# Estimation of error components models with the plm function

#### Yves Croissant

#### 2022-03-03

`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 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 < 0.05

The 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: 0.8601

B. 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: < 0.000000000000000222

# The twoways effect model

The 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 < 0.05

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.

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 < 0.05

# 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.

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 < 0.05

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 B. H. Baltagi and Chang (1994), p. 73). A more detailed analysis of Stata’s Swamy-Arora estimation procedure is given by Cottrell (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 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 < 0.05

The 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 < 0.05

# Nested error component model

This 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 :

- \(\mu_j\) 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 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

# Bibliography

Amemiya, T. 1971. “The Estimation of the Variances in a Variance–Components Model.” International Economic Review 12: 1–13.
Amemiya, Takeshi, and Thomas E MaCurdy. 1986. “Instrumental-Variable Estimation of an Error-Components Model.” Econometrica 54 (4): 869–80.
Balestra, P., and J. Varadharajan–Krishnakumar. 1987. “Full Information Estimations of a System of Simultaneous Equations with Error Components.” Econometric Theory 3: 223–46.
Baltagi, B. H. 1981. “Simultaneous Equations with Error Components.” Journal of Econometrics 17: 21–49.
Baltagi, B. H. 2005. Econometric Analysis of Panel Data. 3rd ed. John Wiley; Sons ltd.
———. 2009. A Companion to Econometric Analysis of Panel Data. John Wiley; Sons ltd.
———. 2013. Econometric Analysis of Panel Data. 5th ed. John Wiley; Sons ltd.
———. 2021. Econometric Analysis of Panel Data. 6th ed. Springer.
Baltagi, B. H., and Y. J. Chang. 1994. “Incomplete Panels: A Comparative Study of Alternative Estimators for the Unbalanced One-Way Error Component Regression Model.” Journal of Econometrics 62: 67–89.
Baltagi, B. H., S. H. Song, and B. C. Jung. 2001. “The Unbalanced Nested Error Component Regression Model.” Journal of Econometrics 101: 357–81.
Breusch, Trevor S, Grayham E Mizon, and Peter Schmidt. 1989. “Efficient Estimation Using Panel Data.” Econometrica 57 (3): 695–700.
Cottrell, A. 2017. “Random Effects Estimators for Unbalanced Panel Data: A Monte Carlo Analysis.” Gretl Working Papers, no. 4. https://EconPapers.repec.org/RePEc:anc:wgretl:4.
Hausman, J. A., and W. E. Taylor. 1981. “Panel Data and Unobservable Individual Effects.” Econometrica 49: 1377–98.
Nerlove, M. 1971. “Further Evidence on the Estimation of Dynamic Economic Relations from a Time–Series of Cross–Sections.” Econometrica 39: 359–82.
Swamy, P. A. V. B., and S. S Arora. 1972. “The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models.” Econometrica 40: 261–75.
Wallace, T. D., and A. Hussain. 1969. “The Use of Error Components Models in Combining Cross Section with Time Series Data.” Econometrica 37 (1): 55–72.