Abstract
This introduction to the plm package is a modified and
extended version of Croissant and Millo
(2008), published in the Journal of Statistical
Software.
Panel data econometrics is obviously one of the main fields in the
statistics profession, but most of the models used are difficult to
estimate with only plain R. plm is a package
for R which intends to make the estimation of linear panel
models straightforward. plm provides functions to estimate
a wide variety of models, to make (robust) inference, and for various
tests.
Panel data econometrics is a continuously developing field. The increasing availability of data observed on cross-sections of units (like households, firms, countries etc.) and over time has given rise to a number of estimation approaches exploiting this double dimensionality to cope with some of the typical problems associated with economic data, first of all that of unobserved heterogeneity.
Timewise observation of data from different observational units has long been common in other fields of statistics (where they are often termed longitudinal data). In the panel data field as well as in others, the econometric approach is nevertheless peculiar with respect to experimental contexts, as it is emphasizing model specification and testing and tackling a number of issues arising from the particular statistical problems associated with economic data.
Thus, while a very comprehensive software framework for (among many
other features) maximum likelihood estimation of linear regression
models for longitudinal data, packages nlme (J. Pinheiro et al. 2007) and lme4
(Bates 2007), is available in the
R ( Development Core Team
(2008)) environment and can be used, e.g., for estimation of
random effects panel models, its use is not intuitive for a practicing
econometrician, and maximum likelihood estimation is only one of the
possible approaches to panel data econometrics. Moreover, economic panel
data sets often happen to be unbalanced (i.e., they have a
different number of observations between groups), which case needs some
adaptation to the methods and is not compatible with those in
nlme. Hence the need for a package doing panel data “from
the econometrician’s viewpoint” and featuring at a minimum the basic
techniques econometricians are used to: random and fixed effects
estimation of static linear panel data models, variable coefficients
models, generalized method of moments estimation of dynamic models; and
the basic toolbox of specification and misspecification diagnostics.
Furthermore, we felt there was a need for automation of some basic
data management tasks such as lagging, summing and, more in general,
applying (in the R sense) functions to the
data, which, although conceptually simple, become cumbersome and
error-prone on two-dimensional data, especially in the case of
unbalanced panels.
This paper is organized as follows: Section linear panel model presents a very short
overview of the typical model taxonomy1. Section software approach discusses the software
approach used in the package. The next three sections present the
functionalities of the package in more detail: data management (Section
managing data and formulae),
estimation (Section model estimation)
and testing (Section tests), giving a short
description and illustrating them with examples. Section plm vs nlme and lme4 compares the approach in
plm to that of nlme and lme4,
highlighting the features of the latter two that an econometrician might
find most useful. Section conclusion
concludes the paper.
The basic linear panel models used in econometrics can be described through suitable restrictions of the following general model:
\[\begin{equation*} y_{it}=\alpha_{it} + \beta_{it}^\top x_{it} + u_{it} \end{equation*}\]
where \(i=1, ..., n\) is the individual (group, country …) index, \(t=1, ..., T\) is the time index and \(u_{it}\) a random disturbance term of mean \(0\).
Of course \(u_{it}\) is not estimable with \(N = n \times T\) data points. A number of assumptions are usually made about the parameters, the errors and the exogeneity of the regressors, giving rise to a taxonomy of feasible models for panel data.
The most common one is parameter homogeneity, which means that \(\alpha_{it}=\alpha\) for all \(i,t\) and \(\beta_{it}=\beta\) for all \(i,t\). The resulting model
\[\begin{equation*} y_{it}=\alpha + \beta^\top x_{it} + u_{it} \end{equation*}\]
is a standard linear model pooling all the data across \(i\) and \(t\).
To model individual heterogeneity, one often assumes that the error term has two separate components, one of which is specific to the individual and doesn’t change over time2. This is called the unobserved effects model:
\[\begin{equation} (\#eq:errcomp) y_{it}=\alpha + \beta^\top x_{it} + \mu_i + \epsilon_{it} \end{equation}\]
The appropriate estimation method for this model depends on the properties of the two error components. The idiosyncratic error \(\epsilon_{it}\) is usually assumed well-behaved and independent of both the regressors \(x_{it}\) and the individual error component \(\mu_i\). The individual component may be in turn either independent of the regressors or correlated.
If it is correlated, the ordinary least squares (OLS) estimator of \(\beta\) would be inconsistent, so it is customary to treat the \(\mu_i\) as a further set of \(n\) parameters to be estimated, as if in the general model \(\alpha_{it}=\alpha_{i}\) for all \(t\). This is called the fixed effects (a.k.a. within or least squares dummy variables) model, usually estimated by OLS on transformed data, and gives consistent estimates for \(\beta\).
If the individual-specific component \(\mu_i\) is uncorrelated with the regressors, a situation which is usually termed random effects, the overall error \(u_{it}\) also is, so the OLS estimator is consistent. Nevertheless, the common error component over individuals induces correlation across the composite error terms, making OLS estimation inefficient, so one has to resort to some form of feasible generalized least squares (GLS) estimators. This is based on the estimation of the variance of the two error components, for which there are a number of different procedures available.
If the individual component is missing altogether, pooled OLS is the most efficient estimator for \(\beta\). This set of assumptions is usually labelled pooling model, although this actually refers to the errors’ properties and the appropriate estimation method rather than the model itself. If one relaxes the usual hypotheses of well-behaved, white noise errors and allows for the idiosyncratic error \(\epsilon_{it}\) to be arbitrarily heteroskedastic and serially correlated over time, a more general kind of feasible GLS is needed, called the unrestricted or general GLS. This specification can also be augmented with individual-specific error components possibly correlated with the regressors, in which case it is termed fixed effects GLS.
Another way of estimating unobserved effects models through removing time-invariant individual components is by first-differencing the data: lagging the model and subtracting, the time-invariant components (the intercept and the individual error component) are eliminated, and the model
\[\begin{equation*} \Delta y_{it}= \beta^\top \Delta x_{it} + \Delta u_{it} \end{equation*}\]
(where \(\Delta y_{it}=y_{it}-y_{i,t-1}\), \(\Delta x_{it}=x_{it}-x_{i,t-1}\) and, from @ref(eq:errcomp), \(\Delta u_{it}=u_{it}-u_{i,t-1}=\Delta \epsilon_{it}\) for \(t=2,...,T\)) can be consistently estimated by pooled OLS. This is called the first-difference or FD estimator. Its relative efficiency, and so reasons for choosing it against other consistent alternatives, depends on the properties of the error term. The FD estimator is usually preferred if the errors \(u_{it}\) are strongly persistent in time, because then the \(\Delta u_{it}\) will tend to be serially uncorrelated.
Lastly, the between model, which is computed on time (group) averages of the data, discards all the information due to intragroup variability but is consistent in some settings (e.g., non-stationarity) where the others are not, and is often preferred to estimate long-run relationships.
Variable coefficients models relax the assumption that \(\beta_{it}=\beta\) for all \(i,t\). Fixed coefficients models allow the coefficients to vary along one dimension, like \(\beta_{it}=\beta_i\) for all \(t\). Random coefficients models instead assume that coefficients vary randomly around a common average, as \(\beta_{it}=\beta+\eta_{i}\) for all \(t\), where \(\eta_{i}\) is a group– (time–) specific effect with mean zero.
The hypotheses on parameters and error terms (and hence the choice of the most appropriate estimator) are usually tested by means of:
Dynamic models and in general lack of strict exogeneity of the regressors, pose further problems to estimation which are usually dealt with in the generalized method of moments (GMM) framework.
These were, in our opinion, the basic requirements of a panel data
econometrics package for the R language and environment.
Some, as often happens with R, were already fulfilled by
packages developed for other branches of computational statistics, while
others (like the fixed effects or the between estimators) were
straightforward to compute after transforming the data, but in every
case there were either language inconsistencies w.r.t. the standard
econometric toolbox or subtleties to be dealt with (like, for example,
appropriate computation of standard errors for the demeaned model, a
common pitfall), so we felt there was need for an “all in one”
econometrics-oriented package allowing to make specification searches,
estimation and inference in a natural way.
Panel data have a special structure: each row of the data corresponds
to a specific individual and time period. In plm the
data argument may be an ordinary data.frame
but, in this case, an argument called index has to be added
to indicate the structure of the data. This can be:
NULL (the default value), it is then assumed that the
first two columns contain the individual and the time index and that
observations are ordered by individual and by time period,The pdata.frame function is then called internally,
which returns a pdata.frame which is a
data.frame with an attribute called index. This attribute
is a data.frame that contains the individual and the time
indexes.
It is also possible to use directly the pdata.frame
function and then to use the pdata.frame in the estimation
functions.
Package plm provides various functions for panel data
estimation, among them:
plm: estimation of the basic panel models and
instrumental variable panel models, i.e., between and
first-difference models and within and random effect models. Models are
estimated internally using the lm function on transformed
data,pvcm: estimation of models with variable
coefficients,pgmm: estimation of generalized method of moments
models,pggls: estimation of general feasible generalized least
squares models,pmg: estimators for mean groups (MG), demeaned MG (DMG)
and common correlated effects MG (CCEMG) for heterogeneous panel
models,pcce: estimators for common correlated effects mean
groups (CCEMG) and pooled (CCEP) for panel data with common
factors,pldv: panel estimators for limited dependent
variables.The interface of these functions is consistent with the
lm() function. Namely, their first two arguments are
formula and data (which should be a
data.frame and is mandatory). Three additional arguments
are common to these functions:
index: this argument enables the estimation functions
to identify the structure of the data, i.e., the individual and
the time period for each observation,effect: the kind of effects to include in the model,
i.e., individual effects, time effects or both3,model: the kind of model to be estimated, most of the
time a model with fixed effects or a model with random effects.The results of these four functions are stored in an object which
class has the same name of the function. They all inherit from class
panelmodel. A panelmodel object contains:
coefficients, residuals,
fitted.values, vcov, df.residual
and call and functions that extract these elements are
provided.
The diagnostic testing interface provides both formula
and panelmodel methods for most functions, with some
exceptions. The user may thus choose whether to employ results stored in
a previously estimated panelmodel object or to re-estimate
it for the sake of testing.
Although the first strategy is the most efficient one, diagnostic
testing on panel models mostly employs OLS residuals from pooling model
objects, whose estimation is computationally inexpensive. Therefore most
examples in the following are based on formula methods,
which are perhaps the cleanest for illustrative purposes.
The feasible GLS methods needed for efficient estimation of unobserved effects models have a simple closed-form solution: once the variance components have been estimated and hence the covariance matrix of errors \(\hat{V}\), model parameters can be estimated as
\[\begin{equation} (\#eq:naive) \hat{\beta}=(X^\top \hat{V}^{-1} X)^{-1} (X^\top \hat{V}^{-1} y) \end{equation}\]
Nevertheless, in practice plain computation of \(\hat{\beta}\) has long been an intractable
problem even for moderate-sized data sets because of the need to invert
the \(N\times N\) \(\hat{V}\) matrix. With the advances in
computer power, this is no more so, and it is possible to program the
“naive” estimator @ref(eq:naive) in R with standard matrix
algebra operators and have it working seamlessly for the standard
“guinea pigs”, e.g., the Grunfeld data. Estimation with a couple of
thousands of data points also becomes feasible on a modern machine,
although excruciatingly slow and definitely not suitable for everyday
econometric practice. Memory limits would also be very near because of
the storage needs related to the huge \(\hat{V}\) matrix. An established solution
exists for the random effects model which reduces the problem to an
ordinary least squares computation.
The estimation methods for the basic models in panel data econometrics, the pooled OLS, random effects and fixed effects (or within) models, can all be described inside the OLS estimation framework. In fact, while pooled OLS simply pools data, the standard way of estimating fixed effects models with, say, group (time) effects entails transforming the data by subtracting the average over time (group) to every variable, which is usually termed time-demeaning. In the random effects case, the various feasible GLS estimators which have been put forth to tackle the issue of serial correlation induced by the group-invariant random effect have been proven to be equivalent (as far as estimation of \(\beta\)s is concerned) to OLS on partially demeaned data, where partial demeaning is defined as:
\[\begin{equation} (\#eq:ldemmodel) y_{it} - \theta \bar{y}_i = ( X_{it} - \theta \bar{X}_{i} ) \beta + ( u_{it} - \theta \bar{u}_i ) \end{equation}\]
where \(\theta=1-[\sigma_u^2 / (\sigma_u^2
+ T \sigma_e^2)]^{1/2}\), \(\bar{y}\) and \(\bar{X}\) denote time means of \(y\) and \(X\), and the disturbance \(v_{it} - \theta \bar{v}_i\) is
homoskedastic and serially uncorrelated. Thus the feasible RE estimate
for \(\beta\) may be obtained
estimating \(\hat{\theta}\) and running
an OLS regression on the transformed data with lm(). The
other estimators can be computed as special cases: for \(\theta=1\) one gets the fixed effects
estimator, for \(\theta=0\) the pooled
OLS one.
Moreover, instrumental variable estimators of all these models may
also be obtained using several calls to lm().
For this reason the three above estimators have been grouped inside the same function.
On the output side, a number of diagnostics and a very general
coefficients’ covariance matrix estimator also benefits from this
framework, as they can be readily calculated applying the standard OLS
formulas to the demeaned data, which are contained inside
plm objects. This will be the subject of subsection inference in the panel model.
The covariance matrix of errors in general GLS models is too generic to fit the quasi-demeaning framework, so this method calls for a full-blown application of GLS as in @ref(eq:naive). On the other hand, this estimator relies heavily on \(n\)–asymptotics, making it theoretically most suitable for situations which forbid it computationally: e.g., “short” micropanels with thousands of individuals observed over few time periods.
R has general facilities for fast matrix computation
based on object orientation: particular types of matrices (symmetric,
sparse, dense etc.) are assigned the relevant class and the additional
information on structure is used in the computations, sometimes with
dramatic effects on performance (see Bates
(2004)) and packages Matrix (see Bates and Maechler (2016)) and
SparseM (see Koenker and Ng
(2016)). Some optimized linear algebra routines are available in
the R package bdsmatrix (see Therneau (2014)) which exploit the particular
block-diagonal and symmetric structure of \(\hat{V}\) making it possible to implement a
fast and reliable full-matrix solution to problems of any practically
relevant size.
The \(\hat{V}\) matrix is
constructed as an object of class bdsmatrix. The peculiar
properties of this matrix class are used for efficiently storing the
object in memory and then by ad-hoc versions of the solve
and crossprod methods, dramatically reducing computing
times and memory usage. The resulting matrix is then used “the naive
way” as in @ref(eq:naive) to compute \(\hat{\beta}\), resulting in speed
comparable to that of the demeaning solution.
