xref: /dports/finance/R-cran-plm/plm/vignettes/A_plmPackage.Rmd

---
output: rmarkdown::html_vignette
bibliography: ../inst/REFERENCES.bib
abstract: |
  This introduction to the `plm` package is a modified
  and extended version of @Croissant: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 and to make (robust) inference.
title: 'Panel data econometrics in R:'
subtitle: 'the plm package'
author:
- name: Yves Croissant
- name: Giovanni Millo
vignette: >
  %\VignetteIndexEntry{Panel data econometrics in R: the plm package}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}

---

# Introduction

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`
[@PINH:BATE:DEBR:SARK:07] and `lme4`
[@BATE:07], is available in the `R` (@R: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,
`apply`ing (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](#linear-panel-model)
presents a very short overview of the typical model
taxonomy^[Comprehensive treatments are to be found in many econometrics textbooks,
e.g., @BALT:05, @BALT:13, @BALT:21 or @WOOL:02, @WOOL:10: the reader is referred to
these, especially to the first 9 chapters of @BALT:05, @BALT:13, @BALT:21.].
Section [software approach](#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](#managing-data-and-formulae)), estimation
(Section [model estimation](#model-estimation)) and testing (Section [tests](#tests)),
giving a short description and illustrating them with
examples. Section [plm vs nlme and lme4](#nlme) 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](#conclusions) concludes the paper.


# The linear panel model{#linear-panel-model}

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 time^[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: $u_{it}=\mu_{i}+\lambda_{t}+\epsilon_{it}$.].
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: *pooling*
tests to check poolability, i.e., the hypothesis that the
same coefficients apply across all individuals; and random effects
tests, comparing the null of spherical residuals with the alternative
of group (time) specific effects in the error term. If the homogeneity
assumption over the $\beta$s is established, the next step is to
establish the presence of unobserved effects. The choice between fixed
and random effects specifications is based on Hausman-type tests,
comparing the two estimators under the null of no significant
difference: if this is not rejected, the more efficient random effects
estimator is chosen.  Even after this step, departures of the error
structure from sphericity can further affect inference, so that either
screening tests or robust diagnostics are needed.
-->


The hypotheses on parameters and error terms (and hence the choice of
the most appropriate estimator) are usually tested by means of:


- *pooling* tests to check poolability, i.e., the hypothesis
  that the same coefficients apply across all individuals,
- if the homogeneity assumption over the coefficients is
  established, the next step is to establish the presence of
  unobserved effects, comparing the null of spherical residuals with
  the alternative of group (time) specific effects in the error term,
- the choice between fixed and random effects specifications is
  based on Hausman-type tests, comparing the two estimators under the
  null of no significant difference: if this is not rejected, the more
  efficient random effects estimator is chosen,
- even after this step, departures of the error structure from
  sphericity can further affect inference, so that either screening
  tests or robust diagnostics are needed.




<!--
Dynamic models in a fixed or random effects setting, 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.
-->

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.

# Software approach{#software-approach}

## Data structure

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,
- a character string, which should be the name of the individual
  index,
- a character vector of length two containing the names of the
  individual and the time index,
- an integer which is the number of individuals (only in case of a
  balanced panel with observations ordered by individual).


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.

## Interface

### Estimation interface

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 both^[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.],
- `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.

<!-- TODO: currently, plm objects do not contain fitted.values -->


### Testing interface

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.

<!--
In some cases where tests are to be done on model objects that be, in
computational terms, more demanding to estimate (and that are also
likely to be already present in the workspace at that stage), an
exception is made to the rule and tests require a `panelmodel`
object argument. This is the case of Hausman tests and of
Wooldridge-type serial correlation tests.
-->

## Computational approach to estimation

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 (quasi--)demeaning framework

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](#inference).


### The object oriented approach to general GLS computations

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 @BATE:04) and packages `Matrix` (see
@BATE:MAEC:2016) and `SparseM` (see @KOEN:NG:2016). Some optimized
linear algebra routines are available in the `R` package `bdsmatrix`
(see @THER:14) 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.


## Inference in the panel model{#inference}


General frameworks for restrictions and linear hypotheses testing are
available in the `R` environment^[See packages `lmtest`
(@HOTH:ZEIL:FARE:CUMM:MILL:MITC:2015) and `car` (@FOX:2016).].
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 $(R \hat{\beta} - r)^\top  [R \hat{V} R^\top ]^{-1} (R
\hat{\beta} - r)$^[Moreover, `coeftest()` provides a compact way of looking at
coefficient estimates and significance diagnostics.].
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 @ZEIL:04; @LUML:ZEIL: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.


