PL80_MVTN: Pitt and Lee (1981) Model III: the multivariate truncated normal panel likelihood

Description

Notes on psfm(model_name = "PL80_MVTN"), which fits the likelihood derived in Appendix 2 of Pitt and Lee (1981) but never used there.

Arguments

Author

David H. Bernstein

Details

The model is $$y_{it} = x_{it}'\beta + u_{it} + v_{it}, \qquad u_{it} \le 0,$$ with the firm's inefficiency vector $u_i = (u_{i1},\dots,u_{iT})'$ drawn from a $T$-variate normal $N(0,\Sigma)$ truncated to the negative orthant, and $v_{it}$ iid $N(0,\sigma_v^2)$ independent of $u$. Unlike model_name = "PL80", which holds inefficiency fixed over time, here it varies across periods and is correlated within a firm -- $\Sigma$ carries that dependence.

Why it was never used. Pitt and Lee derived this likelihood and then set it aside, writing that it “is difficult to evaluate since the quantities $P_0$ and $P(y_i - x_i\beta)$ involve $T$-dimensional numerical integrals”, and estimating Model III by Zellner seemingly-unrelated regression instead. Those quantities are orthant probabilities of a multivariate normal. mnormt::sadmvn() evaluates one in about 3 milliseconds at $T = 6$, so a likelihood evaluation costs roughly $N+1$ of them -- about 0.3 s at $N = 100$, and about 12 s for a whole fit at $N = 80$, $T = 4$. What was intractable in 1981 is merely slow now.

Writing $Q^{-1} = \Sigma^{-1} + I/\sigma_v^2$ and $\mu_i = Q\varepsilon_i/\sigma_v^2$, the per-firm log density is $$\log f(\varepsilon_i) = -\tfrac{T}{2}\log 2\pi - T\log\sigma_v - \tfrac{1}{2}\log|\Sigma| + \tfrac{1}{2}\log|Q| - \log P_0 - \tfrac{1}{2}\left(\tfrac{\varepsilon_i'\varepsilon_i}{\sigma_v^2} - \tfrac{\varepsilon_i'Q\varepsilon_i}{\sigma_v^4}\right) + \log P_i,$$ where $P_0 = \Pr(w \le 0)$ for $w \sim N(0,\Sigma)$ is the truncation constant and $P_i = \Pr(w \le 0)$ for $w \sim N(\mu_i, Q)$.

$\Sigma$ is equicorrelated, not unrestricted. It is parameterized as $\Sigma = \sigma_u^2[(1-\rho)I + \rho \mathbf{1}\mathbf{1}']$, costing two parameters. An unrestricted $\Sigma$ costs $T(T+1)/2$ -- 21 at $T=6$, 55 at $T=10$ -- on top of $\beta$ and $\sigma_v$, every one identified only through orthant probabilities. The equicorrelated form captures what the general $\Sigma$ was introduced for: dependence of a firm's inefficiency across periods. $\rho = 0$ gives inefficiency independent over time; $\rho \to 1$ approaches the time-invariant "PL80" case. Because the form is equicorrelated every matrix quantity above is closed form (Sherman-Morrison), so only the orthant probabilities are numerical.

Requirements and limits. A balanced panel with $T \ge 2$: $\Sigma$ is a single $T \times T$ matrix shared by every firm, and with $T = 1$ there is no cross-period dependence for it to describe. An unbalanced panel is an error pointing at "PL80". Reported parameters are sigv, sigu, rho and the frontier coefficients.

u_hat is the posterior mean of $u$ from the Gaussian part, floored at zero, not $E[u_{it}\mid\varepsilon_i]$: the exact conditional mean of a truncated multivariate normal is another $T$-dimensional integral. It is exact where the truncation does not bind and an approximation where it does.

References

Pitt, M.M. and Lee, L.-F. (1981) 'The measurement and sources of technical inefficiency in the Indonesian weaving industry', Journal of Development Economics, 9(1), pp. 43-64. See Appendix 2 for the likelihood.

Examples

Run this code

# \donttest{
library(sfa)
d <- data_gen_p(t = 4, N = 60, rand = 5, sig_u = 1, sig_v = 0.3, sig_r = 0.2,
                sig_h = 0.4, cons = 0.5, beta1 = 0.5, beta2 = 0.5)
f <- psfm(y_pl_mvtn ~ x1 + x2, model_name = "PL80_MVTN",
          data = as.data.frame(d), individual = "name")
f$out
# }