Econ226_VIIAC

Econ226_VIIAC - 1 VII. Time-varying variances A....

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Unformatted text preview: 1 VII. Time-varying variances A. Introduction to ARCH models y t return on a stock in period t 6 population mean return y t 6 u t Observation: u t is almost impossible to predit E u t | u t " 1 , u t " 2 ,... However: u t 2 does seem to be quite forecastable Question 1: how should we forecast u t 2 ? One answer: autoregression on its own lagged values: u t 2 ? ) 1 u t " 1 2 ) 2 u t " 2 2 C ) m u t " m 2 w t E w t E w t 2 5 2 E w t w A if t p A 2 Question 2: what kind of data- generating process would imply such a forecast? u t h t / t / t L i.i.d. 0,1 (e.g. N 0,1 ) h t ? ) 1 u t " 1 2 ) 2 u t " 2 2 C ) m u t " m 2 Definition: a regression model with Gaussian ARCH m error is characterized by y t x t U * u t u t h t v t v t L i.i.d. N 0,1 h t ? ) 1 u t " 1 2 ) 2 u t " 2 2 C ) m u t " m 2 ARCH autoregressive conditional heteroskedasticity Note: even though u t has a distribution that is conditionally Gaussian, u t | u t " 1 , u t " 2 L N 0, h t , its unconditional distribution is non-Gaussian (fatter tails) 3 parameters of Gaussian ARCH m regression: 2 * U , ) U , ? U estimate by maximum likelihood: ( t " 1 x t , y t " 1 , x t " 1 , y t " 2 , x t " 2 ,... y t | ( t " 1 L N x t U * , h t h t ? ) 1 u t " 1 2 ) 2 u t " 2 2 C ) m u t " m 2 u t y t " x t U * f y t | ( t " 1 ; 2 1 2 = h t exp " y t " x t U * 2 2 h t c 2 ! t 1 T log f y t | ( t " 1 ; 2 choose 2 numerically to maximize c 2 subject to ? u 0, ) j u (e.g., set ) j 5 j 2 ) use first m values of y t and x t for conditioning 4 Although a Gaussian specification for v t is natural starting point, stock returns are better modeled using a Student t y t | ( t " 1 L Student t with 7 2 degrees of freedom conditional mean: E y t | ( t " 1 x t U * conditional variance: E y t " x t U * 2 | ( t " 1 h t log f y t | ( t " 1 ; 2 log 7 1 /2 = 7 /2 7 " 2 " 1/2 " 1 2 log h t " 7 1 2 log 1 y t " x t U * 2 h t 7 " 2 h t ? ) 1 u t " 1 2 ) 2 u t " 2 2 C ) m u t " m 2 5 Issues: (1) covariance-stationary if 1 " ) 1 z " C " ) m z m implies that || z || 1 (2) E u t 2 | u t " 1 ,..., u t " m Sufficient conditions: ? ) j u j 1,..., m ) 1 ) 2 C ) m 1 Why does the conditional variance matter? 1) knowing variance of returns is important for a) assessing risk b) portfolio choice c) options pricing 6 2) even if youre interested in mean only, correctly modeling the variance could matter for a) more accurate hypothesis tests b) more efficient estimates Hamilton, Macroeconomics and ARCH...
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Econ226_VIIAC - 1 VII. Time-varying variances A....

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