A time varying error correction model of price discovery implications for portfolio construction and hedging

Key words: Price Discovery; Time-varying; Error Correction Model; Spot and Futures Markets... We test for time-varying price discovery in no fewer than 17 commodity markets spot and futu

A Time-Varying Error Correction Model of Price

Discovery: Implications for Portfolio

Construction and Hedging

ABSTRACT

We propose a model of time-varying price discovery based on a rolling-window error correction framework We show that price discovery in five commodities is dominated by the spot market, while, in only six commodities, price discovery is dominated by the futures market We consistently discover that for 14 commodities price discovery is time-varying, which has implications for portfolio construction and hedging Our findings, therefore, challenge the well-established view in commodity markets that it is the futures market which dominates the price discovery process

Key words: Price Discovery; Time-varying; Error Correction Model; Spot and Futures Markets

I Introduction

The interest in price discovery, or the lead and lag relationship between any two markets, has been motivated by the work of Hasbrouck (1995) and Gonzalo and Granger (GG, 1995) A feature of the empirical literature on price discovery is that the Hasbrouck and GG measures provide very

consistent results on price discovery; see extensive comparative results in Blanco et al (2005), for

example These methodologies have become popular in several strands of the literature There are

studies on price discovery in commodity spot and futures markets (see, inter alia, Schwarz and Szakmary (1994), Yang et al (2001), Garbade and Silber (1983), and F-Ferretti and Gonzalo

(2010)); there are studies that test for price discovery in the equity or equity options markets (see,

for instance, Muravyev et al (2013)); there are studies that examine the price discovery process in

the stock and CDS spread markets; and there are studies based on the exchange rate market (see

Chen and Gau (2010), Poskitt (2010), Cabrera et al (2009), and Tse et al (2006)) A factor that

is common across these different strands of the literature is that limited attempts have been made

to explore the potential time-varying nature of the price discovery On closer inspection of the literature, we find that with respect to the Hasbrouck measure while some attempt has been made, particularly by allowing for time-varying correlation and or co-variance, nothing of this sort has been attempted when it comes to the GG measure

In this paper we propose a rolling-window-based error correction model to extract varying price discovery coefficients We test for time-varying price discovery in no fewer than 17 commodity markets (spot and futures) using monthly time series data that mostly span the period

time-1977 to 2012 This is not all We also utilise daily data to test the robustness of our results With

an extensive empirical analysis, we discover strong evidence of price discovery in that, for 15 commodities, there is price discovery Of these 15 commodities, in five commodities (cocoa, corn,

the futures market dominates price discovery in only six commodities (canola, copper, crude oil, palladium, soybean meal, and wheat) Moreover, for 14 commodities (canola, cocoa, coffee, copper, corn, gold, soybean oil, soybean yellow, sugar, wheat, palladium, platinum, silver, and soybean meal) there is clear evidence of time-varying price discovery In other words, for these commodities there are cyclical patterns: phases over which spot market dominates price discovery and phases over which price discovery is dominated by the futures market Finally, in an economic significance analysis, we show that time-varying price discovery has implications for portfolio construction and hedging in at least some of the commodity markets

II Motivation: Why is price discovery time-varying?

In this paper, our focus is on price discovery There are strong reasons to believe that price discovery can be time-varying Amongst the simplest of reasons, because price discovery is based

on time series price data, which naturally experience not one but many shocks over their historical time period, one can expect the price discovery process to be time-varying Therefore, it is imperative to understand the factors that affect prices intermittently over time If one considers the literature on equity returns, there is no shortage of reasons why stock prices vary with time The main source of this time-varying nature of stock prices has been attributed to, among others,

business cycles (Andersen et al (2007)) and monetary policy/macroeconomic news (Bernanke and

Kuttner (2005), Garner (1989), Ederington and Lee (1993))

More specifically, let us consider the time-varying behaviour of commodity prices now Amongst empirical evidence, a number of studies document that commodity prices are

characterised by boom and slump phases Cashin et al (2002) consider as many as 36 commodity

price series, and document strong evidence that commodity prices are characterised by booms and

the period from 1957 to 1999, they find that there are, on average, six cycles in the 36 commodity prices This empirical evidence points to the fact that commodity prices are time-varying Perhaps the most famous hypothesis that reinforces the belief that commodity prices are time-varying is the Prebisch and Singer (1950) hypothesis The Prebisch-Singer (PS) hypothesis states that relative commodity prices are steadily decreasing over time That commodity prices are negatively time-varying has been motivated by a number of factors, such as high income elasticity of demand for manufactured goods vis-à-vis primary commodities, productivity differentials between countries, and asymmetric market structures where the industrial sector is characterised by an oligopolistic structure, while primary commodities are generally perfectly competitive (see Kellard and Wohar (2006))

