Use of composite forest commodity price indices for cointegration analysis

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Journal of Forest Economics 15 (2009) 237–260 www.elsevier.de/jfe

Use of composite forest commodity price indices for cointegration analysis Shiv N. Mehrotra, Shashi Kant Faculty of Forestry, University of Toronto, 33 Willcocks Street, Toronto, Ontario, Canada M5S 3B3 Received 13 June 2008; accepted 20 January 2009

Abstract Time series data on forest product prices used in research is frequently the product of temporal, spatial as well as product aggregation. This paper analyzes the implications of the use of composite commodity price indices in cointegration analysis and tests the validity of the assumptions underlying it. It tests for the presence of a common stochastic trend in disaggregated softwood lumber product price series in multiple US markets, a validity condition supported by the Generalized Composite Commodity Theorem (Lewbel, 1996. Aggregation without separability: a generalized composite commodity theorem. The American Economic Review 86(3), 524–543.). The presence of a common stochastic trend in softwood lumber product price series tested is consistently rejected by Johansen’s (1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12(2–3), 231–254.) multivariate cointegration analysis. Together with rejection of non-stationarity property for a significantly large number of price series tested, the results highlight the significance of the assumptions underlying the use of composite forest commodity price indices for cointegration analysis. r 2009 Elsevier GmbH. All rights reserved. JEL classification: C32; Q23 Keywords: Product aggregation; Composite commodity; Generalized Composite Commodity Theorem; Multivariate cointegration analysis Corresponding author.

E-mail address: [email protected] (S.N. Mehrotra). 1104-6899/$ - see front matter r 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.jfe.2009.01.001

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Introduction Cointegration analysis has been applied in forestry research for testing the integration of geographically distinct markets (Jung and Doroodian, 1994; Yin et al., 2002; Nanang, 2000; Shahi et al., 2006), the law of one price (LOP) in a market for a commodity sourced from geographically distinct markets (Hanninen et al., 1997; Hanninen, 1998), the LOP for export prices to geographically distinct markets (Buongiorno and Uusivuori, 1992) and the LOP for the domestic, import and export prices of a commodity (Nyrud, 2002). It has also been applied for the study of price transmission between products at different stages of manufacturing (Zhou and Buongiorno, 2005), to determine long-run equilibrium relationship between variables in a demand and supply system (Sarkar, 1996) or develop a short-run demand–supply market model (Toppinen, 1998) and in a study of long- and shortrun impact on prices of natural catastrophes (Prestemon and Holmes, 2000). Increasing use of this technique has focused attention on the ability to derive empirically useful conclusions, particularly in the context of the limitations imposed by the quality of data used for research. For example, Yin et al. (2002) could not find evidence to support the hypothesis of a single market (spatial integration) for timber (pulpwood and saw timber) in the US south. Yet it is not uncommon for studies to use data aggregated over geographically distinct markets, e.g. Sarkar (1996) treats US and Canada as single markets for a study of softwood lumber trade. The issue of spatial aggregation has also been highlighted by Shahi et al. (2006), which found that aggregate US and Canadian markets for softwood lumber were not integrated but some sets of regional markets in the two countries were integrated. Another form of data aggregation occurs in the temporal dimension. Practically every application of cointegration analysis in forestry research uses secondary price data sourced from various reporting services. Price data reported by reporting services in the US are invariably in the form of temporal arithmetic averages, e.g. monthly, quarterly, or annual average data. The transformation of the statistical properties of time series data as a result of temporal averaging has been widely documented (Working, 1960; Tiao, 1972; Amemiya and Wu, 1972; Granger and Morris, 1976; Stram and Wei, 1986; Wei, 2005). Working (1960) has shown that temporal averaging of a random walk process (a non-stationary time series with a single unit root without moving average (MA) or autoregressive (AR) components) results in the introduction of a MA component in the transformed series. In general, temporal averaging of prices – a form of temporal aggregation widely used for reporting forest product prices – does not impact tests for cointegration relations as long as all the series under investigation have been similarly temporally aggregated (Kirchgassner and Wolters, 1992). A third form of data aggregation occurs over products. For the study of equilibrium relations between time series prices of forest products, homogeneity of the products under investigation is important. For example, several studies apply cointegration analysis to time series price data on softwood lumber (Jung and Doroodian, 1994; Sarkar, 1996; Hanninen, 1998; Nanang, 2000; Shahi et al., 2006).

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Clearly, ‘softwood lumber’ is not a homogenous product category but aggregates several products differentiated by the market on the basis of dimension, grade or other attributes identified by specie, etc. These disaggregated homogenous products may serve as substitutes for each other to varying degrees, but the very fact that they are differently valued by the market contradicts the assumption that they are perfect substitutes and hence could be treated as a single homogenous commodity. Concern regarding heterogeneity of the product under investigation has been recorded by Hanninen et al. (1997), Hanninen (1998) and Shahi et al. (2006), while Nyrud (2002) applies a test derived from the Generalized Composite Commodity Theorem (GCCT) developed by Lewbel (1996), to check the validity of product aggregation. Another important assumption behind product aggregation is that the aggregated series share the same statistical properties, in particular for cointegration analysis, the stationarity property. When this assumption is not true, the results of cointegration analysis with composite commodity value indices do not possess meaningful economic content. The objective of this research is to discuss the assumptions underlying the use of composite commodity price indices, to analyze their implications for research involving cointegration analysis and to test for their validity with empirical data. The multivariate cointegration test (Johansen, 1988) is applied to softwood lumber product data to test the empirical validity of the critical assumption of a common stochastic trend in the aggregated series. The cointegration test results consistently reject the hypothesis of a common stochastic trend for the tested softwood lumber product groups. The assumption of non-stationarity property is also rejected for a significantly large number of softwood lumber product price series tested, a result supported by findings in Yin et al. (2002) for pulpwood price series in US south. Taken together, these results highlight the importance of the assumptions underlying the use of composite commodity data for investigations of cointegration relations for forest products. The next section discusses aggregation of forest product price data and the GCCT. The following section describes the data, followed by presentation of the methodology used for the analysis and the results obtained. The final section discusses the results and concludes with proposals for further research.

