# (solution) Consider the following model, where y t is the quarterly growth rate of US GDP: y t − μ t

Consider the following model, where y t is the quarterly growth rate of US GDP: y t − μ t = φ 1 ( y t − 1 − μ t − 1 ) + ε t , (8.79) where ε t ∼ iid N (0 ,σ 2 ), and μ t is given by m μ t = μ 0 + μ 1 I( S t = 1) + λ I( S t = 0) ) I( S t − j = 1) , (8.80) j =1 where I( A ) is the indicator function for the event A (that is, I( A ) = 1 if A is true, and I( A ) = 0 otherwise), and S t ∈ {0 , 1} is a first-order Markov process with constant transition probabilities given by P ( S t = 0| S t − 1 = 0) = p and P ( S t = 1| S t − 1 = 1) = q . Estimating the parameters in the above model with m = 6 using observations for the period 1954Q1–2003Q4 gives 0 φ � 1 1, � μ 0 μ 0 + � &gt; 0, � μ 1 0, � λ &gt; 0, p = 0 . 94, and � = 0 . 78. � q Describe which features of the US GDP growth are captured by this model. In particular, how would you label the regimes S t = 0 and S t = 1? What is the function of the last component of μ t in (8.80), that is λ I( S t = 0) ), m j =1 I( S t − j = 1)?