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SPURIOUS FACTORS IN DATA WITH LOCAL-TO-UNIT ROOTS

Published online by Cambridge University Press:  31 May 2024

Alexei Onatski Open the ORCID record for Alexei Onatski [Opens in a new window]  and
Chen Wang Open the ORCID record for Chen Wang [Opens in a new window]
Alexei Onatski
Affiliation:
University of Cambridge
Chen Wang*
Affiliation:
University of Hong Kong
*
Address correspondence to Chen Wang, Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong, China, stacw@hku.hk
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Abstract

This paper extends the spurious factor analysis of Onatski and Wang (2021, Spurious factor analysis. Econometrica, 89(2), 591–614.) to high-dimensional data with heterogeneous local-to-unit roots. We find a spurious factor phenomenon similar to that observed in the data with unit roots. Namely, the “factors” estimated by the principal components analysis converge to principal eigenfunctions of a weighted average of the covariance kernels of the demeaned Ornstein–Uhlenbeck processes with different decay rates. Thus, such “factors” reflect the structure of the strong temporal correlation of the data and do not correspond to any cross-sectional commonalities, that genuine factors are usually associated with. Furthermore, the principal eigenvalues of the sample covariance matrix are very large relative to the other eigenvalues, creating an illusion of the “factors”capturing much of the data’s common variation. We conjecture that the spurious factor phenomenon holds, more generally, for data obtained from high frequency sampling of heterogeneous continuous time (or spacial) processes, and provide an illustration.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 38
Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

The authors would like to thank Peter C.B. Phillips, Yixiao Sun, and three anonymous referees for valuable suggestions that deepen our understanding of the spurious factor phenomenon. Chen Wang’s research is supported by the Research Grants Council of Hong Kong (Grant No. ECS 27308219) and the National Natural Science Foundation of China (Grant No. 72033002).

References

REFERENCES

Anselone, P. M. (1967). Collectively compact operator approximations. Technical Report 76, Computer Science Department, Stanford University.Google Scholar
Bai, J., & Ng, S. (2004). A panic attack on unit roots and cointegration. Econometrica , 72(4), 1127–1177.CrossRefGoogle Scholar
Bai, J., & Ng, S. (2013). Principal components estimation and identification of static factors. Journal of Econometrics , 176, 18–29.CrossRefGoogle Scholar
Bai, J., & Ng, S. (2023). Approximate factor models with weaker loadings. Journal of Econometrics , 235(2), 1893–1916.CrossRefGoogle Scholar
Boivin, J., Giannoni, M. P., & Mojon, B. (2009). How has the euro changed the monetary transmission mechanism? In Acemoglu, D., Rogoff, K., and Woodford, M. (Eds.), NBER macroeconomics annual 2008 , vol. 23. University of Chicago Press, pp. 77–126.Google Scholar
Brillinger, D. R. (1981). Time series: Data analysis and theory . Holden Day.Google Scholar
Bykhovskaya, A., & Gorin, V. (2022a). Asymptotics of cointegration tests for high-dimensional VAR( $k$ ). Preprint, arXiv:2202.07150.Google Scholar
Bykhovskaya, A., & Gorin, V. (2022b). Cointegration in large VARs. Annals of Statistics , 50(3), 1593–1617.CrossRefGoogle Scholar
Crump, R. K., & Gospodinov, N. (2022). On the factor structure of bond returns. Econometrica , 90, 295–314.CrossRefGoogle Scholar
Elliott, G. (1998). The robustness of cointegration methods when regressors almost have unit roots. Econometrica , 66, 149–158.CrossRefGoogle Scholar
Elliott, G. (1999). Efficient tests for a unit root when the initial observation is drawn from its unconditional distribution. International Economic Review , 40, 767–783.CrossRefGoogle Scholar
Gohberg, I., & Goldberg, S. (1981). Basic operator theory . Birkhäuser.CrossRefGoogle Scholar
Gonzalo, J., & Pitarakis, J. (2021). Spurious relationships in high-dimensional systems with strong or mild persistence. International Journal of Forecasting , 37, 1480–1497.CrossRefGoogle Scholar
Horn, R. A., & Johnson, C. R. (1985). Matrix analysis . Cambridge University Press.CrossRefGoogle Scholar
Karatzas, I., & Shreve, S. E. (1991). Brownian motion and stochastic calculus . (2nd ed.) Springer.Google Scholar
Kato, T. (1980). Perturbation theory for linear operators . Springer.Google Scholar
Latała, R. (2004). Some estimates of norms of random matrices. Proceedings of the American Mathematical Society , 133, 1273–1282.CrossRefGoogle Scholar
Onatski, A. (2015). Asymptotic analysis of the squared estimation error in misspecified factor models. Journal of Econometrics , 186, 388–406.CrossRefGoogle Scholar
Onatski, A., & Wang, C. (2021). Spurious factor analysis. Econometrica , 89(2), 591–614.CrossRefGoogle Scholar
Perron, P. (1989). The great crash, the oil price shock and the unit root hypothesis. Econometrica , 57, 1361–1401.CrossRefGoogle Scholar
Phillips, P. C. B. (1988). Regression theory for near-integrated time series. Econometrica , 56, 1021–1043.CrossRefGoogle Scholar
Phillips, P. C. B. (1998). New tools for understanding spurious regression. Econometrica , 66, 1299–1325.CrossRefGoogle Scholar
Stock, J. H. (1994). Unit roots, structural breaks and trends. In Engle, R. F. and McFadden, D. L. (Eds.), Handbook of econometrics , vol. 4, ch. 46. Elsevier, pp. 2739–2841.Google Scholar
Stock, J. H., & Watson, M. W. (2016). Factor models and structural vector autoregressions in macroeconomics. In Taylor, J. B. and Uhlig, H. (Eds.), Handbook of macroeconomics , vol. 2A. Elsevier, pp. 415–526.Google Scholar
Uhlig, H. (2009). Comment on “How has the euro changed the monetary transmission mechanism?”. In Acemoglu, D., Rogoff, K., and Woodford, M. (Eds.), NBER macroeconomics annual 2008 , vol. 23. National Bureau of Economic Research, Inc., pp. 141–152.Google Scholar
Watson, M. W. (1994). Vector autoregressions and cointegration. In Engle, R. F. and McFadden, D. L. (Eds.), Handbook of econometrics , vol. 4, ch. 47. Elsevier, pp. 2843–2915.Google Scholar
Zhang, B., Gao, J., & Pan, G. (2020). Estimation and testing for high-dimensional near unit root time series. Working paper 12/20, Monash Econometrics and Business Statistics.CrossRefGoogle Scholar
Zhang, B., Pan, G., & Gao, J. (2018). CLT for largest eigenvalues and unit root testing for high-dimensional nonstationary time series. Annals of Statistics , 46, 2186–2215.CrossRefGoogle Scholar
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