Forest product price data averaging and the generalized composite commodity theorem Since the statistical concept of cointegration is applicable only to integrated data processes (denoted as I(d), where d is the order of integration) it is necessary to test the time series price data for the presence of unit roots. A I(1) process is confirmed when the first difference of a (log-transformed) price series is found to be stationary, i.e. unit root tests reject the null of non-stationarity for the differenced data, while not rejecting it for the levels. This implies that the data could be modeled as a process with a single unit root.

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Theoretical support for a non-stationary model of the price process of a traded commodity in a competitive market is provided by the efficient market hypothesis (Fama, 1970). If markets are ‘informationally efficient’ then prices of assets reflect all information available to the market participants and represent their collective belief about future expectations regarding changes in the prices. Thus, only information that is not known (unpredictable) to the market explains instantaneous price changes, which appear as random shocks. Because empirical asset price series exhibit heteroscedasticity, i.e. the variance is proportional to the magnitude of the price, it is common to apply a logarithmic transformation to the data. Commonly, the logarithm of the non-stationary price process is modeled as a normal variable, which implies that the original price process possesses a lognormal distribution.1 A simple and typical model for log-transformed asset prices is xt ¼ xt1 þ m þ t

(1)

where xt ¼ ln Pt is the natural logarithm of the price Pt of the commodity at time t, m the constant drift and, etN(0, s2) is Gaussian white noise. The model for the stochastic process for xt in Eq. (1) is known as a random walk with drift process. In time series terminology, a random walk model is of a class of autoregressive integrated moving average (ARIMA) processes with the property (0, 1, 0), i.e. the order of autoregression is zero, the order of integration is one and the order of the moving average component is zero. The corresponding model for the original price series is Pt ¼ Pt1 Exp½m þ t  ¼ Pt1 dt

(2)

Thus, the original prices series has a lognormal distribution and dt ¼ Exp[m+et] L(m, se2). Data on most forest product price series used in research is temporally aggregated. Usually the temporal aggregation is accomplished by arithmetic averaging, i.e., the model for the temporally averaged time series is PT ¼

1 ðPmT þ PmT 1 þ    þ PmT mþ1 Þ m

(3)

where m is the order of aggregation and T ¼ t/m.2 The aggregate price variable PT in Eq. (3) represents the sum of m lognormally distributed variables. The problem of finding the distribution of a finite sum (mX2) of lognormal variables arises in a variety of fields. Since there is no recognizable closed form probability density 1

It is also argued that a lognormal representation of the price process, as opposed to a normal representation, accounts for the fact that asset prices cannot take negative values. However, Bouchaud and Potters (2001, p. 10) argue that ‘‘This is however a red-herring argument, since the description of the fluctuations of the price of an financial asset in terms of Gaussian or lognormal statistics is in any case an approximation which is only valid in a certain range. These approximations are totally unadapted to describe extreme risks. Furthermore, even if a price drop of more than 100% is in principle possible for a Gaussian process, the error caused by neglecting such an event is much smaller than that induced by the use of either of these two distributions.’’ 2 Random Lengths and Timber-Mart South, two widely used sources of timber price data in the US confirm the use of arithmetic averaging for aggregation, in personal communication.

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function for the finite sum of lognormal variates, a number of techniques have been developed to approximate the distribution. While some research (e.g. Safak, 1993; Abu-Dayya and Beaulieu, 1994) finds that a suitably parameterized lognormal distribution can provide a reasonable approximation to the distribution of PT, other research recommend other distributions (Milevsky and Posner (1998) recommend the reciprocal gamma distribution). Since ease of modeling is more important to this study than accuracy of approximation, we consider and test for the lognormal distribution to approximate the distribution of PT. Another relevant question relates to the stationarity property of the logarithm of the temporally aggregated series xT ¼ ln PT. Since Eq. (3) has an equivalent representation as (4) PT ¼ PT1 FT Pm1 Qm1j Pm2 Qm2j 1 where FT ¼ j¼0 i¼0 dmTij = j¼0 i¼0 dmT mi þ 1, there is reason to believe that xT should be non-stationary. Eq. (4) retains the lognormal distribution since the denominator and numerator in the term F are composed of sums of products of lognormal variates d. The product of lognormal variates is lognormal. The presence of terms (d) with index mTm up to mT2m+2, which are shocks to the aggregated price for the preceding period (PT1), in the model for xT, i.e. " # m1 m1j X Y xT ¼ xT1 þ ln FT ¼ xT1 þ ln dmTij "  ln

j¼0

m 2 m2j Y X j¼0

# d1 mTmi þ 1

i¼0

(5)

i¼0

suggests that xT is serially correlated with a first-order moving average component. However, the sign and magnitude of the coefficient on the first-order moving average term cannot be determined analytically.3 We used simulation to test for the statistical properties of the log-transformed temporally averaged lognormal process represented by Eq. (5) and compare with a temporally averaged random walk process (as in Working, 1960), using a simple arithmetic average. The commonly used Geometric Brownian Motion (GBM) process with a constant drift and volatility was used as a representative model for lognormal prices. In particular, we used the same series of innovations (etIIDN[0, 1]) to propagate two simulated daily price data series modeled by pffiffiffiffiffi Pt ¼ Pt1 Exp½m Dt þ s Dt t  (6) 3 Past research in forest economics (Washburn and Binkley, 1990a, 1990b) has discussed the application of the results derived by Working (1960) for the arithmetic averages of random walk processes to temporally aggregated (arithmetic averages of) prices of forest products. It must be noted that the Working (1960) derivation is applicable to temporally aggregated prices whose log-transform is modeled by a random walk process, only if the original prices have been transformed to the aggregate by a process of geometric averaging. Working (1960) uses the random walk to model the commodity price itself, as opposed to its logarithmic transform.

