research article low-frequency volatility in china s gold

Research ArticleLow-Frequency Volatility in Chinarsquos Gold Futures Market andIts Macroeconomic Determinants

Song Liu1 Tingfei Tang1 Andrew M McKenzie2 and Yibin Liu3

1College of Economics and Management South China Agricultural University Guangzhou 510642 China2Department of Agricultural Economics and Agribusiness University of Arkansas Fayetteville AR 72701 USA3Department of Economics University of California San Diego CA 92093 USA

Correspondence should be addressed to Andrew M McKenzie mckenzieuarkedu

Received 8 February 2015 Accepted 21 May 2015

Academic Editor Reik Donner

Copyright copy 2015 Song Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We extract low- and high-frequency volatility from Chinarsquos 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 studythe impact ofmacroeconomy on gold futuresmarket and to test for excess volatility Ourmain result is volatility in Chinarsquos Shanghaigold futures market resulting from both macroeconomic fluctuations and investor behaviour Chinese Consumer Price IndexVolatility and US dollar volatility are the two main determinants of low-frequency gold volatility We also find significant evidenceof excess volatility which can in part be explained in terms of loss-aversive investor behaviour

1 Introduction

Theprice of gold has undergone a series of drastic fluctuationssince the 2008 global financial crisis By September 2011gold prices had climbed to an unprecedented high level of$1921 per troy ounce from a level of $682 per troy ouncein October 2008 However gold prices quickly dropped by$400 per troy ounce in merely 20 trading days following thishistoric peak Since 2012 gold price volatility has continuedunabated with many macroeconomic factors such as theglobal economic recovery and appreciating the US dollarmdashaccording to analystsmdashpotentially playing a driving roleHowever extremely large gold price shocks such as a 148decline that occurred over a three-day trading period in April2013 have puzzled both academicians and industry analystsalike [1] As a precious metal gold is widely regarded as botha store of wealth and an inflation hedge and there is muchinterest in the investment community in determining if goldprice volatility is simply excessive noise or is driven by soundmacroeconomic fundamentals

The Efficient Market Hypothesis (EMH) asserts that assetprices only respond to changes in fundamentals [2] Thishas driven a large body of the literature to explore possi-ble macroeconomic variables that can explain gold market

volatility For example Wang et al [3] found that the USdollar crude oil prices and global stock market performancehad a significant impact on Shanghairsquos gold futures marketTully and Lucey [4] similarly looked to macroeconomiceffects to explain gold price shocks They employed anasymmetric powerGARCHmodel (APGARCH) to show thatthe US dollar is the main macroeconomic variable whichinfluences gold In contrast Batten et al [5] showed that goldprice volatility responded primarily to monetary variableslike M2 (broad money) and the inflation rate In additionChristie-David et al [6] and Cai et al [7] reported thatannouncements of macroeconomics news have a significantimpact on gold prices

The methodological approaches used in previous studiessuffer from two noticeable flaws First prior research hasutilized traditional GARCH-type models which set uncondi-tional volatility as a constant and consequently are unlikelyto capture the true dynamics of long-run market volatilityLong-run volatility is equivalent to low-frequency volatilitywhich is assumed to be determined by slowly evolvingmacroeconomic variables while high-frequency or short-runvolatility is mainly attributed to noise

Second the observed frequency of macroeconomic vari-ables does not synchronize with observed gold prices

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 646239 8 pageshttpdxdoiorg1011552015646239

2 Mathematical Problems in Engineering

Macroeconomic variables are generally compiled monthlyor even quarterly while gold prices are reported daily orintradaily Incorporating variables with different frequenciesin the same model creates econometric modeling difficulties

To address these issues Engle and Rangel [8] and Rangeland Engle [9] developed Spline-GARCH and an asymmet-ric Spline-GARCH (ASP-GARCH) model which relax theassumption that unconditional volatility is constantThey usea quadratic spline to isolate the proportion of variation indaily price data that is plausibly caused by macroeconomicvariablesThen using the isolated variation Engle andRangel[8] construct a measure of low-frequency volatility in thesame sampling frequency as macroeconomic data whicheffectively bridges high-frequency commodity price withits low-frequency macroeconomic determinants Karali andPower [10] followed Engle and Rangel [8] and decomposeddaily price volatility into high- and low-frequency compo-nents They found that low-frequency volatility in US goldfutures market responds strongly to changes in the industrialproduction index consumer price index and the US dollarHowever to our knowledge this modelling approach has notbeen used to study factors affecting volatility in Chinarsquos goldfutures market which is the primary objective of this paper

Although much research efforts have been devoted tostudying the macroeconomic determinants of gold pricevolatility it remains unclear whether these macroeconomicdeterminants sufficiently explain the high levels of observedgold market volatility In this paper we examine whetherChinese gold futures market exhibits ldquoexcess volatilityrdquo aterm first coined by LeRoy and Porter [11] and Shiller [12]and attributed to the portion of an assetrsquos volatility thatcannot be explained by fundamentals Evidence of excessvolatility poses a challenge to the EfficientMarket Hypothesis(EMH)mdashthe cornerstone of traditional finance theorymdashfirst outlined in Malkiel and Fama [2] A growing body ofthe literature spanning the last 40 years has accumulatedsignificant empirical evidence of excess volatility in a varietyof financial markets For example De Long and Becht [13]reported excess volatility in German stock market afterthe Second World War Similarly Campbell and Cochrane[14] showed that US stock market volatility is far greaterthan dividend volatility This result indicates that contraryto the EMH movements in dividends cannot be the soledetermining factor of stock market volatility There is alsoconsiderable evidence of excess volatility in Chinese stockand bond markets (eg [15ndash19]) Our paper extends thisliterature by shedding light on whether excess volatility is aprevalent feature of Chinese gold futures market

The EMH and ldquoHomo Economicus Assumptionrdquo fail toprovide us with a widely accepted explanation of excessvolatility in financial market Proponents of behaviouralfinance would argue that classical theories fail because peopleare not rational in the classical economic sense A numberof studies grounded in behavioural finance theorymdashwheremarket actors are assumed to be susceptible to cognitivebiasesmdashhave been relatively successful in explaining excessvolatility in stock markets For example investor sentimenthas been shown to be a good explanation of excess volatility[18 20ndash22] Drawing fromKahneman and Tverskyrsquos prospect

theory Barberis et al [23] found that stock market investorsare loss-aversive and their utility is affected by their previousinvestment returns As a result this form of loss aversionchanges investorsrsquo required risk premium and hence impactsmarket volatility In a similar vein a recent study byWang andHua [24] found that volatility in Chinarsquos copper futures mar-kets correlates with investorsrsquo loss aversion which supportsthe claim that investorsrsquo behaviour affects futures marketsvolatility