Commodity prices are also strongly dependent on business cycle phases and monetary

policy news (see, inter alia, Hong and Sarkar (2008)) As an example, consider Frankel’s (1986)

overshooting theory of commodity prices The main idea of this theory is that commodities are exchanged on fast-moving auction markets Commodity prices, therefore, respond instantaneously

to any shocks that affect commodity markets In response to monetary policy news, for instance, commodity prices in the Frankel model react more than proportionately In other words, monetary policy shocks lead to an overshooting of commodity prices in that they move to new long-run equilibrium Commodity cycles also result from the lag between the initiation of production decisions and the delivery of goods Motivated by this, Mackey (1989) developed a continuous time model for the price dynamics of a single commodity market The two key features of this model are that it accounts: (a) explicitly for the nonlinearities in demand and supply schedules; and (b) for production and storage delays resulting from the market price

In related empirical evidence, studies show that commodity prices are non-stationary, suggesting that shocks, such as those resulting from real interest rates, are responsible for the

changing behaviour of the mean and variance of commodity prices (see Byrne et al (2013)) The

trend behaviour of commodity prices has occupied enough interest to be classified as a strand of the literature on commodity markets Some examples are Zanias (2005), Kellard and Wohar

(2006), Cuddington (1992), Ghoshray (2011), and Kim et al (2003) Zanias (2005) claims that

structural breaks in the data, which effectively contribute to the time-varying nature of prices, are due to productivity growth driven by the rise in commodity prices following the first World War Kellard and Wohar (2006) show that commodity prices do not have a single downward trend; rather, they are best characterised by a shifting trend that also changes sign over time Therefore, what is clear from Kellard and Wohar’s analysis is that while commodity prices are time-varying, this variability comes with an oscillating sign

Specific activities in the spot and futures markets could also affect markets differently and with time Consider the role of speculative trading, for instance Speculative trading in futures markets stabilizes the cash market (Lee and Ohk (1992)) From Friedman (1953), we know that speculation that results from the creation of stock index futures is inversely related to stock return volatility; for a related discussion, see Weller and Yano (1987) Speculation is not a continuous process, it is random; therefore, it should have a time-varying effect on the commodity spot and futures markets regardless of whether the speculation originates from the spot or futures markets

What do we learn from the literature? We learn that commodity prices are time-varying

We also now know that there are multiple sources of time-variation in commodity prices These include consumer income, labour productivity, market structure, monetary policy news, and business cycles in general An associated branch of the literature shows that shocks to commodity

prices have a permanent effect on prices, suggesting that every time commodity markets are exposed to shocks (regardless of the type of shock), prices move to a new long-run equilibrium What are the implications for price discovery? The main implication here is that, because commodity prices are time-varying, the variance should be reflected in a test of price discovery The path of price discovery may change depending not only on the existence of a shock but also

on the magnitude of a shock Shocks in our proposed model are based on error correction terms,

as we explain in the next section Therefore, both the shock(s) and their magnitude are reflected in the error correction terms Therefore, an error correction model of time varying price discovery is ideal to address our proposed research question

III An error correction model of time-varying price discovery

In this section, we propose a rolling window-based error correction model (RW-ECM) of the price discovery process Choosing a sufficiently large initial sample size then using rolling window samples allows us to extract time-varying error correction coefficients from the ECM Therefore,

a RW-ECM allows us to compute price discovery at every point in time beginning from the end

of the chosen initial sample window For example, consider the crude oil market We have monthly data on spot and futures prices beginning in March 1983 and ending September 2012 We choose the initial window of 120 months (10 years), which implies that we first estimate the ECM over the period March 1983 to February 1993 We then re-estimate the ECM over 120 months using a rolling window approach In other words, our next ECM is estimated over the period April 1983

to March 1993, then from May 1983 to April 1993, and so on This process of computing ECMs concludes when the last sample date (September 2012) is absorbed