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pffiffiffiffiffi Pt ¼ Pt1 þ mDt þ s Dtt

(7)

In Eqs. (6) and (7) the symbol Dt denotes a discrete interval of time and s is the constant volatility. Eq. (6) is the GBM model for prices with a random walk representation for its log-transform while Eq. (7) is the random walk model for prices. For models (6) and (7), identical values of P0, m, and s were used. Simulation was carried out for value ranges that could be reasonably associated with asset price series, i.e. 1pP0p1000, 0.01pmp0.9 and 0.01psp0.9, where values for m and s represent annual values. The simulated price series were tested at various levels of aggregation (weekly, monthly, quarterly, annually) and with a number of sample sizes (greater than 5000 daily price observations). The results produced by the simulation exercise, were averaged over 100 iterations and show that a. The results for the logarithm of temporally averaged lognormal model (produced from simulations of model (6)) conformed to the analytical results derived by Working (1960) for temporally averaged processes with a random walk representation. The ratio of the variance of the first difference of data averaged over a period to the corresponding variance of the unaveraged data over the same period was approximately 2/3 (or 0.667). Further, as derived by Working (1960), the differenced temporally averaged data exhibited first-order serial correlation of magnitude converging to +1/4 (or +0.25) and less than significant values for higher order serial correlation. The temporally averaged series produced from simulations of model (7) conformed to Working (1960) results, as expected. b. Application of the Jarque–Bera test for normality to the logarithm of temporally averaged lognormal price data generated by model (6) resulted in rejection of the null of normality (at 5% level of significance) in less than 10% of the iterations. The result of the Jarque–Bera test applied to the temporally averaged price data from model (7) produced a similar result, i.e. the null of normality (at 5% level of significance–95% critical value 5.99) was rejected in less than 10% of the cases. This result provides support for maintaining the normal distribution hypothesis, which implies that the finite sum of lognormal prices can be modeled with a lognormal distribution. c. The Augmented Dickey–Fuller (ADF) test did not reject the null hypothesis of a unit root at 5% level of significance (95% critical value 3.4614, in the presence of trend) in any simulation for the levels of logarithm of the temporally averaged lognormal price data series, while rejecting at for the first difference of the data (95% critical value 3.462, in the presence of trend). This result supports the hypothesis that the log-transformed finite sum of non-stationary lognormal prices can be modeled as a non-stationary process with a unit root. Confirmation of results derived by Working (1960), failure to reject the normal distribution and finding a unit root for the logarithm of the temporal averaged (arithmetic average) lognormal series suggests that the statistical transformation suffered by temporally averaged forest product prices may be trivial for the purpose of investigation of cointegrating relations. Kirchgassner and Wolters (1992)

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investigate the impact of use of temporally aggregated data on Granger-causality tests. It concludes that temporal aggregation does not lead to any major problems when the series are similarly aggregated but non-trivial problems can occur when the series have been temporally aggregated by different means, e.g. when one series is created from monthly averages while another has been created from end-of-month observations. Spatial aggregation is another form of aggregation applied to forest product prices. Determining physical boundaries for markets poses a challenge and it is common to find in research that the concept of a single, integrated market has been applied to different extents of the same geographically defined areas (Sarkar, 1996; Nanang, 2000; Yin et al., 2002; Shahi et al., 2006). It is possible to use the LOP to argue that spatial aggregation of price data from competitive markets does not impact tests for cointegration, i.e., according to the LOP spatially differentiated markets should be integrated and their price processes should be similarly trended. However, Yin et al. (2002), using cointegration analysis, could not confirm a single market for timber (pulpwood and saw timber) in the US south. Similarly, Shahi et al. (2006) could not confirm integration of aggregate US and Canadian markets for softwood lumber but found evidence for integration of some sets of regional markets in the two countries. There is another form of price data aggregation that holds significance for forest research applying cointegration analysis. This aggregation, commonly achieved by the process of arithmetic averaging (simple or weighted), occurs over the contemporaneous prices of homogeneous forest products.4 In other words, the price data used for testing the cointegrating relation represents the value of a composite commodity or price index, comprising of the prices of all the homogenous forest products it aggregates. Thus, in addition to the temporal and spatial aggregation already discussed, the price data may be subjected to another transformation, i.e. of product aggregation. For example, several studies (Jung and Doroodian, 1994; Sarkar, 1996; Hanninen, 1998; Nanang, 2000; Shahi et al., 2006) investigate the cointegration relation of softwood lumber prices. Hanninen (1998) and Shahi et al. (2006) acknowledge that softwood lumber may not represent a homogeneous commodity. Softwood lumber comprises of homogenous products, distinguished by the market on the basis of dimension, grade, etc. The use of composite product indices raises two important concerns. First, it assumes that the aggregated price series share the same stationarity property. This need not be true, that is, some of the aggregated price series could be stationary while others are non-stationary. For example, the hypothesis of presence of a unit root was rejected for 2 of the 13 pulpwood price series tested by Yin et al. (2002). Similarly, the unit root hypothesis was rejected for more than 35% of the softwood lumber price series tested by this study. A cointegration analysis test for LOP or equilibrium relations utilizing composite price indices consisting of a mix of stationary and nonstationary price processes lacks useful economic interpretation. The next concern 4

For example, Statistics Canada reports price indices for softwood lumber aggregated over regions and species, using weighted arithmetic averages.

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posed by composite product indices is with respect to the dependence relation between the aggregated series. From results derived by Granger and Morris (1976), we know that the sum of random walk processes is a random walk.5 Now, if the individual price series aggregated in a price index were independent what would we learn from the result of cointegration analysis about the behavior of the composite value series or its component price series? With independent prices, the result derived from a cointegration test applied to the composite value series may not apply to individual pairs of component price series. On the other hand, if the component price series are not independent, what is the desirable statistical property of the time series data on value of the composite commodity? Before we answer this question, it must first be noted that, typically, the aggregated commodities share significant consumption attributes, e.g. all forest products under the head ‘softwood lumber’ are softwood lumber of different specie, dimensions or grade etc and are likely related by a substitutability relation to each other of varying magnitudes. The existence of substitutability points to dependence between the prices series of these homogenous commodities. It could be argued that since the homogeneous softwood lumber products are substitutes, their prices should be similarly trended. However, since the strength of the substitutability relation is not perfect and could vary significantly across the aggregated products, the existence of a common trend in prices must be proved. A similar argument is made by the Composite Commodity Theorem (CCT) developed by Hicks (1936) and Leontief (1936) and the Generalized Composite Commodity Theorem (Lewbel 1996). The CCT sets out the condition under which it is possible for composite price indices to represent the prices of the group of commodities they aggregate. The CCT requires that the ratio of the price of each commodity (indexed by i) aggregated into a price index (indexed by I) to the contemporaneous value of the price index (yI) must be constant (ci) over time (t), i.e. ci ¼