General frameworks for restrictions and linear hypotheses testing are
available in the R environment4. These are based on
the Wald test, constructed as \(\hat{\beta}^\top \hat{V}^{-1}
\hat{\beta}\), where \(\hat{\beta}\) and \(\hat{V}\) are consistent estimates of \(\beta\) and \(V(\beta)\), The Wald test may be used for
zero-restriction (i.e., significance) testing and, more generally, for
linear hypotheses in the form \((R \hat{\beta}
- r)^\top [R \hat{V} R^\top ]^{-1} (R
\hat{\beta} - r)\)5. To be applicable, the test functions
require extractor methods for coefficients’ and covariance matrix
estimates to be defined for the model object to be tested. Model objects
in plm all have coef() and vcov()
methods and are therefore compatible with the above functions.
In the same framework, robust inference is accomplished substituting
(“plugging in”) a robust estimate of the coefficient covariance matrix
into the Wald statistic formula. In the panel context, the estimator of
choice is the White system estimator. This called for a flexible method
for computing robust coefficient covariance matrices à la White
for plm objects.
A general White system estimator for panel data is:
\[\begin{equation*} \hat{V}_R(\beta)=(X^\top X)^{-1} \sum_{i=1}^n{X_i^\top E_i X_i} (X^\top X)^{-1} \end{equation*}\]
where \(E_i\) is a function of the
residuals \(\hat{e}_{it}, \; t=1,
\dots T\) chosen according to the relevant heteroskedasticity and
correlation structure. Moreover, it turns out that the White covariance
matrix calculated on the demeaned model’s regressors and residuals (both
part of plm objects) is a consistent estimator of the
relevant model’s parameters’ covariance matrix, thus the method is
readily applicable to models estimated by random or fixed effects, first
difference or pooled OLS methods. Different pre-weighting schemes taken
from package sandwich (see Zeileis
(2004); Lumley and Zeileis (2015))
are also implemented to improve small-sample performance. Robust
estimators with any combination of covariance structures and weighting
schemes can be passed on to the testing functions.
The package is now illustrated by application to some well-known examples. It is loaded using
The four data sets used are EmplUK which was used by
M. Arellano and Bond (1991), the
Grunfeld data (Kleiber and Zeileis
2008) which is used in several econometric books, the
Produc data used by Munnell
(1990) and the Wages used by Cornwell and Rupert (1988).
data("EmplUK", package="plm")
data("Produc", package="plm")
data("Grunfeld", package="plm")
data("Wages", package="plm")As observed above, the current version of plm is capable
of working with a regular data.frame without any further
transformation, provided that the individual and time indexes are in the
first two columns, as in all the example data sets but
Wages. If this weren’t the case, an index
optional argument would have to be passed on to the estimating and
testing functions.
## firm year inv value capital
## 1 1 1935 317.6 3078.5 2.8
## 2 1 1936 391.8 4661.7 52.6
## 3 1 1937 410.6 5387.1 156.9
## 4 1 1938 257.7 2792.2 209.2
## 5 1 1939 330.8 4313.2 203.4
## 6 1 1940 461.2 4643.9 207.2## sector emp wage capital output
## 1-1977 7 5.041 13.1516 0.5894 95.7072
## 1-1978 7 5.600 12.3018 0.6318 97.3569
## 1-1979 7 5.015 12.8395 0.6771 99.6083
## 1-1980 7 4.715 13.8039 0.6171 100.5501
## 1-1981 7 4.093 14.2897 0.5076 99.5581
## 1-1982 7 3.166 14.8681 0.4229 98.6151## firm year
## 1 1 1977
## 2 1 1978
## 3 1 1979
## 4 1 1980
## 5 1 1981
## 6 1 1982Two further arguments are logical: drop.index = TRUE
drops the indexes from the data.frame and
row.names = TRUE computes “fancy” row names by pasting the
individual and the time indexes. While extracting a series from a
pdata.frame, a pseries is created, which is
the original series with the index attribute. This object has specific
methods, like summary and as.matrix. The
former indicates the total variation of the variable and the shares of
this variation due to the individual and the time dimensions. The latter
gives the matrix representation of the series, with, by default,
individuals as rows and times as columns.
## total sum of squares: 261539.4
## id time
## 0.980765381 0.009108488
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.104 1.181 2.287 7.892 7.020 108.562## 1976 1977 1978 1979 1980 1981 1982 1983 1984
## 1 NA 5.041 5.600 5.015 4.715 4.093 3.166 2.936 NA
## 2 NA 71.319 70.643 70.918 72.031 73.689 72.419 68.518 NA
## 3 NA 19.156 19.440 19.900 20.240 19.570 18.125 16.850 NA
## 4 NA 26.160 26.740 27.280 27.830 27.169 24.504 22.562 NA
## 5 86.677 87.100 87.000 90.400 89.200 82.700 73.700 NA NA
## 6 0.748 0.766 0.762 0.729 0.731 0.779 0.782 NA NAPanel data estimation requires to apply different transformations to raw series. If \(x\) is a series of length \(nT\) (where \(n\) is the number of individuals and \(T\) is the number of time periods), the transformed series \(\tilde{x}\) is obtained as \(\tilde{x}=Mx\) where \(M\) is a transformation matrix. Denoting \(j\) a vector of one of length \(T\) and \(I_n\) the identity matrix of dimension \(n\), we get:
Between and
between functions perform this operation, the first one
returning a vector of length \(nT\),
the second one a vector of length \(n\),Within function performs this operation.\(d=\left( \begin{array}{ccccccc} 1 & -1 & 0 & 0 & ... & 0 & 0 \\ 0 & 1 & -1 & 0 & ... & 0 & 0 \\ 0 & 0 & 1 & -1 & ... & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & ... & 1 & -1 \\ \end{array} \right)\)
is of dimension \((T-1,T)\).
Note that R’s diff() and lag()
functions don’t compute correctly these transformations for panel data
because they are unable to identify when there is a change in individual
in the data. Therefore, specific methods for pseries
objects have been written in order to handle correctly panel data. Note
that compared to the lag() method for ts
objects, the order of lags are indicated by a positive integer.
Moreover, 0 is a relevant value and a vector argument may be
provided:
## 0 1 2
## 1-1977 5.041 NA NA
## 1-1978 5.600 5.041 NA
## 1-1979 5.015 5.600 5.041
## 1-1980 4.715 5.015 5.600
## 1-1981 4.093 4.715 5.015
## 1-1982 3.166 4.093 4.715Further functions called Between, between
and Within are also provided to compute the between and the
within transformation. The between returns unique values,
whereas Between duplicates the values and returns a vector
which length is the number of observations.
## 1-1977 1-1978 1-1979 1-1980 1-1981 1-1982 1-1983
## NA 0.5590000 -0.5850000 -0.2999997 -0.6220003 -0.9270000 -0.2299998
## 2-1977 2-1978 2-1979
## NA -0.6760020 0.2750010## 1-1977 1-1978 1-1979 1-1980 1-1981 1-1982 1-1983 2-1977 2-1978 2-1979
## NA NA 5.041 5.600 5.015 4.715 4.093 NA NA 71.319## 1-1977 1-1978 1-1979 1-1980 1-1981 1-1982
## 0.6744285 1.2334285 0.6484285 0.3484288 -0.2735715 -1.2005715## 1 2 3 4
## 4.366571 71.362428 19.040143 26.035000## 1 1 1 1 1 1 1 2
## 4.366571 4.366571 4.366571 4.366571 4.366571 4.366571 4.366571 71.362428
## 2 2
## 71.362428 71.362428In some circumstances, standard formulas are not very
useful to describe a model, notably while using instrumental variable
like estimators: to deal with these situations, we use the
Formula package.
The Formula package provides a class which enables to
construct multi-part formula, each part being separated by a pipe sign
(|).
The two formulas below are identical:
In the second case, the . means the previous parts which
describes the covariates and this part is “updated”. This is
particularly interesting when there are a few external instruments.
Several models can be estimated with plm by filling the
model argument:
"within"), the default,"pooling"),"fd"),"between"),"random").The basic use of plm is to indicate the model formula,
the data and the model to be estimated. For example, the fixed effects
model and the random effects model are estimated using:
grun.fe <- plm(inv~value+capital, data = Grunfeld, model = "within")
grun.re <- plm(inv~value+capital, data = Grunfeld, model = "random")Methods to display a summary of the model estimation are available
via summary. For example, for a random model,
the summary method gives information about the variance of
the components of the errors and some test statistics. Random effects of
the estimated model can be extracted via ranef.
## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
##
## Call:
## plm(formula = inv ~ value + capital, data = Grunfeld, model = "random")
##
## Balanced Panel: n = 10, T = 20, N = 200
##
## Effects:
## var std.dev share
## idiosyncratic 2784.46 52.77 0.282
## individual 7089.80 84.20 0.718
## theta: 0.8612
##
## Residuals:
## Min. 1st Qu. Median 3rd Qu. Max.
## -177.6063 -19.7350 4.6851 19.5105 252.8743
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) -57.834415 28.898935 -2.0013 0.04536 *
## value 0.109781 0.010493 10.4627 < 0.0000000000000002 ***
## capital 0.308113 0.017180 17.9339 < 0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 2381400
## Residual Sum of Squares: 548900
## R-Squared: 0.7695
## Adj. R-Squared: 0.76716
## Chisq: 657.674 on 2 DF, p-value: < 0.000000000000000222## 1 2 3 4 5 6
## -9.5242955 157.8910235 -172.8958044 29.9119801 -54.6790089 34.3461316
## 7 8 9 10
## -7.8977584 0.6726376 -28.1393497 50.3144442The fixed effects of a fixed effects model may be extracted easily
using fixef. An argument type indicates how
fixed effects should be computed: in levels by
type = "level" (the default), in deviations from the
overall mean by type = "dmean" or in deviations from the
first individual by type = "dfirst".
## 1 2 3 4 5 6 7 8
## -11.5528 160.6498 -176.8279 30.9346 -55.8729 35.5826 -7.8095 1.1983
## 9 10
## -28.4783 52.1761The fixef function returns an object of class
fixef. A summary method is provided, which prints the
effects (in deviation from the overall intercept), their standard errors
and the test of equality to the overall intercept.
## Estimate Std. Error t-value Pr(>|t|)
## 1 -11.5528 49.7080 -0.2324 0.8164700
## 2 160.6498 24.9383 6.4419 0.0000000009627 ***
## 3 -176.8279 24.4316 -7.2377 0.0000000000113 ***
## 4 30.9346 14.0778 2.1974 0.0292129 *
## 5 -55.8729 14.1654 -3.9443 0.0001129 ***
## 6 35.5826 12.6687 2.8087 0.0054998 **
## 7 -7.8095 12.8430 -0.6081 0.5438694
## 8 1.1983 13.9931 0.0856 0.9318489
## 9 -28.4783 12.8919 -2.2090 0.0283821 *
## 10 52.1761 11.8269 4.4116 0.0000172511647 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In case of a two-ways fixed effect model, argument
effect is relevant in function fixef to
extract specific effect fixed effects with possible values
"individual" for individual fixed effects (default for
two-ways fixed effect models), "time" for time fixed
effects, and "twoways" for the sum of individual and time
fixed effects. Example to extract the time fixed effects from a two-ways
model:
grun.twfe <- plm(inv~value+capital, data=Grunfeld, model="within", effect="twoways")
fixef(grun.twfe, effect = "time")## 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944
## -86.90 -106.10 -127.59 -126.13 -156.37 -131.14 -105.70 -108.04 -129.88 -130.00
## 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954
## -142.58 -118.07 -126.29 -130.62 -160.40 -162.80 -149.38 -151.53 -154.62 -180.43As observed above, the random effect model is obtained as a linear estimation on quasi-demeaned data. The parameter of this transformation is obtained using preliminary estimations.
Four estimators of this parameter are available, depending on the
value of the argument random.method:
"swar": from Swamy and Arora
(1972), the default value,"walhus": from Wallace and
Hussain (1969),"amemiya": from T. Amemiya
(1971),"nerlove": from Nerlove
(1971)."ht": for Hausman-Taylor-type instrumental variable
(IV) estimation, discussed later, see Section Instrumental variable
estimator.For example, to use the amemiya estimator:
The estimation of the variance of the error components are performed
using the ercomp function, which has a method
and an effect argument, and can be used by itself:
## var std.dev share
## idiosyncratic 2644.13 51.42 0.256
## individual 7452.02 86.33 0.721
## time 243.78 15.61 0.024
## theta: 0.868 (id) 0.2787 (time) 0.2776 (total)The default behavior of plm is to introduce individual
effects. Using the effect argument, one may also
introduce:
effect = "time"),effect = "twoways").For example, to estimate a two-ways effect model for the
Grunfeld data:
grun.tways <- plm(inv~value+capital, data = Grunfeld, effect = "twoways",
model = "random", random.method = "amemiya")
summary(grun.tways)## Twoways effects Random Effect Model
## (Amemiya's transformation)
##
## Call:
## plm(formula = inv ~ value + capital, data = Grunfeld, effect = "twoways",
## model = "random", random.method = "amemiya")
##
## Balanced Panel: n = 10, T = 20, N = 200
##
## Effects:
## var std.dev share
## idiosyncratic 2644.13 51.42 0.256
## individual 7452.02 86.33 0.721
## time 243.78 15.61 0.024
## theta: 0.868 (id) 0.2787 (time) 0.2776 (total)
##
## Residuals:
## Min. 1st Qu. Median 3rd Qu. Max.
## -176.9062 -18.0431 3.2697 17.1719 234.1735
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) -63.767791 29.851537 -2.1362 0.03267 *
## value 0.111386 0.010909 10.2102 < 0.0000000000000002 ***
## capital 0.323321 0.018772 17.2232 < 0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 2066800
## Residual Sum of Squares: 518200
## R-Squared: 0.74927
## Adj. R-Squared: 0.74673
## Chisq: 588.717 on 2 DF, p-value: < 0.000000000000000222In the “effects” section of the printed summary of the result, the variance of the three elements of the error term and the three parameters used in the transformation are printed.
Estimations by plm support unbalanced panel models.