It should be kept in mind that while there is no statistical criterion to choose the

rolling-in the coefficients of the model, while too large a wrolling-indow can potentially lead to little variations

in coefficients over time Our choice of 120 months is motivated by these costs However, there is one other statistical consideration that needs to be entertained with our proposal, and we do The ECM is predicated on two statistical conditions: (1) the two variables, which in our case are spot price and futures price, should be unit root non-stationary over the chosen windows; and (2) the two variables should be cointegrated—that is, they should share a long-run relationship over the chosen windows Therefore, our choice of the rolling window is motivated by these two features

of the data After having selected the rolling window, we pre-test for unit roots and cointegration over the rolling samples to ensure that these conditions are met The main implication here is that

in the absence of any selection criteria, our proposal of a ECM requires as a prerequisite that we choose rolling window samples that ensure the two variables are not only unit root non-stationary but are cointegrated Against this background, our error correction model for each rolling window, say 𝑖𝑖, takes the following form:

�𝑞𝑞 −𝐴𝐴12𝑗𝑗𝑗𝑗=1 Δ𝑆𝑆𝑡𝑡−𝑗𝑗+ 𝐴𝐴2𝑗𝑗2 Δ𝐹𝐹𝑡𝑡−𝑗𝑗⎦⎥

⎥

⎤

+ �𝑒𝑒𝑒𝑒𝑓𝑓,𝑡𝑡𝑠𝑠,𝑡𝑡� For simplicity, the constant term is dropped from the long-run cointegrating equation and as we will show later, both 𝑆𝑆𝑡𝑡 and 𝐹𝐹𝑡𝑡 follow a random walk process:

(2) 𝑆𝑆𝑡𝑡 = 𝑆𝑆𝑡𝑡−1+ 𝜂𝜂1𝑡𝑡

(3) 𝐹𝐹𝑡𝑡= 𝐹𝐹𝑡𝑡−1+ 𝜂𝜂2𝑡𝑡

The error terms may be contemporaneously and serially correlated:

(4) 𝑐𝑐𝑐𝑐𝑐𝑐(𝜂𝜂𝑡𝑡1, 𝜂𝜂𝑡𝑡2) = 𝜔𝜔𝑡𝑡

(Blanco et al (2005)) The inclusion of the error correction terms in the model is based on the

assumption that both variables are cointegrated Cointegration implies that at least one market will adjust, in which case this market is inefficient because its price reacts to information contained in another price The concept of cointegration and adjustment of the kind discussed here has been motivated by the Granger representation theorem (Engle and Granger (1987))

Following Gonzalo and Granger (GG, 1995), and see applications in Blanco et al (2005),

one can simply utilise the coefficients of the error correction terms to measure price discovery This can be captured by the following expression:

(7) 𝐺𝐺𝐺𝐺𝑆𝑆 = 𝜆𝜆 𝜆𝜆2

2− 𝜆𝜆1, Here 𝐺𝐺𝐺𝐺𝑆𝑆 represents the price discovery resulting from the spot market for rolling window 𝑖𝑖, where

𝑖𝑖 = 1, … , 𝑛𝑛 with 𝑛𝑛 representing the last window over which the ECM is estimated When in a

two-(orthogonality condition) and 𝐺𝐺𝐺𝐺𝑆𝑆+ 𝐺𝐺𝐺𝐺𝐹𝐹 = 1 (equality condition), where 𝐺𝐺𝐺𝐺𝐹𝐹 is the price discovery resulting from the futures market Since the error correction term in the spot market equation is expected to be negative, and positive in the futures market equation, the GG measure

is expected to be in the [0,1] range This will not be the case, however, if the error correction coefficients appear with incorrect (unexpected) signs In this case, there is no evidence of price discovery; therefore, GG can be outside the [0,1] range Because, as we explained earlier, we use

a fixed window of 120 observations to estimate the ECM and then apply a rolling regression approach, we end up with a GG measure every month from the end of the fixed window estimation period In this way we are able to extract a time-varying GG coefficient

IV Data and Results

A Data

We use monthly time series data on 17 commodity markets These commodities are noted in Table

1 For each commodity, we consider two price series: the spot price and the futures price and compute their returns as the log difference All commodities do not have the same start date although all data run up to September 2012 For 13 commodities the start date is January 1977, while for cotton, canola, crude oil, and natural gas the start dates are January 1979, August 1981, May 1983, and April 1990, respectively It follows that for 13 commodities there are no fewer than