Pi;t yI;t

(8)

or, equivalently ln ci ¼ xi;t  ln yI;t

(9)

The price index (yI, t) represents the common trend in the prices of the commodities aggregated by it since for each i we must have Pi, t ¼ yI, tci. 5

Granger and Morris (1976) derive the statistical properties of the sum of independent ARIMA processes. It shows that if two independent series are ARIMA (p, f, m) and ARIMA (q, h, n) then their sum will be ARIMA (x, g, y) where xpp+q; g ¼ max(f, h); ypmax(p+n+fh, q+m) if fXh; ypmax (p+n, q+m+hf) if hXf. This result is readily extended to the sum of more than two independent series. As a result, the sum of independent random walk processes (ARIMA (0,1,0)) will be a random walk process and the sum of independent IMA(1,1) processes will be a IMA(1,1) process. In general, the sum of nonstationary processes, dependent or independent, is a non-stationary process, unless the sum is a cointegrating relation.

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The constant ci can be seen as the scaling factor. Obviously, the deterministic relation suggested by (8) or (9) is not observed empirically. Empirical relationships between prices of even the closest substitutes are more likely to be characterized by the presence of some noise. Lewbel (1996) argues that this noise is non-trivial for product aggregation. Lewbel (1996) derives the sufficient condition for the validity of a product price aggregate, which requires that ci, t (the added subscript t indicates that it is now allowed to be stochastic) should be independent of yI, t. This result is known as the Generalized Composite Commodity Theorem. Let ln ci, t ¼ ri, t, ln yI, t ¼ RI, t and manipulate Eq. (9) to get xi;t ¼ RI;t þ ri;t

(10)

Thus, according to Eq. (10), the I(1) non-stationary logarithm of the price of a commodity must have a representation as the sum of a trend and another stochastic process that are independent of each other. Lewbel (1996) argues that ‘‘Any ri that is stationary must be uncorrelated with RI’s since the RI’s are non-stationary (sum of non-stationary processes).’’ In a similar vein, Asche et al. (2001) argues that for nonstationary price series, the GCCT is ‘‘equivalent to finding that ri and RI are not cointegratedywhich will always be true if the prices and price index are nonstationary and ri yis stationary.’’ It is significant that model (10) differs from model (1) by the additional term ri, t which is required to be stationary and independent of the non-stationary trend represented by RI, t. To explain model (10) for asset prices it is useful to examine cointegrating relationships. Research into long-term equilibrium relationships through cointegration analysis throws up questions about the statistical model for the data. Consider an asset price with the random walk representation in Eq. (1). An equivalent representation for Eq. (1) is given by xt ¼ x0 þ mt þ

t X

i

(11)

i¼1

P In Eq. (11), x0 is the starting value at an arbitrary time 0. The component it¼ 1ei of Eq. (11) represents the stochastic trend. When we confirm cointegrating relations between two asset price processes with random walkPrepresentations we are finding that the variables share the same stochastic trend it¼ 1ei. Now, unless the asset price series under investigation are exactly identical (overlapping graphs) they must differ in x0+mt, which is the deterministic component. If the asset prices differ only in scale then we could model the two log-transformed series (labeled by subscripts 1 and 2) as ) x1;t ¼ x1;t1 þ m þ t (12) x2;t ¼ x1;t þ d Here, d is the scaling constant (or supplementary drift). Apart from the possible difference in scale, such series would be propagated through identical innovations (et). A hypothetical case for such an economic scenario could be the

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prices of a commodity in geographically separated markets involving a constant proportional transfer cost (additional assumptions regarding absence of competing sources of supply, pricing policy of the producer, etc. would be required). As argued earlier, we do not find identical empirical time series but rather, series that display significant co-movement yet remain sufficiently distinct. This would imply that there is some shock that they do not share while a common stochastic trend holds them together. In the most general form, in the domain of linear models, these series can be modeled as ) x1;t ¼ wt þ u1;t (13) x2;t ¼ wt þ u2;t where wt ¼ wt1+m+zt is a non-stationary process representing the trend while zt, u1, t and u2, t are stationary processes. Thus, we have ) x1;t ¼ wt1 þ m þ zt þ u1;t (14) x2;t ¼ wt1 þ m þ zt þ u2;t which stand in contrast to Eq. (1). Models that decompose economic variables into a trend and a stationary component are known as unobserved component models. The trend represents the permanent component of the variable while the stationary component may represent the cyclical and transient portion (and/or a measurement error). Decomposition models have been considered by Nelson and Plosser (1982), Poterba and Summers (1988) and Fama and French (1988), amongst others. Nelson and Plosser (1982) consider a decomposition which has the form given in model (14) with {zt} and {ut} modeled as independent white noise processes. Another decomposition model considered in the same paper involves an AR(1) model for innovations in wt, i.e. wt ¼ wt1+zt+yzt1, with 1oyo1. Poterba and Summers (1988) and Fama and French (1988) speculate on the presence of weak mean reverting behavior in asset prices and its implications for long- and short-term market efficiency. Poterba and Summers (1988) hypothesize and test for long-term mean reversion in asset prices by using a form of model (14) that allows an AR(1) model for ut, i.e., ut ¼ r1ut1+Zt with {zt} and {Zt}modeled as independent white noise processes. In general, there is no consensus decomposition model and alternate models can be considered for modeling asset prices. To quote Fama and French (1988) ‘‘The model (Poterba and Summers 1988)y. is just one way to represent a mix of random walk and stationary price components.’’ Now, for P the index created from the arithmetic average of forest product prices ðyI;t ¼ ð1=kÞ ki¼1 Pi;t Þ to satisfy the GCCT condition it is necessary that the aggregated prices Pi, t should share the same stochastic trend wI, t ¼ yI, t. To see this, consider the GCCT condition !     k X Pi;t 1 ¼ ln Pi;t  ln yI;t ¼ ln Pi;t  ln Pi;t  ln (15) ri;t ¼ ln yI;t k i¼1

