Instrumental-variable estimators for linear panel models

Two-stage least squares for linear panel models with endogenous regressors: fixed-effects 2SLS, random-effects generalized 2SLS (Balestra and Varadharajan-Krishnakumar) or error-components 2SLS (Baltagi), pooled 2SLS, and the Hausman and Taylor (1981) estimator, which also identifies the effects of time-invariant regressors.

Usage

paniv(
  formula,
  data,
  index,
  model = c("within", "random", "pooling", "hausmantaylor"),
  effect = c("individual", "twoways"),
  inst.method = c("bvk", "baltagi"),
  vcov = c("classical", "cluster"),
  na.action = stats::na.omit
)

Arguments

formula: a two-part formula y ~ regressors | instruments; see the Formula section. lag(x, k), lead(x, k) and diff(x) are taken within each individual on the grid of sorted distinct time values (see "Lags and differences" in panglm(); needs index = c(id, time)); the first waves of each individual get NA and leave the sample, and lag(x, 0:2) gives several lags
data: a data.frame in long format
index: length-1 or length-2 character vector giving the individual (and time) columns
model: one of "within", "random", "pooling" or "hausmantaylor"
effect: "individual", or "twoways" (only for model = "within")
inst.method: instrument set for model = "random": "bvk" (G2SLS, the default, as in plm) or "baltagi" (EC2SLS)
vcov: "classical" or "cluster" (cluster-robust by individual); see vcov.paniv() to recompute after fitting
na.action: missing-data action passed to stats::model.frame()

Value

an object of class "paniv", a list with components coefficients, vcov, vcov_type, residuals (structural residuals y - X b on the original scale, in the panel order), model, effect, inst.method, endogenous (names of the endogenous regressors), instruments (names of the excluded instruments, or for Hausman-Taylor the four variable groups in varlist), first_stage (a data frame with one row per endogenous regressor, or NULL), overid (the Sargan or Hansen test, or NULL when exactly identified), ercomp (variance components and theta for the random and Hausman-Taylor estimators), nobs, n_groups, df.residual, call and formula. Methods: print, summary, coef, vcov, confint, predict, nobs, tidy and glance.

Formula

The formula has two parts separated by |. For model = "pooling", "within" and "random" the second part lists the complete instrument set, as in plm::plm(): exogenous regressors are repeated on both sides and excluded instruments appear only on the right, so that y ~ x1 + x2 | z1 + x2 treats x1 as endogenous, instrumented by z1. Regressors absent from the second part are the endogenous ones.

For model = "hausmantaylor" the conventions of plm::pht() are used: the first part lists all regressors and the second part lists those that are exogenous (uncorrelated with the individual effect), e.g. y ~ x1 + x2 + z1 + z2 | x1 + z1 makes x2 and z2 endogenous. Whether a regressor is time varying or time invariant is detected from the data. Identification requires at least as many exogenous time-varying regressors as endogenous time-invariant ones.

Estimators

"within": 2SLS on within-demeaned regressors and instruments (one-way or, with effect = "twoways", two-way demeaning). Time invariant columns are dropped. The classical covariance uses the degrees of freedom n - k - N (N individual effects), as in plm.
"random": the variance components are estimated in the Swamy-Arora way from within and between 2SLS residuals (the unbalanced-panel degrees of freedom of plm are used, which reduce to the usual ones for balanced panels). The data are then quasi-demeaned with theta_i = 1 - sqrt(s2_nu / (s2_nu + T_i s2_mu)). With inst.method = "bvk" (the default, G2SLS) the instruments are quasi-demeaned as well; with inst.method = "baltagi" (EC2SLS) the instrument matrix is the within and between transforms of the instruments.
"pooling": ordinary 2SLS on the stacked data.
"hausmantaylor": the variance components follow plm::pht(): s2_nu is the within residual variance, and s2_1 comes from a 2SLS regression of the estimated individual effects on the time-invariant regressors, instrumented by the exogenous time-invariant and time-varying regressors. The GLS-transformed regression is then estimated by 2SLS with instruments: within transforms of all time-varying regressors, individual means of the exogenous time-varying regressors, the exogenous time-invariant regressors and a constant. For unbalanced panels s2_mu = (s2_1 - s2_nu) / T_h with T_h the harmonic mean of the T_i, as in plm::pht(). A negative s2_mu is set to zero (with a warning), which reduces the estimator to pooled 2SLS on the same instruments.

In every case the classical covariance is s2 inverse(Xhat'Xhat) with s2 the mean square of the structural residuals of the transformed regression. The cluster covariance is the sandwich B (sum_g u_g u_g') B with u_g = Xhat_g' e_g summed within the panel individual (or the supplied cluster) and the factor G / (G - 1) * (n - 1) / (n - k), the convention of the rest of the package and of plm::vcovHC(type = "sss").

