Nothing
Estimation of error components models with the plm function"
In plm: Linear Models for Panel Data
library("knitr")
opts_chunk$set(message = FALSE, warning = FALSE)
library("texreg")
extract.plm <- function(model, include.rsquared = TRUE, include.adjrs = TRUE,
include.nobs = TRUE, include.ercomp = TRUE, ...) {
s <- summary(model, ...)
coefficient.names <- rownames(coef(s))
coefficients <- coef(s)[ , 1L]
standard.errors <- coef(s)[ , 2L]
significance <- coef(s)[ , 4L]
rs <- s$r.squared[1L]
adj <- s$r.squared[2L]
n <- length(model$residuals)
gof <- numeric()
gof.names <- character()
gof.decimal <- logical()
if (include.ercomp == TRUE){
if (model$args$model == "random"){
se <- sqrt(ercomp(model)$sigma)
gof <- c(gof, se)
gof.names <- c(gof.names, paste("s_", names(se), sep = ""))
gof.decimal <- c(gof.decimal, rep(TRUE, length(se)))
}
}
if (include.rsquared == TRUE) {
gof <- c(gof, rs)
gof.names <- c(gof.names, "R$^2$")
gof.decimal <- c(gof.decimal, TRUE)
}
if (include.adjrs == TRUE) {
gof <- c(gof, adj)
gof.names <- c(gof.names, "Adj.\ R$^2$")
gof.decimal <- c(gof.decimal, TRUE)
}
if (include.nobs == TRUE) {
gof <- c(gof, n)
gof.names <- c(gof.names, "Num.\ obs.")
gof.decimal <- c(gof.decimal, FALSE)
}
tr <- createTexreg(
coef.names = coefficient.names,
coef = coefficients,
se = standard.errors,
pvalues = significance,
gof.names = gof.names,
gof = gof,
gof.decimal = gof.decimal
)
return(tr)
}
setMethod("extract", signature = className("plm", "plm"),
definition = extract.plm)
plm is a very versatile function which enable the estimation of a
wide range of error component models. Those models can be written as
follows :
$$
y_{nt}=\alpha + \beta^\top x_{nt} + \epsilon_{nt} = \alpha + \beta^\top x_{nt} + \eta_n + \mu_t + \nu_{nt}
$$
where $n$ and $t$ are the individual and time indexes, $y$ the
response, $x$ a vector of covariates, $\alpha$ the overall intercept
and $\beta$ the vector of parameters of interest that we are willing
to estimate. The error term $\epsilon_{nt}$ is composed of three elements
(in the two-way case):
- $\eta_n$ is the individual effect,
- $\mu_t$ is the time effect,
- $\nu_{nt}$ is the idiosyncratic error.
Basic use of plm
The first two arguments of plm are, like for most of the estimation
functions of R a formula which describes the model to be estimated
and a data.frame. subset, weights, and na.action are also
available and have the same behavior as in the lm function. Three
more main arguments can be set :
index helps plm to understand the structure of the data : if
NULL, the first two columns of the data are assumed to contain the
individual or the time index. Otherwise, supply the column names of
the individual and time index as a character, e.g., use something like
c("firm", "year") or just "firm" if there is no explicit time
index.effect indicates the effects that should be taken into account ;
this is one of "individual", "time", and "twoways".model indicates the model to be estimated :
"pooling" is the OLS estimation (equivalent to a call to lm),"between" performs the estimation on the individual or time means, "within" (fixed-effects model) on the deviations from the individual
or/and time mean, "fd" on the first differences, and "random" (random-effects model) perform a feasible generalized least
squares estimation which takes into account the correlation induced by the
presence of individual and/or time effects.
The estimation of all but the last model is straightforward, as it
requires only the estimation by OLS of obvious transformations of
the data. The GLS model requires more explanation. In most of the
cases, the estimation is obtained by quasi-differencing the data from
the individual and/or the time means. The coefficients used to perform
this quasi-difference depends on estimators of the variance of the
components of the error, namely $\sigma^2_\nu$, $\sigma^2_\eta$ in
case of individual effects and $\sigma^2_\mu$ in case of time effects.
