To my colleagues at NWU, thank you all for your words of encouragement and support. We run a series of simulations for the GARCH (1,1) model with error terms following a normal distribution.
Introduction
Background
Kellner and R¨osch (2016) analyzed how value at risk (VaR) and expected shortfall (ES) respond to different types of risk model. The danger of working with an inadequate model is known as model risk (Barrieu and Scandolo, 2015).
Problem Statement
When modeling the market risks of financial time series, the spread of the time series changes not only with time, but also intensively in some periods. This means that the range or variance of a financial time series is relatively small in some specific time periods, while it turns out to be relatively large in another time period.
Motivation
Therefore, the error term value distribution reveals leptokurtosis characteristics, which implies that the probability near the mean and tail is greater than the probability of the normal distribution, while the probability of the remainder is less than that of the normal distribution. These models will be considered misspecified in the event that the distribution of the innovations from the given GARCH model is assumed to follow a certain given distribution, whereas a true distribution exists which is different from the assumed distribution (Richmond and Horowitz) , 2015). The analysis is performed on normally distributed simulated data sets and empirical data using a variety of model performance methods and measures.
Aim of the Study
Objectives of the Study
Identify several model performance methods and benchmarks to help us determine the potential impact of model misspecification on model selection and performance.
Method of Investigation
Dissertation Overview
7 Chapter 2: This chapter focuses primarily on the literature on model misspecification in market risk modeling. The discussion of the results is based on the prospect of model misspecification and its impact on model performance.
Introduction
Model risk
Various studies have been conducted on the impact of the risk model in the financial sector. Glasserman and Xu (2014) developed a general approach and specific tools for quantifying model risk and limiting the impact of model errors.
Model misspecification
They used Markov chain reversible jump Monte Carlo simulation methods to determine the model distribution from the data. Edges are treated as a measure of model misspecification, with each edge separately estimating the extent of misspecification in a particular area of the model.
Market risk modelling
They are expected to track and disclose the level of market risk in order to introduce a capital requirement. There are rules for stress testing and identifying the right set of market risk elements.
Value-at-Risk (VaR) and Expected shortfall (ES)(ES)
A risk measure is considered coherent if it satisfies the four axioms established by Artzner et al. The risk measure of the composite distribution should not exceed the sum of the risk measures of the individual distributions, even if the distributions are one hundred percent correlated.
Forecasting Volatility using GARCH (1,1)
We find that the distribution of the error terms has a significant impact on the results of the studies discussed above. In this study, we aim to identify the model misspecification of the GARCH models with Normal, Student-t and GED distributions through analyzes performed on the estimation of key parameters of the .
Chapter summary
Research methodology
- Introduction
- Asset returns
- Simple returns
- Continuously compounded asset returns
- Stylized facts for asset returns
- Preliminary data analysis
- Normality test
- Jarque-Bera (JB)
- Shapiro-Wilk test
- Test for Stationarity
- Augmented Dickey-Fuller (ADF) test
- Phillips Perron (PP) test
- Specification of the modelling techniques
- The Auto Regressive Conditional Heteroskedasticity (ARCH) model
- The Generalized Auto Regressive Conditional Heteroskedas- ticity (GARCH) model
- Parameter estimation
- Maximum likelihood estimation
- Innovations of GARCH (1,1) model
- The Standardized Normal Distribution
- The Standardized Student-t distrubtion
- The Standardized GED
- Auto regressive conditional heteroscedasticity (ARCH) effects(ARCH) effects
- The Ljung-Box test
- The Lagrange multiplier test
- Model selection criteria
- Model accuracy evaluation
- Mean squared error (MSE)
- Root mean squared error (RMSE)
- Mean absolute error (MAE)
- Mean absolute percentage error (MAPE)
- Risk measures: Value-at-Risk (VaR) and Ex- pected Shortfall (ES)pected Shortfall (ES)
- Chapter summary
Depending on the degrees of freedom, the GED is a symmetric distribution that can be both leptokurtic and platykurtic. The GED has thicker tails than the Normal distribution when 0 < κ <2 (leptokurtic) and has thinner tails than the Normal distribution when κ > 2 (platykurtic). We illustrated the mathematical expressions of the ARCH model and its extensions known as the GARCH(1,1) model.
Results and findings
Introduction
Simulated Results
- Simulation study: Scenario 1
- Simulation study: Scenario 2
- Simulation study: Scenario 3
However, when we look in Appendix Table A.1 and Table A.2 at the QQ plots for Scenario 1 of Model 2 for sample sizes of 50, 500 and 5000, we see that the returns appear to be normally distributed with only a few data points deviating from the line of reference. This is also seen in the Appendix, Table A.1 and Table A.2 of the QQ plots for Scenario 1 of Model 3 for sample sizes 500 and 5000, where we observe heavy tails in both QQ plots. The QQ plots in Appendix Table A.1 and Table A.2 for Scenario 1 of Model 4 for the three samples also show no normality for the returns.
The descriptive statistics obtained for the four models of scenario 2 for sample sizes 50, 500 and 5000 are presented in Table 4.12. We can also observe non-normality from the QQ plot for sample size 5000 presented in Appendix Table A.3 and Table A.4. Table 4.22 presents the descriptive statistics obtained for the four models of Scenario 3 for the sample sizes 50, 500 and 5000.