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If wi,tawI,t for all iAI Eq. (15) will be non-stationary because it is the sum of nonstationary variables.6 However, if wi, t ¼ wI, t for all iAI, i.e., all the price series share the same stochastic trend, then "   !# k X 1 ui;t ln Pi;t  ln yI;t ¼ ½wI;t þ ui;t   ln e þ wI;t þ ln k i¼1 !   k X 1 ui;t  ln e (16) ¼ ui;t  ln k i¼1 P It follows that Eq. (16) will be stationary as long as ki¼1 Exp½ui;t  is stationary. To test for the presence of a common stochastic trend in the price series of aggregated homogenous timber products, the following section applies Johansen’s (1995) multivariate cointegration analysis. If a common stochastic trend exists, the cointegrating matrix for k price series will have rank k1. If a common nonstationary trend does not exist, the rank of the cointegrating matrix will be less than k1. For a cointegrating matrix with rank less than k1 but greater than zero the cointegrating relation is not sufficient to justify the aggregation of the price series into a price index, since the hypothesis of a common stochastic trend is not supported. Asche et al. (1999) equates the GCCT condition to a test for LOP and applies it to the world salmon markets. Alternate approaches, derived from the GCCT condition, have been adopted for testing the validity of product aggregation. From Eq. (16) it is clear that the test for a common stochastic trend in the aggregated series is equivalent to a test for stationarity of the ratio series {ri, t}. In fact, if all the aggregated non-stationary price series posses a common stochastic trend, then the logarithm of ratio series created from any two price series included in the composite commodity should be stationary. Nyrud (2002) applies this form of the test for stationarity of ratio series for domestic, import and export prices of pulpwood in Norway. However, tests for stationarity of ratio series are not used in our study because of the low power of unit root tests. As noted in Kirchgassner and Wolters (2007, p. 176) the commonly applied ADF and Phillip–Perron unit root tests suffer from low power ‘‘if, under the alternate hypothesis, the first-order autocorrelation coefficient is close to one, if, for example, 0.95prp1 holds for an AR(1) process. In such situations, i.e. if the mean reverting behavior is only very weakly pronounced, very large sample sizes are necessary to reject the null hypothesis. With economic data, however, such a sample size is rare, at least as long as only monthly, quarterly or even annual data are available.’’ The problem of power applies to other tests of the stationarity property as well (e.g. Dickey–Fuller generalized least squares test and the Zivot–Andrews test). The different tests perform relatively better under certain conditions only and can give contradictory results. The following section describes the data used for the analysis. 6 In the absence of a common non-stationary trend, only a remote coincidence could result in the linear combination of ln Pi, t and ln yI, t producing a stationary process, i.e. the arbitrary combination should turn out to be a cointegrating relation.

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Data description Price data on homogenous softwood lumber products for markets in the US was obtained from Random Length yearbooks (Random Length, 2004, 2006). The data consisted of 144 monthly average (i.e. temporally aggregated) nominal price observations in dollars per thousand board feet for the period January 1995–December 2006. A total of 78 log-transformed price series were tested for stationarity, normality and heteroscedasticity. The unit root hypothesis was rejected (at 95% critical level) for as many 33 price series (details of the price series failing the tests are available from the authors). Based on the identification of distinct markets for softwood lumber in the US provided by the Random Length yearbooks, the selected price data was grouped by markets for creation of composite product indices. Data was grouped by markets to exclude the influence of spatial aggregation. The four groups are: Group Group Group Group

A – Spruce–Pine–Fir, Delivered Boston. B – Spruce–Pine–Fir, Delivered Great Lakes. C – Southern Pine, Eastern. D – Hem-Fir/White-Fir (Inland), Redding Rate.

Table 1 describes the homogeneous softwood lumber products included in the four groups. Figs. 1–4 graph the selected nominal price series by groups. The graphed price series do not provide any evidence of structural breaks, nor is such evidence available from other sources. Shahi et al. (2006) uses softwood lumber price data for the US for the same period and from the same source and considers the impact of the change in trade regimes for softwood lumber trade with Canada-quota regime under the Softwood Lumber Agreement (SLA) with Canada prevailing from 1996 to 2001, followed by the import tariff regime—using the endogenous break Zivot–Andrews unit root test. The study did not find any evidence of structural breaks. Hence, structural breaks were not considered for this analysis. The size of softwood lumber product groups was kept at 3–4 products since with 144 observations higher number of variables could lead to problems of spurious cointegration and low power of cointegration tests (Ho and Sorensen, 1996; Haug, 1996). Nominal data was used for the analysis since the presence of a common inflationary trend that enters the model for nominal prices multiplicatively (as is typically hypothesized) does not impact the tests. If inflation is a stationary process with a deterministic trend, the trend will be captured by the deterministic trend variable in the vector autoregression equation system used for cointegration analysis. If inflation is a I(1) process (see literature on New Keynesian Phillips Curve, e.g. Gali and Gertler, 1999), i.e. it shares the structure of the hypothesized common nonstationary trend in prices, it will not impact the equilibrium cointegrating relations (see Eq. (16)). Moreover, as highlighted by Prestemon et al. (2004) (citing Harvey, 1993; Schnute, 1987), deflation of time series to account for general changes in price

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Table 1. Description of softwood lumber products by composite product group. Group Product code Product description A

Spruce–Pine–Fir, Delivered Boston LAKF Dimension lumber, 2  4, #1&2, LAKG Dimension lumber, 2  6, #1&2, LAKL Studs, 2  480 , PET, kiln dried LAKM Studs, 2  490 , PET, kiln dried

random, kiln dried (prices net) random, kiln dried (prices net) (prices net) (prices net)

B

Spruce–Pine–Fir, Delivered Great Lakes LAKW Dimension lumber, 2  4, #1&2, random, kiln dried (prices net) LAKX Dimension lumber, 2  6, #1&2, random, kiln dried (prices net) LALB Studs, 2  480 , PET, kiln dried (prices net)

C

Southern Pine, Eastern LAGD LAPF LARJ LAGI

D

Dimension Dimension Dimension Dimension

lumber, lumber, lumber, lumber,

2  4, #2, random, kiln dried (prices net f.o.b. mill) 2  4120 , #2, kiln dried (prices net f.o.b. mill) 2  4160 , #2, kiln dried (prices net f.o.b. mill) 2  4, #3, random, kiln dried (prices net f.o.b. mill)

Hem-Fir/White-Fir (Inland), Redding Rate LACB Dimension lumber, 2  4, Std&Btr, random (prices net f.o.b. mill) LACC Dimension lumber, 2  6, #2&Btr, random (prices net f.o.b. mill) LACE Dimension lumber, 2  10, #2&Btr, random (prices net f.o.b. mill) LACF Dimension lumber, 2  12, #2&Btr, random (prices net f.o.b. mill)

Fig. 1. Group-A nominal softwood lumber product prices.

