HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)

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1 HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the forecasts and realized growth are given below:

2 d) The plot of out of sample forecasts are given below:

3 ARMA,) model: 095L)y t = L)ε t with ε t iidn0, σ ) y t = β 0 + β y t + θ ε t + ε t y t = y t 05ε t + ε t a) Stationarity: The stationarity of the ARMA,) process only depends on the AR lag polynomial In this case, the AR) condition that β = 095 < is met; accordingly, the time series process is stationary b) Innite order moving average representation: 095L)y t = L)ε t y t = 095L) L) 05L)ε t = L 095L ε t = L 095L ε t Generally, we have: φl)y t = θl)ε t y t = φl) θl)ε t = ψl)ε t Hence: θl) = φl)ψl) For the ARMA,), this gives: θ 0 + θ L) = φ 0 + φ L) ψ 0 + ψ L + ψ L + ) with θ 0 =, θ = 05, φ 0 =, and φ = 095 Using this identity, I can match the powers of the lag operator as follows: j = 0 : θ 0 = φ 0 ψ 0 j = : θ = φ 0 ψ + φ ψ 0 j : 0 = φ 0 ψ j + φ ψ j Solving for ψ j, I get: ψ 0 = θ0 φ 0 = =, ψ = θ φ φ 0 = θ φ = = 045, ψ = φ ) j θ φ ) = 095) j 045) Accordingly, the innite order moving average representation is as follows: y t = 0 + ε t + 095) j 045)ε t j

4 c) Impulse response function: y t ε t k y t k, y t k, = ψ k = φ ) k θ φ ) = 095) k 045) d) Expectation with respect to F t k : E[y t F t k ] = 0 + E[ε t F t k ] + k 095) j 045)E[ε t j F t k ] + 095) j 045)ε t j = ) j 045)ε t j j=k j=k e) Forecast error k-steps): e k t k = y t yt k k = ε t + k 095) j 045)ɛ t j f) Forecast error k-steps) variance: ) V are k t k F t k) = V arε t + k 095) j 045)ɛ t j F t k ) = σ + k 095) j 045) Distribution: e k t k N 0, σ + k 095) j 045) )) 3 a) Drift sample average of change in log CPI): b) Out-of-Sample Forecasts of log CPI: Table : Out-of-Sample Forecasts of log CPI Forecast 05Q4 06Q 06Q 06Q3 Formula u + y T u + y T 3u + y T 4u + y T Estimate c) Prediction interval of log CPI neglecting parameter uncertainty): Estimate σ with T t y t y t u) = 00867

5 Table : Predictive Intervals Forecast 05Q4 06Q 06Q 06Q3 Formula u + y T ± 96σ u + y T ± 96 σ 3u + y T ± 96 3σ 4u + y T ± 96 4σ 5% % d) Prediction interval of growth of log CPI neglecting parameter uncertainty): Table 3: Predictive Intervals Forecast 05Q4 06Q 06Q 06Q3 Formula u ± 96σ u ± 96 σ 3u ± 96 3σ 4u ± 96 4σ 5% % e) ARMA model for rst dierenced log CPI series: After looking at ACF and PACF, I t an ARMA,) Table 4: ARMA,) Estimates Coecient Standard Error Constant -35E AR) MA) f) Impulse response function: We see that the process fades quickly over time, so it is stationary 3

6 Figure : Impulse Response Functions 4 Write y t = µ + i= ψ iɛ t i + ɛ t, its MA ) representation This exists if the series is stationary a) Best forecast: I expect y k t to be the better forecast of y t+k as it uses one more realization y t ; ie a strictly larger information set) and needs to extrapolate one less step ie, k < k + ) 4

7 b) Claim: yt k yt k+ =white noise yt k yt k+ = E[y t+k F t ] E[y t+k F t ] = k ψ j ε t+k j Hence, given information at t, I have: j=0 E[y k t y k+ t F t ] = ψ k E[ε t F t ] = 0, ie white noise k j=0 ψ j ε t+k j = ψ k ε t with ψ 0 = ) c) Interpretation: The dierence between the two unbiased conditional forecasts is the unexpected part, ie shock, realized between y t and y t ie, ε t scaled by the innite moving average representation coecient) This is white noise conditional on F t ; ie, has expectation of zero at time t ) d) V arψ k ε t ) = ψ k σ 0 if ψ k decays to zero as k increases Since we had assumed the series is stationary, this will always be the case 5 Exact log likelihood for an AR) model: AR) : y t = β 0 + β y t + ε t E[y ] = β0 β, V ary ) = σ β Unconditional: y N β0 β, σ β ) fy ; θ) = π σ lnfy ; θ)) = lnπ) σ ln ) β β exp y β 0 β ) σ β ) y β 0 β σ β ) Conditional: y t y t N β 0 + β y t, σ ) fy t y t ; θ) = exp π σ yt β0 βyt ) σ ) lnfy t y t ; θ)) = lnπ) lnσ ) yt β0 βyt ) σ Exact Log Likelihood: Lθ) = T ln fy t y t ; θ)) + ln fy ; θ)) t= 5

8 = T T lnπ) lnσ ) T t= y t β 0 β y t ) σ σ ln ) β y β 0 β ) σ β 6 a) σ + θ ) k = 0 Covy t, y t k ) = E[ɛ t θɛ t )ɛ t k θɛ t k )} = σ θ k = Covariance Matrix: σ + θ ) i j = 0 Σ ij = σ θ i j = 0 k 0 i j b) Likelihood Function: fy µ, Σ) = π) T/ Σ / exp y µ) Σ y µ) Conditional Log Likelihood: Ly µ, Σ) = T logπ) logσ) y µ) Σ y µ) 7 x i = ψ i ε i ψ i = ω + αx i x i = ωε i + αx i ε i fx i F i ; θ) = ψ i exp xi ψ i ) a) Conditional log likelihood conditional on x for sample N): The vector of parameters θ contains ω α The log likelihood can be written as follows: Lθ) = N lnψ i ) N x i ψ i = N lnω + αx i ) N ωε i+αx i ε i 6

9 b) First order conditions: [ω] : N + N [α] : N x i x i ) x ix i ) + N = 0 N = 0 N x i ) x ix i ) = N x i = N c) Re-expressing the rst order conditions in terms of ε: N ψ [ω] : iε i = N N ε i = N ) 0 = N ε i [α] : N ψ iε ix i ) x i = N ε ix i N x i = N 0 = N ε i )x i ω identies the level/constant, while α is identifying the slope coecient based on the orthogonality condition N ε i )x i = 0 7

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