429 monthly observations, while for the remaining four commodities the sample size ranges from

270 observations (natural gas) to 405 observations (cotton) All data were obtained from the Commodity Research Bureau (CRB) data CD The futures price data are adjusted for contract rollovers

Some preliminary observations of the data are presented in Table 1 In panel A we report commonly used descriptive statistics for commodity spot returns, while in panel B the corresponding statistics are reported for commodity futures returns In particular, we report the mean, coefficient of variation, skewness, kurtosis, and the Ljung-Box (1978) Q-statistic at the lag

of 12, which examines the no autocorrelation hypothesis

A brief account of these statistics is useful to demonstrate the difference among commodities The following observations, in particular, are noteworthy:

• Mean spot returns of three commodities are negative and for the remaining 14 commodities mean returns fall in the range [0.02, 0.60] The commodities futures returns follow a very similar pattern, although, in only two commodities returns appear to be less than zero

• The most volatile commodities are coffee, soybean yellow, and cotton Most commodities appear to have a leptokurtic distribution with a negative skew Only a small number (five)

of commodities have a positive skewness

• The ADF test applied to the returns of spot and futures series suggests that the null hypothesis of a unit root is comfortably rejected for all commodities at the 1% level Therefore, as expected, returns are stationary

• When we consider the null hypothesis of no autocorrelation, we find that the null is only rejected (at the 5% level) for six commodities, namely, corn, copper, coffee, crude oil, gold, and natural gas, while in the futures market it is rejected for eight commodities (cocoa, gold, copper, soybean yellow, soybean meal, cotton, crude oil, and natural gas)

B Preliminary evidence

In order to estimate rolling window ECMs, we need to first establish that spot and futures prices are unit root non-stationary and both price variables are cointegrated With our data set, for 13 commodities we have 308 rolling regressions, while for cotton, canola, crude oil and natural gas

we have 284, 253, 232, and 149 rolling regressions, respectively As a next step, for each of the samples for rolling regressions, we have to ascertain that prices are not only unit root non-stationary but are also cointegrated

We begin with an application of the unit root non-stationarity test Our approach is to apply

a rolling Augmented Dickey and Fuller (ADF, 1979) test to both the spot price and futures price The ADF test examines the null hypothesis that there is a unit root while the alternative hypothesis

is that the price variable is stationary The model includes an intercept and a time trend Lags of the dependent variable are used to control for any serial correlation in the model The optimal lag length is chosen by applying the Schwarz Information Criterion (SIC) We set the maximum lag length to eight and then obtain the optimal lag length based on the SIC For each commodity’s price series, we extract the ADF t-statistic for each sample of rolling regression For most of the commodity price series, there is strong evidence that the price series in each rolling regression model is characterised by a unit root At this point, we should emphasise the observation that, with the ADF test-statistics, there is clear evidence of variations over time This is implying nothing but the randomness of shocks that impact these commodity price series In other words, for some commodity prices the test statistics at some points in time are too far away from the 5% critical value, while at other times they are very close to the 5% critical value The plots of time-varying ADF test-statistics are available upon request

Similarly, we notice that at some times, evidence suggests that prices are more stationary than non-stationary Consider some examples Of the 308 rolling samples used for corn, at around the 235 sample, both spot and futures prices behave in a stationary manner For cocoa, notice that spot and futures prices behave in a stationary manner during some of the early rolling samples For the cocoa spot price, around the 148th rolling sample, some evidence of stationarity is noticed, while for the futures prices around the 125th to 130th samples some evidence of stationary prices are found With copper spot and futures prices stationary prices are found around the 125th to 130thsamples With the rest of the commodity prices, similar evidence of random stationary prices are found at some point in time

In Table 2, we also provide a summary of the ADF test results In particular, we report the percentage of times (out of all samples of rolling windows) the null hypothesis is rejected at the 5% level of significance We find that natural gas is the only commodity for which the majority of the rolling windows reject the null; around 83% of rolling windows for spot prices and around 93% of the rolling windows for futures prices are found to be stationary Cotton is another commodity for which greater evidence of stationary prices are found; with respect to the spot price, the null is rejected for 25% of rolling windows while with respect to the futures price the null is rejected for around 45% of rolling windows For silver and crude oil, around 25-26% of the rolling windows suggest stationary prices For nine commodities the rejection rate is less than 10% and for another three commodities the rejection rate is less than 15% In Table 3, we summarise the findings further We report the range of the t-test statistics across all rolling regressions per commodity The test statistics are only for regressions where the null is not rejected