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Fig. 2. Group-B nominal softwood lumber product prices.

Fig. 3. Group-C nominal softwood lumber product prices.

level ‘‘may create a time series process more complex than either of the combined original series.’’ Sample autocorrelation coefficients (ri, where the subscript indexes the lag order) were calculated up to 20 lags for the logarithm of price series in levels and first difference. Appendix A presents a partial listing of the calculated sample autocorrelation coefficients. The 95% critical value bounds are calculated as pffiffiffi 1:96= n (Brockwell and Davis, 1991, p. 222) where n is the number of observations. The sample autocorrelation coefficients for the levels of the

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Fig. 4. Group-D nominal softwood lumber product prices.

log-transformed price series start at values close to positive one and exhibit gradual decline at higher lags, while the partial autocorrelation coefficients have a significant value at lag one and insignificant (or insignificant number of weakly significant) values for higher lags. The autocorrelogram of an AR(1) process for a variable x of form xt ¼ m+axt1+et (m and |ajo1 are constants and {et} is a white noise process) is generated by ri ¼ ai, while its partial autocorrelation function at lag one equals r1 and zero for higher lags. Thus the observed pattern of sample autocorrelation and partial autocorrelation coefficient values is consistent with an AR(1) or a unit root process (a ¼ 1). On the other hand, the first-differenced series display no (or insignificant number of weakly) significant sample autocorrelation coefficients and partial autocorrelation coefficients. Taken together with the pattern of sample autocorrelation coefficients for the levels of data, this result suggests the presence of a unit root. This is because a differenced AR(1) process is another AR(1) process (Dxt ¼ m+(a1)xt1+et, whereD is the first difference operator) while a differenced unit root process is the sum of a constant and white noise (Dxt ¼ m+et). Note that the autocorrelograms in general do not display the expected first-order serial correlation in temporally averaged time series, derived by Working (1960) and replicated in the simulation exercise with lognormal price model described earlier. In particular, the absence of significant first-order sample autocorrelation coefficients in the majority of first-differenced series (Table 6) is contrary to the expected positive autocorrelation coefficient value of approximately 0.25 for the first lag. The Augmented Dickey–Fuller (ADF) test for presence of a unit root was used to supplement the information on stationarity property of the price series obtained from the autocorrelograms. Since the presence of a linear trend in the data could not be ruled out by visual inspection of the price series plots, the ADF test was applied

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by subjecting the log-transformed data series {xt} to an ordinary least squares (OLS) regression of the form Dxt ¼ a þ ð1  rÞbt  rxt1 þ y1 Dxt1 þ    þ yk Dxtk

(17)

In Eq. (17), D represents the first difference operator, a represents the intercept, b is the coefficient on the time trend t and yk are coefficients on lagged values of the differenced series. The null hypothesis is that the data series has a unit root and is tested for H0:r ¼ 1 against the alternative HA:ro1. The Schwartz Bayesian Criteria (SBC) was used to select the appropriate lag-length (k) for the regression equation. The SBC was used since, along with the Hannan–Quinn criteria, it estimates the true order of the process consistently, while the Akaike Information Criteria (AIC) overestimates the true order asymptotically (Kirchgassner and Wolters, 2007, p. 57).The 95% critical value of 3.447 (in the presence of a linear trend) for the t-test of the test statistic r^ is taken from MacKinnon (1991, p. 275). The null of unit root could not be rejected for levels of all selected series. However, the null of unit root was rejected when the test was applied to the first-differenced series. This result supports the hypothesis regarding presence of a unit root in all price series. The results from the fitted regression could not reject the presence of a significant trend in all price series. The residual errors from the fitted regression equation (Eq. (17)) to the levels of data were subjected to tests for normality and heteroscedasticity. Table 2 summarizes the results of ADF, normality and heteroscedasticity tests for the selected price series. The following section describes the Johansen multivariate cointegration test methodology and presents the results of the analysis.

Methodology and results To test for the presence of a common non-stationary trend, the multivariate maximum likelihood procedure developed in Johansen (1988) was used. The procedure estimates a k dimensional VAR (p) (vector autoregressive) model of form (this discussion follows Kirchgassner and Wolters, 2007, pp. 218–224): Xt ¼

p X

Aj X tj þ Dt þ U t ;

t ¼ 1; . . . ; T

(18)

j¼1

In Eq. (18), Ut represents a normally distributed k dimensional white noise process, D represents the vector of deterministic terms, and Aj, j ¼ 1, 2, y, p, are k  k dimensional parameter matrices. Here, Xt represents the k dimensional vector of variables included in the system (the logarithm of homogenous product prices in a group). The SBC information criterion was used to determine the appropriate lag length for variables to be included in Xt. Reparametrization of Eq. (18) yields the

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Table 2. Results of unit root, normality and heteroscedasticity tests. Product ADF test ADF test first codes levels difference LACB

3.10

7.94

LACC

3.18

8.84

LACE

2.97

8.47

LACF

3.20

9.01

LAGD

2.18

11.48

LAGI

2.71

6.89

LAKF

2.73

11.42

LAKG

2.88

11.46

LAKL

3.36

11.20

LAKM

3.25

9.47

LAKW

2.82

11.60

LAKX

2.85

11.25

LALB

3.41

11.10

LAPF

2.48

11.39

LARJ

2.40

10.47

Jarque–Bera test for ARCH test for normality heteroscedasticity

Significant trend

0.37 (0.83) 0.23 (0.89) 0.16 (0.92) 0.27 (0.87) 10.07 (0.01) 5.45 (0.07) 2.35 (0.31) 1.74 (0.42) 2.90 (0.24) 1.37 (0.51) 1.92 (0.38) 0.84 (0.66) 1.40 (0.50) 5.80 (0.06) 5.67 (0.06)

