Econ226_VIID

Econ226_VIID - 1 VII Time-varying variances A Introduction...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 VII. Time-varying variances A. Introduction to ARCH models B. Extensions C. Markov-switching GARCH D. Stochastic volatility GARCH family: y t x t U * u t u t h t v t v t L i.i.d. 0, 1 (e.g. N 0,1 ) h t h u t " 1 , u t " 2 , ... Implication: the difference between the realized value y t and its conditional expectation x t U * is the only information useful for forecasting the variance h t 2 Stochastic volatility: Some latent variables in addition to u t " j contribute to h t Example: y t exp h t /2 v t h t 6 C h t " 1 " 6 @1 t v t 1 t L i.i.d. N , 1 1 argument in favor of stochastic vol: more natural and flexible argument in favor of GARCH: ultimately our forecast E u t 2 | x t , y t " 1 , x t " 1 , y t " 2 , ... will be some function of x t , y t " 1 , x t " 1 , y t " 2 ,.. . so why not take this function as a primitive of the model? 3 Note sv model above implies y t 2 exp h t v t 2 log y t 2 h t log v t 2 log y t 2 6 h t " 6 log v t 2 For 8 t h t " 6 this is a state-space model of the form 8 t C8 t " 1 @1 t log y t 2 6 8 t log v t 2 problem: log v t 2 is not Normally distributed Gibbs sampler blocks: (1) @ 2 | C , 6 , h , y gamma prior for @ " 2 implies gamma posterior 4 (2) C | @ 2 , 6 , h , y easy way: Normal prior for C | @ 2 implies Normal posterior However: authors want to restrict C " 1, 1 and use unconditional distribution of first observation prior: C ' L Beta ) 1 , ) 2 ® p C ' - C ' ) 1 " 1 1 " C ' ) 2 " 1 C 2 C ' " 1 ® C " 1, 1 ¡ C ' 1 C /2 ¢ 5 ® p C - 1 C 2 a 1 " 1 1 " C 2 ) 2 " 1 C " 1, 1 otherwise h 1 | 6 , C , @ 2 L N 6 , @ 2 / 1 " C 2 h t | h t " 1 , h t " 2 ,. .., h 1 , 6 , C , @ 2 L N 6 C h t " 1 " 6 , @ 2 p h | 6 , C , @ 2 - 1 ¡ @ 2 / 1 " C 2 ¢ 1/2 exp " h 1 " 6 2 2 @ 2 / 1 " C 2 " !...
View Full Document

This note was uploaded on 03/02/2012 for the course ECON 226 taught by Professor Jameshamilton during the Winter '09 term at UCSD.

Page1 / 19

Econ226_VIID - 1 VII Time-varying variances A Introduction...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online