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1 X-12-ARIMA Band-PassDECOMP HP X-12-ARIMADECOMP HPBeveridge and Nelson DECOMP
2
3 X-12-ARIMA Band-PassHodrick and PrescottHP DECOMPBeveridge and Nelson M CD X ARIMA DECOMP
4 HP Band-PassDECOMP
5 Kiyotaki and Moore GhyselsMiron
6 GDPGDPGDP GDPGDP Haltmaier
7 CPI CPI Band Pass Hodrick and PrescottHP Beveridge and Nelson Blanchard and Quah
8 AIC Beveridge and Nelson DECOMP X-12-ARIMA Band-PassHP DECOMP Beveridge and Nelson Band-PassDECOMP X-12-ARIMA X-12-ARIMA TC t S t D t I t t TC t
9 S t I t TC t t S t I t S t +I t I t S t S t TC t I t I t t TC t S t I t ARIMA Band-Pass Band-Pass Band-Pass M2+CD
10 Hodrick and PrescottHP g t c t g t { MIN T T [( ) ( ) g ] } c g g g g 2 2 T t + λ t t 1 t 1 t 2 t t= 1 t = 1 t = 1 λ λλ λ λ DECOMP DECOMPKitagawa and Gersch T t V t S t D t ε t T t m T t v t v t τ BBT t T t DECOMP
11 AIC mm DECOMPAR AIC Beveridge and Nelson ARIMAMA MAMAt t+s s s+ E t p t + s = a s + p t + 1 i t 0 i= 1 i = 2 s+ 2 + i t t β ε β ε 1+ β 2 + i ε i = 3 p a 0 ε p t s+ 1 s + i t i t t s t + i β ε β ε 1+ β 2 + i ε i = 3 = 1 i = 2 s+ 2 µ t DECMPAICHP DECOMPAIC HP
12 CPI M2+CDP P volatilecpim2+cd CPIP PCPI CPIPM2+CD GDP P GDP X-12-ARIMA M2+CD
13
14
15
16 M2+CD P volatile CPI P CPIM2+CDIIP M2+CD volatilecpi M2+CD
17 ADFAugumented Dicky-Fulle n P = γ P + β P + ε t t 1 i t + i 1 t i= 2 (1) = α 0 P t = α 0 P t + γ P n + β P + ε t 1 i t + i 1 t i= 2 + γ Pt 1 + α 1 t + β i Pt + i + ε 1 t n i= 2 (2) (3) α α t ε ADFγ γ γ < CPIM2+CDP ADF CPI M2+CDP ADF near unit root X-12-ARIMA Ghysels and Perron M2+CD
18 -12-ARIMA X-12-ARIMA CPI M2+CDP
19
20
21
22
23 CPIM2+CDP CPIM2+CD Band-Pass CPI X-12-ARIMA XARIMA Band-PassM2+CDP
24 X-12-ARIMA
25 X-12-ARIMA
26 X-12-ARIMA
27 sinλ j t cosλ j t cosλ j t α j n 12 / p t = T α + j t ( α j cos λ β j sin λ t ) j j 1 T p t n 12 / = T α ( α j cos λ j t + β sin t ) j λ + α j j 1 t ( 1) n+1 T CPIBand-Pass CPI ARIMA Band-Pass MCD
28 X-12-ARIMA
29 Hodrick and PrescottHP HP Pλ=1 CPIM2+CDIIP Band-Pass M2+CDCPI M2+CDCPI HP P volatile volatilityhp CPIλ
30 X-12-ARIMA
31 X-12-ARIMA
32 X-12-ARIMA DECOMP CPM2+CDP Band-Pass AR DECOMP DECOMPAR
33 CPM2+CD M2+CD DECOMP P CPI DECOMP CPIP A ARvolatileAR AIC AR ARAIC ARAICAR M2+CDAICAR AR AICAR DECOMPAR
34 DECOMP
35
36 Beveridge and Nelson CPI CPI M2+CD PCPIM2+CDP Beveridge and NelsonARIMA CPI M2+CD P CPI ARIMA
37 X-12-ARIMA
38 X-12-ARIMA
39 CPI X-12-ARIMA ARIMA CPI ARIMA
40 ARIMA CPI
41 Band-Pas Band-Pass ARIMA Band-Pass ARIMA Band-Pass
42 Hodrick and Prescott HP HP HP DECOMP DECOMP CPI HP HP
43 DECOMP DECOMP Beveridge and Nelson Beveridge and Nelson CPI ARIMACPI ARIMA
44 X-12-ARIMA Band-PassHP DECOMPBeveridge and Nelson CPIM2+CDP X-12-ARIMA Band-PassDECOMP
45 CPI DECOMPBand-Pass Band-Pass ARIMA DECOMPBand-Pass
46
47 DECOMP X ARIMA DECOM DECOMPWhite Noise DECOMPWhite NoisAR