In this paper we use an ASP-GARCH model to extractlow-frequency volatility in Chinarsquos gold futures market andits response to changes in macroeconomic variables We findevidence of excess volatility in the market which we explainin a behavioural framework following Barberis et alrsquos [23] lossaversion modelling approach Our main contributions are asfollows (1) we present the first empirical study to extractlow-frequency volatility in Chinarsquos gold futures market (2)

we are able to explain a portion of low-frequency volatilityin terms of macroeconomic news and (3) we find significantevidence of excess volatility in Chinarsquos gold futures marketwhich we at least in part explain in terms of investor lossaversive behaviour

2 Modelling Approach andHypotheses Testing

21 ASP-GARCH Model The assumption underlying almostall traditional GARCH models is that volatility is meanreverting to a constant level and unconditional volatility isconstant However the mean reverting assumption has beengreatly challenged by observation of real market movementsFor example it is widely recognized that volatility is higherduring recessions and following ldquonewsrdquo announcements [8]Traditional GARCH model does not adequately capture thelow-frequency changes in unconditional volatility whereas itis generally useful in modelling high-frequency conditionalvolatility To address this issue Engle and Rangel [8] proposeda semiparametric Spline-GARCH model which relaxes theassumption that volatility is mean reverting to a constantlevel To better understand how the Spline-GARCH modelworks it is helpful for us to briefly review the traditionalGARCHmodel Bollerslev [25] proposed theGARCHmodel

119903119905

minus 119864119905minus1119903119905

= radicℎ119905120576119905 (1)

ℎ119905

= 120603 + 1205721205762119905minus1 + 120573ℎ

119905minus1 (2)

120576119905

| 119868119905minus1 sim 119873 (0 1) (3)

where 119903119905is the investment return at time 119905 the expectation

119864119905minus1 is conditional on an information set 119868

119905minus1at time 119905 minus 1 120576

119905

is the innovation term assumed to be distributed with mean0 and variance 1 again conditioned upon the information set119868119905minus1

at time 119905minus1 ℎ119905denotes the conditional variance of returns

for period 119905 and ℎ119905is a function of past errors and squared

errors and variances observed at time 119905minus1The terms120603120572 and120573 are the estimated coefficients on these conditional varianceARCH andGARCH terms Equation (1) is themean equationand (2) is the conditional variance equation

Mathematical Problems in Engineering 3

Following Engle and Rangel [8] and focusing on thelong-run properties of the model the conditional varianceequation can be rewritten in terms of the unconditionalvariance

ℎ119905

= 1205902

+ 120572 (1205762119905minus1 minus 120590

2) + 120573 (ℎ

119905minus1 minus 1205902) (4)

where 1205902 = 120603(1minus 120572 minus 120573) is the unconditional variance How-ever thismodel designed to capture conditional volatility failsto model long-term trends in unconditional volatility When120572+120573 lt 1 the unconditional variance reverts to itsmean value1205902 at a geometric rate of120572+120573Therefore as noted byEngle and

Rangel [8] for a long horizon 119879 the 119879-days-ahead volatilityforecast will be the same constant 120590 no matter whether theforecast is made at day 119905 or at day 119905 minus 119896 119896 gt 0 To better studythe dynamics of financial time series a model that allowsunconditional volatility 120590

2 to vary slowly over time is muchneeded to capture the low-frequency component of volatility

The Spline-GARCH model can decompose daily pricevolatility into high- and low-frequency components and thusis able to capture high-frequency news or noise componentand also a low-frequency component incorporating marketreactions to macroeconomic events

119903119905

= radic119892119905120591119905120576119905 (5)

ℎ119905

= 119892119905120591119905 (6)

119892119905

= (1minus 120572 minus 120573) + 120572 (1199032119905minus1

120591119905minus1

) + 120573119892119905minus1 (7)

120591119905

= 119888 exp(1199080119905 +

119896

sum119894=1

119908119894((119905 minus 119905

119894minus1)+

)2) (8)

where 119903119905is a time series of zero-mean white noise residuals

from a conditional mean regression of asset prices Theterm 119892

119905is the high-frequency component of the conditional

variance while 120591119905is the low-frequency component 120576

119905is

distributed as iid normal (0 1) 120572 is the ARCH term 120573 isthe GARCH term 120591

119905is the quadratic exponential time spline

119888 is a constant 1199080119905 is a time trend in the low-frequencyvolatility sum

119896

119894=1 119908119894((119905 minus 119905

119894minus1)+

)2 is a low-order quadratic spline

and (119905 minus 119905119894minus1)+= max(119905 minus 119905

119894minus1 0) The coefficients 119908119894control

the sharpness of the cycles described by the spline while thenumber of cycles is determined by the number of knots 119896which divides the sample into 119896 equal parts 1 lt 1199051 lt

1199052 lt sdot sdot sdot lt 119905119896

lt 119905 The number of cycles increases with119896 and the duration of each cycle shortens as 119896 increasesThe value of 119896 is selected based on a comparison of theAkaike Information Criterion for each specification In themodel unconditional volatility is the same as low-frequencyvolatility 119864(119903

2119905) = 119864(119892

119905)120591119905

= 120591119905 The Spline-GARCH model

uses an exponential quadratic spline to approximate thetime-varying unconditional volatility by generating a smoothcurve that describes the low-frequency component whichis likely to result from changes in macroeconomic variablesEquation (7) models the high-frequency volatility Equation(8) nonparametrically estimates the low-frequency volatility

Rangel and Engle [9] proposed the ASP-GARCH modelwhich we apply in this paper and which extends the Spline-GARCH model In the ASP-GARCH model the high-frequency component 119892

119905is different from its counterpart in

Spline-GARCHmodel Instead the 119892119905component is modeled

as in theGJR-GARCHmodel developed byGlosten et al [26]

119892119905

= (1minus 120572 minus 120573 minusV2

) + 120572 (1199032119905minus1

120591119905minus1

) + V(1199032119905minus1

120591119905minus1

) 119868119903119905minus1lt0

+ 120573119892119905minus1

(9)

Equation (9) measures the leverage effect in high-frequencyvolatility which is not feasible to estimate from (7) Theleverage effect is observed when negative price shocks cause alarger change in volatility than do positive price shocks 119868

119903119905minus1lt0

is an indicator function of negative returns If V is significantlydifferent from 0 there is an asymmetric impact from priorpositive versus negative price shocks on current volatility

From (9) we obtain the high-frequency volatility in thegold futures market From (8) we are able to extract thedaily low-frequency volatility component 120591

119905 which can be

used to estimate 120591119905(119910119905) a volatility function explained by a

subset of macroeconomic variables 119910119905 However a problem

with this approach is that macroeconomic data relevant togold markets are observed monthly whereas we estimatelow-frequency volatility from daily gold prices Thereforeto proceed we need to construct a monthly low-frequencyvolatility series LV