Diagnostics

For each endogenous regressor, the first-stage regression of its transformed values on the transformed instruments is compared with the regression on the transformed included exogenous regressors only. The result reports the partial R-squared of the excluded instruments and the F statistic for their joint significance (with the absorbed fixed effects counted in the residual degrees of freedom); for vcov = "cluster" a cluster-robust Wald F is reported as well. These first-stage statistics are not computed for "hausmantaylor", whose instruments are built from the regressors themselves. When the model is overidentified, the Sargan statistic n e'P_W e / e'e (classical) or the cluster-robust Hansen J statistic of the two-step efficient GMM estimator (cluster) tests the validity of the overidentifying restrictions; it is chi-squared with L - k degrees of freedom under the null.

Validation

Numerical accuracy is checked in the package tests and in the validation suite; see the validation article on the package website.

References

Baltagi, B. H. (1981). Simultaneous equations with error components. Journal of Econometrics, 17, 189-200.

Balestra, P. and Varadharajan-Krishnakumar, J. (1987). Full information estimations of a system of simultaneous equations with error component structure. Econometric Theory, 3, 223-246.

Hausman, J. A. and Taylor, W. E. (1981). Panel data and unobservable individual effects. Econometrica, 49, 1377-1398.

Examples

# Simulated panel of 500 patients over 4 waves. Private insurance is
# endogenous for doctor visits (a time-varying health shock drives both)
# and is instrumented by an employer offer of insurance.
set.seed(1)
N <- 500; TT <- 4
d <- data.frame(id = rep(seq_len(N), each = TT), wave = rep(seq_len(TT), N))
alpha <- rep(rnorm(N), each = TT)       # unobserved frailty
s0 <- rep(rnorm(N), each = TT)          # family background
need <- rnorm(N * TT)                   # time-varying health shock
d$school <- round(12 + 2 * s0 + alpha)  # correlated with the frailty
d$female <- rep(rbinom(N, 1, 0.5), each = TT)
d$offer <- rbinom(N * TT, 1, plogis(s0))
d$income <- rnorm(N * TT) + 0.5 * alpha
d$insured <- as.numeric(d$offer + need + rnorm(N * TT) > 0.5)
d$visits <- 1 + 0.6 * d$insured + 0.2 * d$income + alpha + need + rnorm(N * TT)

# the within estimator is biased upwards; FE-2SLS is not (true value 0.6)
coef(panglm(visits ~ insured + income, data = d, index = c("id", "wave"),
            model = "within"))
#>  insured   income 
#> 1.697161 0.175453 
fe2sls <- paniv(visits ~ insured + income | offer + income, data = d,
                index = c("id", "wave"), model = "within", vcov = "cluster")
summary(fe2sls)
#> 
#> Call:
#> paniv(formula = visits ~ insured + income | offer + income, data = d, 
#>     index = c("id", "wave"), model = "within", vcov = "cluster")
#> 
#> Estimator: fixed-effects 2SLS 
#> n = 2000  groups = 500  vcov: cluster 
#> Endogenous: insured 
#> Excluded instruments: offer 
#> 
#> Coefficients:
#>         Estimate Std. Error z value Pr(>|z|)    
#> insured  0.86178    0.36261   2.377   0.0175 *  
#> income   0.17291    0.03735   4.629 3.67e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> First stage (excluded instruments):
#>         partial.R2     F df1  df2  p.value F.robust  p.robust
#> insured    0.04144 64.76   1 1498 1.71e-15    66.21 3.252e-15
#> Residual df: 1498 
#> 

# Hausman-Taylor: the effect of schooling (time invariant, correlated
# with the frailty) on health costs, identified by the exogenous
# time-varying employer offer; income is endogenous (true values 0.4,
# 0.2, 0.3, 0.1)
d$cost <- 2 + 0.4 * d$offer + 0.2 * d$income + 0.3 * d$female +
  0.1 * d$school + alpha + rnorm(N * TT)
ht <- paniv(cost ~ offer + income + female + school | offer + female,
            data = d, index = c("id", "wave"), model = "hausmantaylor")
summary(ht)
#> 
#> Call:
#> paniv(formula = cost ~ offer + income + female + school | offer + 
#>     female, data = d, index = c("id", "wave"), model = "hausmantaylor")
#> 
#> Estimator: Hausman-Taylor 
#> n = 2000  groups = 500  vcov: classical 
#> Exogenous time-varying:   offer 
#> Endogenous time-varying:  income 
#> Exogenous time-invariant: female 
#> Endogenous time-invariant: school 
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  2.29707    0.43720   5.254 1.49e-07 ***
#> offer        0.47878    0.05873   8.152 3.59e-16 ***
#> income       0.24645    0.02615   9.425  < 2e-16 ***
#> female       0.24550    0.09953   2.467   0.0136 *  
#> school       0.07426    0.03699   2.008   0.0447 *  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Variance components: idiosyncratic = 0.9919, individual = 0.9807
#> theta: 0.5508 
#> Residual df: 1995 
#>