The main implication here is that while, generally, there is extremely strong evidence that commodity prices are non-stationary, some (albeit limited) cases of stationary samples of prices

cannot be ruled out Even amongst the evidence of non-stationary prices, the ADF test statistics that are less than the CV, the variation is very much time-dependent However, in terms of progressing to the next stage of testing for cointegration, it is best to drop natural gas, and entertain caution when considering results obtained from cotton in particular

To test for cointegration, we propose a rolling window-based Johansen (1991) trace test The trace test examines the hypothesis that the system of equations contains, at most, 𝑣𝑣 cointegrating vectors The trace test is calculated as 𝜆𝜆(𝑣𝑣) = −𝑇𝑇 ∑𝑛𝑛 𝑙𝑙𝑐𝑐𝑙𝑙(1 − 𝜆𝜆𝑖𝑖)

no cointegration (𝑣𝑣 = 0) is rejected against the alternative of 𝑣𝑣 = 1, and the null hypothesis of 𝑣𝑣 ≤

1 against the alternative of 𝑣𝑣 = 2 These time-varying plots are available upon request The results are as follows Except for cotton and natural gas, where at least one of the price series is mostly stationary (that is, stationary in most of the rolling samples), the null hypothesis of a zero cointegrating vector is rejected in most rolling samples for most of the commodities Take some examples For eight commodities the null is rejected in over 90% of the rolling samples, while for another three commodities the null is rejected in over 80% of the rolling samples Soybean oil and canola are the only two commodities for which the null is rejected in less than 50% of the rolling samples, suggesting weak evidence for cointegration Although when we test the null hypothesis

of, at most, one cointegrating vector against the alternative of two vectors, except for two commodities (copper and gold), the null is rejected in favour of two cointegrating vectors in more

than 50% of the rolling samples On the whole, then, we find reasonable evidence that the spot and futures price series for most commodities contain at least one cointegrating vector

C Results on price discovery

We now turn to results on price discovery From our preceding analysis of unit roots and cointegration for rolling samples, it seems clear that natural gas and cotton are the two commodities which depart from the prerequisite that spot and futures prices should be non-stationary Prices seem to violate this condition in most rolling samples, therefore, while we do report the price discovery results for these two commodities here, just to get a feel of it, they need to be interpreted with the care they deserve In fact, as it turns out, we do not pay much attention to these results

In Table 4 we report a summary of the price discovery results In the last column, we report the number of rolling samples/regressions and, in the second column, we report the average (across the total number of samples) price discovery coefficients Standard deviations appear in column 3, followed by the 95% confidence interval for the average price discovery coefficient, while in columns 5 and 6 we report the minimum and maximum price discovery coefficients, respectively The following features of the results deserve mention First, the mean price discovery falls in the range [0.4287, 0.5602]; the coefficient is lowest for palladium and highest for soybean yellow The null hypothesis that the GG coefficient is equal to zero is rejected at the 1% level for all commodities Second, for only five commodities (cocoa, corn, platinum, soybean yellow, and soybean oil), the average GG coefficient exceeds 0.5, suggesting that price discovery is dominated

by the commodity spot market To confirm this, we also test the null hypothesis that the GG coefficient for these five commodities is equal to 0.5 The t-test statistic turns out to be 5.7, 6.3, 4.9, 16.7, and 4.4 for cocoa, corn, platinum, soybean oil, and soybean yellow, respectively This

suggests that we can comfortably reject (at the 1% level) the null suggesting that the GG coefficient

is indeed greater than 0.5 for these commodities For seven commodities, the average GG coefficient is less than 0.5, suggesting that it is the futures market which dominates price discovery For coffee, silver, and gold, on the other hand, the coefficient is around 0.5, suggesting that both markets contribute equally to price discovery Third, we notice that the rolling samples reveal volatile price discovery coefficients when price discovery is compared across markets, suggesting nothing but the fact that commodity markets are heterogeneous with respect to price discovery.1