119898which can be matched seamlessly with

macroeconomic variables

LV119898

= radic1

119873119898

119873119898

sum119889=1

120591119889119898

(10)

where 120591119889119898

is the low-frequency volatility ofmonth119898 and day119889 119873119898is the number of trading days in month 119898

For any given 119896 we can estimate the Spline-GARCHmodel which follows a normal distribution by maximumlikelihood estimation The log likelihood function is given as

119871 = minus12

119879

sum119905=1

(log (120591119905119892119905) +

1199032119905

120591119905119892119905

) (11)

Once we have monthly low-frequency volatility data forChinarsquos gold futuresmarket we can proceed to test the impactof macroeconomic indicators on our low-frequency volatilityseries Specifically we estimate the following multivariablelinear regression where low-frequency volatility is the depen-dent variable and relevant macroeconomic variables areindependent variables

LV119898

= 119886 +

119899

sum119894=1

119887119894119911119894119898

+ V1DF119898 + V2DW119898 + V3DS119898

+ 119890119896119898

(12)

where LV119898is the low-frequency volatility in the gold futures

market 119911119894119898

is the volatility of macroeconomic variable 119894 inmonth m (the specific macroeconomic variables considered


in the study are explained in Section 3) and 119890119896119898

is theresidual term To account for seasonality DF

119898 DW

119898 and

DS119898are binary dummy variables for fall winter and spring

respectively For example DF119898

takes a value of one if 119898

is September October or November and zero otherwiseDW119898

takes a value of one if 119898 is December January orFebruary and zero otherwise DS

119898takes a value of one if 119898

is March April or May and zero otherwise 119886 is a constantthat captures low-frequency volatility in summer months 119887

119894

is the coefficient with respect to the macroeconomic variable119894 V1 V2 and V3 are the coefficients that measure seasonalityAlthough we believe that (12) provides an adequate measureof the influence of our chosen macroeconomic variables onlow-frequency gold volatility it should be noted that we didnot conduct an extensive analysis to determine potentialcorrelations between macroeconomic variables

22 Hypotheses Testing Following Engle and Rangel [8] andRangel and Engle [9] we assume that low-frequency volatilityis largely caused by changes in fundamentals while high-frequency volatility is largely driven by noise So onemeasureof ldquoexcess volatilityrdquo is the ratio of low-frequency volatility tototal volatility in a market

119862 =Var [log (120591

119905)]

Var [log (119892119905120591119905)]

(13)

where 119862 is the ratio of low-frequency volatility to totalvolatility If 119862 lt 1 total volatility is greater than thevolatility caused by fluctuations in fundamentals supportingthe existence of excess volatility in the market Of coursethis is a somewhat crude measure as some high-frequencyvolatility is probably driven by short-term ldquonewsrdquo relevant togold market fundamentals

Our empirical results presented in Section 3 show strongevidence of excess volatility in Chinese gold futures marketmdashin the sense that high-frequency volatility comprises a muchlarger component of total volatility than low-frequencyvolatility This result begs the question what is the causeof such ldquoexcess volatilityrdquo We turn to behavioural financetheory to answer this question and follow Barberis et al [23]and test for the impact of investor loss aversion on excessvolatility in Chinarsquos gold futures market Under this approachwe assume that 119911

119905represents the previous dayrsquos modified

investment return for a long futures position Note that weassume that long-futures tradersmdashwho will profit from anincrease in gold pricesmdashare likely to be more susceptible toldquoloss aversionrdquo than short-futures traders who profit fromshort-term falls in gold prices This assumption is consistentwith the notion that institutional trading firms wishing tocapture returns associated with rising gold prices and whotake a long-term investment position in gold incorporatelong gold futures positions in their investment portfolios toreplicate cash gold Irwin and Sanders [27] note that investorscan gain exposure to commodity price increases throughcommodity index funds that hold long commodity futurespositions As such these firms (long-futures traders) are akinto traditional stockmarket investors and are particularly sen-sitive to investment losses However our modelling approach

is able to discern if short-futures trader ldquoloss aversionrdquo alsocontributes to gold price volatility The return is modified inthe sense that it is compared with an historical benchmarkreturn for the market to incorporate a behavioural framingeffect which is assumed to influence investor behaviour Itis formally defined as 119911

119905= 120595119905119878119905 where 120595

119905is the historical

benchmark price of gold while 119878119905is the current price of gold

at time 119905 119911119905depends on the historical performance of gold

prices and can be either gt1 (when 120595119905

gt 119878119905and there is

previous investment loss) lt1 (when 120595119905

lt 119878119905and there is

previous investment gain) or =1 (when 120595119905

= 119878119905) 120595119905varies

with current gold price but at a lower rate Equation (14)models the dynamics of 119911

119905

119911119905+1 = 120578 (119911

119905

119877

119877119905+1

) + (1minus 120578) 120578 isin (0 1) (14)

where 119877 is a fixed parameter which sets the median of 119911119905over

the time period of our data set to 1 In other words the chanceof investment loss or gain over our data set is 50-50 119877 isaverage return 119877

119905+1 is the return at time 119905 + 1 If there is aninvestment gain 119877

119905+1 gt 119877 and 119911 declines conversely if thereis an investment loss then 119877

119905+1 lt 119877 and 119911 increases Theterm 120578 measures investorrsquos memory The closer 120578 is to 0 thecloser 120595

119905is to the current price of gold 119878

119905 This is indicative

of the case when investors have a short memory and they aregenerally not influenced by previous gain or loss Howeverwhen 120578 is closer to 1 120595

119905moves slowly and investors have a

long memory of their investment performanceTo assess the impact of long-futures investor loss aversion

on high-frequency volatility we regress

119892119905

= 119888 + V11199111015840

119905minus1119868+

119905minus1 + V21199111015840

119905minus1119868minus

119905minus1 + 120576119905 (15)

where 119892119905is the high-frequency component of the volatility in

Chinarsquos gold futuresmarket that we estimate from (9) 1199111015840119905is the

modifiedmdashtaking account of the historical benchmark priceof goldmdashdaily log return of a long-futures gold investmentat time 119905 If 119911

1015840

119905is positive (indicating a modified investment

loss) 119868+ takes the value of 1 and otherwise takes the value 0 if1199111015840119905is negative (indicating amodified investment gain) then 119868minus

takes the value of 1 and takes the value 0 otherwise If V1 andV2 are statistically significant both previous gains and lossesto a long-futures position cause high-frequency volatilitymdashwhich we associate with excess volatilitymdashin Chinarsquos goldfutures market If V1 gt 0 prior loss intensifies volatility IfV2 lt 0 prior gain increases volatility Note that a prior lossfrom the perspective of a long-futures trader is equivalent toa prior gain of the same magnitude from the perspective ofa short-futures trader and vice versa By further comparingthe absolute values of V1 and V2 we are able to determinethe asymmetric impact of previous investment gainloss toa long-futures position on excess volatility in Chinarsquos goldfutures market

