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):
plmThe 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 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"). 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")), Wallace and
Hussain (1969),"swar" (equivalent to
models = c("within", "Between")), Swamy and Arora (1972),"amemiya" (equivalent to
models = c("within")), T. Amemiya
(1971),"nerlove", which is a specific method which doesn’t fit
to the quadratic form methodology described above (Nerlove (1971)) 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
T. Amemiya (1971) estimator of the
individual component of the variance under-estimates as the
time-invariant covariates disappear in the within regression. In this
case, Hausman and Taylor (1981) 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 B. H. Baltagi (2013), p. 21; B. H. Baltagi (2021), 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)")##
## =================================================================================================
## ols between within walhus amemiya swar
## -------------------------------------------------------------------------------------------------
## value 0.11556 *** 0.13465 ** 0.11012 *** 0.10979 *** 0.10978 *** 0.10978 ***
## (0.00584) (0.02875) (0.01186) (0.01052) (0.01048) (0.01049)
## capital 0.23068 *** 0.03203 0.31007 *** 0.30818 *** 0.30808 *** 0.30811 ***
## (0.02548) (0.19094) (0.01735) (0.01717) (0.01718) (0.01718)
## -------------------------------------------------------------------------------------------------
## R^2 0.81241 0.85777 0.76676 0.76941 0.76954 0.76950
## Adj. R^2 0.81050 0.81713 0.75311 0.76707 0.76720 0.76716
## Num. obs. 200 10 200 200 200 200
## s_idios 53.74518 52.76797 52.76797
## s_id 87.35803 83.52354 84.20095
## =================================================================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05The estimated variance can be extracted using the ercomp
function. For example, for the amemiya model :
ercomp(amemiya)## var std.dev share
## idiosyncratic 2784.46 52.77 0.285
## individual 6976.18 83.52 0.715
## theta: 0.8601B. H. Baltagi (2013), p. 27; B. H. Baltagi (2021), 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)## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
##
## Call:
## plm(formula = log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp,
## data = Produc, model = "random", random.method = "swar",
## random.dfcor = 3)
##
## Balanced Panel: n = 48, T = 17, N = 816
##
## Effects:
## var std.dev share
## idiosyncratic 0.001454 0.038137 0.175
## individual 0.006838 0.082691 0.825
## theta: 0.8888
##
## Residuals:
## Min. 1st Qu. Median 3rd Qu. Max.
## -0.1067230 -0.0245520 -0.0023694 0.0217333 0.1996307
##
## Coefficients:
## Estimate Std. Error z-value Pr(>|z|)
## (Intercept) 2.13541100 0.13346149 16.0002 < 0.00000000000000022 ***
## log(pcap) 0.00443859 0.02341732 0.1895 0.8497
## log(pc) 0.31054843 0.01980475 15.6805 < 0.00000000000000022 ***
## log(emp) 0.72967053 0.02492022 29.2803 < 0.00000000000000022 ***
## unemp -0.00617247 0.00090728 -6.8033 0.00000000001023 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 29.209
## Residual Sum of Squares: 1.1879
## R-Squared: 0.95933
## Adj. R-Squared: 0.95913
## Chisq: 19131.1 on 4 DF, p-value: < 0.000000000000000222The two-ways effect model is obtained by setting the
effect argument to "twoways". B. H. Baltagi (2013) pp. 51-53; B. H. Baltagi (2021), 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)##
## ==========================================================
## Wallace-Hussain Swamy-Arora Amemiya
## ----------------------------------------------------------
## (Intercept) -57.81705 * -57.86538 * -63.89217 *
## (28.63258) (29.39336) (30.53284)
## value 0.10978 *** 0.10979 *** 0.11145 ***
## (0.01047) (0.01053) (0.01096)
## capital 0.30807 *** 0.30819 *** 0.32353 ***
## (0.01719) (0.01717) (0.01877)
## ----------------------------------------------------------
## s_idios 55.33298 51.72452 51.72452
## s_id 87.31428 84.23332 89.26257
## s_time 0.00000 0.00000 15.77783
## R^2 0.76956 0.76940 0.74898
## Adj. R^2 0.76722 0.76706 0.74643
## Num. obs. 200 200 200
## ==========================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05The 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.
B. H. Baltagi (2009) 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)##
## ==========================================================
## Wallace-Hussain Swamy-Arora Amemiya
## ----------------------------------------------------------
## (Intercept) 2.39200 *** 2.36350 *** 2.85210 ***
## (0.13833) (0.13891) (0.18502)
## log(pcap) 0.02562 0.01785 0.00221
## (0.02336) (0.02332) (0.02469)
## log(pc) 0.25781 *** 0.26559 *** 0.21666 ***
## (0.02128) (0.02098) (0.02438)
## log(emp) 0.74180 *** 0.74490 *** 0.77005 ***
## (0.02371) (0.02411) (0.02584)
## unemp -0.00455 *** -0.00458 *** -0.00398 ***
## (0.00106) (0.00102) (0.00108)
## ----------------------------------------------------------
## s_idios 0.03571 0.03429 0.03429
## s_id 0.08244 0.08279 0.15390
## s_time 0.01595 0.00984 0.02608
## R^2 0.92915 0.93212 0.85826
## Adj. R^2 0.92880 0.93178 0.85756
## Num. obs. 816 816 816
## ==========================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05Two difficulties arise with unbalanced panels :
B. H. Baltagi (2021), B. H. Baltagi (2013), and B. H. Baltagi (2009) present results of the
estimation of the Swamy and Arora (1972)
model with the Hedonic data set. B.
