8.2.1 Fixed Coefficients

The heterogeneous panel model is:

(8.1)

where are individual‐specific parameters and is a vector of explanatory variables.

If the pooling assumption is relaxed and one does not want to make any other assumption about how the are generated, and if the dimension permits, one can simply estimate a separate vector of coefficients for each regression.

Individual slope parameters can be estimated (‐consistently) by least squares as:

(8.2)

This can be accomplished by subsetting the data and running OLS; more efficient functionality is provided in plm through the function pvcm, leaving the model argument at the default value of 'within'.

8.2.2 Random Coefficients

Estimating separate regressions negates the advantages of panel datasets in that degrees of freedom are greatly reduced with respect to the pooled data. If s are treated as fixed, there will be parameters to estimate with observations. Random coefficients specifications allow instead for cross‐sectional variability while still reaping the benefits of pooling.

8.2.2.1 The Swamy Estimator

Swamy (1970) proposed a model with all individual‐specific coefficients. In this case, we have:

where homoscedasticity of is not assumed and , or . The model is then rewritten as:

with . The model errors can be heteroscedastic (in particular because we did not impose homoscedasticity of ) and the errors of each individual are correlated as containing the same parameter vector . For the th individual, the error covariance is then:

and being uncorrelated by hypothesis, we have:

For the whole sample, is a block diagonal matrix, each block being equal to .

OLS estimation of this model is inefficient, not taking into account the heteroscedasticity and the correlation of errors. The model can be efficiently estimated by generalized least squares by computing and then applying OLS to the variables transformed by pre‐multiplying them by . Given that the latter is a block diagonal matrix, the same result is obtained by pre‐multiplying each individual's data by the corresponding block . The generalized least squares method is clearly infeasible because is unknown, but it can be made operational by employing an estimate thereof from a consistent model. This amounts to estimating and the elements of the matrix, or in total parameters.

To this end, we start by estimating each individual model by OLS. We then have:

A natural estimator of is then:

The estimates are then averaged:

The estimation of is based on the expression , which, developing and regrouping terms, can be written:

The usefulness of this expression is in writing as a linear combination of uncorrelated random variates, which considerably simplifies the computation of the variance of as all covariances are zero. We then have:

Finally, regrouping terms:

We then have:

which gives the estimator of :

8.2.2.2 The Mean Groups Estimator

Under less restrictive parametric assumptions than those of the Swamy model, assuming only exogeneity of the regressors and independently sampled errors, the average can be estimated by the simpler mean groups (MG) method

(8.3)

and its dispersion, in a nonparametric fashion, through the empirical covariance of the individual :

(8.4)

which is in fact the simplified version of the Swamy covariance seen above. In the context of the Swamy model, it is biased but ‐consistent and, differently from the original, always non‐negative definite; as such, it has been suggested by Swamy (1970) himself as an alternative for cases when his parametric covariance is not. In general, it can be shown that the MG estimator is a special case with equal GLS weighting of the Swamy estimator, to which it converges as grows sufficiently large (Hsiao and Pesaran, 2008). The function pmg performs mean groups estimation by default (model='mg').

Dynamic Mean Groups

Importantly, Pesaran and Smith (1995) consider the MG estimator in dynamic models of the type

(8.5)

and show that, unlike aggregated or pooled regressions, it provides consistent estimates of both coefficients and standard errors. Considering the full parameter vector , they observe that, while for fixed the estimator is biased of order , the individual regressions (8.2) become consistent estimators of as diverges. Hence the MG estimator of the average parameter vector is consistent for both and (see the discussion in Hsiao and Pesaran, 2008). Explicit calculation of the individual parameters' covariance as in (8.4) in turn provides a consistent estimate of .

8.2.3 Testing for Poolability

Heterogeneous estimators relax the assumption made in the error components model, which imposes homogeneity of all model parameters (but the intercept) across individuals. Under this assumption, one can estimate a single model for the whole sample, at most including individual‐specific constant terms. This restriction, which is usually called poolability, can be tested by comparing the estimation results from the different approaches. Furthermore, one can impose the further restriction of no individual‐specific intercepts.