3 Empirical Analysis

31 Extracting Low-Frequency Volatility First we constructmonthly low-frequency volatility using daily gold futures


00000

00002

00004

00006

00008

00010

2008 2009 2010 2011 2012 2013

LVOLHVOL

Vola

tility

leve

l

Year

Figure 1 High- and low-frequency volatility in Chinarsquos gold futuresmarket Note LVOL is low-frequency volatility while HVOL meanshigh-frequency volatility

price data We use daily settlement price of the main contractin Shanghairsquos gold futures market ranging from January 92008 to December 31 2013 In total there are 1423 datapoints The source of the data is Resset Financial Database(httpwwwressetcncn) To ensure stationarity we use thelog return of gold futures

119877119905

= 100times ln(119875119905

119875119905minus1

) (16)

where 119877119905is the log return at time 119905 and 119875

119905is the daily

settlement price at time 119905We obtain a white noise residual series 119903

119905from regressing

119877119905on its unconditional mean as in the following equation

119877119905

= 120583 + 119903119905 (17)

We then apply the ASP-GARCHmodel ((5) (6) (8) and (9))to the series 119903

119905and report the results in Table 1 As notedmost

of the coefficients are statistically significant at 5 level Webase our selection of the number of knots by AIC and theoptimal number of knots is 10 Hence there are 10 cycles inlow-frequency volatility in Chinarsquos gold futures market from2008 to 2013 Given our relatively small timeframewe assumethat the cycles are of equal length which is consistent withKarali and Power [10] who found 8 cycles of equal length forgold over the 2006ndash09 period Figure 1 charts both high- andlow-frequency volatility in Chinarsquos gold futures market It isobvious from Figure 1 that gold futures price is most volatilein October 2008 March 2010 December 2011 and June 2013Out of these 4 time periods October 2008 which is at theheight of global financial crisis witnessed the most volatilegold futures price Our ASP-GARCH volatility estimates areconsistent with observed volatility in the market and in-sample model predictions capture the unprecedented pricespikes

32 Macroeconomic Determinants of Low-Frequency Volatil-ity Now that low-frequency volatility has been estimated

Table 1 Estimates fromASP-GARCHmodel in Chinarsquos gold futuresmarket

Coefficients SE120572 00774

lowastlowastlowast00212

120573 07908lowastlowastlowast 00436

119888 14448119890 minus 004lowastlowastlowast 51803119890 minus 005

1199080

00103lowastlowastlowast

28767119890 minus 003

1199081

minus22463119890 minus 005lowastlowastlowast 44678119890 minus 006

1199082


1199083

12517119890 minus 004lowastlowastlowast

74997119890 minus 006

1199084

minus12422119890 minus 004lowastlowastlowast

69961119890 minus 006

1199085

71531119890 minus 005lowastlowastlowast 11471119890 minus 005

1199086


1199087


35234119890 minus 005

1199088


1199089

minus12909119890 minus 004lowastlowast 56524119890 minus 005

11990810

minus23390119890 minus 005 82915119890 minus 005

V 00227 00294

Log likelihood 55941610

AIC minus78510

Note lowast lowast lowast lowastlowast denote significance at the 1 and 5 levels respectively SEis the covariance based standard error of estimated coefficients

from the ASP-GARCH model we proceed with estimatingthe impact that macroeconomic variables may have on low-frequency volatility To bridge daily low-frequency volatilitydata with monthly macroeconomic indicators we use (10)to construct monthly low-frequency volatility series LV

119898

which is in turn regressed uponmacroeconomic and seasonaldummy variables as in (12)

Gold price is influenced by both supply and demandand macroeconomic conditions Potential macroeconomicvariables include but are not limited to Gross DomesticProduct (GDP) inflation rate United States dollar USDexchange rate interest rateM2 unemployment rate oil pricestock indices prices of substitutes (eg silver) and politicalrisk We exclude GDP from our study since GDP is reportedquarterly and consequently we are only able to obtain 24 datapoints of GDP from 2008 to 2013 Instead we use ConsumerConfidence Index and Industrial Production (both compiledmonthly) as proxies for GDP Industrial Production is animportant economic indicator of production and industrythe Consumer Confidence Index (CCI) is a compound indexwhich incorporates information on employment incomeprice interest rate and so forth and reflects peoplersquos confi-dence in and expectation of the economy It is worth notingthat the unemployment rate in China is reported quarterlyand so again we choose to omit this variable from ouranalysis Nevertheless we believe thatCCI is a plausible proxyfor unemployment rate since CCI takes into account peoplersquosemployment as well Lastly we exclude political risk from theregression since it is extremely difficult to quantify

To sum up we choose the following monthly Chinesemacroeconomic variables based upon relevance and avail-ability Chinese Consumer Price Index Volatility (CPIVOL)Chinese Industrial Production Volatility (IPVOL) Chinese


Table 2 Estimates from regression of low-frequency volatility in Chinarsquos gold futures market on volatility of macroeconomic variables

Regression on Chinese macroeconomic volatility variables Regression on US macroeconomic volatility variablesVariable Coefficients SE Variable Coefficients SECPIVOL 4053267lowastlowastlowast 0497055 CPIVOL 3481405lowastlowastlowast 0508096

IPVOL minus0000414 0000596 IPVOL 910E minus 06lowast 534119864 minus 06

IRVOL 232119864 minus 05 178119864 minus 05 IRVOL minus0000141lowast 721119864 minus 05

FXVOL 0105768lowastlowastlowast 0039564 FXVOL 0126571lowastlowastlowast 0043275

M2VOL 0118330 0111885 M2VOL 552E minus 05lowast 309119864 minus 05

CCIVOL minus0100860 0352567 CCIVOL 1040476 2065703

Intercept minus478119864 minus 05 347119864 minus 05 Intercept minus193119864 minus 05 263119864 minus 05

Spring 237119864 minus 05 220119864 minus 05 Spring 183119864 minus 05 196119864 minus 05

Winter 268119864 minus 05 215119864 minus 05 Winter 132119864 minus 06 193119864 minus 05

Fall 958119864 minus 06 212119864 minus 05 Fall minus897119864 minus 06 195119864 minus 05

Adjusted 1198772 0626325 Adjusted 1198772 0655592

119865-statistic 13850280lowastlowastlowast 119865-statistic 16016770lowastlowastlowast