A limitation of averaging the price discovery coefficients across the various rolling samples

is that we end up losing quite a bit of information on price discovery In other words, if there are indeed time periods over which price discovery is dominated by one market over another, and vice versa, we have simply ignored this The cost of doing so can be substantial because evidence of price discovery seems to be time-varying, as can be observed from Figure 1 There are some commodities for which time-varying price discovery is obvious In Table 5 we identify those commodities where there are clear phases of alternating price discovery processes between spot and futures markets For crude oil, evidence is clear that it is the futures market that dominates price discovery Crude oil together with natural gas and cotton, for which limited evidence of unit root was discovered, are excluded from further analysis In all, 14 commodities make the list of those where clear evidence exists that price discovery is time-varying and alternates between spot and futures markets For cocoa, corn, sugar, wheat, palladium, soybean yellow, and silver there is evidence of one phase where the spot market dominates price discovery, and one phase where the futures market dominates price discovery For coffee and soybean oil there is evidence of two phases of spot market and two phases of futures market dominance of price discovery For canola,

copper, and soybean meal there is evidence of two phases over which spot market dominates price discovery, and one phase over which the futures market is dominant, while for platinum evidence suggests one phase of spot market and two phases of futures market dominance of price discovery

D Are the different phases of price discovery related to specific events?

In this section we attempt to examine whether the different phases of price discovery dominated

by either the spot or futures market are related to any specific events A summary of events relating

to these phases of a market’s dominance for selected commodities are contained in Table 5 Generally, we find that for most of the phases some events can be associated with the dominance

of price discovery Consider some examples to demonstrate this point The dominance of the canola spot market is associated with a sharp rise in demand for canola and the persistent rise in energy prices, while the dominance of the futures market coincides with the Agricultural Improvement and Reform Act (US) and the Gulf War With regard to coffee, the dominance of the spot market coincides with agreements and disagreement on coffee export quotas, while the dominance of the futures market coincides with a 35% decline in yield in Ethiopia, the seventh largest producer of coffee in the world With respect to activities on the copper market, we find that the dominance of the spot market coincides with stagnant world demand and rising inventories, while the futures market dominance is associated with the Chilean mine strike To wheat, in this market while the spot market dominance is associated with the prolonged drought

in Australia and the late spring frost in the US which damaged emerging crops, the futures market dominance only associates itself with the Australian drought

E Robustness test results on price discovery using daily data

For the robustness test we use time series 5-day daily data on 16 commodities The commodities

do not all have the same start date although all data is up to 12 October 2012 For 13 commodities the start date is 05 January 1982, while for cotton, canola, and crude oil the start dates are 04 January 1984, 06 August 1986, and 17 May 1988, respectively To estimate the ECM, we choose the initial window of 1306 days (5 years) For example, consider the crude oil market We have 5-day daily data on spot and futures prices beginning 16 May 1983 and ending 12 October 2012 We choose the initial window of 1306 days (5 years), which implies that we first estimate the ECM over the period 16 May 1983 to 17 May 1988 We then re-estimate the ECM over 1306 days using

a rolling window approach In other words, our next ECM is estimated over the period 17 May

1983 to 18 May 1988, then from 18 May 1983 to 19 May 1988, and so on This process of computing ECMs concludes when the last sample date (12 October 2012) is absorbed

In Table 6 we report a summary of the price discovery results based on daily data First, the average price discovery falls in the range [0.2885, 0.5418]; the coefficient is lowest for canola and highest for soybean oil Second, for only four commodities (crude oil, platinum, sugar, and soybean oil), the average GG coefficient exceeds 0.5, suggesting that price discovery is dominated

by the commodity spot market For nine commodities, the average GG coefficient is less than 0.5, suggesting that it is the futures market which dominates price discovery For gold, palladium, and soybean yellow, on the other hand, the coefficient is around 0.5, suggesting that both markets contribute to price discovery equally Third, we notice that the rolling samples reveal volatile price discovery coefficients when price discovery is compared across markets, suggesting nothing other than the fact that commodity markets are heterogeneous with respect to price discovery (see Figure 2)

V An Economic Significance Analysis of Price Discovery: Implications for portfolio

construction and hedging

We do not know anything about the relationship between price discovery, portfolio construction and hedging in commodity markets Therefore, the goal of this section is two-fold First, we want

to examine optimal portfolio weights and test the effectiveness of hedging in the commodity markets Second, we want to establish any potential relationship between time-varying price discovery and the corresponding time-varying portfolio weights and hedging ratios These objectives stand out as they have not been investigated before The logic is simple As a result, it

is surprising that it has not been explored to-date In the previous section, we established that in some commodities price discovery takes place in the spot market, while in others it takes place in the futures market, and, regardless of the source, price discovery is time-varying Therefore, the question is: does time-varying price discovery have implications for portfolio allocation and hedging in commodity markets?

