How regular are directional movements in commodity and asset prices? A Wald test

Journal of Empirical Finance 38 (2016) 290–306

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Journal of Empirical Finance journal homepage: www.elsevier.com/locate/jempfin

How regular are directional movements in commodity and asset prices? A Wald test Atle Oglend a, * , Tore Selland Kleppe b a b

University of Stavanger, Department of Industrial Economics, Norway University of Stavanger, Department of Mathematics and Natural Sciences, Norway

A R T I C L E

I N F O

Article history: Received 19 October 2015 Received in revised form 10 June 2016 Accepted 1 July 2016 Available online 19 July 2016 Keywords: Commodity prices Asset prices Duration dependence Efficiency Markov chain

A B S T R A C T This paper derives a Wald test to evaluate whether up/down movements in prices follow a two-state first-order time-homogenous Markov chain. Probabilities that prices, separated by up to k periods, move in the same direction are derived and compared to empirical probabilities using a Wald statistic. The hypothesis is evaluated for 48 monthly commodity prices and five major stock price indices. Nominal commodity prices show evidence of symmetric momentum in up and down movements. Stock indices have momentum in up movements, with a positive trend due to more frequent up movements. The testing reveals fundamental differences between commodity and asset prices. Several commodities show evidence against the null hypothesis, while none of the stock indices reject the hypothesis. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Consider the following hypothesis: if the price went up (or down) this period, the probability of going up (or down) again the next period is fixed and independent of the history of price movements beyond the current state. If this does not hold, there is a degree of regularity to the unfolding of price movements such as duration dependence, regular cyclicality or higher order lag dependence. If it holds, it is sufficient for weak-form market efficiency in a market that pays/deducts a fixed amount for given price increases/decreases. It means directional prices movements can be modeled as a first order Markov chain with constant transition probabilities. We develop a Wald test to evaluate this hypothesis. The test looks at the patterns in probabilities of directional price movements as the distance between observations increases, and then compares implied probabilities under the null hypothesis with the empirical probabilities. The test can be used to reveal patterns in directional price movements such as trending or regular cyclicality. The test is similar to the Runs test for independence of runs in the data, but rather than focusing on the distribution of runs, it focuses on the pattern in probabilities of directional price movements. The test is applied to evaluate the asset like nature of commodity prices. In recent years, commodity prices have displayed large swings around relatively high price levels. This is not isolated to specific commodity groups, but is found in everything from meats and crops to energy and metals (Natanelov et al., 2011; Trostle, 2010; Trostle et al., 2011). This has spurred some

* Corresponding author. E-mail address: [email protected] (A. Oglend).

http://dx.doi.org/10.1016/j.jempfin.2016.07.001 0927-5398/© 2016 Elsevier B.V. All rights reserved.

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debate regarding the nature of commodity price movements (Calvo, 2008; Carter et al., 2011; Lombardi et al., 2011; Nazlioglu and Soytas, 2011; Trostle, 2010), including the role of commodity investments funds in commodity markets (Cheung and Miu, 2010; Domanski and Heath, 2007; Gorton et al., 2013; Irwin and Sanders, 2010; Sanders and Irwin, 2010). Investment funds go long in commodity futures contracts to gain exposure to commodity markets risk factors. Commodities are treated as financial assets, valued by the cumulative changes in futures prices, equivalent to a non-divided paying stock. It is well known that commodities have both an asset and a consumption value. A commodity in storage is a capital asset, valued principally by cumulative price changes (making allowance for the cost of carry and convenience yield). Consumption value ensures that prices will not appreciate or depreciate indefinitely. A consistently high economic rent will lead to increased investments and expansion of supply. Cash prices will revert to normalize economic rents. In addition, capacity constraints on storage and the inability to borrow future commodities implies limits-to-arbitrage that bounds the directional price movements. Systematic variations in use demand, seasonal production restrictions and sluggish supply responses also contribute to some degree of regularity in price movements different from pricing of financial assets. For a commodity valued as an asset, the principal value of the commodity is determined by price returns. For a commodity valued as a consumption good, the price level is the key determinant of value. In the former case, prices are more likely to contain a permanent component, a unit-root, ensuring a fixed return over time. In the latter case, prices are more likely to be mean-reverting, ensuring normalization of economic rents, although the equilibrium price might evolve similar to a stochastic trend (this is the case for the popular two-factor model of commodity prices in Schwartz (1997)). Commodity prices often fail to reject a unit root using conventional tests. Wang and Tomek (2007) argue that this is due to tests with weak power against more realistic price processes. They find that allowing for stationary non-linear alternatives, the evidence of unit-roots become much weaker. Similarly, Balagtas and Holt (2009) find that for 16 out of 24 commodities, the linear unit-root null is rejected in favor of a non-linear smooth transition autoregression representation. This is consistent with commodity price theory (see for example Deaton and Laroque (1992)), which predicts that prices should follow a non-linear autoregressive threshold process. Commodity prices are bounded and globally stationary (in absence of permanent changes to preferences or technology), following a renewal process where historical shock dependencies vanish when prices cross upper (stock-out) or lower (capacity constraint) threshold levels. When preferences, technology or market structures change, unitroot tests need to differentiate a stochastic trend from structural changes to parameters, a procedure that becomes increasingly difficult as the number of structural changes increase. Kellard and Wohar (2006) show that commodity price trends are not well represented by a single slope, but rather by a shifting trend that changes signs. Enders and Holt (2012) investigates commodities from the World Bank Pink Sheets database, modeling commodity prices with a smooth shifting-mean autoregressive process to account for structural changes. The authors find some evidence that most prices revert to a smoothly shifting mean. In terms of trends in commodity prices, Harvey et al. (2010) finds evidence of a significant downward trend for 11 of 25 commodities analyzed using over four centuries of data. Others find similar results, but trend analysis remains sensitive to the specification of testing equations (Kellard and Wohar, 2006). We test the Markov chain hypothesis on a large set of monthly commodity prices spanning up to forty years. The data consists of prices of 48 major commodities, including 11 different meat products. We also test five major stock indices for comparison to the commodities. The empirical analysis reveals fundamental differences between commodity prices and assets. Almost all commodity prices display symmetric momentum in both up and down price movements. This means that if the price has moved up (down), the probability is greater than one-half that it will move up (down) again the following period. For the financial indices, we find only momentum in upwards movements. Only a few of the commodity prices show evidence of trending in nominal prices, and as the distance between price observations increase the probability of up or down movements become independent of past states, indicating strong mixing in consistent with a renewal process. Stock indices show strong evidence of a positive trend (the unconditional probability of increasing prices is approximately 60%). The expected return in stocks appears to be due to more frequent up movements in asset prices, not larger relative up movements. Despite finding positive momentum in commodity prices, several of the commodities analyzed show evidence against the null of constant and price history independent probabilities of directional price movements. All stock indices fail to reject the hypothesis, consistent with weak-form market efficiency in directional movements. The paper is structured as follows. Section 2 develops goes through the hypothesis and develops the Wald test. In Section 3, we investigate the size and power of the test. Section 4 presents the empirical results for commodity and financial data. Finally, Section 4 offers some concluding remarks. 2. Directional price movements and weak-form market efficiency Let y1:n+1 = (y1 , y2 , . . . , yn+1 ) be a sequence of equally spaced discrete price observations. We are concerned with testing the following hypothesis H0 : prob (Dyt+1 > 0|Dyt > 0, y1:t ) = 1 − k1 , prob (Dyt ≤ 0|Dyt ≤ 0, y1:t ) = 1 − k2 . The hypothesis can also be stated in terms of the hazard function for price movements. If we distinguish expansions from contractions, H0 implies constant hazards h(ti ) = k1 and h(ti ) = k2 for expansions and contractions respectively (ti is the duration of spell i). The hazard function gives the conditional probability that a spell will end given it has persisted for ti periods.

