X-12-ARIMA Band-PassDECOMP HP X-12-ARIMADECOMP HPBeveridge and Nelson DECOMP
X-12-ARIMA Band-PassHodrick and PrescottHP DECOMPBeveridge and Nelson M CD X ARIMA DECOMP
HP Band-PassDECOMP
Kiyotaki and Moore GhyselsMiron
GDPGDPGDP GDPGDP Haltmaier
CPI CPI Band Pass Hodrick and PrescottHP Beveridge and Nelson Blanchard and Quah
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
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
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
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
CPI M2+CDP P volatilecpim2+cd CPIP PCPI CPIPM2+CD GDP P GDP X-12-ARIMA M2+CD
M2+CD P volatile CPI P CPIM2+CDIIP M2+CD volatilecpi M2+CD
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
-12-ARIMA X-12-ARIMA CPI M2+CDP
CPIM2+CDP CPIM2+CD Band-Pass CPI X-12-ARIMA XARIMA Band-PassM2+CDP
X-12-ARIMA
X-12-ARIMA
X-12-ARIMA
sinλ j t cosλ j t cosλ j t α j n 12 / p t = T α + j t + 0 2 ( α j cos λ β j sin λ t ) j j 1 T p t n 12 / = T α 0 + 2 ( 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
X-12-ARIMA
Hodrick and PrescottHP HP Pλ=1 CPIM2+CDIIP Band-Pass M2+CDCPI M2+CDCPI HP P volatile volatilityhp CPIλ
X-12-ARIMA
X-12-ARIMA
X-12-ARIMA DECOMP CPM2+CDP Band-Pass AR DECOMP DECOMPAR
CPM2+CD M2+CD DECOMP P CPI DECOMP CPIP A ARvolatileAR AIC AR ARAIC ARAICAR M2+CDAICAR AR AICAR DECOMPAR
DECOMP
Beveridge and Nelson CPI CPI M2+CD PCPIM2+CDP Beveridge and NelsonARIMA CPI M2+CD P CPI ARIMA
X-12-ARIMA
X-12-ARIMA
CPI X-12-ARIMA ARIMA CPI ARIMA
ARIMA CPI
Band-Pas Band-Pass ARIMA Band-Pass ARIMA Band-Pass
Hodrick and Prescott HP HP HP DECOMP DECOMP CPI HP HP
DECOMP DECOMP Beveridge and Nelson Beveridge and Nelson CPI ARIMACPI ARIMA
X-12-ARIMA Band-PassHP DECOMPBeveridge and Nelson CPIM2+CDP X-12-ARIMA Band-PassDECOMP
CPI DECOMPBand-Pass Band-Pass ARIMA DECOMPBand-Pass
DECOMP X ARIMA DECOM DECOMPWhite Noise DECOMPWhite NoisAR
HP HP λ λ Band-Pass HP DECOMP DECOMP
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 0 + 2 ( α cosλ t t j j + β sin λ j j ) α j 1 Fourier representationα j β j (5) T αβ
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
j λ π π π π π π
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
Band-Pass π ππ j α j β j Band-Pass
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
ARV ARV t nar V n t = a iv + ν t i 2 t i = 1 2 2 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
DECOMP AIC DECOMP MADECOMP MAAR AR DECOMP seasonal dip seasonal dip AIC DECOMP AICAIC AIC X-12-ARIMA DECOMP AIC
~
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.151-174. Blanchard, O. and D. Quah, The Dynamic Effects of Aggregate Demand and Supply Disturbances, American Economic Review, 79, 1989, pp.655-673. 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.168-172., and Perron, P., The Effect of Seasonal Adjustment Filters on Tests for a Unit Root, Journal of Econometrics, 55, 1993, pp.57-98. Haltmaier, J., Inflation-Adjusted Potential Output, Board of Governors of the Federal Reserve System, International Finance Discussion Papers, 561, 1996.
Hodrick, R. and E., Prescott, Post-war U.S. Business Cycles: An Investigation, Working Paper, Carnegie-Mellon University, 1980. 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, 378-389. Kiyotaki, N., and J., Moore, Credit Cycles, NBER Working Paper, 5083, 1995. Miron, J., The Economics of Seasonal Cycles, MIT Press, 1996.