Forecasting Volatility of Gold Price Using Markov Regime Switching and Trading Strategy

In this paper, we forecast the volatility of gold prices using Markov Regime Switching GARCH (MRS-GARCH) models. These models allow volatility to have different dynamics according to unobserved regime variables. The main purpose of this paper is to find out whether MRS-GARCH models are an improvement on the GARCH type models in terms of modeling and forecasting gold price volatility. The MRS-GARCH is best performance model for gold price volatility in some loss function. Moreover, we forecast closing prices of gold price to trade future contract. MRS-GARCH got the most cumulative return same GJR model.


Introduction
The characteristic that all financial markets have in common is uncertainty, which is related to their short and longterm price state.This feature is undesirable for the investor but it is also unavoidable whenever the financial market is selected as the investment tool.The best that one can do is to try to reduce this uncertainty.Financial market forecasting (or Prediction) is one of the instruments in this process.
The financial market forecasting task divides researchers and academics into two groups: those who believe that we can devise mechanisms to predict the market and those who believe that the market is efficient and whenever new information comes up the market absorbs it by correcting itself, thus there is no space for prediction.Furthermore they believe that the financial market follows a random walk, which implies that the best prediction you can have about tomorrow's value is today's value.
In time series, a financial price transformated to log return series for stationary process which look like white noise.Mehmet [1] said financial returns have three characteristics.First is volatility clustering that means large changes tend to be followed by large changes and small changes tend to be followed by small changes.Second is fat tailedness (excess kurtosis) which means that financial returns often display a fatter tail than a standard normal distribution and the third is leverage effect which means that negative returns result in higher volatility than positive returns of the same size.
The generalized autoregressive conditional heteroskedasticity (GARCH) models mainly capture three characteristics of financial returns.The development of GARCH type models was started by Engle [2].Engle introduced ARCH to model the heteroskedasticity by relating the conditional variance of the disturbance term to the linear combination of the squared disturbances in the recent past.Bollerslev [3] generalized the ARCH (GARCH) model by modeling the conditional variance to depend on its lagged values as well as squared lagged values of disturbance.The Exponential GARCH (EGARCH) model proposed by Nelson [4] to cope with the skewness of ten encountered in financial returns, led to GJR-GARCH which was introduced independently by Glosten, Aganathan, and Runkle [5] to account for the leverage effect.
Hamilton and Susmel [6] stated that the spurious high persistence problem in GARCH type models can be solved by combining the Markov Regime Switching (MRS) model with ARCH models (SWARCH).The idea behind regime switching models is that as market conditions change, the factors that influence volatility also change.Nowaday some researchers have development of GARCH model, as well as the benefit of using GARCH model [1, [7][8][9].
Gold is a precious metal which is also classed as a commodity and a monetary asset.Gold has acted as a multifaceted metal through the centuries, possessing similar characteristics to money in that it acts as a store of wealth, a medium of exchange and a unit of value.Gold has also played an important role as a precious metal with sig-nificant portfolio diversification properties.Gold is used in industrial components, jewellery and as an investment asset.The quantity of gold required is determined by the quantity demanded for industry investment and jewellery use.Therefore an increase in the quantity demanded by the industry will lead to an increase in the price of the metal.
The changing price of gold can also be the result of a change in the Central Bank's holding of these precious metals.In addition, changes in the rate of inflation, currency markets, political harmony, equity markets, and producer and supplier hedging, all affect the price equilibrium.
Gold futures is an alternative investment tool which relies on the gold price movement.The investors can benefit from the gold futures investment by making profit from both directions, either up or down, which is like stock index futures trading.In addition, gold futures can also hedge against gold price fluctuations or stock market volatility due to the negative correlation to the stock market.This will provide a greater opportunity to make profit when the stock market declines during an economic downturn.
Gold futures in Thailand are futures contracts which rely on gold bullion with a purity of 96.5% due its popularity among buyers nationwide for gold physical trading.Gold futures trade in implement cash settlement method with no need of physical delivery.
Edel Tully, et al. [10] has investigated the Asymmmetric power GARCH model has to capture the dynamics of the gold market.Results suggest that the APGARCH model provides the most adequate description for the gold price.
In this paper, we use GARCH, EGARCH, GJR-GARCH and MRS-GARCH models to forecast the volatility of gold prices and to compare their performance.Moreover we shall use this estimated volatility to forecast the closing price of gold.Finally, we apply the forecasting price of gold to trading in gold future contracts with a maturity date of August 2011 (GF10Q11).
In the next section, we present the MRS-GARCH model.Estimation and in-sample evaluation results are given in Section 3. In Section 4, statistical loss functions are described and out-of-sample forecasting performance of various models is discussed.In Section 5 we apply the forecasting price to the gold price for trading in future contracts.The conclusion is given in Section 6.