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As suggested by Lancaster (1979), a general hazard function can be stated as h (ti ) = k (a + 1) tia , where k is a scaling parameter and a the duration dependence parameter. An a greater (lower) than zero implies positive (negative) duration dependence. With no duration dependence (a = 0), price spells (consecutive movements of prices in one direction) are exponentially distributed, nested in the general Weibull distribution. H0 can be tested by a likelihood ratio test for a = 0. Cochran and Defina (1995) and Harman and Zuehlke (2004) use a similar approach to investigate duration dependence in asset prices in light of speculative bubbles. Duration dependence has also been tested for the business cycle (see for instance Castro (2010)), as well as in labor economics in studies of spells of unemployment (Kiefer, 1988). For commodities, Roberts (2009) investigates duration dependence in metal price cycles, while Labys (2006) tests for duration dependence in 21 commodity prices. Testing for duration dependence by means of a specific hazard function is suitable when departures from H0 are nested within h(ti ). However, there are a several relevant departures from H0 not nested in a specific hazard function. A weakly stationary process violates H0 since k1 and k2 depends on price levels, and thus cumulative price changes. A series with deterministic cyclicality violates H0 since k1 and k2 depends on t. None of these departures from H0 are nested within h (ti ) = k (a + 1) tia . One way around this problem is to extend the hazard function by making it dependent on known covariates to directional price movements. We take a different approach by studying the implication of H0 on the binary series of directional price movements. We derive the probabilities that prices, separated by i periods, move in the same direction under the null hypothesis. We then construct a statistic that compares these implied probabilities to the empirical probabilities. In a simulation experiment we show that this statistic has power against departures from H0 such as deterministic cyclicality and higher-order autoregressive dependence, while still retaining power, although weaker than the hazard function tests, against departures from H0 nested in the a hazard function. In addition, we also demonstrate how the patterns in the empirical probabilities can reveal relevant information on the form of dependence in prices, such as long-run cycles or seasonality. Other tests for H0 exist. Since H0 means durations are exponentially distributed, a typical approach is to use some goodnessof-fit tests such as the Shapiro and Wilk (1972) or Stephens (1978) tests. Our test deviates from these methods by focusing on the binary series of directional price movements rather than duration data. In this sense our approach is related to Markov switching, or regime switching, models. In these models, directional price movements are modeled as the result of a Markov process that switches between states of up or down movements. Some examples of this approach applied to the business cycle can be found in Hamilton (1989), Durland and McCurdy (1994), Kim and Nelson (1998) and Lam (2004). Our test can be used to evaluate the validity of applying a constant transition probability matrix for regime switching models. 2.1. Directional price movements under H0 Consider the binary variable st , where st = 1 if yt > yt−1 and zero otherwise (we do not consider samples where prices are constant over periods). Under H0 , st evolves as a first-order Markov chain with transition probabilities prob(st+1 = 1|st = 0) = k1 and prob(st+1 = 0|st = 1) = k2 . Given k1 , k2 ∈ (0, 1), the Markov chain is irreducible and aperiodic, having a unique stationary distribution. We proceed by defining additional binary variables that take a unit value if prices, separated by i periods, move in the same direction, and zero otherwise. For a distance i between observations at time t this binary variable is ci,t = st st−i +(1−st )(1−st−i ).   Let mi = E(ci,t ) and si,j2 = cov ci,t , cj,t . The mean mi is the expected probability, under the stationary distribution, of prices now and i periods ago moving in the same direction, mi =





         k2 k1 ci − 1 ci − 1 + 1 − k2 , Pi , Pi p  = 1 − k1 11 22 c−1 k1 + k 2 c−1 k1 + k 2

(1)

c = 1 − k 1 − k2 , p=

k2 , k1 + k 2

k1 , k1 + k 2

where P is the [2 × 2] transition matrix for the vector [st , 1 − st ] , (P i )jk is the row j column k value of the matrix raised to the power i, c is the non-unit eigenvalue of P, and p is the stationary distribution of [st , 1 − st ]. If k1 = k2 the transition matrix is symmetric and lim mi = 0.5. In this case, price movements will satisfy strong mixing - as the distance between observations i→∞

increases, the probability of moving up or down becomes independent of the past state. Under H0 , mi converges to its limit value (as i increases) at a geometrical rate. For negative eigenvalues (c < 0), mi oscillates toward this limit value. The covariance between ci,t and cj,t , i ≥ j, is   si,j2 = E ci,t cj,t − mi mj , =

         Pi , Pi P i−j P i−j p − mi mj , i ≥ j. 11

11

22

22

(2)

As for the mean, the covariance is completely defined by k1 and k2 under H0 . We collect the covariances in the [k × k] covariance matrix Y, and the mean values in the [k × 1] vector m. When k1 and k2 are not one-half, the random vector

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ct = [c1,t , c2,t , . . . , ck,t ] is a correlated sequence. Let C l be the auto-covariance of ct at lag l such that the long-run covariance  matrix is Y∗ = Y + ∞ l=−∞, =0 Cl . The sample counterpart of mi is ⎧ ⎫ n−k n−k ⎬ 1  1 ⎨ ˆi= ci,t = st st+i + (1 − st ) (1 − st+i )) , m ( ⎭ n−k n−k ⎩ t=1

(3)

t=1

for i = 1, 2, . . . , k. From the multivariate central limit theorem for correlated sequences, the vector of sample means is limit Gaussian, 

  ˆ − m → N (0, Y∗ ) . n−k m

(4)

d

ˆ i calculates the relative frequency of same directional price movements when observations are separated The sample mean m by i periods. The implied mean mi calculates this relative frequency under the null hypothesis that directional price movements ˆ i from mi forms the basis for can be modeled as a first-order Markov chain with constant transition probabilities. Deviations of m testing the null hypothesis. Section 2.3 derives the specific test statistics applied in this paper. This is a portmanteau test where the alternative hypothesis remains unspecified. In the simulation experiment in Section 3 we investigate the size and power of the test under some different alternative hypotheses. Calculating the exact long-run covariance matrix Y∗ can be done by summing the probabilities of paths of st satisfying ci,t cj,t+l = 1. This procedure is not practical when i + j + l grows large due to the curse of dimensionality (the number of possible paths of st relevant to calculate E(ci,t cj,t+1 ) grows by 2i+j+l ). An alternative is to apply some long-run variance estimator. However, as the probabilities k1 and k2 differ from one-half, the auto-covariance can become large. Sample uncertainty in the estimator of Y∗ can then lead to determinants of Y∗ very close to zero. Consequently, any test statistic that is inversely proportional to Y∗ becomes highly sensitive to the sample uncertainty in the Y∗ estimator. A practical approach to derive test statistics based on Eq. (4) is to set Y∗ = Y and use Monte Carlo methods. The Monte Carlo procedure used for our test statistic is described in Section 2.3. 2.2. Estimating transition probabilities As is well known, a consistent and efficient estimate of k = [k1 , k2 ] under H0 can be found by counting the relative frequency of consecutive versus broken up/down movements of prices. This is the maximum likelihood estimator (MLE), which follows   from basic MLE theory. Let n1,1 = nt=2 st−1 st be the number of two-period price increases, n1,0 = nt=2 st−1 (1 − st ) the number n of decreasing then increasing prices, n0,0 = t=2 (1 − st−1 )(1 − st ) the number of two-period decreasing prices, and n0,1 = n t=2 (1 − st−1 )st the number of decreasing then increasing prices. We then have n = n1,1 + n1,0 + n0,1 + n0,0 + 1, and the MLEs kˆ 1 = n1,0 /(n1,1 + n1,0 ), kˆ 2 = n0,1 /(n0,0 + n0,1 ), with associated Fisher Information

I(k) =

1 k1 + k 2



k2 k1 (1−k1 )

0

0 k1 k2 (1−k2 )

 ,

where it follows from standard MLE theory that

 √ ˆ n k − k0 → N(0, I−1 (k0 )). d

2.2.1. Testing for momentum When directional price movements contain momentum, the probability of increasing (decreasing) price today is greater if price increased (decreased) the previous period. Full momentum in both directions implies that k1 < 0.5 and k2 < 0.5. Momentum gives rise to price cycles. The expected duration of such cycles are 1/k1 + 1/k2 , where 1/k1 and 1/k2 are the expected duration of up and down movements respectively. This is only valid under H0 when cycles are purely random. Partial momentum (ki < 0.5, kj ≥ 0.5) where i, j = 1, 2 means momentum in only one direction (up or down), and can be indicative of trending behavior. We will test for full and partial momentum in the empirical section below.

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2.2.2. Testing for trending The presence of a price trend means cumulative upward movements differ from cumulative downward movements. Let  n+1 zup,n = n+1 t=2 (yt − yt−1 ) st−1 and zdown,n = − t=2 (yt − yt−1 ) (1 − st−1 ) be the cumulative up and down movements observed in the data. With no trending we should observe that lim zup,n /zdown,n = 1. Let nup,n = n1,1 + n0,1 and ndown,n = n0,0 + n1,0 be n→∞

the number of up and down movements. The criteria becomes nup z¯ up zup,n p1 z¯ up k2 z¯ up = = = , zdown,n ndown z¯ down p2 z¯ down k1 z¯ down where z¯ up and z¯ down are the average magnitudes of up and down movements. A sufficient condition for no trending is k1 = k2 and z¯ up = z¯ down . The first condition can be tested using a likelihood ratio test whereas the second condition can be tested using for instance Welsh’s t-test for equality of means. This approach to trend analysis provides a decomposition of the trend in terms of number of up/down movements and the magnitude of up/down movements. Note that k1 = k2 and z¯ up = z¯ down is not a necessary condition for no trending. It is for instance possible that there are more up movements in prices, but that the magnitude of down movements are larger such that their asymmetry cancels out and lim zup,n /zdown,n = 1. n→∞

2.3. A test statistic for H0 We now derive a test statistics to evaluate H0 . We do  this  by comparing the distance of the implied mean mi under the  null ˆ i for all i = 1, 2, . . . , k. Let m kˆ be the implied mean of ct calculated using Eq. (1), and let Y kˆ the from the empirical mean m     implied covariance matrix, calculated using Eq. (2). Furthermore, let Y∗ kˆ be the long-run covariance matrix, and kˆ = kˆ 1 , kˆ 2 the MLE of the transition probabilities.      1 n−k ˆ . Testing H0 is done in a portmanteau-test manner by evaluating Define the sample average g kˆ = n−k t=1 ct − m k ˆ to what degree g(k) deviates from zero. Specifically, we consider the Wald statistics J∗ (k) =



       n − kg kˆ Y∗ kˆ −1 n − kg kˆ .

