Threshold effects in price transmission: the case of Brazilian wheat, maize, and soya prices.

Threshold effects in nonstationary series presuppose the existence of a "long-run" equilibrium relationship between prices. Testing for cointegration along with estimation and inference about the cointegrating parameters ([beta]) is now commonplace in the econometrics literature. The most common method for testing the hypothesis of no cointegration and estimating [beta] is based on full information maximum likelihood (FIML), outlined in Johansen (1995). Under the assumption that the variables are cointegrated, a vector autoregression (VAR) is estimated by minimizing the determinant of the error covariance matrix [omega] from equation (9). However, the bivariate VAR requires a "rank restriction" dictating that there can only be one stochastic trend driving both variables.

The majority of work that tests for thresholds has, as a first step, ignored the existence of thresholds when estimating the long-run equilibrium ([beta]) and testing for cointegration. Threshold adjustment toward this equilibrium has generally been modeled as a second step. This process is not optimal where threshold effects exist, since the likelihood function upon which the long-run estimates are based depends on the threshold parameters. With regard to testing for cointegration, non-parametric methods such as those outlined in Beirens (1997) can be employed. However, FIML estimation and testing for thresholds is relatively difficult. As we have already noted, the likelihood function of threshold models are jagged (nondifferentiable). This is problematic on two levels. First, optimization of the likelihood is difficult because derivative methods cannot be used. Second, inference is greatly complicated where the second derivatives of the likelihood do not exist. Furthermore, as we mentioned above, equation (9) presents the possibility that identification problems arise in the case that [lambda] = 0.

Hansen and Seo (2002) propose algorithms that can maximize the threshold likelihood function, along with a classical "fixed regressor" bootstrap that can be used to test for threshold effects. This test is based on the restriction [[pi].sub.i,u] = [[pi].sub.i,l] where, in effect, the threshold parameters are being treated as fixed at their maximum likelihood values. Because the thresholds are estimated, not set a priori, a Wald, Lagrange multiplier (LM) or likelihood ratio (LR) test, which is constructed by treating the threshold parameters as given, will be misleading if conventional critical values are used. Similarly, the standard errors of the error correction parameters computed using these fixed threshold parameters would also be likely to understate the true variance. Bootstrap procedures can overcome these problems. However, the following drawbacks apply:

* Threshold models imply inequality constraints for nonperversity that are difficult to enforce within a classical framework.

* The tests proposed in Hansen and Seo (2002) are computationally expensive and problematic due to the jagged likelihood function. These can be overcome in simple models, but the problems become more severe in more richly parameterized systems such as those being suggested here.

A Bayesian analysis can circumvent these problems by employing "Gibbs sampling" and "Metropolis-Hastings" (M-H) algorithms. Descriptions of these algorithms can be found in Bauwens, Lubrano, and Richard (1999) and we will not repeat them here. The jagged nature of the likelihood creates few problems when using these algorithms. Moreover, while these methods are computationally intensive, they are no more arduous than classical bootstrap procedures. Inequality restrictions on parameters such as [theta] can be enforced in a simple way within the Bayesian setting. However, it should be acknowledged that if the parameter value is close to the boundary of the prior, then in small samples, the posterior mean might tend to overstate the distance of the estimate from that boundary. Bayesian methodologies have an elegant approach to drawing inferences, however, the practical application of this methodology can be problematic and computationally intensive.

The parameters in the VECM in equation (9) may be partitioned into two sets ([[THETA].sub.1], [[THETA].sub.2]) with [[THETA].sub.2] = ([beta], [lambda], [theta]) and [[THETA].sub.1] being all other parameters in the VECM. The priors used for [[THETA].sub.1] can be specified as conjugate priors (Normal-Wishart) as in the standard Bayesian linear regression. Accordingly, the posterior distributions of [[THETA].sub.1] conditional on [[THETA].sub.2] can be generated as in the case of a system of linear regressions (see Chib and Greenberg 1995a), and Gibbs Sampling can be employed (with the priors being set as relatively noninformative). The remaining parameters were set following Bauwens, Lubrano, and Richard (1999), which results in a posterior for [beta] that has finite first and second moments within a standard cointegration setting. Therefore, this prior was adopted for [beta] along with the prior for [theta], specified as a positive constant over the interval (0, 1) and zero otherwise. Finally, the prior for [lambda] was also constant, but with an indicator variable specifying that at least 20% of the observations for [e.sub.t] were required to be either outside or inside the thresholds. The procedures used here employ four M-H algorithms initialized at different points. The sequence of [beta] parameters generated by each of the four algorithms should converge to the same point if the algorithms are functioning correctly.

Before applying the generalized TECM to real world data we conducted a Monte Carlo experiment. Two sets of Monte Carlo data were generated. In each case a Bivariate VAR of order one was specified as the data generating process. However, in only one of the simulated data sets were thresholds present. In both cases, the simulated data were generated using a cointegrating parameter [beta] = 1, a zero coefficient on the cointegrating time trend. The two data sets differed only in the treatment of the threshold parameter, [lambda], the attractor indicator [theta] and the speed of adjustment parameters [[pi].sub.il] and [[pi].sub.iu]. In the threshold case, the threshold parameter, [lambda], was set equal to 1, and out-of-threshold adjustment was toward the edge of the threshold, [theta] = 1. The within-threshold adjustment parameter was zero [[pi].sub.i,l] = 0, but out-of-threshold adjustment parameter was [[pi].sub.i,u] = 1 (in both price equations). This series then follows the Band-TAR model of equation (7). In the second case, where threshold behavior was excluded, the VAR was again specified using a cointegrating parameter, [beta], set equal to 1, a zero coefficient on the cointegrating time trend, but here both [lambda] and [theta] were set to zero and speed of adjustment parameters [[pi].sub.i,l] and [[pi].sub.i,u] were both set equal to 1.

The results of the experiments, using the Bayesian estimating algorithm described above, applied to the threshold and nonthreshold Monte Carlo data are not presented here in full. To summarize, the generalized model produced posterior distributions (p.ds) for the cointegrating parameter (-[beta]) centered on 0.94 and 0.91 for the threshold and nonthreshold case, respectively, and in both cases produced a p.ds for the time trend centered close to zero. The model was able to detect an asymmetric adjustment, in both directions ([[pi].sub.i,u]--[[pi].sub.i,l], negative and truncated at zero, and for [[pi].sub.2,u]--[[pi].sub.2,l], positive and truncated at zero) in the threshold case but not in the nonthreshold case (although the latter result was less conclusive in one direction). The model also reported modes of the p.ds of the threshold parameters [lambda] and [theta], at 0.9 and 0.9 in the threshold case and 0.5 and 0.22 in the nonthreshold case. These results were typical, and the posterior odds were the same if either further lags were introduced into the models, or their cointegrating parameters were varied within a range of 0.5 to 1.5. Taken together then, Model III does appear to be able to detect threshold behavior when it is present in the date generating process.

The contention in this article is that [theta] = 1 (the Band-TAR) is more likely to be the correct specification of a threshold model. However, the posterior distributions of the Eq-TAR specification should be similar to the Band-TAR, with the exception that the p.ds for 0 should have the mass around zero rather than 1.

Empirical Section

Data