The most common technique used to estimate these variance is to use
the following result :
$$
\frac{\mbox{E}(\epsilon^\top W \epsilon)}{N(T-1)} = \sigma_\nu^2
$$
and
$$
\frac{\mbox{E}(\epsilon^\top B \epsilon)}{N} = T \sigma_\eta^2 + \sigma_\nu^2
$$
where $B$ and $W$ are respectively the matrices that performs the
individual (or time) means and the deviations from these
means. Consistent estimators can be obtained by replacing the unknown
errors by the residuals of a consistent preliminary estimation and by
dropping the expecting value operator. Some degree of freedom
correction can also be introduced. plm calls the general function
ercomp to estimate the variances. Important arguments to ercomp
are:
models indicates which models are estimated in order to calculate
the two quadratic forms ; for example c("within", "Between"). Note that
when only one model is provided in models, this means that the same
residuals are used to compute the two quadratic forms.dfcor indicates what kind of degrees of freedom correction is
used : if 0, the quadratic forms are divided by the number of
observations, respectively $N\times T$ and $N$ ; if 1, the
numerators of the previous expressions are used ($N\times (T-1)$ and
$N$) ; if 2, the number of estimated parameters in the preliminary
estimate $K$ is deducted. Finally, if 3, the unbiased version is
computed, which is based on much more complex computations, which
relies on the calculus of the trace of different cross-products
which depends on the preliminary models used.method is an alternative to the models argument; it is one of :
"walhus" (equivalent to setting models = c("pooling")),
@WALL:HUSS:69,"swar" (equivalent to models = c("within", "Between")), @SWAM:AROR:72,"amemiya" (equivalent to models = c("within")), @AMEM:71,"nerlove", which is a specific method which doesn't fit to the
quadratic form methodology described above (@NERLO:71) and uses an
within model for the variance estimation as well,"ht" is an slightly modified version of "amemiya": when there
are time-invariant covariates, the @AMEM:71 estimator of the
individual component of the variance under-estimates as the
time-invariant covariates disappear in the within regression. In
this case, @HAUS:TAYL:81 proposed to regress the estimation of
the individual effects on the time-invariant covariates and use
the residuals in order to estimate the components of the
variance.
Note that for plm, the arguments are random.models, random.dfcor, and
random.method and correspond to arguments models, method, and
random.dfcor of function ercomp with the same values as above, respectively.
To illustrate the use of plm, we use examples reproduced in @BALT:13, p. 21;
@BALT:21, p. 31, table 2.1 presents EViews' results of the estimation on the
Grunfeld data set :
library("plm")
data("Grunfeld", package = "plm")
ols <- plm(inv ~ value + capital, Grunfeld, model = "pooling")
between <- update(ols, model = "between")
within <- update(ols, model = "within")
walhus <- update(ols, model = "random", random.method = "walhus", random.dfcor = 3)
amemiya <- update(walhus, random.method = "amemiya")
swar <- update(amemiya, random.method = "swar")
Note that the random.dfcor argument is set to 3, which means that
the unbiased version of the estimation of the error components is
used. We use the texreg package to present the results :
library("texreg")
screenreg(list(ols = ols, between = between, within = within,
walhus = walhus, amemiya = amemiya, swar = swar),
digits = 5, omit.coef = "(Intercept)")
The estimated variance can be extracted using the ercomp function. For
example, for the amemiya model :
ercomp(amemiya)
@BALT:13, p. 27; @BALT:21, p. 31 presents the Stata estimation of the Swamy-Arora
estimator ; the Swamy-Arora estimator is the same if random.dfcor is
set to 3 or 2 (the quadratic forms are divided by $\sum_n T_n - K - N$
and by $N - K - 1$), so I don't know what is the behaviour of Stata for the
other estimators for which the unbiased estimators differs from the
simple one.
data("Produc", package = "plm")
PrSwar <- plm(log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp, Produc,
model = "random", random.method = "swar", random.dfcor = 3)
summary(PrSwar)
The twoways effect model
The two-ways effect model is obtained by setting the effect argument
to "twoways". @BALT:13 pp. 51-53; @BALT:21, pp. 61-62, tables 3.1-3.3, presents
EViews' output for the Grunfeld data set.
Grw <- plm(inv ~ value + capital, Grunfeld, model = "random", effect = "twoways",
random.method = "walhus", random.dfcor = 3)
Grs <- update(Grw, random.method = "swar")
Gra <- update(Grw, random.method = "amemiya")
screenreg(list("Wallace-Hussain" = Grw, "Swamy-Arora" = Grs, "Amemiya" = Gra), digits = 5)
The estimated variance of the time component is negative for the
Wallace-Hussain as well as the Swamy-Arora models and plm sets it to 0.