# Managing data and formulae{#dataformula}

The package is now illustrated by application to some well-known
examples. It is loaded using

```{r echo=FALSE,results='hide'}
options(prompt= "R> ", useFancyQuotes = FALSE, scipen = 999)
library("knitr")
opts_chunk$set(message = FALSE, warning = FALSE)
```

```{r echo=TRUE, results='hide'}
library("plm")
```

The four data sets used are `EmplUK` which was used by
@AREL:BOND:91, the `Grunfeld` data [@KLEI:ZEIL:08]
which is used in several econometric books, the `Produc` data
used by @MUNN:90 and the `Wages` used by @CORN:RUPE:88.

```{r }
data("EmplUK", package="plm")
data("Produc", package="plm")
data("Grunfeld", package="plm")
data("Wages", package="plm")
```


## Data structure

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.

```{r setdata1}
head(Grunfeld)
E <- pdata.frame(EmplUK, index=c("firm","year"), drop.index=TRUE, row.names=TRUE)
head(E)
head(attr(E, "index"))
```

Two 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.

```{r }
summary(E$emp)
head(as.matrix(E$emp))
```
## Data transformation

Panel 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:

- the between transformation: $P=\frac{1}{T}I_n\otimes jj'$
  returns a vector containing the individual means. The
  `Between` and `between` functions perform this
  operation, the first one returning a vector of length $nT$, the
  second one a vector of length $n$,
- the within transformation: $Q=I_{nT}-P$ returns a vector
  containing the values in deviation from the individual means. The
  `Within` function performs this operation.
- the first difference transformation $D=I_n \otimes d$ where

$d=\left(
\begin{array}{ccccccc}
1 & -1 & 0 & 0 & ... & 0 & 0 \\
0 &  1 & -1 & 0 & ... & 0 & 0 \\
0 & 0 & 1 & -1 & ... & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & ... & 1 & -1 \\
\end{array}
\right)$

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:

```{r }
head(lag(E$emp, 0:2))
```

Further 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.

```{r }
head(diff(E$emp), 10)
head(lag(E$emp, 2), 10)
head(Within(E$emp))
head(between(E$emp), 4)
head(Between(E$emp), 10)
```


## Formulas

In some circumstances, standard `formula`s 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:
```{r results='hide'}
emp ~ wage + capital | lag(wage, 1) + capital
emp ~ wage + capital | . -wage + lag(wage, 1)
```

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.

# Model estimation{#modelestimation}


## Estimation of the basic models with plm


Several models can be estimated with `plm` by filling the
`model` argument:

- the fixed effects model (`"within"`), the default,
- the pooling model (`"pooling"`),
- the first-difference model (`"fd"`),
- the between model (`"between"`),
- the error components model (`"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:

```{r fe_re}
grun.fe <- plm(inv~value+capital, data = Grunfeld, model = "within")
grun.re <- plm(inv~value+capital, data = Grunfeld, model = "random")
```
Methods to display a sumamry 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`.
```{r summary_re}
summary(grun.re)
ranef(grun.re)
```

The 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"`.

```{r }
fixef(grun.fe, type = "dmean")
```

The `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.

```{r }
summary(fixef(grun.fe, type = "dmean"))
```

In 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:

```{r }
grun.twfe <- plm(inv~value+capital, data=Grunfeld, model="within", effect="twoways")
fixef(grun.twfe, effect = "time")
```

## More advanced use of plm

### Random effects estimators

As 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 @SWAM:AROR:72, the default value,
- `"walhus"`: from @WALL:HUSS:69,
- `"amemiya"`: from @AMEM:71,
- `"nerlove"`: from @NERLO:71.
- `"ht"`: for Hausman-Taylor-type instrumental variable (IV) estimation,
discussed later, see Section [Instrumental variable estimator](#instrumental-variable-est).

