IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1910.03821.html
   My bibliography  Save this paper

Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

Author

Listed:
  • Matteo Barigozzi
  • Matteo Luciani

Abstract

We study estimation of large Dynamic Factor models implemented through the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother. We prove that as both the cross-sectional dimension, $n$, and the sample size, $T$, diverge to infinity: (i) the estimated loadings are $\sqrt T$-consistent, asymptotically normal and equivalent to their Quasi Maximum Likelihood estimates; (ii) the estimated factors are $\sqrt n$-consistent, asymptotically normal and equivalent to their Weighted Least Squares estimates. Moreover, the estimated loadings are asymptotically as efficient as those obtained by Principal Components analysis, while the estimated factors are more efficient if the idiosyncratic covariance is sparse enough.

Suggested Citation

  • Matteo Barigozzi & Matteo Luciani, 2019. "Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm," Papers 1910.03821, arXiv.org, revised Sep 2024.
    • Handle: RePEc:arx:papers:1910.03821
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1910.03821
    File Function: Latest version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Roberto S. Mariano & Yasutomo Murasawa, 2003. "A new coincident index of business cycles based on monthly and quarterly series," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(4), pages 427-443.
    2. Juan Antolin-Diaz & Thomas Drechsel & Ivan Petrella, 2017. "Tracking the Slowdown in Long-Run GDP Growth," The Review of Economics and Statistics, MIT Press, vol. 99(2), pages 343-356, May.
    3. Giannone, Domenico & Reichlin, Lucrezia & Small, David, 2008. "Nowcasting: The real-time informational content of macroeconomic data," Journal of Monetary Economics, Elsevier, vol. 55(4), pages 665-676, May.
    4. Forni, Mario & Lippi, Marco, 2001. "The Generalized Dynamic Factor Model: Representation Theory," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1113-1141, December.
    5. Danny Quah & Thomas J. Sargent, 1993. "A Dynamic Index Model for Large Cross Sections," NBER Chapters, in: Business Cycles, Indicators, and Forecasting, pages 285-310, National Bureau of Economic Research, Inc.
    6. Gourieroux,Christian & Monfort,Alain, 1995. "Statistics and Econometric Models 2 volume set," Cambridge Books, Cambridge University Press, number 9780521478373, July.
    7. Forni, Mario & Hallin, Marc & Lippi, Marco & Zaffaroni, Paolo, 2017. "Dynamic factor models with infinite-dimensional factor space: Asymptotic analysis," Journal of Econometrics, Elsevier, vol. 199(1), pages 74-92.
    8. Sundberg,Rolf, 2019. "Statistical Modelling by Exponential Families," Cambridge Books, Cambridge University Press, number 9781108701112, September.
    9. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    10. Doz, Catherine & Giannone, Domenico & Reichlin, Lucrezia, 2011. "A two-step estimator for large approximate dynamic factor models based on Kalman filtering," Journal of Econometrics, Elsevier, vol. 164(1), pages 188-205, September.
    11. Jushan Bai & Serena Ng, 2002. "Determining the Number of Factors in Approximate Factor Models," Econometrica, Econometric Society, vol. 70(1), pages 191-221, January.
    12. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    13. Laura Coroneo & Domenico Giannone & Michele Modugno, 2016. "Unspanned Macroeconomic Factors in the Yield Curve," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(3), pages 472-485, July.
    14. Fiorentini, Gabriele & Galesi, Alessandro & Sentana, Enrique, 2018. "A spectral EM algorithm for dynamic factor models," Journal of Econometrics, Elsevier, vol. 205(1), pages 249-279.
    15. Bai, Jushan & Ng, Serena, 2007. "Determining the Number of Primitive Shocks in Factor Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 52-60, January.
    16. Forni, Mario & Giannone, Domenico & Lippi, Marco & Reichlin, Lucrezia, 2009. "Opening The Black Box: Structural Factor Models With Large Cross Sections," Econometric Theory, Cambridge University Press, vol. 25(5), pages 1319-1347, October.
