Problem 42 (AR(1) model fitting) Let Et i.i.d. N(0, 1). Consider the finite AR(1)-process, X₁=0 Xt=aXt-1 + Et with a < 1

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Problem 42 Ar 1 Model Fitting Let Et I I D N 0 1 Consider The Finite Ar 1 Process X 0 Xt Axt 1 Et With A 1 1 (78.47 KiB) Viewed 15 times

Problem 42 (AR(1) model fitting) Let Et i.i.d. N(0, 1). Consider the finite AR(1)-process, X₁=0 Xt=aXt-1 + Et with a < 1. a) Write a function in R that estimates a by the OLS method. b) Write a function in R that estimates a by the maximum likelihood method. Hint: The joint density of X₁,..., X₁ does not factorize because the random variables are not independent. But you may use that the joint density can be factorized using conditional densities: fxxX₁ (x,x₁) =fX₁|Xt-1,-,X₁ (x₁|x₁-1₁, ₁)ƒxt-1 X₁ (₁-1,...,x₁) =fx|Xt_1...,X_(2t|t_1,•••,1)fX_l|Xt_2!.,Xı(@t−1|2t_l,, 1) x fxt-2,...,X₁ (It-2,...,₁) =... ⠀ t =fx₁ (₁) IIfX₁|Xi-1...X₁ (Xi |Œi—1¹, . . . , ₁). i=2 To minimize the negative log-likelihood you can use the R function nlminb. c) Use your functions to estimate an AR(1) model for the time series AR1.csv in moodle.