Working Paper Series. An Empirical Investigation of Direct and Iterated Multistep Conditional Forecasts. Michael W. McCracken and Joseph McGillicuddy

Transcription

1 RESEARCH DIVISION Working Paper Series An Empirical Investigation of Direct and Iterated Multistep Conditional Forecasts Michael W. McCracken and Joseph McGillicuddy Working Paper A November 2017 FEDERAL RESERVE BANK OF ST. LOUIS Research Division P.O. Box 442 St. Louis, MO The views expressed are those of the individual authors and do not necessarily reflect official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors. Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to Federal Reserve Bank of St. Louis Working Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

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6 y t x t

7 ( ) ( ) ( yt 0 b yt 1 = 0 c x t x t 1 ) ( et + v t ρ ), (y t, x t ) t y t+1 x t+1 ŷ c t,1 = ŷ u t,1 + ρ(ˆx c t,1 ˆx u t,1) = bx t + ρ(x t+1 cx t ) c u y MSE ŷt,1 u ˆx c t,1 = x t+1 e IMS t,1 = y t+1 ŷt,1 c E(eIMS t,1 ) 2 = 1 ρ 2 y t x t y t = γx t + ε t ŷ c t,1 = γˆx c t,1 = (bc + ρ(1 c 2 ))x t+1

8 e DMS t,1 = y t+1 ŷ c t,1 E(e DMS t,1 ) 2 = 1 ρ 2 + (b cρ) 2 (b cρ) 2 > 0 x t x t = αx t 2 +η t y x α = c 2 x 1 + c 2 ρ ŷ c t,1 = ŷ u t,1 + ˆρ(ˆx c t,1 ˆx u t,1) = bx t + ρ 1 + c 2 (x t+1 c 2 x t 1 ) E(e IMS t,1 ) 2 = 1 + ρ 2 2ρ2 1+c 2 E(e IMS t,1 ) 2 E(e DMS t,1 ) 2 2ρ 2 (1 1 1+c 2 ) < (b ρc)2 b = ρ 2 c = ρ 2ρ 2 (1 1 ) < 0 1+ρ 2 ρ = ρ 0 ρ 1 T T + 1 T ŷ c T,1 = ŷ u T,1 + ρ 0 (ˆx c T,1 ˆx u T,1) = bx T + ρ 0 (x T +1 cx T )

9 ŷ c T,1 = γˆx c T,1 = (bc + ρ 0 (1 c 2 ))x T +1 E(e IMS T,1 )2 = 1 + ρ 2 0 2ρ 0ρ 1 E(e DMS T,1 ) 2 = 1 + b 2 + ρ 2 0 (1 c2 ) 2bcρ 1 2ρ 0 ρ 1 (1 c 2 ) b > ρ 0 c E(e IMS t,1 ) 2 E(e DMS t,1 ) 2 2cρ 1 < b + ρ 0 c b = 0 c (0, 1) ρ 0 ( 1, 0) ρ 1 (0, 1) DMS Z t = (Y t, X t ) Y t+h y t Y t x t X t y x

10 Ŷ t,h c h Y t+h t = R,..., T h z t = C + A(L)z t 1 + ε t z = (y, x) ε = (ε y, ε x ) A(L) = p 1 j=0 A jl j h y t+h ŷ c t,h = ŷu t,h + 1 i h ˆγ i,t (x t+i ˆx u t,i) ˆγ i,t  i,t ˆΣ t = (t 2p 1) 1 t 1 s=1 ˆε s+1ˆε s+1 ŷu t,h ˆxu t,i y x Y t+h ŷ c t,i i = 1,..., h Y ŷ Ŷt,h c t,h c Y t I(0) = Y t + h i=1 ŷc t,i Y t I(1) Y t + h Y t + h i i=1 j=1 ŷc t,j Y t I(2) t = α + β j y t h j + δ j x t h j + y (h) p 1 j=0 y (h) t = Y t p 1 j=0 Y t Y t h Y t Y t h h Y t 1 i h Y t I(0) Y t I(1) Y t I(2) γ i x t h+i + ε t h y (h) t+h ŷ c(h) t,h p 1 = ˆα + j=0 p 1 ˆβ j,t y t j + ˆδ j,t x t j + j=0 1 i h ˆγ i,t x t+i Y t+h Y ŷ c(h) t,h Y t I(0) Ŷt,h c = Y t + ŷ c(h) t,h Y t I(1) Y t + h Y t + ŷ c(h) t,h Y t I(2)

11 p t AIC BIC p {0,..., 12} y x h = h = h x t+1,..., x t+h x t+h t = R,..., T h s = 1,.., t R = h T = R = h T = h

12 y x y x (y, x)

13 2, 000 y x y x 4, 000 [h 1.5] h = 3 y x x y

14 2, % h % h

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19 h ε i t,h i = c, u y t+h α 0 α 1 ˆε u t,h = ĝ t,h α + error ĝ t,h = (1, ŷu t,h ) α = (α 0, α 1 ) h k E(ε u t,h )2 E(ε c t,h )2 = k A i Σ γ i Ψ j Σ 1/2

20 j Σ 1/ R = Ψ 1 Σ 1/2 Σ 1/2 0 0 Σ 1/2 0 Ψ h 1 Σ 1/2 Ψ h 2 Σ 1/2 Ψ 1 Σ 1/2 Σ 1/2 R R k = ι R R ( R R ) 1 RR ι ι (ˆε u t,h )2 (ˆε c t,h )2 ˆk T = α + error ˆk T k A i Σ α Âi,t ˆΣ t

21 5% = 450 y x z 5% 25% %

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23 100(BIC(pre GM) BIC(GM))/ BIC(pre GM) h = 12 h = 3 24 h = 12 h = 3 24

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28 +h +h h 1.5 h = 3

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34 p = 4 Real, h=12 Financial, h= A 2 1B 3B 4 1A 1A: 14% 1B: 53% 2: 33% 3A: 0% 3B: 0% 4: 0% A: 0% 1B: 47% 2: 3% 3A: 1% 3B: 29% 4: 20% DMS Percent Improvement A: 6% 1B: 13% 2: 1% 3A: 4% 3B: 28% 4: 48% Nominal, h= Nominal, h=24 1A: 11% 1B: 47% 2: 23% 3A: 0% 3B: 11% 4: 9% DMS Percent Improvement IMS Percent Improvement IMS Percent Improvement (x1 x2)/ x1 x1 x2 ±40

35 x x t 2 x t log(x t ) log(x t ) 2 log(x t )

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38 x x t 2 x t log(x t ) log(x t ) 2 log(x t )