    17. Catherine Doz & Domenico Giannone & Lucrezia Reichlin, 2012. "A Quasi–Maximum Likelihood Approach for Large, Approximate Dynamic Factor Models," The Review of Economics and Statistics, MIT Press, vol. 94(4), pages 1014-1024, November.
    18. Matteo Luciani, 2015. "Monetary Policy and the Housing Market: A Structural Factor Analysis," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(2), pages 199-218, March.
    19. Luciana Juvenal & Ivan Petrella, 2015. "Speculation in the Oil Market," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(4), pages 621-649, June.
    20. Li, Kunpeng & Li, Qi & Lu, Lina, 2018. "Quasi maximum likelihood analysis of high dimensional constrained factor models," Journal of Econometrics, Elsevier, vol. 206(2), pages 574-612.
    21. Sundberg,Rolf, 2019. "Statistical Modelling by Exponential Families," Cambridge Books, Cambridge University Press, number 9781108476591, September.
    22. Donald Rubin & Dorothy Thayer, 1982. "EM algorithms for ML factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(1), pages 69-76, March.
    23. Antonello D’ Agostino & Domenico Giannone, 2012. "Comparing Alternative Predictors Based on Large‐Panel Factor Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 74(2), pages 306-326, April.
    24. J. B. Taylor & Harald Uhlig (ed.), 2016. "Handbook of Macroeconomics," Handbook of Macroeconomics, Elsevier, edition 1, volume 2, number 2.
    25. Jushan Bai & Kunpeng Li, 2016. "Maximum Likelihood Estimation and Inference for Approximate Factor Models of High Dimension," The Review of Economics and Statistics, MIT Press, vol. 98(2), pages 298-309, May.
    26. Boivin, Jean & Ng, Serena, 2006. "Are more data always better for factor analysis?," Journal of Econometrics, Elsevier, vol. 132(1), pages 169-194, May.
    27. Luciani, Matteo, 2014. "Forecasting with approximate dynamic factor models: The role of non-pervasive shocks," International Journal of Forecasting, Elsevier, vol. 30(1), pages 20-29.
    28. Mario Forni & Marc Hallin & Marco Lippi & Lucrezia Reichlin, 2000. "The Generalized Dynamic-Factor Model: Identification And Estimation," The Review of Economics and Statistics, MIT Press, vol. 82(4), pages 540-554, November.
    29. Michael E. Tipping & Christopher M. Bishop, 1999. "Probabilistic Principal Component Analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 611-622.
    30. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178.
    31. Jungbacker, B. & Koopman, S.J. & van der Wel, M., 2011. "Maximum likelihood estimation for dynamic factor models with missing data," Journal of Economic Dynamics and Control, Elsevier, vol. 35(8), pages 1358-1368, August.
    32. repec:hal:journl:peer-00844811 is not listed on IDEAS
    33. R. H. Shumway & D. S. Stoffer, 1982. "An Approach To Time Series Smoothing And Forecasting Using The Em Algorithm," Journal of Time Series Analysis, Wiley Blackwell, vol. 3(4), pages 253-264, July.
    34. Harvey,Andrew C., 1991. "Forecasting, Structural Time Series Models and the Kalman Filter," Cambridge Books, Cambridge University Press, number 9780521405737, September.
    35. Matteo Luciani & Lorenzo Ricci, 2014. "Nowcasting Norway," International Journal of Central Banking, International Journal of Central Banking, vol. 10(4), pages 215-248, December.
    36. Barigozzi, Matteo & Cho, Haeran & Fryzlewicz, Piotr, 2018. "Simultaneous multiple change-point and factor analysis for high-dimensional time series," Journal of Econometrics, Elsevier, vol. 206(1), pages 187-225.
    37. Marta Bańbura & Michele Modugno, 2014. "Maximum Likelihood Estimation Of Factor Models On Datasets With Arbitrary Pattern Of Missing Data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 29(1), pages 133-160, January.
    38. Daniela M. Witten & Robert Tibshirani, 2009. "Covariance‐regularized regression and classification for high dimensional problems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 615-636, June.
    39. Bai, Jushan & Liao, Yuan, 2016. "Efficient estimation of approximate factor models via penalized maximum likelihood," Journal of Econometrics, Elsevier, vol. 191(1), pages 1-18.