Our approach is as follows We begin with a portfolio analysis From this analysis, following Kroner and Ng (1998), we obtain a portfolio that minimizes risk without reducing

expected returns The weight �𝑤𝑤𝑠𝑠𝑓𝑓,𝑡𝑡� of the spot (s) market in a one dollar portfolio of spot and futures (f) commodities at time t is given by:

(8) 𝑤𝑤𝑠𝑠𝑓𝑓,𝑡𝑡 =ℎ ℎ𝑓𝑓,𝑡𝑡− ℎ𝑠𝑠𝑓𝑓,𝑡𝑡

𝑠𝑠,𝑡𝑡− 2ℎ𝑠𝑠𝑓𝑓,𝑡𝑡+ ℎ𝑓𝑓,𝑡𝑡 The time-varying conditional variance of commodity spot �ℎ𝑠𝑠,𝑡𝑡� and commodity futures �ℎ𝑓𝑓,𝑡𝑡�, and the conditional covariance �ℎ𝑠𝑠𝑓𝑓,𝑡𝑡� are extracted from estimating a bivariate GARCH model similar to the one proposed by Baillie and Myers (1991) The only difference is that because we find that for all commodities (over the full sample period) spot and futures prices are cointegrated,

the mean equation of the bivariate GARCH model includes an error correction term, as in Kroner and Sultan (1993) Therefore, we have a bivariate error correction GARCH model, whose

innovations, say {𝑒𝑒𝑡𝑡}𝑡𝑡=1𝑇𝑇 = �𝑒𝑒𝑠𝑠,𝑡𝑡�𝑡𝑡=1𝑇𝑇 + �𝑒𝑒𝑓𝑓,𝑡𝑡�𝑡𝑡=1𝑇𝑇 where subscript 𝑠𝑠 and 𝑓𝑓 represent innovations resulting from the spot and futures error correction equations, respectively, are modelled as:

(9) 𝑒𝑒𝑡𝑡|Ω𝑡𝑡−1~𝑁𝑁(0, ℎ𝑡𝑡), ℎ𝑡𝑡 = �ℎℎ𝑠𝑠,𝑡𝑡 ℎ𝑓𝑓,𝑡𝑡

𝑓𝑓,𝑡𝑡 ℎ𝑠𝑠𝑓𝑓,𝑡𝑡� (10) vec(ℎ𝑡𝑡) = 𝐶𝐶 + 𝐴𝐴 vec(𝑒𝑒𝑡𝑡−1𝑒𝑒𝑡𝑡−1′ ) + 𝐵𝐵 vec(ℎ𝑡𝑡−1) From here, the conditional minimum variance hedge ratio, say 𝐻𝐻𝐻𝐻, at time 𝑡𝑡 is simply ℎ𝑠𝑠𝑓𝑓,𝑡𝑡⁄ℎ𝑓𝑓,𝑡𝑡

Our key results are contained in Table 7 A range of results are reported here We begin with a note on the unit root properties of time-varying price discovery (reported in column 2) and

of the portfolio weight (unreported) This information is important as the mean equation of the GARCH model requires variables to be stationary We find that the unit root null is strongly rejected for the portfolio weight series with regard to all commodities; however, the same cannot

be concluded for the time-varying price discovery series The null is not rejected at the 5% level for all commodities except corn, cocoa, palladium, and natural gas Therefore, for those commodities where price discovery is non-stationary, the variable enters the regression model in first difference form Now, we consider the statistic of most importance, the portfolio weight, which is simply averaged over time and reported in the table The optimal holding of spot and futures differs from commodity to commodity For wheat and natural gas the optimal holding of spot in a one dollar portfolio of spot and futures is around 42 cents The corresponding optimal portfolio holdings for crude oil and silver are around 48 cents and 52 cents in spot and futures, respectively For most commodities (corn, gold, palladium, platinum, soybean yellow, soybean meal, sugar, and canola) the optimal portfolio is between 54-56 cents in favour of the spot market