Suppose the time series y t is generated according to the following threshold process y t = r − δ + ε t if q t ≤ c δ + ε t if q t > c with δ > 0, ε t is a white noise series with E[ ε t ] = 0 and E[ ε 2 = σ 2 ] for all t , and t ε the threshold variable q t is i.i.d. standard normally distributed. Furthermore, ε t and q t are independent. Derive an expression (in terms of the parameters δ , σ 2 and c ) for the following ε characteristics of the time series y t : the unconditional mean μ y = E[ y t ], the unconditional variance γ0(y) = E[(yt − E[yt ])2], and the first-order autocorrelation ρ 1 ( y ) = γ 1 ( y ) /γ 0 ( y ), where γ 1 ( y ) is the first- order autocovariance of y t , that is, γ 1 ( y ) = E[( y t − E[ y t ])( y t − 1 − E [ y t − 1 ])]. Hint: Use the fact that the covariance between two random variables X and Z can be written as E[( X − E[ X ])( Z − E[ Z ])] = E[ XZ ] − E[ X ]E[ Z ]. Interpret these expressions, and discuss how they behave as a function of the parameters.
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