43, 2, Forecasting Based Upon Cointegration Analysis and Its Applications Taku Yamamoto 80 The cointegration analysis has been a major

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1 43, 2, Forecasting Based Upon Cointegration Analysis and Its Applications Taku Yamamoto 80 The cointegration analysis has been a major topic in time series analysis in the field of economics since the mid-1980s. This paper gives a brief review on issues associated with forecasting in a cointegrated process, in particular, with forecasting accuracy. As an application, it concerns with forecasting of mortality rate based upon the life table. It discusses merits and problems associated with forecasting based upon cointegration analysis. : Box and Jenkins (1970) ARIMA (autoregressive integrated moving average) 1973 integration 70 Fuller (1976) Dickey and Fuller (1979) ( yamamoto.taku@nihon-u.ac.jp)

2 VAR (vector auroregressive) (cointegration) Granger (1981) Engle and Granger (1987) 20 Granger 1) 2) 3) (1) (2) (3) ) 2005 (2006) Johansen (1995) 2) Phillips (1998) Elliott (2006) 3)

3 m (vector moving-average: VMA) (1 L)y t = µ + C(L)ε t = µ + C i ε t i (2.1) L y t = [y 1t, y 2t,..., y mt ] µ C i m m {sc s } s=0 ε t (m 1) iid(0, Σ) 4 r β m r β C(1) = 0 {y t } µ µ Mortality Data Base 4 i=0 2.2 h T h T + h y T +h h h t T h y T +h = hµ + C j ε T +t + y T + C T t+j ε t (2.2) t=1 j=0 t=1 j=1

4 (0 109 ) 2 T 2 2 ŷ SY S T +h ŷ SY S T +h = hµ + y T + T t=1 j=1 h C T t+j ε t (2.3) ê SY T +h S (2.2) 2 ê SY S T +h = y T +h ŷ SY S T +h = h h t C j ε T +t (2.4) t=1 j=0 ê SY T +h S (Mean Squared Error: MSE) h h tr MSE(ê SY S T +h ) tr(e(ê SY S SY S T +h ê T +h )) = O(h) (2.5)

5 319 MSE T +h i lim h j=0 C j = C(1), (2.6) Engle and Yoo (1987) 4) lim h β {ŷt SY +h S (hµ + y T )} = 0 (2.7) 2.7 β ê SY T +h S β ê SY T +h S MSE lim tr h MSE(β ê SY T +h S ) = β Qβ < Q 2.3 5) MSE A ŷ A T +h ê A = y T +h ŷt A +h MSE tr MSE(ê A ) = tr(e(ê A T +hê A T +h)) 2 tr MSE tr MSE(ê A T +h ) tr MSE(ê B T +h ) ê B T +h B 3. 4) β µ 0 5) Christoffersen and Diebold (1998)

6 Engle and Yoo (1987) (2.1) 2 VEC (vector error correction) y t = αβ y t 1 + m + ε t (3.1) α 2 1 m 2 1 VEC VAR VAR A = I 2 + αβ y t = Ay t 1 + m + ε t 2 Engle and Yoo (1987) VEC Engle and Granger (1987) 2 VAR 20 h tr MSE tr MSE(ê V AR T +h ) tr MSE(ê V EC T +h ) (3.2) 1 Engle and Yoo (1987) Lin and Tsay (1996) 3.2 Christoffersen and Diebold (1998) VEC 1 ARIMA a 1 ARIMA Granger and Morris (1976) (1 L)y a,t = µ a + θ a,i u a,t i, (3.3) {u a,t i } θ a,i h h h+1 ŷa,t ARIMA +h = hµ a + y a,t + θ a,j u a,t + θ a,j u a,t 1 + j=1 i=0 = hµ a + y a,t + θ a (L)u a,t (a = 1, 2,, m) j=2

7 321 m ŷ ARIMA T +h = hµ + y T + Θ(L)u T (3.4) 1 ARIMA ŷ ARIMA T +h ê ARIMA T +h = y T +h ŷ ARIMA T +h (3.5) Christoffersen and Diebold (1998) 3.1 tr MSE(ê ARIMA t+h ) lim h tr MSE(ê V t+h EC) = 1 (3.6) 3.3 Chigira and Yamamoto (2012) (a) (misspecified) ŷ t+h MISS µ ŷ MISS T +h = µh + y T + v T +h (3.7) v T +h m 1 T tr E(v T +h v T +h ) = O(1) Chigira and Yamamoto (2012) 3.2 tr MSE(ê MISS T +h lim ) h tr MSE(ê SY T +h S) = 1 (3.8) 3.1 µh

