Assignment_2_solutions_Complete - Problem Set 2 Volatility...

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Problem Set 2: Volatility Models Question 1 Consider the following ARCH(1) model: y t = μ + u t u t = v t σ t v t N (0 , 1) σ 2 t = α 0 + α 1 u 2 t - 1 Suppose that you have data available up to and including time t . Write down the ex- pressions for the forecasts of σ 2 for periods t + 1 , t + 2 , and t + 3 . If you can, obtain a general expression for any horizon t + s . Solution σ 2 t +1 | t = E α 0 + α 1 u 2 t | Ω t = α 0 + α 1 u 2 t σ 2 t +2 | t = E α 0 + α 1 u 2 t +1 | Ω t = α 0 + α 1 E u 2 t +1 | Ω t = α 0 + α 1 E σ 2 t +1 v 2 t +1 | Ω t = α 0 + α 1 E σ 2 t +1 | Ω t E v 2 t +1 | Ω t = α 0 + α 1 E σ 2 t +1 | Ω t = α 0 + α 1 σ 2 t +1 | t 1
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σ 2 t +3 | t = E α 0 + α 1 u 2 t +2 | Ω t = α 0 + α 1 E u 2 t +2 | Ω t = α 0 + α 1 E σ 2 t +2 v 2 t +2 | Ω t = α 0 + α 1 E σ 2 t +2 | Ω t E v 2 t +2 | Ω t = α 0 + α 1 E σ 2 t +2 | Ω t = α 0 + α 1 σ 2 t +2 | t = α 0 + α 1 ( α 0 + α 1 σ 2 t +1 | t ) = α 0 + α 0 α 1 + α 2 1 σ 2 t +1 | t More generally: σ 2 t + s | t = α 0 s - 1 X i =1 α i - 1 1 + α s - 1 1 σ 2 t +1 | t 2
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Question 2 Consider the following ARCH(2) model: y t = μ + u t u t = v t σ t v t N (0 , 1) σ 2 t = α 0 + α 1 u 2 t - 1 + α 2 u 2 t - 2 Suppose that you have data available up to and including time t . Write down the ex- pressions for the forecasts of σ 2 for periods t + 1 , t + 2 , and t + 3 . Solution σ 2 t +1 | t = E α 0 + α 1 u 2 t + α 2 u 2 t - 1 | Ω t = α 0 + α 1 u 2 t + α 2 u 2 t - 1 σ 2 t +2 | t = E α 0 + α 1 u 2 t +1 + α 2 u 2 t | Ω t = α 0 + α 1 E u 2 t +1 | Ω t + α 2 u 2 t = α 0 + α 1 E σ 2 t +1 v 2 t +1 | Ω t + α 2 u 2 t = α 0 + α 1 E σ 2 t +1 | Ω t E v 2 t +1 | Ω t + α 2 u 2 t = α 0 + α 1 E σ 2 t +1 | Ω t + α 2 u 2 t = α 0 + α 1 σ 2 t +1 | t + α 2 u 2 t σ 2 t +3 | t = E α 0 + α 1 u 2 t +2 + α 2 u 2 t +1 | Ω t = α 0 + α 1 E u 2 t +2 | Ω t + α 2 E u 2 t +1 | Ω t = α 0 + α 1 E σ 2 t +2 v 2 t +2 | Ω t + α 2 E σ 2 t +1 v 2 t +1 | Ω t = α 0 + α 1 E σ 2 t +2 | Ω t E v 2 t +2 | Ω t + α 2 E σ 2 t +1 | Ω t E v 2 t +1 | Ω t = α 0 + α 1 σ 2 t +2 | t + α 2 σ 2 t +1 | t = α 0 + α 1 ( α 0 + α 1 σ 2 t +1 | t + α 2 u 2 t ) + α 2 σ 2 t +1 | t = α 0 + α 0 α 1 + α 1 α 2 u 2 t + ( α 2 1 + α 2 ) σ 2 t +1 | t 3
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Question 3 For this question you will use returns data on the S & P 500 “Sp500.mat" and the EUR/USD exchange rate “EURUSD.mat", available on CANVAS. Part 1: Consider the returns on the S & P 500 1. Estimate a GARCH(1,1) and a GJR-GARCH(1,1,1) to the (simple) returns of the S & P 500 series. You need to load the dataset sp500. To compute the simple return of S & P 500 series, you can do: ret = ( sp 500(2 : end ) - sp 500(1 : end - 1)) ./sp 500(1 : end - 1); You can use the function tarch.m to run a GARCH model or a GJR-GARCH model. Read the documentation of tarch.m carefully. A simple usage of tarch.m follows the equation below: [ PARAMETERS ] = tarch ( EPSILON, P, O, Q ) EPSILON is the return series you would like to analysis.
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