Short- and Long-Run Determinants of Commodity Price Volatility

Introduction

The volatility of commodity prices was exceptionally high during the commodity bull-and-bear cycle of 2006–2009 (10). For example, the volatility of corn and wheat futures prices, historically averaging 19.7% and 22.2%, respectively, reached record high levels of 30%–50% from 2006–2011 (28; and authors' calculations).¹ Sustained periods of high price volatility create important business difficulties. For example, previous research suggests that the availability of forward contracting (e.g., through grain elevators) sharply decreased in some regions, and the increased size of futures margin calls led to a significant number of bankruptcies (22). Moreover, options, whose premium increases with volatility, have become much more costly to use when managing risk. The challenges associated with high price volatility indicate the importance of identifying its determinants for producers, processors, and policy-makers. The body of literature on commodity price volatility is vast, but mainly comprised of reduced-form time series models that focus on high-frequency determinants, rather than structural economic variables. This is because more structural models have been found to explain only a small fraction of the observed volatility (26; 2).

The main contribution of this paper is that it estimates a model of common and commodity-specific, high- and low-frequency factors to study the determinants and dynamics of price volatility in commodity futures markets. To bridge high-frequency commodity price volatility with its lower-frequency macroeconomic determinants, we use a two-stage approach. First, in the high-frequency data we isolate the proportion of variation that is plausibly due to macroeconomic variables. Then, using the isolated variation, we construct a measure of low-frequency volatility in the same sampling frequency as the macroeconomic data to use in seemingly unrelated regression (SUR) estimation. Our estimation framework follows 14, who show how a spline-GARCH model can decompose daily price volatility into high- and low-frequency components, the latter plausibly being driven by slowly-changing common and commodity-specific macroeconomic factors. A rich body of literature has investigated the macroeconomic determinants of commodity prices (24; 26; 6; 4; 7; 1; 10). Despite the evidence that macroeconomic forces affect prices, the R-squares reported by these studies are typically low. In contrast, the literature on how fundamentals affect volatility is smaller, but it has been established that volatility is time-varying (20), highly persistent (18), and that, at least for grains and oilseeds, it is affected by supply and demand inflexibilities (17).

This paper examines the following questions: (1) What are the common and commodity-specific determinants of volatility, and are the effects asymmetrical? (2) Is the effect of factors statistically the same within or across commodity groups? (3) Does modelling the low-frequency component (capturing the effect of macroeconomic forces) improve high-frequency volatility estimation? (4) Are the findings sensitive to whether the time period is a low- or high-volatility regime?

Model

Modelling volatility is of great interest to researchers and practitioners in finance and agricultural finance. The body of literature has grown since the development of the GARCH framework, in which volatility is modelled as being conditional on the information available at a specific time period (see 8; 27). 23, for example, show that a GARCH model significantly improves option pricing compared with the Black-Scholes benchmark. Volatility models use high-frequency data, that is, daily or intra-daily “tick” price observations. The effect of macroeconomic indicators is generally ignored in these models, as the data are sampled only monthly or quarterly. Recently, however, 14 proposed a model in which daily volatility is the product of a high-frequency news component and a low-frequency component reflecting market reactions to macroeconomic events, such as changes in inflation, real gross domestic product (GDP), or industrial production. The latter component is captured using a quadratic spline to provide a smooth, nonlinear long-run trend in the volatility time series, and can be aggregated to a monthly frequency for use, for example, in regressions over a set of macroeconomic variables.

The low-frequency component of volatility, τ_t(z_t), governs how macroeconomic state variables z_t affect asset price returns through a multiplicative structure. In our study, τ_t is sampled at the same frequency as the daily futures price data, and serves as a proxy for the macroeconomic variables z_t, which are sampled monthly.

The GARCH model, including the low-frequency component τ_t, is then written as follows:

(1)

(2)

(3)

where r_t is a time series of zero-mean, white noise residuals from a conditional mean regression of asset prices, for example, price log-returns, g_t is the high-frequency component of the conditional variance, α is the ARCH term, β is the GARCH term, ε_t is distributed as i.i.d. (0,1), typically either Normal or Student-t, c is a constant, w₀t is a time trend in the low-frequency volatility, is a low-order quadratic spline, and . The coefficients w_i are parameters governing the sharpness of the cycles described by the spline, while the number of cycles is determined by the number of knots k, the latter of which was selected based on a comparison of the Akaike Information Criterion for each specification.

