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endogCopulaPanel

Package website: https://ashgreat.github.io/endogCopulaPanel/

endogCopulaPanel implements the fixed effects Gaussian copula estimator for panel data by Haschka (2022, Journal of Marketing Research). The estimator corrects for endogenous regressors in linear fixed effects panel models without instrumental variables. The data are transformed by forward orthogonal deviations within each panel, and the regression coefficients are estimated jointly with the Gaussian copula correlation between the regression error and each endogenous regressor by maximum likelihood. Standard errors come from a bootstrap that resamples whole panels.

The package builds on endogCopula, which supplies the shared formula parsing, marginal CDF estimators and diagnostic helpers used across the whole toolbox.

Installation 

# install.packages("remotes")
remotes::install_github("ashgreat/endogCopulaPanel")

Usage

The exported function is CopRegPANEL(). The model is specified with the same two part formula used across the toolbox, y ~ endogenous | exogenous, with the endogenous regressors before the bar and the exogenous regressors after it. lag(), lead() and diff() can be used inside the formula and are panel aware, so they never look across a panel boundary.

The panel structure is declared through the index argument, index = c("id", "time"). The panel identifier comes first, both identifier columns must be numeric, and data should hold the untransformed panel data, because CopRegPANEL() applies the forward orthogonal deviations transformation itself.

By default the transformed regression has no intercept, because a fixed effects transformation removes anything time invariant and the structural intercept is absorbed into the individual effects. Set intercept = TRUE only alongside a full set of time dummies.

library(endogCopulaPanel)

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

fit <- CopRegPANEL(y ~ x | z, data = d, index = c("id", "year"),
                    nboots = 15, verbose = FALSE)
fit

This is the real output of the example above, run with nboots = 15 to keep it fast. Use more bootstrap replicates in practice. The default is 199.

Panel copula MLE (Haschka 2022)

Call:
CopRegPANEL(formula = y ~ x | z, data = d, index = c("id", "year"),
    nboots = 15, verbose = FALSE)

Balanced panel: n = 30, T = 6, observations = 180

Coefficients:
     x       z
0.4755  0.9448

summary(fit) adds standard errors, the copula correlation between each endogenous regressor and the error, a likelihood ratio and a bootstrap Wald test of no endogeneity, and R-squared:

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 ***

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

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

Checking the identifying assumptions

Identification of the copula correction rests on distributional assumptions that are partly testable. validity(fit), an S3 method re-exported from endogCopula, checks three things: non-normality of each endogenous regressor after the panel transformation, since identification needs that non-normality; normality of the error of the transformed model, which the likelihood assumes directly; and how much the copula-corrected standard errors are inflated relative to the uncorrected within estimator, where a large inflation factor signals weak identification.

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.
  • Park, S. and S. Gupta (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31(4), 567-586.

License

MIT (c) Ashwin Malshe