Hi-Stat Discussion Paper Series No.228 経済時系列分析と単位根検定 : これまでの発展と今後の展望黒住英司 December 27 Hitotsubashi University Research Unit for Statistical Analysis i

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1 経済時系列分析と単位根検定 : これまでの発展と今後の Title 展望 Author(s) 黒住, 英司 Citation Issue Date Type Technical Report Text Version publisher URL Right Hitotsubashi University Repository

2 Hi-Stat Discussion Paper Series No.228 経済時系列分析と単位根検定 : これまでの発展と今後の展望黒住英司 December 27 Hitotsubashi University Research Unit for Statistical Analysis in Social Sciences A 21st-Century COE Program Institute of Economic Research Hitotsubashi University Kunitachi, Tokyo, Japan

3 ADF

4 1. 3 AR ADF ADF 2 ADF 1

5 y t y t = μ z t + x t, φ(l)x t = u t (t =1, 2,,T). (1) z t u t i.i.d.(,σ 2 ) φ(l) L p φ(l) =1 φ 1 L φ 2 L 2 φ p L p x,x 1,,x 1 p T x t y t μ z t p (AR(p)) z t z t z t =1( ) z t =[1,t] ( ) y t μ z t ( x t = y t μ z t ) ( ) y t (1) x t φ(z) = 1 H : φ(1) = v.s. H 1 : φ(1) > (2) φ(1) > φ(1) < φ() = 1 (, 1) ( ) 2

6 I(1) (an integrated process of order 1) ( ) I() μ z t = μ (1) y t = μ + x t, φ(l)x t = u t φ(l) y t y t = c + φ 1 y t φ p y t p + u t (3) c = φ(1)μ Δy t = c + ρy t 1 + ψ 1 Δy t ψ p 1 Δy t p+1 + u t. (4) Δ =1 L p ρ = φ i 1, i=1 p ψ i = φ j (i =1, 2,,p 1) j=i+1 ρ = φ(1) c = φ(1)μ (2) (4) H : ρ = c = v.s. H 1 : ρ< (5) ( ) : Δy t = ψ 1 Δy t ψ p 1 Δy t p+1 + u t, ( ) : y t = c + φ 1 y t φ p y t p + u t, y t y t = μ + μ 1 t + x t, φ(l)x t = u t 3

7 y t y t = c + c 1 t + φ 1 y t φ p y t p + u t (6) c = φ(1)μ φ (1)μ 1 c 1 = φ(1)μ 1 (3) (4) Δy t = c + c 1 t + ρy t 1 + ψ 1 Δy t ψ p 1 Δy t p+1 + u t (7) (2) (7) H : ρ = c 1 = v.s. H 1 : ρ< (8) ( ) : Δy t = c + ψ 1 Δy t ψ p 1 Δy t p+1 + u t, ( ) : y t = c + c 1 t + φ 1 y t φ p y t p + u t, y t Dickey and Fuller (1981) (5) (8) c c 1 ρ = H : ρ = v.s. H 1 : ρ< (9) 2.2. ADF (9) H ρ 4

8 (4) Δy t y t 1 Δy t 1,, Δy t p+1 y t 1 ˆρ μ (7) Δy t y t 1 Δy t 1,, Δy t p+1 y t 1 ˆρ τ Dickey and Fuller (1979) Said and Dickey (1984) Phillips (1987) T ˆρ μ d 1 2 (B2 (1) 1) B(1) 1 B(s)ds 1 B2 (s)ds ( 1, Tˆρ τ B(s)ds)2 d 1 2 (B2 (1) 1) + A D. (1) B(r) r 1 ( 1 A =12 sb(s)ds 1 1 )( 1 B(s)ds B(s)ds 1 ) B(1) B(1) B(s)ds, 1 ( ( 1 ) 2 D = B 2 (s)ds 12 sb(s)ds) +12 B(s)ds sb(s)ds 4 B(s)ds T ˆρ μ T ˆρ τ ρ t (4) (7) ρ t t ADF μ t ADF τ t ADF μ d 1 2 (B2 (1) 1) B(1) 1 B(s)ds 1 B2 (s)ds (, t ADF τ 1 B(s)ds)2 d 1 2 (B2 (1) 1) + A D (11) (4) (7) 2 T ˆρ μ T ˆρ τ t ADF μ t ADF τ ADF(augmented Dickey-Fuller) p =1 AR(1) DF(Dickey-Fuller) t ADF t 2.3. ADF-GLS ADF ADF Elliott, Rothenberg 5

