Principal Components and Factor Analysis. A Comparative Study
A comparison between Principal Component Analysis (PCA) and Factor Analysis (FA) is performed both theoretically and empirically for a random matrix X:(n x p) , where n is the number of observations and both coordinates may be very large. The comparison surveys the asymptotic properties of the factor scores, of the singular values and of all other elements involved, as well as the characteristics of the methods utilized for detecting the true dimension of X. In particular, the norms of the FA scores, whichever their number, and the norms of their covariance matrix are shown to be always smaller and to decay faster as n goes to infinity. This causes the FA scores, when utilized as regressors and/or instruments, to produce more efficient slope estimators in instrumental variable estimation. Moreover, as compared to PCA, the FA scores and factors exhibit a higher degree of consistency because the difference between the estimated and their true counterparts is smaller, and so is also the corresponding variance. Finally, FA usually selects a much less encumbering number of scores than PCA, greatly facilitating the search and identification of the common components of X.
|Date of creation:||13 Oct 2011|
|Date of revision:|
|Contact details of provider:|| Postal: Ludwigstraße 33, D-80539 Munich, Germany|
Web page: https://mpra.ub.uni-muenchen.de
More information through EDIRC
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Bair, Eric & Hastie, Trevor & Paul, Debashis & Tibshirani, Robert, 2006. "Prediction by Supervised Principal Components," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 119-137, March.
- Chamberlain, Gary & Rothschild, Michael, 1983.
"Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets,"
Econometric Society, vol. 51(5), pages 1281-304, September.
- Chamberlain, Gary & Rothschild, Michael, 1982. "Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets," Scholarly Articles 3230355, Harvard University Department of Economics.
- Gary Chamberlain & Michael Rothschild, 1982. "Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets," NBER Working Papers 0996, National Bureau of Economic Research, Inc.
- Mario Forni & Marc Hallin & Marco Lippi & Lucrezia Reichlin, 2004.
"The generalised dynamic factor model: consistency and rates,"
ULB Institutional Repository
2013/10133, ULB -- Universite Libre de Bruxelles.
- Forni, Mario & Hallin, Marc & Lippi, Marco & Reichlin, Lucrezia, 2004. "The generalized dynamic factor model consistency and rates," Journal of Econometrics, Elsevier, vol. 119(2), pages 231-255, April.
- Lucia Alessi & Matteo Barigozzi & Marco Capasso, 2009. "A Robust Criterion for Determining the Number of Factors in Approximate Factor Models," Working Papers ECARES 2009_023, ULB -- Universite Libre de Bruxelles.
- 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.
- Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-54, July.
- Jushan Bai & Serena Ng, 2000.
"Determining the Number of Factors in Approximate Factor Models,"
Econometric Society World Congress 2000 Contributed Papers
1504, Econometric Society.
- Jushan Bai & Serena Ng, 2002. "Determining the Number of Factors in Approximate Factor Models," Econometrica, Econometric Society, vol. 70(1), pages 191-221, January.
- Jushan Bai & Serena Ng, 2000. "Determining the Number of Factors in Approximate Factor Models," Boston College Working Papers in Economics 440, Boston College Department of Economics.
- 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.
- Mario Forni & Luca Gambetti, 2008.
"The dynamic e ects of monetary policy: A structural factor model approach,"
Center for Economic Research (RECent)
026, University of Modena and Reggio E., Dept. of Economics "Marco Biagi".
- Forni, Mario & Gambetti, Luca, 2010. "The dynamic effects of monetary policy: A structural factor model approach," Journal of Monetary Economics, Elsevier, vol. 57(2), pages 203-216, March.
- Forni, Mario & Gambetti, Luca, 2008. "The Dynamic Effects of Monetary Policy: A Structural Factor Model Approach," CEPR Discussion Papers 7098, C.E.P.R. Discussion Papers.
- James H. Stock & Mark W. Watson, 2005. "Implications of Dynamic Factor Models for VAR Analysis," NBER Working Papers 11467, National Bureau of Economic Research, Inc.
- Jushan Bai, 2003. "Inferential Theory for Factor Models of Large Dimensions," Econometrica, Econometric Society, vol. 71(1), pages 135-171, January.
- Bernanke, Ben S. & Boivin, Jean, 2003.
"Monetary policy in a data-rich environment,"
Journal of Monetary Economics,
Elsevier, vol. 50(3), pages 525-546, April.
- Connor, Gregory & Korajczyk, Robert A., 1986. "Performance measurement with the arbitrage pricing theory : A new framework for analysis," Journal of Financial Economics, Elsevier, vol. 15(3), pages 373-394, March.
- Seung C. Ahn & Alex R. Horenstein, 2013. "Eigenvalue Ratio Test for the Number of Factors," Econometrica, Econometric Society, vol. 81(3), pages 1203-1227, 05.
- Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:35486. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Joachim Winter)
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 references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link 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 profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.