Panel copula correction for endogenous regressors with fixed effects

Maximum-likelihood estimator for linear panel models with individual fixed effects and endogenous regressors correlated with the structural error, following Haschka (2022). A forward orthogonal deviations (FOD) transformation removes the fixed effects, and the regression coefficients, the copula correlation(s) between the endogenous regressors and the error, and the error variance are then estimated jointly by maximum likelihood. Standard errors come from a bootstrap that resamples whole panels. This is a distinct model class from the cross-sectional endogCopula estimators reached through copreg(), in the same way that plm is separate from lm; CopRegPANEL() is not registered with copreg().

Usage

CopRegPANEL(
  formula,
  data,
  index,
  cdf = "kde.plugin",
  ties = "max",
  intercept = FALSE,
  nboots = 199,
  method = c("BFGS", "Nelder-Mead"),
  start = NULL,
  maxit = 5000L,
  subset = NULL,
  contrasts = NULL,
  parallel = FALSE,
  ncores = NULL,
  verbose = interactive()
)

Arguments

formula: A two-part formula y ~ endogenous | exogenous, as for the cross-sectional endogCopula estimators. Anything lm() accepts is allowed, plus the panel-aware lag(), lead() and diff().
data: A data.frame with the untransformed panel data.
index: A character vector naming the panel identifier and, usually, the time variable in data: c("id", "time"). With only an identifier the existing row order within each panel is taken as the time order.
cdf: Marginal CDF estimator applied to the regressors that enter the copula: one of "kde.silverman", "kde.cv", "kde.plugin" (the default here), "ecdf.fixed", "ecdf.adj", "rank.n" or "rank.n1".
ties: How ties are handled in the CDF estimate: "max" (the counting function, the default) or "average" (midranks).
intercept: Logical, FALSE by default. A fixed-effects transformation removes anything time invariant, so an intercept in the transformed regression is identified only alongside a full set of time dummies, where it stands for the reference period; it is not the structural intercept, which is absorbed into the individual effects.
nboots: Number of panel bootstrap replicates used for the standard errors; a single number of at least 2.
method: Optimiser for the likelihood: "BFGS" (the default), which falls back to Nelder-Mead automatically if it fails to converge, or "Nelder-Mead" directly.
start: Optional numeric vector of starting values for the optimiser.
maxit: Maximum number of iterations for the optimiser.
subset: An optional logical or index vector selecting rows of data, as in lm().
contrasts: An optional list of contrasts passed to model.matrix().
parallel: How the bootstrap is parallelised: FALSE (the default), TRUE (chooses multicore on Unix and snow elsewhere), "multicore" or "snow".
ncores: Number of worker processes when parallel is not FALSE; NULL uses one less than the number of detected cores.
verbose: Logical; print progress messages during maximisation and the bootstrap. Defaults to interactive().

Value

An object of class "copregpanel": a list with, among other elements,

coefficients, std.error, vcov: the regression coefficients of the transformed model and their bootstrap standard errors and covariance matrix
rho, rho.se: the copula correlation(s) between the normal score of each endogenous regressor and that of the error, with bootstrap standard errors
sigma2, sigma2.se: the error variance of the transformed model and its bootstrap standard error
fixef: the estimated individual (panel) effects
fitted.structural, residuals.structural: fitted values and residuals of the structural model y = alpha_i + x'beta + z'delta + e on the original rows
fitted.transformed, residuals.transformed: fitted values and residuals of the FOD-transformed model that the likelihood treats as normal
logLik, AIC, BIC, lr.test, wald.test: the maximised log-likelihood, information criteria, and a likelihood-ratio and a bootstrap Wald test of rho = 0
within.coefficients, within.std.error, se.ratio: the uncorrected within-estimator benchmark on the same bootstrap resamples, and the ratio of the copula to the within standard errors
boot: the raw bootstrap draws
diagnostics: identification diagnostics underlying validity.copregpanel
r.squared: transformed, structural, within, between and overall R-squared

and further elements recording the call, the data on both scales, the panel index, sample sizes and the terms/formula used to fit the model.