9 and Stock (1996, ERS ) ADF-GLS ADF-GLS POI(point optimal invariant) POI H 1 H l : ρ = v.s. H l 1 : ρ = ρ(θ) = θ T (θ>). α 1 ρ = ρ = θ /T θ ρ(θ) y =[y 1,y 2,,y T ] f(y; θ) Neyman-Pearson L(θ; α) f(y; θ) f(y;) k α ρ(θ) k α P θ =(L(θ; α) k α )=α ϕ(θ, θ ; α) P θ (L(θ; α) k α ) θ θ ϕ(θ, θ ; α) θ θ ρ(θ) POI POI ρ(θ) ρ(θ ) POI ρ(θ) θ θ = θ ϕ(θ,θ ; α) 6

10 1: POI ϕ(θ,θ ; α) θ POI 1 θ = θ1 POI θ1 θ2 POI POI King (1983) Tanaka (1996) POI ϕ(θ, θ ; α) 5% θ POI θ ERS POI θ =7 θ =13.5 POI ERS POI ADF ADF 7

11 ADF-GLS (1) y t z t (2) y qd t { { y qd y1 : t =1 t = y t ay t 1 : t 2, z1 : t =1 zqd t = z t az t 1 : t 2 a =1 7/T a =1 13.5/T z qd t ˆμ qd (3) y t x qd t = y t ˆμ qdz t (4) x qd t ADF (t ) Δx qd t OLS ρ t t GLS μ ) = ρx qd t 1 + ψ 1Δx qd t ψ p 1Δx qd t p+1 + e t ( ) t GLS τ ( ERS t GLS μ t GLS μ d t GLS τ 1 2 (B2 (1) 1) 2 B2 (s)ds, t GLS τ d 1 2 (V 2 (1,θ) 1) 2 V 2 (s, θ)ds. V (r, θ) =B(r) r{λb(1) + 3(1 λ) 1 sb(s)ds} λ =(1 θ)/(1 θ + θ2 /3) θ =7 θ =13.5 t GLS μ ADF-GLS t GLS τ 2 ADF ADF-GLS ADF-GLS ADF ADF-GLS 8

12 ADF ADF-GLS ADF ADF-GLS c (i) c (ii) 2: ADF ADF-GLS 2.4. Box and Jenkins (1971) p q (ARMA(p, q)) y t = μ z t + x t, φ(l)x t = ϑ(l)u t. p φ(l) 1 ϑ(l) q φ(l) ϑ(l) φ(l) (ARIMA(p 1, 1,q)) (MA) ARIMA ARIMA Hall (1989) Pantula and Hall (1991) Ahn (1993) ARIMA MA ϑ(l) 1 ϑ(l) AR( ) φ (L)x t = u t, φ (L) φ(l)ϑ 1 (L) = φ i L i, (12) i= 9

13 AR( ) AR(p) ADF Berk (1974) Said and Dickey (1984) Ng and Perron (1995) p T ARIMA y t = μ z t + x t, (1 φl)x t = w t, w t. (13) Wold w t w t = ψ i u t i, ψ =1, i=1 ψi 2 < i=1 w t MA( ) Brillinger(1981) MA w t (12) AR( ) (13) AR( ) AR (4) (7) ADF ADF-GLS Ng and Perron (1995) (1) T Schwartz (1989) { ( ) } T 1/4 p =int 4 1 p =int { 12 ( ) } T 1/4 1 (2) AIC BIC p (3) p p p = p φ p t AR( p) φ p t AR( p 1) AR( p 1) 1

14 φ p 1 t AR( p 2) p Ng and Perron (1995) 3 Ng and Perron (21) MA Ng and Perron (21) AIC BIC (MIC) MIC =lnˆσ p 2 + C T (τ T (p)+p 1). T p ˆρ û p,t (4) (7) OLS ˆσ 2 p = 1 T p T û 2 p,t, t= p+1 τ T (p) = ˆρ2 T t= p+1 ( x qd t 1 )2 ˆσ 2 p AIC MAIC C T =2 BIC MBIC C T = ln(t p) Ng and Perron (21) MA MAIC ADF-GLS ADF Müller and Elliott (23) ADF-GLS ADF 11

15 AR(1) y t = μ z t + x t, x t = φx t 1 + u t. (14) φ =1 t x t = x + u i i=1 x z t x t x t = ρ t x + ρ t i u i i=1 ρ t x Müller and Elliott (23) x F (x ) N(,kσ 2 ) ϕ (φ,k) ϕ (φ,k) = arg ϕ max P (ϕ rejects φ = φ,x ) df (x ) ϕ ϕ (φ,k) k = ϕ(φ, ) x = k Müller and Elliott (23) ADF ADF-GLS ADF-GLS k = ADF k AR(1) (14) T = 1 3 ( 3(i-a)) ADF-GLS ADF 5 ( 3(i-b))ADF-GLS ADF ( 3(ii)) 12