For example, to use the `amemiya` estimator:

```{r }
grun.amem <- plm(inv~value+capital, data=Grunfeld,
                 model="random", random.method="amemiya")
```

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:

```{r }
ercomp(inv~value+capital, data=Grunfeld, method = "amemiya", effect = "twoways")
```

### Introducing time or two-ways effects

The default behavior of `plm` is to introduce individual
effects. Using the `effect` argument, one may also introduce:

- time effects (`effect = "time"`),
- individual and time effects (`effect = "twoways"`).

For example, to estimate a two-ways effect model for the
`Grunfeld` data:

```{r 2RE-amemiya}
grun.tways <- plm(inv~value+capital, data = Grunfeld, effect = "twoways",
                  model = "random", random.method = "amemiya")
summary(grun.tways)
```

In 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.

### Unbalanced panels

Estimations by `plm` support unbalanced panel models.

The following example is using data used by @HARR:RUBI:78 to
estimate an hedonic housing prices function. It is reproduced in
@BALT:CHAN:94, table 2 (and in @BALT:05, pp. 172/4; @BALT:13, pp. 195/7
tables 9.1/3).

```{r hedonic}
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)
```
Measures 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 @AHRE:PINC:81 where for both 1 represents
balanced data and the more unbalanced the data the lower the value.

```{r hedonic-punbal}
punbalancedness(Hed)
```

### Instrumental variable estimators{#instrumental-variable-est}


All 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  @BALE:VARA:87, the default value,
- `"baltagi"`, from @BALT:81,
- `"am"`, from @AMEM:MACU:86,
- `"bms"`, from @BREU:MIZO:SCHM:89.

An illustration is in the following example from
@BALT:05, p. 120; @BALT:13, p. 137; @BALT:21, p. 165, table 7.3 ("G2SLS").

```{r 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)
```

The Hausman-Taylor model (see @HAUS:TAYL:81) may be estimated
with the `plm`^[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)`.]
function by setting parameters `random.method = "ht"` and
`inst.method = "baltagi"` like in
the example below. The following replicates @BALT:05, pp. 129/30;
@BALT:13, pp. 145/6, tables 7.4/5; @BALT:21, pp. 174/5 tables 7.5/6:

```{r hausman-taylor}
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)
```


## Variable coefficients model

The `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 different
model is estimated for each individual (or time period). In the second
case, the Swamy model (see @SWAM:70) model is estimated. It is a generalized
least squares model which uses the results of the previous
model. 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:

```{r grunfeld.within}
grun.varw <- pvcm(inv~value+capital, data=Grunfeld, model="within")
grun.varr <- pvcm(inv~value+capital, data=Grunfeld, model="random")
summary(grun.varr)
```

## Generalized method of moments estimator

The generalized method of moments is mainly used in panel data
econometrics to estimate dynamic models [@AREL:BOND:91; @HOLT:NEWE:ROSE:88].

\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 @ANDE:HSIA:81). The GMM estimator uses the fact that
the number of valid instruments is growing with $t$:

- $t=3$: $y_1$,
- $t=4$: $y_1,y_2$,
- $t=5$: $y_1,y_2,y_3$.

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.


@BLUN:BOND:98 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 `NULL`, `"individual"`
(the default), or `"twoways"`. In the first case, the model is
estimated in levels. In the second case, 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 @AREL:BOND:91. Employment is
explained by past values of employment (two lags), current and first
lag of wages and output and current value of capital.

```{r gmm}
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)
```

The following example is from @BLUN:BOND:98. 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 proposed by
@WIND:05. For all pgmm models, `robust = TRUE` is the default (but set
in this example explicitly).

```{r gmm2}
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)
```

## General FGLS models

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 @WOOL:02 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 @GREE:03 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" effects^[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.].

The "random effect" general FGLS is estimated by:

```{r pggls}
zz <- pggls(log(emp)~log(wage)+log(capital), data=EmplUK, model="pooling")
summary(zz)
```

The fixed effects `pggls` (see @WOOL:02, p. 276) is based
on the estimation of a within model in the first step; the rest follows as
above. It is estimated by:

```{r }
zz <- pggls(log(emp)~log(wage)+log(capital), data=EmplUK, model="within")
```

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.


# Tests{#tests}

As sketched in Section [linear panel model](#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](#robust).