Note The left panel of the table presents results of regression of low-frequency Chinese gold volatility on volatility of Chinese macroeconomic variables theright panel presents results of regression of low-frequency Chinese gold volatility on volatility of US macroeconomic variables (except for CPIVOL) SE is thecovariance based standard error of estimated coefficientslowast lowast lowast and lowast denote significance at the 1 and 10 levels respectively

Interbank Lending Rate Volatility (IRVOL) US Dollar For-eign Exchange Rate Index Volatility (FXVOL) ChineseMoney Supply Volatility (M2VOL) and Chinese ConsumerConfidence Volatility (CCIVOL) These monthly volatilitiesare estimated over the January 2008 to December 2013period yielding 72 observations for each series The sourceis Zhongjinwang Statistical Database (httpdbceigovcnpageDefaultaspx) and Economy Prediction System (EPS)Statistical Database (httpwwwepsnetcomcn)

The estimates of (12) are reported in Table 2 The lefthalf of Table 2 reports the results of the regression of low-frequency Chinese gold market volatility on the volatility ofChinese macroeconomic variables while the right half ofTable 2 provides a comparison regression of low-frequencyChinese gold market volatility on volatility of comparableUS macroeconomic variables Given the global influenceof the US economy we wanted to test whether the sameset of macroeconomic variables from the US are able toexplain the low-frequency volatility in Chinarsquos gold futuresmarket Thus to specify (12) in terms of US macroeconomicvariable volatilities we simply replace all of our Chinesemacroeconomic volatility variablesmdashexcept for the ChineseConsumer Price Index Volatility (CPIVOL)mdashwith variablesfrom the US We relabel our US variable volatilities asUS Industrial Production Volatility (USIPVOL) US Inter-bank Lending Rate Volatility (USIRVOL) US Money Sup-ply Volatility (USM2VOL) and US Consumer ConfidenceVolatility (USCCIVOL) First with respect to our Chinesemacroeconomic results the coefficients of CPIVOL andFXVOL are positive and statistically significant at 1 levelHowever all other coefficients are insignificant Also we findno evidence of seasonality in Chinarsquos gold futures marketOur macroeconomic variables explain 63 of variation inlow-frequency Chinese gold market volatility In sum low-frequency volatility in Chinarsquos gold futures market is mainly

driven by volatility in CPIVOL and FXVOL This may beexplained by the fact that gold is commonly used by investorsas an inflation hedge Specifically investorsrsquo demand for goldis higher when they expect higher inflation in the future andthis in turn results in higher levels of low-frequency goldmarket volatilityMoreover gold price is quoted and traded inUS dollars Generally speaking gold price rises when the USdollar weakensThis explains why low-frequency volatility ofgold covaries closely with volatility in the US dollar Basedon our findings CPIVOL and the US dollar can be used assignals of the potential risk in gold futures market Investorsin Chinarsquos gold futures market should pay close attention tomovements in CPIVOL and the US dollar

Turning to our US macroeconomic results presented inthe right half of Table 2 we again find that volatility coef-ficients with respect to CPIVOL and FXVOL are again sta-tistically significant at 99 confidence levelmdashconsistent withour Chinesemacroeconomic variable specificationHoweverother US macroeconomic variablesmdashwith the exception ofCCIVOLmdashare also statistically significant at 90 confidencelevel We find that an increase in US interest rate volatilityslightly lowers the low-frequency volatility in Chinarsquos goldfutures market In addition an increase in Industrial Pro-duction and M2 volatility in the US increases low-frequencyvolatility in Chinarsquos gold futures market by a small amount

Combining the results from both regressions our pre-liminary finding is that industrial production and monetarypolicy in the US have an impact on Chinarsquos gold futuresmarket by affecting its low-frequency volatility Converselythere is no evidence that Chinarsquos own industrial outputand monetary policy directly influence its own gold futuresmarket A plausible explanation is that gold price is largelyaffected by the US dollar which is determined mainly by USFederal Reserversquos monetary policies Industrial Productionis a key signal of the macrotrend in industrial output


and economic development In addition we would arguethat economic conditions may affect the demand for goldthrough peoplersquos expectation about future economic growthand wealth Therefore it is not surprising that IndustrialProduction in the US could affect low-frequency volatility inChinarsquos gold futures market

However an interesting question arising from our resultsis why is there no evidence of Chinarsquos Industrial Productionaffecting Chinarsquos gold market One possible explanationcould be that US (international) gold market volatilityspillovers dominate Chinarsquos gold futures market Along theselines because of the dominant role played by the US in globalfinancial markets US macroeconomic variables are closelywatched by gold traders all over the world It is plausible thatChinese investors in Chinese gold futures market also lookto the US economy when making their tradinginvestmentdecisions

33 Testing Excess Volatility in Chinarsquos Gold Futures Mar-ket Our regression results show that the variance of low-frequency volatility is largely explained by the selectedmacroeconomic variables This finding lends support tothe claim that low-frequency volatility mirrors changesin macroeconomic fundamentals However the questionremains as to whether changes in macroeconomic funda-mentals associated with low-frequency volatility explain totalgold market volatility Recall that if high-frequency volatilitydominates low-frequency volatility then this would suggestthat Chinese gold futuresmarket exhibits ldquoexcess volatilityrdquomdashthe component of overall volatility that cannot be explainedby fundamentals Of course we acknowledge that our linearregression model is likely not a complete description of allmarket fundamentals pertinent to the gold market With thisin mind we turn to our ldquoexcess volatilityrdquo results

We estimate 119862 from (13)mdashthe ratio of low-frequencyvolatility to total volatilitymdashto be 07802 This shows thatmacroeconomic fundamentals account for 7802 of theoverall volatility in Chinarsquos gold futures market This in turnillustrates that there is a considerable proportion of volatilitythat is not due to changes in fundamentals a key signal ofexcess volatility

We also calculate the same 119862 ratio with respect toLondonrsquos gold market In this case we find that fundamentaldriven low-frequency volatility accounts for 93 of overallvolatility considerably higher than its Chinese counterpartWe tentatively conjecture that there are probably higher levelsof short-term speculation and irrationalitymdashin the classicalEMHsensemdashamong traders inChinarsquos gold futures given thatLondon is a more maturedeveloped market However it isimportant to qualify our results by acknowledging that not allinfluences on tradersrsquo activities (eg political risk) can be rea-sonably quantified We surmise that behavioural factors mayplay an important role in determining volatility in Chinarsquosgoldmarketmdasha hypothesis that we turn to in the next section

34 Behavioural Explanation of Excess Volatility FollowingBarberis et al [23] we set 120578 in (14) to 09 and estimate amodified long-futures returns time series 119911 which measures

Table 3 Asymmetric impact of investment gainloss on excessivevolatility in Chinarsquos gold futures market

Variable Coefficients SE 119905-statistics 119901 value119888 088472 0016013 5524815 00000V1

