Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 646239, 8 pages

http://dx.doi.org/10.1155/2015/646239

## Low-Frequency Volatility in China’s Gold Futures Market and Its Macroeconomic Determinants

^{1}College of Economics and Management, South China Agricultural University, Guangzhou 510642, China^{2}Department of Agricultural Economics and Agribusiness, University of Arkansas, Fayetteville, AR 72701, USA^{3}Department of Economics, University of California, San Diego, CA 92093, USA

Received 8 February 2015; Accepted 21 May 2015

Academic Editor: Reik Donner

Copyright © 2015 Song Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We extract low- and high-frequency volatility from China’s Shanghai gold futures market using an asymmetric Spline-GARCH (ASP-GARCH) model. We then regress monthly low-frequency volatility on selected monthly macroeconomic indicators to study the impact of macroeconomy on gold futures market and to test for excess volatility. Our main result is volatility in China’s Shanghai gold futures market resulting from both macroeconomic fluctuations and investor behaviour. Chinese Consumer Price Index Volatility and US dollar volatility are the two main determinants of low-frequency gold volatility. We also find significant evidence of excess volatility, which can in part be explained in terms of loss-aversive investor behaviour.

#### 1. Introduction

The price of gold has undergone a series of drastic fluctuations since the 2008 global financial crisis. By September 2011, gold prices had climbed to an unprecedented high level of $1921 per troy ounce from a level of $682 per troy ounce in October 2008. However, gold prices quickly dropped by $400 per troy ounce in merely 20 trading days following this historic peak. Since 2012 gold price volatility has continued unabated with many macroeconomic factors such as the global economic recovery and appreciating the US dollar—according to analysts—potentially playing a driving role. However, extremely large gold price shocks such as a 14.8% decline that occurred over a three-day trading period in April 2013 have puzzled both academicians and industry analysts alike [1]. As a precious metal, gold is widely regarded as both a store of wealth and an inflation hedge and there is much interest in the investment community in determining if gold price volatility is simply excessive noise or is driven by sound macroeconomic fundamentals.

The Efficient Market Hypothesis (EMH) asserts that asset prices only respond to changes in fundamentals [2]. This has driven a large body of the literature to explore possible macroeconomic variables that can explain gold market volatility. For example, Wang et al. [3] found that the US dollar, crude oil prices, and global stock market performance had a significant impact on Shanghai’s gold futures market. Tully and Lucey [4] similarly looked to macroeconomic effects to explain gold price shocks. They employed an asymmetric power GARCH model (APGARCH) to show that the US dollar is the main macroeconomic variable which influences gold. In contrast, Batten et al. [5] showed that gold price volatility responded primarily to monetary variables like M2 (broad money) and the inflation rate. In addition, Christie-David et al. [6] and Cai et al. [7] reported that announcements of macroeconomics news have a significant impact on gold prices.

The methodological approaches used in previous studies suffer from two noticeable flaws. First, prior research has utilized traditional GARCH-type models which set unconditional volatility as a constant and consequently are unlikely to capture the true dynamics of long-run market volatility. Long-run volatility is equivalent to low-frequency volatility which is assumed to be determined by slowly evolving macroeconomic variables, while high-frequency or short-run volatility is mainly attributed to noise.

Second, the observed frequency of macroeconomic variables does not synchronize with observed gold prices. Macroeconomic variables are generally compiled monthly or even quarterly while gold prices are reported daily or intradaily. Incorporating variables with different frequencies in the same model creates econometric modeling difficulties.

To address these issues, Engle and Rangel [8] and Rangel and Engle [9] developed Spline-GARCH and an asymmetric Spline-GARCH (ASP-GARCH) model which relax the assumption that unconditional volatility is constant. They use a quadratic spline to isolate the proportion of variation in daily price data that is plausibly caused by macroeconomic variables. Then, using the isolated variation, Engle and Rangel [8] construct a measure of low-frequency volatility in the same sampling frequency as macroeconomic data, which effectively bridges high-frequency commodity price with its low-frequency macroeconomic determinants. Karali and Power [10] followed Engle and Rangel [8] and decomposed daily price volatility into high- and low-frequency components. They found that low-frequency volatility in US gold futures market responds strongly to changes in the industrial production index, consumer price index, and the US dollar. However, to our knowledge, this modelling approach has not been used to study factors affecting volatility in China’s gold futures market, which is the primary objective of this paper.