(5)

When kˆ is known it follows from Eq. (5) that J∗ (k) → w 2 (k). In most cases kˆ must be estimated. The MLE uses st and st−1 as d

ˆ From the expressions for m ˆ 1 = m1 . ˆ 1 and m1 and the MLE estimator kˆ we have that for n01 = n10 , m data when estimating k. Furthermore, n01 will only differ from n10 by a maximum of one (specifically when the first and last observation of st are not the ˆ 1 and m1 will differ by an amount that tends to zero asymptotically.1 As such, when kˆ is estimated using same). In this case m the MLE, J∗ (k) is close to w 2 (k − 1) for most reasonable sample sizes under the null. The test statistics becomes similar to an over-identification restriction test.  As discussed above, when Y∗ kˆ cannot be practically or robustly derived (for instance if k is large or the sample small)     using either direct calculation or a long-run variance estimator, the preferred approach is to set Y∗ kˆ = Y kˆ in Eq. (5) and derive the finite sample distribution   of the statistics  under the null using Monte Carlo simulations (parametric bootstrap). Let J(k) be the statistics J∗ (k) with Y kˆ replacing Y∗ kˆ . For a given k, the Monte Carlo procedure is as follows:     1. For price sequence y1:n+1 estimate kˆ using the ML-estimator. Derive m kˆ and Y kˆ and calculateJ(k) 2. For i from 1 to nboot ,  n (a) Simulate binary sequence st,i t=1 given kˆ data using the MLE estimator (b) Estimate kˆi on  the simulated   (c) Derive m kˆ and Y kˆ using Eqs. (2) and (3) (d) Calculate and store Ji (k) nboot  3. Compare J(k) to the empirical distribution of Ji (k) i=1 . Reject H0 for large J(k) defined according to some suitable  nboot percentile of Ji (k) i=1 . ˆ and so the parametric bootstrap preferred over non-parametric The complete distribution of st under H0 is known given k, bootstrap, which would require some form of block-bootstrapping to incorporate the time series dependence in st under H0 . The simulated null distribution is conditioned on the kˆ estimate. Sample uncertainty in kˆ under the null is accounted for by reestimating kˆ on the simulated data.

1

The implied probability at k = 1 can be written as m1 =

Since the ratio

n1,0 n0,1

n1,1 +n0,0

n1,0 n0,1

 n

(n1,1 +n1,0 )+(n0,0 +n0,1 ) n1,0 0,1

ˆ 1 will tend to m1 . tends to one as n increases, m

ˆ1 = , while the empirical probability can be written as m

n00 +n11 n00 +n11 +n01 +n10

.

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Table 1 Simulation results for parametric process 1. h(1) = 0.5 a0 a0 a0 a0

= 0.0 = 0.1 = 0.2 = 0.3

(1)

J(2)

J(6)

J(12)

J(24)

J(36)

Jmax

Weib. haz.

Reg. test

0.04 0.22 0.62 0.91

0.05 0.12 0.38 0.72

0.04 0.09 0.26 0.58

0.03 0.04 0.16 0.44

0.04 0.08 0.18 0.35

0.03 0.17 0.47 0.84

0.04 0.30 0.74 0.98

0.04 0.14 0.45 0.82

0.07 0.14 0.33 0.72

0.06 0.15 0.36 0.63

0.04 0.10 0.22 0.48

0.03 0.09 0.17 0.33

0.03 0.07 0.16 0.29

0.03 0.10 0.34 0.64

0.06 0.18 0.57 0.91

0.07 0.16 0.40 0.66

h(1) = 0.25 a0 a0 a0 a0

= 0.0 = 0.1 = 0.2 = 0.3

Note: Simulation results are for 250 simulated binary sequences of length 500 for each specified process and nboot = 1000 for the bootstrapped statistics. Random number draws are the same for each specified process.

An important researcher choice when applying the test is the choice of lag length k. This is a choice over how far back we are testing for regularity in the directional price movements. The J(k) test is a portmanteau test similar to the popular Ljung-Box or Box-Pierce Q(k) statistics for testing whether data is independently distributed. As is well known, choosing a low k risks missing relevant dependence in the data, while choosing a high k reduces the power of the test. For the Q(k) statistics, Hyndman and Athanasopoulos (2014) suggests k = 10 for non-seasonal data and k = 2m for seasonal data, where m is the seasonal span. In the simulation experiment below and in the empirical analysis we consider different k values. To reduce the sensitivity of the test to k we also consider a Sup test over different k values up to some kmax . The Sup test 2 requires standardizing J(k). We consider two standardizations.  The first applies the w distribution mean and variance such that the standardized statistics is J(1) (k) = (J(k) − (k − 1)) / 2 (k − 1). The second standardizes J(k) by the mean and standard nboot  from the Monte Carlo analysis. We denote this J(2) (k). We then have the deviation of the empirical of Ji (k) i=1  distribution  ( j)

sup statistics Jmax = sup J( j) (k) for j = {1, 2}. The maximum is considered up to some specified upper bound kmax . To apply k ( j)

the test we estimate Jmax on the actual data of interest and reject H0 for large values relative to some suitable percentile of the ( j)

empirical distribution of Ji,max

nboot i=1

as derived from the parametric bootstrap. The actual data statistics J(2) (k) is standardized

according to the mean and standard deviation of the empirical distribution of J(2) (k) from the Monte Carlo analysis. ˆ i for i = 1, 2, . . . , k along with An alternative approach to evaluate the regularity in price movements is to plot the estimated m ˆ i . This would be equivalent to the commonly applied method the implied probabilities mi and some confidence interval for m of plotting the autocorrelation of regression residuals to investigate the profile of residual serial correlation. Peaks in ci at for instance seasonal frequencies would be indicative of seasonality. We do this for a selection of commodities in Section 4.2. 3. Simulation studies In this section we evaluate the power and size of the J(k) test when confronted with different departures from H0 . The objective is to evaluate the performance of the test on a set of data generating processes that are somewhat reasonable for commodity prices. We consider the following data generating processes for directional price movements: 1. Weibull hazard function: h (ti ) = k (a + 1) tia , for a = {0, 0.1, 0.2, 0.3} and k such that h(1) = k(a − 1) = 0.5 and h(1) = k(a − 1) = 0.25 2. Parametric price process with deterministic cyclicality: yt+1 = b1 yt + b2 sin(2pt/18) + et , where et ∼ N(0, 1) and b1 = {1, 0.95}, b2 = {0, 0.1, 0.25, 0.5} 3. Parametric price process with higher order autoregressive dependence: yt+1 = b1 yt + b2 yt−5 + et , where et ∼ N(0, 1) and b1 = {1, 0.95}, b2 = {0, −0.05, −0.1, −0.15} For process 1, when a = 0 the hazard is constant (equal to k) and H0 is satisfied. For a > 0 there is positive duration dependence and the hazard increases with the duration of spells. For parametric processes 2 and 3, H0 is satisfied when b1 = 1 and b2 = 0, whereby prices follow a random walk with zero drift (e.g. a constant hazard function with k = 0.5). A b2 different from zero is associated with increasing deterministic cyclicality (process 2), and increasing higher order autoregressive dependence (process 3). For each process and specification we simulate 250 series’ of length 500 and proceed to evaluate the statistics at their respective 5% nominal critical values. The J(k) statistics is evaluated using the Monte Carlo procedure discussed above with nboot = 1000. To investigate the sensitivity of the test to k we consider some different lag values k = {2, 6, 12, 24, 36}. (1) (2) We also evaluate the Sup test statistics Jmax and Jmax with kmax = 36. We also consider some alternative tests for H0 . One alternative is to estimate the parametric model st = c0 + c1 st−1 + c2 st−1 dt−1 + error, where dt−1 denotes consecutive periods spent moving upwards. A test for H0 is a test for the restriction c2 = 0. This test was discussed in Ohn et al. (2004) where they show that testing c2 = 0 is equivalent to testing whether the

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Table 2 Simulation results for parametric process 2. b 1 = 1. 0 b2 b2 b2 b2

= 0.0 = 0.1 = 0.25 = 0.5

(1)

J(2)

J(6)

J(12)

J(24)

J(36)

Jmax

Weib. haz.

