02.„o“φiflì„㙃fic†j

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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.

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