Essay About Covariance Matrices Of Return Series And Autoregressive Conditional Heteroscedasticity

Generalized Autoregressive Conditional Heteroscedasticity (garch) and Stochastic Volatility (sv) Models
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AbstractReturn models and covariance matrices of return series have been studied. In particular, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) and Stochastic Volatility (SV) models are compared with respect to their fore-casting accuracy when applied to intraday return series. SV models are found to be considerably more accurate and more consistent in accuracy in forecasting.Covariance matrices formed from Gaussian and GARCH return series, and in particular, return series auto-correlated as an AR(1) process, have been studied. In the case of Gaussian returns, the largest eigenvalue is found to approximately follow a gamma distribution also when the returns are auto-correlated. Expres-sions relating the mean and the variance of the asymptotic Gaussian distribution of the matrix elements are derived. In the case of GARCH returns, both the largest and the smallest eigenvalues of the correlation matrix are seen to increase with increasing auto-correlation. The matrix elements are found to follow Levy distributions with di erent Levy indices for the diagonal and the non-diagonal elements.Localization of eigenvectors of correlation matrices of returns from GARCH processes has been investigated. It is found that the localization is reduced as the auto-correlation is increased. Quantitatively, the number of localized eigenvectors decreases approximately as a quadratic function with the auto-correlation strength, i.e. the autoregressive coe cient of the AR(1) process.Contents1Introduction52Return Models72.1Case Study: Nordea 15-minute Returns  . . . . . . . . . . . . . .102.1.1GARCH Model . . . . . . . . . . . . . . . . . . . . . . . .112.1.2Stochastic Volatility Model  . . . . . . . . . . . . . . . . .122.1.3  Comparison of the Forecasts  . . . . . . . . . . . . . . . .162.2Case study: Volvo 30-minute Returns  . . . . . . . . . . . . . . .182.3Unconditional Distribution Functions of SV Models . . . . . . . .202.3.1The Simpli ed model  . . . . . . . . . . . . . . . . . . . .212.3.2The General model . . . . . . . . . . . . . . . . . . . . . .232.3.3  Relation to Conditional Distribution Functions . . . . . .2823AcronymsACF Auto-correlation Function. 11AR Autoregressive. 10, 31, 33, 51ARIMA Integrated Autoregressive Moving Average. 9, 12, 20, 28ARMA Autoregressive Moving Average. 10, 20CDF Cummulative Distribution Function. 26, 38GARCH Generalized Autoregressive Conditional Heteroscedasticity. 1, 6{8, 10, 11, 16{20, 32, 37, 38, 46, 50, 51, 53, 54, 56, 58{61IPR Inverse Participation Ratio. 44MGF Moment Generating Function. 25, 26MLE Maximum Likelihood Estimate. 12, 23, 26PDF Probability Density Function. 21, 25, 26, 28, 31, 38, 44{48, 65, 67, 71, 72