Reg. test

0.06 0.07 0.08 0.17

0.05 0.05 0.05 0.13

0.05 0.05 0.09 0.56

0.03 0.02 0.08 0.67

0.04 0.06 0.09 0.78

0.05 0.07 0.06 0.66

0.08 0.06 0.08 0.14

0.08 0.07 0.05 0.02

0.08 0.08 0.07 0.14

0.09 0.06 0.05 0.12

0.08 0.08 0.14 0.68

0.04 0.06 0.10 0.72

0.05 0.07 0.12 0.82

0.08 0.10 0.10 0.71

0.10 0.11 0.09 0.10

0.08 0.10 0.08 0.03

b1 = 0.95 b2 b2 b2 b2

= 0.0 = 0.1 = 0.25 = 0.5

Note: Simulation results are for 250 simulated binary sequences of length 500 for each specified process and nboot = 1000 for the bootstrapped statistics.

density of durations is geometric. We also evaluate H0 by estimating Weibull and log-Logistic hazard functions as in McQueen and Thorley (1994). The log-Logistic model makes hazard probabilities monotone in duration and imposes probabilities on the unit line. The log-Logistic model is h(ti ) = {1 + exp(−k − alog(ti ))} −1 . As for the Weibull function, the hazard is constant when a = 0. We use hazard model likelihood functions for discrete data as in Harman and Zuehlke (2004) and McQueen and Thorley (1994). For a given a hazard h(t ), when durations ti are discretely valued the log-likelihood for a sample of n durations  function ti −1 i  ti is lhaz (k, a ) = ni=1 log h (ti ) j=1 (1 − h (j)) . We test a = 0 using the likelihood ratio 2(lhaz (k, a) − lhaz (k|a = 0)), which is asymptotically w 2 (1) under H0 . All tests are considered at size 0.05. The tables report only results for the Weibull hazard test since the log-Logistic test gave almost identical results. Also, the (1) (2) (1) Jmax and Jmax Sup tests gave very similar results so only the Jmax test results are reported (results for the other tests are available on request). Table 1 shows the results for the first process. When the departure from H0 is nested in the Weibull hazard function, the hazard function tests (both the Weibull and log-Logistic) have highest power. The J(k) test power drops as k increases. The dependence in the hazard function declines geometrically with k so that probabilities of equal directional price movements far (1) away are not very informative, reducing the power of the test when k increases. The Jmax test has weaker power than the J(2) test, but better than the tests with higher lag order. When the initial hazard decreases (lower panel, h(1) = 0.25), the power of all tests decline while the ranking remains the same. With lower hazard, there are fewer turning points and hence fewer spells in the data to evaluate the hypothesis by. Table 2 shows results for process 2. In the upper panel b1 = 1 (unit-root) while in the lower panel b1 = 0.95 (weak seasonal stationarity). Increasing b2 is associated with increasing deterministic cyclicality. When b2 is 0.1, 0.25 and 0.5 respectively, the cyclicality accounts for 0.5%, 3% and 11.1% of the unconditional variance in prices when b1 = 0.95.2 All tests have weak power when b2 = 0.1 and 0.25. The J(k) power in general increases when k increases. The dependence in the data occurs at higher lag orders than in process 1, and increasing k allows the test to incorporate this information. Only for b2 = 0.5 do we observe any substantial power against H0 for the J(k) test. The Weibull hazard test (and the regression test) show overall weak power. If we increase the length of the sample from 500 to 1000 in the simulation experiment (results not shown in the table but available on request), the power of all tests, except the regression based test, increases. Table 3 shows results for process 3. Again, in the upper panel b1 = 1 while for the lower panel b1 = 0.95. We see similar results as for process two with the J(k) test having highest power. The power of the J(k) test is greatest for the intermediate value k = 12. The recursive dependence is at lag five, with historic dependence decreasing as the distance between observations increases. At high k values more noise relative to signal enters, and so the power of the test drops. The Weibull hazard test (1) show overall quite weak power, and the regression based test shows no power at all. The Jmax Sup test offers similar power for processes 1, 2 and 3, suggesting a relative good compromise in power when no prior on the form of dependence in the data is assumed. We have not addressed the theoretical consistency of the J(k) test to departures from H0 . In Appendix B, we provide a discussion of the consistency of the test to departures from H0 nested in the Weibull hazard function, and a weakly stationary first order autoregressive process for price movements. The J(k) test derives its power from estimates of mi (the implied probabilities ˆ i (the empirical probabilities). The estimates of mi are calcuof same directional price movement under H0 ) deviating from m lated using the MLE of k, and are based purely on information between adjacent directional price movements. It assumes this ˆ i derive the fraction of actual same directional price information is sufficient to fully describe the dynamics of st . The estimates m movements in the data. When the history of price movements beyond the most recent matters to the next price movement, the ˆ i will differ. For instance, the AR(1) process with autoregressive parameter between zero and one will have estimates of mi and m more broken consecutive price movements than same ones due to its mean reverting property. As such, the MLE k will tend to

2

The unconditional variance of y is

1

(1−b12 )

+

b22

2 1−b12

(

)

.

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297

Table 3 Simulation results for parametric process 3. b 1 = 1. 0 b2 b2 b2 b2

= 0.0 = −0.05 = −0.1 = −0.15

(1)

J(2)

J(6)

J(12)

J(24)

J(36)

Jmax

Weib. haz.

Reg. test

0.06 0.09 0.16 0.23

0.05 0.09 0.20 0.54

0.05 0.12 0.40 0.84

0.03 0.09 0.32 0.77

0.04 0.12 0.28 0.66

0.05 0.09 0.34 0.73

0.08 0.07 0.16 0.30

0.08 0.06 0.06 0.06

0.08 0.07 0.09 0.14

0.08 0.10 0.23 0.49

0.08 0.14 0.40 0.82

0.04 0.10 0.28 0.68

0.05 0.11 0.26 0.54

0.08 0.11 0.31 0.67

0.10 0.07 0.09 0.12

0.08 0.05 0.03 0.02

b1 = 0.95 b2 b2 b2 b2

= 0.0 = −0.05 = −0.1 = −0.15

Note: Simulation results are for 250 simulated binary sequences of length 500 for each specified process and nboot = 1000 for the bootstrapped statistics.

be greater than 0.5. This leads to an oscillating converging sequence of mi in i. Oscillating probabilities of same directional price ˆ i sequence will not be oscillating. movements as distance increases is not a property of this process, and so the estimated m The simulation experiments here are by no means exhaustive. There are other departures from H0 and other tests for H0 that could be considered. One interesting departure from H0 we have not explored is possible structural changes in the transition probabilities. This would violate H0 but as of yet it is unclear whether the test has any power to detect this. The results however do show that the J(k) test has power against conventional duration dependence as well as deviations from H0 reasonable for commodity prices. The experiment shows that the much applied hazard function tests has weak power when departures from H0 are not nested within a general hazard function specification. Furthermore, the experiment shows that the lag value k with highest power depends on the form of dependence in the alternative hypothesis. This is well known problem for simi(1) (2) lar portmanteau tests, such as the Ljung-Box Q(k) test. The Jmax Sup test (and Jmax ) reduces this dependence on the alternative hypothesis while still retaining relatively high power.

Table 4 Transition probability estimates and testing, part I. k1

k2

k1 = 0.5

k2 = 0.5

k1 = k2 = 0.5

k1 = k2

Est. (S.E)

Est. (S.E)

p-Value

p-Value

p-Value

p-Value

Energy Brent oil Natural Gas US Natural Gas EU

0.43 (0.04) 0.45 (0.06) 0.24 (0.03)

0.46 (0.04) 0.42 (0.06) 0.32 (0.04)

0.03 0.33 0.00

0.21 0.08 0.00

0.04 0.13 0.00

0.52 0.61 0.19

Beverages Coffee Arabica Coffee Robusta Tea, Sri-Lanka Tea, India Cocoa

0.42 (0.04) 0.47 (0.04) 0.43 (0.04) 0.50 (0.05) 0.48 (0.04)

0.40 (0.04) 0.41 (0.04) 0.41 (0.04) 0.37 (0.03) 0.41 (0.04)

0.01 0.39 0.02 0.94 0.49

0.00 0.00 0.01 0.00 0.00

0.00 0.01 0.00 0.00 0.01

0.62 0.18 0.77 0.00 0.15

Prec. metals Gold Platinum Silver Metals Aluminium Copper Lead Nickel Tin Zinc

0.42 (0.04) 0.39 (0.04) 0.44 (0.04)

0.42 (0.04) 0.44 (0.04) 0.44 (0.04)

0.01 0.00 0.06

0.02 0.10 0.05

0.00 0.00 0.03

0.92 0.21 0.97

0.55 (0.05) 0.36 (0.03) 0.42 (0.04) 0.42 (0.04) 0.34 (0.03) 0.40 (0.04)

0.44 (0.04) 0.41 (0.04) 0.44 (0.04) 0.35 (0.03) 0.36 (0.03) 0.44 (0.04)

