Application of ARMA and GARCH models to the daily gold and silver exchange prices in US dollar

Table of content

List of illustrations

Abstract

1.Introduction

2.Data

3.Methodology
3.1. Properties of an ARMA model
3.2. Properties of a GARCH model

4.Data Analysis

5.Examining gold close data
5.1. Gold log - returns
5.2. ACF and PACF of the gold log-returns
5.3. Applying ARMA-models on gold log-returns
5.4. Testing the residuals of the gold log – returns
5.5. Normality assumption of the residuals of the gold log - returns
5.6. Heteroscedasticity of the gold log - returns
5.7. Using GARCH to model the time series of the gold log - returns
5.8. Testing for skewed t-distribution of the residuals in a GARCH - model

6.Examining the silver close data
6.1. Silver log-returns & ACF/PACF of the silver return data
6.2. Testing ARMA models on the silver log-returns
6.3. Independence assumption of the residuals of the silver log-return data
6.4. Normality assumption of the residuals of the silver log-return data
6.5. Heteroscedasticity of the residuals of the silver log-return data
6.6. Examining GARCH - models on the silver log-return data
6.7. Testing skewed t-distribution and independence assumption of residuals of the silver log – return data

7. Forecasting gold and silver returns

8. Drawbacks of the research

9. Conclusion

10. References

Appendix 1: Testing of models - AR (1); MA (1); ARMA (1,1) for gold log-returns

Appendix 2: Overfitting approach with MA (2); MA (3) for gold log-returns

Appendix 3: Testing of models - AR (1); MA (1); ARMA (1,1) for silver log-returns

Appendix 4: Overfitting approach with MA (2); MA (3) for gold log-returns

Appendix 5: fitted GARCH(1,1)-model skewed t-distributed with mean for silver returns

Appendix 6: Attempt of forecasting the gold and silver returns

Appendix 7: R codes used to examine the gold return data

Appendix 8: R codes used to examine the gold return data

List of illustrations

Figure 1: Daily close prices of gold and silver

Figure 2: General statistics about the gold and silver close prices

Figure 3: Gold returns and gold log-returns

Figure 4: ACF and PACF of the gold close log-returns

Figure 5: ACF and PACF of the squared gold close log-returns

Figure 6: ACF and Box- Ljung test of the estimated residuals of the MA (1) – model for gold return data

Figure 7: Histogram of estimated residuals

Figure 8: QQ-plot of estimated residuals

Figure 9: Kolmogorrov-Smirnov test of estimated residuals

Figure 10: ACF oft the squared and absolute residuals

Figure 11: Optimal parameters and information criteria of the GARCH (1,1) – fitted model for gold returns

Figure 12: Optimal parameters and information criteria of the GARCH (1,1) – fitted model with mean and skewed t-distribution for gold returns

Figure 13: Plotted GARCH (1,1) fitted model with mean and skewed t-distribution

Figure 14: Histogram and Kolmogorov-Smirnov test of residuals for GARCH – fitted model

Figure 15: ACF of residuals and squared residuals of the GARCH – fitted model

Figure 16: Silver returns and silver log-returns

Figure 17: ACF and PACF of the silver close log-returns

Figure 18: ACF and PACF of squared silver log-returns

Figure 19: ACF and Box- Ljung test of the estimated residuals of the AR (1) – model for silver returns

Figure 20: Histogram of residuals & QQ-plot of estimated residuals

Figure 21: Kolmogorov-Smirnov test of estimated residuals

Figure 22: ACF oft the squared and absolute residuals (silver)

Figure 23: Optimal parameters and information criteria of the GARCH (1,1) – fitted model for silver returns

Figure 24: Optimal parameters and information criteria of the GARCH (1,1) – fitted model with t-Distribution for silver returns

Figure 25: Plotted GARCH (1,1) fitted model with skewed t-distribution for silver log-returns

Figure 26: Histogram and Kolmogorov-Smirnov test of residuals for GARCH – fitted model

Figure 27: ACF of residuals and squared residuals of the GARCH – fitted model for silver return data

Abstract

This paper shows the development of the gold and silver close prices from 2001 – 2015. The ARMA model introduced by Box and Jenkins as well as the GARCH model introduced by Bollerslev are used to examine the properties of both time series as well as to disclose, which model fits the data best.