Markov Regime Switching of GARCH Model
The GARCH (1,1) model for the series of the returns can be written as where 0 1 1 0, 0 and 0 The parameters of the GARCH model are generally considered as constants.But the movement of financial returns between recession and expansion is different, and may result in differences in volatility.Gray [11] extended the GARCH model to the MRS-GARCH model in order to capture regime changes in volatility with unobservable state variables.It was assumed that those unobservable state variables satisfy the first order Markov Chain process.
The MRS-GARCH model with only two regimes can be represented as follows:

Forecasting Volatility
In MRS-GARCH model with two regimes, Klaassen [12] forecast volatility for k-step-ahead by using the recursive method as in the standard GARCH model where is a k positive integer.In order to compute the multi-step-ahead volatility forecasts, we firstly compute a weighted average of the multi-step-ahead volatility forecasts in each regime where the weights are the prediction probability ).Since there is no serial correlation in the returns, the k-step-ahead volatility forecast at time T depends on information at time T − 1.Let  , T T k h  denote the time T aggregated volatility forecasts for the next k steps.It can be calculated as follows: (See, for example Marcucci [9], page 8) where indicates the  -step-ahead volatileity forecast in regime i made at time T and can be calculated recursively as follows: Also, in generally the prediction probability in ( 5) is computed as , where P defined in (4) and

 
1 1 Pr will be discussed in (11).Lastly, we compute expectation part , where Similarly, we computed in the second term of right hand side in (7) such that substitutes both ( 8) and ( 10) to (7) such that In the next step, we will compute those regime probabilities in (9).Note that when lities are based on information up to time t, we describe this as filtered probability ( the regime probabi   compute the regime probabilities, we deno In order to te We shall compute reg istributions f ime probabilities recursively by following two steps (Kim and Nelson, [13], page 63): Step 1: Given

 
Pr S j F  at the end of the ti Since the current regime ) only depends on the regi ( t S en me one period ago ( 1 Step 2: At the end of the time t, when observed return at is joint density of returns and rved regime at s unobse tate i for 1, 2 i  variables can be written as follows: ted as follows: Pr .
Then, all regime probabilities can be computed by iterating these two steps.Pr S i F  .These are given by n and conditio r ance g-likeli ursively similar to that in

Forecasting Price
We forecast financial price at k-step-ahead with MRS-GARCH models.Denote is k-step-ahead foreca Given initial values for regime probabilities, conditional mea nal va i in each regime, the parameters of the MRS-GARCH model can be obtained by maximizing numerically the log-likelihood function.The lo hood function is constructed rec GARCH models , ˆt t k r  sting logarithm return of financial price at time t depend on 1 t F  .We compute as: where Forecasting financial price one-step-ahead, we use ( 12) and (1) combine in log-return of financial price is

3.
prices of gold price 1

Data
The data set used in this study is the daily closed pe not constant over time and exhibits volatility clustering with large changes in indices often followed by large changes, and small changes often followed by small changes.
Descriptive statistics of t r are represented in Table 1.As Table 1 shows, t r has a positive average return of 0.074%.The daily standard deviation is 1.537%.The series also displays a negative skewness of −0.102 and an excess kurtosis of 9.457.These values indicate that the returns are not normally distributed, namely it has fatter tails because skewness does not equal zero and is greater than 3. Also, the Jarque-Bera test 1 statistic of 2107.620confirms the non-normality of t r .And the

Empirical Methodol
This empirical part adopts GARCH type and MRS-ARCH (1,1) models to estimate the volatility of the t P .GARCH type models that will be considered as GARCH (1,1), EGARCH(1,1) and GJR-GARCH (1,1).In order to account for the fat tails feature of financial returns, we consider three different distributions for the innovations: Normal (N), Student-t (t) and Generalized Error Distributions (GED).