Although much research efforts have been devoted to studying the macroeconomic determinants of gold price volatility, it remains unclear whether these macroeconomic determinants sufficiently explain the high levels of observed gold market volatility. In this paper we examine whether Chinese gold futures market exhibits “excess volatility,” a term first coined by LeRoy and Porter [11] and Shiller [12] and attributed to the portion of an asset’s volatility that cannot be explained by fundamentals. Evidence of excess volatility poses a challenge to the Efficient Market Hypothesis (EMH)—the cornerstone of traditional finance theory—first outlined in Malkiel and Fama [2]. A growing body of the literature spanning the last 40 years has accumulated significant empirical evidence of excess volatility in a variety of financial markets. For example, De Long and Becht [13] reported excess volatility in German stock market after the Second World War. Similarly, Campbell and Cochrane [14] showed that US stock market volatility is far greater than dividend volatility. This result indicates that, contrary to the EMH, movements in dividends cannot be the sole determining factor of stock market volatility. There is also considerable evidence of excess volatility in Chinese stock and bond markets (e.g., [15–19]). Our paper extends this literature by shedding light on whether excess volatility is a prevalent feature of Chinese gold futures market.

The EMH and “Homo Economicus Assumption” fail to provide us with a widely accepted explanation of excess volatility in financial market. Proponents of behavioural finance would argue that classical theories fail because people are not rational in the classical economic sense. A number of studies grounded in behavioural finance theory—where market actors are assumed to be susceptible to cognitive biases—have been relatively successful in explaining excess volatility in stock markets. For example, investor sentiment has been shown to be a good explanation of excess volatility [18, 20–22]. Drawing from Kahneman and Tversky’s prospect theory, Barberis et al. [23] found that stock market investors are loss-aversive and their utility is affected by their previous investment returns. As a result, this form of loss aversion changes investors’ required risk premium and hence impacts market volatility. In a similar vein, a recent study by Wang and Hua [24] found that volatility in China’s copper futures markets correlates with investors’ loss aversion, which supports the claim that investors’ behaviour affects futures markets volatility.

In this paper we use an ASP-GARCH model to extract low-frequency volatility in China’s gold futures market and its response to changes in macroeconomic variables. We find evidence of excess volatility in the market, which we explain in a behavioural framework following Barberis et al.’s [23] loss aversion modelling approach. Our main contributions are as follows: we present the first empirical study to extract low-frequency volatility in China’s gold futures market; we are able to explain a portion of low-frequency volatility in terms of macroeconomic news; and we find significant evidence of excess volatility in China’s gold futures market, which we at least in part explain in terms of investor loss aversive behaviour.

#### 2. Modelling Approach and Hypotheses Testing

##### 2.1. ASP-GARCH Model

The assumption underlying almost all traditional GARCH models is that volatility is mean reverting to a constant level and unconditional volatility is constant. However, the mean reverting assumption has been greatly challenged by observation of real market movements. For example, it is widely recognized that volatility is higher during recessions and following “news” announcements [8]. Traditional GARCH model does not adequately capture the low-frequency changes in unconditional volatility whereas it is generally useful in modelling high-frequency conditional volatility. To address this issue Engle and Rangel [8] proposed a semiparametric Spline-GARCH model which relaxes the assumption that volatility is mean reverting to a constant level. To better understand how the Spline-GARCH model works, it is helpful for us to briefly review the traditional GARCH model. Bollerslev [25] proposed the GARCH model:where is the investment return at time , the expectation is conditional on an information set at time , is the innovation term assumed to be distributed with mean 0 and variance 1, again conditioned upon the information set at time , denotes the conditional variance of returns for period , and is a function of past errors and squared errors and variances observed at time . The terms , , and are the estimated coefficients on these conditional variance ARCH and GARCH terms. Equation (1) is the mean equation and (2) is the conditional variance equation.

Following Engle and Rangel [8], and focusing on the long-run properties of the model, the conditional variance equation can be rewritten in terms of the unconditional variance:where is the unconditional variance. However, this model designed to capture conditional volatility fails to model long-term trends in unconditional volatility. When , the unconditional variance reverts to its mean value at a geometric rate of . Therefore, as noted by Engle and Rangel [8], for a long horizon , the -days-ahead volatility forecast will be the same constant no matter whether the forecast is made at day or at day , . To better study the dynamics of financial time series, a model that allows unconditional volatility to vary slowly over time is much needed to capture the low-frequency component of volatility.