@BALT:09 pp. 60-62, presents EViews' output for the Produc data.
data("Produc", package = "plm")
Prw <- plm(log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp, Produc,
model = "random", random.method = "walhus",
effect = "twoways", random.dfcor = 3)
Prs <- update(Prw, random.method = "swar")
Pra <- update(Prw, random.method = "amemiya")
screenreg(list("Wallace-Hussain" = Prw, "Swamy-Arora" = Prs, "Amemiya" = Pra), digits = 5)
Unbalanced panels
Two difficulties arise with unbalanced panels :
- There are no obvious denominators for the quadratic forms of the
residuals that are used to estimate the components of the
variance. The strategy is then to compute the expected value and
equate it to the actual quadratic forms. Detailed formula are
omitted here, they depend on the preliminary estimator.
- For the one-way effect model, the estimator is still obtained by
applying OLS on demeaned data (the individual and the time
means are now deducted) for the within model and on quasi-demeaned
data for the random effects model ; this is not the case for the
two-ways effects model.
@BALT:21, @BALT:13, and @BALT:09 present results of the estimation of the
@SWAM:AROR:72 model with the Hedonic data set. @BALT:13, p. 195;
@BALT:21, p. 237, table 9.1, presents the Stata output and @BALT:09,
p. 211 presents EViews' output. EViews' Wallace-Hussain estimator is
reported in @BALT:09, p. 210.
data("Hedonic", package = "plm")
form <- mv ~ crim + zn + indus + chas + nox + rm +
age + dis + rad + tax + ptratio + blacks + lstat
HedStata <- plm(form, Hedonic, model = "random", index = "townid",
random.models = c("within", "between"))
HedEviews <- plm(form, Hedonic, model = "random", index = "townid",
random.models = c("within", "Between"))
HedEviewsWH <- update(HedEviews, random.models = "pooling")
screenreg(list(EViews = HedEviews, Stata = HedStata, "Wallace-Hussain" = HedEviewsWH),
digits = 5, single.row = TRUE)
The difference is due to the fact that Stata uses a between regression
on $N$ observations while EViews uses a between regression on $\sum_n
T_n$ observations, which are not the same on unbalanced panels. Note
the use of between with or without the B capitalized ("Between" and
"between") in the random.models argument. plm's default is to use
the between regression with $\sum_n T_n$ observations when setting
model = "random", random.method = "swar". The default employed is what the
original paper for the unbalanced one-way Swamy-Arora estimator
defined (in @BALT:CHAN:94, p. 73). A more detailed analysis of Stata's
Swamy-Arora estimation procedure is given by @COTT:2017.
Instrumental variable estimators
All of the models presented above may be estimated using instrumental
variables (IV). The instruments are specified using two- or three-part
formulas, each part being separated by a | sign :
- the first part contains the covariates,
- the second part contains the "double-exogenous" instruments, i.e.,
variables that can be used twice as instruments, using their within
and the between transformation,
- the third part contains the "single-exogenous" instruments, i.e.,
variables for which only the within transformation can be used as
instruments, those variables being correlated with the individual
effects.
The instrumental variables estimator used is indicated with the
inst.method argument:
"bvk", from @BALE:VARA:87, the default value : in this case, all the
instruments are introduced in quasi-differences, using the same
transformation as for the response and the covariates,"baltagi", from @BALT:81, the instruments of the second part are
introduced twice by using the between and the within transformation and
instruments of the third part are introduced with only the within
transformation,"am", from @AMEM:MACU:86, in addition to the instrument set of "baltagi",
the within transformation of the variables of the second part for each period
are also included as instruments,"bms", from @BREU:MIZO:SCHM:89, in addition to the instrument set of
"baltagi", the within transformation of the variables of the second
and the third part for each period are included as instruments.
The various possible values of the inst.method argument are not relevant
for fixed effect IV models as there is only one method for this type of IV
models but many for random effect IV models.
The instrumental variable estimators are illustrated in the following example
from @BALT:05, pp. 117/120; @BALT:13, pp. 133/137; @BALT:21, pp. 162/165, tables
7.1, 7.3.
data("Crime", package = "plm")
crbalt <- 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", inst.method = "baltagi")
crbvk <- update(crbalt, inst.method = "bvk")
crwth <- update(crbalt, model = "within")
crbe <- update(crbalt, model = "between")
screenreg(list(FE2SLS = crwth, BE2SLS = crbe, EC2SLS = crbalt, G2SLS = crbvk),
single.row = FALSE, digits = 5, omit.coef = "(region)|(year)",
reorder.coef = c(1:16, 19, 18, 17))
The Hausman-Taylor model (@HAUS:TAYL:81) may be estimated
with the plm function by setting argument random.method = "ht" and
inst.method = "baltagi".
