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

The model introduced in this article, that encompasses both the Eq-TAR and Band-TAR models as special cases, does introduce additional parameters over and above those in either the Eq-TAR or Band-TAR models. Classical maximum likelihood (ML) estimation of threshold models is far from straight-forward. The generalized models employed here become highly problematic to estimate when using the type of algorithms suggested by Hansen and Seo (2002). (3) These difficulties arise from two sources; first, the jagged and potentially multimodal nature of the likelihood function complicates optimization and also prevents inference based on derivative methods (this is demonstrated in Hansen and Seo 2002); second, some threshold models have parameters that are unidentified should the other parameters take certain values. The first of these difficulties can be surmounted using multidimensional grid search techniques, including conditional iterative searches, which are now computationally feasible. However, within a classical setting inferential problems remain in respect to the identification of parameters in some cases.

Bayesian approaches to the estimation of threshold error correction models are advantageous in this respect since they do not rely on a differentiable likelihood function and identification is much less of an issue. Within the Bayesian framework the nonidentification of some parameters at certain points in the parameter space does not prohibit the mapping of posterior distributions (see Bauwens, Lubrano, and Richard 1999, p. 41). While Bayesian approaches have been used in the context of threshold models before, (Bauwens, Lubrano, and Richard 1999) applications are few and none have been applied to the type of generalized threshold models estimated here.

The article proceeds by introducing standard error correction models and threshold versions of these models in the next section. However, we assume that readers have a working knowledge of the basics of unit root and cointegration econometrics. The third section makes the case for the use of Bayesian estimation and illustrates the approach in a Monte Carlo setting. We subsequently implement both ML and Bayesian approaches using the Brazilian, U.S., and Argentine commodity price data.

Long-Run Behavior and Thresholds

The most popular method used to model the relationship between the prices of similar goods in spatially separated markets recently has been the "cointegration" approach. This approach assumes that each of the prices share a "stochastic trend." Therefore, the prices of a homogeneous commodity in two separate countries (or markets), defined at time t in countries A and B, respectively, [p.sub.A,t] and [p.sub.B,t], are assumed to have a "long-run equilibrium" relationship that takes the form

(1) [p.sub.At], = [beta][p.sub.B,t]

(where prices may potentially be logged). This relationship will not hold exactly if, for any reason, there are delays in returning to the long-run equilibrium following some short-run shock or incident. Therefore a "long-run disequilibrium" term, defined as

(2) [[epsilon].sub.t] = ([p.sub.A,t] = [beta][p.sub.B,t]),

describes the distance from long-run equilibrium at time t. Once a suitable lag structure for [epsilon] is defined, this disequilibrium term provides a means of accounting for short-run adjustment back to equilibrium following a shock in a previous period.

For a cointegrating relationship between the two prices to hold requires that the disequilibrium term in equation (2) does not itself have a trend (4) (for a more formal statement, see Hatanaka 1996, p. 150). However, by defining the "long-run disequilibrium" term as

(3) [[epsilon].sub.t] = ([p.sub.A,t] = [beta][p.sub.B,t] - [[beta].sub.0]

which is assumed to be stationary, equation (3) can then be used to develop the "error correction" framework. This framework assumes, for a single-period lag, that there is a linear adjustment mechanism of the form

(4) [F.sup.0.sub.i]([e.sub.t-1]) = [[pi].sub.i][e.sub.t-1]

for i = A, B, where [[pi].sub.i] defines the speed at which the ith price series returns to equilibrium so that:

(5) [DELTA][p.sub.i,t-1] = [F.sup.0.sub.i]([e.sub.t-1]) + [u.sub.t],

where [u.sub.t] is a stationary error with moments that do not depend on past values of the long-run disequilibrium term, [e.sub.t].

Threshold cointegration allows for values of [[pi].sub.A] and/or [[pi].sub.B] that depend on the value of [e.sub.t-1], such that speed of adjustment back to equilibrium may be a function of the lagged distance from equilibrium.