The following example is using data used by Harrison and Rubinfeld (1978) to estimate an hedonic housing prices function. It is reproduced in B. H. Baltagi and Chang (1994), table 2 (and in B. H. Baltagi (2005), pp. 172/4; B. H. Baltagi (2013), pp. 195/7 tables 9.1/3).
data("Hedonic", package = "plm")
Hed <- plm(mv~crim+zn+indus+chas+nox+rm+age+dis+rad+tax+ptratio+blacks+lstat,
data = Hedonic, model = "random", index = "townid")
summary(Hed)## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
##
## Call:
## plm(formula = mv ~ crim + zn + indus + chas + nox + rm + age +
## dis + rad + tax + ptratio + blacks + lstat, data = Hedonic,
## model = "random", index = "townid")
##
## Unbalanced Panel: n = 92, T = 1-30, N = 506
##
## Effects:
## var std.dev share
## idiosyncratic 0.01696 0.13025 0.562
## individual 0.01324 0.11505 0.438
## theta:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2505 0.5483 0.6284 0.6141 0.7147 0.7976
##
## Residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.62902 -0.06712 -0.00156 -0.00216 0.06858 0.54973
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) 9.685866695 0.197510264 49.0398 < 0.00000000000000022 ***
## crim -0.007411967 0.001047812 -7.0738 0.00000000000150795 ***
## zn 0.000078877 0.000650012 0.1213 0.9034166
## indus 0.001556340 0.004034911 0.3857 0.6997051
## chasyes -0.004424737 0.029211764 -0.1515 0.8796041
## nox -0.005842506 0.001245183 -4.6921 0.00000270431168602 ***
## rm 0.009055167 0.001188629 7.6182 0.00000000000002573 ***
## age -0.000857873 0.000467933 -1.8333 0.0667541 .
## dis -0.144418433 0.044093739 -3.2753 0.0010557 **
## rad 0.095983935 0.026610945 3.6069 0.0003098 ***
## tax -0.000377396 0.000176926 -2.1331 0.0329190 *
## ptratio -0.029475776 0.009069842 -3.2499 0.0011546 **
## blacks 0.562775469 0.101973789 5.5188 0.00000003412743874 ***
## lstat -0.291074917 0.023927306 -12.1650 < 0.00000000000000022 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 987.94
## Residual Sum of Squares: 8.9988
## R-Squared: 0.99091
## Adj. R-Squared: 0.99067
## Chisq: 1199.5 on 13 DF, p-value: < 0.000000000000000222Measures for the unbalancedness of a panel data set or the data used
in estimated models are provided by function
punbalancedness. It gives the measures \(\gamma\) and \(\nu\) from Ahrens
and Pincus (1981) where for both 1 represents balanced data and
the more unbalanced the data the lower the value.
## gamma nu
## 0.4715336 0.5188292All of the models presented above may be estimated using instrumental
variables. The instruments are specified at the end of the formula after
a | sign (pipe).
The instrumental variables estimator used is indicated with the
inst.method argument:
"bvk", from Balestra and
Varadharajan–Krishnakumar (1987), the default value,"baltagi", from B. H. Baltagi
(1981),"am", from Takeshi Amemiya and
MaCurdy (1986),"bms", from Trevor S. Breusch,
Mizon, and Schmidt (1989).An illustration is in the following example from B. H. Baltagi (2005), p. 120; B. H. Baltagi (2013), p. 137; B. H. Baltagi (2021), p. 165, table 7.3 (“G2SLS”).
data("Crime", package = "plm")
cr <- plm(lcrmrte ~ lprbarr + lpolpc + lprbconv + lprbpris + lavgsen +
ldensity + lwcon + lwtuc + lwtrd + lwfir + lwser + lwmfg + lwfed +
lwsta + lwloc + lpctymle + lpctmin + region + smsa + factor(year)
| . - lprbarr - lpolpc + ltaxpc + lmix,
data = Crime, model = "random")
summary(cr)## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
## Instrumental variable estimation
## (Balestra-Varadharajan-Krishnakumar's transformation)
##
## Call:
## plm(formula = lcrmrte ~ lprbarr + lpolpc + lprbconv + lprbpris +
## lavgsen + ldensity + lwcon + lwtuc + lwtrd + lwfir + lwser +
## lwmfg + lwfed + lwsta + lwloc + lpctymle + lpctmin + region +
## smsa + factor(year) | . - lprbarr - lpolpc + ltaxpc + lmix,
## data = Crime, model = "random")
##
## Balanced Panel: n = 90, T = 7, N = 630
##
## Effects:
## var std.dev share
## idiosyncratic 0.02227 0.14924 0.326
## individual 0.04604 0.21456 0.674
## theta: 0.7457
##
## Residuals:
## Min. 1st Qu. Median 3rd Qu. Max.
## -0.7485357 -0.0709883 0.0040648 0.0784455 0.4756273
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) -0.4538501 1.7029831 -0.2665 0.789852
## lprbarr -0.4141383 0.2210496 -1.8735 0.060998 .
## lpolpc 0.5049461 0.2277778 2.2168 0.026634 *
## lprbconv -0.3432506 0.1324648 -2.5913 0.009563 **
## lprbpris -0.1900467 0.0733392 -2.5913 0.009560 **
## lavgsen -0.0064389 0.0289407 -0.2225 0.823935
## ldensity 0.4343449 0.0711496 6.1047 0.00000000103 ***
## lwcon -0.0042958 0.0414226 -0.1037 0.917403
## lwtuc 0.0444589 0.0215448 2.0636 0.039060 *
## lwtrd -0.0085579 0.0419829 -0.2038 0.838476
## lwfir -0.0040305 0.0294569 -0.1368 0.891166
## lwser 0.0105602 0.0215823 0.4893 0.624630
## lwmfg -0.2018020 0.0839373 -2.4042 0.016208 *
## lwfed -0.2134579 0.2151046 -0.9923 0.321029
## lwsta -0.0601232 0.1203149 -0.4997 0.617275
## lwloc 0.1835363 0.1396775 1.3140 0.188846
## lpctymle -0.1458703 0.2268086 -0.6431 0.520131
## lpctmin 0.1948763 0.0459385 4.2421 0.00002214292 ***
## regionwest -0.2281821 0.1010260 -2.2586 0.023905 *
## regioncentral -0.1987703 0.0607475 -3.2721 0.001068 **
## smsayes -0.2595451 0.1499718 -1.7306 0.083518 .
## factor(year)82 0.0132147 0.0299924 0.4406 0.659500
## factor(year)83 -0.0847693 0.0320010 -2.6490 0.008074 **
## factor(year)84 -0.1062027 0.0387893 -2.7379 0.006183 **
## factor(year)85 -0.0977457 0.0511681 -1.9103 0.056097 .
## factor(year)86 -0.0719451 0.0605819 -1.1876 0.235004
## factor(year)87 -0.0396595 0.0758531 -0.5228 0.601081
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 30.169
## Residual Sum of Squares: 12.419
## R-Squared: 0.5923
## Adj. R-Squared: 0.57472
## Chisq: 542.478 on 26 DF, p-value: < 0.000000000000000222The Hausman-Taylor model (see Hausman and
Taylor (1981)) may be estimated with the plm6 function
by setting parameters random.method = "ht" and
inst.method = "baltagi" like in the example below. The
following replicates B. H. Baltagi (2005),
pp. 129/30; B. H. Baltagi (2013),
pp. 145/6, tables 7.4/5; B. H. Baltagi
(2021), pp. 174/5 tables 7.5/6:
ht <- plm(lwage ~ wks + south + smsa + married + exp + I(exp ^ 2) +
bluecol + ind + union + sex + black + ed |
bluecol + south + smsa + ind + sex + black |
wks + married + union + exp + I(exp ^ 2),
data = Wages, index = 595,
model = "random", random.method = "ht", inst.method = "baltagi")
summary(ht)## Oneway (individual) effect Random Effect Model
## (Hausman-Taylor's transformation)
## Instrumental variable estimation
## (Baltagi's transformation)
##
## Call:
## plm(formula = lwage ~ wks + south + smsa + married + exp + I(exp^2) +
## bluecol + ind + union + sex + black + ed | bluecol + south +
## smsa + ind + sex + black | wks + married + union + exp +
## I(exp^2), data = Wages, model = "random", random.method = "ht",
## inst.method = "baltagi", index = 595)
##
## Balanced Panel: n = 595, T = 7, N = 4165
##
## Effects:
## var std.dev share
## idiosyncratic 0.02304 0.15180 0.025
## individual 0.88699 0.94180 0.975
## theta: 0.9392
##
## Residuals:
## Min. 1st Qu. Median 3rd Qu. Max.
## -12.643736 -0.466002 0.043285 0.524739 13.340263
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) 2.912726279 0.283652215 10.2687 < 0.00000000000000022 ***
## wks 0.000837403 0.000599732 1.3963 0.16263
## southyes 0.007439837 0.031955005 0.2328 0.81590
## smsayes -0.041833367 0.018958129 -2.2066 0.02734 *
## marriedyes -0.029850749 0.018979963 -1.5728 0.11578
## exp 0.113132791 0.002470954 45.7851 < 0.00000000000000022 ***
## I(exp^2) -0.000418865 0.000054598 -7.6718 0.00000000000001696 ***
## bluecolyes -0.020704707 0.013780948 -1.5024 0.13299
## ind 0.013603930 0.015237366 0.8928 0.37196
## unionyes 0.032771447 0.014908437 2.1982 0.02794 *
## sexfemale -0.130923610 0.126658988 -1.0337 0.30129
## blackyes -0.285747871 0.155701854 -1.8352 0.06647 .
## ed 0.137943957 0.021248489 6.4919 0.00000000008473689 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 243.04
## Residual Sum of Squares: 4163.6
## R-Squared: 0.60945
## Adj. R-Squared: 0.60833
## Chisq: 6891.87 on 12 DF, p-value: < 0.000000000000000222The pvcm function enables the estimation of variable
coefficients models. Time or individual effects are introduced if
argument effect is fixed to "time" or
"individual" (the default value).
Coefficients are assumed to be fixed if model="within"
or random if model="random". In the first case, a separate
OLS model is estimated for each individual (or time period). In the
second case, Swamy’s random coefficients model (see Swamy (1970)) model is estimated. It is a
generalized least squares model which uses the results of separate OLS
models. Denoting \(\hat{\beta}_i\) the
vectors of coefficients obtained for each individual, we get:
\[\begin{equation*} \hat{\beta}=\left(\sum_{i=1}^n \left(\hat{\Delta}+\hat{\sigma}_i^2(X_i^\top X_i)^{-1}\right)^{-1}\right)\left(\hat{\Delta}+\hat{\sigma}_i^2(X_i^\top X_i)^{-1}\right)^{-1}\hat{\beta}_i \end{equation*}\]
where \(\hat{\sigma}_i^2\) is the unbiased estimator of the variance of the errors for individual \(i\) obtained from the preliminary estimation and:
\[\begin{equation*} \hat{\Delta}=\frac{1}{n-1}\sum_{i=1}^n\left(\hat{\beta}_i-\frac{1}{n}\sum_{i=1}^n\hat{\beta}_i\right) \left(\hat{\beta}_i-\frac{1}{n}\sum_{i=1}^n\hat{\beta}_i\right)^\top -\frac{1}{n}\sum_{i=1}^n\hat{\sigma}_i^2(X_i^\top X_i)^{-1} \end{equation*}\]
If this matrix is not positive-definite, the second term is dropped.
With the Grunfeld data, we get:
grun.varw <- pvcm(inv~value+capital, data=Grunfeld, model="within")
grun.varr <- pvcm(inv~value+capital, data=Grunfeld, model="random")
summary(grun.varr)## Oneway (individual) effect Random coefficients model
##
## Call:
## pvcm(formula = inv ~ value + capital, data = Grunfeld, model = "random")
##
## Balanced Panel: n = 10, T = 20, N = 200
##
## Residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -211.486 -32.321 -4.283 9.048 12.714 579.216
##
## Estimated mean of the coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) -9.629285 17.035040 -0.5653 0.5718946
## value 0.084587 0.019956 4.2387 0.00002248 ***
## capital 0.199418 0.052653 3.7874 0.0001522 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Estimated variance of the coefficients:
## (Intercept) value capital
## (Intercept) 2344.24402 -0.6852340 -4.0276612
## value -0.68523 0.0031182 -0.0011847
## capital -4.02766 -0.0011847 0.0244824
##
## Total Sum of Squares: 474010000
## Residual Sum of Squares: 2194300
## Multiple R-Squared: 0.99537
## Chisq: 38.8364 on 2 DF, p-value: 0.0000000036878
## Test for parameter homogeneity: Chisq = 901.43 on 27 DF, p-value: < 0.000000000000000222A \(\chi^2\) test of homogeneous parameters of the random coefficients model is printed in the last line of the summary output.
The unconditioned best linear unbiased single estimations for the
coefficient of the single panel-effect specific regressions associated
with the random coefficients model can be extracted from the returned
object, for the example in grun.varr$single.coefs, while
single variance-covariance matrices and standard errors are available in
grun.varr$single.vcov and
grunvarr$single.std.error, respectively.
The generalized method of moments is mainly used in panel data econometrics to estimate dynamic models (M. Arellano and Bond 1991; Holtz–Eakin, Newey, and Rosen 1988).
\[\begin{equation*} y_{it}=\rho y_{it-1}+\beta^\top x_{it}+\mu_i+\epsilon_{it} \end{equation*}\]
The model is first differenced to get rid of the individual effect:
\[\begin{equation*} \Delta y_{it}=\rho \Delta y_{it-1}+\beta^\top \Delta x_{it}+\Delta \epsilon_{it} \end{equation*}\]
Least squares are inconsistent because \(\Delta \epsilon_{it}\) is correlated with \(\Delta y_{it-1}\). \(y_{it-2}\) is a valid, but weak instrument (see Anderson and Hsiao (1981)). The GMM estimator uses the fact that the number of valid instruments is growing with \(t\):
For individual \(i\), the matrix of instruments is then:
\[\begin{equation*} W_i=\left( \begin{array}{ccccccccccccc} y_1 & 0 & 0 & 0 & 0 & 0 & ... & 0 & 0 & 0 & 0 & x_{i3} \\ 0 & y_1 & y_2 & 0 & 0 & 0 & ... & 0 & 0 & 0 & 0 & x_{i4} \\ 0 & 0 & 0 & y_1 & y_2 & y_3 & ... & 0 & 0 & 0 & 0 & x_{i5} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & ... & ... & ... & y_1 & y_2 & ... & y_{t-2} & x_{iT-2} &\\ \end{array} \right) \end{equation*}\]
The moment conditions are: \(\sum_{i=1}^n W_i^\top e_i(\beta)\) where \(e_i(\beta)\) is the vector of residuals for individual \(i\). The GMM estimator minimizes:
\[\begin{equation*} \left(\sum_{i=1}^n e_i(\beta)^\top W_i\right) A \left(\sum_{i=1}^n W_i^\top e_i(\beta)\right) \end{equation*}\]
where \(A\) is the weighting matrix of the moments.
One-step estimators are computed using a known weighting matrix. For the model in first differences, one uses:
\[\begin{equation*} A^{(1)}=\left(\sum_{i=1}^n W_i^\top H^{(1)}W_i\right)^{-1} \end{equation*}\]
with:
\[\begin{equation*} H^{(1)}=d^\top d=\left( \begin{array}{ccccc} 2 & -1 & 0 & ... & 0\\ -1 & 2 & -1 & ... & 0\\ 0 & -1 & 2 & ... & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & -1 & 2\\ \end{array} \right) \end{equation*}\]
Two-steps estimators are obtained using \(H^{(2)}_i=\sum_{i=1}^n e^{(1)}_i e^{(1)\top }_i\) where \(e_i^{(1)}\) are the residuals of the one step estimate.