8 Engle and Yoo (1987) Cristoffersen and Diebold (1998) 6) (b) A (2.1) µ C i (i = 0, 1,... ) ˆµ A ĈA i (i = 0, 1,... ) ˆµ A µ Var (ˆµ A ) = O(1/T ) ĈA i (i = 0, 1,... ) C i (i = 0, 1,... ) tr MSE[ĈA i C i ] = O(1) h ŷ A T +h = hˆµ A + y T + ê A T +h = y T +h ŷt A +h = (µ ˆµ A )h }{{} P t=1 j=0 T h t=1 j=1 } {{ } Q Ĉ A T i jε i h h t T C j ε T +t t=1 j=1 h (C T t+j ĈA T t+j)ε t } {{ } R = P + Q + R (3.9) T h E[P P ] = MSE[ˆµ A ]h 2 = O(h 2 ), E[QQ ] = E[RR ] = O(h), E[P R ] = o(h) (3.9) 1 P = (µ ˆµ A )h MSE h h ê A T +h MSE h 2 P tr MSE(e A T +h)/h 2 tr(e(ê A T +hê A T +h))/h 2 MSE[ˆµ A ] 6) Engle and Yoo (1987) Chigira and Yamamoto (2012, p. 358)

9 ˆµ A ˆµ B A B µ ê A T +h êb T +h h tr MSE(ê A T +h lim ) h tr MSE(ê B T +h ) = tr MSE[ˆµA ] tr MSE[ˆµ B = ] Chigira and Yamamoto (2012) VEC Engle and Yoo (1987) Engle and Granger (1987) 2 Johansen (1991) VEC 3.5 (1) ( h 30 (2)

10 (3) 4. (2013) 7) ) Bell (1997) Darkiewicz and Hoedemakers (2004) 9) Lee and Carter (1992) (1) t a w at (a = 1, 2,..., m, t = 1, 2,..., T ) y at = log(w at ) 7) (2013) Lee and Carter (1992) MTV 8) (2008) 9) Arlt et al. (2010) Carter (2010)

11 MTV m VEC MTV multivariate time series variance component MTV Kariya (1987) Chigira and Yamamto (2009) (i) ˇy t = y t ˆγ ˆµ trend t ˆγ ˆµ trend OLS (ii) {ˇy t } 2 ] ] B (m r) = [b 1... b m r B (r) = [b m r+1... b m b 1,..., b m π 1 π m B (m r)ˇy t I(1) B (r)ˇy t I(0) (4.1) 4.4 (iii) 1 (m r) B (m r)ˇy T +h ARIMA(p,1,q) B (m r)ˇy T +h (m r + 1) m B (r)ˇy T +h ARMA(p,q) B (r)ˇy T +h (iv) [B (m r), B (r) ] ˇy T +h (v) y T +h ŷ MT V T +h = (T + h)ˆµ trend + ˆγ + B (m r) B (m r)ˇy T +h + B (r) B (r)ˇy T +h (4.2) MTV 2 (1) m O(m) VEC O(m 2 ) T

12 (2) MTV MTV (1) Lee-Carter (2) 4.3 Lee-Carter Lee and Carter (1992) (LC) (i) ỹ t = y t ȳ ȳ = T t=1 y t/t (ii) {ỹ t } 1 f 1 1 {x 1t } x 1t = f 1ỹ t (t = 1, 2,..., T ) MTV {ỹ t } {x 1t } (iii) {x 1t } x 1t = α + x 1,t 1 + u t (4.3) 1 Lee and Carter (1992) (data generation process: DGP) (iv) {x 1t = f 1ỹ t } ˆα = 1 T 1 T t=2 x 1t = 1 T 1 (v) x 1t h T f 1 ỹ t = f 1 y t=2 ˆx 1,T +h = ˆαh + x 1,T = f 1 yh + f 1ỹ T (vi) y t h f 1

13 327 ŷ LC T +h = f 1ˆx 1,T +h + ȳ = f 1 (ˆαh + x 1,T ) + ȳ = f 1 (f 1 yh + f 1ỹ T ) + ȳ = f 1 f 1 yh + y LC T y LC T = f 1 f 1ỹ T + ȳ LC y T LC (2013 2) ŷt LC +h = f 1 f 1 yh + yt LC µh + y LC T T f 1 f 1 y µ LC 4 (1) µ (2) (4.3) (iv) (3) (4.3) LC 1 Girosi and King (2007) LC (1) y LC T y T T LC (2) ( (3) LC {y t } 1

14 Monte Carlo (a) VEC (Vector Error Correction) DGP (data generating process) y t = αβ y t 1 + µ + ε t, ε t NID(0, I m ) (4.4) α β (m r) µ (m 1) m = 30 r = 27 T = 50, 200 h = 1, 2,..., 5, 10, 20, 30, 40, ) (b) MTV (ii) (4.1) ˇy t 1 m r I(1) m r + 1 m I(0) Johansen (1991) m = 10 Chigira and Yamamoto (2009) Kwiatkowski et al. (KPSS) (1992) {b iˇy t} (i = 1, 2,..., m) 11) Kurozumi and Tanaka (2010) b 1ˇy t,..., b mˇy t 1 1 b 1ˇy t H 0 : b 1ˇy t I(0) vs. H 1 : b 1ˇy t I(1) H 0 : r = m vs. H 1 : r m 1 H 0 r = m H b 1ˇy t H 0 : b 2ˇy t I(0) vs. H 1 : b 2ˇy t I(1) H 0 : r = m 1 vs. H 1 : r m 2 H 0 r = m 1 H ) (2013) m = 3 r = 2 11) MTV Chigira and Yamamoto (2009) Phillips and Perron (1988)