The following example is from @BALT:05, p. 130; @BALT:13, pp. 145-7,
tables 7.4-7.6; @BALT:21, pp. 174-6 , tables 7.5-7.7.
data("Wages", package = "plm")
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 + exp + I(exp^2) + union,
data = Wages, index = 595,
inst.method = "baltagi", model = "random",
random.method = "ht")
am <- update(ht, inst.method = "am")
bms <- update(ht, inst.method = "bms")
screenreg(list("Hausman-Taylor" = ht, "Amemiya-MaCurdy" = am,
"Breusch-Mizon-Schmidt" = bms),
digits = 5, single.row = FALSE)
Nested error component model
This section shows how the nested error component model as per
@BALT:SONG:JUNG:01 can be estimated. The model is given by :
$$
y_{nt}=\alpha + \beta^\top x_{jnt} + u_{jnt} = \alpha + \beta^\top x_{jnt} + \mu_{j} + \nu_{jn} + \epsilon_{jnt}
$$
where $n$ and $t$ are the individual and time indexes and $j$ is the group index in
which the individuals are nested. The error $u_{jnt}$ consists of three components :
- $\mu_j$ is the group effect,
- $\nu_{jn}$ the nested effect of the individual nested in group $j$
- $\epsilon_{jnt}$ is the idiosyncratic error.
In the estimated examples below (replication of @BALT:SONG:JUNG:01, p. 378,
table 6; @BALT:21, p. 248, table 9.1), states are nested within regions.
The group index is given in the 3rd position of the index argument to
pdata.frame or to plm directly and plm's argument effect is set to
"nested":
data("Produc", package = "plm")
swar <- plm(form <- log(gsp) ~ log(pc) + log(emp) + log(hwy) + log(water) + log(util) + unemp,
Produc, index = c("state", "year", "region"), model = "random", effect = "nested", random.method = "swar")
walhus <- update(swar, random.method = "walhus")
amem <- update(swar, random.method = "amemiya")
screenreg(list("Swamy-Arora" = swar, "Wallace-Hussain" = walhus, "Amemiya" = amem), digits = 5)
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library("knitr") opts_chunk$set(message = FALSE, warning = FALSE)
library("texreg") extract.plm <- function(model, include.rsquared = TRUE, include.adjrs = TRUE, include.nobs = TRUE, include.ercomp = TRUE, ...) { s <- summary(model, ...) coefficient.names <- rownames(coef(s)) coefficients <- coef(s)[ , 1L] standard.errors <- coef(s)[ , 2L] significance <- coef(s)[ , 4L] rs <- s$r.squared[1L] adj <- s$r.squared[2L] n <- length(model$residuals) gof <- numeric() gof.names <- character() gof.decimal <- logical() if (include.ercomp == TRUE){ if (model$args$model == "random"){ se <- sqrt(ercomp(model)$sigma) gof <- c(gof, se) gof.names <- c(gof.names, paste("s_", names(se), sep = "")) gof.decimal <- c(gof.decimal, rep(TRUE, length(se))) } } if (include.rsquared == TRUE) { gof <- c(gof, rs) gof.names <- c(gof.names, "R$^2$") gof.decimal <- c(gof.decimal, TRUE) } if (include.adjrs == TRUE) { gof <- c(gof, adj) gof.names <- c(gof.names, "Adj.\ R$^2$") gof.decimal <- c(gof.decimal, TRUE) } if (include.nobs == TRUE) { gof <- c(gof, n) gof.names <- c(gof.names, "Num.\ obs.") gof.decimal <- c(gof.decimal, FALSE) } tr <- createTexreg( coef.names = coefficient.names, coef = coefficients, se = standard.errors, pvalues = significance, gof.names = gof.names, gof = gof, gof.decimal = gof.decimal ) return(tr) } setMethod("extract", signature = className("plm", "plm"), definition = extract.plm)
plm is a very versatile function which enable the estimation of a
wide range of error component models. Those models can be written as
follows :
$$ y_{nt}=\alpha + \beta^\top x_{nt} + \epsilon_{nt} = \alpha + \beta^\top x_{nt} + \eta_n + \mu_t + \nu_{nt} $$
where $n$ and $t$ are the individual and time indexes, $y$ the response, $x$ a vector of covariates, $\alpha$ the overall intercept and $\beta$ the vector of parameters of interest that we are willing to estimate. The error term $\epsilon_{nt}$ is composed of three elements (in the two-way case):
- $\eta_n$ is the individual effect,
- $\mu_t$ is the time effect,
- $\nu_{nt}$ is the idiosyncratic error.