Blundell and Bond (1998) show that with weak hypothesis on the data generating process, supplementary moment conditions exist for the equation in level:
\[ y_{it} = \gamma y_{it-1}+\mu_i+\eta_{it} \]
More precisely, they show that \(\Delta y_{it-2}=y_{it-2}-y_{it-3}\) is a valid instrument. The estimator is obtained using the residual vector in difference and in level:
\[ e^+_i=(\Delta e_i, e_i) \]
and the matrix of augmented moments:
\[ Z_i^+=\left( \begin{array}{ccccc} Z_i & 0 & 0 & ... & 0 \\ 0 & \Delta y_{i2} & 0 & ... & 0 \\ 0 & 0 & \Delta y_{i3} & ... & 0 \\ 0 & 0 & 0 & ... & \Delta y_{iT-1} \end{array} \right) \]
The moment conditions are then
\[\begin{eqnarray*} \left(\sum_{i=1}^n Z_i^{+\top} \left(\begin{array}{c}\bar{e}_i(\beta)\\ e_i(\beta)\end{array}\right)\right)^\top = \left(\sum_{i=1}^n y_{i1} \bar{e}_{i3},\sum_{i=1}^n y_{i1}\bar{e}_{i4},\sum_{i=1}^n y_{i2}\bar{e}_{i4}, ..., \right.\\ \left. \sum_{i=1}^n y_{i1} \bar{e}_{iT}, \sum_{i=1}^n y_{i2} \bar{e}_{iT}, ...,\sum_{i=1}^n y_{iT-2} \bar{e}_{iT}, \sum_{i=1}^n \sum_{t=3}^T x_{it} \bar{e}_{it}\right.\\ \left.\sum_{i=1}^n e_{i3} \Delta y_{i2}, \sum_{i=1}^n e_{i4} \Delta y_{i3}, ... , \sum_{i=1}^n e_{iT} \Delta y_{iT-1} \right)^\top \end{eqnarray*}\]
The GMM estimator is provided by the pgmm function. By
using a multi-part formula, the variables of the model and the lag
structure are described.
In a GMM estimation, there are “normal instruments” and “GMM instruments”. GMM instruments are indicated in the second part of the formula. By default, all the variables of the model that are not used as GMM instruments are used as normal instruments, with the same lag structure; “normal” instruments may also be indicated in the third part of the formula.
The effect argument is either "individual"
or "twoways" (the default). In the first case, the model is
the model is estimated in first differences to get rid of the
individuals effects. In the last case, the model is estimated in first
differences and time dummies are included.
The model argument specifies whether a one-step or a
two-steps model is requested ("onestep" or
"twosteps").
The following example is from M. Arellano and Bond (1991), table 4, column b. Employment is explained by past values of employment (two lags), current and first lag of wages and output and current value of capital.
emp.gmm <- pgmm(log(emp)~lag(log(emp), 1:2)+lag(log(wage), 0:1)+log(capital)+
lag(log(output), 0:1) | lag(log(emp), 2:99),
data = EmplUK, effect = "twoways", model = "twosteps")
summary(emp.gmm, robust = FALSE)## Twoways effects Two-steps model Difference GMM
##
## Call:
## pgmm(formula = log(emp) ~ lag(log(emp), 1:2) + lag(log(wage),
## 0:1) + log(capital) + lag(log(output), 0:1) | lag(log(emp),
## 2:99), data = EmplUK, effect = "twoways", model = "twosteps")
##
## Unbalanced Panel: n = 140, T = 7-9, N = 1031
##
## Number of Observations Used: 611
## Residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.6190677 -0.0255683 0.0000000 -0.0001339 0.0332013 0.6410272
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## lag(log(emp), 1:2)1 0.474151 0.085303 5.5584 0.0000000272221654 ***
## lag(log(emp), 1:2)2 -0.052967 0.027284 -1.9413 0.0522200 .
## lag(log(wage), 0:1)0 -0.513205 0.049345 -10.4003 < 0.00000000000000022 ***
## lag(log(wage), 0:1)1 0.224640 0.080063 2.8058 0.0050192 **
## log(capital) 0.292723 0.039463 7.4177 0.0000000000001191 ***
## lag(log(output), 0:1)0 0.609775 0.108524 5.6188 0.0000000192269973 ***
## lag(log(output), 0:1)1 -0.446373 0.124815 -3.5763 0.0003485 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Sargan test: chisq(25) = 30.11247 (p-value = 0.22011)
## Autocorrelation test (1): normal = -2.427829 (p-value = 0.01519)
## Autocorrelation test (2): normal = -0.3325401 (p-value = 0.73948)
## Wald test for coefficients: chisq(7) = 371.9877 (p-value = < 0.000000000000000222)
## Wald test for time dummies: chisq(6) = 26.9045 (p-value = 0.0001509)The following example is from Blundell and
Bond (1998). The “sys” estimator is obtained using
transformation = "ld" for level and difference. The
robust argument of the summary method enables
to use the robust covariance matrix, in case of a two-steps model with
the small-sample correction proposed by Windmeijer (2005). For all pgmm models,
robust = TRUE is the default (but set in this example
explicitly).
z2 <- pgmm(log(emp) ~ lag(log(emp), 1)+ lag(log(wage), 0:1) +
lag(log(capital), 0:1) | lag(log(emp), 2:99) +
lag(log(wage), 2:99) + lag(log(capital), 2:99),
data = EmplUK, effect = "twoways", model = "onestep",
transformation = "ld")
summary(z2, robust = TRUE)## Twoways effects One-step model System GMM
##
## Call:
## pgmm(formula = log(emp) ~ lag(log(emp), 1) + lag(log(wage), 0:1) +
## lag(log(capital), 0:1) | lag(log(emp), 2:99) + lag(log(wage),
## 2:99) + lag(log(capital), 2:99), data = EmplUK, effect = "twoways",
## model = "onestep", transformation = "ld")
##
## Unbalanced Panel: n = 140, T = 7-9, N = 1031
##
## Number of Observations Used: 1642
## Residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.7530341 -0.0369030 0.0000000 0.0002882 0.0466069 0.6001503
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## lag(log(emp), 1) 0.935605 0.026295 35.5810 < 0.00000000000000022 ***
## lag(log(wage), 0:1)0 -0.630976 0.118054 -5.3448 0.0000000905012861 ***
## lag(log(wage), 0:1)1 0.482620 0.136887 3.5257 0.0004224 ***
## lag(log(capital), 0:1)0 0.483930 0.053867 8.9838 < 0.00000000000000022 ***
## lag(log(capital), 0:1)1 -0.424393 0.058479 -7.2572 0.0000000000003952 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Sargan test: chisq(100) = 118.763 (p-value = 0.097096)
## Autocorrelation test (1): normal = -4.808434 (p-value = 0.0000015212)
## Autocorrelation test (2): normal = -0.2800133 (p-value = 0.77947)
## Wald test for coefficients: chisq(5) = 11174.82 (p-value = < 0.000000000000000222)
## Wald test for time dummies: chisq(7) = 14.71138 (p-value = 0.039882)General FGLS estimators are based on a two-step estimation process: first an OLS model is estimated, then its residuals \(\hat{u}_{it}\) are used to estimate an error covariance matrix more general than the random effects one for use in a feasible-GLS analysis. Formally, the estimated error covariance matrix is \(\hat{V}=I_n \otimes \hat{\Omega}\), with \[\hat{\Omega}=\sum_{i=1}^n \frac{\hat{u}_{it} \hat{u}_{it}^\top }{n} \] (see Wooldridge (2002) 10.4.3 and 10.5.5).
This framework allows the error covariance structure inside every
group (if effect = "individual") of observations to be
fully unrestricted and is therefore robust against any type of
intragroup heteroskedasticity and serial correlation. This structure, by
converse, is assumed identical across groups and thus general FGLS is
inefficient under groupwise heteroskedasticity. Cross-sectional
correlation is excluded a priori.
Moreover, the number of variance parameters to be estimated with \(N=n\times T\) data points is \(T(T+1)/2\), which makes these estimators particularly suited for situations where \(n>>T\), as e.g., in labour or household income surveys, while problematic for “long” panels, where \(\hat{V}\) tends to become singular and standard errors therefore become biased downwards.
In a pooled time series context (effect = "time"),
symmetrically, this estimator is able to account for arbitrary
cross-sectional correlation, provided that the latter is time-invariant
(see Greene (2003) 13.9.1–2, pp. 321–2).
In this case serial correlation has to be assumed away and the estimator
is consistent with respect to the time dimension, keeping \(n\) fixed.
The function pggls estimates general FGLS models, with
either fixed or “random” effects7.
The “random effect” general FGLS is estimated by:
## Oneway (individual) effect General FGLS model
##
## Call:
## pggls(formula = log(emp) ~ log(wage) + log(capital), data = EmplUK,
## model = "pooling")
##
## Unbalanced Panel: n = 140, T = 7-9, N = 1031
##
## Residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1.80696 -0.36552 0.06181 0.03230 0.44279 1.58719
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) 2.023480 0.158468 12.7690 < 0.00000000000000022 ***
## log(wage) -0.232329 0.048001 -4.8401 0.000001298 ***
## log(capital) 0.610484 0.017434 35.0174 < 0.00000000000000022 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Total Sum of Squares: 1853.6
## Residual Sum of Squares: 402.55
## Multiple R-squared: 0.78283The fixed effects pggls (see Wooldridge (2002), p. 276) is based on the
estimation of a within model in the first step; the rest follows as
above. It is estimated by:
The pggls function is similar to plm in
many respects. An exception is that the estimate of the group covariance
matrix of errors (zz$sigma, a matrix, not shown) is
reported in the model objects instead of the usual estimated variances
of the two error components.
As sketched in Section linear panel model, specification testing in panel models involves essentially testing for poolability, for individual or time unobserved effects and for correlation between these latter and the regressors (Hausman-type tests). As for the other usual diagnostic checks, we provide a suite of serial correlation tests, while not touching on the issue of heteroskedasticity testing. Instead, we provide heteroskedasticity-robust covariance estimators, to be described in subsection robust covariance matrix estimation.
pooltest tests the hypothesis that the same coefficients
apply to each individual. It is a standard F test, based on the
comparison of a model obtained for the full sample and a model based on
the estimation of an equation for each individual. The first argument of
pooltest is a plm object. The second argument
is a pvcm object obtained with model="within".
If the first argument is a pooling model, the test applies to all the
coefficients (including the intercepts), if it is a within model,
different intercepts are assumed.
To test the hypothesis that all the coefficients in the
Grunfeld example, excluding the intercepts, are equal, we
use :
znp <- pvcm(inv ~ value + capital, data = Grunfeld, model = "within")
zplm <- plm(inv ~ value + capital, data = Grunfeld, model = "within")
pooltest(zplm, znp)##
## F statistic
##
## data: inv ~ value + capital
## F = 5.7805, df1 = 18, df2 = 170, p-value = 0.0000000001219
## alternative hypothesis: unstabilityThe same test can be computed using a formula as first argument of
the pooltest function:
plmtest implements Lagrange multiplier tests of
individual or/and time effects based on the results of the pooling
model. Its main argument is a plm object (the result of a
pooling model) or a formula.
Two additional arguments can be added to indicate the kind of test to
be computed. The argument type is one of:
"honda": Honda (1985),
the default value,"bp": T. S. Breusch and Pagan
(1980),"kw": King and Wu
(1997)8,"ghm": Gourieroux, Holly, and
Monfort (1982).The effects tested are indicated with the effect
argument (one of "individual", "time", or
"twoways"). The test statistics implemented are also
suitable for unbalanced panels.9
To test the presence of individual and time effects in the
Grunfeld example, using the Gourieroux, Holly, and Monfort (1982) test, we
use:
g <- plm(inv ~ value + capital, data=Grunfeld, model="pooling")
plmtest(g, effect="twoways", type="ghm")##
## Lagrange Multiplier Test - two-ways effects (Gourieroux, Holly and
## Monfort)
##
## data: inv ~ value + capital
## chibarsq = 798.16, df0 = 0.00, df1 = 1.00, df2 = 2.00, w0 = 0.25, w1 =
## 0.50, w2 = 0.25, p-value < 0.00000000000000022
## alternative hypothesis: significant effectsor
pFtest computes F tests of effects based on the
comparison of the within and the pooling model. Its main arguments are
either two plm objects (a pooling and a within model) or a
formula.
gw <- plm(inv ~ value + capital, data=Grunfeld, effect="twoways", model="within")
gp <- plm(inv ~ value + capital, data=Grunfeld, model="pooling")
pFtest(gw, gp)##
## F test for twoways effects
##
## data: inv ~ value + capital
## F = 17.403, df1 = 28, df2 = 169, p-value < 0.00000000000000022
## alternative hypothesis: significant effectsphtest computes the Hausman test (at times also called
Durbin–Wu–Hausman test) which is based on the comparison of two sets of
estimates (see Hausman (1978)).
Its main arguments are two panelmodel objects or a
formula. A classical application of the Hausman test for panel data is
to compare the fixed and the random effects models:
gw <- plm(inv ~ value + capital, data = Grunfeld, model="within")
gr <- plm(inv ~ value + capital, data = Grunfeld, model="random")
phtest(gw, gr)##
## Hausman Test
##
## data: inv ~ value + capital
## chisq = 2.3304, df = 2, p-value = 0.3119
## alternative hypothesis: one model is inconsistentThe command also supports the auxiliary-regression-based version as
described in, e.g., Wooldridge (2010)
Sec.10.7.3 by using the formula interface and setting argument
test = "aux". This auxiliary-regression-based version can
be robustified by specifying a robust covariance estimator as a function
through the argument vcov:
##
## Regression-based Hausman test, vcov: vcovHC
##
## data: inv ~ value + capital
## chisq = 8.2998, df = 2, p-value = 0.01577
## alternative hypothesis: one model is inconsistentA model with individual effects has composite errors that are serially correlated by definition. The presence of the time-invariant error component10 gives rise to serial correlation which does not die out over time, thus standard tests applied on pooled data always end up rejecting the null of spherical residuals11. There may also be serial correlation of the “usual” kind in the idiosyncratic error terms, e.g., as an AR(1) process. By “testing for serial correlation” we mean testing for this latter kind of dependence.
For these reasons, the subjects of testing for individual error components and for serially correlated idiosyncratic errors are closely related. In particular, simple (marginal) tests for one direction of departure from the hypothesis of spherical errors usually have power against the other one: in case it is present, they are substantially biased towards rejection. Joint tests are correctly sized and have power against both directions, but usually do not give any information about which one actually caused rejection. Conditional tests for serial correlation that take into account the error components are correctly sized under presence of both departures from sphericity and have power only against the alternative of interest. While most powerful if correctly specified, the latter, based on the likelihood framework, are crucially dependent on normality and homoskedasticity of the errors.