Balke and Fomby (1997) outlined three distinct adjustment mechanisms, although we only consider two of those here. The first adjustment mechanism we consider (for some positive constant, [lambda] > 0) is, Model I, the Eq-TAR representation:

(6) [F.sup.I.sub.i]([e.sub.t-1]) = [[pi].sub.i,u] [e.sub.t-1] if [absolute value of [e.sub.t-1]] > [lambda]. = [[pi].sub.i,l] [e.sub.t-1] if [absolute value of [e.sub.t-1]] < [lambda]

where the subscripts u and l denote the without and within-threshold behavior, or adjustment speed, respectively. Model I has an interval of attraction, or threshold band, of a width dictated by [lambda]. If prices lie outside equilibrium they are attracted to the center of the threshold interval whether they begin their reversion from inside or outside the threshold boundary. However, the speed of reversion to equilibrium may, potentially, differ depending on whether prices are outside or inside the interval if [[pi].sub.il] [not equal to] [[pi].sub.iu] . Model I is a threshold model of the type employed in Hansen and Seo (2002).

The second adjustment mechanism we consider here is, Model II, the Band-TAR representation:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Model II also has the same interval of attraction, defined by [lambda], as Model I, but when prices are outside of their thresholds, they are attracted to the edge of the threshold band rather than to the middle of the threshold interval, as is the case in Model I. However, if prices are inside their thresholds, they may be attracted to the middle of the interval in a similar manner to Model I if [[pi].sub.il] [not equal to] 0.

The rationale behind threshold models suggests that the adjustment parameters should be Asymmetric (5) ([[pi].sub.iu] [not equal to] [[pi].sub.il]), indicating different "within-threshold" and "out-of-threshold" response rates. There are, therefore, potentially two separate "regimes," or adjustment speeds, dependent on whether prices lie inside or outside their thresholds. Additionally, consideration of positive transfer costs and arbitrage conditions would suggest that the adjustment regime should be Nonperverse, ([absolute value of [[pi].sub.il]] < [absolute value of [[pi].sub.iu]]), with adjustment speeds being slower within the threshold than without. It is sometimes further conjectured that [[pi].sub.il] = 0. However, these restrictions are not formally required.

A generalized model, Model III, encompassing both I and II, can be constructed as:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Within Model III, if [theta] = 0, the Eq-TAR model is obtained and if [theta] = 1, the Band-TAR model is obtained. Allowing [theta] to take any value within the unit interval provides a more general framework within which to analyze threshold behavior. While [lambda] sets the interval over which differing speeds of adjustment occur, the interval of attraction, when outside the threshold interval, is set by the value [theta][lambda]. There are two possible economic interpretations of this case. First, if transfer costs are made up of both fixed and variable components then equilibrium reversion may overshoot the threshold limit from without but within-threshold behavior may utilize the full threshold band defined by [lambda]. Secondly, that trade, initiated during an out-of-threshold episode, may be subject to shipping duration lag times and might present a similar overshoot into the threshold band. Of course, ambiguous results might emerge if transfer costs are themselves non-stationary. In particular, if we believe that transfer costs, which are unobserved here, are indeed nonstationary, then both the Eq-TAR and Band-TAR models should be respecified. One potentially useful generalization, which is beyond the scope of this article, might be to estimate [lambda] as a time dependent coefficient in a random parameter context.

Modeling the dynamics of price adjustment may require further lags than specified in equation (5). The adjustment mechanisms in equations (6), (7), and (8) can therefore be embodied within a vector error correction model (VECM). Letting [y'.sub.t] = ([p.sub.At], [p.sub.Bt]), the VECM can be expressed as

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where all the [u.sub.t] are assumed to be independently and normally distributed with a covariance matrix [omega]. The model specified in equation (9) is employed in the empirical section. This specification is a generalized version of the VECM now commonly employed in the price transmission literature, (i.e., that which employs the adjustment mechanism [F.sup.0] in equation (4)). However, as we have already outlined, it also offers a more general and flexible characterization of threshold models than those previously employed.

Estimation