3102509 1491269 2080448 00000V2

minus1762709 1840754 minus957601 00000Adjusted 119877

2 023440 119865-statistic 2183726lowastlowastlowast

Note lowast lowast lowast denotes significance at the 1 level SE is standard error ofestimated coefficients

the investment gain or loss against a historical benchmarkWe then regress the high-frequency volatility component 119892

119905

on 119911 using the specification in (15)Our results are reported in Table 3 As seen from Table 3

all the parameters are statistically significant at 1 level Wefind that both V

1and V

2are significant showing that excess

volatility in the market results from prior gains and losses toa long-futures position Given the nature of futures marketsan alternative way of looking at this is to say that prior lossesto both long- and short-futures positions impact gold futuresvolatility However our Waldrsquos test results indicate that thereis an asymmetric impact with prior losses to long-futurespositions having a larger effect on excess gold volatility thanprior losses to short-futures positions |V

1| gt |V

2| This

is consistent with our assumption that long-futures traders(predominantly long-term institutional investment firms) aremore sensitive to losses than short-traders According to ourresults investorsrsquo prior investment return is able to explain234 of the variance in excess volatility (adjusted 119877-squarereported in Table 3)

4 Conclusions

This study uses the ASP-GARCH model to extract bothlow- and high-frequency volatility from Chinarsquos gold futuresmarket Our low-frequency volatility measures are regressedon a list of selected macroeconomic variables to determinethe extent to which the market follows ldquotraditionalrdquo rationaleconomic theory We also test for and find significant evi-dence of excess volatility in the market which we explain inthe light of behavioural finance theory

Our main conclusions can be summarized as followsFirst we find excess volatility in Chinarsquos gold futures marketwhich cannot purely be explained in terms of market fun-damentals In comparison to Londonrsquos long established goldmarket Chinarsquos gold futures market is more susceptible tospeculation

Second loss aversion is an important factor contributingto excess volatility Investorsrsquo prior investment performanceleads to changes in the degree of loss aversion which exertsa great impact on investorsrsquo following decisions Moreoverinvestment loss from the perspective of a long futures traderhas a greater impact on excess volatility than does investmentgain

Third volatility in Chinarsquos gold futures market resultsfrom both fundamentals and short-term speculativebehaviour With respect to fundamentals Chinarsquos domestic


Consumer Price Index Volatility (CPIVOL) and US dollarvolatility are the major movers of the gold market whereasChinarsquos Industrial Production interest rate and M2volatilities do not have a significant impact We also find thatUS Industrial Production interest rates and M2 volatilitiesare significant factors in explaining volatility in Chinarsquosgold futures market We argue that such a phenomenonimplies that Chinese gold futures price movements areinfluenced by the changes in US fundamentals On a finalnote we acknowledge that likely covariance between themacroeconomic variables qualifies the extent to which eachvariable should be considered as truly independent driversof Chinese gold futures volatility

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is supported by China Scholarship Foundation(201208440325)

References

[1] J Farchy and D McCrum ldquoGold hit by sharpest tumble in 30yearsrdquo Financial Times 2013

[2] B GMalkiel and E F Fama ldquoEfficient capital markets a reviewof theory and empirical workrdquo The Journal of Finance vol 25pp 383ndash417 1970

[3] W Wang H Bu and F Lu ldquoAn empirical study on volatility inChinarsquos gold futuresmarket under financial crisisrdquoManagementReview vol 2 pp 77ndash83 2009

[4] E Tully and B M Lucey ldquoA power GARCH examination of thegold marketrdquo Research in International Business and Financevol 21 no 2 pp 316ndash325 2007

[5] J A Batten C Ciner and B M Lucey ldquoThe macroeconomicdeterminants of volatility in preciousmetalsmarketsrdquoResourcesPolicy vol 35 no 2 pp 65ndash71 2010

[6] R Christie-David M Chaudhry and T Koch ldquoDo macroeco-nomic news releases affect gold and silver pricesrdquo Journal ofEconomics and Business vol 52 pp 405ndash421 2000

[7] J Cai Y-L Cheung and M C S Wong ldquoWhat moves the goldmarketrdquo Journal of Futures Markets vol 21 no 3 pp 257ndash2782001

[8] R F Engle and J G Rangel ldquoThe spline-GARCH model forlow-frequency volatility and its global macroeconomic causesrdquoReview of Financial Studies vol 21 no 3 pp 1187ndash1222 2008

[9] J G Rangel and R F Engle ldquoThe factor-spline-GARCHmodelfor high and low frequency correlationsrdquo Journal of Business ampEconomic Statistics vol 30 no 1 pp 109ndash124 2012

[10] B Karali and G J Power ldquoShort- and long-run determinantsof commodity price volatilityrdquo American Journal of AgriculturalEconomics vol 95 no 3 pp 724ndash738 2013

[11] S F LeRoy and R D Porter ldquoThe present-value relation testsbased on implied variance boundsrdquo Econometrica vol 49 no 3pp 555ndash574 1981

[12] R J Shiller ldquoDo stock prices move too much to be justifiedby subsequent changes in dividendsrdquoThe American EconomicReview vol 71 no 3 pp 421ndash436 1981

[13] J B De Long and M Becht ldquoExcess volatility and the Germanstock market 1876ndash1990rdquo NBER Working Papers no 4054National Bureau of Economic Research 1992

[14] J Y Campbell and J H Cochrane ldquoBy force of habit aconsumption-based explanation of aggregate stock marketbehaviorrdquo Journal of Political Economy vol 107 no 2 pp 205ndash251 1999

[15] J He and Y Huo ldquoInvestor behavior asset price and stockmarket volatilityrdquo Nankai Economic Studies vol 2 pp 62ndash672004

[16] C Xu and H Song ldquoExcess volatility in Chinarsquos closed-endfundsrdquo Economic Research Journal vol 3 pp 33ndash44 2005

[17] J Xu ldquoExcess volatility in Chinarsquos stock-a marketrdquo Journal ofFinancial Research vol 8 pp 94ndash111 2010

[18] H ZhouWWu andY Zhou ldquoInvestor sentiment and volatilityin Chinarsquos stock marketrdquo Shanghai Economic Review vol 4 pp3ndash13 2012

[19] J Bao and J Pan ldquoBond illiquidity and excess volatilityrdquo Reviewof Financial Studies vol 26 no 12 pp 3068ndash3103 2013

[20] W F de Bondt and R Thaler ldquoDoes the stock market overre-actrdquoThe Journal of Finance vol 40 no 3 pp 793ndash805 1985

[21] J Pontiff ldquoExcess volatility and closed-end fundsrdquo The Ameri-can Economic Review vol 87 no 1 pp 155ndash169 1997

