Estimation of error components models with the plm function

library("knitr") opts_chunk$set(message = FALSE, warning = FALSE)
library("texreg") extract.plm <- function(model, include.rsquared = TRUE, include.adjrs = TRUE, include.nobs = TRUE, include.ercomp = TRUE, ...) { s <- summary(model, ...) coefficient.names <- rownames(coef(s)) coefficients <- coef(s)[, 1] standard.errors <- coef(s)[, 2] significance <- coef(s)[, 4] rs <- s$r.squared[1] adj <- s$r.squared[2] n <- length(model$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 follow :

$$ y{nt}=\alpha + \beta^\top x{nt} + \epsilon{nt} = \alpha + \beta^\top x{nt} + \eta_n + \mut + \nu{nt} $$

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 is composed of three elements :

  • $\eta_n$ is the individual effect,
  • $\mu_t$ is the time effect,
  • $\epsilon_{nt}$ 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 "indiviudal", "time" and "twoways".
  • model indicates the model to be estimated : "pooling" is just the OLS estimation (equivalent to a call to lm), "between" performs the estimation on the individual or time means, "within" on the deviations from the individual or/and time mean, "fd" on the first differences and "random" 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").
  • 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 deduced. 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" - which is equivalent to models = c("pooling"), @WALL:HUSS:69,
    • "swar" - models = c("within", "Between"), @SWAM:AROR:72,
    • "amemiya" - models = c("within"), @AMEM:71,
    • "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 is under-estimated 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,
    • "nerlove", which is a specific method which doesn't fit to the methodology described above (@NERLO:71).

Note that when only one model is provided in models, this means that the same residuals are used to compute the two quadratic forms.

To illustrate the use of plm, we use examples reproduced in @BALT:13. Table 2.1 on page 21 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, 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 p. 44-46, 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 p. 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 deduced) 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:13 and @BALT:09 present results of the estimation of the @SWAM:AROR:72 model with the Hedonic data set. @BALT:13, p. 174, presents the Stata output as @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 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 using the between and the within transformation as those of the third are only introduced with the within transformation,
  • "am", from @AMEM:MACU:86, the within transformation of the variables of the second parts for each period are also included as instruments,
  • "bmc", from @BREU:MIZO:SCHM:89, the within transformation of the variable of the second and of the third parts 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:2005, p. 120/ @BALT:13, p. 137.

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") screenreg(list(FE2SLS = crwth, EC2SLS = crbalt, G2SLS = crbvk), single.row = TRUE, 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". The following example is from @BALT:2005, p. 130 and @BALT:13, p. 146.

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") screenreg(list("Hausman-Taylor" = ht, "Amemiya-MaCurdy" = am), digits = 5, single.row = TRUE)

Nested error component model

@BALT:SONG:JUNG:01

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"), 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)

Bibliography