H. Baltagi (2013), p. 195; B. H. Baltagi
(2021), p. 237, table 9.1, presents the Stata output and B. H. Baltagi (2009), p. 211 presents EViews’
output. EViews’ Wallace-Hussain estimator is reported in B. H. Baltagi (2009), 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)##
## ======================================================================================
## EViews Stata Wallace-Hussain
## --------------------------------------------------------------------------------------
## (Intercept) 9.68587 (0.19751) *** 9.67780 (0.20714) *** 9.68443 (0.19922) ***
## crim -0.00741 (0.00105) *** -0.00723 (0.00103) *** -0.00738 (0.00105) ***
## zn 0.00008 (0.00065) 0.00004 (0.00069) 0.00007 (0.00066)
## indus 0.00156 (0.00403) 0.00208 (0.00434) 0.00165 (0.00409)
## chasyes -0.00442 (0.02921) -0.01059 (0.02896) -0.00565 (0.02916)
## nox -0.00584 (0.00125) *** -0.00586 (0.00125) *** -0.00585 (0.00125) ***
## rm 0.00906 (0.00119) *** 0.00918 (0.00118) *** 0.00908 (0.00119) ***
## age -0.00086 (0.00047) -0.00093 (0.00046) * -0.00087 (0.00047)
## dis -0.14442 (0.04409) ** -0.13288 (0.04568) ** -0.14236 (0.04439) **
## rad 0.09598 (0.02661) *** 0.09686 (0.02835) *** 0.09614 (0.02692) ***
## tax -0.00038 (0.00018) * -0.00037 (0.00019) * -0.00038 (0.00018) *
## ptratio -0.02948 (0.00907) ** -0.02972 (0.00975) ** -0.02951 (0.00919) **
## blacks 0.56278 (0.10197) *** 0.57506 (0.10103) *** 0.56520 (0.10179) ***
## lstat -0.29107 (0.02393) *** -0.28514 (0.02385) *** -0.28991 (0.02391) ***
## --------------------------------------------------------------------------------------
## s_idios 0.13025 0.13025 0.14050
## s_id 0.11505 0.12974 0.12698
## R^2 0.99091 0.99029 0.99081
## Adj. R^2 0.99067 0.99004 0.99057
## Num. obs. 506 506 506
## ======================================================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05The 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 B. H. Baltagi
and Chang (1994), p. 73). A more detailed analysis of Stata’s
Swamy-Arora estimation procedure is given by Cottrell (2017).
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 instrumental variables estimator used is indicated with the
inst.method argument:
"bvk", from Balestra and
Varadharajan–Krishnakumar (1987), 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 B. H. Baltagi
(1981), 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 Takeshi Amemiya and
MaCurdy (1986), 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 Breusch, Mizon, and
Schmidt (1989), 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 B. H. Baltagi (2005), pp. 117/120; B. H. Baltagi (2013), pp. 133/137; B. H. Baltagi (2021), 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))##
## ===================================================================
## FE2SLS BE2SLS EC2SLS G2SLS
## -------------------------------------------------------------------
## lprbarr -0.57551 -0.50294 * -0.41293 *** -0.41414
## (0.80218) (0.24062) (0.09740) (0.22105)
## lpolpc 0.65753 0.40844 * 0.43475 *** 0.50495 *
## (0.84687) (0.19300) (0.08970) (0.22778)
## lprbconv -0.42314 -0.52477 *** -0.32289 *** -0.34325 **
## (0.50194) (0.09995) (0.05355) (0.13246)
## lprbpris -0.25026 0.18718 -0.18632 *** -0.19005 **
## (0.27946) (0.31829) (0.04194) (0.07334)
## lavgsen 0.00910 -0.22723 -0.01018 -0.00644
## (0.04899) (0.17851) (0.02702) (0.02894)
## ldensity 0.13941 0.22562 * 0.42903 *** 0.43434 ***
## (1.02124) (0.10247) (0.05485) (0.07115)
## lwcon -0.02873 0.31400 -0.00748 -0.00430
## (0.05351) (0.25910) (0.03958) (0.04142)
## lwtuc 0.03913 -0.19894 0.04545 * 0.04446 *
## (0.03086) (0.19712) (0.01979) (0.02154)
## lwtrd -0.01775 0.05356 -0.00814 -0.00856
## (0.04531) (0.29600) (0.04138) (0.04198)
## lwfir -0.00934 0.04170 -0.00364 -0.00403
## (0.03655) (0.30562) (0.02892) (0.02946)
## lwser 0.01859 -0.13543 0.00561 0.01056
## (0.03882) (0.17365) (0.02013) (0.02158)
## lwmfg -0.24317 -0.04200 -0.20414 * -0.20180 *
## (0.41955) (0.15627) (0.08044) (0.08394)
## lwfed -0.45134 0.14803 -0.16351 -0.21346
## (0.52712) (0.32565) (0.15945) (0.21510)
