Model components for fitted models with plm

Yves Croissant

2024-04-01

plm tries to follow as close as possible the way models are fitted using lm. This relies on the following steps, using the formula-data with some modifications:

compute internally the model.frame by getting the relevant arguments (formula, data, subset, weights, na.action and offset) and the supplementary argument,
extract from the model.frame the response y (with pmodel.response) and the model matrix X (with model.matrix),
call the (non-exported) estimation function plm.fit with X and y as arguments.

Panel data has a special structure which is described by an index argument. This argument can be used in the pdata.frame function which returns a pdata.frame object. A pdata.frame can be used as input to the data argument of plm. If the data argument of plm is an ordinary data.frame, the index argument can also be supplied as an argument of plm. In this case, the pdata.frame function is called internally to transform the data.

Next, the formula, which is the first and mandatory argument of plm is coerced to a Formula object.

model.frame is then called, but with the data argument in the first position (a pdata.frame object) and the formula in the second position. This unusual order of the arguments enables to use a specific model.frame.pdata.frame method defined in plm.

As for the model.frame.formula method, a data.frame is returned, with a terms attribute.

Next, the X matrix is extracted using model.matrix. The usual way to do so is to feed the function with two arguments, a formula or a terms object and a data.frame created with model.frame. lm uses something like model.matrix(terms(mf), mf) where mf is a data.frame created with model.frame. Therefore, model.matrix needs actually one argument and not two and we therefore wrote a model.matrix.pdata.frame which does the job ; the method first checks that the argument has a term attribute, extracts the terms (actually the formula) and then computes the model’s matrix X.

The response y is usually extracted using model.response, with a data.frame created with model.frame as first argument, but it is not generic. We therefore created a generic called pmodel.response and provide a pmodel.response.pdata.frame method. We illustrate these features using a simplified (in terms of covariates) example with the SeatBelt data set:

library("plm")
data("SeatBelt", package = "pder")
SeatBelt$occfat <- with(SeatBelt, log(farsocc / (vmtrural + vmturban)))
pSB <- pdata.frame(SeatBelt)

We start with an OLS (pooling) specification:

formols <- occfat ~ log(usage) + log(percapin)
mfols <- model.frame(pSB, formols)
Xols <- model.matrix(mfols)
y <- pmodel.response(mfols)
coef(lm.fit(Xols, y))

##   (Intercept)    log(usage) log(percapin) 
##     7.4193570     0.1657293    -1.1583712

which is equivalent to:

coef(plm(formols, SeatBelt, model = "pooling"))

##   (Intercept)    log(usage) log(percapin) 
##     7.4193570     0.1657293    -1.1583712

Next, we use an instrumental variables specification. Variable usage is endogenous and instrumented by three variables indicating the law context: ds, dp, and dsp.

The model is described using a two-parts formula, the first part of the RHS describing the covariates and the second part the instruments. The following two formulations can be used:

formiv1 <- occfat ~ log(usage) + log(percapin) | log(percapin) + ds + dp + dsp
formiv2 <- occfat ~ log(usage) + log(percapin) | . - log(usage) + ds + dp + dsp

The second formulation has two advantages:

in the common case when a lot of covariates are instruments, these covariates don’t need to be indicated in the second RHS part of the formula,
the endogenous variables clearly appear as they are proceeded by a - sign in the second RHS part of the formula.

The formula is coerced to a Formula, using the Formula package. model.matrix.pdata.frame then internally calls model.matrix.Formula in order to extract the covariates and instruments model matrices:

mfSB1 <- model.frame(pSB, formiv1)
X1 <- model.matrix(mfSB1, rhs = 1)
W1 <- model.matrix(mfSB1, rhs = 2)
head(X1, 3) ; head(W1, 3)

##    (Intercept) log(usage) log(percapin)
## 8            1 -0.7985077      9.955748
## 9            1 -0.4155154      9.975622
## 10           1 -0.4155154     10.002110