In the variable coefficients framework, unrestricted estimation consists in estimating by OLS one different model for each individual. The sum of squared residuals is then: . For this model, degrees of freedom are: . The restricted model to compare to can be either pooled OLS ( with degrees of freedom) or the within model ( with degrees of freedom), depending on whether the absence of individual effects is imposed or not. The test statistic is then (taking the within specification as the restricted model):

This takes the form of a well‐known stability test (known as the Chow test) distributed under as an F with and degrees of freedom.

The function performing this kind of test is called pooltest. One possible usage is to provide two models, one estimated separately for each individual, and either an OLS or a within model. In the first case, all parameters are supposed constant under , including the constant terms. The unrestricted model is estimated by the function pvcm. As seen above, this function allows to estimate two different models, depending on the parameter model; here, the appropriate value for this argument is 'within' (the other possible choice being illustrated in the next section).

8.3.1 The Common Factor Model

Consider the factor‐augmented panel model

where is the cross‐sectional index and the time index. is a vector of observed, strictly exogenous regressors including a 1 and is a vector of unobserved, cross‐sectionally invariant common factors.

Such structure is capable of generating cross‐sectional correlation in case of a similar, albeit not identical, response across countries to modifications in the common factors, measured by the factor loadings . The common factors are allowed to be correlated with the regressors, as is most likely to be the case, so their effect comes both through factor loadings and through the indirect effect on the observed regressors. The common factors are also allowed to be nonstationary. Moreover, the remainder error term is allowed to be spatially correlated as in

where is the generic element of an spatial weights matrix in which nonzero elements correspond to pairs of spatially close observation units (e.g., regions sharing a common border, or below a given distance threshold); so that each error is correlated with a weighted average of the errors in close‐by observations according to the parameter .²

The two kinds of error dependence induced by omitted common factors and by spatial error correlation have serious consequences on the properties of estimators if they are neglected. The former induces cross‐sectional correlation of a pervasive type, not dying out with distance, characterized by Pesaran and Tosetti (2011) as strong; moreover, if the omitted common factors are correlated with the regressors, the latter become endogenous and estimators become inconsistent. The latter type of dependence, dubbed weak because it dies out with distance, has less serious consequences on estimation but can still cause inefficiency (and hence inconsistent standard errors and invalid inference); moreover, as discussed in the next section, it weakens consistency in the particular case of spurious panel regression. Estimators able to control for the strong kind of dependence, as it turns out, are consistent in the presence of weak dependence as well.

In the special case of only one factor with uniform factor loadings , the common factor model becomes a time fixed effects model, which can be estimated either by OLS with time dummies or by the appropriate within estimator, i.e., OLS on cross‐sectionally demeaned data.

8.3.2 Common Correlated Effects Augmentation

The principle of common correlated effects (CCE) augmentation of Pesaran (2006) is based on the idea that, for large , the factors can be approximated by cross‐sectional averages of the response and regressors. Following the original paper (see also Holly et al., 2010), consider the model:

(8.6)

where both the (composite) error and the regressors are generated by linear combinations of the unobserved, cross‐sectionally invariant factors :

(8.7)

(8.8)

Substituting (8.7) in (8.6) and combining the result with (8.8), we get:

(8.9)

where and

Taking cross‐section averages of (8.9),

so that, if is invertible, the common factors can be written as:

If as and , then

Following this line of reasoning, Pesaran (2006) shows that the cross‐sectional averages of the response () and regressors () are ‐consistent estimators of the unobserved common factors and can therefore be used as observable proxies thereof. Augmenting the regression with these averages is known as the common correlated effects (CCE) principle. CCE estimators can be used to consistently estimate the individual slope parameters by applying least squares to the augmented regression

where .

The estimator for each individual slope coefficient can then be written compactly as

with , contains: the matrix of cross‐sectional averages , ; and a deterministic component comprising individual intercept and time trend (Pesaran, 2006, p. 974). The average is then estimated by the MG method,

This estimator is known as , for “common correlated effects mean groups.”