## Tests of poolability

`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
:

```{r }
znp <- pvcm(inv~value+capital, data=Grunfeld, model="within")
zplm <- plm(inv~value+capital, data=Grunfeld, model="within")
pooltest(zplm, znp)
```

The same test can be computed using a formula as first argument of the
`pooltest` function:

```{r results='hide'}
pooltest(inv~value+capital, data=Grunfeld, model="within")
```

## Tests for individual and time effects


`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"`: @HOND:85, the default value,
- `"bp"`: @BREU:PAGA:80,
- `"kw"`: @KING:WU:97^[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.],
- `"ghm"`: @GOUR:HOLL:MONF:82.

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.^[The `"bp"` test
for unbalanced panels was derived in @BALT:LI:90, the `"kw"` test for
unbalanced panels in @BALT:CHAN:LI:98. The `"ghm"` test and the `"kw"`
test were extended to two--way effects in @BALT:CHAN:LI:92. For a
concise overview of all these statistics see @BALT:13 Sec. 4.2,
pp. 68--76 (for balanced panels) and Sec. 9.5, pp. 200--203 (for
unbalanced panels) or @BALT:21, Sec. 4.2, pp. 81-84 (balanced), Sec. 9.6,
pp. 243-246 (unbalanced).]

To test the presence of individual and time effects in the
`Grunfeld` example, using the @GOUR:HOLL:MONF:82 test,
we use:

```{r }
g <- plm(inv ~ value + capital, data=Grunfeld, model="pooling")
plmtest(g, effect="twoways", type="ghm")
```

or

```{r results='hide'}
plmtest(inv~value+capital, data=Grunfeld, effect="twoways", type="ghm")
```



`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.

```{r }
gw <- plm(inv ~ value + capital, data=Grunfeld, effect="twoways", model="within")
gp <- plm(inv ~ value + capital, data=Grunfeld, model="pooling")
pFtest(gw, gp)
```

```{r results='hide'}
pFtest(inv~value+capital, data=Grunfeld, effect="twoways")
```

## Hausman test

`phtest` computes the Hausman test which is based on the
comparison of two sets of estimates (see @HAUS:78). 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:

```{r }
gw <- plm(inv~value+capital, data=Grunfeld, model="within")
gr <- plm(inv~value+capital, data=Grunfeld, model="random")
phtest(gw, gr)
```

## Tests of serial correlation{#serialcor}


A model with individual effects has composite errors that are serially
correlated by definition. The presence of the time-invariant error
component^[Here we treat fixed and random effects alike, as components
of the error term, according with the modern approach in econometrics
(see @WOOL:02, @WOOL:10).] 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 residuals^[Neglecting
time effects may also lead to serial correlation in residuals (as
observed in @WOOL:02 10.4.1).].  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.

### Unobserved effects test

The unobserved effects test *à la Wooldridge*
(see @WOOL:02 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 @WOOL:02, "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 @MUNN:90:

```{r wtest}
pwtest(log(gsp)~log(pcap)+log(pc)+log(emp)+unemp, data=Produc)
```

### Locally robust tests for serial correlation or random effects

The 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
@BALT:LI:91 and @BALT:LI:95 and is implemented as an
option in `pbsytest`:

```{r pbsytestJoint}
pbsytest(log(gsp)~log(pcap)+log(pc)+log(emp)+unemp, data=Produc, test="j")
```

Rejection 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:YOON:01 (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
test^[$LM_3$ in @BALT:LI:95.].  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
@BALT:LI:95 (see `pbltest`).

The serial correlation version is the default:

```{r pbsytestAR}
pbsytest(log(gsp)~log(pcap)+log(pc)+log(emp)+unemp, data=Produc)
```

The BSY test for random effects is implemented in the one-sided
version^[Corresponding to $RSO^*_{\mu}$ in the original paper.],
which takes heed that the variance of the random effect
must be non-negative:

```{r pbsytestRE}
pbsytest(log(gsp)~log(pcap)+log(pc)+log(emp)+unemp, data=Produc, test="re")
```

### Conditional LM test for AR(1) or MA(1) errors under random effects

@BALT:LI:91 and @BALT:LI:95 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"`:

```{r pbltest}
pbltest(log(gsp)~log(pcap)+log(pc)+log(emp)+unemp,
        data=Produc, alternative="onesided")
```

As 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](#nlme)).

### General serial correlation tests

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 @WOOL:02, 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](#software-approach)).