0.19 0.00 0.01 0.02 0.00 0.00

0.05 0.01 0.07 0.00 0.00 0.05

0.06 0.00 0.01 0.00 0.00 0.00

0.02 0.25 0.67 0.18 0.63 0.47

Cereals Soybeans Barley Corn Sorghum Rice Wheat

0.40 (0.04) 0.42 (0.04) 0.41 (0.04) 0.40 (0.04) 0.37 (0.03) 0.42 (0.04)

0.39 (0.04) 0.44 (0.04) 0.40 (0.04) 0.43 (0.04) 0.34 (0.03) 0.40 (0.04)

0.00 0.01 0.00 0.00 0.03 0.02

0.00 0.08 0.00 0.02 0.00 0.00

0.00 0.01 0.00 0.00 0.00 0.00

0.80 0.61 0.97 0.62 0.42 0.67

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4. Empirical analysis We now test the hypothesis for a selection of prices. We have data on 48 major commodities spanning various commodity groups. The data is monthly from 01.1975 to 05.2015, although not all commodities cover the entire range. Most prices are from the World Bank Pink Sheet database. Some of the food commodities are from the FAO database (http://www.fao.org/economic/ est/prices). Metal prices outside the precious metals are from the London Metal Exchange, collected using DataStream. For comparison we consider five major stock indices over the same period (S&P500, Dow-Jones Industrial average, JP Nikkei 225, Hang Seng and the NASDAQ composite index). Detailed information on the data can be found in the Appendix. A relevant question is whether to use nominal or real values. For real values we have to choose a normalizing series. Popular choices are the US Producer Price Index or the US Consumer Price Index. The data we consider are all expressed in US dollars, however most of the commodity prices represent globally traded, produced and consumed goods. As such it is not immediately clear what real prices should reflect. By normalizing we also run the risk of introducing non-representative trends or cycles in the data. Because of this we have chosen to consider the raw data with no filtering applied. The results in this section hence apply to nominal prices in US dollar. 4.1. Transition probabilities, momentum and trending We start by estimating transition probabilities k = [k1 , k2 ] using the MLE. Tables 4 and 5 show the results. The first two columns report the transition density estimates along with asymptotic standard errors in parenthesis. The third and fourth columns report p-values for likelihood-ratio tests of k1 = 0.5 and k2 = 0.5. Rejection implies momentum given k lower than one-half. The fifth column tests whether both probabilities can be set to one-half. The last column tests the symmetry of the transition matrix. Failing to reject symmetry suggests strong mixing in directional price movements. Table 4 shows results for energy, beverages, precious metals, metals and cereals. Almost all commodities have estimates of k below one-half. Those that do not are not statistically different from one-half. This suggests momentum in price movements. Table 5 Transition probability estimates and testing, part II. k1

k2

k1 = 0.5

k2 = 0.5

k1 = k2 = 0.5

k 1 = k2

Est. (S.E)

Est. (S.E)

p-Value

p-Value

p-Value

p-Value

Fats and oils Coconut oil Groundnut oil Palm oil Soybean oil Copra

0.41 (0.04) 0.33 (0.03) 0.40 (0.04) 0.45 (0.04) 0.42 (0.04)

0.42 (0.04) 0.29 (0.03) 0.41 (0.04) 0.41 (0.04) 0.37 (0.03)

0.00 0.00 0.00 0.10 0.01

0.01 0.00 0.00 0.01 0.00

0.00 0.00 0.00 0.01 0.00

0.78 0.28 0.79 0.46 0.30

Meats Chicken, US Chicken, Brazil Sheep, N.Z. Lamb, London Lamb, N.Z. Pork, US Pig meat, US Bovine meat, Aus. Bovine meat, US Salmon, Norway Shrimp, Mexico

0.26 (0.03) 0.41 (0.05) 0.42 (0.04) 0.40 (0.04) 0.34 (0.04) 0.47 (0.05) 0.52 (0.06) 0.41 (0.05) 0.47 (0.06) 0.39 (0.04) 0.39 (0.04)

0.34 (0.03) 0.50 (0.07) 0.42 (0.04) 0.36 (0.04) 0.44 (0.05) 0.44 (0.04) 0.57 (0.07) 0.43 (0.05) 0.52 (0.06) 0.40 (0.04) 0.25 (0.02)

0.00 0.04 0.02 0.00 0.00 0.40 0.57 0.02 0.46 0.00 0.00

0.00 0.92 0.02 0.00 0.19 0.08 0.10 0.08 0.67 0.00 0.00

0.00 0.12 0.00 0.00 0.00 0.15 0.23 0.02 0.69 0.00 0.00

0.06 0.19 0.97 0.43 0.08 0.53 0.44 0.73 0.41 0.95 0.00

Other Sugar Tobacco Fishmeal Cotton Rubber Bananas Oranges Dairy Tapioca

0.42 (0.04) 0.37 (0.03) 0.31 (0.03) 0.30 (0.03) 0.39 (0.04) 0.51 (0.05) 0.41 (0.04) 0.31 (0.04) 0.44 (0.06)

0.41 (0.04) 0.45 (0.04) 0.31 (0.03) 0.29 (0.03) 0.41 (0.04) 0.50 (0.05) 0.44 (0.04) 0.25 (0.03) 0.36 (0.05)

0.01 0.00 0.00 0.00 0.00 0.75 0.01 0.00 0.25

0.00 0.18 0.00 0.00 0.01 0.90 0.09 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.94 0.01 0.00 0.01

0.86 0.07 0.99 0.79 0.63 0.75 0.49 0.27 0.25

Stock price indices S&P500 Dow Jones JP Nikkei Hang Seng NASDAQ

0.39 (0.04) 0.43 (0.04) 0.45 (0.04) 0.41 (0.04) 0.38 (0.04)

0.60 (0.06) 0.62 (0.06) 0.56 (0.05) 0.58 (0.05) 0.56 (0.05)

0.00 0.01 0.09 0.00 0.00

0.01 0.00 0.08 0.03 0.08

0.00 0.00 0.05 0.00 0.00

0.00 0.00 0.01 0.00 0.00

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299

Table 6 p-Values for test of H0 , part I. (1)

J(2)

J(6)

J(12)

J(24)

J(36)

Jmax

Weib. haz.

Reg. test

Energy Brent oil Natural Gas US Natural Gas EU

0.476 0.868 0.000

0.516 0.922 0.000

0.376 0.180 0.000

0.122 0.325 0.004

0.061 0.212 0.002

0.146 0.337 0.000

0.130 0.811 0.000

0.055 0.453 0.983

Beverages Coffee Arabica Coffee Robusta Tea, Sri-Lanka Tea, India Cocoa

0.435 0.848 0.619 0.251 0.412

0.265 0.549 0.877 0.078 0.071

0.215 0.451 0.036 0.000 0.331

0.162 0.217 0.038 0.000 0.731

0.174 0.278 0.013 0.000 0.856

0.360 0.514 0.090 0.000 0.277

0.238 0.427 0.245 0.755 0.743

0.388 0.422 0.212 0.029 0.555

Prec. metals Gold Platinum Silver

0.388 0.643 0.643

0.457 0.157 0.095

0.020 0.010 0.086

0.130 0.091 0.465

0.152 0.076 0.308

0.088 0.029 0.153

0.425 0.504 0.766

0.492 0.669 0.910

Metals Aluminium Copper Lead Nickel Tin Zinc

0.814 0.624 0.795 0.827 0.993 0.519

0.021 0.587 0.200 0.938 0.900 0.317

0.012 0.661 0.350 0.381 0.978 0.668

0.026 0.945 0.448 0.040 0.337 0.096

0.057 0.987 0.652 0.020 0.632 0.234

0.015 0.946 0.579 0.120 0.732 0.301

0.467 0.590 0.561 0.907 0.609 0.241

0.714 0.890 0.507 0.561 0.204 0.911

Cereals Soybeans Barley Corn Sorghum Rice Wheat

0.268 0.527 0.542 0.813 0.325 0.552

0.603 0.079 0.582 0.975 0.210 0.036

0.927 0.135 0.735 0.354 0.137 0.090

0.866 0.022 0.746 0.154 0.139 0.018

0.856 0.002 0.570 0.015 0.074 0.012

0.763 0.023 0.864 0.100 0.269 0.046

0.308 0.366 0.645 0.995 0.478 0.537

0.374 0.741 0.782 0.565 0.725 0.120

Note: p-values for the test statistic J(k) derived by parametric bootstrap under the null. Based on nboot =10000 simulated series’ with sample lengths equal to the actual data-series tested. The Weibull hazard column p-value is a LR-test for a = 0 in the hazard function h (ti ) = k (a − 1) tia where ti are samples of duration lengths for up or down movements. The regression test tests the exclusion of dt −1 in the regression st = c0 + c1 st −1 + c2 st −1 dt −1 + error, where dt −1 denotes consecutive periods spent in an expansion phase, and st = 1 if price move up and zero if down.The p-values are from a two-sided t-test.