GARCH Type Models Table 3 presents an est
of the results for GARCH type models.It is clear from the table that almost all parameter estimates including  in GARCH type models wever, the asymmetry are highly significant at 1%. Ho effect term  in EGARCH models is significantly dif- where  is the asymmetry param pture leverage effect.eter to ca 6 one whe is greater than zero and dicates u pected n tive returns implying higher conditional variance as compared o same size positive returns.All models display strong zero at least at 95% confidence level.But 0 t persistence in volatility ranging from 0.9654 to 0.9741 unless EGARCH models are very low, that is, volatility is likely to remain high over several price periods once it increases.

Markov Regime Switching GARCH Models
Estimation results and summary statistics of MRS-GARCH models are mates in RS-GAR are sig cantly d rent from  and 1  are insignificant in some states.All models display strong n esian Information Criteria s of M persistence in volatility, that is, volatility is likely to remain high over several price periods once it increases.

In-Sample Evaluation
We use various goodness-of-fit statistics to compare volatility models.These statistics are Akaike Informatio Criteria (AIC) Schwarz Bay (SBIC) and Log-likelihood (LOGL) values.In Table 5, MRS-GARCH models.

Forecasting Volatility in Out-of-Sample
In this section, we investigate the ability of MRS-GARCH and GARCH type models to forecast volatility of Gold prices from out-of-sample.
In Table 6, we present the result of loss function of out-of-sample with forecasting volatility for one day ahead, and we found the EGARCH and MRS-GARCH models perform best.

Trading Future Contract with Forecast Volatility a
The aim apply of thi g differ s study is t ent evaluate t o the v he profitability of tility o in models ola f gold prices.We assumed the market is a perfect market and the p si scribed below.
price with maturity date at August 2011) to trade in o contract with day by day and we did not include settlement, return do not include cost price i.e. margin, fee charged.The net daily rate of return for long position is computed as follows: The net daily rate of return on close position is computed as follows: Table 7 shows that the cumulative of return with Markov Regime Switching the GARCH-N model and the JR-N model give cumulative of return more than the G 7 Loss functions: Copyright © 2012 SciRes.JMF ) and GJR-GARCH (1,1) in terms of MRS-GARCH (1,1) (1,1), GJR-GARCH (1,1) models.All models are estimated under three distributional assumptions which are Nor-mal, Student-t and GED.Moreover, Student-t distribution which takes different degrees of freedom in each regime is considered for MRS-GARCH models.
We first analyze in-sample performance of various volatility models to determine the best form of the volatility model over the period 4/01/2007 through 30/08/2011.As expected, volatility is not constant over time and exhibits volatility clustering showing large changes in the price of an asset often followed by large changes, and small changes often followed by small changes.
Descriptive statistics of return series are represented by returns with fatter tails.The Augmented Dickey-Fuller test indicates gold price log returns are stationary.Serial correlation in the gold price confirms it is nonstationary but serial log returns of gold price are stationary.Serial correlation in the squared returns suggests conditional heteroskedasticity.This empirical part adopts GARCH type and MRS-GARCH models to estimate the volatility of the gold price.In order to account for fat tailed features of financial returns, we consider three different distributions for the innovations.Almost all parameter estimates in GARCH type models are highly significant at 1%.Most parameter estimates in MRS-GARCH are significantly different from zero at least at 95% confidence level.However, the results of goodness-of-fit statistics and loss functions for all volatility models show different results.
The trading details we have used describe forecasts of closed price of gold price between 1/08/2011-30/08/2011 and trading in gold future contract (GF10Q11).We found the cumulative returns with the Markov Regime Switching GARCH-N (MRS-GARCH-N) model and the GJR-N model give us higher cumulative returns than the other models when we use m = 30.
For further study, three or four volatility regimes setting can be considered rather than two-volatility regimes.Also, using Markov Regime Switching with other volatility models e rmance of MR -GAR H mo els ca be h ged in ture fo ong an short osition .
Figure 1.Graph of p losed ices ( log returns series (r t the p d 08/2011.
Augmented Dickey-Fuller test 2 of −35.873 indicates that is stationary.The autocorrelation functions (ACF) test the significance level of autocorrelation in Table 2, when we apply Ljung and Box Q-test.The null hypothesis of the test is that there is no serial correlation in the return series up to the specified lag.

Table 4 . Summary result
22 and P-value for LBQ test in parentheses.Std.err is standard error.