48 HP HP λ λ Band-Pass HP DECOMP DECOMP
49 time domain frequency domain p p 1, p 2 p t T T π Tn n = (T)/ 2 T n = (T-1)/ 2 T n λ j = 2πj / T j = 1,2,...,n λ sinλ t cosλ j t p t n 12 / p t = T α + t + t (4) 0 2 ( α cosλ β sin λ j j j j ) T j 1 n t 12 / p = + ( n+1 1) t T α ( α cosλ t t j j + β sin λ j j ) α j 1 Fourier representationα j β j (5) T αβ
50 α = ( 2 / T ) p cos λ j t j 12 / T t= 1 t (6) T 12 j p sin λ t j t = 1 β = ( 2 / T ) / t (7) t λ j α β PS T = α 2 + β 2 = ( 2 / T ) p cos t λjt + t = 1 PS j j j 2 T t =1 p sin t λj t 2 (8) nj=1,2,...,nπj Tλ j π j0 RATS VersionTent Window Tn λ j
51 j λ π π π π π π
52 Band-Pass Band-PassBand Band-Pass α j β j j α = ( 2 / T ) p cos λ j t = 0 j 12 / T t= 1 t (6) β = ( 2 / T ) / T p sin λ t = 0 12 j t j t = 1 (7) λ j = 2πj / T j = 1,2,...,n π π ππ π πππ
53 Band-Pass π ππ j α j β j Band-Pass
54 DECOMP DECOMPKitagawa and Gersch DECOMP DECOMP P t = T t + V t + S t +D t +ε t PTVARS Dε T T t (1 B) m T t = ν 1t ν 1t N(0, τ 2 1 ) BT t = T t-1 DECOMPDECOMPOSITION m
55 ARV ARV t nar V n t = a iv + ν t i 2 t i = ν 2 t N(0, τ ) (3) AR S qs t (1 B q )S t = 0 q 1 BS i = 0 (5) i= 0 t q 1 BS i t = i= 0 ν 3 t ν 3 N(0, τ ) t 2 3 (6) D td t β i i=1,,7 7 β it = 0 i=1 (7) tid it D t D t 7 β i t i= 1 6 = * D i t = β ( * i t D * i t ) β i t D i t i 1 6 D 7t i= 1 (8) qq=4q DECOMP
56 DECOMP AIC DECOMP MADECOMP MAAR AR DECOMP seasonal dip seasonal dip AIC DECOMP AICAIC AIC X-12-ARIMA DECOMP AIC
57 ~
58 ISM Research Memorandum DECOMP DECOMP IMES Discussion Paper X-12-ARIMA Harvey, A., Time Series Models, Philip Allan Publishers Limited, 1981 Beveridge, S. and C. Nelson, A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycles, Journal of Monetary Economics, Vol.7, 1981, pp Blanchard, O. and D. Quah, The Dynamic Effects of Aggregate Demand and Supply Disturbances, American Economic Review, 79, 1989, pp Ghysels, E., A Study Toward a Dynamic Theory of Seasonality for Economic Time Series, Journal of the American Statistical Association, 83, No.401, 1988, pp , and Perron, P., The Effect of Seasonal Adjustment Filters on Tests for a Unit Root, Journal of Econometrics, 55, 1993, pp Haltmaier, J., Inflation-Adjusted Potential Output, Board of Governors of the Federal Reserve System, International Finance Discussion Papers, 561, 1996.
59 Hodrick, R. and E., Prescott, Post-war U.S. Business Cycles: An Investigation, Working Paper, Carnegie-Mellon University, Kitagawa, G. and W., Gersch, A Smoothness Priors State Space Modeling of Time Series with Trend and Seasonality, Journal of the American Statistical Association, 79, No. 386, 1984, Kiyotaki, N., and J., Moore, Credit Cycles, NBER Working Paper, 5083, Miron, J., The Economics of Seasonal Cycles, MIT Press, 1996.
60
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