[22] S Lin and Q Yu ldquoLimited rationality animal spirit and marketcollapse an experimental study on investor sentiment andtrading behaviorrdquo Economic Research Journal vol 8 pp 115ndash1272010

[23] N Barberis M Huang and T Santos ldquoProspect theory andasset pricesrdquo Quarterly Journal of Economics vol 116 no 1 pp1ndash53 2001

[24] Y Wang and R Hua ldquoInvestor behavior and futures marketvolatility based on OLG model and high-frequency datardquoChinese Journal of Management Science vol 1 pp 91ndash101 2012

[25] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986

[26] L RGlosten R Jagannathan andD E Runkle ldquoOn the relationbetween the expected value and the volatility of the nominalexcess return on stocksrdquo The Journal of Finance vol 48 no 5pp 1779ndash1801 1993

[27] S H Irwin and D R Sanders ldquoIndex funds financializationand commodity futuresmarketsrdquoApplied Economic Perspectivesand Policy vol 33 no 1 pp 1ndash31 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Macroeconomic variables are generally compiled monthlyor even quarterly while gold prices are reported daily orintradaily Incorporating variables with different frequenciesin the same model creates econometric modeling difficulties

To address these issues Engle and Rangel [8] and Rangeland Engle [9] developed Spline-GARCH and an asymmet-ric Spline-GARCH (ASP-GARCH) model which relax theassumption that unconditional volatility is constantThey usea quadratic spline to isolate the proportion of variation indaily price data that is plausibly caused by macroeconomicvariablesThen using the isolated variation Engle andRangel[8] construct a measure of low-frequency volatility in thesame sampling frequency as macroeconomic data whicheffectively bridges high-frequency commodity price withits low-frequency macroeconomic determinants Karali andPower [10] followed Engle and Rangel [8] and decomposeddaily price volatility into high- and low-frequency compo-nents They found that low-frequency volatility in US goldfutures market responds strongly to changes in the industrialproduction index consumer price index and the US dollarHowever to our knowledge this modelling approach has notbeen used to study factors affecting volatility in Chinarsquos goldfutures market which is the primary objective of this paper

Although much research efforts have been devoted tostudying the macroeconomic determinants of gold pricevolatility it remains unclear whether these macroeconomicdeterminants sufficiently explain the high levels of observedgold market volatility In this paper we examine whetherChinese gold futures market exhibits ldquoexcess volatilityrdquo aterm first coined by LeRoy and Porter [11] and Shiller [12]and attributed to the portion of an assetrsquos volatility thatcannot be explained by fundamentals Evidence of excessvolatility poses a challenge to the EfficientMarket Hypothesis(EMH)mdashthe cornerstone of traditional finance theorymdashfirst outlined in Malkiel and Fama [2] A growing body ofthe literature spanning the last 40 years has accumulatedsignificant empirical evidence of excess volatility in a varietyof financial markets For example De Long and Becht [13]reported excess volatility in German stock market afterthe Second World War Similarly Campbell and Cochrane[14] showed that US stock market volatility is far greaterthan dividend volatility This result indicates that contraryto the EMH movements in dividends cannot be the soledetermining factor of stock market volatility There is alsoconsiderable evidence of excess volatility in Chinese stockand bond markets (eg [15ndash19]) Our paper extends thisliterature by shedding light on whether excess volatility is aprevalent feature of Chinese gold futures market

The EMH and ldquoHomo Economicus Assumptionrdquo fail toprovide us with a widely accepted explanation of excessvolatility in financial market Proponents of behaviouralfinance would argue that classical theories fail because peopleare not rational in the classical economic sense A numberof studies grounded in behavioural finance theorymdashwheremarket actors are assumed to be susceptible to cognitivebiasesmdashhave been relatively successful in explaining excessvolatility in stock markets For example investor sentimenthas been shown to be a good explanation of excess volatility[18 20ndash22] Drawing fromKahneman and Tverskyrsquos prospect

theory Barberis et al [23] found that stock market investorsare loss-aversive and their utility is affected by their previousinvestment returns As a result this form of loss aversionchanges investorsrsquo required risk premium and hence impactsmarket volatility In a similar vein a recent study byWang andHua [24] found that volatility in Chinarsquos copper futures mar-kets correlates with investorsrsquo loss aversion which supportsthe claim that investorsrsquo behaviour affects futures marketsvolatility

In this paper we use an ASP-GARCH model to extractlow-frequency volatility in Chinarsquos gold futures market andits response to changes in macroeconomic variables We findevidence of excess volatility in the market which we explainin a behavioural framework following Barberis et alrsquos [23] lossaversion modelling approach Our main contributions are asfollows (1) we present the first empirical study to extractlow-frequency volatility in Chinarsquos gold futures market (2)

we are able to explain a portion of low-frequency volatilityin terms of macroeconomic news and (3) we find significantevidence of excess volatility in Chinarsquos gold futures marketwhich we at least in part explain in terms of investor lossaversive behaviour

2 Modelling Approach andHypotheses Testing

21 ASP-GARCH Model The assumption underlying almostall traditional GARCH models is that volatility is meanreverting to a constant level and unconditional volatility isconstant However the mean reverting assumption has beengreatly challenged by observation of real market movementsFor example it is widely recognized that volatility is higherduring recessions and following ldquonewsrdquo announcements [8]Traditional GARCH model does not adequately capture thelow-frequency changes in unconditional volatility whereas itis generally useful in modelling high-frequency conditionalvolatility To address this issue Engle and Rangel [8] proposeda semiparametric Spline-GARCH model which relaxes theassumption that volatility is mean reverting to a constantlevel To better understand how the Spline-GARCH modelworks it is helpful for us to briefly review the traditionalGARCHmodel Bollerslev [25] proposed theGARCHmodel

119903119905

minus 119864119905minus1119903119905

= radicℎ119905120576119905 (1)

ℎ119905

= 120603 + 1205721205762119905minus1 + 120573ℎ

119905minus1 (2)

120576119905

| 119868119905minus1 sim 119873 (0 1) (3)

where 119903119905is the investment return at time 119905 the expectation

119864119905minus1 is conditional on an information set 119868

119905minus1at time 119905 minus 1 120576

119905

is the innovation term assumed to be distributed with mean0 and variance 1 again conditioned upon the information set119868119905minus1

at time 119905minus1 ℎ119905denotes the conditional variance of returns

for period 119905 and ℎ119905is a function of past errors and squared

errors and variances observed at time 119905minus1The terms120603120572 and120573 are the estimated coefficients on these conditional varianceARCH andGARCH terms Equation (1) is themean equationand (2) is the conditional variance equation



ℎ119905

= 1205902

+ 120572 (1205762119905minus1 minus 120590

2) + 120573 (ℎ

119905minus1 minus 1205902) (4)





119903119905

= radic119892119905120591119905120576119905 (5)