The purpose of the spline is to modify the conditional variance equation to allow the unconditional variance to slowly change over time, affected by macroeconomic variables z_t. As the latter are only sampled at a monthly frequency, they cannot be directly included in the high-frequency (daily) dataset. The spline therefore lets τ_t reflect the effect of z_t, and later in this paper we empirically show that monthly low-frequency volatility (constructed from daily τ_t) is explained by macroeconomic variables z_t. As the sampling frequency prevents the direct inclusion of macroeconomic variables, we do not impose structure on how the macroeconomic indicators affect volatility. However, using a spline to model the low-frequency component implies a hidden structural assumption that macro effects cannot cause a sharp jump in low-frequency volatility. This appears reasonable because sharp jumps in volatility are captured through the high-frequency component. Standard GARCH restrictions are imposed to ensure positivity of the conditional variance (α≥0 and β≥0) and covariance-stationarity (α+β<1). As Engle and Rangel note, in this model the low-frequency volatility equals the unconditional volatility: .

To explain macroeconomic determinants of volatility, low-frequency volatility (LV_m) and realized volatility (RV_m) are sampled for months m=(1,⋯,M), in the same frequency as are macroeconomic variables. Realized volatility, RV_m, is defined as the square root of the sum of squared daily price returns (or residuals). For a specific month m, we use all N_m business-day observations d=(1,⋯,N_m), such that . Then, using τ_t, the low-frequency component of daily volatility obtained from the spline-GARCH model, we construct low-frequency volatility, LV_m, as the monthly sample average of .

Data and Data Processing

We analyze 11 different futures markets that can be categorized into five commodity groups: grain (corn, soybeans, wheat); livestock (live cattle, lean hogs); energy (crude oil, natural gas, heating oil); precious metals (gold, silver); and industrial metals (copper). Futures price data present several advantages over cash price data. For example, they are available at a higher sampling frequency (daily rather than weekly), the futures contract is standardized (e.g. commodity grade), price discovery usually occurs in futures markets (9; 31), and futures are used as tools for hedging and for capitalizing on commodity price fluctuations.

For each commodity, we construct time series of daily settlement prices of the first three nearby contracts from April 1990 through November 2009 by rolling over contracts.² To compare the determinants of price volatility during low- and high-volatility regimes, we split our data into two sample periods: 1990–2005 and 2006–2009. For each commodity in each sample period, we combine price series for the first three nearby futures contracts and estimate the following model:

(4)

where and F_i,k,t is the commodity k's i^th nearby futures price on day t. Thus, there are three price observations for each commodity on day t, resulting in contemporaneous correlation among the same-day observations. To correct for this correlation and use all three price observations on any given day, we apply the generalized least squares (GLS) method of 19.³ Briefly, the steps involve: (1) estimating the above regression model by ordinary least squares (OLS); (2) using the OLS residuals to compute sample means of associated squared residuals to obtain variance estimates of the first, second, and third nearby futures contracts, and to compute sample means of the products of related residuals to obtain estimates of covariances between first and second nearby, between first and third nearby, and between second and third nearby contracts; (3) computing the Cholesky decomposition of the variance-covariance matrix obtained in the previous step; (4) applying a GLS transformation to the data using the Cholesky decomposition; (5) estimating the regression model with the transformed variables via OLS; (6) eliminating insignificant regressors and repeating the same procedure with the new set of regressors. This way, we obtain contemporaneously-uncorrelated GLS residuals for each commodity to use in the spline-GARCH estimation.

To explain long-run economic determinants of volatility, we regress monthly low-frequency volatility obtained from the spline-GARCH estimation on both the fundamental macroeconomic variables that are common across commodities and the commodity-specific variables. Common macroeconomic variables include percentage changes in “Consumer Price Index for All Urban Consumers: All Items,” and “Industrial Production Index,” and the spread between the 10-year and 2-year “Treasury Constant Maturity Rate” obtained from ArchivaL Federal Reserve Economic Data (ALFRED). ALFRED is released by the Economic Research Division of the Federal Reserve Bank of St. Louis, and contains data on major economic variables available at the time of the release (without any revisions). All these variables are recorded monthly.⁴ We also include percentage changes in “Trade Weighted Exchange Index: Broad” obtained from the Board of Governors of the Federal Reserve System, which represents a weighted average of the foreign exchange value of the U.S. dollar against the currencies of a broad group of major U.S. trading partners.