In addition, all commodities except Tea India and Aluminium fail to reject symmetry of the transition matrix indicating strong mixing in commodity price movements. Table 5 shows the results for fats and oils, meats, other commodities and the stock price indices. We again see almost all estimates of k below one-half. Similarly, evidence is again strong for a symmetric transition matrix (it is rejected for Chicken US, Lamb NZ, Shrimp Mexico and Tobacco with p-values below 0.1). The overall test results for the commodities are consistent across commodity groups. Commodities show strong evidence of symmetric momentum in both directions with strong mixing in directional price movements. For the stock price indices we find k1 < 0.5 and k2 > 0.5. Up movements tend to persist (momentum) while down movements revert. This likely reflects positive trending in asset prices. To answer whether a trend is present we also have to evaluate the average magnitude of up and down movements. Using Welch’s t-test for equality of means, eight out of the 48 commodities reject equality of means. That not more reject is perhaps surprising given prices are bounded from below at zero, but the result is consistent with lack of significant trending in nominal commodity prices over the sample period. The prices that show evidence of trending are: Copra (positive trend, equal frequency of up and down movements, larger average size of up movements), Chicken US (positive trend, higher likelihood of up movements, larger average size of up movements), Lamb NZ (positive trend, higher likelihood of up movements, equal average size of up and down movements), Tobacco (positive trend, higher likelihood of up movements, equal average size of up and down movements), Fishmeal (equal frequency of up and down movements, larger average size of up movements), Dairy (equal frequency of up and down movements, larger average size of up movements), and Gold (equal frequency of up and down movements, larger average size of up movements). For three of the commodities (Tea India, Shrimp Mexico and Aluminium) the results are indeterminate as the likelihood of down movements are larger, but up movements have a higher average magnitude). It is worth remarking that testing for a deterministic linear trend using an OLS regression finds that only one of the commodities rejected the hypothesis of no linear trend (Chicken US). For the stock indices we cannot reject equal magnitude of directional price movements. This suggests a positive trend in the asset indices, and positive returns due to more frequent up movements, not greater magnitude of up movements. The t-test results where evaluated at the 10% critical values and are available on request.

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Table 7 p-Values for test of H0 , part II. (1)

J(2)

J(6)

J(12)

J(24)

J(36)

Jmax

Weib. haz.

Reg. test

Fats and oils Coconut oil Groundnut oil Palm oil Soybean oil Copra

0.471 0.310 0.014 0.027 0.241

0.005 0.749 0.021 0.289 0.001

0.054 0.700 0.173 0.223 0.001

0.003 0.051 0.116 0.503 0.000

0.002 0.011 0.015 0.600 0.000

0.014 0.046 0.039 0.085 0.001

0.660 0.086 0.092 0.702 0.536

0.645 0.621 0.594 0.660 0.332

Meats Chicken, US Chicken, Brazil Sheep, N.Z. Lamb, London Lamb, N.Z. Pork, US Pig meat, US Bovine meat, Aus. Bovine meat, US Salmon, Norway Shrimp, Mexico

0.573 0.209 0.986 0.008 0.989 0.923 0.179 0.039 0.434 0.241 0.621

0.075 0.486 0.986 0.014 0.040 0.988 0.298 0.102 0.519 0.406 0.796

0.000 0.791 0.778 0.043 0.005 0.013 0.110 0.177 0.805 0.564 0.905

0.000 0.676 0.183 0.013 0.000 0.003 0.171 0.118 0.069 0.700 0.049

0.000 0.635 0.309 0.006 0.000 0.000 0.098 0.203 0.061 0.586 0.000

0.000 0.647 0.518 0.007 0.000 0.007 0.338 0.136 0.241 0.703 0.004

0.150 0.289 0.451 0.784 0.922 0.834 0.695 0.054 0.673 0.788 0.111

0.936 0.921 0.623 0.563 0.238 0.147 0.839 0.351 0.714 0.118 0.595

Other Sugar Tobacco Fishmeal Cotton Rubber Bananas Oranges Dairy Tapioca

0.735 0.007 0.110 0.178 0.456 0.933 0.254 0.190 0.983

0.530 0.001 0.691 0.419 0.192 0.011 0.001 0.501 0.973

0.928 0.000 0.239 0.502 0.062 0.015 0.000 0.364 0.575

0.879 0.000 0.641 0.362 0.063 0.018 0.000 0.081 0.757

0.736 0.000 0.855 0.465 0.194 0.002 0.000 0.002 0.722

0.844 0.000 0.404 0.481 0.103 0.021 0.000 0.023 0.918

0.626 0.000 0.081 0.413 0.404 0.241 0.140 0.504 0.777

0.580 0.975 0.168 0.018 0.478 0.090 0.238 0.112 0.581

Stock price indices S&P500 Dow Jones JP Nikkei Hang Seng NASDAQ

0.776 0.875 0.596 0.982 0.639

0.338 0.899 0.170 0.966 0.277

0.540 0.878 0.202 0.846 0.694

0.505 0.838 0.292 0.803 0.467

0.538 0.852 0.617 0.858 0.650

0.751 0.991 0.370 0.918 0.525

0.234 0.507 0.197 0.255 0.244

0.392 0.103 0.848 0.793 0.534

Note: p-values for the test statistic J(k) derived by parametric bootstrap under the null. Based on nboot =10000 simulated series’ with sample lengths equal to the actual data-series tested. The Weibull hazard column p-value is a LR-test for a = 0 in the hazard function h(ti ) = k(a −1)tia where ti are samples of duration lengths for up or down movements. The regression test tests the exclusion of dt −1 in the regression st = c0 + c1 st −1 + c2 st −1 dt −1 + error, where dt −1 denotes consecutive periods spent in an expansion phase, and st = 1 if price move up and zero if down.The p-values are from a two-sided t-test.

4.2. Regularity in directional price movements We now test H0 . As in the simulation experiment we use the Weibull and log-Logistic hazard function tests, the regression (1) (2) based test, the J(k) test statistics and the Jmax and Jmax . For the J(k) statistics we use the same k values as in the simulation (1) (2) experiment, k = {2, 6, 12, 24, 36}. The Jmax and Jmax test have kmax = 36. All J(k) tests where evaluated based on simulated test statistics under the null with the procedure described in Section 2.3. As in the simulation experiment, results for the log-Logistic (2) (1) and Jmax statistics are not reported in the tables due to almost identical results to the Weibull and Jmax tests respectively. Tables 6 and 7 show the p-values for testing H0 for all the tests. For the commodities most rejection occurs at the higher k values. Using a p-value of 0.1, seven out of 48 commodities reject H0 when k = 2. When k is 6, 12, 24 and 36 we have 17, 18, (1) 22 and 26 rejections respectively. The Jmax test rejects H0 for 23 out of the 48 commodities. Both hazard function tests reject for six of the 48 commodities. The regression based test reject only for three out of 48 (p-values are here from a two-sided t-test). Historic price movements matter for commodities. Most of the departures from H0 in commodities are not due to deviations from non-duration dependence as specified by the hazard function, but due to higher lag-order dependence in the directional movements. The results suggest that up/down movements in commodity prices cannot be modeled as a first order Markov chain with fixed transition probabilities. There is a greater degree of regularity to the directional commodity price movements than what is summarized in the most recent observation. This is not the case for the stock price indices since no test rejects H0 . Information on whether the asset price index went up six or twelve months ago provides no additional information on whether it will go up or down this month. The J(k) statistics can be plotted for different values of k to gain some insights into the pattern of price movements. In Fig. 1 we ˆ i for k ranging from one to 36 months as well as the J(k) statistics for k from 2 to 36 months. This plot empirical probabilities m ˆ i when price movements are separated in is done for Crude oil, Pork US and the S&P 500. The solid line in the left panels show m time by values shown on the x-axis. The dotted line shows the implied probability under the null, mi . The two dashed lines show

A. Oglend, T. Kleppe / Journal of Empirical Finance 38 (2016) 290–306

301

Fig. 1. Probabilities of same directional price movement (left) and J(k) statistics (right), some selected prices. Note: For the left panels, the solid black line refers to the empirical mi values plotted as a function of i, the distance between observations. The solid grey line is the implied mi value under H0 with MLE estimates for k. Dotted grey lines show implied 95% confidence intervals under the null. For the right panels, the solid black line is the J(k) statistics, the solid grey line is the 5% bootstrapped critical value, the dotted grey line is the 5% w 2 (k − 1) critical value.

