(rho). |
| kdensity | plots a kernel density plot of the cross correlations, see twoway kdensity. 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. |
| name(string) | saves the kdensity. |
| heatplot[(absolute options_heatplot)] | draws a heatplot of the cross-correlations. options_heatplot are options to be passed to heatplot. absolute uses the absolute values of the cross-correlations. |
| contour[(absolute options_contour)] | draws a contour plot of the cross-correlations. options_contour are options to be passed to twoway contour. absolute uses the absolute values of the cross-correlations. |
| noadjust | do not remove cross-section specific means. This was the default in versions prior 2.3. |
| seed(integer) | sets the seed for the weighted CD test. |
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. Estimating a simple panel version of the Solow model and run the CD test afterwards:
reg d.log_rgdpo log_hc log_ck log_ngd
xtcd2
Predicting the error terms after reg, leads to the same result:
reg d.log_rgdpo log_hc log_ck log_ngd
predict res, residuals
xtcd2 res
The test statistic is 36.34 and the p-value is 0, therefore rejecting the null hypothesis of weak cross sectional dependence.
To draw a density plot with the cross correlations the kdensity option is used:
xtcd2 res, kdensity
The CD test statistic is known to diverge if many periodic specific parameters are used (Juodis, Reese, 2021). Unit specific rademacher weights can be applied to prevent this behaviour by using the option cdw:
xtcd2 res, cdw
To reduce the dependence of the weighted CD test statistic, the test can be repeatedly performed with different weights using the reps() option:
xtcd2 res, cdw reps(20)
To improve the power of the weighted CD test, the Power Enhancement Approach can be applied by using the pea option:
xtcd2 res, pea
Testing the variable log_rgdpo for cross sectional dependence reads:
xtcd2 log_rgdpo, noestimation
The xtdcce2 package includes xtcse2 which estimates the exponent of cross-sectional dependence. The syntax is:
xtcse2 [varlist] [if] [, pca(integer) standardize nocenter nocd
RESsidual Reps(integer) size(real) tuning(real) lags(integer) ]
Data has to be xtset before using xtcse2; see tsset. varlist may contain time-series operators, see tsvarlist. If varlist if left empty, xtcse2 predicts residuals from the last estimation command, see predict. xtcse2 uses an expectation–maximization (EM) algorithm to impute missing values if the panel is unbalanced.
xtcse2 estimates the exponent of cross-sectional dependence in a panel with a large number of observations over time (T) and cross-sectional units (N). The estimation method follows Bailey, Kapetanios, Pesaran (2016,2019) (henceforth BKP). A variable or a residual is cross-sectional dependent if it inhibits an across cross-sectional units common factor.
xtcse2 estimates the strength of the factor, for a residual or one or more variables. It outputs a standard error and confidence interval in the usual estimation output fashion, however it does not show a t or z statistic and p-value. Generally speaking strong cross-sectional dependence occurs if alpha is above 0.5. Testing this is done by a separate test of weak cross-sectional dependence. Therefore a confidence interval is more informative when estimating alpha.
xtcse2 is intend to support the decision whether to include cross-sectional averages when using xtdcce2 and accompanies xtcd2 in testing for weak cross-sectional dependence. As a default it uses xtcd2 to test for weak cross-sectional dependence. For a discussion of xtdcce2 and xtcd2 see Ditzen (2018,2019).
In case of unbalanced panels an Expected Maximisation algorithm taken from xtnumfac is used. xtcd2 imputes values independently from xtcse2 and therefore results can differ.
For the following assume a general factor model with m factors:
x(i,t) = sum(j=1,m) b(j,i) f(j,t) + u(i,t)
i = 1,...,N and t = 1,...,T
where x(i,t) depends on unobserved m common factors f(j,t) with loading b(j,i) and a cross sectionally independent error term u(i,t). The time dimension (T) and the number of cross-sectional units (N) increases to infinity; (N,T) -> infinity.
Chudik et al (2011) specify the factors as weak or strong using a constant 0<=alpha<=1 such that:
lim N^(-alpha) sum(j=1,m) abs(b(j,i)) = K < infinity.
The type of dependence of the factors and thus the series then depends on the characteristics of b(j,i):
| alpha | dependence |
|---|---|
| alpha = 0 | weak |
| 0 < alpha < 0.5 | semi weak |
| 0.5 <= alpha < 1 | semi strong |
| alpha = 1 | strong |
Weak cross-sectional dependence can be thought of as the following: Even if the number of cross-sectional units increases to infinity, the sum of the effect of the common factors on the dependent variable remain constant. In the case of strong cross-sectional dependence, the sum of the effect of the common factors becomes stronger with an increase in the number of cross-sectional units.
In an estimation ignoring (semi-) strong dependence in the dependent or independent variables can cause an omitted variable bias and therefore lead to inconsistent estimates. Pesaran (2015) proposes a test to test for weak cross-sectional dependence, see xtcd2. Pesaran (2006) and Chudik, Peasaran (2015) develop a method to estimate models with cross-sectional dependence by adding time averages of the dependent and independent variables (cross-sectional averages). This estimator is implemented in Stata by xtdcce2.
xtcse2 estimates alpha in the equation above. An alpha above 0.5 implies strong cross-sectional dependence and the appropriate when using a variable is required.