Yes

3.46 (0.06) 5.15 (0.023) 1.46 (0.23) 2.90 (0.09) 1.99 (0.16) 0.10 (0.75) 0.62 (0.43) 1.54 (0.22) 0.15 (0.70) 1.86 (0.17) 0.58 (0.45) 1.36 (0.24) 0.12 (0.73) 1.10 (0.30) 0.58 (0.45)

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

95% critical value for ADF test 3.447. Figures in brackets are p-values.  Denotes significant values at 95% critical level.

vector error correction (VEC) model

DX t ¼ PX t1 þ

p1 X

Gj DX tj þ Dt þ U t

j¼1

with P ¼ I 

Pp

j¼1 Aj

and Gj ¼¼ 

Pp

i¼jþ1 Ai ,

j ¼ 1,2,y, p1

(19)

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The matrix P represents the long-run relation between the variables. Since the matrix P must have reduced rank rok (a full rank would mean that the system consists of stationary variables), it can be decomposed P ¼ a b0

ðkxkÞ

(20)

ðkxrÞ ðrxkÞ

Both matrices, a and b, have rank r and b0 Xt1 are r stationary linear combinations which guarantee that the equations in system (19) are balanced. The columns of the matrix b contain r linearly independent cointegrating vectors. If r ¼ k1, the system contains only one common non-stationary trend, which implies that all the variables in the system are pair-wise integrated. To determine the cointegration rank Johansen (1988) proposes a likelihood ratio-based trace test: TrðrÞ ¼ T

k X

lnð1  l^ i Þ

(21)

i¼rþ1

Table 3. Results of trace and maximal eigenvalue tests for cointegrating rank. Null hypothesis

Trace test statistics

Critical value (95%)

Group A r¼0 rp1 rp2 rp3

Spruce–Pine–Fir, 38.66 18.60 11.29 6.40

Delivered Boston 31.79 74.94 25.42 36.29 19.22 17.69 12.39 6.40

63.00 42.34 25.77 12.39

Group B r¼0 rp1 rp2

Spruce–Pine–Fir, 30.38 18.41 6.46

Delivered Great Lakes 25.42 55.25 19.22 24.87 12.39 6.46

42.34 25.77 12.39

Group C r¼0 rp1 rp2 rp3

Southern Pine, Eastern 45.96 31.79 22.93 25.42 19.07 19.22 6.23 12.39

94.20 48.24 25.30 12.39

63.00 42.34 25.77 12.39

Group D r¼0 rp1 rp2 rp3

Hem-Fir/White-Fir (Inland), Redding Rate 34.65 31.79 80.19 23.33 25.42 45.54 13.64 19.22 22.21 8.58 12.39 8.58

63.00 42.34 25.77 12.39

 Denotes significant value.

Maximal eigenvalue statistic

Critical value (95%)

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The null hypothesis for the trace tests is H0:rpc against the alternative hypothesis HA:r4c, where cis a number less than or equal to k. In Eq. (21) l^ i represent the estimated eigenvalues. An alternate likelihood ratio-based test is the maximal eigenvalue test (lmax) which analyzes whether there are r or r+1 cointegration vectors: lmax ðr; r þ 1Þ ¼ T lnð1  l^ rþ1 Þ

(22)

The null hypothesis for this test is H0:r ¼ c and the alternate hypothesis is HA:r ¼ c+1. Critical values for both these tests are taken from Johansen (1995, p. 214). The specification of deterministic components of the VEC model is a critical part of Johansen’s multivariate cointegration approach. Johansen (1994) shows that improper specification results in substantial loss in power of the test procedure. Since a significant time trend is found in all price series, the VEC model used in this study is specified with unrestricted intercepts and restricted trends. This model allows the individual price series to have trends by not restricting the intercept term to zero, while the trend entering the cointegrating relation implies that no implicit assumption is made about links between the deterministic growth patterns across the series. The results of the trace and maximal eigenvalue tests for the five groups are presented in Table 3. Table 3 shows that the cointegrating analysis rejects a rank of k1 for the cointegrating matrix for all products groups. Both the trace and maximal eigenvalue tests reject the hypothesis of existence of a common stochastic trend in all four groups.

Conclusions The typical time series data on prices of forest products, used in cointegration analysis, is the product of several transformations. These transformations include temporal, spatial and product aggregation. This study tests empirical data for validity of product aggregation or composite forest commodity price indices. Temporal aggregation does not influence cointegration analysis as long as the investigated data series are similarly temporally aggregated (Kirchgassner and Wolters, 1992). However, since the arithmetic temporal averaging of lognormally distributed prices involves a finite sum of lognormal variates, the probability density of which does not have a closed form, simulation was utilized to test for the statistical properties. Results for a simulated non-stationary lognormal process with a random walk representation for its log-transform did not reject lognormality or non-stationarity for its temporally (arithmetic) averaged transform, while confirming

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other properties derived by Working (1960) for temporally averaged random walk processes. However, tests on the temporally averaged empirical prices of softwood lumber products did not produce a significant first-order autocorrelation coefficient value, as expected from results derived in Working (1960) and the simulation exercise, in almost all cases. This result suggests that alternate models to the simple random walk should be considered for the log-transform of price processes of softwood lumber products and this could be the subject of future research. The principal justification offered for the validity of product aggregation is a close substitutability relation between the aggregated products. Thus, the underlying assumption for product aggregation is that prices of close substitutes in a market will be similarly trended and that an investigation of equilibrium relations with composite commodities is a joint investigation of the equilibrium relations of individual products comprising the composite commodity. However, the very fact that the market price-differentiates the products that form the composite commodity calls for an investigation of the validity of the close substitutability assumption. The argument for presence of a common trend is also derived from the GCCT condition for validity of composite commodities. With non-stationary prices the presence of a common stochastic trend can be tested by cointegration analysis. Testing for validity of composite softwood lumber price indices, the Johansen multivariate cointegration test rejects the presence of a common non-stationary trend for all four cases tested. This result raises questions regarding the validity of treatment of softwood lumber as a composite commodity for the purpose of cointegration analysis. Taken together with similar results questioning the validity of spatial aggregation in Yin et al. (2002), these results stress the importance of the common stochastic trend assumption. Equally significant is the rejection of the unit root hypothesis for a significantly high number of tested softwood lumber price series. If some aggregated price series in a composite product index are stationary, cointegration analysis cannot provide meaningful results. The assumption that all the aggregated price series are non-stationary is challenged by these findings. While the rejection of nonstationarity for tested softwood lumber price series did not conform to any easily identifiable pattern or product groupings, the reasons for and implications of existence of stationary as well as non-stationary price series could be the subject of future research. In conclusion, the results of the analysis presented in this paper highlight the importance of the assumptions supporting the use of composite forest product indices for cointegration analysis.