In plm we provide a number of joint, marginal and
conditional ML-based tests, plus some semiparametric alternatives which
are robust vs. heteroskedasticity and free from distributional
assumptions.
The unobserved effects test à la Wooldridge (see Wooldridge (2002) 10.4.4), is a semiparametric test for the null hypothesis that \(\sigma^2_{\mu}=0\), i.e. that there are no unobserved effects in the residuals. Given that under the null the covariance matrix of the residuals for each individual is diagonal, the test statistic is based on the average of elements in the upper (or lower) triangle of its estimate, diagonal excluded: \(n^{-1/2} \sum_{i=1}^n \sum_{t=1}^{T-1} \sum_{s=t+1}^T \hat{u}_{it} \hat{u}_{is}\) (where \(\hat{u}\) are the pooled OLS residuals), which must be “statistically close” to zero under the null, scaled by its standard deviation: \[W=\frac{ \sum_{i=1}^n \sum_{t=1}^{T-1} \sum_{s=t+1}^T \hat{u}_{it} \hat{u}_{is} }{ [{ \sum_{i=1}^n ( \sum_{t=1}^{T-1} \sum_{s=t+1}^T \hat{u}_{it} \hat{u}_{is} } )^2 ]^{1/2} }\]
This test is (\(n\)-) asymptotically distributed as a standard normal regardless of the distribution of the errors. It does also not rely on homoskedasticity.
It has power both against the standard random effects specification, where the unobserved effects are constant within every group, as well as against any kind of serial correlation. As such, it “nests” both random effects and serial correlation tests, trading some power against more specific alternatives in exchange for robustness.
While not rejecting the null favours the use of pooled OLS, rejection may follow from serial correlation of different kinds, and in particular, quoting Wooldridge (2002), “should not be interpreted as implying that the random effects error structure must be true”.
Below, the test is applied to the data and model in Munnell (1990):
##
## Wooldridge's test for unobserved individual effects
##
## data: formula
## z = 3.9383, p-value = 0.00008207
## alternative hypothesis: unobserved effectThe presence of random effects may affect tests for residual serial
correlation, and the opposite. One solution is to use a joint test,
which has power against both alternatives. A joint LM test for random
effects and serial correlation under normality and
homoskedasticity of the idiosyncratic errors has been derived by B. Baltagi and Li (1991) and B. Baltagi and Li (1995) and is implemented as
an option in pbsytest:
##
## Baltagi and Li AR-RE joint test
##
## data: formula
## chisq = 4187.6, df = 2, p-value < 0.00000000000000022
## alternative hypothesis: AR(1) errors or random effectsRejection of the joint test, though, gives no information on the direction of the departure from the null hypothesis, i.e.: is rejection due to the presence of serial correlation, of random effects or of both?
Bera, Sosa–Escudero, and Yoon (2001)
(hereafter BSY) derive locally robust tests both for individual random
effects and for first-order serial correlation in residuals as
“corrected” versions of the standard LM test (see plmtest).
While still dependent on normality and homoskedasticity, these are
robust to local departures from the hypotheses of,
respectively, no serial correlation or no random effects. The authors
observe that, although suboptimal, these tests may help detecting the
right direction of the departure from the null, thus complementing the
use of joint tests. Moreover, being based on pooled OLS residuals, the
BSY tests are computationally far less demanding than likelihood-based
conditional tests.
On the other hand, the statistical properties of these “locally
corrected” tests are inferior to those of the non-corrected counterparts
when the latter are correctly specified. If there is no serial
correlation, then the optimal test for random effects is the
likelihood-based LM test of Breusch and Godfrey (with refinements by
Honda, see plmtest), while if there are no random effects
the optimal test for serial correlation is, again, Breusch-Godfrey’s
test12.
If the presence of a random effect is taken for granted, then the
optimal test for serial correlation is the likelihood-based conditional
LM test of B. Baltagi and Li (1995) (see
pbltest).
The serial correlation version is the default:
##
## Bera, Sosa-Escudero and Yoon locally robust test
##
## data: formula
## chisq = 52.636, df = 1, p-value = 0.0000000000004015
## alternative hypothesis: AR(1) errors sub random effectsThe BSY test for random effects is implemented in the one-sided version13, which takes heed that the variance of the random effect must be non-negative:
##
## Bera, Sosa-Escudero and Yoon locally robust test (one-sided)
##
## data: formula
## z = 57.914, p-value < 0.00000000000000022
## alternative hypothesis: random effects sub AR(1) errorsB. Baltagi and Li (1991) and B. Baltagi and Li (1995) derive a Lagrange
multiplier test for serial correlation in the idiosyncratic component of
the errors under (normal, heteroskedastic) random effects. Under the
null of serially uncorrelated errors, the test turns out to be identical
for both the alternative of AR(1) and MA(1) processes. One- and
two-sided versions are provided, the one-sided having power against
positive serial correlation only. The two-sided is the default, while
for the other one must specify the alternative option to
"onesided":
##
## Baltagi and Li one-sided LM test
##
## data: log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp
## z = 21.69, p-value < 0.00000000000000022
## alternative hypothesis: AR(1)/MA(1) errors in RE panel modelAs usual, the LM test statistic is based on residuals from the
maximum likelihood estimate of the restricted model (random effects with
serially uncorrelated errors). In this case, though, the restricted
model cannot be estimated by OLS anymore, therefore the testing function
depends on lme() in the nlme package for
estimation of a random effects model by maximum likelihood. For this
reason, the test is applicable only to balanced panels.
No test has been implemented to date for the symmetric hypothesis of
no random effects in a model with errors following an AR(1) process, but
an asymptotically equivalent likelihood ratio test is available in the
nlme package (see Section plm versus nlme
and lme4).
A general testing procedure for serial correlation in fixed effects (FE), random effects (RE) and pooled-OLS panel models alike can be based on considerations in Wooldridge (2002), 10.7.2.
Recall that plm model objects are the result of OLS
estimation performed on “demeaned” data, where, in the case of
individual effects (else symmetric), this means time-demeaning for the
FE (within) model, quasi-time-demeaning for the RE
(random) model and original data, with no demeaning at all,
for the pooled OLS (pooling) model (see Section software approach).
For the random effects model, Wooldridge (2002) observes that under the null of homoskedasticity and no serial correlation in the idiosyncratic errors, the residuals from the quasi-demeaned regression must be spherical as well. Else, as the individual effects are wiped out in the demeaning, any remaining serial correlation must be due to the idiosyncratic component. Hence, a simple way of testing for serial correlation is to apply a standard serial correlation test to the quasi-demeaned model. The same applies in a pooled model, w.r.t. the original data.
The FE case needs some qualification. It is well-known that if the
original model’s errors are uncorrelated then FE residuals are
negatively serially correlated, with \(cor(\hat{u}_{it},
\hat{u}_{is})=-1/(T-1)\) for each \(t,s\) (see Wooldridge (2002) 10.5.4). This correlation
clearly dies out as T increases, so this kind of AR test is applicable
to within model objects only for T “sufficiently large”14. On the
converse, in short panels the test gets severely biased towards
rejection (or, as the induced correlation is negative, towards
acceptance in the case of the one-sided DW test with
alternative="greater"). See below for a serial correlation
test applicable to “short” FE panel models.
plm objects retain the “demeaned” data, so the procedure
is straightforward for them. The wrapper functions pbgtest
and pdwtest re-estimate the relevant quasi-demeaned model
by OLS and apply, respectively, standard Breusch-Godfrey and
Durbin-Watson tests from package lmtest:
##
## Breusch-Godfrey/Wooldridge test for serial correlation in panel models
##
## data: inv ~ value + capital
## chisq = 42.587, df = 2, p-value = 0.0000000005655
## alternative hypothesis: serial correlation in idiosyncratic errorsThe tests share the features of their OLS counterparts, in particular
the pbgtest allows testing for higher-order serial
correlation, which might turn useful, e.g., on quarterly data.
Analogously, from the point of view of software, as the functions are
simple wrappers towards bgtest and dwtest, all
arguments from the latter two apply and may be passed on through the
ellipsis (the ... argument).
For the reasons reported above, under the null of no serial correlation in the errors, the residuals of a FE model must be negatively serially correlated, with \(cor(\hat{\epsilon}_{it}, \hat{\epsilon}_{is})=-1/(T-1)\) for each \(t,s\). Wooldridge suggests basing a test for this null hypothesis on a pooled regression of FE residuals on themselves, lagged one period: \[\hat{\epsilon}_{i,t}=\alpha + \delta \hat{\epsilon}_{i,t-1} + \eta_{i,t}\] Rejecting the restriction \(\delta = -1/(T-1)\) makes us conclude against the original null of no serial correlation.
The building blocks available in plm make it easy to
construct a function carrying out this procedure: first the FE model is
estimated and the residuals retrieved, then they are lagged and a
pooling AR(1) model is estimated. The test statistic is
obtained by applying the above restriction on \(\delta\) and supplying a
heteroskedasticity- and autocorrelation-consistent covariance matrix
(vcovHC with the appropriate options, in particular
method="arellano")15.
##
## Wooldridge's test for serial correlation in FE panels
##
## data: plm.model
## F = 312.3, df1 = 1, df2 = 889, p-value < 0.00000000000000022
## alternative hypothesis: serial correlationThe test is applicable to any FE panel model, and in particular to “short” panels with small \(T\) and large \(n\).
In the context of the first difference model, Wooldridge (2002), 10.6.3 proposes a serial
correlation test that can also be seen as a specification test to choose
the most efficient estimator between fixed effects (within)
and first difference (fd).
The starting point is the observation that if the idiosyncratic errors of the original model \(u_{it}\) are uncorrelated, the errors of the (first) differenced model16 \(e_{it} \equiv u_{it}-u_{i,t-1}\) will be correlated, with \(cor(e_{it}, e_{i,t-1})=-0.5\), while any time-invariant effect, “fixed” or “random”, is wiped out in the differencing. So a serial correlation test for models with individual effects of any kind can be based on estimating the model \[\hat{u}_{i,t}= \delta \hat{u}_{i,t-1} + \eta_{i,t}\] and testing the restriction \(\delta = -0.5\), corresponding to the null of no serial correlation. Drukker (2003) provides Monte Carlo evidence of the good empirical properties of the test.
On the other extreme (see Wooldridge
(2002) 10.6.1), if the differenced errors \(e_{it}\) are uncorrelated, as by definition
\(u_{it} = u_{i,t-1}
+ e_{it}\), then \(u_{it}\) is a
random walk. In this latter case, the most efficient estimator is the
first difference (fd) one; in the former case, it is the
fixed effects one (within).
The function pwfdtest allows testing either hypothesis:
the default behaviour h0="fd" is to test for serial
correlation in first-differenced errors:
##
## Wooldridge's first-difference test for serial correlation in panels
##
## data: plm.model
## F = 1.5251, df1 = 1, df2 = 749, p-value = 0.2172
## alternative hypothesis: serial correlation in differenced errorswhile specifying h0="fe" the null hypothesis becomes no
serial correlation in original errors, which is similar to the
pwartest.
##
## Wooldridge's first-difference test for serial correlation in panels
##
## data: plm.model
## F = 131.55, df1 = 1, df2 = 749, p-value < 0.00000000000000022
## alternative hypothesis: serial correlation in original errorsNot rejecting one of the two is evidence in favour of using the
estimator corresponding to h0. Should the truth lie in the
middle (both rejected), whichever estimator is chosen will have serially
correlated errors: therefore it will be advisable to use the
autocorrelation-robust covariance estimators from the subsection robust covariance matrix estimation in inference.
Next to the more familiar issue of serial correlation, over the last years a growing body of literature has been dealing with cross-sectional dependence (henceforth: XSD) in panels, which can arise, e.g., if individuals respond to common shocks (as in the literature on factor models) or if spatial diffusion processes are present, relating individuals in a way depending on a measure of distance (spatial models).
The subject is huge, and here we touch only some general aspects of
misspecification testing and valid inference. If XSD is present, the
consequence is, at a minimum, inefficiency of the usual estimators and
invalid inference when using the standard covariance matrix17. The
plan is to have in plm both misspecification tests to
detect XSD and robust covariance matrices to perform valid inference in
its presence, like in the serial dependence case. For now, though, only
misspecification tests are included.
The function pcdtest implements a family of XSD tests
which can be applied in different settings, ranging from those where
\(T\) grows large with \(n\) fixed to “short” panels with a big
\(n\) dimension and a few time periods.
All are based on (transformations of–) the product-moment correlation
coefficient of a model’s residuals, defined as
\[ \hat{\rho}_{ij}=\frac{\sum_{t=1}^T \hat{u}_{it} \hat{u}_{jt}}{(\sum_{t=1}^T \hat{u}^2_{it})^{1/2} (\sum_{t=1}^T \hat{u}^2_{jt})^{1/2} } \]
i.e., as averages over the time dimension of pairwise correlation coefficients for each pair of cross-sectional units.
The Breusch-Pagan (T. S. Breusch and Pagan 1980) LM test, based on the squares of \(\rho_{ij}\), is valid for \(T \rightarrow \infty\) with \(n\) fixed; defined as
\[LM=\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} T_{ij} \hat{\rho}_{ij}^2\]
where in the case of an unbalanced panel only pairwise complete observations are considered, and \(T_{ij}=min(T_i,T_j)\) with \(T_i\) being the number of observations for individual \(i\); else, if the panel is balanced, \(T_{ij}=T\) for each \(i,j\). The test is distributed as \(\chi^2_{n(n-1)/2}\). It is inappropriate whenever the \(n\) dimension is “large”. A scaled version, applicable also if \(T \rightarrow \infty\) and then \(n \rightarrow \infty\) (as in some pooled time series contexts), is defined as
\[SCLM=\sqrt{\frac{1}{n(n-1)}} ( \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} T_{ij} \hat{\rho}_{ij}^2 -1 )\]
and distributed as a standard normal (see M. H. Pesaran (2004)).
A bias-corrected scaled version, \(BCSCLM\), for the fixed effect model with individual effects only is also available which is simply the \(SCLM\) with a term correcting for the bias (Badi H. Baltagi, Feng, and Kao (2012))18. This statistic is also asymptotically distributed as standard normal. \[BCSCLM=\sqrt{\frac{1}{n(n-1)}} ( \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} T_{ij} \hat{\rho}_{ij}^2 -1)-\frac{n}{2(T-1)}\]
Pesaran’s (M. H. Pesaran (2004), M. Hashem Pesaran (2015)) \(CD\) test
\[CD=\sqrt{\frac{2}{n(n-1)}} ( \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sqrt{T_{ij}} \hat{\rho}_{ij} )\]
based on \(\rho_{ij}\) without squaring (also distributed as a standard normal) is appropriate both in \(n\)– and in \(T\)–asymptotic settings. It has remarkable properties in samples of any practically relevant size and is robust to a variety of settings. The only big drawback is that the test loses power against the alternative of cross-sectional dependence if the latter is due to a factor structure with factor loadings averaging zero, that is, some units react positively to common shocks, others negatively.