Bailey, Kapetanios and Pesaran (2016) [BKP] propose a method for the estimation of the exponent. This section summarizes their approach, a careful reading of the assumptions and theorems is strongly encouraged.
BKP derive a bias-adjusted estimator for alpha in a panel with N_g cross-sectional units (see Eq. 13):
alpha = 1 + 1/2 ln(sigma_x^2)/ln(N_g) - 1/2 ln(mu^2)/ln(N_g) - 1/2 cn / [N_g * ln(N_g) * sigma_x^2]
where sigma_x^2 is the variance of the cross-sectional averages. mu^2 is average variance of significant regression coefficients of x(i,t) on standardized cross-sectional averages with a pre specified size of the test. cn is the variance of scaled errors from a regression of the x(i,t) on its first K(PC) principle components. The number of principle components can be set using the option pca(integer). The default is to use the first 4 principle components.
xtcse2 outputs a standard error for alpha and a confidence interval in the usual Stata estimation fashion. A t- or z-test statistic with p-value is however omitted, because the test is done by the test for weak cross-sectional dependence (CD-test), see xtcd2. xtcse2 automatically calculates the CD-test statistic and posts its results. For the estimation of alpha a confidence interval is therefore more informative.
The calculation of the standard error of alpha follows the equation B47, Section VI of the online appendix of BKP, available here:
sigma(alpha) = [1/T V(q) + 4/N^(alpha) S]^(1/2) * 1/2 * 1/ln(N)
V(q) is the regression standard error over the square of the sum of q coefficients of an AR(q) process of the square of the deviation of standardized cross-sectional averages. q is the third root of T. S is the squared sum divided by N^(alpha-1) of OLS coefficients of x(it) on standardized cross-sectional averages sorted according to their absolute value.
In the case of estimating the exponent of cross-sectional dependence in residuals Bailey, Kapetanios and Pesaran (2019) propose to use pair-wise correlations to estimate the exponent. For the calculation, only significant correlations are taken into account. The exponent is estimated according to (Eq 25 in BKP 2019):
alpha = ln(tau' delta tau) / [2 ln(N)]
where tau is a Nx1 vector of ones and delta is a matrix which contains the significant pair-wise correlations. For the significance, the size of the test and a tuning parameter need to be set a priori. xtcse2 uses a size of 10% and a tuning parameter of 0.5 as a default. Both can be changed with the options size() and tuning().
In the case of a panel with weakly exogenous regressors, the pair-wise correlations are based on recursive residuals, see BKP 2019, section 5.2. xtcse2 allows for this if the option lags() is used.
BKP 2019 do not derive a closed form solution for standard errors. Therefore standard errors and confidence intervals are calculated using a simple bootstrap, where the cross-sectional units are replaced with replacement. This approach is outlined in BKP 2019 section 5.3.
| Options | Description |
|---|---|
| pca(integer) | sets the number of principle components for the calculation of cn. Default is to use the first 4 components. |
| standardize | standardizes variables. |
| nocenter | do not center variables. |
| nocd | suppresses test for cross-sectional dependence using xtcd2. |
| size(real) | size of the test. Default is 10% (0.1). |
| ressidual | estimates the exponent of cross-sectional depdendence in residuals, following BKP 2019. |
| tuning(real) | tuning parameter for estimation of the exponent in residuals. Default is 0.5. |
| reps(integer) | number of repetitions for bootstrap for calculation of standard error and confidence interval for exponent in residuals. Default is 0. |
| lags(integer) | number of lags (or training period) for calculation of recursive residuals when estimating the exponent after a regression with weakly exogenous regressors. |
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%).
Before running the growth regression the exponent of the cross-sectional dependence for the variables is estimated:
xtcse2 d.log_rgdpo L.log_rgdpo log_hc log_ck log_ngd.
All variables are highly cross-sectional dependent with alphas close or even above 1. Therefore an estimation method taking cross-sectional dependence is required. xtdcce2 is uses such an estimation method by adding cross-sectional averages to the model. After running xtdcce2 it is possible to use xtcse2 to estimate the strength of the exponent of the residual using the option residuals.
xtdcce2 log_rgdpo L.log_rgdpo log_ck log_ngd log_hc , cr(log_rgdpo log_ck log_ngd log_hc) .
xtcse2, res
xtcse2 automatically predicts the residuals using predict (predict after xtdcce2). The CD statistic is still in a rejection region, therefore the residuals exhibit strong cross-sectional dependence.
The estimated model above is mis-specified as it is a dynamic model, but no lags of the cross-sectional averages are added. The number of lags should be in the region of T^(1/3), so with 47 periods 3 lags are added. Then xtcse2 is used to estimate alpha again, this time the CD test is omitted:
xtdcce2 log_rgdpo L.log_rgdpo log_ck log_ngd log_hc , cr(log_rgdpo log_ck log_ngd log_hc) cr_lags(3) .
xtcse2 ,nocd residual lags(3) reps(200)
The value of the CD test statistic is 1.32 and in a non-rejection region. The estimate of alpha is considerably small the confidence interval does not overlap with 0.5
As a second exercise the first row of Table 1. in BKP is reproduced. The data is available on Pesaran's webpage and for download here .