For commodity-specific variables, we consider inventories for storable commodities. Even though gold and silver are storable, we could not obtain complete and reliable inventory data on these commodities. For corn, soybeans, and wheat we use the stocks-to-use ratio computed with the series of “Ending Stocks” and “Total Use” published in World Agricultural Supply and Demand Estimates (WASDE) reports released monthly by the World Agricultural Outlook Board of USDA. For live cattle, we compute monthly percentage changes in the number of cattle on feed released in Cattle on Feed reports by the USDA National Agricultural Statistics Service (NASS). For lean hogs, we interpolate quarterly percentage changes in the number of all hogs and pigs, computed using Hogs and Pigs reports released quarterly by the NASS, into monthly series with a step function. For inventories in energy markets, we use percentage changes in the monthly series of “U.S. Crude Oil Ending Stocks Excluding SPR,” “U.S. Natural Gas Underground Storage Volume,” and “U.S. Ending Stocks of Distillate Oil” series from the Energy Information Administration. Finally, for copper we compute percentage changes in the monthly series of “Stocks of Refined Copper in the United States” from the American Bureau of Metal Statistics.

Spline-GARCH Estimation

GLS-detrended residuals are first fitted to a conditional mean equation. Specifically, serial correlation is removed through a univariate ARMA filter, where the number of AR and MA terms is selected using likelihood ratio (LR) tests.⁵ After being filtered through the conditional mean equation, the residuals r_t are fitted to the spline-GARCH model (1)–(3). The results presented in table 1 describe the GARCH model parameter estimates, robust standard errors (computed using the Bollerslev-Wooldridge method) and tests of significance, the preferred distribution of innovations (with degrees of freedom ν for the Student-t), the optimal number of spline knots k, and the half-life of shocks (in weeks) implied by the model.

The optimal number of knots ranges from 5 to 15 in the sample period 1990–2005, and ranges from 5 to 16 in the period 2006–2009. A smaller number of knots implies fewer but longer cycles in the low-frequency volatility component. For the period 1990–2005, volatility for energy commodities was clearly less cyclical than it was for agricultural commodities and gold, but this difference does not hold for the period 2006–2009. Spectral analysis on the spline cycles does not reveal economically meaningful findings.⁶ Likelihood ratio tests are used to obtain the most suitable error specification. A model using Normal errors is estimated as a restricted case against a model with Student-t errors, since the Gaussian Normal distribution is obtained from the Student-t as a limiting case (degrees of freedom ). In all cases, the restriction of Normal errors is rejected by the LR test (see table 1). Similarly, for all commodities we cannot reject the GARCH(1,1) case as a restriction of higher-order GARCH specifications.

To evaluate the persistence of volatility shocks, their half-life is computed for all commodities in both sample periods and reported as the parameters λ and λ^G for the estimates from the spline-GARCH and standard GARCH models, respectively.⁷ Estimated using a standard GARCH model, volatility half-life in 1990–2005 ranges from a low of 6.8 weeks (wheat) to a high of 85.5 weeks (silver). However, when estimated using the spline-GARCH model, estimates of half-life drop substantially, ranging from a low of 0.1 week (wheat) to a high of 8.2 weeks (silver). From 2006–2009, half-life computed from a standard GARCH model ranges from 5.8 weeks (natural gas) to 54.8 weeks (lean hogs). Estimated from a spline-GARCH model, half-life drops to a range from 0.1 week (soybeans) to 5.7 weeks (heating oil). Indeed, volatility persistence computed from the spline-GARCH model is lower for all commodities in both sample periods. As noted by 14, allowing the unconditional variance to be time-varying through the τ_t term reveals shocks to be much less persistent than a benchmark GARCH model would suggest.⁸ This argument can be related to the work by 21, who showed that evidence of integrated conditional variance (i.e. permanent shocks, modeled as IGARCH) was spurious and disappeared once structural shifts in the unconditional variance were accounted for. Lastly, it appears that persistence increased for energy and decreased for metals during the commodity boom and bust cycle, but overall there is no clear evidence that persistence is greater for certain commodity groups.

To assess the improvement in goodness-of-fit provided by the spline-GARCH model, several standard metrics from the literature are considered (e.g. 16).⁹ First, as an in-sample tracking metric we compute the Root Mean Squared Error (RMSE) of the prediction error from using GARCH-implied conditional volatility (i.e. conditional standard deviation) to predict a proxy of realized volatility (i.e. the absolute value of GLS-detrended returns). We find that using the spline-GARCH model improves the fit for all commodities in both sample periods, reducing the prediction RMSE on average by 6.0% in period 1990–2005 and by 5.3% in period 2006–2009. Second, we compare R² from regressions of realized volatility on conditional volatility provided by each model. We find that R² increases for all commodities in both sample periods, with the exception of lean hogs. On average, R² increases from 4.6% to 5.3% in the first sample period, and increases from 6.3% to 7.7% in the second sample period. The magnitudes of the goodness-of-fit metrics are comparable to 16 findings for equities. Third, we verify that both the model-implied unconditional volatility and conditional volatility match the sample unconditional volatility (i.e. standard deviation of returns or GLS residuals). Both GARCH models perform roughly as well in this regard. Fourth and last, using Kolmogorov and Ljung-Box tests, we confirm that the standardized innovations from the GARCH models are well-behaved (i.i.d. white noise) and both models perform well. Overall, the spline-GARCH model is a clear improvement over a standard GARCH model for goodness-of-fit and in-sample tracking.