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ˆ i under the null (derived from simulating ci under the null following the procedure plus/minus two standard deviations of m described in Section 2.3). The right panels shows the J(k) statistics along with 5% critical values for each k (the solid line is the bootstrapped 5% critical value while the dotted line shows the w 2 (k − 1) 5% critical values. For Brent oil (top), we observe a higher probability of being in the same price phase for prices separated by around 10 and 30 months.3 We observe that once k reaches around 30, the J(k) statistics exceeds the 5% critical value. The Pork US price shows an outcome indicative of seasonality. The probability peaks at seasonal frequencies (especially one and three years). This is also clear in the J(k) statistics which jumps up when the seasonal span is reached. The S&P500 probabilities show no clear pattern. The history does not contain any significant additional information. Note also that for the S&P500 data, the implied probability mi converges in i to a value greater than 0.5, which is due to the asymmetric transition matrix. A final thing to note is that the w 2 (k − 1) critical values do not deviate substantially from the simulated critical values. When the k estimates do not deviate substantially from 0.5 the correlation in the random   vector ct will not be too high, and the long run variance will be similar to the covariance matrix Y. The sample moments g kˆ will be close to i.i.d. under H0 and J(k) will be close to w 2 (k − 1). 5. Concluding remarks We have derived a Wald test to evaluate whether up/down movements in prices can be modeled by a two-state firstorder time-homogenous Markov chain. The validity of the hypothesis means price movements lack duration dependence. It is sufficient for weak-form market efficiency in a market that pays/deducts a fixed amount for given price increases/decreases. Rejecting the hypothesis suggests some degree of regularity to directional price movements, such as mean reversion or deterministic cyclicality, more in line with commodities priced as consumption goods rather than financial assets. The MLE estimates of the transition probabilities in the Markov chain are used to derive the implied probabilities that prices, separated by up to some k periods, move in the same direction under the null hypothesis. We then evaluate the null hypothesis by comparing these probabilities to the empirical probabilities. The resulting Wald statistics is asymptotically w(k) when the transition density is known, and asymptotically w(k − 1) when transition densities are derived using the maximum likelihood estimator. When transition densities differ from one-half, the data used to derive the statistics is a correlated sequence. The exact long-run covariance matrix under the null can be calculated exactly by summing different probabilities of paths in the binary sequence of directional price movements. This procedure becomes computationally demanding when k is large. In addition, sample uncertainty in any long-run variance estimator can lead to determinants close to zero when there is substantial autocovariance. We propose a parametric bootstrapping procedure to derive the distribution of the Wald statistics under the null. To apply the test, a lag length k must be specified. As is well known for similar portmanteau tests, such as the Ljung-Box Q(k) test, the power of the test at any k is sensitive to the form of dependence in the alternative hypothesis. To reduce this dependence we suggest a Sup test based on the maximum of the standardized Wald statistics over a range of k values. In a simulation experiment we evaluated the power of the test for different departures from the null hypothesis. The test displays good power relative to tests that estimate the hazard function when departures from the null are not nested in a specified hazard function. The test also has reasonably good power when departures from the null are nested within a hazard function, although the power in these cases is weaker than the hazard function likelihood ratio tests. The experiment also shows that the Sup tests provide a reasonable compromise in power when the form of dependence in the alternative hypothesis is not known. We applied the test to a set of 48 monthly commodity prices and five major stock indices. The tests were applied to nominal prices with no pre-filters applied. Estimates of transition probabilities suggest that commodity price movements have symmetric momentum in both up and down movements: if the price goes up (down) today, it is more likely to go up (down) again the following period. The symmetry of the transition matrix suggests strong mixing in directional commodity price movements. The tests do not reveal any strong evidence of trending in nominal commodity prices (only eight of the 48 commodities show clear evidence for positive trends). Since the analysis was performed on nominal prices, this suggests most real commodity prices have a negative trend over the sample period. The results on commodities differ notably from the stock indices. Stock indices display asymmetric transition probabilities with momentum only for up movements. We fail to reject equality in the average size of up/down movements, suggestive of a positive trend where positive index returns due to more frequent up movements, not bigger relative up movements. The Markov chain null hypothesis is rejected for several of the commodities. Overall, the evidence suggests that directional price movements in commodities are not well represented by a first order Markov chain. For several commodities the probability that price will go up or down today conditional on it having gone up or down a year or more ago is not consistent with what a first-order Markov process predicts. This is notably different from the stock indices, which cannot reject the null hypothesis for any test. By plotting empirical probabilities we also show how patterns in price movements such as seasonality and long run cycles can be revealed in the commodity prices. Commodity prices behave quite differently from efficient asset prices. There is more dependence in price movements separated by more than one period than a first order Markov chain can explain. As such, standard no-arbitrage asset pricing models based on linear first-order Markov processes for price movements does not appear sufficient for pricing of commodities.

3 Some care should be taken when interpreting these figures, for 36 independent tests we would expect between one to two rejections purely by chance if use a p-value of 0.05. For k1 and k2 different from one-half, the tests are however not independent under the null.

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Appendix A. Data description Label

Data range

Description

Brent oil Natural Gas US Natural Gas EU Coffee Arabica

01.1979–05.2015 01.1997–05.2015 01.1997–05.2015 01.1975–05.2015

Coffee Robusta

01.1975–05.2015

Tea, Sri-Lanka Tea, India Cocoa

01.1975–05.2015 01.1975–05.2015 01.1977–05.2015

Gold Platinum Silver

01.1975–05.2015 01.1975–05.2015 01.1975–05.2015

Aluminium Copper Lead Nickel Tin Zinc Soybeans Barley Corn Sorghum Rice

02.1979–07.2014 01.1975–05.2015 02.1980–05.2015 02.1980–05.2015 01.1975–05.2014 01.1975–05.2014 01.1975–05.2015 01.1975–05.2015 01.1975–05.2015 01.1975–05.2015 01.1975–05.2015

Wheat

01.1975–05.2015

Coconut oil Groundnut oil Palm oil Soybean oil Copra Chicken, US

01.1975–05.2015 01.1975–05.2015 01.1975–05.2015 01.1975–05.2015 01.1975–05.2015 01.1980–05.2015

Chicken, Brazil Sheep, N.Z.

01.1996–12.2014 01.1980–05.2015

Lamb, London Lamb, N.Z. Pork, US Pig meat, US Bovine meat, Aus. Bovine meat, US Salmon, Norway Shrimp, Mexico

01.1980–04.2015 01.1990–12.2014 01.1980–04.2015 12.1990–11.2014 01.1990–1.2014 05.1990–12.2013 01.1980–04.2015 01.1980–05.2015

Sugar

01.1975–05.2015

Tobacco Fishmeal

01.1975–05.2015 01.1979–05.2015

Cotton

01.1975–05.2015

Rubber

01.1975–05.2015

Bananas

01.1975–05.2015

Oranges Dairy

01.1980–04.2015 01.1990–12.2014

Tapioca S&P500 Dow Jones JP Nikkei Hang Seng NASDAQ

01.1990–03.2007 01.1975–05.2015 01.1975–05.2015 01.1975–04.2015 01.1975–05.2015 01.1975–03.2015