16 1 ADF (x=) ADF-GLS (x=) 1 ADF (x=5) ADF-GLS (x=5) phi phi (i-a) (x =) (i-b) (x =5) 1 ADF (x=) ADF-GLS (x=) 1 ADF (x=5) ADF-GLS (x=5) phi phi (ii-a) (x =) (ii-b) (x =5) 3: ρ t Elliott and Müller (26) ˆQ m = q m + q m 1 ( ŷm T ) 2 + q m 3 ŷ m ŷm T T + q m 4 Tt=1 (ŷm t 1 ) 2 T 2. m = μ ŷ μ t =(y t T s= y s /T )/ˆω ˆω 2 x t AR(1) 13

17 (2π ) q μ = 1, qμ 1 = 4.95, qμ 2 =9.5, qμ 3 =.9, qμ 4 = 1 m = τ ŷ τ t ŷμ t 1 t q τ = 15, q τ 1 = 7.127, q τ 2 =12.166, q τ 3 = 3.31, q τ 4 = 225 Harvey and Leybourne (25) ADF ADF-GLS Harvey and Leybourne (26) ADF Elliott and Müller (26) ˆQ m Müller and Elliott (23) 2.5. ADF ADF-GLS Phillips and Perron (1988) (13) DF w t DF Phillips-Perron ADF Perron and NG (1996) OLS Pantula, Gonzalez-Farias and Fuller (1994) 14

18 AR AR WS (weighted symmetric) So and Shin (1999) y t 1 Cauchy Shin and So (21) ADF Wald Dickey and Fuller (1981) Schmidt and Phillips (1992) ADF Müller and Elliott (23) ADF Nelson and Plosser (1982) 4 common stock prices ( ) 1929 ( ) Perron (1989) Perron (1989) ( ) A: y t = μ a + μb DU t + μ 1 t + x t, x t = φx t 1 + w t, 15

19 : B: y t = μ + μ a 1 t + μb 1 DT t + x t, x t = φx t 1 + w t, C: y t = μ a + μb DU t + μ a 1 t + μb 1 DT t + x t, x t = φx t 1 + w t, w t T B DU t = { : t T B 1 : t>t B,DT t = { : t T B t T B : t>t B,DT t = { : t T B t : t>t B φ =1 φ <1 A B C TB +1 y t AO (additive outlier) y t IO (innovational outlier) AO A B C y t 16

20 { A: yt = c a + dd(t B ) t + y t 1 + w t, A: y t = c a + cb DU t + c 1 t + w t, { B: yt = c + c 1 DU t + y t 1 + w t, B: y t = c + c a 1 t + cb 1 DT t + w t, { C: yt = c + dd(tb ) t + C 1 DU t + y t 1 + w t, C: y t = c a + cb DU t + c a 1 t + cb 1 DT t + w t, D(TB ) t t = TB +1 1 Perron (1989) y t Perron (1989) T (λ = T B /T ) Park and Sung (1994) A C Perron (199) Perron and Vogelsang (1992b) 3.2. (1) Perron (1989) Christiano (1992) Banerjee, Lumsdaine and Stock (1992) Zivot and Andrews (1992) Banerjee, Lumsdaine and Stock (1992) Zivot and Andrews (1992) T B y t ADF t ρ (λ) λ = T B /T λ Λ 17

21 λ Λ t ρ (λ) inf λ Λ t ρ Λ Zivot and Andrews (1992) Λ=[2/T, (T 1)/T ] Perron (1997) c b c b 1 t t ˆλ c t ρ (ˆλ c ) Amsler and Lee (1995) Perron and Vogelsang (1992a) 2.1 Perron (1989) Perron (1989) 3.3. (2) Leybourne, Mills and Newbold (1998) ( ) ADF Perron (1989) 18

22 Perron (1989) Leybourne, Mills and Newbold (1998) Vogelsang and Perron (1998) Hatanaka and Yamada (1999) Zivot and Andews (1992) Vogelsang and Perron (1998) Hatanaka and Yamada (1999) Zivot and Andrews (1992) ˆλ t ρ (ˆλ) Harvey, Leybourne and Newbold (21) Lee and Strazicich (21), Carrion-i-Silvestre and Sansó (26) Perron and Rodríguez (23) ADF-GLS Liu and Rodrǵuez (26) Saikkonen and Lütkepohl (22) 3.4. Perron (1989) Lumsdaine and Papell (1997) Kim, Leybourne and Newbold (2) 2 19

