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 for (weakly) stationary stochastic processes and the GARCH-model from Bollerslev to model heteroscedastic time series. The results, which were obtained with the help of the statistics package R, 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.
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, which provide economic and physical safety for the investors during times of political and economic crises as well as equity market crashes. This phenomenon could be observed in the 2008 financial crisis, where other mineral prices fell, but only the gold price increased by 6%. 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. Hence, gold was found to be uncorrelated with other types of assets, which leads to advantages for an investor in an era of globalization.
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 and an efficient modelling of the volatility of the prices with GARCH models was further inspected by many researchers.
Table of Contents
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
Research Objectives & Topics
This paper examines the development of gold and silver close prices from 2001 to 2015 to identify the best statistical models for capturing their volatility and time series properties. The research aims to evaluate if ARMA and GARCH models can effectively describe the returns of these assets and whether these models can provide reliable forecasts.
- Analysis of gold and silver price volatility
- Application of ARMA models for autocorrelation
- Modeling of heteroscedasticity using GARCH
- Testing for skewed t-distribution of residuals
- Evaluation of forecasting performance in financial time series
Excerpt from the Book
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 lag t.
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).
Summary of Chapters
1. Introduction: Outlines the historical importance of gold and silver as financial hedges and introduces the use of ARMA and GARCH models for analysis.
2. Data: Presents the daily closing price datasets for gold and silver from 2001 to 2015 and describes their general trends and volatility.
3. Methodology: Defines the mathematical frameworks of ARMA and GARCH models used to address time series properties like stationarity and volatility clustering.
4. Data Analysis: Provides basic descriptive statistics for the gold and silver price data.
5. Examining gold close data: Details the process of modeling gold log-returns, including testing for stationarity, autocorrelation, and heteroscedasticity, and applying the GARCH model.
6. Examining the silver close data: Repeats the modeling process for silver, evaluating model fit and testing residual assumptions.
7. Forecasting gold and silver returns: Discusses the challenges and results of attempting to predict future returns for the two assets.
8. Drawbacks of the research: Acknowledges limitations regarding model selection, heuristic approaches, and the difficulty of forecasting volatile financial data.
9. Conclusion: Summarizes the findings, noting that while ARMA models were largely unsuitable, GARCH models with skewed t-distributions successfully captured the assets' heteroscedasticity.
Keywords
Gold, Silver, Financial Time Series, ARMA Model, GARCH Model, Volatility, Heteroscedasticity, Log-returns, Autocorrelation, Skewed t-distribution, Statistical Modeling, R Package, Forecasting, Market Efficiency, Price Volatility
Frequently Asked Questions
What is the core focus of this research paper?
The paper focuses on modeling the price development and volatility of gold and silver from 2001 to 2015 using financial econometric methods.
Which central themes are explored?
The central themes include asset price trends, the stationarity of log-returns, autocorrelation in financial data, and the modeling of time-varying variance (volatility).
What is the primary goal of the study?
The primary goal is to determine whether standard models like ARMA and more advanced models like GARCH can effectively fit the return data of gold and silver.
Which scientific methods are employed?
The study employs the statistics package R to implement Box-Jenkins ARMA models and Bollerslev’s GARCH models, along with Ljung-Box and Kolmogorov-Smirnov tests.
What does the main body cover?
It covers data preparation, diagnostic testing of residuals, model parameter estimation, and the testing of assumptions like normality and heteroscedasticity for both precious metals.
Which keywords best characterize this work?
Key terms include GARCH modeling, financial time series, gold and silver returns, and volatility clustering.
Does the paper successfully model gold returns with ARMA?
No, the research concludes that ARMA models were not suitable for gold returns as the coefficients were not significantly different from zero.
How is heteroscedasticity addressed for these assets?
Heteroscedasticity is addressed by successfully fitting GARCH (1,1) models, which account for the changing variance over time.
What is the conclusion regarding the long-term price increase?
The study concludes that the logarithm of gold prices increases linearly, while for silver, the evidence for a significant linear increase is weaker.
- Citation du texte
- Van Anh Hoang (Auteur), 2016, Application of ARMA and GARCH models to the daily gold and silver exchange prices in US dollar, Munich, GRIN Verlag, https://www.grin.com/document/319861