The default version of the test is "cd" yielding
Pesaran’s \(CD\) test. These tests are
originally meant to use the residuals of separate estimation of one
time-series regression for each cross-sectional unit, so this is the
default behaviour of pcdtest.
##
## Pesaran CD test for cross-sectional dependence in panels
##
## data: inv ~ value + capital
## z = 5.3401, p-value = 0.00000009292
## alternative hypothesis: cross-sectional dependenceIf a different model specification (within,
random, …) is assumed consistent, one can resort to its
residuals for testing19 by specifying the relevant
model type. The main argument of this function may be
either a model of class panelmodel or a
formula and a data.frame; in the second case,
unless model is set to NULL, all usual
parameters relative to the estimation of a plm model may be
passed on. The test is compatible with any consistent
panelmodel for the data at hand, with any specification of
effect. E.g., specifying effect = "time" or
effect = "twoways" allows to test for residual
cross-sectional dependence after the introduction of time fixed effects
to account for common shocks.
##
## Pesaran CD test for cross-sectional dependence in panels
##
## data: inv ~ value + capital
## z = 4.6612, p-value = 0.000003144
## alternative hypothesis: cross-sectional dependenceIf the time dimension is insufficient and model=NULL,
the function defaults to estimation of a within model and
issues a warning.
A local variant of the \(CD\) test, called \(CD(p)\) test (M. H. Pesaran 2004), takes into account an appropriate subset of neighbouring cross-sectional units to check the null of no XSD against the alternative of local XSD, i.e. dependence between neighbours only. To do so, the pairs of neighbouring units are selected by means of a binary proximity matrix like those used in spatial models. In the original paper, a regular ordering of observations is assumed, so that the \(m\)-th cross-sectional observation is a neighbour to the \((m-1)\)-th and to the \((m+1)\)-th. Extending the \(CD(p)\) test to irregular lattices, we employ the binary proximity matrix as a selector for discarding the correlation coefficients relative to pairs of observations that are not neighbours in computing the \(CD\) statistic. The test is then defined as
\[CD=\sqrt{\frac{1}{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} w(p)_{ij}}} ( \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} [w(p)]_{ij} \sqrt{T_{ij}}\hat{\rho}_{ij} )\]
where \([w(p)]_{ij}\) is the \((i,j)\)-th element of the \(p\)-th order proximity matrix, so that if \(h,k\) are not neighbours, \([w(p)]_{hk}=0\) and \(\hat{\rho}_{hk}\) gets “killed”; this is easily seen to reduce to formula (14) in Pesaran (M. H. Pesaran 2004) for the special case considered in that paper. The same can be applied to the \(LM\), \(SCLM\), and \(BCSCLM\) tests.
Therefore, the local version of either test can be computed
supplying an \(n \times n\) matrix (of
any kind coercible to logical), providing information on
whether any pair of observations are neighbours or not, to the
w argument. If w is supplied, only
neighbouring pairs will be used in computing the test; else,
w will default to NULL and all observations
will be used. The matrix needs not really be binary, so commonly used
“row-standardized” matrices can be employed as well: it is enough that
neighbouring pairs correspond to nonzero elements in w 20.
Below, first an overview is provided which tests are implemented per
functions. A theoretical treatment is given for a few of those tests
later on. The package plm offers several panel unit root
tests contained in three functions:
purtest (Levin-Lin-Chu test, IPS test, several
Fisher-type tests, Hadri’s test),cipstest (cross-sectionally augmented IPS test),
andphansitest (Simes’ test).While purtest implements various tests which can be
selected via its test argument, cipstest and
phansitest are functions for a specific test each.
Function purtest offers the following tests by setting
argument test to:
"levinlin" (default), for the Levin-Lin-Chu test (Levin, Lin, and Chu (2002)), see below for a
theoretical exposition (Levin-Lin-Chu
test)),"ips", for Im-Pesaran-Shin (IPS) test by Im, Pesaran, and Shin (2003), see below for a
theoretical exposition (Im-Pesaran-Shin test)),"madwu", is the inverse \(\chi^2\) test by Maddala and Wu (1999), also called P test by
Choi (2001),"Pm", is the modified P test proposed by Choi (2001) for large N,"invnormal", is the inverse normal test (Choi (2001)),"logit", is the logit test (Choi
(2001)),"hadri", for Hadri’s test (Hadri
(2000)).The tests in purtest are often called first generation
panel unit root tests as they do assume absence of cross-sectional
correlation; all these, except Hadri’s test, are based on the estimation
of augmented Dickey-Fuller (ADF) regressions for each time series. A
statistic is then computed using the t-statistics associated with the
lagged variable. I a different manner, the Hadri residual-based LM
statistic is the cross-sectional average of individual KPSS statistics
(Kwiatkowski et al. (1992)), standardized
by their asymptotic mean and standard deviation. Among the tests in
purtest, "madwu", "Pm",
"invormal", and "logit" are Fisher-type
tests.21
purtest returns an object of class
"purtest" which contains details about the test performed,
among them details about the individual regressions/statistics for the
test. Associated summary and print.summary
methods can be used to extract/display the additional information.
Function cipstest implements Pesaran’s (M. Hashem Pesaran (2007)) cross-sectionally
augmented version of the Im-Pesaran-Shin panel unit root test and is a
so-called second-generation panel unit root test.
Function phansitest implements the idea of Hanck (2013) to apply Simes’ testing approach
for intersection of individual hypothesis tests to panel unit root
testing, see below for a more thorough treatment of Simes’ approach for intersecting hypotheses.
We consider the following model:
\[ y_{it} = \delta y_{it-1} + \sum_{L=1}^{p_i} \theta_i \Delta y_{it-L}+\alpha_{mi} d_{mt}+\epsilon_{it} \]
The unit root hypothesis is \(\rho = 1\). The model can be rewritten in difference:
\[ \Delta y_{it} = \rho y_{it-1} + \sum_{L=1}^{p_i} \theta_i \Delta y_{it-L}+\alpha_{mi} d_{mt}+\epsilon_{it} \]
So that the unit-root hypothesis is now \(\rho = 0\).
Some of the unit-root tests for panel data are based on preliminary results obtained by running the above Augmented Dickey-Fuller (ADF) regression.
First, we have to determine the optimal number of lags \(p_i\) for each time-series. Several possibilities are available. They all have in common that the maximum number of lags have to be chosen first. Then, \(p_i\) can be chosen by using:
The ADF regression is run on \(T-p_i-1\) observations for each individual, so that the total number of observations is \(n\times \tilde{T}\) where \(\tilde{T}=T-p_i-1\)
\(\bar{p}\) is the average number of lags. Call \(e_{i}\) the vector of residuals.
Estimate the variance of the \(\epsilon_i\) as:
\[ \hat{\sigma}_{\epsilon_i}^2 = \frac{\sum_{t=p_i+1}^{T} e_{it}^2}{df_i} \]
Then, as per Levin, Lin, and Chu (2002), compute artificial regressions of \(\Delta y_{it}\) and \(y_{it-1}\) on \(\Delta y_{it-L}\) and \(d_{mt}\) and get the two vectors of residuals \(z_{it}\) and \(v_{it}\).
Standardize these two residuals and run the pooled regression of \(z_{it}/\hat{\sigma}_i\) on \(v_{it}/\hat{\sigma}_i\) to get \(\hat{\rho}\), its standard deviation \(\hat{\sigma}({\hat{\rho}})\) and the t-statistic \(t_{\hat{\rho}}=\hat{\rho}/\hat{\sigma}({\hat{\rho}})\).
Compute the long run variance of \(y_i\) :
\[ \hat{\sigma}_{yi}^2 = \frac{1}{T-1}\sum_{t=2}^T \Delta y_{it}^2 + 2 \sum_{L=1}^{\bar{K}}w_{\bar{K}L}\left[\frac{1}{T-1}\sum_{t=2+L}^T \Delta y_{it} \Delta y_{it-L}\right] \]
Define \(\bar{s}_i\) as the ratio of the long and short term variance and \(\bar{s}\) the mean for all the individuals of the sample
\[ s_i = \frac{\hat{\sigma}_{yi}}{\hat{\sigma}_{\epsilon_i}} \]
\[ \bar{s} = \frac{\sum_{i=1}^n s_i}{n} \]
\[ t^*_{\rho}=\frac{t_{\rho}- n \bar{T} \bar{s} \hat{\sigma}_{\tilde{\epsilon}}^{-2} \hat{\sigma}({\hat{\rho}}) \mu^*_{m\tilde{T}}}{\sigma^*_{m\tilde{T}}} \]
follows a normal distribution under the null hypothesis of stationarity. \(\mu^*_{m\tilde{T}}\) and \(\sigma^*_{m\tilde{T}}\) are given in table 2 of the original paper and are also available in the package.
An example how the Levin-Lin-Chu test is performed with
purtest using a lag of 2 and intercept and a time trend as
exogenous variables in the ADF regressions is:
data("HousePricesUS", package = "pder")
lprice <- log(pdata.frame(HousePricesUS)$price)
(lev <- purtest(lprice, test = "levinlin", lags = 2, exo = "trend"))##
## Levin-Lin-Chu Unit-Root Test (ex. var.: Individual Intercepts and
## Trend)
##
## data: lprice
## z = -1.2573, p-value = 0.1043
## alternative hypothesis: stationarity## Levin-Lin-Chu Unit-Root Test
## Exogenous variables: Individual Intercepts and Trend
## User-provided lags
## statistic: -1.257
## p-value: 0.104
##
## lags obs rho trho p.trho sigma2ST sigma2LT
## 1 2 26 -0.092065357 -1.66309731 0.767613204 0.0003143120 0.0004013788
## 4 2 26 -0.124093984 -1.29563385 0.888755668 0.0010950144 0.0014736172
## 5 2 26 -0.104647566 -1.10814627 0.926357866 0.0007296044 0.0007451534
## 6 2 26 -0.219022744 -2.94312106 0.148774635 0.0007716609 0.0048254402
## 8 2 26 -0.052471794 -0.95375744 0.948405601 0.0006375257 0.0028152736
## 9 2 26 -0.181914333 -2.73331072 0.222919642 0.0021489671 0.0064455696
## 10 2 26 -0.232215125 -3.37321191 0.054989191 0.0005566400 0.0024147067
## 11 2 26 -0.356452679 -4.35943612 0.002479709 0.0008542529 0.0045574510
## 12 2 26 0.279936991 1.83482002 0.999998365 0.0004172617 0.0012951914
## 13 2 26 -0.062610441 -0.84216587 0.960499065 0.0003168316 0.0002981994
## 16 2 26 -0.159254884 -2.29683734 0.435109226 0.0007437190 0.0010203969
## 17 2 26 -0.237065476 -4.05050006 0.007367490 0.0005512405 0.0009463645
## 18 2 26 -0.140788644 -2.08598977 0.553093684 0.0005079423 0.0005697978
## 19 2 26 -0.099218199 -1.83853581 0.686000079 0.0008756343 0.0020399374
## 20 2 26 -0.046049208 -0.85174237 0.959567857 0.0003914722 0.0011128571
## 21 2 26 -0.102633777 -1.81503721 0.697696805 0.0004063481 0.0002345858
## 22 2 26 -0.115700485 -1.72146553 0.741996511 0.0010094113 0.0047169602
## 23 2 26 -0.218251170 -2.90863990 0.159598845 0.0010530905 0.0031254005
## 24 2 26 -0.293126134 -3.65827755 0.025157448 0.0004297907 0.0012860757
## 25 2 26 -0.107476475 -2.33427946 0.414643019 0.0008718077 0.0083545213
## 26 2 26 -0.135655633 -2.06664416 0.563904448 0.0009443422 0.0007705242
## 27 2 26 -0.005168776 -0.06565125 0.995432637 0.0007059964 0.0017093982
## 28 2 26 -0.101736562 -1.02991147 0.938382427 0.0008552994 0.0003212357
## 29 2 26 -0.106917037 -1.44344289 0.848353136 0.0004659842 0.0004528307
## 30 2 26 -0.143955051 -1.60594256 0.791068910 0.0016589513 0.0022706981
## 31 2 26 -0.093688191 -1.92279670 0.642427798 0.0004025885 0.0009260538
## 32 2 26 -0.313691108 -2.30732500 0.429359754 0.0009640889 0.0019245305
## 33 2 26 -0.151599029 -2.54586869 0.305755540 0.0011446680 0.0072973098
## 34 2 26 -0.113830637 -2.06152082 0.566761534 0.0008514932 0.0055913854
## 35 2 26 -0.220363663 -1.72391205 0.740887358 0.0005138068 0.0007015982
## 36 2 26 -0.211779244 -3.98522621 0.009148144 0.0011004960 0.0062947120
## 37 2 26 -0.161851244 -2.19906397 0.489557509 0.0002334460 0.0001298656
## 38 2 26 -0.222507555 -1.45762738 0.843900682 0.0048242337 0.0019584921
## 39 2 26 -0.119405321 -2.41405422 0.372087172 0.0004737829 0.0009459741
## 40 2 26 -0.066956522 -0.94176615 0.949844273 0.0011477969 0.0044950987
## 41 2 26 -0.107420235 -2.09998836 0.545259606 0.0009669881 0.0033414294
## 42 2 26 -0.211640785 -3.51839705 0.037371037 0.0004302930 0.0020971822
## 44 2 26 -0.160491538 -2.32116399 0.421790841 0.0016866384 0.0067053791
## 45 2 26 0.013957358 0.21048073 0.998138553 0.0002474183 0.0001310960
## 46 2 26 -0.125206819 -1.36523187 0.871040437 0.0014782610 0.0005893232
## 47 2 26 -0.146576570 -2.35613125 0.402831448 0.0002851628 0.0001796349
## 48 2 26 -0.106184312 -1.41243370 0.857717409 0.0006722417 0.0029218865
## 49 2 26 -0.110328029 -2.31986075 0.422503260 0.0007413810 0.0032547907
## 50 2 26 -0.336849990 -3.07534065 0.112195130 0.0015064070 0.0017150678
## 51 2 26 -0.219041498 -2.26882562 0.450564573 0.0004175437 0.0010455238
## 53 2 26 -0.249921002 -2.67545341 0.246874228 0.0008780514 0.0016600448
## 54 2 26 -0.092856496 -1.36610183 0.870804641 0.0011141161 0.0011656778
## 55 2 26 -0.119379994 -2.19438511 0.492186948 0.0008204933 0.0007526503
## 56 2 26 -0.094196887 -1.53868461 0.816472629 0.0016308380 0.0062631397This test by Im, Pesaran, and Shin (2003) does not require that \(\rho\) is the same for all the individuals. The null hypothesis is still that all the series have an unit root, but the alternative is that some may have a unit root and others have different values of \(\rho_i <0\).
