Economists often view a close association between prices of similar
goods in spatially or vertically separated markets, a concept closely
associated with the Law of One Price (LOP), as being a sign of
competition and the efficient functioning of markets. Observing closely
related prices might, however, also reflect oligopolistic, collusive, or
price fixing behavior. An extensive literature has developed on
evaluating spatial market integration to assess the degree to which
shocks in one market are transmitted into spatially separate markets.
The interest for the economist is often, as noted by Barrett and Li
(2002), concerned with the concept of Pareto efficiency since prices
form the appropriate signaling mechanism of relative scarcity which
ensures that producers appropriately specialize and that resources are
optimally used. The question of price transmission also has important
distributional implications, with the pass-through of policy and
nonpolicy changes determining the extent to which different
constituencies gain or lose.
The issues of integration and efficiency, in the context of
spatially separated markets, has attracted much attention in the
literature and are often linked to concerns over the impact of market
liberalization across developed, less developed, and transition country
economies alike (Baulch 1997). Early work in this area used cross-market
price correlation or simple regression-based tests to assess the degree
of market integration. More recently the recognition that price series
are often nonstationary has led to widespread use of cointegration
techniques following Ardeni (1989). However, these relatively simple
Granger causality and cointegration approaches to the problem have been
criticized on the grounds that they ignore the potentially important
role played by transfer costs, such as transport and transactions costs
(McNew and Fackler 1997; Fackler and Goodwin 2001; Barrett 2001; Barrett
and Li 2002), they assume a linear relationship between prices which is
inconsistent with discontinuous trade (Baulch 1997), and possess only
weak power to discriminate between integrated and independent markets.
The LOP states that the price of identical goods in spatially
separated markets should be the same after conversion to a common
currency. The mechanism by which the LOP is maintained is that of
spatial arbitrage. Should the prices of identical products differ in two
markets then, in the absence of transport and transaction costs, rents
to arbitrage exist which ensure that traders move product from surplus,
low price, markets toward deficit, high price, markets until such rents
are exhausted and the LOP holds once more. Nevertheless, in the majority
of the literature, asymmetries in adjustment, poor spatial transmission
of prices and deviation from the LOP have often been linked to high
transport or transaction costs, protection, market barriers or some
other form of imperfect competition (e.g., Kinnucan and Forker 1987;
Ward 1982; Pick, Karrenbrock, and Carman 1990). Such imperfections,
however, require that the spatial arbitrage conditions be modified to
take explicit account of these costs since they drive a wedge between
prices observed in different locations. Consider two spatially separated
markets, A and B, that trade a single homogeneous good. Denote the
transfer costs in time t between market A and B as [k.sup.AB.sub.t] and
contemporaneous prices, expressed in a common currency, in each
respective market as [P.sup.A.sub.t] and [P.sup.B.sub.t]. Rents to
arbitrage are present, and trade occurs from markets A to B, for
example, for as long as [P.sup.A.sub.t] + [k.sup.AB.sub.t] [less than or
equal to] [P.sup.B.sub.t]. (1) Arbitrage rents disappear and trade
ceases when [P.sup.A.sub.t] + [k.sup.AB.sub.t] > [P.sup.B.sub.t],
however, only when [P.sup.A.sub.t] + [k.sup.AB.sub.t] [greater than or
equal to] [P.sup.B.sub.t] can these two markets be said to be integrated
since [P.sup.A.sub.t] + [k.sup.AB.sub.t] < [P.sup.B.sub.t] can only
hold in the long term in the absence of trade or if trade fails to
address the relative abundance of goods in either market because of the
relative size of each market.
It can be seen from these arguments that the transmission of price
signals between spatially segregated markets may, if it occurs, exhibit
a nonlinear form. Price comovement might, under these arbitrage
conditions, be "equilibrium restoring" when price
differentials exceed transfer costs to traders while when the price
differentials fall short of transfer costs, prices are not equilibrium
restoring. This case could lead to a switch in regime between periods of
trade and nontrade. However, if some proportions of traders'
transfer costs are fixed then it is possible that some form of, somewhat
slower, equilibrium restoring process may still be expected within the
"threshold," or "neutral," band defined by k. The
insight that there may be bands and asymmetries in price adjustment
means there is a need for new approaches.
The most common approach used in the recent literature makes use of
threshold effects, as one manifestation of poor transmission, to take
account of transactions costs, asymmetries and nonlinearities (e.g.,
Abdulai 2000). One strand of the threshold literature has focused on
asymmetric adjustment, whereby prices might adjust differently depending
on whether they are above or below equilibrium (see, e.g., Granger and
Lee 1989; Kinnucan and Forker 1987; Mohanty, Peterson, and Kruse 1996).