The Spline-GARCH model can decompose daily price volatility into high- and low-frequency components and thus is able to capture high-frequency news or noise component and also a low-frequency component incorporating market reactions to macroeconomic events:where is a time series of zero-mean, white noise residuals from a conditional mean regression of asset prices. The term is the high-frequency component of the conditional variance while is the low-frequency component. is distributed as i.i.d. normal . is the ARCH term; is the GARCH term. is the quadratic exponential time spline, is a constant, is a time trend in the low-frequency volatility, is a low-order quadratic spline, and . The coefficients control the sharpness of the cycles described by the spline while the number of cycles is determined by the number of knots , which divides the sample into equal parts: . The number of cycles increases with and the duration of each cycle shortens as increases. The value of is selected based on a comparison of the Akaike Information Criterion for each specification. In the model, unconditional volatility is the same as low-frequency volatility: . The Spline-GARCH model uses an exponential quadratic spline to approximate the time-varying unconditional volatility by generating a smooth curve that describes the low-frequency component, which is likely to result from changes in macroeconomic variables. Equation (7) models the high-frequency volatility. Equation (8) nonparametrically estimates the low-frequency volatility.

Rangel and Engle [9] proposed the ASP-GARCH model which we apply in this paper and which extends the Spline-GARCH model. In the ASP-GARCH model, the high-frequency component is different from its counterpart in Spline-GARCH model. Instead the component is modeled as in the GJR-GARCH model developed by Glosten et al. [26]:Equation (9) measures the leverage effect in high-frequency volatility, which is not feasible to estimate from (7). The leverage effect is observed when negative price shocks cause a larger change in volatility than do positive price shocks. is an indicator function of negative returns. If is significantly different from 0, there is an asymmetric impact from prior positive versus negative price shocks on current volatility.

From (9), we obtain the high-frequency volatility in the gold futures market. From (8), we are able to extract the daily low-frequency volatility component , which can be used to estimate , a volatility function explained by a subset of macroeconomic variables . However, a problem with this approach is that macroeconomic data relevant to gold markets are observed monthly whereas we estimate low-frequency volatility from daily gold prices. Therefore, to proceed we need to construct a monthly low-frequency volatility series which can be matched seamlessly with macroeconomic variables:where is the low-frequency volatility of month and day . is the number of trading days in month .

For any given , we can estimate the Spline-GARCH model which follows a normal distribution by maximum likelihood estimation. The log likelihood function is given asOnce we have monthly low-frequency volatility data for China’s gold futures market, we can proceed to test the impact of macroeconomic indicators on our low-frequency volatility series. Specifically, we estimate the following multivariable linear regression where low-frequency volatility is the dependent variable and relevant macroeconomic variables are independent variables:where is the low-frequency volatility in the gold futures market, is the volatility of macroeconomic variable in month* m* (the specific macroeconomic variables considered in the study are explained in Section 3), and is the residual term. To account for seasonality, , , and are binary dummy variables for fall, winter, and spring, respectively. For example, takes a value of one if is September, October, or November and zero otherwise. takes a value of one if is December, January, or February and zero otherwise. takes a value of one if is March, April, or May and zero otherwise. is a constant that captures low-frequency volatility in summer months; is the coefficient with respect to the macroeconomic variable . , , and are the coefficients that measure seasonality. Although we believe that (12) provides an adequate measure of the influence of our chosen macroeconomic variables on low-frequency gold volatility, it should be noted that we did not conduct an extensive analysis to determine potential correlations between macroeconomic variables.

##### 2.2. Hypotheses Testing

Following Engle and Rangel [8] and Rangel and Engle [9], we assume that low-frequency volatility is largely caused by changes in fundamentals while high-frequency volatility is largely driven by noise. So, one measure of “excess volatility” is the ratio of low-frequency volatility to total volatility in a market: where is the ratio of low-frequency volatility to total volatility. If , total volatility is greater than the volatility caused by fluctuations in fundamentals, supporting the existence of excess volatility in the market. Of course this is a somewhat crude measure as some high-frequency volatility is probably driven by short-term “news” relevant to gold market fundamentals.