23 1 Hatanaka and Yamada (1999) Lee and Strazicich (23) Ng and Perron (21) MAIC AR Elliott and Müller (23) Elliott and Müller (26) 2 ADF ADF ADF-GLS ADF-GLS Ng and Perron (21) MAIC Elliott and Müller (26) Harvey and Leybourne (25) 2

24 ADF ADF-GLS m m m Hatanaka and Yamada (1999) 1 TAR (threshold AR) STAR (smooth transition AR) [1] Ahn, S. K. (1993). Some Tests for Unit Roots in Autoregressive-Integrated-Moving Average Models with Deterministic Trends. Biometrika 8, [2] Amsler, C. and J. Lee (1995). An LM Test for a Unit Root in the Presence of a Structural Change. Econometric Theory 11,

25 [3] Banerjee, A., R. L. Lumsdaine and J. H. Stock (1992). Recursive and Sequential Tests of the Unit-Root and Trend-Break Hypothesis: Theory and International Evidence. Journal of Business and Economic Statistics 1, [4] Berk, K. N. (1974). Consistent Autoregressive Spectral Estimates. Annals of Statistics 2, [5] Box, G. E. P. and G. M. Jenkins (1976). Time Series Analysis: Forecasting and Control (2nd. ed.). Honden-Day, San Francisco. [6] Brillinger, D. R. (1981). Time Series Data Analysis and Theory. Holden-Day, San Francisco. [7] Carrion-i-Silvestre, J. L. and A. Sansó (26). Joint Hypothesis Specification for Unit Root Tests with a Structural Break. Econometrics Journal 9, [8] Christiano, L. J. (1992). Searching for a Break in GNP. Journal of Business and Economic Statistics 1, [9] Dickey, D. A. and W. A. Fuller (1979). Distribution of the Estimators for Autoregressive Time Series With a Unit Root. Journal of the American Statistical Association 74, [1] Dickey, D. A. and W. A. Fuller (1981). Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root. Econometrica 49, [11] Elliott, G. and U. K. Müller (26). Minimizing the impact of the initial condition on testing for unit roots. Econometrica 71, [12] Elliott, G., T. J. Rothenberg and J. Stock (1996). Efficient Tests for an Autoregressive Unit Root. Econometrica 64, [13] Hall, P. (1989). Testing for a Unit Root in the Presence of Moving Average Errors. Biometrika 76,

26 [14] Harvey, D. I. and S. J. Leybourne (25). On testing for unit roots and the initial condition. Econometrics Journal 8, [15] Harvey, D. I. and S. J. Leybourne (26). Power of a Unit-Root test and the Initial Condition. Journal of Time Series Analysis 27, [16] Harvey, D., S. J. Leybourne and P. Newbold (21). Innovational Outlier Unit Root Tests with an Endogenously Determined Break in Level. Oxford Bulletin of Economics and Statistics 63, [17] Harvey, D. I., S. J. Leybourne and P. Newbold (21). Innovational Outlier Unit Root Tests with an Endogenously Determined Break in Level. Oxford Bulletin of Economics and Statistics 63, [18] Hatanaka, M. and K. Yamamda (1999). A Unit Root Test in the Presence of Structural Changes in I(1) and I() Models. In R. F. Engle and H. White (eds.), Cointegration, causality, and forecasting: A festschrift in honour of Clive W. J. Granger, [19] Kim, T-H., S. J. Leybourne and P. Newbold (2). Spurious Rejections by Perron Tests in the Presence of a Break. Oxford Bulletin of Economics and Statistics 62, [2] King, M. L. (1983). Testing for Moving Average Regression Disturbances. Australian Journal of Statistics 25, [21] Lee, J. and M. C. Strazicich (21). Break Point Estimation and Spurious Rejections with Endogenous Unit Root Tests. Oxford Bulletin of Economics and Statistics 63, [22] Lee, J. and M. C. Strazicich (23). Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks. Review of Economics and Statistics 85, [23] Leybourne, S. J., T. C. Mills and P. Newbold (1998). Spurious Rejections by Dickey- Fuller Tests in the Presence of a Break under the Null. Journal of Econometrics 87,