This can also be seen in Appendix Table A.5 and Table A.6, the Model 3 QQ plots for sample size 5000, where non-normality is observed in the QQ plot. The Scenario 3 results obtained for the Jarque-Bera test and Shapiro-Wilk test are shown in Table 4.23. Looking at Table 4.28, we see that the normal distribution was consistent in producing the lowest AIC and BIC values for the majority of sample sizes.
Empirical results
- Samsung Electronics stock’s results
- Bitcoin-USD cryptocurrency results
- BAAA yield results
The return series of the real datasets were calculated using the continuously compounded return. This is confirmed by the Jarque-Bera and Shapiro-Wilk probability values in Table 4.33, as the hypothesis tests rejected the normality of the Samsung return series at a 5% significance level. We tested for ARCH effects in the return series using the Ljung-Box and LM tests, with the results obtained shown in Table 4.35.
We see the presence of heavy tails from Figure 4.4 the QQ plot of Bitcoin return series. BAAA yield yield series show that the Jarque-Bera probability value in Table 4.45 is greater than 5%. Thus, we do not reject the null hypothesis at a 5% significance level and conclude that BAAA return-return series follow a normal distribution.
Qormaatni yaada (hypothesis tests) lamaan sadarkaa barbaachisummaa %5 irratti dhaabbataa ta’uu dhabuu tartiiba bu’aa oomishaa BAAA ni dide.
Chapter Summary
Goodness-of-fit checks were performed to ensure that the model described the data adequately using log-likelihood. After selecting a model and checking its fit, model accuracy was examined using measures of accuracy (MSE and RMSE) and measures of risk (VaR and ES). The results revealed that if the chosen model does not adequately describe your data, model misspecification will certainly be present.
Conclusions and Recommendations
Introduction
Objectives and Conclusions
- Objective 1: Explore the GARCH model with error terms assuming Normal, Student-t and Generalized Er-
- Objective 2: Identify several model performance meth- ods as well as performance measures, to help us in
- Objective 3: Evaluate how violation of model assump- tions impact the prediction accuracy and model per-
The return series simulated from the GARCH (1,1) model was set to be normally distributed. We expected this result since the return series was simulated using normally distributed innovations. The distribution of Bitcoin return series had heavy tails (leptokurtic), and the distribution of Samsung returns had thinner tails than the Normal distribution (platykurtic).
A model mismatch was also found in the Scenario 3 example, where an analyst identified an inappropriate model, the GARCH (1,1) with GED innovations at a sample size of 500, as the model that best describes the return series. The GARCH (1,1) with Normal, Student-t, and GED innovations and varying sample sizes were fitted to the return series of each real data set. The GARCH (1,1) model with normally distributed error terms was identified as the optimal model for the BAAA return sequences.
The analysis of the results showed that the distribution of the series of returns affects the decision of which innovation is the most appropriate to use.
Limitations of the study and future studies
2001b), 'Empirical Properties of Asset Returns: Stylized Facts and Statistics. 2013), “Predicting Value at Risk and Expected Shortfall Using Partially Integrated Conditional Volatility Models: International Evidence”, International Review of Financial Analysis Estimator Distribution for Autoregressive Time Series with Unit Root”, Journal of the American Statistical Association 74 (366a) Measuring Market Risk, John Wiley and Sons. 1982), "Autoregressive Conditional Heteroscedasticity with Variance Estimates of UK Inflation", Econometrica: Journal of the Econometric Society, p. 2010), Fundamentals of Financial Markets and Institutions, Pearson/Addison-Wesley. 2011), Supervisory Guidance on Model Risk Management, Board of Governors of the Federal Reserve System, Washington, DC. 1977), "The Valuation of Corporate Liabilities as Compound Options", Journal of Financial and Quantitative Analysis, p. 2002), "Information Content of Implied Volatility Indices for Predicting Volatility and Market Risk", Available at SSRN. Consequences of model misspecification for maximum likelihood estimation with missing data', Econometrics Forecasting accuracy for arch models and garch (1, 1) family: Which model does best capture the volatility of the swedish stock market Structured ambiguity and model misspecification', Journal of Economic Theory p. 1997), 'Bank Capital Requirements for Market Risk: The. 2006), "Stationarity and Stability of Underwriting Profits in Property Liability Insurance: Part i", The Journal of Risk Finance.
A new approach', Econometrica: Journal of the Econometric Society p. 2006), 'A further critique of stochastic volatility models garch/arma/var/evt and related approaches', Applied Mathematics and Computer Science. 1997), Managing Financial Institutions: A Contemporary Perspective, Irwin New York. 2005), 'Mean squared error', Encyclopedia of Biostatistics Quantifying model risk caused by model misspecification', Journal of Applied Statistics p. 2020), 'A comparison of linear and nonlinear garch models for forecasting the volatility of selected emerging countries', Journal of Advances in Management Research. 2017), “On the Implementation of Asymmetric Variable Models for Market Risk Management and Forecasting”, Journal of Applied Finance and Banking a), “Interpretive Issues Related to Changes in the Market Risk Framework”. 2011b), Revisions to the Basel II Market Risk Framework:.
2011), “Forecasting usd/mur exchange rate volatility using a garch (1, 1) model with ged and student'st errors”, University of Mauritius Research Journal, Maximum Likelihood Estimation of Misspecified Models, Econometrica: Journal of the Econometric Society p. 1982), 'Behavioral theories of dispersion and misspecification of travel demand models', Transportation Research Part B:. 2017), “The 'fbasics' Package”, Rmetrics-Markets and Basic Statistics.