Our empirical results, presented in Section 3, show strong evidence of excess volatility in Chinese gold futures market—in the sense that high-frequency volatility comprises a much larger component of total volatility than low-frequency volatility. This result begs the question what is the cause of such “excess volatility”? We turn to behavioural finance theory to answer this question and follow Barberis et al. [23] and test for the impact of investor loss aversion on excess volatility in China’s gold futures market. Under this approach we assume that represents the previous day’s modified investment return for a long futures position. Note that we assume that long-futures traders—who will profit from an increase in gold prices—are likely to be more susceptible to “loss aversion” than short-futures traders, who profit from short-term falls in gold prices. This assumption is consistent with the notion that institutional trading firms wishing to capture returns associated with rising gold prices, and who take a long-term investment position in gold, incorporate long gold futures positions in their investment portfolios to replicate cash gold. Irwin and Sanders [27] note that investors can gain exposure to commodity price increases through commodity index funds that hold long commodity futures positions. As such, these firms (long-futures traders) are akin to traditional stock market investors and are particularly sensitive to investment losses. However, our modelling approach is able to discern if short-futures trader “loss aversion” also contributes to gold price volatility. The return is modified in the sense that it is compared with an historical benchmark return for the market to incorporate a behavioural framing effect, which is assumed to influence investor behaviour. It is formally defined as , where is the historical benchmark price of gold while is the current price of gold at time . depends on the historical performance of gold prices and can be either >1 (when and there is previous investment loss), <1 (when and there is previous investment gain), or =1 (when ). varies with current gold price but at a lower rate. Equation (14) models the dynamics of :where is a fixed parameter which sets the median of over the time period of our data set to 1. In other words, the chance of investment loss or gain over our data set is 50-50. is average return; is the return at time . If there is an investment gain, , and declines; conversely if there is an investment loss, then , and increases. The term measures investor’s memory. The closer is to 0, the closer is to the current price of gold . This is indicative of the case when investors have a short memory and they are generally not influenced by previous gain or loss. However, when is closer to 1, moves slowly and investors have a long memory of their investment performance.

To assess the impact of long-futures investor loss aversion on high-frequency volatility we regress where is the high-frequency component of the volatility in China’s gold futures market that we estimate from (9). is the modified—taking account of the historical benchmark price of gold—daily log return of a long-futures gold investment at time . If is positive (indicating a modified investment loss), takes the value of 1 and otherwise takes the value 0; if is negative (indicating a modified investment gain), then takes the value of 1 and takes the value 0 otherwise. If and are statistically significant, both previous gains and losses to a long-futures position cause high-frequency volatility—which we associate with excess volatility—in China’s gold futures market. If , prior loss intensifies volatility. If , prior gain increases volatility. Note that a prior loss from the perspective of a long-futures trader is equivalent to a prior gain of the same magnitude from the perspective of a short-futures trader and vice versa. By further comparing the absolute values of and , we are able to determine the asymmetric impact of previous investment gain/loss to a long-futures position on excess volatility in China’s gold futures market.

#### 3. Empirical Analysis

##### 3.1. Extracting Low-Frequency Volatility

First, we construct monthly low-frequency volatility using daily gold futures price data. We use daily settlement price of the main contract in Shanghai’s gold futures market ranging from January 9, 2008, to December 31, 2013. In total, there are 1423 data points. The source of the data is Resset Financial Database (http://www.resset.cn/cn/). To ensure stationarity, we use the log return of gold futures:where is the log return at time and is the daily settlement price at time .

We obtain a white noise residual series from regressing on its unconditional mean as in the following equation:We then apply the ASP-GARCH model ((5), (6), (8), and (9)) to the series and report the results in Table 1. As noted, most of the coefficients are statistically significant at 5% level. We base our selection of the number of knots by AIC and the optimal number of knots is 10. Hence, there are 10 cycles in low-frequency volatility in China’s gold futures market from 2008 to 2013. Given our relatively small timeframe we assume that the cycles are of equal length, which is consistent with Karali and Power [10] who found 8 cycles of equal length for gold over the 2006–09 period. Figure 1 charts both high- and low-frequency volatility in China’s gold futures market. It is obvious from Figure 1 that gold futures price is most volatile in October 2008, March 2010, December 2011, and June 2013. Out of these 4 time periods, October 2008, which is at the height of global financial crisis, witnessed the most volatile gold futures price. Our ASP-GARCH volatility estimates are consistent with observed volatility in the market, and in-sample model predictions capture the unprecedented price spikes.