Copula-based nonlinear modeling of the law of one price for lumber products

Distinctions between tests of LOP and spatial market integration are not especially meaningful. In both cases, the economic phenomena being evaluated (spatial market arbitrage) is identical. A survey of both strands of literature can be found in Fackler and Goodwin (2001).

Patton (2006) allowed for time variation in the conditional joint distribution of the returns on the Deutsche mark/US dollar and Japanese Yen/US dollar exchange rates by allowing the parameter(s) of a given copula to vary through time. Smith et al. (2011) examined linkages among logarithmic prices in regional Australian electricity markets but do not explicitly model the nonlinear error correction adjustment process that is standard in spatial arbitrage models and that we consider here. Reboredo (2011) also examined the comovement of crude oil prices using copulas, though his approach is also fundamentally different from that considered here in that it does not directly consider the error correction process commonly applied in models of spatial price linkages.

We are agnostic as to the specific linkage among prices within any transactions cost band and allow this relationship to be determined within the context of the overall joint distribution function. Alternative nonlinear dependencies across the span of the marginals can be accommodated within the range of copula functions that we consider. That said, we expect to see a stronger “error correction” type of relationship for bigger deviations from parity.

See, for example, Taylor (2001), who applies regime switching, time series models of this form to empirical tests of purchasing power parity—an aggregate version of the LOP.

A note of explanation regarding our nomenclature may be helpful in avoiding confusion. Here and below, we refer to values of \((p^i_{t}-p^j_{t})\) as the “price differential” and values of \(\Delta (p^i_{t}-p^j_{t})\) as the “differenced price differential.”.

A specification that is often referred to as an “iceberg” model, reflecting the fact that the value of the commodity melts away via a proportionally lower price as it is shipped.

The behavior underlying spatial price linkages is likely to be discrete—representing the two states of trade/no trade. However, in that empirical evaluations of such models almost always involve some degree of aggregation, the patterns of adjustment may be of a more smooth nature and therefore may favor the STAR-type models.

As we discuss in greater detail below, the interpretations of the relationships in 3 and 4 are not fully analogous within the context of joint distribution functions since temporal correlation may have important implications for the interpretation of the implied density.

For details on construction and properties of copulas, see, among others, Joe (1997) and Nelsen (2006). Much of the work on copulas has been motivated by their applicability to the issues in risk management, insurance and financial economics (see, among others, Rodriguez (2003), Cherubini et al. (2004), Hu (2006), Patton (2006), and Jondeau and Rockinger (2006)). Extensive surveys on the use of copulas in economics and finance are provided by Patton (2009, 2012) and Patton and Fan (2014).

Specifically, we consider goodness of fit for the following eighteen parametric copulas: the Gaussian, t, Clayton, Gumbel, Frank, Joe, Clayton–Gumbel, Joe–Gumbel, Joe–Clayton, and the Joe–Frank copulas. We also consider 180-degree rotations of the asymmetric copulas, which include all of the preceding except for the Gaussian and t copulas.

A copula is rotated \(180^{\circ }\) by using \(1-u^x_i\) in place of \(u^x_i\), where \(u^x_i\) is the quantile corresponding to the marginal distribution for \(x_i\). Such rotations (by \(90^{\circ }\) or \(270^{\circ }\)) can accommodate inverse correlations in copulas that can only represent positive correlation in the unrotated form. Likewise, tail dependencies can be reversed by rotating the copula \(180^{\circ }\).

As a reviewer has noted, comparisons of goodness-of-fit criteria across alternative specifications provide a heuristic basis for comparing alternative specifications. Our approach is reinforced by a consideration of Cramér von Mises (CvM) and Kolmogorov–Smirnov (KS) tests of the specifications.

Estimation and inferences were accomplished using the “COPULA” and “NLP” procedures of SAS and the “copula” and “VineCopula” packages of the R language. Details are available in Chvosta et al. (2011), Schepsmeier et al. (2015), and Yan (2012). Excellent overviews of the R packages and implementation issues are presented by Yan (2007) and Czado (2011).

OSB is engineered by using waterproof and heat-cured resins and waxes and consists of rectangular-shaped wood strands that are arranged in oriented layers. OSB is produced in long, continuous mats which are then cut into panels of varying sizes. In this regard, OSB is similar to plywood, although OSB is generally considered to have more uniformity than plywood and is, moreover, cheaper to produce.

Nonstationarity tests for the individual prices are not presented here but are available on request. The cointegration relationship is evaluated using the residuals from \(ln(P^{i}_{t}) = \alpha + \beta ln(P^{j}_{t}) + \epsilon _t\). Note that a test of stationarity of the price differentials is equivalent to restricting \(\alpha =0\) and \(\beta =1\).

Deviation half-lives are given by \(ln(0.5)/ln(1-\beta )\).

In the interest of space and in light of the BIC values, we do not present estimates of the STAR model. These estimates are available from the authors on request.

Standard normal marginals are used only to illustrate the dependencies in the joint distributions. We use the empirical marginals to evaluate the nature of price adjustment below.

Mean relationships were estimated by simulating data from the estimated copula and the nonparametric marginal densities. Splines and high-ordered polynomial expressions were then fit to the simulated data. Note that simulation of the joint distributions using nonparametric density estimates involved generating random, dependent draws from the univariate values and then using a nonparametric estimate of the inverse cdf. A fine grid was defined for the kernel density estimates of the density, and then, quantiles were matched to the randomly drawn uniform variates to generate values of the two dependent variables \((\Delta (p^i_{t}-p^j_{t}),(p^j_{t-1}-p^i_{t-1}))\).

Derivatives are derived by solving the bivariate relationship suggested by the copula for one variable in terms of the other. Splines and high-ordered polynomials were then used to parameterize this relationship, and derivatives are obtained by differentiating this polynomial. This provides an accurate approximation of the patterns of price responses that can be compared to conventional linear models.