The test is based on the average of the student statistic of the \(\rho\) obtained for each individual:
\[ \bar{t}=\frac{1}{n}\sum_{i=1}^n t_{\rho i} \]
The statistic is then:
\[ z = \frac{\sqrt{n}\left(\bar{t}- E(\bar{t})\right)}{\sqrt{V(\bar{t})}} \]
\(\mu^*_{m\tilde{T}}\) and \(\sigma^*_{m\tilde{T}}\) are given in table 2 of the original paper and are also available in the package.
An example of the IPS test with purtest with the same
settings as in the previously performed Levin-Lin-Chu test is:
##
## Im-Pesaran-Shin Unit-Root Test (ex. var.: Individual Intercepts and
## Trend)
##
## data: lprice
## Wtbar = 0.76622, p-value = 0.7782
## alternative hypothesis: stationarityA different approach to panel unit root testing can be drawn from the general Simes’ test for intersection of individual hypothesis tests (Simes 1986). Hanck (2013) suggests to apply the approach for panel unit root testing: The tests works by combining p-values from single hypothesis tests (individual unit root tests) with a global (intersected) hypothesis and controls for the multiplicity in testing. Thus, it works “on top” of any panel unit root test which yield a p-value for each individual series. Unlike most other panel unit root tests, this approach allows to discriminate between individuals for which the individual H0 (unit root present for individual series) is rejected/is not rejected and requires a pre-specified significance level. Further, the test is robust versus general patterns of cross-sectional dependence.
The function phansitest for this test takes as main
input object either a numeric containing p-values of individual tests or
a "purtest" object as produced by function
purtest which holds a suitable pre-computed panel unit root
test (one that produces p-values per individual series). The
significance level is set by argument alpha (default 5 %).
The function’s return value is a list with detailed evaluation of the
applied Simes test. The associated print method gives a verbal
evaluation.
The following examples shows both accepted ways of input, the first example replicates Hanck (2013), table 11 (left side), who applied some panel unit root test for a Purchasing Power Parity analysis per country (individual H0 hypotheses per series) to get the individual p-values and then used Simes’ approach for testing the global (intersecting) hypothesis for the whole panel.
### input is numeric (p-values), replicates Hanck (2013), Table 11 (left side)
pvals <- c(0.0001,0.0001,0.0001,0.0001,0.0001,0.0001,0.0050,0.0050,0.0050,
0.0050,0.0175,0.0175,0.0200,0.0250,0.0400,0.0500,0.0575,0.2375,0.2475)
countries <- c("Argentina","Sweden","Norway","Mexico","Italy","Finland","France",
"Germany","Belgium","U.K.","Brazil","Australia","Netherlands",
"Portugal","Canada", "Spain","Denmark","Switzerland","Japan")
names(pvals) <- countries
h <- phansitest(pvals)
print(h)##
## Simes Test as Panel Unit Root Test (Hanck (2013))
##
## H0: All individual series have a unit root
## HA: Stationarity for at least some individuals
##
## Alpha: 0.05
## Number of individuals: 19
##
## Evaluation:
## H0 rejected (globally)
##
## Individual H0 rejected for 10 individual(s) (integer id(s)):
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10## Argentina Sweden Norway Mexico Italy Finland
## TRUE TRUE TRUE TRUE TRUE TRUE
## France Germany Belgium U.K. Brazil Australia
## TRUE TRUE TRUE TRUE FALSE FALSE
## Netherlands Portugal Canada Spain Denmark Switzerland
## FALSE FALSE FALSE FALSE FALSE FALSE
## Japan
## FALSERobust estimators of the covariance matrix of coefficients are provided, mostly for use in Wald-type tests, and this section provides some basics and examples. A more comprehensive exposition of the theory and the capabilities that come with the plm package is given in Millo (2017).
vcovHC estimates three “flavours” of White’s
heteroskedasticity-consistent covariance matrix22 (known as the
sandwich estimator). Interestingly, in the context of panel
data the most general version also proves consistent vs. serial
correlation.
All types assume no correlation between errors of different groups
while allowing for heteroskedasticity across groups, so that the full
covariance matrix of errors is \(V=I_n \otimes
\Omega_i; i=1,..,n\). As for the intragroup error
covariance matrix of every single group of observations,
"white1" allows for general heteroskedasticity but no
serial correlation, i.e.
\[\begin{equation} (\#eq:omegaW1) \Omega_i= \left[ \begin{array}{c c c c} \sigma_{i1}^2 & \dots & \dots & 0 \\ 0 & \sigma_{i2}^2 & & \vdots \\ \vdots & & \ddots & 0 \\ 0 & ... & ... & \sigma_{iT}^2 \\ \end{array} \right] \end{equation}\]
while "white2" is "white1" restricted to a
common variance inside every group, estimated as \(\sigma_i^2=\sum_{t=1}^T{\hat{u}_{it}^2}/T\),
so that \(\Omega_i=I_T \otimes
\sigma_i^2\) (see Greene (2003),
13.7.1–2 and Wooldridge (2002), 10.7.2;
"arellano" (see ibid. and the original ref. Manuel Arellano (1987)) allows a fully general
structure w.r.t. heteroskedasticity and serial correlation:
\[\begin{equation} (\#eq:omegaArellano) \Omega_i= \left[ \begin{array}{c c c c c} \sigma_{i1}^2 & \sigma_{i1,i2} & \dots & \dots & \sigma_{i1,iT} \\ \sigma_{i2,i1} & \sigma_{i2}^2 & & & \vdots \\ \vdots & & \ddots & & \vdots \\ \vdots & & & \sigma_{iT-1}^2 & \sigma_{iT-1,iT} \\ \sigma_{iT,i1} & \dots & \dots & \sigma_{iT,iT-1} & \sigma_{iT}^2 \\ \end{array} \right] \end{equation}\]
The latter is, as already observed, consistent w.r.t. timewise correlation of the errors, but on the converse, unlike the White 1 and 2 methods, it relies on large \(n\) asymptotics with small \(T\).
The fixed effects case, as already observed in Section tests of serial correlation on serial correlation,
is complicated by the fact that the demeaning induces serial correlation
in the errors. The original White estimator ("white1")
turns out to be inconsistent for fixed \(T\) as \(n\) grows, so in this case it is advisable
to use the "arellano" version (see Stock and Watson (2008)).
The errors may be weighted according to the schemes proposed by J. G. MacKinnon and White (1985) and Cribari–Neto (2004) to improve small-sample performance23.
The main use of vcovHC (and the other
variance-covariance estimators provided in the package
vcovBK, vcovNW, vcovDC,
vcovSCC) is to pass it to plm’s own functions like
summary, pwaldtest, and phtest or
together with testing functions from the lmtest and
car packages. All of these typically allow passing the
vcov or vcov. parameter either as a matrix or
as a function (see also Zeileis (2004)).
If one is happy with the defaults, it is easiest to pass the function
itself24:
re <- plm(inv~value+capital, data = Grunfeld, model = "random")
summary(re, vcov = vcovHC) # gives usual summary output but with robust test statistics## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
##
## Note: Coefficient variance-covariance matrix supplied: vcovHC
##
## Call:
## plm(formula = inv ~ value + capital, data = Grunfeld, model = "random")
##
## Balanced Panel: n = 10, T = 20, N = 200
##
## Effects:
## var std.dev share
## idiosyncratic 2784.46 52.77 0.282
## individual 7089.80 84.20 0.718
## theta: 0.8612
##
## Residuals:
## Min. 1st Qu. Median 3rd Qu. Max.
## -177.6063 -19.7350 4.6851 19.5105 252.8743
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) -57.834415 23.449626 -2.4663 0.01365 *
## value 0.109781 0.012984 8.4551 < 0.00000000000000022 ***
## capital 0.308113 0.051889 5.9379 0.000000002887 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 2381400
## Residual Sum of Squares: 548900
## R-Squared: 0.7695
## Adj. R-Squared: 0.76716
## Chisq: 78.7096 on 2 DF, p-value: < 0.000000000000000222##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -57.834415 23.449626 -2.4663 0.01365 *
## value 0.109781 0.012984 8.4551 < 0.00000000000000022 ***
## capital 0.308113 0.051889 5.9379 0.000000002887 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Another approach is to compute the covariance matrix inside the call and pass it on:
summary(re, vcov = vcovHC(re, method="white2", type="HC3"))
coeftest(re, vcovHC(re, method="white2", type="HC3"), df = Inf)For some tests, e.g., for multiple model comparisons by
waldtest, one should always provide a function25. In
this case, optional parameters are provided as shown below (see also
Zeileis (2004), p. 12):
## Wald test
##
## Model 1: inv ~ value + capital
## Model 2: inv ~ value
## Res.Df Df Chisq Pr(>Chisq)
## 1 197
## 2 198 -1 87.828 < 0.00000000000000022 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Moreover, linearHypothesis from package car
may be used to test for linear restrictions:
##
## Linear hypothesis test:
## 2 value - capital = 0
##
## Model 1: restricted model
## Model 2: inv ~ value + capital
##
## Note: Coefficient covariance matrix supplied.
##
## Res.Df Df Chisq Pr(>Chisq)
## 1 198
## 2 197 1 3.4783 0.06218 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A specific methods are also provided for pcce and
pgmm objects, for the latter vcovHC provides a
robust covariance matrix with the small-sample adjustment as proposed by
Windmeijer (2005) for generalized method
of moments estimators.
The models termed panel by the econometricians have
counterparts in the statistics literature on mixed models (or
hierarchical models, or models for longitudinal data),
although there are both differences in jargon and more substantial
distinctions. This language inconsistency between the two communities,
together with the more complicated general structure of statistical
models for longitudinal data and the associated notation in the
software, is likely to scare some practicing econometricians away from
some potentially useful features of the R environment, so
it may be useful to provide here a brief reconciliation between the
typical panel data specifications used in econometrics and the general
framework used in statistics for mixed models26.
R is particularly strong on mixed models’ estimation,
thanks to the long-standing nlme package (see J. Pinheiro et al. (2007)) and the more recent
lme4 package, based on S4 classes (see Bates (2007))27. In the following we
will refer to the more established nlme to give some
examples of “econometric” panel models that can be estimated in a
likelihood framework, also including some likelihood ratio tests. Some
of them are not feasible in plm and make a useful
complement to the econometric “toolbox” available in R.
Econometrics deal mostly with non-experimental data. Great emphasis is put on specification procedures and misspecification testing. Model specifications tend therefore to be very simple, while great attention is put on the issues of endogeneity of the regressors, dependence structures in the errors and robustness of the estimators under deviations from normality. The preferred approach is often semi- or non-parametric, and heteroskedasticity-consistent techniques are becoming standard practice both in estimation and testing.
For all these reasons, although the maximum likelihood framework is
important in testing28 and sometimes used in estimation as well,
panel model estimation in econometrics is mostly accomplished in the
generalized least squares framework based on Aitken’s Theorem and, when
possible, in its special case OLS, which are free from distributional
assumptions (although these kick in at the diagnostic testing stage). On
the contrary, longitudinal data models in nlme and
lme4 are estimated by (restricted or unrestricted) maximum
likelihood. While under normality, homoskedasticity and no serial
correlation of the errors OLS are also the maximum likelihood estimator,
in all the other cases there are important differences.
The econometric GLS approach has closed-form analytical solutions
computable by standard linear algebra and, although the latter can
sometimes get computationally heavy on the machine, the expressions for
the estimators are usually rather simple. ML estimation of longitudinal
models, on the contrary, is based on numerical optimization of nonlinear
functions without closed-form solutions and is thus dependent on
approximations and convergence criteria. For example, the “GLS”
functionality in nlme is rather different from its
“econometric” counterpart. “Feasible GLS” estimation in plm
is based on a single two-step procedure, in which an inefficient but
consistent estimation method (typically OLS) is employed first in order
to get a consistent estimate of the errors’ covariance matrix, to be
used in GLS at the second step; on the converse, “GLS” estimators in
nlme are based on iteration until convergence of two-step
optimization of the relevant likelihood.
The fixed/random effects terminology in econometrics is often recognized to be misleading, as both are treated as random variates in modern econometrics (see, e.g., Wooldridge (2002) 10.2.1). It has been recognized since Mundlak’s classic paper (Mundlak (1978)) that the fundamental issue is whether the unobserved effects are correlated with the regressors or not. In this last case, they can safely be left in the error term, and the serial correlation they induce is cared for by means of appropriate GLS transformations. On the contrary, in the case of correlation, “fixed effects” methods such as least squares dummy variables or time-demeaning are needed, which explicitly, although inconsistently29, estimate a group– (or time–) invariant additional parameter for each group (or time period).
Thus, from the point of view of model specification, having fixed effects in an econometric model has the meaning of allowing the intercept to vary with group, or time, or both, while the other parameters are generally still assumed to be homogeneous. Having random effects means having a group– (or time–, or both) specific component in the error term.
In the mixed models literature, on the contrary, fixed effect indicates a parameter that is assumed constant, while random effects are parameters that vary randomly around zero according to a joint multivariate normal distribution.
So, the FE model in econometrics has no counterpart in the mixed
models framework, unless reducing it to OLS on a specification with one
dummy for each group (often termed least squares dummy
variables, or LSDV model) which can trivially be estimated by OLS.
The RE model is instead a special case of a mixed model where only the
intercept is specified as a random effect, while the “random” type
variable coefficients model can be seen as one that has the same
regressors in the fixed and random sets. The unrestricted generalized
least squares can in turn be seen, in the nlme framework,
as a standard linear model with a general error covariance structure
within the groups and errors uncorrelated across groups.
To reconcile the two terminologies, in the following we report the
specification of the panel models in plm according to the
general expression of a mixed model in Laird-Ware form (see the web appendix to Fox 2002) and the
nlme estimation commands for maximum likelihood estimation
of an equivalent specification30.
A general representation for the linear mixed effects model is given in Laird and Ware (1982).
\[ \begin{array}{rcl} y_{it} & = & \beta_1 x_{1ij} + \dots + \beta_p x_{pij} \\ & & b_1 z_{1ij} + \dots + b_p z_{pij} + \epsilon_{ij} \\ b_{ik} & \sim & N(0,\psi^2_k), \phantom{p} Cov(b_k,b_{k'}) = \psi_{kk'} \\ \epsilon_{ij} & \sim & N(0,\sigma^2 \lambda_{ijj}), \phantom{p} Cov(\epsilon_{ij},\epsilon_{ij'}) = \sigma^2 \lambda_{ijj'} \\ \end{array} \]
where the \(x_1, \dots x_p\) are the fixed effects regressors and the \(z_1, \dots z_p\) are the random effects regressors, assumed to be normally distributed across groups. The covariance of the random effects coefficients \(\psi_{kk'}\) is assumed constant across groups and the covariances between the errors in group \(i\), \(\sigma^2 \lambda_{ijj'}\), are described by the term \(\lambda_{ijj'}\) representing the correlation structure of the errors within each group (e.g., serial correlation over time) scaled by the common error variance \(\sigma^2\).