The covariance matrix is estimated nonparametrically, on the basis of the empirical covariance of the individual coefficients, just like in the MG case:

(8.10)

Unlike other estimators, the CCE is (‐) consistent for any fixed, unknown number of possibly nonstationary common factors. Being robust to strong forms of cross‐sectional dependence, the CCE estimator is also robust to weak ones such as spatial correlation (see Pesaran and Tosetti, 2011). Moreover, the CCE strategy has proved most effective in a number of simulation studies, e.g., Coakley et al. (2006), Pesaran and Tosetti (2011), Kapetanios et al. (2011).

8.3.2.1 CCE Mean Groups vs. CCE Pooled

Estimation by the CCE principle can be performed either leaving parameters free to vary, as above, or imposing parameter homogeneity (but maintaining heterogeneity in intercepts, factor loadings, and possibly time trends), which leads to the CCEP (pooled) estimator

(8.11)

and is to be preferred on efficiency grounds when the underlying assumption that is reasonable. It must be observed that the CCEP estimator, although imposing , still allows individual factor loadings to differ.

The standard pooled or heterogeneous estimators can be seen an special cases of this more general formulation where augmentation is eliminated or reduced: pooled OLS as CCEP with , individual fixed effects as CCEP with containing only individual dummies. The mean groups (MG) estimator can in turn be seen as where .

8.3.2.2 Computing the CCEP Variance

According to Pesaran (2006, 5.2), the variance of the CCEP estimator can be computed in two different ways, depending on whether the assumption of parameter homogeneity is imposed here as well (homogeneous estimator) or not (heterogeneous, or nonparametric, estimator).

The heterogeneous version (Pesaran, 2006, Th. 3) is based again on the nonparametric estimate of the individual coefficients' covariance. Defining

and

the estimator is

(8.12)

This estimator is consistent under quite general conditions as regards the rate of growth of vs and the distribution of individual parameters; it is the one that fares best in the original papers' simulation study and the one the author recommends to use. It is therefore the default method in the pcce function.

Nevertheless, strictly speaking, (8.12) is not appropriate under complete homogeneity. Pesaran (2006, Th. 4) presents an alternative, which is appropriate for large panels (i.e., if as ). The latter, which is presented in detail in Pesaran (2006, p. 988), is based on the nonparametric kernel‐smoothed estimator of Newey and West (see 5.1.1.3) and can be calculated using standard methods. Analogously, and again in large‐ settings, the familiar clustering estimator can be applied. In fact, being idempotent, the CCEP estimator in (8.11) can be seen as OLS on the transformed variables ; hence methods for robust covariances can be applied to pcce objects the same way they are to plm ones – e.g., those representing a within model. From a software viewpoint, the pcce function is compliant with both vcovNW and vcovHC.

8.4.1 Unit Root Testing: Generalities

Detecting unit roots has become a central subject in macroeconometrics. The techniques employed are adaptations from the time series literature to the panel case. We will begin by reviewing the main results regarding time series.

Consider a variable generated by an autoregressive process of order one:

The vector of explanatory variables may contain an intercept, a linear trend, and different explanatory variables. To keep things simple, in the following we will assume , so that follows a “pure” autoregressive process. As regards the error (which in this context is often called the innovation), we will assume that it has mean zero and standard deviation . By recursive substitution, one has:

If is deterministic and the are not correlated, the variance of can be written:

If , we have:

On the other hand, if , so that the variance grows to infinity with ; the series is then nonstationary and is said to have a unit root. The presence of unit roots poses various problems, first and foremost that of spurious regressions. In the presence of a unit root, a series presents a peculiar sort of trend that is not deterministic but stochastic, and the presence of such trends in two series containing unit roots may induce an artificial correlation between them. In Figure 8.2 we present two autoregressive series with respectively and . We see how in the former case the autoregressive process translates into correlation between successive values of ; in particular, if then is more likely to be negative than positive. However, the curve representing the realization of the process crosses the horizontal axis frequently. On the other hand, in the case of a unit root, one can clearly detect the presence of a stochastic trend (in this case, on the rise): only changes sign once, and most of its realizations are positive.

