Consider the Switching AR(1) model y t = ( μ 0 + φ 0 y t − 1 ) + ( μ 1 + φ 1 y t − 1 ) s t + ε t for t = 1 ,..., T with ε t ∼ N (0 ,σ 2 ), where s t is a latent binary random variable with exp( δ + γ y t − 1 ) Pr[ s t = 1] = F ( δ + γ y t − 1 ) = 1 + exp( δ . + γ y t − 1 ) What is/are the key difference(s) between the Switching AR(1) model and the LSTAR(1) model given by y t = ( μ 0 + φ 0 y t − 1 ) + ( μ 1 + φ 1 y t − 1 ) F ( δ + γ y t − 1 ) + ε t for t = 1 ,..., T with ε t ∼ N (0 ,σ 2 ) or are both models the same? Derive the unbiased 1-step ahead forecasts of y T + 1 made at time T for the Switching AR(1) model and for the LSTAR(1) model. Consider the Switching AR(1) model with exp( δ + γ s t − 1 ) Pr[ s t = 1] = 1 + exp( δ . + γ s t − 1 ) Show that this model is the same as a Markov Switching AR(1) model where Pr[ s t = 1| s t − 1 = 1] = p and Pr[ s t = 0| s t − 1 = 0] = q . Express p and q in terms of δ and γ . Suppose that you have observed yt for t = 1,..., T . A recession is defined as a period where for at least 2 consecutive periods st = 1. Suppose that ST = 0. Derive the probability in terms of p and q that the period [T + 1, T + 3] contains a recession.
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