Infinite-Dimensional Autoregressive Systems and the Generalized Dynamic Factor Model
[Systèmes Autorégressifs de Dimension Infinie et le Modèle à Facteurs Dynamiques Généralisé]
Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 71-81.

Le Modèle à Facteurs Dynamiques Généralisé est habituellement représenté comme une moyenne mobile infinie des chocs communs à laquelle s’ajoutent des composantes idiosyncratiques. Dans cet article, nous expliquons que cette représentation n’est que la solution (sous forme réduite) d’un système d’équations plus profond, structural, de dimension infinie, et contenant à la fois une partie autorégressive et une partie à moyenne mobile. Nous livrons des conditions qui garantissent que les solutions de tels systèmes sont faiblement stationnaires et nous analysons leur décomposition en composantes communes et idiosyncratiques. Enfin, nous établissons des liens avec la mémoire longue qui est engendrée par agrégation.

The Generalized Dynamic Factor Model is usually represented as an infinite-dimensional moving average of the common shocks plus idiosyncratic components. Here I argue that such representation can only be the solution (reduced form) of a deeper, structural, infinite-dimensional system of equations which contain both an autoregressive and a moving average part. I give conditions for the solutions of such infinite-dimensional systems to make sense and be weakly stationary, and study their decomposition into common and idiosyncratic components. Interesting links to long memory as generated by aggregation are also shown.

Keywords: Infinite-Dimensional Datasets, Generalized Dynamic Factor Models
Mot clés : Ensembles de données de dimension infinie, Modèle à Facteurs Dynamiques Généralisé 
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Lippi, Marco. Infinite-Dimensional Autoregressive Systems and the Generalized Dynamic Factor Model. Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 71-81. http://www.numdam.org/item/JSFS_2012__153_1_71_0/

[1] Bai, J. Inferential Theory for Factor Models of Large Dimensions, Econometrica, Volume 71 (2003) no. 1, pp. 135-171 http://ideas.repec.org/a/ecm/emetrp/v71y2003i1p135-171.html | MR | Zbl

[2] Bai, J.; Ng, S. Determining the Number of Factors in Approximate Factor Models, Econometrica, Volume 70 (2002) no. 1, pp. 191-221 http://ideas.repec.org/a/ecm/emetrp/v70y2002i1p191-221.html | MR | Zbl

[3] Chudik, A.; Pesaran, M. H. Infinite-dimensional VARs and factor models, Journal of Econometrics, Volume 163 (2011) no. 1, pp. 4-22 http://ideas.repec.org/a/eee/econom/v163y2011i1p4-22.html | MR | Zbl

[4] di Mauro, F.; Smith, L. V.; Dees, S.; Pesaran, M. H. Exploring the international linkages of the euro area: a global VAR analysis, Journal of Applied Econometrics, Volume 22 (2007) no. 1, pp. 1-38 http://ideas.repec.org/a/jae/japmet/v22y2007i1p1-38.html | MR

[5] Dunford, N.; Schwartz, J. T. Linear Operators, Part I, Wiley, New York, 1988 | Zbl

[6] Dupor, B. Aggregation and irrelevance in multi-sector models, Journal of Monetary Economics, Volume 43 (1999) no. 2, pp. 391-409 http://ideas.repec.org/a/eee/moneco/v43y1999i2p391-409.html

[7] Forni, M.; Giannone, D.; Lippi, M.; Reichlin, L. Opening The Black Box: Structural Factor Models With Large Cross Sections, Econometric Theory, Volume 25 (2009) no. 05, pp. 1319-1347 http://ideas.repec.org/a/cup/etheor/v25y2009i05p1319-1347_09.html | MR | Zbl

[8] Forni, M.; Hallin, M.; Lippi, M.; Reichlin, L. The Generalized Dynamic-Factor Model: Identification And Estimation, The Review of Economics and Statistics, Volume 82 (2000) no. 4, pp. 540-554 http://ideas.repec.org/a/tpr/restat/v82y2000i4p540-554.html

[9] Forni, M.; Lippi, M. The Generalized Dynamic Factor Model: Representation Theory, Econometric Theory, Volume 17 (2001) no. 06, pp. 1113-1141 http://ideas.repec.org/a/cup/etheor/v17y2001i06p1113-1141_17.html | MR | Zbl

[10] Granger, C. W. J. Long memory relationships and the aggregation of dynamic models, Journal of Econometrics, Volume 14 (1980) no. 2, pp. 227-238 http://ideas.repec.org/a/eee/econom/v14y1980i2p227-238.html | MR | Zbl

[11] Horvath, M. Sectoral shocks and aggregate fluctuations, Journal of Monetary Economics, Volume 45 (2000) no. 1, pp. 69-106 http://ideas.repec.org/a/eee/moneco/v45y2000i1p69-106.html

[12] Hansen, L. P.; Sargent, T. J. Formulating and estimating dynamic linear rational expectations models, Journal of Economic Dynamics and Control, Volume 2 (1980) no. 1, pp. 7-46 http://ideas.repec.org/a/eee/dyncon/v2y1980i2p7-46.html | MR

[13] Pesaran, M.H.; Chudik, A. Econometric Analysis of High Dimensional VARs Featuring a Dominant Unit (2010) no. 1024 http://ideas.repec.org/p/cam/camdae/1024.html (Cambridge Working Papers in Economics)

[14] Stock, J. H.; Watson, M. W. Macroeconomic Forecasting Using Diffusion Indexes, Journal of Business & Economic Statistics, Volume 20 (2002a) no. 2, p. 147-62 http://ideas.repec.org/a/bes/jnlbes/v20y2002i2p147-62.html | MR

[15] Stock, J. H.; Watson, M. W. Forecasting Using Principal Components From a Large Number of Predictors, Journal of the American Statistical Association, Volume 97 (2002b) no. 460, pp. 1167-1179 | MR | Zbl

[16] Zaffaroni, P. Contemporaneous aggregation of linear dynamic models in large economies, Journal of Econometrics, Volume 120 (2004) no. 1, pp. 75-102 http://ideas.repec.org/a/eee/econom/v120y2004i1p75-102.html | MR | Zbl