Basic use of plm
The first two arguments of plm are, like for most of the estimation
functions of R a formula which describes the model to be estimated
and a data.frame. subset, weights, and na.action are also
available and have the same behavior as in the lm function. Three
more main arguments can be set :
indexhelpsplmto understand the structure of the data : ifNULL, the first two columns of the data are assumed to contain the individual or the time index. Otherwise, supply the column names of the individual and time index as a character, e.g., use something likec("firm", "year")or just"firm"if there is no explicit time index.effectindicates the effects that should be taken into account ; this is one of"individual","time", and"twoways".modelindicates the model to be estimated :"pooling"is the OLS estimation (equivalent to a call tolm),"between"performs the estimation on the individual or time means,"within"(fixed-effects model) on the deviations from the individual or/and time mean,"fd"on the first differences, and"random"(random-effects model) perform a feasible generalized least squares estimation which takes into account the correlation induced by the presence of individual and/or time effects.
The estimation of all but the last model is straightforward, as it requires only the estimation by OLS of obvious transformations of the data. The GLS model requires more explanation. In most of the cases, the estimation is obtained by quasi-differencing the data from the individual and/or the time means. The coefficients used to perform this quasi-difference depends on estimators of the variance of the components of the error, namely $\sigma^2_\nu$, $\sigma^2_\eta$ in case of individual effects and $\sigma^2_\mu$ in case of time effects.
The most common technique used to estimate these variance is to use the following result :
$$ \frac{\mbox{E}(\epsilon^\top W \epsilon)}{N(T-1)} = \sigma_\nu^2 $$
and
$$ \frac{\mbox{E}(\epsilon^\top B \epsilon)}{N} = T \sigma_\eta^2 + \sigma_\nu^2 $$
where $B$ and $W$ are respectively the matrices that performs the
individual (or time) means and the deviations from these
means. Consistent estimators can be obtained by replacing the unknown
errors by the residuals of a consistent preliminary estimation and by
dropping the expecting value operator. Some degree of freedom
correction can also be introduced. plm calls the general function
ercomp to estimate the variances. Important arguments to ercomp
are:
modelsindicates which models are estimated in order to calculate the two quadratic forms ; for examplec("within", "Between"). Note that when only one model is provided inmodels, this means that the same residuals are used to compute the two quadratic forms.dfcorindicates what kind of degrees of freedom correction is used : if0, the quadratic forms are divided by the number of observations, respectively $N\times T$ and $N$ ; if1, the numerators of the previous expressions are used ($N\times (T-1)$ and $N$) ; if2, the number of estimated parameters in the preliminary estimate $K$ is deducted. Finally, if3, the unbiased version is computed, which is based on much more complex computations, which relies on the calculus of the trace of different cross-products which depends on the preliminary models used.methodis an alternative to themodelsargument; it is one of :"walhus"(equivalent to settingmodels = c("pooling")), @WALL:HUSS:69,"swar"(equivalent tomodels = c("within", "Between")), @SWAM:AROR:72,"amemiya"(equivalent tomodels = c("within")), @AMEM:71,"nerlove", which is a specific method which doesn't fit to the quadratic form methodology described above (@NERLO:71) and uses an within model for the variance estimation as well,"ht"is an slightly modified version of"amemiya": when there are time-invariant covariates, the @AMEM:71 estimator of the individual component of the variance under-estimates as the time-invariant covariates disappear in the within regression. In this case, @HAUS:TAYL:81 proposed to regress the estimation of the individual effects on the time-invariant covariates and use the residuals in order to estimate the components of the variance.
Note that for plm, the arguments are random.models, random.dfcor, and
random.method and correspond to arguments models, method, and
random.dfcor of function ercomp with the same values as above, respectively.