To illustrate the importance of the spurious regression problem, we perform a short simulation exercise; we draw two autoregressive series independently, regress one on the other, and recover the t‐statistic corresponding to the null hypothesis . This hypothesis is true by construction; therefore, in a normal context the t‐statistic should not reject (i.e., be roughly less than 2) in 95% of cases. Let us begin by illustrating this result for . To this end, we employ two functions: code generates an autoregressive series, tstat performs OLS estimation, and recovers the t‐statistic:

 autoreg <- function(rho = 0.1, T = 100){
  e <- rnorm(T)
  for (t in 2:(T)) e[t] <- e[t] + rho *e[t-1]
  e
}
tstat <- function(rho = 0.1, T = 100){
  y <- autoreg(rho, T)
  x <- autoreg(rho, T)
  z <- lm(y ˜ x)
  coef(z)[2] / sqrt(diag(vcov(z))[2])
}
result <- c()
R <- 1000
for (i in 1:R) result <- c(result, tstat(rho = 0.2, T = 40))
quantile(result, c(0.025, 0.975))
  2.5%  97.5%
-2.114  1.990
prop.table(table(abs(result) > 2))

FALSE  TRUE
0.943 0.057 

We can see how the empirical quantiles are very close to their expected values and the share of false positives is in the region of 5%. Let us now do the same with two series, each containing a unit root:

 result <- c()
R <- 1000
for (i in 1:R) result <- c(result, tstat(rho = 1, T = 40))
quantile(result, c(0.025, 0.975))
  2.5%  97.5%
-9.158  8.227
prop.table(table(abs(result) > 2))

FALSE  TRUE
0.379 0.621 

Judging by the usual t‐statistic, in two thirds of cases one would conclude in favor of a significant relationship between our two independently generated variables.

It is therefore crucial to detect the presence of unit roots in time series data; otherwise, there are considerable chances to obtain falsely significant results. To this end, it is simplest to write the equation of the autoregressive process subtracting to both sides. One has then:

The unit root test then becomes a zero restriction test for the coefficient associated to in the model where the regressand is . One might want to use a classic t‐statistic, obtained dividing by its standard error. Setting vs , one will then reject the unit root hypothesis at the 5% level if the statistic is less than .

 R <- 1000
T <- 100
result <- c()
for (i in 1:R){
  y <- autoreg(rho=1, T=100)
  Dy <- y[2:T] - y[1:(T-1)]
  Ly <- y[1:(T-1)]
  z <- lm(Dy ˜ Ly)
  result <- c(result, coef(z)[2] / sqrt(diag(vcov(z))[2]))
} 

In Figure 8.3 we depict a histogram of the realizations of the t‐statistic, superposing a normal density curve:

One can easily see that employing classic inference procedures to detect the presence of unit roots is unwarranted, as the t‐statistic follows a distribution that is very far from the normal. Employing the usual critical value of , one has here:

 prop.table(table(result < -1.64))

FALSE  TRUE
0.542 0.458 

which leads to reject the true hypothesis of a unit root one half of the times. To perform the Dickey‐Fuller test, one needs specific critical values that are not those of the normal (or the t) distribution. The test can be performed augmenting the auxiliary model with a constant and/or a deterministic trend; lags of can also be added in order to clean out any possible autocorrelation of .

The regression between two series both containing a unit root is only appropriate if they present a long‐term structural relationship. One speaks then of co‐integration. More precisely, we will say that two variables and are cointegrated if there exists such that:

where is stationary, i.e., it does not have unit roots. A simple cointegration test can then be performed as follows:

1. verify whether and have unit roots with a Dickey‐Fuller test,
2. if they both do, then estimate a model of on and recover the residuals ,
3. do a Dickey‐Fuller test on : if the unit root hypothesis is rejected, then and are cointegrated and the regression of on is meaningful; otherwise, and are integrated but not cointegrated, and the regression of on will be spurious.

8.4.2 First Generation Unit Root Testing

The classical test for unit roots is usually called ADF for “augmented Dickey‐Fuller”. Many extensions of this test have been proposed to adapt it to a panel data setting.