Crude oil, UK Brent 38’ API. ($/bbl) Natural Gas (U.S.), spot price at Henry Hub, Louisiana. ($/mmbtu) Natural Gas (Europe), average import border price and a spot price component. ($/mmbtu) Coffee (ICO), International Coffee Organization indicator price, other mild Arabicas, average New York and Bremen/Hamburg markets. ($/kg) Coffee (ICO), International Coffee Organization indicator price, Robustas, average New York and Le Havre/Marseilles markets, ex-dock. ($/kg) Tea (Colombo auctions), Sri Lankan origin, all tea, arithmetic average of weekly quotes. ($/kg) Tea (Kolkata auctions), leaf, include excise duty, arithmetic average of weekly quotes. ($/kg) Cocoa (ICCO), International Cocoa Organization daily price, average of the first three positions on the terminal markets of New York and London, nearest three future trading months. ($/kg) Gold (UK), 99.5% fine, London afternoon fixing, average of daily rates. ($/troy oz) Platinum (UK), 99.9% refined, London afternoon fixing. ($/troy oz) Silver (UK), 99.9% refined, London afternoon fixing; prior to July 1976 Handy & Harman. Grade prior to 1962 unrefined silver. ($/troy oz) LME-Aluminium 99.7% Cash. ($/mt) LME-Copper Grade A Cash. ($/mt) Lead, 99.97% pure, LME spot price, c.i.f.European Ports. ($/mt) Nickel, melting grade, LME spot price, c.i.f. European ports. ($/mt) LME-Tin 99.85% Cash. ($/mt) LME-SHG Zinc 99.995% Cash. ($/mt) Soybeans (US), c.i.f. Rotterdam. ($/mt) Barley (US) feed, no. 2, spot. ($/mt) Maize (US), no. 2, yellow, f.o.b. US Gulf ports. ($/mt) Sorghum (US), no. 2 milo yellow, f.o.b. Gulf ports. ($/mt) Rice (Thailand), 5% broken, white rice (WR), milled, indicative price based on weekly surveys of export transactions, government standard, f.o.b. Bangkok. ($/mt) Wheat (US), no. 1, hard red winter, ordinary protein, export price delivered at the US Gulf port for prompt or 30 days shipment. ($/mt) Coconut oil (Philippines/Indonesia), bulk, c.i.f. Rotterdam. ($/mt) Groundnut oil (any origin), c.i.f. Rotterdam. ($/mt) Palm oil (Malaysia), 5% bulk, c.i.f. N. W. Europe. ($/mt) Soybean oil (Any origin), crude, f.o.b. ex-mill Netherlands. ($/mt) Copra (Philippines/Indonesia), bulk, c.i.f. N.W. Europe. ($/mt) Meat, chicken (US), broiler/fryer, whole birds, 2-1/2 to 3 pounds, U.S.D.A.grade “A” , ice-packed, Georgia Dock preliminary weighted average, wholesale. ($/kg) Poultry Meat,Brazil, export value for chicken, f.o.b., Aves & Ovos: Boletin Mensal. ($/mt) Meat, sheep (New Zealand), frozen whole carcasses Prime Medium (PM) wholesale, Smithfield, London. ($/kg) Lamb, frozen carcass Smithfield London, US cents per pound. (US cents/pound) Ovine Meat, New Zealand, Lamb 17.5 kg cwt, export price, Meat & Livestock, Australia. ($/mt) Swine (pork), 51–52% lean Hogs, U.S. price. (US cents/pound) Pig Meat, USA, pork, frozen product, export unit value, U.S.D.A.: U.S. trade exports. ($/mt) Bovine Meat, Australia: Cow 90CL export prices to the USA, FAS, Meat & Livestock, Australia. ($/mt) dfdBovine Meat, USA, beef export, export unit value, U.S.D.A.: U.S. trade exports. ($/mt) Fish (salmon), Farm Bred Norwegian Salmon, export price. ($/kg) Shrimp, (Mexico), west coast, frozen, white, no. 1, shell-on, headless, 26 to 30 count per pound, wholesale price at New York. ($/kg) Sugar (world), International Sugar Agreement (ISA) daily price, raw, f.o.b. and stowed at greater Caribbean ports. ($/kg) Tobacco (any origin), unmanufactured, general import , c.i.f., US. ($/mt) Fishmeal (any origin), 64–65%, c&f Bremen, estimates based on wholesale price, beginning 2004; previously c&f Hamburg. ($/mt) Cotton (Cotton Outlook “CotlookA index”), middling 1-3/32 in., traded in Far East, C/F beginning 2006; previously Northern Europe, c.i.f. ($/kg) Rubber (any origin), Ribbed Smoked Sheet (RSS) no. 1, in bales, Rubber Traders Association (RTA), spot, New York ($/kg) Bananas (Central & South America), major brands, free on truck (f.o.t.) Southern Europe, including duties; prior to October 2006, f.o.t. Hamburg. ($/kg) Oranges (Mediterranean exporters) navel, European Union indicative import price, c.i.f. Paris. ($/kg) Dairy_Butter, European and Oceania average indicative export prices, f.o.b, average of mid-point of price ranges reported bi-weekly by Dairy Market News (U.S.D.A.). ($/mt) Tapioca, hard pellets, f.o.b. Bangkok, The Tapioca Trade Association (TTTA). ($/mt) S&P500 index Dow Jones Industrial Average index JP Nikkei 225 Index Hang Seng Index (Hong Kong) US NASDAQ Composite Index

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Appendix B. Notes on test consistency √     √   n − kg kˆ Y kˆ −1 n − kg kˆ to diverges      1 n−k ˆ with single a when H0 does not hold and the sample size n increases. The vector of moments is g kˆ = n−k t=1 ct − m k element For the Wald test to be consistent we need the Wald statistics Jn (k) =

  gi kˆ =

n−k 1  (st st−i + (1 − st )(1 − st−i )) − mi , n−k t=1

ˆ To establish consistency for some alternative hypothesis we need from Eq. (1) using the MLE estimate k. where mi is calculated     to show that gi kˆ diverges in n for any i for non-singular Y kˆ . B.1. Weibull Hazard function process Assume the binary sequence of directional price movements is generated by the Weibull hazard function h (ti ) = k (a + 1) tia . The hazard function gives the conditional probability that a spell will end given it has persisted for ti periods. This hazard function is symmetric, it does not distinguish between downward or upwards price spells. Let prob(ti = k) be the proba a bility of a k period spell. The limiting MLE estimator is then k(∞) = ∞ k=1 prob (ti = k) k (a + 1) k . This is the limiting fraction ( ∞) of broken price spells. For limiting implied probability m2 we have from Eq. (1), (∞)

m2

= 1 − 2k(∞) + 2k(∞) 2 ,

ˆ (∞) is the limiting fraction of equal price movements lagged by two periods. Since 1 − k(a + The limiting empirical probability m 2 1)ka is the probability that we will continue the movement the next period when it has persisted for k periods, this is given by ˆ (∞) = m 2

∞ 

  prob (ti = k) 1 + k2 (a + 1)2 ka − k (a + 1) (k + 1)a − k (a + 1) ka + k2 (a + 1)2 ka (k + 1)a .

k=1

(∞) (∞) ˆ (∞) ˆ (∞) ˆ (∞) which is For a = 0, we have that m2 = m = m 2 . For a not zero m2 2 . This is due to the presence of the k + 1 term in m2 (∞) not present in m2 . This term is due to the  hazard dependence on the history of the price spell. As such, the moment condition ˆ will diverge for a not zero. The Y kˆ is non-singular as long as the hazards are bounded away from zero and one. g(k)

B.2. Parametric weakly stationary Gaussian AR(1) process Consider the Gaussian AR(1) process y1 = by0 + 41 , where 41 ∼ N(0, 1). We suppress the t subscript for notational ease. Assume that 0 < b < 1. The unconditional distribution of y is

y ∼ N 0,

1 1 − b2

 ,

while the conditional distribution of the first difference Dy1 is Dy1 |y0 ∼ N((b − 1) y0 , 1) The probability of a price increase at time two ( y2 > 0) conditional on y1 with y0 integrated out is  prob ( y2 > 0| y1 ) =

[1 − V (− (b − 1) (y0 + y1 ))]

⎛ ⎞  ⎜ y0 ⎟ 1 − b 2 0 ⎝   ⎠ dy0 , 1 − b2



where V and 0 is the standard normal CDF and PDF. When y1 > 0, the probability is lower than 0.5 that y2 > 0 because of mean reversion. As such prob( y2 > 0| y1 > 0) = prob(s2 = 1|s1 = 1) < 0.5. Since there are more broken consecutive directional movements than same ones we have that k(∞) > 0.5. Applying this under H0 leads to a negative eigenvalue c of the

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transition probability matrix of st . As such, the implied probability mi will be oscillating and converging in i. The AR(1) process considered here however has monotone probabilities in i. Note that   prob ( y3 > 0| y1 ) =

[1 − V (− (b − 1) (y0 + y1 + y2 ))] ⎞ ⎛   ⎜ y ⎟ 0 0 ( y2 − (b − 1) (y0 + y1 )) 1 − b2 0 ⎝   ⎠ d y2 dy0 2 1−b When y1 > 0 the price change y2 will have more of its probability mass on negative values due to mean reversion. This balances out the effect of a positive values of y1 on y3 in such a way that prob( y3 > 0| y1 > 0) > prob( y2 > 0| y1 > 0). ˆ (∞) ˆ (∞) converges to 0.5 as i tends to infinite due to stationarity. Since m(∞) ˆ (∞) As such m < mˆ (∞) < mˆ (∞) is 1 2 3 . . . < mn , where mi i   (∞) ˆ is not, the limiting moment condition g kˆ will diverge for 0 < b < 1. For b = 1, the process is a random oscillating and m i

(∞) ˆ (∞) walk with no drift and mi = 0.5 and m = 0.5 for all i. i (∞) The values mi are derived using transition probabilities estimated on information from the relationship between current and one period lagged price movements. It assumes a Markov property of directional movements, i.e.

prob(sn+2 = 1|sn+1 = 1, sn = 1, . . . , s1 = 1) = prob(sn+2 = 1|sn+1 = 1). ˆ (∞) The empirical m values are based on the observed relationship between current and i period lagged price movements. i When the Markov property does not hold, the relationship between current and i period lagged price movements can not (∞) ˆ (∞) be derived from the relationship between current and one period lagged price movements, and m and mi will produce i different estimates for some i. While the Markov property for the AR(1) process holds for the conditional distribution of price levels or price differences, it does not hold for the conditional distribution of directional price movements. Note that

 prob ( yn+2 > 0| yn+1 , yn ) =

[1 − V (− (b − 1) (yn + yn+1 + yn ))]

 prob ( yn+2 > 0| yn+1 ) =

[1 − V (− (b − 1) (yn+1 + yn+1 ))]



⎛ ⎞  ⎜ yn−1 ⎟ 1 − b 2 0 ⎝   ⎠ dyn−1 1 − b2 ⎛ ⎞



 ⎜ yn ⎟ 1 − b 2 0 ⎝   ⎠ dyn 1 − b2

which are not the same unless yn is fixed at zero (or b = 1, but in that case the stationary distribution of y does not exist).

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