To illustrate the use of plm, we use examples reproduced in @BALT:13, p. 21;
@BALT:21, p. 31, table 2.1 presents EViews' results of the estimation on the
Grunfeld data set :
library("plm") data("Grunfeld", package = "plm") ols <- plm(inv ~ value + capital, Grunfeld, model = "pooling") between <- update(ols, model = "between") within <- update(ols, model = "within") walhus <- update(ols, model = "random", random.method = "walhus", random.dfcor = 3) amemiya <- update(walhus, random.method = "amemiya") swar <- update(amemiya, random.method = "swar")
Note that the random.dfcor argument is set to 3, which means that
the unbiased version of the estimation of the error components is
used. We use the texreg package to present the results :
library("texreg") screenreg(list(ols = ols, between = between, within = within, walhus = walhus, amemiya = amemiya, swar = swar), digits = 5, omit.coef = "(Intercept)")
The estimated variance can be extracted using the ercomp function. For
example, for the amemiya model :
ercomp(amemiya)
@BALT:13, p. 27; @BALT:21, p. 31 presents the Stata estimation of the Swamy-Arora
estimator ; the Swamy-Arora estimator is the same if random.dfcor is
set to 3 or 2 (the quadratic forms are divided by $\sum_n T_n - K - N$
and by $N - K - 1$), so I don't know what is the behaviour of Stata for the
other estimators for which the unbiased estimators differs from the
simple one.
data("Produc", package = "plm") PrSwar <- plm(log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp, Produc, model = "random", random.method = "swar", random.dfcor = 3) summary(PrSwar)
The twoways effect model
The two-ways effect model is obtained by setting the effect argument
to "twoways". @BALT:13 pp. 51-53; @BALT:21, pp. 61-62, tables 3.1-3.3, presents
EViews' output for the Grunfeld data set.
Grw <- plm(inv ~ value + capital, Grunfeld, model = "random", effect = "twoways", random.method = "walhus", random.dfcor = 3) Grs <- update(Grw, random.method = "swar") Gra <- update(Grw, random.method = "amemiya") screenreg(list("Wallace-Hussain" = Grw, "Swamy-Arora" = Grs, "Amemiya" = Gra), digits = 5)
The estimated variance of the time component is negative for the
Wallace-Hussain as well as the Swamy-Arora models and plm sets it to 0.
@BALT:09 pp. 60-62, presents EViews' output for the Produc data.
data("Produc", package = "plm") Prw <- plm(log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp, Produc, model = "random", random.method = "walhus", effect = "twoways", random.dfcor = 3) Prs <- update(Prw, random.method = "swar") Pra <- update(Prw, random.method = "amemiya") screenreg(list("Wallace-Hussain" = Prw, "Swamy-Arora" = Prs, "Amemiya" = Pra), digits = 5)
Unbalanced panels
Two difficulties arise with unbalanced panels :
- There are no obvious denominators for the quadratic forms of the residuals that are used to estimate the components of the variance. The strategy is then to compute the expected value and equate it to the actual quadratic forms. Detailed formula are omitted here, they depend on the preliminary estimator.
- For the one-way effect model, the estimator is still obtained by applying OLS on demeaned data (the individual and the time means are now deducted) for the within model and on quasi-demeaned data for the random effects model ; this is not the case for the two-ways effects model.
@BALT:21, @BALT:13, and @BALT:09 present results of the estimation of the
@SWAM:AROR:72 model with the Hedonic data set. @BALT:13, p. 195;
@BALT:21, p. 237, table 9.1, presents the Stata output and @BALT:09,
p. 211 presents EViews' output. EViews' Wallace-Hussain estimator is
reported in @BALT:09, p. 210.
data("Hedonic", package = "plm") form <- mv ~ crim + zn + indus + chas + nox + rm + age + dis + rad + tax + ptratio + blacks + lstat HedStata <- plm(form, Hedonic, model = "random", index = "townid", random.models = c("within", "between")) HedEviews <- plm(form, Hedonic, model = "random", index = "townid", random.models = c("within", "Between")) HedEviewsWH <- update(HedEviews, random.models = "pooling") screenreg(list(EViews = HedEviews, Stata = HedStata, "Wallace-Hussain" = HedEviewsWH), digits = 5, single.row = TRUE)
The difference is due to the fact that Stata uses a between regression
on $N$ observations while EViews uses a between regression on $\sum_n
T_n$ observations, which are not the same on unbalanced panels. Note
the use of between with or without the B capitalized ("Between" and
"between") in the random.models argument. plm's default is to use
the between regression with $\sum_n T_n$ observations when setting
model = "random", random.method = "swar". The default employed is what the
original paper for the unbalanced one-way Swamy-Arora estimator
defined (in @BALT:CHAN:94, p. 73). A more detailed analysis of Stata's
Swamy-Arora estimation procedure is given by @COTT:2017.