1. Introduction

It is widely known in the financial world that both equities, silver and gold have a long history of serving as a hedge against inflation, political risk and currency exchange risk (Blose, 1996), which provide economic and physical safety for the investors during times of political and economic crises as well as equity market crashes (Bolgorian & Gharli, 2001; Tully & Lucey, 2007). This phenomenon could be observed in the 2008 financial crisis, where other mineral prices fell, but only the gold price increased by 6% (Shafiee & Topal, 2010; Solt & Swanson, 1981). Moreover, researchers also show that gold and dollar seem to be negatively related, as in times, when the dollar was weak the price for gold increases (Pukthuanthong & Roll, 2011). Hence, gold was found to be uncorrelated with other types of assets, which leads to advantages for an investor in an era of globalization (Baur & Lucey, 2010).

As gold and silver assets seem to play an important role for investors, it is of great necessity to monitor its prices and the volatility of the time series. The autoregressive moving average models (ARMA) and the generalized autoregressive heteroscedasticity (GARCH) models became popular for academics and practitioners and led to a fundamental change to the approach of examining financial data. The ARMA models have been further extended (Bauwens & Laurent, 2006) and an efficient modelling of the volatility of the prices with GARCH models was further inspected by many researchers (Francq & Zakoian, 2011).

This paper deals with the development of the gold and silver prices from January 2001 until January 2015 and introduces the ARMA - model from Box & Jenkins (1971) for (weakly) stationary stochastic processes and the GARCH - model from Bollerslev (1986) to model heteroscedastic time series. The results, which were obtained with the help of the statistics package R (see Appendix 6 and 7), are presented in section 5 and 6 respectively. Besides, a forecast of the prices for both assets is made in section 7, the limitations of the research are presented in section 8 and section 9 concludes with a summary of the findings.

2. Data

The close prices of gold and silver per ounce in US dollar are examined from January, 3rd 2001 until January, 23th 2015, where prices of 3636 trading dates could be obtained. Figure 1 shows the original daily close prices of both assets from 2001 to 2015 in a line chart.

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Figure 1: Daily close prices of gold and silver

The charts show a steady increase of the gold and silver prices from 2001 to 2013. The starting price for gold in January, 3rd 2001 was 267,77 US dollar per ounce. The price increased constantly until 2011, where it peaked around 1800 US dollar per ounce. Afterwards the gold price decrease till 2015 and stayed at the price of approximately 1300 US dollar per ounce. In general, an upward trend of the gold price from 2001 till 2015 could be observed.

The silver prices follow a similar trend. The starting price for silver in January 3rd, 2001 was much lower than the gold price with 4,49 US dollar per ounce with a peak at approximately 48 US dollar per ounce in 2011. In comparison with the gold price, the price of silver per ounce had a stronger decrease from 2011 until 2015, where the price in January 2015 remained approximately at 18 US dollar per ounce.

The main difference of both assets are that the growth and decrease in price are different and the silver price seems to have a higher volatility rate during the period of 2001 to 2015.

3. Methodology

Real time series have various properties such as excess kurtosis and skewness, volatility clustering or fat-tailedness. Thus, it is difficult to find a model, which fits the real data best. It is assumed that the time series of gold and silver are at least weakly stationary, where the mean and the variance of a stochastic process do not depend on t as well as the autocovariance between Xt and Xt+τ only depend on lagt.

The close return data of gold and silver were examined with using the statistics package R. The returns were fitted in an ARMA model introduced by Box and Jenkins in 1971 and in the GARCH model introduced by Bollerslev (1986).

3.1. Properties of an ARMA model

The ARMA model is used to explain the auto-correlation in the time series and consists of the autoregressive (AR) process and the moving average (MA) process, where a linear process in the white noise exist. The autoregressive process of order p (AR(p)) exists if:

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And the AR (p) process is weakly stationary if:

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The moving average of order q (MA (q)) exists if:

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Therefore an ARMA model of order p, q exists if:

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3.2. Properties of a GARCH model

The GARCH model is used to describe time series, where volatility clustering seems to be evident. In these time-series periods of low volatility as well as periods of strong volatility can be observed. Hence, the conditional variance can change over time. A GARCH (p, q) process has the following property:

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And a GARCH (p,q) process is only stationary if:

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4. Data Analysis

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Figure 2: General statistics about the gold and silver close prices

Figure 2 presents the general statistics of the close gold and silver prices. The minimum gold price during the 2001 – 2015 period was 255.3 US dollar per ounce and the maximum price, was 1900 US dollar per ounce. The mean of the gold price during the period 2001 - 2015 period was 863.2 US dollar per ounce. In comparison, the minimum price for silver during the period of 2001 – 2015 was 4.04 US dollar per ounce, the maximum price amounted 48.42 dollar per ounce and the mean price of silver was 15.12 US dollar per ounce.