8.4.2.1 Preliminary Results

Some of these tests are obtained by applying separate ADF tests to every individual in the sample. To perform these preliminary tests, one shall choose the number of lags and the relevant set of deterministic variables , which can be either , (an intercept), or (an intercept and a time trend).

(8.13)

This choice can be based on a number of criteria:

• the Schwarz information criterion (SIC),
• the Akaike information criterion (AIC),
• the Hall method, consisting in adding as many lags as there are significant ones.

The regression is performed on observations for each individual, which leads to in total, with , being the average number of lags. The variance of the residuals for individual is estimated by:

(8.14)

with the degrees of freedom of the regression.

8.4.2.2 Levin‐Lin‐Chu Test

Levin et al. (2002) proposed the first panel unit root test. In order to perform it, one must run two preliminary regressions: respectively, of and of as functions of and , obtaining two residual vectors denoted respectively by and .

These two residuals are then normalized dividing them by the estimated standard error (equation 8.14). The estimator of is obtained by regressing on for the whole sample. Its standard deviation and t‐statistic are denoted respectively by and .

The long‐term variance of is estimated by:

where is the truncation lag parameter and are the sample covariance weights, which depend on the choice of kernel.

Calling the ratio between the long‐term and the short‐term variance for the ‐th individual and the sample average thereof, Levin et al. (2002) show that the statistic:

is normally distributed under the null hypothesis of a unit root. and can be found in the original paper.

8.4.2.3 Im, Pesaran and Shin Test

One of the drawbacks of the Levin et al. (2002) test is that the alternative hypothesis holds that , but at the same time it is the same for all individuals. The test proposed by Im et al. (2003) (IPS) overtakes this limitation: the null hypothesis is still for all individuals, but the alternative is now that can be different across individuals, provided that at least for some of them. The IPS test takes the form of a simple average of the t‐statistics for from the individual ADF regressions (8.13):

The IPS statistic follows a nonstandard distribution, and must be therefore compared with values tabulated ad hoc. Alternatively, it can be standardized with mean and variance and given in the Im et al. (2003) paper. The test statistic is then :

which, under the null of a unit root, is normally distributed.

8.4.2.4 The Maddala and Wu Test

Maddala and Wu (1999) proposed a similar test, again not imposing homogeneity of under the alternative. Instead of the t‐statistics, it is based on combining the critical values obtained from the individual ADF tests. The test statistic is then simply:

and, under the null of a unit root for all individuals, it is distributed as a with degrees of freedom.

8.4.3 Second Generation Unit Root Testing

The above panel unit root tests do all rest on the hypothesis of absence of cross‐sectional correlation. When, after the turn of the millennium, the panel data literature started recognizing how pervasive cross‐sectional correlation is in applications and progressed toward the development of consistent methods in its presence, the above assumption started to be seen as too restrictive. The tests assuming no cross‐sectional correlation became known under the collective name of “first‐generation” panel unit root tests, to distinguish them from the new breed of testing procedures that was emerging. These new panel unit root tests, sharing the quality of being consistent in the face of cross‐sectional correlation, were dubbed “second generation” to distinguish them from the former and are currently most often employed in applications.

The reference framework for cross‐sectionally correlated panels is, as discussed above, the common factor model. A number of cross‐correlation‐compliant panel unit root procedures have been devised in this framework based on various defactoring procedures. One of the most popular second‐generation tests, due to Pesaran (2007), takes the approach of controlling for the common factors, instead of trying to eliminate them; it does so in the CCE framework, by augmenting the auxiliary regressions through cross‐sectional averages of the response and regressors. The individual ADF regressions are augmented with the cross‐sectional averages of lagged levels and differences of the individual series:

(8.15)

The individual ADF regressions are therefore denoted “cross‐sectionally augmented ADF” (CADF) regressions; the resulting individual CADF statistics can in principle be combined as described above, forming the basis for either a “cross‐sectionally augmented IPS” (CIPS) or a Maddala‐Wu test. However, the limiting distributions for the latter do not apply anymore in the absence of cross‐sectional independence; for this reason, Pesaran (2007) tabulated critical values for the CIPS test for the three different cases where the auxiliary CADF regressions contain an intercept, a deterministic trend, or none of the above.