H19 - H19 HW10 (due Apr. 10) Problem 1 Give 1-step and...

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H19 HW10 (due Apr. 10) Problem 1 Give 1-step and 2-step volatility forecasts for the ARCH(1) model with 01 1.5, 0.9 α == in terms of the known values of 2 t X and 2 t σ . Problem 2 Give a 3-step volatility forecast for an ARCH(2) model with 2 1, 0.5, 0.2 αα = in terms of the known values of 2 t X and 2 t . Problem 3 Give 1-step and 2-step volatility forecasts for the GARCH (1,1) model with 011 1.5, 0.6, 0.3 αβ === in terms of the known values of 2 t X and 2 t . __________________________________________________________________________________________________ A generalized autoregressive conditional heteroscedastic (GARCH) model with order and is defined as (1 ) p (0 ) q σε = tt X t 2 and 22 2 2 1 1 1 β −− = + +⋅⋅⋅+ + p t p t XX q t q , (1) where 0 i and 0 j are constants, {}~ I ID ( 0 , 1 ) ε t , and t is independent of {, for all t . 1 tk Xk } A stochastic process {} t X defined by the equations above is called a GARCH (,) p q process. Formally, 2 t X follows an ARMA (, ) p qq model, where max{ , } p qp ∨= q : 2 2 00 11 1 1 () = = =+ + + + + ∑∑ pq p q q t i ti j t j t i i ti t jt j ij i j X Xe X e e , (2) where 0 ++ pj qj for , and .
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H19 - H19 HW10 (due Apr. 10) Problem 1 Give 1-step and...

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