    40. Domenico Giannone & Lucrezia Reichlin & David Small, 2008. "Nowcasting: the real time informational content of macroeconomic data releases," ULB Institutional Repository 2013/6409, ULB -- Universite Libre de Bruxelles.
    41. Stock, J.H. & Watson, M.W., 2016. "Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 415-525, Elsevier.
    42. J. G. Booth & J. P. Hobert, 1999. "Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 265-285.
    43. Harvey, Andrew C. & Delle Monache, Davide, 2009. "Computing the mean square error of unobserved components extracted by misspecified time series models," Journal of Economic Dynamics and Control, Elsevier, vol. 33(2), pages 283-295, February.
    44. Stock J.H. & Watson M.W., 2002. "Forecasting Using Principal Components From a Large Number of Predictors," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1167-1179, December.
    45. Reichlin, Lucrezia & Giannone, Domenico & Small, David, 2005. "Nowcasting GDP and Inflation: The Real Time Informational Content of Macroeconomic Data Releases," CEPR Discussion Papers 5178, C.E.P.R. Discussion Papers.
    46. Hallin, Marc & Liska, Roman, 2007. "Determining the Number of Factors in the General Dynamic Factor Model," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 603-617, June.
    47. Doukhan, Paul & Neumann, Michael H., 2007. "Probability and moment inequalities for sums of weakly dependent random variables, with applications," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 878-903, July.
    48. Jushan Bai, 2003. "Inferential Theory for Factor Models of Large Dimensions," Econometrica, Econometric Society, vol. 71(1), pages 135-171, January.
    49. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    50. Jushan Bai & Peng Wang, 2015. "Identification and Bayesian Estimation of Dynamic Factor Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(2), pages 221-240, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Danilo Cascaldi-Garcia & Matteo Luciani & Michele Modugno, 2024. "Lessons from nowcasting GDP across the world," Chapters, in: Michael P. Clements & Ana Beatriz Galvão (ed.), Handbook of Research Methods and Applications in Macroeconomic Forecasting, chapter 8, pages 187-217, Edward Elgar Publishing.
    2. Filippo Pellegrino, 2021. "Factor-augmented tree ensembles," Papers 2111.14000, arXiv.org, revised Jun 2023.
    3. Poncela, Pilar & Ruiz, Esther & Miranda, Karen, 2021. "Factor extraction using Kalman filter and smoothing: This is not just another survey," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1399-1425.
    4. Danilo Cascaldi-Garcia & Matteo Luciani & Michele Modugno, 2024. "Lessons from nowcasting GDP across the world," Chapters, in: Michael P. Clements & Ana Beatriz Galvão (ed.), Handbook of Research Methods and Applications in Macroeconomic Forecasting, chapter 8, pages 187-217, Edward Elgar Publishing.
    5. Linton, O. B. & Tang, H. & Wu, J., 2022. "A Structural Dynamic Factor Model for Daily Global Stock Market Returns," Janeway Institute Working Papers camjip:2214, Faculty of Economics, University of Cambridge.
    6. Junfan Mao & Zhigen Gao & Bing-Yi Jing & Jianhua Guo, 2024. "On the statistical analysis of high-dimensional factor models," Statistical Papers, Springer, vol. 65(8), pages 4991-5019, October.
    7. Diego Fresoli & Pilar Poncela & Esther Ruiz, 2024. "Dealing with idiosyncratic cross-correlation when constructing confidence regions for PC factors," Papers 2407.06883, arXiv.org.
    8. Hie Joo Ahn & Matteo Luciani, 2021. "Relative prices and pure inflation since the mid-1990s," Finance and Economics Discussion Series 2021-069, Board of Governors of the Federal Reserve System (U.S.).
    9. Linton, O. B. & Tang, H. & Wu, J., 2022. "A Structural Dynamic Factor Model for Daily Global Stock Market Returns," Cambridge Working Papers in Economics camjip:2214, Faculty of Economics, University of Cambridge.
    10. Linton, O. B. & Tang, H. & Wu, J., 2022. "A Structural Dynamic Factor Model for Daily Global Stock Market Returns," Cambridge Working Papers in Economics 2237, Faculty of Economics, University of Cambridge.
