On model selection problems in terms of prediction mean squared error and interpretaion of AIC (slides)
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1 Applications in Econometrics and Finance by Long Memory Processes :30-16:10
2 Table of Contents PART1 PART2 PART3
3 PART1
4 1 {y t } ρ(h) = ( h ) = Cov[y t y t+h ]/ Var[y t ] (yt y)(y t+h y) ρ(h) = ( h S) = (yt y) Example of plot: value of rho(h) lag h
5 2 A Short Memory Time Series A Long Memory Time Series Series : SM Series : LM (in months) (in months)
6 3 A Seasonal Time Series
7 4 SLM1 SLM Series : SLM1 Series : SLM (in months) (in months)
8 MeanBr TrendBr VarBr MemoryBr
9 6
10 7 US GNP US CPI asset pricing model exchange rate interest rate
11 [1] Baillie, R.T. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, [2] Maddala, G. S. and Kim, I.M. (1998) Unit roots, cointegration, and structural change (Themes in modern econometrics). Cambridge: Cambridge University Press. [3] (2003), 8 ( ), ,
12 PART2
13 8 p t GDP yt n GDP w t r t n = 103 p t, y n t, w t p t, yt n National Account Statistics, Japanese Cabinet Office web page ( w t, r t International Financial Statistics (2004), International Monetary Fund.
14 9 (p t, y n t, w t, r t ) p t, y n t, w t 20 r t I(1) Pt Ytn D1Pt D1Ytn Wt Rt D1Wt D1Rt
15 10 p t, yt n, w t, r t S r t I(1) p t, yt n, w t Series : D1Pt Series : D1Ytn Series : D1Wt Series : D1Rt
16 11 F I(d) I(1) (1 L)X t = ε t I(2) (1 L) 2 X t = ε t F I(d) (1 L) d X t = ε t, d F I(1) = I(1), F I(2) = I(2) d < 0.5 F I(d) 0.5 < d F I(d)
17 12 S F I(d), d = 0.25, 0.45, 0.75, 1.00 Series : d0.25 Series : d Series : d0.75 Series : d
18 13 S p t, y n t, w t, F I(0.40) Series : D1Pt Series : D1Ytn Series : D1Wt Series : FId
19 14 S 2 p t, 2 y n t, 2 w t, F I(0.40) Series : D2Pt Series : D2Ytn Series : D2Wt Series : D1FId
20 15 S p t, yt n, w t I(1) F I(d) 0 d < F I(d 1) Series : D2Pt 1 ρ(1) = d/(1 d). 1.0 d < 0.5, 0.33 < ρ(1) 0.50 Series : D2Wt Series : D2Ytn Series : D1FId
21 16 20 S Series : D1Pt[ - c(1:20)] Series : D1Ytn[ - c(1:20)] Series : D1Wt[ - c(1:20)] Series : D1Rt[ - c(1:20)]
22 17 PART2 r t I(1) 1 p t, yt n, w t F I(d) F I(d) I(0) d Fractional cointegration I(1) cointegration
23 PART3
24 18 SARFIMA (1 L) d 0 (1 L s ) d s y t = ε t. s s = 4 12 (1 L s ) d s SARIMA d 0 d s SARFIMA d 0 d s
25 19 SARFIMA d 0 = d s = Monthly Data s Yearly Data s
26 20 SARFIMA SARFIMA (1 L) d 0 (1 L s ) d s (y t µ) = u t (u t SARMA) Katyama (2007, Hitotsubashi J. of Econ.) SARIMA SARFIMA.
27 21 HP 10,,,,, 7 3 SARIMA SARIMA SARFIMA
28 22 Japanese Total Power Consumption *10^7 2.1*10^7 2.2*10^7 2.3*10^7 2.4*10^7 Jan 1996 Jan 1998 Jan 2000 Jan 2002 Jan 2004 Time in months
29 23 S Series : A Series : B (in months) Series : C (in months) Series : D (in months) (in months)
30 24 step1. SARFIMA SARIMA step2. step3. AIC BIC step4. SARFIMA 1 ID50 (1 L) (y t ) = ( L L 24 )ε t, y t = (1 L)(1 L 12 )x t, and bσ =
31 25 ID50 Plots of Data, Forecasts, and Forecast Intervals 2*10^7 2.4*10^7 Jan 1996 Jan 1998 Jan 2000 Jan 2002 Jan 2004 Jan 2006 Time in months Enlarged Plots of above 2*10^7 2.3*10^7 2.6*10^7 Jan 4 Jan 5 Jan 6 Jan 7 Time in months
32 26 SARFIMA SARIMA SARFIMA SARIFIMA ID50 SARFIMA ID50 SARIMA 3 ID
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CONTENTS P.3 P.5 P.7 P.13 P.21 P.31 P.33 P.3 P.3 P.3 1 2 3 4 4 1 5 6 2 7 8 2 2 9 10 2 2 1000 800 600 400 200 0 3 4 5 11 12 13 14 15 16 17 18 25 1 8 15 22 29 1 18 25 1 6 17 6 8 11 12 3 13 14 15 16 3 311
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