Macroeconomic Determinants of Low-Frequency Volatility

To study the effects of common- and commodity-specific variables on low-frequency volatility, we estimate the following model in an SUR framework:

(5)

(6)

where LV_k,m is the low-frequency volatility of commodity k in month m, %ΔCPI_m is the percentage change in consumer price index in month m, %ΔIP_m is the percentage change in industrial production index in month m, %ΔFX_m is the percentage change in trade-weighted foreign exchange index in month m, TSpread_m is the difference between 10-year and 2-year constant maturity rate in month m, and %ΔINV_k,m is the percentage change in inventory level of commodity k in month m. The indicator variables , , , take the value of one if the percentage change in the associated variable is nonnegative, and zero otherwise, while the indicator variables , , , take the value of one if the percentage change in the associated variable is negative, and zero otherwise. For corn, soybeans, and wheat the variable %ΔINV_k,m refers to the stocks-to-use ratio in month m rather than the percentage change in inventories. Moreover, DS_m is a summer dummy variable which takes the value of one if m is July, August, or September, and zero otherwise; DF_m is a fall dummy variable which takes the value of one if m is October, November, or December, and zero otherwise; DW_m is a winter dummy variable which takes the value of one if m is January, February, or March, and zero otherwise. We use this simple specification to account for seasonality because the commonly used alternatives, such as sinusoidal or polynomial functions, require higher-frequency observations. If inventory data are unavailable for a commodity, inventory terms are naturally excluded from the regression. Given K=11 commodities, M=189 months in the first sample period and M=47 months in the second, a total of 2,079 and 517 observations, respectively, are available in the SUR system. Results from the SUR estimation are presented in table 2. The table reports coefficient estimates and standard errors in brackets.¹⁰

Conclusions

Economic theory suggests that commodity price volatility is affected by macroeconomic forces as well as by high-frequency market news. However, because macroeconomic indicators are sampled only monthly or quarterly, while prices are sampled daily or intra-daily, most empirical studies model volatility as being determined only by high-frequency news. In contrast, the present work estimates a model of high- and low-frequency volatility components for eleven commodities over the period 1990–2009, and relates the low-frequency component to data on macroeconomic indicators. To this end, we build on the spline-GARCH framework of 14, who studied international equities. A GLS-detrending procedure is used to obtain contemporaneously-uncorrelated price residuals to avoid the “splicing bias” associated with futures data, and a SUR framework is used to test several hypotheses on the macroeconomic determinants of commodity volatility.

We find that computing monthly volatility from the low-frequency component of volatility, rather than using realized volatility, reveals more clearly the effect of macroeconomic indicators, which supports 14 results on equities. Moreover, the effect of macroeconomic variables has been greater during the commodity bull-and-bear cycle than prior to it. Furthermore, a greater number of macro effects are common across commodities from 1990–2005, while they tend to be commodity-specific from 2006–2009. We also find that using a spline-GARCH model improves goodness-of-fit for daily volatility estimation and tracking, and reveals that volatility persistence is much lower than a standard GARCH model would imply.

For most commodities, low-frequency volatility is affected by changes in inflation, industrial production, inventories, and the long-term and short-term interest rate spread, especially during periods of high volatility. The effect is asymmetrical in some cases. Volatility for most commodities increases with both positive and negative changes in inflation and the industrial production index, and also with the interest rate spread. We also find that the effect of macroeconomic indicators generally does not vary within commodity groups, with the exception of inventories and the interest rate spread. Further research could investigate the model's implications for commodity option-pricing and price risk management, as well as for accuracy in long-term volatility forecasting (e.g. 13).

The authors acknowledge constructive comments from Wally Thurman, two anonymous reviewers, and the editor. The authors also gratefully acknowledge financial support from NIFA (formerly CSREES) Hatch project RI-9262, Institut de Finance Mathématique de Montréal (IFM2), and the Centre Inter-universitaire de Recherche sur le Risque, les Politiques Économiques et l'Emploi (CIRPEE).