For the random effects model, @WOOL:02 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 @WOOL:02 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"^[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 @BALT:LI:95 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
@BHAR:FRAN:NARE:82, a (modified) version for unbalanced and/or
non-consecutive panels is implemented in `pbnftest` as is Baltagi-Wu's
LBI statistic (for both see @BALT:WU:99).].  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`:

```{r generalAR}
pbgtest(grun.fe, order = 2)
```

The 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).


### Wooldridge's test for serial correlation in "short" FE panels

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"`)^[see subsection [robust covariance matrix estimation](#robust).].

```{r pwartest}
pwartest(log(emp) ~ log(wage) + log(capital), data=EmplUK)
```

The test is applicable to any FE panel model, and in particular to
"short" panels with small $T$ and large $n$.



### Wooldridge's first-difference-based test

In the context of the first difference model,
@WOOL:02, 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 model^[Here, $e_{it}$ for notational simplicity (and as in
Wooldridge): equivalent to $\Delta \epsilon_{it}$ in the general notation of
the paper.]
$e_{it} \equiv
u_{it}-u_{i,t-1}$ 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.  @DRUK:03
provides Monte Carlo evidence of the good empirical properties of the
test.

On the other extreme (see @WOOL:02 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:

```{r pwfdtest1}
pwfdtest(log(emp) ~ log(wage) + log(capital), data=EmplUK)
```

while specifying `h0="fe"` the null hypothesis becomes no serial
correlation in *original* errors, which is similar to the `pwartest`.

```{r pwfdtest2}
pwfdtest(log(emp) ~ log(wage) + log(capital), data=EmplUK, h0="fe")
```

Not 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](#robust) in inference.


## Tests for cross-sectional dependence

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
matrix^[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 @DEHO:SARA:06).].
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.


### CD and LM-type tests for global cross-sectional dependence

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 [@BREU:PAGA:80] 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 @PESA:04).

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 (@BALT:FENG:KAO:12)^[The unbalanced version of this statistic uses
max(Tij) for T in the bias-correction term.]. 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 (@PESA:04, @PESA:15) $CD$ 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`.

```{r pcdtest1}
pcdtest(inv~value+capital, data=Grunfeld)
```

If a different model specification (`within`, `random`, ...)
is assumed consistent, one can resort to its residuals for
testing^[This is also the only solution when the time dimension's length is
insufficient for estimating the heterogeneous model.]
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.

```{r pcdtest2}
pcdtest(inv~value+capital, data=Grunfeld, model="within")
```



If the time dimension is insufficient and `model=NULL`, the
function defaults to estimation of a `within` model and issues a
warning.


### CD(p) test for local cross-sectional dependence

A *local* variant of the $CD$ test, called $CD(p)$ test
[@PESA:04], 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 [@PESA:04] 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` ^[The very comprehensive package `spdep` for spatial dependence analysis
(see @BIVA:08) 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 `nb`s
through `nblag`.].

## Panel unit root tests

### Overview of functions for panel unit root testing


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), and
- `phansi` (Simes' test).

While `purtest` implements various tests which can be selected via its `test`
argument, `cipstest` and `phansi` 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:CHU:02), see below for
              a theoretical exposition ([Levin-Lin-Chu test](#levinlin))),
- `"ips"`, for Im-Pesaran-Shin (IPS) test by @IM:PESAR:SHIN:03, see below for
              a theoretical exposition ([Im-Pesaran-Shin test](#ips))),
- `"madwu"`, is the inverse $\chi^2$ test by @MADDA:WU:99, also called
            P test by @CHOI:01,
- `"Pm"`, is the modified P test proposed by @CHOI:01 for large N,
- `"invnormal"`, is the inverse normal test (@CHOI:01),
- `"logit"`, is the logit test (@CHOI:01),
- `"hadri"`, for Hadri's test (@HADR:00).

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
(@KWIA:PHIL:SCHM:SHIN:92), standardized
by their asymptotic mean and standard deviation.
Among the tests in `purtest`, `"madwu"`, `"Pm"`, `"invormal"`, and `"logit"` are
Fisher-type tests.^[The individual p-values for the Fisher-type tests
are approximated as described in @MACK:96 if the package
`urca` (@PFAFF:08) is available, otherwise as described in @MACK:94.]

`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 (@pes07) 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 `phansi` implements the idea of @HANCK:13 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](#phansi).


### Preliminary results

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 Schwarz information criterion (SIC) (also known as Bayesian information criterion (BIC)),
- the Akaike information criterion (AIC),
- the Hall's method, which consist in removing the higher lags while
  they are not significant.