Threshold behavior and asymmetric adjustment are distinct concepts.
However, Abdulai (2000) also distinguishes between threshold models of
an asymmetric and a symmetric type, the former being where the reaction
to positive price shocks differs from that to a negative shock, but both
types allow for asymmetric within- and out-of-threshold adjustment. It
is the latter case that interests us here. Under such circumstances, and
within a range, markets may be effectively separated, in that trade does
not occur, although still integrated according to the modified LOP
definition. Only when prices are outside of a threshold, will price
changes in one market be transmitted to another market. This type of
threshold model corresponds closely to those introduced by Balke and
Fomby (1997) and developed by Hansen and Seo (2002) and Seo (2003).
These articles postulate that the existence of transaction costs
prevents investors realizing an investment opportunity and apply
threshold cointegration to the term structure of interest rates. Goodwin
and Piggot (2001) and Sephton (2003) make use of similar models to these
in the context of price transmission.
The threshold autoregressive (TAR) and momentum threshold
autoregressive (MTAR) models in Granger and Lee (1999), Enders and
Granger (1998), and Escribano and Pfann (1997) have been the most
popular threshold models. These allow for negative shocks, or deviations
from equilibrium, to have different effects from those that are
positive. They are related to, but distinct from, the models suggested
by Balke and Fomby (1997), Hansen and Seo (2002), and Seo (2003).
If data on transport and other transactions costs were available to
the price analyst then it would, as Baulch (1997) states, be a
relatively simple arithmetic exercise to determine the "threshold
band" within which trade would not be profitable. However, such
data are rarely available. Also, it may be that there exist some costs
faced by traders that are fixed. The partitioning of the various costs
in k into fixed and variable components is likely to be arbitrary.
Furthermore, as noted by Barrett and Li (2002), the potential for
transfer costs to be nonstationary places important restrictions on this
type of approach. However, typical estimates of such cost from
"Structure, Conduct and Performance" studies are rarely
available for the frequency and duration of available price series to
enable the analyst to investigate the potential relationship further.
The attractiveness, therefore, of employing models that allow the
readily available price data to "speak for themselves" is
evident.
This article introduces and implements a generalization of the
symmetric version of the Hansen and Seo (2002) threshold autoregression
model (TAR), which embodies both the Equilibrium-TAR (Eq-TAR) and
Band-TAR models discussed in Balke and Fomby (1997). (2) While the
Eq-TAR model follows conventional practice and assumes that it is the
center of the threshold interval that forms the point of attraction from
both outside and inside the interval, the Band-TAR allows the outer
boundary of the threshold band to be that point of attraction from
without. This distinction is important for inference and for subsequent
analysis. If the data support the Band-TAR model, then price analysts
would do better to look for mechanisms other than notional
"long-run equilibrium" between two prices with which forecast
price movements in a given series when that series lies within the
threshold band.
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
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
The focus of this article is with price transmission onto Brazilian
markets for wheat, soybeans, and maize. In the case of wheat, this
corresponds to import price transmission; Brazil is among the
world's top five importers of wheat. Virtually all this wheat comes
from Brazil's neighbor Argentina, making this the most relevant
market from which to measure price transmission. In overall terms, the
United States is the world's largest exporter of wheat, providing
an alternative estimate of a suitable "world" price. Brazil is
the second largest exporter of soybeans, a crop that forms Brazil's
most important export commodity. Our concern here is principally with
the extent to which world market prices are transmitted to Brazilian
exporters. Brazil's chief rival in the world market is the United
States, so we use U.S. prices as an estimate of the relevant world
price. Finally, in the case of maize, Brazil has changed from being a
net importer to a net exporter (from the beginning of 2001). However,
the volume of maize traded across Brazil's borders has typically
been small, never exporting more than 750,000 metric tones (mt) and
rarely importing more than 200,000 mt per quarter over the period of
study. Since the United States is the world's largest producer and
exporter of maize and Argentina is its largest regional trading partner
in this commodity, we consider price transmission from both of these
important markets.
The data used in this study are monthly Brazilian, U.S., and
Argentine prices for wheat, maize, and soya. The monthly frequency has
been chosen because it is the highest frequency that is available over
the window of interest. Furthermore, as noted by Barrett and Li (2002),
monthly data limit the potential problem of aggregation biases
associated with the use of lower frequency data. In addition, the
expectation that traders are likely to react to price signals, at least
partially, within thirty days favors the use of monthly data. However,
we note that shipping times, particularly important between the United
States and the two South American countries, make higher frequency data
less desirable for this purpose.
The domestic wheat, maize, and soya prices for Brazil were as
received by producers, (Getulio Vargas Founda