    11. Poncela, Pilar & Ruiz, Esther, 2020. "A comment on the dynamic factor model with dynamic factors," Economics Discussion Papers 2020-7, Kiel Institute for the World Economy (IfW Kiel).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Poncela, Pilar & Ruiz, Esther & Miranda, Karen, 2021. "Factor extraction using Kalman filter and smoothing: This is not just another survey," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1399-1425.
    2. Matteo Barigozzi & Matteo Luciani, 2017. "Common Factors, Trends, and Cycles in Large Datasets," Finance and Economics Discussion Series 2017-111, Board of Governors of the Federal Reserve System (U.S.).
    3. Matteo Barigozzi, 2023. "Quasi Maximum Likelihood Estimation of High-Dimensional Factor Models: A Critical Review," Papers 2303.11777, arXiv.org, revised May 2024.
    4. Catherine Doz & Peter Fuleky, 2019. "Dynamic Factor Models," Working Papers 2019-4, University of Hawaii Economic Research Organization, University of Hawaii at Manoa.
    5. Stock, J.H. & Watson, M.W., 2016. "Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 415-525, Elsevier.
    6. Barigozzi, Matteo & Lippi, Marco & Luciani, Matteo, 2021. "Large-dimensional Dynamic Factor Models: Estimation of Impulse–Response Functions with I(1) cointegrated factors," Journal of Econometrics, Elsevier, vol. 221(2), pages 455-482.
    7. Matteo Barigozzi & Marc Hallin, 2023. "Dynamic Factor Models: a Genealogy," Papers 2310.17278, arXiv.org, revised Jan 2024.
    8. Simona Delle Chiaie & Laurent Ferrara & Domenico Giannone, 2022. "Common factors of commodity prices," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(3), pages 461-476, April.
    9. Kihwan Kim & Norman Swanson, 2013. "Diffusion Index Model Specification and Estimation Using Mixed Frequency Datasets," Departmental Working Papers 201315, Rutgers University, Department of Economics.
    10. Matteo Barigozzi & Marco Lippi & Matteo Luciani, 2016. "Non-Stationary Dynamic Factor Models for Large Datasets," Finance and Economics Discussion Series 2016-024, Board of Governors of the Federal Reserve System (U.S.).
    11. Barigozzi, Matteo & Hallin, Marc & Luciani, Matteo & Zaffaroni, Paolo, 2024. "Inferential theory for generalized dynamic factor models," Journal of Econometrics, Elsevier, vol. 239(2).
    12. Juho Koistinen & Bernd Funovits, 2022. "Estimation of Impulse-Response Functions with Dynamic Factor Models: A New Parametrization," Papers 2202.00310, arXiv.org, revised Feb 2022.
    13. Pilar Poncela & Esther Ruiz, 2016. "Small- Versus Big-Data Factor Extraction in Dynamic Factor Models: An Empirical Assessment," Advances in Econometrics, in: Dynamic Factor Models, volume 35, pages 401-434, Emerald Group Publishing Limited.
    14. Jianqing Fan & Kunpeng Li & Yuan Liao, 2020. "Recent Developments on Factor Models and its Applications in Econometric Learning," Papers 2009.10103, arXiv.org.
    15. Hindrayanto, Irma & Koopman, Siem Jan & de Winter, Jasper, 2016. "Forecasting and nowcasting economic growth in the euro area using factor models," International Journal of Forecasting, Elsevier, vol. 32(4), pages 1284-1305.
    16. Bai, Jushan & Liao, Yuan, 2016. "Efficient estimation of approximate factor models via penalized maximum likelihood," Journal of Econometrics, Elsevier, vol. 191(1), pages 1-18.
    17. Doz, Catherine & Giannone, Domenico & Reichlin, Lucrezia, 2011. "A two-step estimator for large approximate dynamic factor models based on Kalman filtering," Journal of Econometrics, Elsevier, vol. 164(1), pages 188-205, September.
    18. Jin, Sainan & Miao, Ke & Su, Liangjun, 2021. "On factor models with random missing: EM estimation, inference, and cross validation," Journal of Econometrics, Elsevier, vol. 222(1), pages 745-777.
    19. Karim Barhoumi & Olivier Darné & Laurent Ferrara, 2014. "Dynamic factor models: A review of the literature," OECD Journal: Journal of Business Cycle Measurement and Analysis, OECD Publishing, Centre for International Research on Economic Tendency Surveys, vol. 2013(2), pages 73-107.
    20. Marcellino, Massimiliano & Sivec, Vasja, 2016. "Monetary, fiscal and oil shocks: Evidence based on mixed frequency structural FAVARs," Journal of Econometrics, Elsevier, vol. 193(2), pages 335-348.

    More about this item

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1910.03821. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.