After the data is loaded, reshaped (it comes in a matrix) and renamed as variable gdp, the option standardize is used to standardize the variable as done in BKP:
xtcse2 gdp , standardize.
Ahn, S. C., & Horenstein, A. R. 2013. Eigenvalue ratio test for the number of factors. Econometrica, 81(3), 1203–1227.
Bailey, N., G. Kapetanios and M. H. Pesaran. 2016. Exponent of cross-sectional dependence: estimation and inference. Journal of Applied Econometrics 31: 929-960.
Bailey, N., G. Kapetanios and M. H. Pesaran. 2019. Exponent of Cross-sectional Dependence for Residuals. Sankhya B. The Indian Journal of Statistics: 81(4) p. 46-102.
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
Blackburne, E. F., and M. W. Frank. 2007. Estimation of nonstationary heterogeneous panels. Stata Journal 7(2): 197-208.
Chudik, A., M. H. Pesaran and E. Tosetti. 2011. Weak and strong cross-section dependence and estimation of large panels. The Econometrics Journal 14(1):C45–C90.
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.
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Ditzen, J. 2021. Estimating long run effects and the exponent of cross-sectional dependence: an update to xtdcce2. The Stata Journal 21:3.
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Jan Ditzen (Free University of Bolzano-Bozen)
Email: jan.ditzen@unibz.it
Web: www.jan.ditzen.net
I am grateful to Achim Ahrens, Arnab Bhattacharjee, David M. Drukker, Markus Eberhardt, Tullio Gregori, Sebastian Kripfganz, Erich Gundlach, Sean Holly, Kyle McNabb, Vasilis Sarafidis 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. All remaining errors are my own.
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.
Please cite as follows:
Ditzen, J. 2018. xtdcce2: Estimating dynamic common correlated effects in Stata. The Stata Journal, 18:3, 585 - 617.
or
Ditzen, J. 2021. Estimating long run effects and the exponent of cross-sectional dependence: an update to xtdcce2. The Stata Journal 21:3.
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
Version 4.7 - June 2024
- bug fixes when using predict with xtdcce2fast.
- factor variables for option cr().
- added rank classifier and information criteria to select CSA.
Version 4.6 - January 2024
- fixed bug when using cr() and absorb()
- fixed bug in rcce and unbalanced panels
Version 4.5 - November 2023
- supports reghdfe v.6.12
Version 4.4 - May 2023
- fixed bug in trend option
Version 4.3 - May 2023
- fixed bug in var/cov when using pooled coefficients
Version 4.2 - May 2023
- fixed bug in xtcd2 and xtcse2
Version 4.1 - March 2023
- fixed bug when using different lag lengths for CSA
Version 4.0 - February 2023
- bootstrap support
- added option mgmissing
- added option rcce
- added option fast2
- fixed error when pooled and ardl used.
Version 3.0 - August 2021
- several small bug fixes
- improved support for factor variables
- fix for mm_which2
- message for large panels
- error in calculation for variances of cross-sectional unit specific coefficients
- fix predict program: partial now only in-sample and bug fixed when xb2 and reportc was used (thanks to Tullio Gregori for the pointers).
- added global and local cross-sectional averages
- improved output
- xtdcce2fast
- speed improvements
Version 1.35 to Version 2.0
- Bug fix in calculation of minimal T dimension, added option nodimcheck.
- Speed improvements (thanks to Achim Ahrens for the suggestions).
- Bug fixes for jackknife (thanks to Collin Rabe for the pointer).
- Bug fix in predict and if (thanks for Deniey A. Purwanto and Tullio Gregoi for the pointers).
- Bug fix if binary variable used and constant partialled out.
- Bug fixed in calculation of R2, added adjusted R2 for pooled and MG regressions.
- Newey West and Westerlund, Petrova, Norkute standard errors for pooled regressions.
- invsym for rank deficient matrices.
- Added xtcse2 support.
Version 1.33 to Version 1.34
- small bug fixes in code and help file.
Version 1.32 to Version 1.33
- bug in if statements fixed.
- noomitted added, bug in cr(all) fixed.
- added option "replace" and "cfresiduals" to predict.
- CS-DL and CS-ARDL method added.
- Output as in Stata Journal Version.
Version 1.31 to Version 1.32
- bug number of groups fixed
- predict, residual produced different results within xtdcce2 and after if panel unbalanced or trend used (thanks to Tullio Gregori for the pointer).
- check for rank condition.
- several bugs fixed.
Version 1.2 to Version 1.31
- code for regression in Mata
- corrected standard errors for pooled coefficients, option cluster not necessary any longer. Please rerun estimations if used option pooled()
- Fixed errors in unbalanced panel
- option post_full removed, individual estimates are posted in e(bi) and e(Vi)
- added option ivslow.
- legacy control for endogenous_var(), exogenous_var() and residuals().