15 329. m 1 m 1 b m 1ˇy t H 0 : b m 1ˇy t I(0) vs. H 1 : b m 1ˇy t I(1) H 0 : r = 2 vs. H 1 : r 1 H 0 r = 2 H 0 m m m b mˇy t H 0 : b mˇy t I(0) vs. H 1 : b mˇy t I(1) H 0 : r = 1 vs. H 1 : r = 0 H 0 r = 1 H 0 r = 0 ˆr 1 1 (ˆr) ( r = 27) (a) (b) (r 29) T \ˆr T \ˆr T ˆr 27 T ˆr 1 Breitung (2002) 12) 12) (2013) B

16 Trace MSE T = 50 h ratio (MTV) ratio (LC) T = 200 h ratio (MTV) ratio (LC) (c) 3 MTV LC ARIMA trace MSE ratio(mtv) = V tr MSE(ŷMT T +h ) tr MSE(ŷLC T +h tr MSE(ŷT ARIMA, ratio(lc) = ) +h ) tr MSE(ŷT ARIMA +h ) ARIMA 1 ARIMA T = 50 MTV ARIMA LC ARIMA T = 200 MTV ARIMA LC T = 50 ARIMA LC (1) T tr MSE MTV LC (a)

17 331 3 Trace MSE ) m = 30, ˆr = 27 h ratio (MTV) ratio (LC) m = (T = 58) ) 3 ˆr = 27 (1) MTV 1 ARIMA (2) LC ARIMA trace MSE 3 (3) MTV LC (b) Human Mortality Database m = (T = 210) h = 1, 2,..., 10, 20, 30, 40, 50 14) 4 3 ˆr = 26 MTV LC 4 Trace MSE ) (m = 30, ˆr = 26) h ratio (MTV) ratio (LC) ) (2013) ) (2013)

18 (0 109 ) ) Human Mortality Database LC 5. 15)

19 333 MTV Lee-Carter MTV Arlt, J., Arltova, M., Basta, M. and Langhamrova, J. (2010). Cointegrated Lee-Carter mortality forecasting method, COMPSTAT 2010, Bell, W. R. (1997). Comparing and assessing time series methods for forecasting age specific demographic rates, J. Off. Stat., 13, Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco. Breitung, J. (2002). Nonparametric tests for unit roots and cointegration, J. Econom., 108, Carter, L. R. (2010). Long-run relationships in differential U.S. mortality forecasts by race and sex: Tests for co-integration, in Ageing in Advanced Industrial States: Riding the Age Waves Volume 3, Tuljapurkar, S., Ogawa, N. and Gauthier, A. H. eds., Springer, Chigira, H. and Yamamoto, T. (2009). Forecasting in large cointegrated processes, J. Forecast., 28, Chigira, H. and Yamamoto, T. (2012). The effect of estimating parameters on long-term forecasts for cointegrated systems, J. Forecast., 31, (2013). Lee-Carter ( ). Christoffersen, P. F. and Diebold, F. X. (1998). Cointegration and long-horizon forecasting, J. Bus. Econ. Stat., 16, Darkiewicz, G. and Hoedemakers, T. (2004). How the co-integration analysis can help in mortality forecasting, manuscript, Actuarial Science Research Group, Catholic University of Leuven. Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root, J. Am. Stat. Assoc., 74,

20 Elliott, G. (2006). Forecasting with trending data, in Handbook of Economic Forecasting, Chapter 11 in Elliott, G. et al. eds., North-Holland. Engle, R. F. and Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation and testing, Econometrica, 55, Engle, R. F. and Yoo, S. (1987). Forecasting and testing in cointegrated systems, J. Econom., 35, Fuller, W. A. (1976). Introduction to Statistical Time Series, John Wiley & Sons. Girosi, F. and King, G. (2007). Understanding the Lee-Carter mortality forecasting method, unpublished manuscript, Center for Basic Research in the Social Sciences, Harvard University. Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification, J. Econom., 16, Granger, C. W. J. and Morris, M. J. (1976). Time series modelling and interpretation, J. R. Stat. Soc., Ser. A, 139, Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at or (data downloaded on 22/08/2012). Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models, Econometrica, 59, Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press. Kariya, T. (1987). MTV model and its application to the prediction of stock prices, Proceedings of the Second International Tampere Conference in Statistics, (2008). 21 I 8 Kurozumi, E. and Tanaka, S. (2010). Reducing the size distortion of the KPSS test, J. Time Ser. Anal., 31, Kwiatkowski, D., Phillips, P. C. B., Schmidt, P. and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root, J. Econom., 54, Lee, R. D. and Carter, L. A. (1992). Modeling and forecasting U.S. mortality, J. Am. Stat. Assoc., 87, Lin, J. L. and Tsay, R. (1996). Co-integration constraint and forecasting: An empirical examination, J. Appl. Econom., 11, Phillips, P. C. B. (1998). Impulse response and forecast error variance asymptotics in nonstationary VARs, J. Econom., 83, Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression, Biometrika, 75, (2006)

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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 }