5. Examining gold close data

5.1. Gold log - returns

In order to be able to model the real data sets of the gold and silver returns, it is important to consider the stationarity property of the time-series. Thus, when computing the return of the gold and silver close prices, it is crucial to remove the non-stationarity property of the daily data by computing the log returns instead of the closing prices. Then the mean of the returns is fixed at zero and the volatility of the prices fluctuate around the mean. Figure 3 shows the computed close returns and log-returns of the gold close prices during the 3636 trading days.

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Figure 3: Gold returns and gold log-returns

Both ways of calculating the returns do not differ much in their results, which can be seen above. It is evident that the returns of all 3636 trading days for gold, fluctuate around the mean zero. Not only is the volatility of the returns high, as there are spikes for positive returns but also high negative outliers are evident.

5.2. ACF and PACF of the gold log-returns

The autocorrelation function (ACF) and the partial autocorrelation (PACF) of the gold close log-returns and the squared gold close log-returns can be seen in figure 4.

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Figure 4: ACF and PACF of the gold close log-returns

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Figure 5: ACF and PACF of the squared gold close log-returns

The above ACF and PACF of the gold close log-returns show that the lags are close to zero and almost all of them lie in the 95% confidence band. It is arguable that significant autocorrelation is evident. Only a few lags like lag 6 and lag 25 do not lie in the 95% confidence band. Moreover, it is noticeable that the autocorrelations decrease with increasing lag, but an exponential decay cannot be observed (see figure 5). Hence, it is necessary to perform a general modelling approach of ARMA models to find out, whether autocorrelation of the data is evident and which ARMA model fits best to the gold close returns data.

5.3. Applying ARMA-models on gold log-returns

Different models were tested by the researcher, with narrowing down to the AR (1) – model, MA (1) – model and ARMA (1,1) model for the gold close returns. The estimated parameters form R of these three models can be found in Appendix 1.

Due to testing the gold log-return data with the general modelling approach, results show that the estimated parameters for the AR(1)-model are α1 = -0,0205 and σ2= 1,376. However the |t(α1)| value = 1,23, which is smaller than the value of 1,96. This indicates that the coefficient is not significantly different from zero with a confidence interval of 95%.

Testing for the MA (1)-model, shows that the parameters are α2 = -0,0210 and σ2 = 1,376, and |t(α2)| = 1,25 (<1,96), thus the MA coefficient is not significantly different from zero with a confidence interval of 95%.

The same results could be observed for the ARMA (1,1) – model, where satisfying parameters could not be obtained. Here the parameters are 0,1767 and α1 = -0,1981 and σ2= 1,376, |t(α1)| for AR (1) = 10,77; for MA (1) = 0,445 (< 1,96). The coefficients for the ARMA (1,1) – model are also not significantly different from zero.

According to the testing results, it can be concluded that all three models are not suitable for modelling gold close returns, as the t-values are too small and all coefficients do not seem to be significant. Unfortunately, a right ARMA model could not be selected. However, the model with the lowest AIC (Akaike’s information criterion), which estimates the quality of each model relative to other models, is chosen. The MA (1) – model has the lowest AIC and is considered as the “best” model in comparison to the other models. Therefore, a continuation with the examination of the gold close returns with the MA (1) – model was done by the researcher. An overfitting approach with the MA (2) – model and the MA (3) - model was made. Nonetheless, a better model could not be detected, as all parameters from testing were not significant (See Appendix 2).

5.4. Testing the residuals of the gold log – returns

The independence assumption was checked, with estimating the residuals from the MA (1) model. The ACF of the estimated residuals (see figure 6) shows that most autocorrelations lie in the 92% confidence band and are different from zero. However, a few spurious autocorrelations are evident, for example at lag 6 and 25.

Additionally, the Ljung-Box test statistic shows that the p-values are arguably large with p = 0,08946 at lag 6 and p = 0,1342 at lag 25. It can either be concluded that significant autocorrelations of the residuals are evident at both lags or it is possible to argue that at lag 6 and at lag 25 spurious autocorrelations exists.

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Figure 6: ACF and Box- Ljung test of the estimated residuals of the MA (1) – model for gold return data

5.5. Normality assumption of the residuals of the gold log - returns

To check, whether the residuals of the MA (1) – model are normally distributed, it is adequate using a histogram with the estimated normal density of the residuals or a QQ plot. This is shown in figure 7 and 8.

