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Statistics in finance

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This article considers Markov chain simulation and statistical analysis of high‐dimensional financial time series. In particular, we discuss Markov chain Monte Carlo methods, for example, Gibbs sampling and Metropolis‐Hasting algorithm, and multivariate volatility models with applications in finance. Real examples are used to demonstrate statistical applications of the methods discussed in risk management and volatility estimation. WIREs Comp Stat 2011 3 289–315 DOI: 10.1002/wics.168

Figure 1.

Time plot of monthly S&P 500 index from 1962 to 2009: (a) log level and (b) log return in percentage.

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Figure 2.

Density functions of prior and posterior distributions of parameters in a stochastic volatility model for the monthly log returns of the S&P 500 index. The dashed line denotes prior density and the solid line the posterior density, which is based on results of Gibbs sampling with 2000 iterations. See the text for more details.

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Figure 3.

Time plots of fitted volatilities for monthly log returns of the S&P 500 index from 1962 to 2009. The lower panel shows the posterior means of a Gibbs sampler with 2000 iterations. The upper panel shows the results of a Gaussian GARCH(1,1) model.

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Figure 4.

Density functions of log(\documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\chi_1^2$\end{document}), solid line, and that of a mixture of seven‐normal distributions, dashed line. Results are based on 100,000 observations.

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Figure 5.

Estimated volatility of monthly log returns of the S&P 500 index from January 1962 to November 2004 using stochastic volatility models: (a) with leverage effect (b) without leverage effect.

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Figure 6.

(a) Time plot of the monthly log returns, in percentages, of GE stock from 1926 to 1999. (b) Time plot of the posterior probability of being in State 2 based on results of the last 2000 iterations of a Gibbs sampling with 5000 + 2000 total iterations. The model used is a two‐state Markov switching GARCH‐M model.

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Figure 7.

Histograms of the risk premium and transition probabilities of a two‐state Markov switching GARCH‐M model for the monthly log returns of GE stock from 1926 to 1999. The results are based on the last 2000 iterations of a Gibbs sampling with 5000 + 2000 total iterations.

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Figure 8.

Histograms of volatility parameters of a two‐state Markov switching GARCH‐M model for the monthly log returns of GE stock from 1926 to 1999. The results are based on the last 2000 iterations of a Gibbs sampling with 5000 + 2000 total iterations.

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Figure 9.

Time plots of the persistent parameter αi1 + αi2 of a two‐state Markov switching GARCH‐M model for the monthly log returns of GE stock from 1926 to 1999. The results are based on the last 2000 iterations of a Gibbs sampling with 5000 + 2000 total iterations.

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Figure 10.

Fitted volatility series for the monthly log returns of GE stock from 1926 to 1999: (a) the squared log returns, (b) the GARCH‐M model in Eq. (28) and (c) the two‐state Markov switching GARCH‐M model in Eq. (26).

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Figure 11.

Time plot of daily stock returns from January 2, 2001 to December 31, 2009: (a) Exxon‐Mobil and (b) International Business Machines.

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Figure 12.

Time plot of standardized returns for the selected five stocks in Table 4. The standardization is based on the fitted GARCH(1, 1) − t model.

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Figure 13.

Time plots of selected correlations obtained from the dynamic conditional correlation model in Eq. (43).

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Figure 14.

Scatterplots between five uniform marginal distributions of stock returns. The GARCH(1, 1) − t models in Table 4 are used to standardize the returns.

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