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Midterm 1 Answer - (g | θ 2 |< 1(h y t = μ(1 θ 2 L ε...

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Economics 467: Economic Forecasting State University of New York at Binghamton Department of Economics Spring 2010 Midterm I — Answers 1. (a) Stationary (b) Stationary (c) Stationary (d) Stationary (e) Stationary (f) Stationary (g) Stationary (h) Nonstationary (i) Stationary (j) Nonstationary (k) Stationary 2. (a) MA (2) (b) Bi-annual (twice per year) (c) E ( y t ) = E ( μ + ε t + θ 2 ε t 2 ) = μ (d) γ 0 = E h ( y t μ ) 2 i = E h ( ε t + θ 2 ε t 2 ) 2 i = E ¡ ε 2 t + 2 θ 2 ε t ε t 2 + θ 2 2 ε 2 t 2 ¢ = σ 2 ¡ 1 + θ 2 2 ¢ (e) γ 1 = E [( y t μ ) ( y t 1 μ )] = E [( ε t + θ 2 ε t 2 ) ( ε t 1 + θ 2 ε t 3 )] = 0, γ 2 = E [( y t μ ) ( y t 2 μ )] = E [( ε t + θ 2 ε t 2 ) ( ε t 2 + θ 2 ε t 4 )] = θ 2 σ 2 , γ j = E [( y t μ ) ( y t j μ )] = E [( ε t + θ 2 ε t 2 ) ( ε t j + θ 2 ε t j 1 )] = 0 j > 2 (f) ρ 1 = γ 1 γ 0 = 0, ρ 2 = γ 2 γ 0 = θ 2 σ 2 σ 2 ( 1+ θ 2 2 ) = θ 2 ( 1+ θ 2 2 ) , ρ j = γ j γ 0 = 0 j > 2. The ACF has a spike equal to θ 2 ( 1+ θ 2 2 ) at lag 2 and all other lags are equal to zero.
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Unformatted text preview: (g) | θ 2 | < 1 (h) y t = μ + (1 + θ 2 L ) ε t ⇒ y t (1+ θ 2 L ) = μ (1+ θ 2 ) + ε t which is an AR ( ∞ ) 3 . ( a ) T his m etho d o f s u btractin g the series at lag 4 (for quarterly data) is known as seasonally di f er-encing. (b) Given that we have an AR (2) we cannot start at 1960:Q1 because we would need values for y at 1959:Q4 and 1959:Q3. Thus we must start at 1960:Q3. (c) Yes. Each of the AR roots are less than one. (d) No. One of the MA roots is greater than one. 1...
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