ℎ119905

= 119892119905120591119905 (6)

119892119905


120591119905minus1

) + 120573119892119905minus1 (7)

120591119905

= 119888 exp(1199080119905 +

119896

sum119894=1

119908119894((119905 minus 119905

119894minus1)+

)2) (8)





119905is




119896

119894=1 119908119894((119905 minus 119905

119894minus1)+







2119905) = 119864(119892

119905)120591119905







119892119905


) + 120572 (1199032119905minus1

120591119905minus1

) + V(1199032119905minus1

120591119905minus1

) 119868119903119905minus1lt0

+ 120573119892119905minus1

(9)


119903119905minus1lt0



119905 which can be






LV119898

= radic1

119873119898

119873119898

sum119889=1

120591119889119898

(10)

where 120591119889119898



119871 = minus12

119879

sum119905=1

(log (120591119905119892119905) +

1199032119905

120591119905119892119905

) (11)


LV119898

= 119886 +

119899

sum119894=1

119887119894119911119894119898

+ V1DF119898 + V2DW119898 + V3DS119898

+ 119890119896119898

(12)


market 119911119894119898





119898 DW

119898 and








119894



119862 =Var [log (120591

119905)]

Var [log (119892119905120591119905)]

(13)






119905= 120595119905119878119905 where 120595









= 119878119905) 120595119905varies


119905

119911119905+1 = 120578 (119911

119905

119877

119877119905+1

) + (1minus 120578) 120578 isin (0 1) (14)












119892119905

= 119888 + V11199111015840

119905minus1119868+

119905minus1 + V21199111015840


119905minus1 + 120576119905 (15)




1015840







00000

00002

00004

00006

00008

00010

2008 2009 2010 2011 2012 2013

LVOLHVOL

Vola

tility

leve

l

Year



119877119905

= 100times ln(119875119905

119875119905minus1

) (16)


119905is the daily




119877119905

= 120583 + 119903119905 (17)










1199080


28767119890 minus 003

1199081


1199082


1199083


74997119890 minus 006

1199084


69961119890 minus 006

1199085


1199086


1199087


35234119890 minus 005

1199088


1199089


11990810


V 00227 00294


AIC minus78510



119898

































3102509 1491269 2080448 00000V2





119905



1and V



1| gt |V

2| This


4 Conclusions









Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ119905

= 1205902

+ 120572 (1205762119905minus1 minus 120590

2) + 120573 (ℎ

119905minus1 minus 1205902) (4)





119903119905

= radic119892119905120591119905120576119905 (5)

ℎ119905

= 119892119905120591119905 (6)

119892119905


120591119905minus1

) + 120573119892119905minus1 (7)

120591119905

= 119888 exp(1199080119905 +

119896

sum119894=1

119908119894((119905 minus 119905

119894minus1)+

)2) (8)





119905is




119896

119894=1 119908119894((119905 minus 119905

119894minus1)+







2119905) = 119864(119892

119905)120591119905







119892119905


) + 120572 (1199032119905minus1

120591119905minus1

) + V(1199032119905minus1

120591119905minus1

) 119868119903119905minus1lt0

+ 120573119892119905minus1

(9)


119903119905minus1lt0



119905 which can be






LV119898

= radic1

119873119898

119873119898

sum119889=1

120591119889119898

(10)

where 120591119889119898



119871 = minus12

119879

sum119905=1

(log (120591119905119892119905) +

1199032119905

120591119905119892119905

) (11)


LV119898

= 119886 +

119899

sum119894=1

119887119894119911119894119898

+ V1DF119898 + V2DW119898 + V3DS119898

+ 119890119896119898

(12)


market 119911119894119898





119898 DW

119898 and








119894



119862 =Var [log (120591

119905)]

Var [log (119892119905120591119905)]

(13)






119905= 120595119905119878119905 where 120595









= 119878119905) 120595119905varies


119905

119911119905+1 = 120578 (119911

119905

119877

119877119905+1

) + (1minus 120578) 120578 isin (0 1) (14)












119892119905

= 119888 + V11199111015840

119905minus1119868+

119905minus1 + V21199111015840


119905minus1 + 120576119905 (15)




1015840







00000

00002

00004

00006

00008

00010

2008 2009 2010 2011 2012 2013

LVOLHVOL

Vola

tility

leve

l

Year



119877119905

= 100times ln(119875119905

119875119905minus1

) (16)


119905is the daily




119877119905

= 120583 + 119903119905 (17)










1199080


28767119890 minus 003

1199081


1199082


1199083


74997119890 minus 006

1199084


69961119890 minus 006

1199085


1199086


1199087


35234119890 minus 005

1199088


1199089


11990810


V 00227 00294


AIC minus78510



119898

































3102509 1491269 2080448 00000V2





119905



1and V



1| gt |V

2| This


4 Conclusions









Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898 DW

119898 and








119894



119862 =Var [log (120591

119905)]

Var [log (119892119905120591119905)]

(13)






119905= 120595119905119878119905 where 120595









= 119878119905) 120595119905varies


119905

119911119905+1 = 120578 (119911

119905

119877

119877119905+1

) + (1minus 120578) 120578 isin (0 1) (14)












119892119905

= 119888 + V11199111015840

119905minus1119868+

119905minus1 + V21199111015840


119905minus1 + 120576119905 (15)




1015840







00000

00002

00004

00006

00008

00010

2008 2009 2010 2011 2012 2013

LVOLHVOL

Vola

tility

leve

l

Year



119877119905

= 100times ln(119875119905

119875119905minus1

) (16)


119905is the daily




119877119905

= 120583 + 119903119905 (17)










1199080


28767119890 minus 003

1199081


1199082


1199083


74997119890 minus 006

1199084


69961119890 minus 006

1199085


1199086


1199087


35234119890 minus 005

1199088


1199089


11990810


V 00227 00294


AIC minus78510



119898

































3102509 1491269 2080448 00000V2





119905



1and V



1| gt |V

2| This


4 Conclusions









Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00000

00002

00004

00006

00008

00010

2008 2009 2010 2011 2012 2013

LVOLHVOL

Vola

tility

leve

l

Year



119877119905

= 100times ln(119875119905

119875119905minus1

) (16)


119905is the daily




119877119905

= 120583 + 119903119905 (17)










1199080


28767119890 minus 003

1199081


1199082


1199083


74997119890 minus 006

1199084


69961119890 minus 006

1199085


1199086


1199087


35234119890 minus 005

1199088


1199089


11990810


V 00227 00294


AIC minus78510



119898

































3102509 1491269 2080448 00000V2





119905



1and V



1| gt |V

2| This


4 Conclusions









Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































3102509 1491269 2080448 00000V2





119905



1and V



1| gt |V

2| This


4 Conclusions









Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article low-frequency volatility in china s gold

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