Details

Data are supplied untransformed: data holds the original panel, and CopRegPANEL() performs the forward orthogonal deviations transformation itself. lag(), lead() and diff() may be used inside formula exactly as any other function; they are panel aware and never look across a panel boundary. Time dummies (and any other regressor that is identical across panels once transformed) stay in the regression but are dropped from the copula, because after the transformation a full set of them carries no independent variation and would leave the correlation matrix of the copula data singular. The bootstrap used for the standard errors resamples whole cross-sectional units (panels), not individual rows, which is what preserves the within-panel dependence and keeps the standard errors valid under serial correlation and heteroskedasticity.

References

Haschka, R. E. (2022). Handling endogenous regressors using copulas: A generalization to linear panel models with fixed effects and correlated regressors. Journal of Marketing Research 59(4), 861-880.

Arellano, M. (1993). On the testing of correlated effects with panel data. Journal of Econometrics 59, 87-97.

Goncalves, S. and L. Kilian (2004). Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 89-120.

Examples

set.seed(1)
N <- 30L; Time <- 6L
d <- data.frame(id = rep(seq_len(N), each = Time),
                 year = rep(seq_len(Time), times = N))
alpha <- rep(rnorm(N), each = Time)
e <- rnorm(N * Time)
d$x <- exp(rnorm(N * Time) + 0.5 * e)   # endogenous: correlated with e
d$z <- rnorm(N * Time)                  # exogenous
d$y <- alpha + 0.5 * d$x + d$z + e

# \donttest{
fit <- CopRegPANEL(y ~ x | z, data = d, index = c("id", "year"),
                    nboots = 15, verbose = FALSE)
summary(fit)
#> 
#> Panel copula MLE (Haschka 2022)
#> 
#> Call:
#> CopRegPANEL(formula = y ~ x | z, data = d, index = c("id", "year"), 
#>     nboots = 15, verbose = FALSE)
#> 
#> Panel: index = (id, year)
#>   30 cross-sectional units, T = 6
#>   180 observations, 150 after the forward orthogonal deviations transformation
#>   no constant in the transformed regression: the structural intercept is
#>   time invariant and goes with the individual effects
#> 
#> Residuals of the structural model (y - alpha_i - x'beta - z'delta):
#>      Min       1Q   Median       3Q      Max 
#> -2.64558 -0.63148 -0.07251  0.50013  2.47346 
#> 
#> Residuals of the transformed model, which the likelihood treats as normal:
#>      Min       1Q   Median       3Q      Max 
#> -2.89808 -0.56351  0.01295  0.78300  2.61122 
#> 
#> Coefficients:
#>   Estimate Std. Error z value Pr(>|z|)    
#> x  0.47548    0.07064   6.731 1.69e-11 ***
#> z  0.94479    0.12728   7.423 1.14e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Dependence parameters: rho(P*, xi*) is the correlation between the normal
#>   score of an endogenous regressor and that of the error, and sigma^2 is the
#>   variance of the error of the transformed model. rho = 0 means no endogeneity.
#>              Estimate Std. Error z value Pr(>|z|)   
#> rho(x*, xi*)   0.4469     0.1502   2.975  0.00293 **
#> sigma2         0.9559     0.1245                    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> No endogeneity, all rho = 0:
#>   likelihood ratio  chi-squared = 6.277 on 1 df, p = 0.01223
#>   bootstrap Wald   chi-squared = 8.849 on 1 df, p = 0.002933
#> 
#> R-squared:
#>   transformed model 0.762   structural model 0.828
#>   within 0.783   between 0.3704   overall 0.6684
#>   within, between and overall are squared correlations excluding the
#>   individual effects.
#> 
#> Log-likelihood -192.7 on 4 parameters;  AIC 393.4,  BIC 405.4
#> Standard errors from 15 panel bootstrap replicates (cross-sectional units
#>   resampled, not rows); cdf = "kde.plugin", ties = "max"; optimiser BFGS.
#> Pr(>|z|) in both tables: Wald test using the normal approximation with
#>   the bootstrap standard error.
#> 
#> --- Identification diagnostics ------------------------------------
#> 
#> Non-normality of the transformed endogenous regressors
#> (small p = non-normal, which is what identifies the model):
#>      AD      AD p      KS p
#> x 7.074 2.193e-17 0.0008258
#> 
#> Collinearity of the copula data (omega near 0 = weakly identified):
#>   corr(P, C) omega
#> x     0.9094 0.182
#> 
# }