The pooling specification in plm is equivalent
to a classical linear model (i.e., no random effects regressor and
spherical errors: \(b_{iq}=0 \phantom{p}
\forall i,q, \phantom{p} \lambda_{ijj}=\sigma^2\) for \(j=j'\), \(0\) else). The within one is the
same with the regressors’ set augmented by \(n-1\) group dummies. There is no point in
using nlme as parameters can be estimated by OLS which is
also ML.
In the Laird and Ware notation, the RE specification is a model with only one random effects regressor: the intercept. Formally, \(z_{1ij}=1 \phantom{p}\forall i,j, \phantom{p} z_{qij}=0 \phantom{p} \forall i, \forall j, \forall q \neq 1\) \(\lambda_{ij}=1\) for \(i=j\), \(0\) else). The composite error is therefore \(u_{ij}=1b_{i1} + \epsilon_{ij}\). Below we report coefficients of Grunfeld’s model estimated by GLS and then by ML:
library(nlme)
reGLS <- plm(inv~value+capital, data=Grunfeld, model="random")
reML <- lme(inv~value+capital, data=Grunfeld, random=~1|firm)
coef(reGLS)## (Intercept) value capital
## -57.8344149 0.1097812 0.3081130## (Intercept) value capital
## -57.8644245 0.1097897 0.3081881Swamy’s variable coefficients model (Swamy 1970) has coefficients varying randomly (and independently of each other) around a set of fixed values, so the equivalent specification is \(z_{q}=x_{q} \phantom{p} \forall q\), i.e. the fixed effects and the random effects regressors are the same, and \(\psi_{kk'}=\sigma_\mu^2 I_N\), and \(\lambda_{ijj}=1\), \(\lambda_{ijj'}=0\) for \(j \neq j'\), that’s to say they are not correlated.
Estimation of a mixed model with random coefficients on all regressors is rather demanding from the computational side. Some models from our examples fail to converge. The below example is estimated on the Grunfeld data and model with time effects.
vcm <- pvcm(inv~value+capital, data=Grunfeld, model="random", effect="time")
vcmML <- lme(inv~value+capital, data=Grunfeld, random=~value+capital|year)
coef(vcm)## (Intercept) value capital
## -18.5538638 0.1239595 0.1114579## (Intercept) value capital
## -26.3558395 0.1241982 0.1381782This specification actually entails separate estimation of \(T\) different standard linear models, one
for each group in the data, so the estimation approach is the same: OLS.
In nlme this is done by creating an lmList
object, so that the two models below are equivalent (output
suppressed):
The general, or unrestricted, feasible GLS (FGLS), pggls
in the plm nomenclature, is equivalent to a model with no
random effects regressors (\(b_{iq}=0
\phantom{p} \forall i,q\)) and an error covariance structure
which is unrestricted within groups apart from the usual requirements.
The function for estimating such models with correlation in the errors
but no random effects is gls().
This very general serial correlation and heteroskedasticity structure is not estimable for the original Grunfeld data, which have more time periods than firms, therefore we restrict them to firms 4 to 6.
sGrunfeld <- Grunfeld[Grunfeld$firm %in% 4:6, ]
ggls <- pggls(inv~value+capital, data=sGrunfeld, model="pooling")
gglsML <- gls(inv~value+capital, data=sGrunfeld,
correlation=corSymm(form=~1|year))
coef(ggls)## (Intercept) value capital
## 1.19679342 0.10555908 0.06600166## (Intercept) value capital
## -2.4156266 0.1163550 0.0735837The within case is analogous, with the regressor set augmented by \(n-1\) group dummies.
Finally, amongst the many possible specifications estimable with
nlme, we report a couple cases that might be especially
interesting to applied econometricians.
Linear models with groupwise structures of time-dependence31 may be
fitted by gls(), specifying the correlation structure in
the correlation option32:
Grunfeld$year <- as.numeric(as.character(Grunfeld$year))
lmAR1ML <- gls(inv~value+capital,data=Grunfeld,
correlation=corAR1(0,form=~year|firm))and analogously the random effects panel with, e.g., AR(1) errors
(see B. H. Baltagi (2005); B. H. Baltagi (2013); B.
H. Baltagi (2021), ch. 5), which is a very common specification
in econometrics, may be fit by lme specifying an additional
random intercept:
reAR1ML <- lme(inv~value+capital, data=Grunfeld,random=~1|firm,
correlation=corAR1(0,form=~year|firm))The regressors’ coefficients and the error’s serial correlation coefficient may be retrieved this way:
## (Intercept) value capital
## -40.27650822 0.09336672 0.31323330## Phi
## 0.823845Significance statistics for the regressors’ coefficients are to be
found in the usual summary object, while to get the
significance test of the serial correlation coefficient one can do a
likelihood ratio test as shown in the following.
A likelihood ratio test for serial correlation in the idiosyncratic
residuals can be done as a nested models test, by anova(),
comparing the model with spherical idiosyncratic residuals with the more
general alternative featuring AR(1) residuals. The test takes the form
of a zero restriction test on the autoregressive parameter.
This can be done on pooled or random effects models alike. First we report the simpler case.
We already estimated the pooling AR(1) model above. The GLS model
without correlation in the residuals is the same as OLS, and one could
well use lm() for the restricted model. Here we estimate it
by gls().
## Model df AIC BIC logLik Test L.Ratio p-value
## lmML 1 4 2400.217 2413.350 -1196.109
## lmAR1ML 2 5 2094.936 2111.352 -1042.468 1 vs 2 307.2813 <.0001The AR(1) test on the random effects model is to be done in much the same way, using the random effects model objects estimated above:
## Model df AIC BIC logLik Test L.Ratio p-value
## reML 1 5 2205.851 2222.267 -1097.926
## reAR1ML 2 6 2094.802 2114.501 -1041.401 1 vs 2 113.0496 <.0001A likelihood ratio test for random effects compares the specifications with and without random effects and spherical idiosyncratic errors:
## Model df AIC BIC logLik Test L.Ratio p-value
## lmML 1 4 2400.217 2413.350 -1196.109
## reML 2 5 2205.851 2222.267 -1097.926 1 vs 2 196.366 <.0001The random effects, AR(1) errors model in turn nests the AR(1) pooling model, therefore a likelihood ratio test for random effects sub AR(1) errors may be carried out, again, by comparing the two autoregressive specifications:
## Model df AIC BIC logLik Test L.Ratio p-value
## lmAR1ML 1 5 2094.936 2111.352 -1042.468
## reAR1ML 2 6 2094.802 2114.501 -1041.401 1 vs 2 2.134349 0.144whence we see that the Grunfeld model specification doesn’t seem to need any random effects once we control for serial correlation in the data.
With plm we aim at providing a comprehensive package
containing the standard functionalities that are needed for the
management and the econometric analysis of panel data. In particular, we
provide: functions for data transformation; estimators for pooled,
random and fixed effects static panel models and variable coefficients
models, general GLS for general covariance structures, and generalized
method of moments estimators for dynamic panels; specification and
diagnostic tests. Instrumental variables estimation is supported. Most
estimators allow working with unbalanced panels. While among the
different approaches to longitudinal data analysis we take the
perspective of the econometrician, the syntax is consistent with the
basic linear modeling tools, like the lm function.
On the input side, formula and data
arguments are used to specify the model to be estimated. Special
functions are provided to make writing formulas easier, and the
structure of the data is indicated with an index
argument.
On the output side, the model objects (of the new class
panelmodel) are compatible with the general restriction
testing frameworks of packages lmtest and car.
Specialized methods are also provided for the calculation of robust
covariance matrices; heteroskedasticity- and correlation-consistent
testing is accomplished by passing these on to testing functions,
together with a panelmodel object.
The main functionalities of the package have been illustrated here by applying them on some well-known data sets from the econometric literature. The similarities and differences with the maximum likelihood approach to longitudinal data have also been briefly discussed.
While retaining responsibility for any error, we thank Jeffrey Wooldridge, Achim Zeileis and three anonymous referees for useful comments. We also acknowledge kind editing assistance by Lisa Benedetti.
Comprehensive treatments are to be found in many econometrics textbooks, e.g., B. H. Baltagi (2005), B. H. Baltagi (2013), B. H. Baltagi (2021) or Wooldridge (2002), Wooldridge (2010): the reader is referred to these, especially to the first 9 chapters of B. H. Baltagi (2005), B. H. Baltagi (2013), B. H. Baltagi (2021).↩︎
For the sake of exposition we are considering only the individual effects case here. There may also be time effects, which is a symmetric case, or both of them, so that the error has three components: \(u_{it}=\mu_{i}+\lambda_{t}+\epsilon_{it}\).↩︎
Although in most models the individual and time effects
cases are symmetric, there are exceptions: estimating the
first-difference model on time effects is meaningless because
cross-sections do not generally have a natural ordering, so trying
effect = "time" stops with an error message as does
effect = "twoways" which is not defined for
first-difference models.↩︎
See packages lmtest (Hothorn et al. (2015)) and car
(Fox (2016)).↩︎
Moreover, coeftest() provides a compact way
of looking at coefficient estimates and significance diagnostics.↩︎
Function pht is a deprecated way to
estimate this type of model:
ht <- pht(lwage~wks+south+smsa+married+exp+I(exp^2)+ bluecol+ind+union+sex+black+ed | sex+black+bluecol+south+smsa+ind, data=Wages,index=595).↩︎
The “random effect” is better termed “general FGLS” model, as in fact it does not have a proper random effects structure, but we keep this terminology for general language consistency.↩︎
NB: Oneway King-Wu ("kw") statistics
("individual" and "time") coincide with the
respective Honda statistics ("honda"); however, the twoway
statistics of "kw" and "honda" differ.↩︎
The "bp" test for unbalanced panels was
derived in B. H. Baltagi and Li (1990),
the "kw" test for unbalanced panels in Badi Baltagi, Chang, and Li (1998). The
"ghm" test and the "kw" test were extended to
two–way effects in B. H. Baltagi, Chang, and Li
(1992). For a concise overview of all these statistics see B. H. Baltagi (2013) Sec. 4.2, pp. 68–76 (for
balanced panels) and Sec. 9.5, pp. 200–203 (for unbalanced panels) or
B. H. Baltagi (2021), Sec. 4.2, pp. 81-84
(balanced), Sec. 9.6, pp. 243-246 (unbalanced).↩︎
Here we treat fixed and random effects alike, as components of the error term, according with the modern approach in econometrics (see Wooldridge (2002), Wooldridge (2010)).↩︎
Neglecting time effects may also lead to serial correlation in residuals (as observed in Wooldridge (2002) 10.4.1).↩︎
\(LM_3\) in B. Baltagi and Li (1995).↩︎
Corresponding to \(RSO^*_{\mu}\) in the original paper.↩︎
Baltagi and Li derive a basically analogous
T-asymptotic test for first-order serial correlation in a FE panel model
as a Breusch-Godfrey LM test on within residuals (see B. Baltagi and Li (1995) par. 2.3 and formula
12). They also observe that the test on within residuals can be used for
testing on the RE model, as “the within transformation [time-demeaning,
in our terminology] wipes out the individual effects, whether fixed or
random”. Generalizing the Durbin-Watson test to FE models by applying it
to fixed effects residuals is documented in Bhargava, Franzini, and Narendranathan (1982), a
(modified) version for unbalanced and/or non-consecutive panels is
implemented in pbnftest as is Baltagi-Wu’s LBI statistic
(for both see Badi H. Baltagi and Wu
(1999)).↩︎
see subsection robust covariance matrix estimation.↩︎
Here, \(e_{it}\) for notational simplicity (and as in Wooldridge): equivalent to \(\Delta \epsilon_{it}\) in the general notation of the paper.↩︎
This is the case, e.g., if in an unobserved effects model when XSD is due to an unobservable factor structure, with factors that are uncorrelated with the regressors. In this case the within or random estimators are still consistent, although inefficient (see De Hoyos and Sarafidis (2006)).↩︎
The unbalanced version of this statistic uses max(Tij) for T in the bias-correction term.↩︎
This is also the only solution when the time dimension’s length is insufficient for estimating the heterogeneous model.↩︎
The very comprehensive package spdep for
spatial dependence analysis (see Bivand
(2008)) contains features for creating, lagging and manipulating
neighbour list objects of class nb, that can be
readily converted to and from proximity matrices by means of the
nb2mat function. Higher orders of the \(CD(p)\) test can be obtained by lagging the
corresponding nbs through nblag.↩︎
The individual p-values for the Fisher-type tests are
approximated as described in James G. MacKinnon
(1996) if the package urca (Pfaff (2008)) is available, otherwise as
described in James G. MacKinnon (1994).↩︎
See Halbert White (1980) and H. White (1984).↩︎
The HC3 and HC4 weighting schemes are computationally expensive and may hit memory limits for \(nT\) in the thousands, where on the other hand it makes little sense to apply small sample corrections.↩︎
For coeftest set df = Inf to
have the coefficients’ tests be performed with standard normal
distribution instead of t distribution as we deal with a random effects
model here. For these types of models, the precise distribution of the
coefficients estimates is unknown.↩︎
Joint zero-restriction testing still allows providing
the vcov of the unrestricted model as a matrix, see the
documentation of package lmtest.↩︎
This discussion does not consider GMM models. One of the basic reasons for econometricians not to choose maximum likelihood methods in estimation is that the strict exogeneity of regressors assumption required for consistency of the ML models reported in the following is often inappropriate in economic settings.↩︎
The standard reference on the subject of mixed models
in S/R is J. C.
Pinheiro and Bates (2000).↩︎
Lagrange Multiplier tests based on the likelihood principle are suitable for testing against more general alternatives on the basis of a maintained model with spherical residuals and find therefore application in testing for departures from the classical hypotheses on the error term. The seminal reference is T. S. Breusch and Pagan (1980).↩︎
For fixed effects estimation, as the sample grows (on the dimension on which the fixed effects are specified) so does the number of parameters to be estimated. Estimation of individual fixed effects is \(T\)– (but not \(n\)–) consistent, and the opposite.↩︎
In doing so, we stress that “equivalence” concerns only
the specification of the model, and neither the appropriateness nor the
relative efficiency of the relevant estimation techniques, which will of
course be dependent on the context. Unlike their mixed model
counterparts, the specifications in plm are, strictly
speaking, distribution-free. Nevertheless, for the sake of exposition,
in the following we present them in the setting which ensures
consistency and efficiency (e.g., we consider the hypothesis of
spherical errors part of the specification of pooled OLS and so
forth).↩︎
Take heed that here, in contrast to the usual meaning of serial correlation in time series, we always speak of serial correlation between the errors of each group.↩︎
Note that the time index is coerced to numeric before the estimation.↩︎