Instrumental variable estimators
All of the models presented above may be estimated using instrumental
variables (IV). The instruments are specified using two- or three-part
formulas, each part being separated by a | sign :
- the first part contains the covariates,
- the second part contains the "double-exogenous" instruments, i.e., variables that can be used twice as instruments, using their within and the between transformation,
- the third part contains the "single-exogenous" instruments, i.e., variables for which only the within transformation can be used as instruments, those variables being correlated with the individual effects.
The instrumental variables estimator used is indicated with the
inst.method argument:
"bvk", from @BALE:VARA:87, the default value : in this case, all the instruments are introduced in quasi-differences, using the same transformation as for the response and the covariates,"baltagi", from @BALT:81, the instruments of the second part are introduced twice by using the between and the within transformation and instruments of the third part are introduced with only the within transformation,"am", from @AMEM:MACU:86, in addition to the instrument set of"baltagi", the within transformation of the variables of the second part for each period are also included as instruments,"bms", from @BREU:MIZO:SCHM:89, in addition to the instrument set of"baltagi", the within transformation of the variables of the second and the third part for each period are included as instruments.
The various possible values of the inst.method argument are not relevant
for fixed effect IV models as there is only one method for this type of IV
models but many for random effect IV models.
The instrumental variable estimators are illustrated in the following example from @BALT:05, pp. 117/120; @BALT:13, pp. 133/137; @BALT:21, pp. 162/165, tables 7.1, 7.3.
data("Crime", package = "plm") crbalt <- 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", inst.method = "baltagi") crbvk <- update(crbalt, inst.method = "bvk") crwth <- update(crbalt, model = "within") crbe <- update(crbalt, model = "between") screenreg(list(FE2SLS = crwth, BE2SLS = crbe, EC2SLS = crbalt, G2SLS = crbvk), single.row = FALSE, digits = 5, omit.coef = "(region)|(year)", reorder.coef = c(1:16, 19, 18, 17))
The Hausman-Taylor model (@HAUS:TAYL:81) may be estimated
with the plm function by setting argument random.method = "ht" and
inst.method = "baltagi".
The following example is from @BALT:05, p. 130; @BALT:13, pp. 145-7,
tables 7.4-7.6; @BALT:21, pp. 174-6 , tables 7.5-7.7.
data("Wages", package = "plm") 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 + exp + I(exp^2) + union, data = Wages, index = 595, inst.method = "baltagi", model = "random", random.method = "ht") am <- update(ht, inst.method = "am") bms <- update(ht, inst.method = "bms") screenreg(list("Hausman-Taylor" = ht, "Amemiya-MaCurdy" = am, "Breusch-Mizon-Schmidt" = bms), digits = 5, single.row = FALSE)
Nested error component model
This section shows how the nested error component model as per @BALT:SONG:JUNG:01 can be estimated. The model is given by :
$$ y_{nt}=\alpha + \beta^\top x_{jnt} + u_{jnt} = \alpha + \beta^\top x_{jnt} + \mu_{j} + \nu_{jn} + \epsilon_{jnt} $$ where $n$ and $t$ are the individual and time indexes and $j$ is the group index in which the individuals are nested. The error $u_{jnt}$ consists of three components :
- $\mu_j$ is the group effect,
- $\nu_{jn}$ the nested effect of the individual nested in group $j$
- $\epsilon_{jnt}$ is the idiosyncratic error.
In the estimated examples below (replication of @BALT:SONG:JUNG:01, p. 378,
table 6; @BALT:21, p. 248, table 9.1), states are nested within regions.
The group index is given in the 3rd position of the index argument to
pdata.frame or to plm directly and plm's argument effect is set to
"nested":
data("Produc", package = "plm") swar <- plm(form <- log(gsp) ~ log(pc) + log(emp) + log(hwy) + log(water) + log(util) + unemp, Produc, index = c("state", "year", "region"), model = "random", effect = "nested", random.method = "swar") walhus <- update(swar, random.method = "walhus") amem <- update(swar, random.method = "amemiya") screenreg(list("Swamy-Arora" = swar, "Wallace-Hussain" = walhus, "Amemiya" = amem), digits = 5)
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