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Figure 7: Histogram of estimated residuals

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Figure 8: QQ-plot of estimated residuals

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Figure 9: Kolmogorrov-Smirnov test of estimated residuals

The histogram shows the distribution of the residuals in comparison with a normal distribution. When monitoring the histogram as well as the QQ-plot, it is clearly evident that the residuals are not normally distributed. The distribution seems to be right-skewed with a larger tail on the left side. The QQ-plot shows that the black points are not scattered around the red line. Besides, the Kolmogorov-Smirnov shows that the p-value for the residuals is almost zero, which indicates that the residuals are clearly not normally distributed.

5.6. Heteroscedasticity of the gold log - returns

To check whether the variance is constant over time it is necessary to compute an ACF plot of the estimated squared residuals and absolute residuals (figure 11). The ACF of the estimated squared residuals of the gold log-returns clearly has significant correlations and the ACF of the estimated absolute residuals also has significant correlations. This indicates that the data have significant heteroscedasticity.

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Figure 10: ACF oft the squared and absolute residuals

5.7. Using GARCH to model the time series of the gold log - returns

As heteroscedasticity in the data seems to be evident, which is shown in figure 11, GARCH models are applied to model that the variance is not constant over time.

First the specified GARCH (1,1) – model, which shows model specifications like the estimated parameters, the log-likelihood, the information criteria and the test-statistics, is used to estimate the specifications for the gold log-returns. The optimal parameters and the information criteria for the fitted GARCH (1,1) – model for the gold returns are shown in figure 11:

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Figure 11: Optimal parameters and information criteria of the GARCH (1,1) – fitted model for gold returns

The p-values and the t-values of all parameters of the fitted GARCH (1,1) – model are greater than 2, so all parameter estimates are significantly different from zero to a 5% significance. This indicates, that the GARCH (1,1) model is a suitable model for modelling gold-log returns.

Further, through a general modelling approach, it could be estimated that the classical mean model (ARMA (0,0) - GARCH (1,1) with mean and skewed t-distribution is a better model for modelling the gold log-returns, as the properties of the model fits the data best. Furthermore, the Bayesian information criterion is lower and confirms the conclusion (see figure 13):

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Figure 12: Optimal parameters and information criteria of the GARCH (1,1) – fitted model with mean and skewed t-distribution for gold returns

As the mean of the fitted GARCH (1,1) – model with skewed t-distribution is significantly different from zero, it could also be concluded, that the logarithm of gold prices is significantly increasing linearly on the long period.

Figure 14 visualizes the fitted GARCH (1,1) – model with mean and skewed t-distribution to the gold log-returns. The plot again confirms that the fitted GARCH – model is able to illustrate the time-series of the gold log - returns well, as the returns could be fitted to the model in the 90% / 99% bound. It could be observed that only a few returns could not be represented with the fitted mean GARCH (1,1) – model with skewed t-distribution.

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Figure 13: Plotted GARCH (1,1) fitted model with mean and skewed t-distribution

5.8. Testing for skewed t-distribution of the residuals in a GARCH - model

As the presented GARCH (1,1) – model with mean and skewed t-distribution fits the real return data of gold, it is also necessary to check the skewed t-distribution assumption of the residuals of the fitted GARCH model. As seen in figure 15 the histogram of the residuals as well as the Kolmogorov-Smirnov test for the residuals were computed. Both show that the residuals are not normally distributed as the distribution of the residuals seem to be slightly skewed to the right and the p-values from the Kolmogorov-Smirnov test are equally to zero, which indicates that the residuals are not normally distributed, but the assumption that the residuals are t-distributed and skewed (to the right) therefore can be hold true.

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Figure 14: Histogram and Kolmogorov-Smirnov test of residuals for GARCH – fitted model

Moreover, testing the residuals and the squared residuals show that the independence assumption of the residuals is satisfied. The autocorrelation function of both show that most of the autocorrelations lie within the 95% confidence band.

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Figure 15: ACF of residuals and squared residuals of the GARCH – fitted model

6. Examining the silver close data

6.1. Silver log-returns & ACF/PACF of the silver return data

For the log-returns of the silver prices, different results could be obtained. Figure 17 shows the close and the log-returns of silver.

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Figure 16: Silver returns and silver log-returns

The returns of the silver prices are likewise highly volatile, as there are periods of positive returns and periods of highly negative returns. When computing the ACF and PACF of the log-returns of the silver prices (see figure 18), it could be concluded, that the lags are close to zero and almost all of them lie in the 95% confidence band and the autocorrelations decrease with increasing lag. It could only be observed that lag 11 and 17 do not lie in the 95% confidence band. So ARMA – models are also tested on the silver log-returns to examine, whether autocorrelations are existent.

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