27 [24] Liu, H. and G. Rodrǵuez (26). Unit Root Tests and Structural Change When the Initial Observation is Drawn from its Unconditional Distribution. Econometrics Journal 9, [25] Lumsdaine, R. L. and D. H. Papell (1997). Multiple Trend Breaks and the Unit-Root Hypothesis. Review of Economics and Statistics 79, [26] Müller, U. K. and G. Elliott (23). Tests for Unit Roots and the Initial Condition. Journal of Econometrics 135, [27] Nelson, C. R. and C. I. Plosser (1982). Trends and Random Walks in Macroeconomic Time Series. Journal of Monetary Economics 1, [28] Ng, S. and P. Perron (1995). Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag. Journal of the American Statistical Society 9, [29] Ng, S. and P. Perron (21). Lag Length Selection and the Construction of Unit Root Tests With Good Size and Power. Econometrica 69, [3] Pantula, S. G., G. Gonzales-Farias and W. A. Fuller (1994). A Comparison of Unit-Root Test Criteria. Journal of Business and Economic Statistics 12, [31] Pantula, S. G. and A. Hall (1991). Testing for Unit Roots in Autoregressive Moving Average Models: An Instrumental Variable Approach. Journal of Econometrics 48, [32] Park, J. Y. and J. Sung (1994). Testing for Unit Roots in Models with Structural Change. Econometric Theory 1, [33] Perron, P. (1989). The Great Crash, the Oil Price shock, and the Unit Root Hypothesis. Econometrica 57, (Erratum, 61, ). [34] Perron, P. (199). Testing for a Unit Root in a Time Series with a Changing Mean. Journal of Business and Economic Statistics 8,

28 [35] Perron, P. (1997). Further Evidence on Breaking Trend Functions in Macroeconomic Variables. Journal of Econometrics 8, [36] Perron, P. and S. Ng (1996). Useful Modifications to Some Unit Root Tests with Dependent Errors and Their Local Asymptotic Properties. Review of Economic Studies 63, [37] Perron, P. and G. Rodríguez (23). GLS Detrending, Efficient Unit Root Tests and Structural Change. Journal of Econometrics 115, [38] Perron, P. and T. J. Vogelsang (1992a). Nonstationarity and Level Shifts with an Application to Purchasing Power Parity. Journal of Business and Economic Statistics 1, [39] Perron, P. and T. J. Vogelsang (1992b). Testing for a Unit Root in a Time Series with a Changing Mean: Corrections and Extensions. Journal of Business and Economic Statistics 1, [4] Phillips, P. C. B. (1987). Time Series Regression with a Unit Root. Econometrica 55, [41] Phillips, P. C. B. and P. Perron (1988). Testing for a Unit Root in Time Series Regression. Biometrika 75, [42] Said, E. S. and D. A. Dickey (1984). Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order. Biometrika 71, [43] Saikkonen, P. and H. Lütkepohl (22). Testing for a Unit Root in a Time Series with a Level Shift at Unknown Time. Econometric Theory 18, [44] Schmidt, P. and P. C. B. Phillips (1992). LM Tests for a Unit Root in the Presence of Deterministic Trends. Oxford Bulletin of Economics and Statistics 54, [45] Schwart, G. W. (1989). Tests for Unit Roots: A Monte Carlo Investigation. Journal of Business and Economic Statistics 7,

29 [46] Shin, D. W. and B. S. So (21). Recursive Mean Adjustment for Unit Root Tests. Journal of Time Series Analysis 22, [47] So, B. S. and D. W. Shin (1999). Cauchy Estimators for Autoregressive Processes with Applications to Unit Root Tests and Confidence Intervals. Econometric Theory 15, [48] Tanaka, K. (1996). Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. Wiley, New York. [49] Vogelsang, T. J. and P. Perron (1998). Additional Tests for a Unit Root Allowing for a Break in the Trend Function at an Unknown Time. International Economic Review 39, [5] Zivot, E. and W. K. Andrews (1992). Further Evidence on the Great Crash, the Oil- Price Shock, and the Unit Root Hypothesis. Journal of Business and Economic Statistics 1,

seminar0220a.dvi

seminar0220a.dvi 1 Hi-Stat 2 16 2 20 16:30-18:00 2 2 217 1 COE 4 COE RA E-MAIL: ged0104@srv.cc.hit-u.ac.jp 2004 2 25 S-PLUS S-PLUS S-PLUS S-code 2 [8] [8] [8] 1 2 ARFIMA(p, d, q) FI(d) φ(l)(1 L) d x t = θ(l)ε t ({ε t }

Hi-Stat Discussion Paper Series No.228 経済時系列分析と単位根検定 : これまでの発展と今後の展望 黒住英司 December 27 Hitotsubashi University Research Unit for Statistical Analysis i

Hi-Stat Discussion Paper Series No.228 経済時系列分析と単位根検定 : これまでの発展と今後の展望黒住英司 December 27 Hitotsubashi University Research Unit for Statistical Analysis i