sfa (version 1.2.0)

TIC: Choosing between non-nested stochastic frontier specifications

Description

Model selection for stochastic frontier models when the candidates are not nested and none of them need be correctly specified. TIC is Takeuchi's information criterion, a version of AIC whose penalty is estimated from the data rather than assumed equal to the number of parameters. vuong is Vuong's pairwise test, which returns a p-value and is allowed to conclude that two specifications are indistinguishable.

Usage

TIC(object, detail = FALSE)
vuong(object1, object2, correction = c("none", "aic", "tic"), level = 0.05)

Value

TIC returns a single numeric value, or with detail = TRUE a list with components TIC, AIC, logLik, df, penalty and ratio.

vuong returns an object of class "sfa_vuong" with components statistic, p.value, n, lr, lr_adjusted, omega, correction, penalty, critical, level, models, favoured and exact_null. Lower TIC is better; favoured is the name of the winning model_name, or "neither".

Arguments

object, object1, object2

Objects of class "sfareg", fitted with keep_objective = TRUE (see ‘Details’).

detail

If FALSE (the default) TIC returns a single number. If TRUE it returns a list also holding the log-likelihood, the parameter count, the estimated penalty and their ratio.

correction

Bias correction applied to the log-likelihood difference before it is standardised. "none" (the default) is the statistic whose \(N(0,1)\) limit Vuong established. "aic" subtracts the parameter counts and "tic" the Takeuchi penalties; both shift the mean of the statistic and neither has an established null distribution, so their p-values are flagged as indicative.

level

Two-sided size used for the reported critical value and the verdict.

Details

Why not AIC. sfa offers many model_names for the same data, and AIC assumes the fitted model is correctly specified -- exactly the assumption in doubt when choosing between "NHN", "NE", "NG" and "NTN". Most such pairs are not nested either, so the ordinary likelihood ratio test has no chi-square limit.

What TIC does. Takeuchi's criterion replaces AIC's penalty \(d\) with \(\mathrm{tr}[H(\hat\theta) I(\hat\theta)^{-1}]\), where \(I\) is the sample Fisher information and \(H\) the outer product of the per-observation scores. Under correct specification the information matrix equality gives \(H = I\), the trace collapses to \(d\), and TIC equals AIC. The gap between penalty and df -- reported as ratio under detail = TRUE -- is therefore a readable diagnostic in its own right: a ratio near one says the distributional assumption is not doing visible damage.

What vuong does. It tests \(H_0\) that the two models are equally close to the truth in the Kullback-Leibler sense. Neither model has to be correct. With \(m_i\) the difference of per-observation log-likelihoods, the statistic is \(n^{-1/2}\sum_i m_i / \hat\sigma\), asymptotically standard normal. Because it is a test rather than a criterion it can return "neither", which is frequently the honest answer and is not available from AIC.

Requirement. Both need per-observation log-likelihoods, so the fits must be made with keep_objective = TRUE. This is supported by sfm (for "NHN", "NE", "NR", "NG", "NNAK", "THT", "NTN", "NHN_Z", "NE_Z", "NU", "NGE", "NLN", "NW", "tHN" and "TSL") and by psfm. vuong additionally requires both fits to use the same observations, which it checks by sample size and warns about when the data arguments differ.

Estimators that maximise nothing. psfm's "GTRE_SEQ1", "GTRE_SEQ2" and "SSFE", and ivsfm's "C2SLS", are moment-based or FE and have no log-likelihood, so no likelihood-based criterion applies to them and both functions refuse rather than returning a number built on NA.

References

Lai, H.-p. and Huang, C.J. (2010). Likelihood ratio tests for model selection of stochastic frontier models. Journal of Productivity Analysis, 34(1), 3--13.

Takeuchi, K. (1976). Distribution of informational statistics and a criterion of model fitting. Suri-Kagaku (Mathematical Sciences), 153, 12--18.

Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57(2), 307--333.

See Also

sfm, skewness_test, sfa_diagnostics, AIC

Examples

# \donttest{
set.seed(42)
n  <- 400
x1 <- rnorm(n); x2 <- rnorm(n)
d  <- data.frame(y = 1 + 0.5 * x1 + 0.5 * x2 +
                     rnorm(n, 0, 0.4) - abs(rnorm(n)),
                 x1 = x1, x2 = x2)

a <- sfm(y ~ x1 + x2, model_name = "NHN", data = d, keep_objective = TRUE)
b <- sfm(y ~ x1 + x2, model_name = "NE",  data = d, keep_objective = TRUE)

## Lower is better, and comparable across non-nested models.
TIC(a)
TIC(b)

## Is the penalty near the parameter count? If so, TIC has little to add.
TIC(a, detail = TRUE)$ratio

## And a test, which may decline to pick a winner.
vuong(a, b)
# }