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:

$$
\hat{\sigma}_{\epsilon_i}^2 = \frac{\sum_{t=p_i+1}^{T} e_{it}^2}{df_i}
$$



### Levin-Lin-Chu model{#levinlin}

Then, as per @LEVIN:LIN:CHU:02, 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:
```{r levinlin}
data("HousePricesUS", package = "pder")
lprice <- log(pdata.frame(HousePricesUS)$price)
(lev <- purtest(lprice, test = "levinlin", lags = 2, exo = "trend"))
summary(lev) ### gives details
```


### Im-Pesaran-Shin (IPS) test{#ips}

This test by @IM:PESAR:SHIN:03 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:
```{r ips}
purtest(lprice, test = "ips", lags = 2, exo = "trend")
```



### Simes' approach: intersecting hypotheses{#phansi}

A different approach to panel unit root testing can be drawn from the general
Simes' test for intersection of individual hypothesis tests [@SIMES:86].
@HANCK:13 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 `phansi` 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:13, 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.
```{r phansi1}
### 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 <- phansi(pvals)
print(h)
h$rejected # logical indicating the individuals with rejected individual H0
```

```{r phansi2, results='hide'}
### input is a (suitable) purtest object / different example
y <- data.frame(split(Grunfeld$inv, Grunfeld$firm))
obj <- purtest(y, pmax = 4, exo = "intercept", test = "madwu")
phansi(obj, alpha = 0.06) # test with significance level set to 6 %
```


## Robust covariance matrix estimation{#robust}

Robust 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 @mil17b.

`vcovHC` estimates three "flavours" of White's heteroskedasticity-consistent
covariance matrix^[See @WHIT:80 and @WHIT:84b.] (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 @GREE:03, 13.7.1--2 and
@WOOL:02, 10.7.2; `"arellano"` (see ibid. and the original ref. @AREL:87) 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](#serialcor) 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 @STOC:WATS:08).

The errors may be weighted according to the schemes proposed by
@MACK:WHIT:85 and @CRIB:04 to improve small-sample performance^[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.].

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 @ZEIL:04). If one
is happy with the defaults, it is easiest to pass the function
itself^[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.]:

```{r vcovHC1}
re <- plm(inv~value+capital, data = Grunfeld, model = "random")
summary(re, vcov = vcovHC) # gives usual summary output but with robust test statistics

library("lmtest")
coeftest(re, vcovHC, df = Inf)
```


else one may do the covariance computation inside the call, thus passing on a matrix:

```{r vcovHC2, results='hide'}
summary(re, vcov = vcovHC(re, method="white2", type="HC3"), df = Inf)
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 function^[Joint zero-restriction testing still
allows providing the `vcov` of the unrestricted model as a matrix, see
the documentation of package `lmtest`.].  In this case, optional
parameters are provided as shown below (see also @ZEIL:04, p. 12):

```{r waldtest-vcovHC}
waldtest(re, update(re, . ~ . -capital),
         vcov=function(x) vcovHC(x, method="white2", type="HC3"))
```

Moreover, `linearHypothesis` from package `car` may be used to test
for linear restrictions:

```{r car-vcovHC}
library("car")
linearHypothesis(re, "2*value=capital", vcov. = vcovHC)
```

A specific methods are also provided for `pcce` and `pgmm` objects, for the latter
`vcovHC` provides the robust covariance matrix proposed by
@WIND:05 for generalized method of moments estimators.


# plm versus nlme and lme4{#nlme}

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 models^[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.].

`R` is particularly strong on mixed models' estimation, thanks to the
long-standing `nlme` package (see @PINH:BATE:DEBR:SARK:07) and the
more recent `lme4` package, based on S4 classes (see @BATE:07)^[The
standard reference on the subject of mixed models in `S`/`R` is
@PINH:BATE:00.].  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`.

## Fundamental differences between the two approaches

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 testing^[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 @BREU:PAGA:80.]  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.

## Some false friends

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., @WOOL:02 10.2.1). It has been
recognized since Mundlak's classic paper (@MUND:78) 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 inconsistently^[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.], 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 (viz. time period, or both) (often termed
 *least squares dummy variables*, or LSDV model) which can
 trivially be estimated by OLS.
-->

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.



## A common taxonomy

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:02] and the `nlme` estimation commands for maximum likelihood
estimation of an equivalent specification^[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).].

### The Laird-Ware representation for mixed models

A general representation for the linear mixed effects model is given
in @LAIR:WARE:82.

$$
\begin{array}{rcl}
y_{it} & = & \beta_1 x_{1ij} +  \dots + \beta_p x_{pij} \\
 &  & b_1 z_{1ij} +  \dots + b_p z_{pij} + \epsilon_{ij} \\
b_{ik} & \sim & N(0,\psi^2_k), \phantom{p} Cov(b_k,b_{k'}) = \psi_{kk'} \\
\epsilon_{ij} & \sim & N(0,\sigma^2 \lambda_{ijj}), \phantom{p} Cov(\epsilon_{ij},\epsilon_{ij'}) = \sigma^2 \lambda_{ijj'} \\
\end{array}
$$

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 $\sigma^2$.

### Pooling and Within

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.

### Random effects

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:

```{r re2}
library(nlme)
reGLS <- plm(inv~value+capital, data=Grunfeld, model="random")

reML <- lme(inv~value+capital, data=Grunfeld, random=~1|firm)

coef(reGLS)

summary(reML)$coefficients$fixed
```

### Variable coefficients, "random"

Swamy's variable coefficients model [@SWAM:70] 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.

```{r vcmrand}
vcm <- pvcm(inv~value+capital, data=Grunfeld, model="random", effect="time")

vcmML <- lme(inv~value+capital, data=Grunfeld, random=~value+capital|year)

coef(vcm)

summary(vcmML)$coefficients$fixed
```

### Variable coefficients, "within"

This 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):


```{r vcmfixed}
vcmf <- pvcm(inv~value+capital, data=Grunfeld, model="within", effect="time")

vcmfML <- lmList(inv~value+capital|year, data=Grunfeld)
```

### General FGLS

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.

```{r gglsre}
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)

summary(gglsML)$coefficients
```

The *within* case is analogous, with the regressor set augmented
by $n-1$ group dummies.

## Some useful "econometric" models in nlme
Finally, amongst the many possible specifications estimable with
`nlme`, we report a couple cases that might be especially
interesting to applied econometricians.

### AR(1) pooling or random effects panel

Linear models with groupwise structures of time-dependence^[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*.]  may be fitted by `gls()`, specifying the correlation
structure in the `correlation` option^[note that the time index is
coerced to numeric before the estimation.]:


```{r lmAR1}
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 @BALT:05; @BALT:13; @BALT:21, ch. 5), which is a very common specification
in econometrics, may be fit by `lme` specifying an additional
random intercept:

```{r reAR1}
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:

```{r fetchcoefs}
summary(reAR1ML)$coefficients$fixed
coef(reAR1ML$modelStruct$corStruct, unconstrained=FALSE)
```

Significance 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.

### An LR test for serial correlation and one for random effects

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()`.


```{r LRar}
lmML <- gls(inv~value+capital, data=Grunfeld)
anova(lmML, lmAR1ML)
```


The 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:

```{r LRarsubRE}
anova(reML, reAR1ML)
```

A likelihood ratio test for random effects compares the specifications
with and without random effects and spherical idiosyncratic errors:


```{r LRre}
anova(lmML, reML)
```


The 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:

```{r LRresubAR}
anova(lmAR1ML, reAR1ML)
```

whence we see that the Grunfeld model specification doesn't seem to need any
random effects once we control for serial correlation in the data.


# Conclusions{#conclusions}

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.

<!--
We plan to expand the methods in this paper to systems of equations
and to the estimation and testing of models with autoregressive
errors. Additions to the misspecification testing toolbox and
covariance estimators robust vs. cross-sectional correlation are also
in the offing. Lastly, conditional visualization features in the
`R` environment seem to offer a promising toolbox for visual
diagnostics, which is another subject for future work.
-->

We plan to expand the methods in this paper to systems of equations
and to the estimation of models with autoregressive errors. Addition
of covariance estimators robust vs. cross-sectional correlation are
also in the offing. Lastly, conditional visualization features in the
`R` environment seem to offer a promising toolbox for visual
diagnostics, which is another subject for future work.


# Acknowledgments {-}

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.

# Bibliography {-}

 <!-- Local IspellDict: english --> <!-- Local IspellPersDict: ~/emacs/.ispell-english -->