## lwsta -0.01875 -0.20309 -0.05405 -0.06012
## (0.28082) (0.29815) (0.10568) (0.12031)
## lwloc 0.26326 0.04444 0.16305 0.18354
## (0.31239) (0.49436) (0.11964) (0.13968)
## lpctymle 0.35112 -0.09472 -0.10811 -0.14587
## (1.01103) (0.19180) (0.13969) (0.22681)
## smsayes -0.08050 -0.22515 -0.25955
## (0.14423) (0.11563) (0.14997)
## lpctmin 0.16890 ** 0.18904 *** 0.19488 ***
## (0.05270) (0.04150) (0.04594)
## (Intercept) -1.97714 -0.95380 -0.45385
## (4.00081) (1.28397) (1.70298)
## -------------------------------------------------------------------
## R^2 0.44364 0.87385 0.59847 0.59230
## Adj. R^2 0.32442 0.83729 0.58115 0.57472
## Num. obs. 630 90 630 630
## s_idios 0.14924 0.14924
## s_id 0.21456 0.21456
## ===================================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05The Hausman-Taylor model (Hausman and Taylor
(1981)) may be estimated with the plm function by
setting argument random.method = "ht" and
inst.method = "baltagi". The following example is from
B. H. Baltagi (2005), p. 130; B. H. Baltagi (2013), pp. 145-7, tables 7.4-7.6;
B. H. Baltagi (2021), 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)##
## ===================================================================
## Hausman-Taylor Amemiya-MaCurdy Breusch-Mizon-Schmidt
## -------------------------------------------------------------------
## (Intercept) 2.91273 *** 2.92734 *** 1.97944 ***
## (0.28365) (0.27513) (0.26724)
## wks 0.00084 0.00084 0.00080
## (0.00060) (0.00060) (0.00060)
## southyes 0.00744 0.00728 0.01467
## (0.03196) (0.03194) (0.03188)
## smsayes -0.04183 * -0.04195 * -0.05204 **
## (0.01896) (0.01895) (0.01891)
## marriedyes -0.02985 -0.03009 -0.03926 *
## (0.01898) (0.01897) (0.01892)
## exp 0.11313 *** 0.11297 *** 0.10867 ***
## (0.00247) (0.00247) (0.00246)
## exp^2 -0.00042 *** -0.00042 *** -0.00049 ***
## (0.00005) (0.00005) (0.00005)
## bluecolyes -0.02070 -0.02085 -0.01539
## (0.01378) (0.01377) (0.01374)
## ind 0.01360 0.01363 0.01902
## (0.01524) (0.01523) (0.01520)
## unionyes 0.03277 * 0.03248 * 0.03786 *
## (0.01491) (0.01489) (0.01486)
## sexfemale -0.13092 -0.13201 -0.18027
## (0.12666) (0.12660) (0.12639)
## blackyes -0.28575 -0.28590 -0.15636
## (0.15570) (0.15549) (0.15506)
## ed 0.13794 *** 0.13720 *** 0.22066 ***
## (0.02125) (0.02057) (0.01985)
## -------------------------------------------------------------------
## s_idios 0.15180 0.15180 0.15180
## s_id 0.94180 0.94180 0.94180
## R^2 0.60945 0.60948 0.60686
## Adj. R^2 0.60833 0.60835 0.60572
## Num. obs. 4165 4165 4165
## ===================================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05This section shows how the nested error component model as per B. H. Baltagi, Song, and Jung (2001) 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 :
In the estimated examples below (replication of B. H. Baltagi, Song, and Jung (2001), p. 378,
table 6; B. H. Baltagi (2021), 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)##
## ==========================================================
## Swamy-Arora Wallace-Hussain Amemiya
## ----------------------------------------------------------
## (Intercept) 2.08921 *** 2.08165 *** 2.13133 ***
## (0.14570) (0.15035) (0.16014)
## log(pc) 0.27412 *** 0.27256 *** 0.26448 ***
## (0.02054) (0.02093) (0.02176)
## log(emp) 0.73984 *** 0.74164 *** 0.75811 ***
## (0.02575) (0.02607) (0.02661)
## log(hwy) 0.07274 *** 0.07493 *** 0.07211 **
## (0.02203) (0.02235) (0.02363)
## log(water) 0.07645 *** 0.07639 *** 0.07616 ***
## (0.01386) (0.01387) (0.01402)
## log(util) -0.09437 *** -0.09523 *** -0.10151 ***
## (0.01677) (0.01677) (0.01705)
## unemp -0.00616 *** -0.00615 *** -0.00584 ***
## (0.00090) (0.00091) (0.00091)
## ----------------------------------------------------------
## s_idios 0.03676 0.03762 0.03676
## s_id 0.06541 0.06713 0.08306
## s_gp 0.03815 0.05239 0.04659
## R^2 0.97387 0.97231 0.96799
## Adj. R^2 0.97368 0.97210 0.96776
## Num. obs. 816 816 816
## ==========================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05