Appendix A. Sample autocorrelation coefficients of log-transformed price series See Tables 4–7.

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Table 4. Sample autocorrelation coefficient values for levels of natural logarithm of price series.

LACB LACC LACE LACF LAGD LAGI LAKF LAKG LAKL LAKM LAKW LAKX LALB LAPF LARJ

r1

r2

r3

r4

r5

r6

0.9105 0.8992 0.8976 0.8811 0.9032 0.9459 0.8994 0.8924 0.8451 0.8872 0.8932 0.8945 0.8398 0.8966 0.9104

0.7931 0.7768 0.7914 0.7480 0.8056 0.8537 0.7982 0.7843 0.6846 0.7328 0.7883 0.7860 0.6740 0.7862 0.8094

0.6854 0.6828 0.7012 0.6480 0.7149 0.7547 0.7115 0.7010 0.5486 0.5396 0.7011 0.7010 0.5353 0.6803 0.7132

0.5766 0.5965 0.6131 0.5719 0.6112 0.6587 0.6003 0.5923 0.3924 0.4393 0.5520 0.5943 0.3755 0.5650 0.6072

0.4941 0.5340 0.5347 0.5302 0.5336 0.5771 0.5152 0.5098 0.2843 0.3284 0.5022 0.5117 0.2657 0.4836 0.5151

0.4327 0.4956 0.4615 0.4962 0.4689 0.5117 0.4495 0.4556 0.2396 0.2648 0.4361 0.4558 0.2177 0.4281 0.4450

95% critical value 0.1633.  Denotes significant value.

Table 5. Sample partial autocorrelation coefficient values for levels of natural logarithm of price series.

LACB LACC LACE LACF LAGD LAGI LAKF LAKG LAKL LAKM LAKW LAKX LALB LAPF LARJ

r1

r2

r3

r4

r5

r6

0.9105 0.8992 0.8976 0.8811 0.9032 0.9459 0.8994 0.8924 0.8451 0.8872 0.8932 0.8945 0.8398 0.8966 0.9104

0.2099 0.1662 0.0736 0.1267 0.0552 0.3898 0.0562 0.0591 0.1031 0.2549 0.0469 0.0708 0.1061 0.0904 0.1133

0.0173 0.0947 0.0237 0.0772 0.0167 0.0123 0.0197 0.0615 0.0089 0.0006 0.0291 0.0573 0.0059 0.0379 0.0240

0.0925 0.0499 0.0453 0.0294 0.1238 0.0229 0.1803 0.1769 0.1627 0.1512 0.1784 0.1638 0.1667 0.1125 0.1173

0.1034 0.0908 0.0003 0.1113 0.0818 0.0728 0.0796 0.0820 0.0718 0.1191 0.0730 0.0759 0.0705 0.1063 0.0247

0.0174 0.0594 0.0258 0.0077 0.0120 0.0289 0.0240 0.0536 0.1208 0.0709 0.0210 0.0493 0.1039 0.0647 0.0574

95% critical value 0.1633.  Denotes significant value.

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Table 6. Sample autocorrelation coefficient values for first differenced natural logarithm of price series.

LACB LACC LACE LACF LAGD LAGI LAKF LAKG LAKL LAKM LAKW LAKX LALB LAPF LARJ

r1

r2

r3

r4

r5

r6

0.1478 0.0992 0.0024 0.0626 0.0153 0.4227 0.0012 0.0041 0.0093 0.1181 0.0163 0.0145 0.0114 0.0067 0.0656

0.0538 0.1408 0.0613 0.1353 0.0315 0.0940 0.0699 0.1130 0.0771 0.0360 0.0798 0.1091 0.0860 0.0238 0.0161

0.0353 0.0247 0.0193 0.0617 0.1221 0.0217 0.1265 0.1233 0.0736 0.0544 0.1251 0.1060 0.0687 0.0879 0.1043

0.1579 0.1278 0.0635 0.1340 0.1269 0.0850 0.1340 0.1358 0.1558 0.1736 0.1295 0.1121 0.1525 0.1599 0.0605

0.1118 0.1157 0.0092 0.0093 0.0058 0.1103 0.0934 0.1004 0.1958 0.1937 0.0880 0.1242 0.1807 0.0855 0.0879

0.0175 0.0061 0.0662 0.0804 0.0298 0.0919 0.0826 0.0405 0.0141 0.0018 0.0878 0.0985 0.0104 0.0639 0.0839

95% critical value 0.1633.  Denotes significant value.

Table 7. Sample partial autocorrelation coefficient values for first differenced natural logarithm of price series.

LACB LACC LACE LACF LAGD LAGI LAKF LAKG LAKL LAKM LAKW LAKX LALB LAPF LARJ

r1

r2

r3

r4

r5

r6

0.1478 0.0992 0.0024 0.0626 0.0153 0.4227 0.0012 0.0041 0.0093 0.1181 0.0163 0.0145 0.0114 0.0067 0.0656

0.0773 0.1521 0.0613 0.1398 0.0317 0.1030 0.0699 0.1130 0.0772 0.0746 0.0801 0.1093 0.0862 0.0239 0.0205

0.0569 0.0067 0.0197 0.0444 0.1212 0.0263 0.1270 0.1233 0.0755 0.0790 0.1232 0.1107 0.0713 0.0876 0.1072

0.1823 0.1521 0.0677 0.1495 0.1266 0.1175 0.1427 0.1358 0.1657 0.2131 0.1357 0.1320 0.1641 0.1608 0.0764

0.0524 0.0917 0.0122 0.0068 0.0006 0.0316 0.0741 0.1004 0.1843 0.1143 0.0711 0.0956 0.1674 0.0835 0.0749

0.0203 0.0159 0.0757 0.0398 0.0543 0.0371 0.0516 0.0405 0.0144 0.0363 0.0534 0.0694 0.0197 0.0826 0.0891

95% critical value 0.1633.  Denotes significant value.

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