##    (Intercept) log(percapin) ds dp dsp
## 8            1      9.955748  0  0   0
## 9            1      9.975622  1  0   0
## 10           1     10.002110  1  0   0

For the second (and preferred formulation), the dot argument should be set and is passed to the Formula methods. . has actually two meanings:

all available covariates,
the previous covariates used while updating a formula.

which correspond respectively to dot = "seperate" (the default) and dot = "previous". See the difference between the following two examples:

library("Formula")
head(model.frame(Formula(formiv2), SeatBelt), 3)

##       occfat log(usage) log(percapin) state year farsocc farsnocc usage
## 8  -3.788976 -0.7985077      9.955748    AK 1990      90        8  0.45
## 9  -3.904837 -0.4155154      9.975622    AK 1991      81       20  0.66
## 10 -3.699611 -0.4155154     10.002110    AK 1992      95       13  0.66
##    percapin unemp  meanage   precentb   precenth  densurb   densrur
## 8     21073  7.05 29.58628 0.04157167 0.03252657 1.099419 0.1906836
## 9     21496  8.75 29.82771 0.04077293 0.03280357 1.114670 0.1906712
## 10    22073  9.24 30.21070 0.04192957 0.03331731 1.114078 0.1672785
##         viopcap    proppcap vmtrural vmturban fueltax lim65 lim70p mlda21 bac08
## 8  0.0009482704 0.008367458     2276     1703       8     0      0      1     0
## 9  0.0010787370 0.008940661     2281     1740       8     0      0      1     0
## 10 0.0011257068 0.008366873     2005     1836       8     1      0      1     0
##    ds dp dsp
## 8   0  0   0
## 9   1  0   0
## 10  1  0   0

head(model.frame(Formula(formiv2), SeatBelt, dot = "previous"), 3)

##       occfat log(usage) log(percapin) ds dp dsp
## 8  -3.788976 -0.7985077      9.955748  0  0   0
## 9  -3.904837 -0.4155154      9.975622  1  0   0
## 10 -3.699611 -0.4155154     10.002110  1  0   0

In the first case, all the covariates are returned by model.frame as the . is understood by default as “everything”.

In plm, the dot argument is internally set to previous so that the end-user doesn’t have to worry about these subtleties.

mfSB2 <- model.frame(pSB, formiv2)
X2 <- model.matrix(mfSB2, rhs = 1)
W2 <- model.matrix(mfSB2, rhs = 2)
head(X2, 3) ; head(W2, 3)

##    (Intercept) log(usage) log(percapin)
## 8            1 -0.7985077      9.955748
## 9            1 -0.4155154      9.975622
## 10           1 -0.4155154     10.002110

##    (Intercept) log(percapin) ds dp dsp
## 8            1      9.955748  0  0   0
## 9            1      9.975622  1  0   0
## 10           1     10.002110  1  0   0

The IV estimator can then be obtained as a 2SLS estimator: First, regress the covariates on the instruments and get the fitted values:

HX1 <- lm.fit(W1, X1)$fitted.values
head(HX1, 3)

##    (Intercept) log(usage) log(percapin)
## 8            1 -1.0224257      9.955748
## 9            1 -0.5435055      9.975622
## 10           1 -0.5213364     10.002110

Next, regress the response on these fitted values:

coef(lm.fit(HX1, y))

##   (Intercept)    log(usage) log(percapin) 
##     7.5641209     0.1768576    -1.1722590

The same can be achieved in one command by using the formula-data interface with plm:

coef(plm(formiv1, SeatBelt, model = "pooling"))

##   (Intercept)    log(usage) log(percapin) 
##     7.5641209     0.1768576    -1.1722590

or with the ivreg function from package AER (or with the newer function ivreg in package ivreg superseding AER::ivreg()):

coef(AER::ivreg(formiv1, data = SeatBelt))

##   (Intercept)    log(usage) log(percapin) 
##     7.5641209     0.1768576    -1.1722590