Forecasting Large Datasets with Reduced Rank Multivariate Models

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and vec( ^ O 0 ) = gO vec( ^ A) (33) where ^ A = ( ^ A1; ^ A2; :::; ^ Ap) and A = (A1; A2; :::; Ap). Therefore, gO(:) and gK(:) de…ne the singular value decomposition operator. By theorems 5.6 and 5.8 of Chatelin (1983) gO(:) and gK(:) are bounded, continuous and di¤erentiable and therefore admit a …rst order Taylor expansion. Therefore, and vec( ^ K 0 ) vec(K 0 ) = @g0 K @A vec( ^ A) vec(A) @g0 K @A + op(T a ) (34) vec( ^ O 0 ) vec(O 0 ) = @g0 O @A vec( ^ A) vec(A) @g0 O @A + op(T a ) (35) By theorem 1 every element of vec( ^ A) vec(A) is op(T a ). The number of columns of @g0 K @A and @g0 O @A are of the order N 2 . Thus, each element of vec( ^ O 0 ) vec(O 0 ) and vec( ^ K 0 ) vec(K 0 ) is a linear combination of possibly all elements of vec( ^ A) vec(A) . It then follows that each element of vec( ^ O 0 ) vec(O 0 ) and vec( ^ K 0 ) vec(K 0 ) is op(T a+2b )- consistent. References [1] Ahn, S. K., and Reinsel, G. C., 1988, “Nested Reduced-Rank Autoregressive Models for Time Series”, Journal of the American Statistical Association, 83, 849–856. [2] An H.Z, Chen Z.G., Hannan, E.J., 1982, “Autocorrelation, Autoregression and Autoregressive Approximation”, The Annals of Statistics, Vol. 10, No. 3, pp. 926- 936. [3] Bai, J. and Ng, S., 2006. “Con…dence Intervals for Di¤usion Index Forecasts and Inference for Factor-Augmented Regressions”, Econometrica, 74(4), 1133-1150. [4] Bai, J. and Ng, S., 2009, “Boosting Di¤usion Indices”, Journal of Applied Econometrics, (2009). [5] Banbura M., Giannone D. and Reichlin, L., 2007, “Bayesian VARs with Large Panels”, CEPR working paper no.6326 [6] Breiman, L., 1996, “Bagging Predictors”, Machine Learning, 36, 105-139. [7] Buhlmann, P. 2006, “Boosting for High-Dimensional Linear Models”, Annals of Statistics 54:2, 559–583. [8] Buhlmann, P. and B. Yu, 2002, “Analyzing Bagging”, Annals of Statistics, 30, 927-961. 24
[9] Camba-Mendez, G., Kapetanios, G., Smith, R.J., and Weale, M.R., 2003, “Tests of Rank in Reduced Rank Regression Models”, Journal of Business and Economic Statistics, 21, 145-155. [10] Canova, F., and Ciccarelli, M., 2004. “Forecasting and Turning Point Predictions in a Bayesian Panel VAR Model”, Journal of Econometrics, Elsevier, vol. 120(2), pages 327-359, June. [11] Clements M.P. and Hendry D.F., 1999, “Forecasing Nonstationary Economic Time Series”, MIT press [12] De Mol, C., Giannone, D. and Reichlin, L., 2006, “Forecasting Using a Large Number of Predictors. Is Bayesian Regression a Valid Alternative to Principal Components?”, ECB Working paper no.700. [13] Deistler, M. and Hannan, E. J., 1988, “The Statistical Theory of Linear Systems”, Wiley. [14] Doan, T., R. Litterman, and C. A. Sims, 1984, “Forecasting and Conditional Projection Using Realistic Prior Distributions”, Econometric Reviews, 3, 1-100. [15] Eickmeier, S. and Ziegler, C., 2006, “How Good are Dynamic Factor Models at Forecasting Output and In‡ation? A meta-analytic approach”, Discussion Paper Series 1: Economic Studies, 42, Deutsche Bundesbank, Research Centre. [16] Forni, M., Hallin, M., Lippi, M. & Reichlin, L., 2000, “The Generalized Factor Model: Identi…cation and Estimation”, The Review of Economics and Statistics 82(4), 540-554. [17] Geweke, J., 1996, “Bayesian Reduced Rank Regression in Econometrics”, Journal of Econometrics 75, 121–146. [18] Giacomini R., White H., 2006, “Tests of Conditional Forecast Accuracy”, Econometrica, Vol. 74, pp. 1545-1578. [19] Hamilton, J.D., Waggoner, D.F. and Zha, T. 2007, “Normalization in Econometrics”, Econometric Reviews, 26 pp.221–252. [20] Hoogerheide, L.F., and Van Dijk, H.K., 2001, “Comparison of the Anderson-Rubin Test for Overidenti…cation and the Johansen Test for Cointegration”, Econometric Institute report EI 2001-04, Erasmus University Rotterdam. [21] Inoue, Atsushi and Kilian, Lutz, 2004, “Bagging Time Series Models”, CEPR Discussion Paper No. 4333. [22] Kadiyala, K. R., and S. Karlsson, 1997, “Numerical Methods for Estimation and Inference in Bayesian VAR-Models”, Journal of Applied Econometrics, 12(2), 99– 132. 25
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Page 6 and 7: 1 Introduction Forecasting future d
Page 8 and 9: sectoral data on industrial product
Page 10 and 11: i = 1; :::; p. An interesting speci
Page 12 and 13: In this paper we implement a Normal
Page 14 and 15: with and being respectively M r and
Page 16 and 17: 2.5 Multivariate Boosting (MB) The
Page 18 and 19: of boosting iterations m is a much
Page 20 and 21: and very likely to create problems
Page 22 and 23: multivariate benchmark is the basel
Page 24 and 25: ur with gains relatively smaller (b
Page 26 and 27: eduction produces additional gains
Page 30 and 31: [23] Kleibergen, F., van Dijk H.K.,
Page 32 and 33: Table 2: Relative WTMSFE vs AR(p*)
Page 34 and 35: Table 4: BVAR in Di¤erences and BV
Page 36 and 37: Table 6: Relative WTMSFE vs AR(p*)

Forecasting Large Datasets with Reduced Rank Multivariate Models

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