Alexi Onatski has an interesting recent paper,

"Asymptotic Analysis of the Squared Estimation Error in Misspecified Factor Models." There's also an

Appendix.

Four interesting cases have emerged in the literature, corresponding to two types of data-generating process (exact factor structure -- diagonal idiosyncratic covariance matrix vs. approximate factor structure -- non-diagonal idiosyncratic covariance matrix) and two modes of asymptotic analysis (strong factor structure vs. weak -- see Alexi's paper for the technical definitions, but you can imagine).

Much recent work focuses on approximate factor structure and strong factor asymptotics. The

classic work of Bai and Ng (2002), for example, is in that tradition. Alexi instead focuses on weak factor asymptotics. Crucially and compellingly, moreover, he focuses on selecting the number of factors \(p\) for best estimation of the common component, since estimation of the common component is typically the goal in factor modeling.

Let's get a bit more precise. The DGP is the usual approximate factor model,

$$

X=\Lambda F^{\prime }+e,

$$where \(X\) is an \(n\times T\) matrix of data, \(\Lambda\) is an \(n\times r\) matrix of factor loadings, \(F\) is a \(T\times r\) matrix of factors and \(e\) is an \(n\times T\) matrix of idiosyncratic terms.

We want to select \(p\), the number of factors, to get the best principal-component estimate, \(\hat{\Lambda}_{1:p}\hat{F}_{1:p}^{\prime }\), of the common component \(\Lambda F^{\prime }\) under quadratic loss. That is, the objective is minimization (over time and space) of

$$

L_{p}=\ tr \left[ (\hat{\Lambda}_{1:p}\hat{F}_{1:p}^{\prime }-\Lambda

F^{\prime })(\hat{\Lambda}_{1:p}\hat{F}_{1:p}^{\prime }-\Lambda F^{\prime

})^{\prime }\right] /\left( nT\right).

$$Among many other things, Alexi shows that under weak-factor asymptotics the optimal number of factors is not generally the "true" number!

All told, I find highly compelling the move to loss functions explicitly based on divergence between the true and estimated common component. I'm a little less sure how I feel about the move to weak-factor asymptotics, as my gut tells me that the common component in many macroeconomic and financial environments is driven by a few strong factors, and not much else. We'll see. In any event Alexi's contribution is refreshing, original, and valuable.

-----

By the way, I first saw the paper at the

SoFiE Lugano conference (with the Swiss Finance Institute (SFI), and Labex Louis Bachelier, "Large-Scale Factor Models in Finance," generously hosted by The Faculty of Economics of the Università della Svizzera Italiana, Lugano, Switzerland). The title of Alexi's talk was "Asymptotic Analysis of the Squared Estimation Error in Misspecified Factor Models," but the actual paper is the one cited above.

Here's the Lugano program FYI, as there were lots of other interesting papers as well.

**Invited Session 1 (Chair: E. Renault)**
R. Korajczyk (Northwestern University): Small-sample Properties of Factor Mimicking Portfolio Estimates (with Zhuo Chen and Gregory Connor)
**Contributed Session 1: Factor Models and Asset Pricing (Chair: F. Trojani)**
S. Ahn, A. Horenstein, N. Wang: Beta Matrix and Common Factors in Stock Returns, Paper
T. Chordia, A. Goyal, J. Shanken: Cross-Sectional Asset Pricing with Individual Stocks: Betas vs. Characteristics, Slides
P. Gagliardini, E. Ossola, O. Scaillet: Time-Varying Risk Premium in large Cross-Sectional Equity Datasets- Paper, Slides
**Poster Session **
E. Andreou, E. Ghysels: What Drives the VIX and the Volatility Risk Premium?
T. Berrada, S. Coupy: It Does Pay to Diversify
S. Darolles, S. Dubecq, C. Gouriéroux: Contagion Analysis in the Banking Sector
D. Karstanje, M. van der Wel, D. van Dijk: Common Factors in Commodity Futures Curves
P. Maio, D. Philip: Macro factors and the cross-section of stock returns, Paper

**Contributed Session 2: Dynamic Factor Models (Chair: M. Deistler)**
G. Fiorentini, E. Sentana: Dynamic Specification Tests for Dynamic Factor Models- Paper, Slides
M. Forni, M. Hallin, M. Lippi, P. Zaffaroni: One-Sided Representations of Generalized Dynamic Factor Models
**Invited Session 2 (Chair: E. Ghysels)**
C. Gourieroux (CREST and University of Toronto): Positional Portfolio Management (with P. Gagliardini and M. Rubin)
**Contributed Session 3: Systemic Risk (Chair: S. Darolles)**
J. Boivin, M. P. Giannoni, D. Stevanovic: Dynamic Effects of Credit Shocks in a Data-Rich Environment
S. Giglio, B. Kelly, S. Pruitt, X. Quiao: Systemic Risk and the Macroeconomy: An Empirical Evaluation
B. Schwaab, S. J. Koopman, A. Lucas: Modeling Global Financial Sector Stress and Credit Market Dislocation
**Invited Session 3 (Chair: F. Diebold)**
Alexei Onatski (University of Cambridge): Loss-Efficient Selection of the Number of Factors
**Contributed Session 4: Model Specification (Chair: O. Scaillet)**
M. Carrasco, B. Rossi: In-sample Inference and Forecasting in Misspecified Factor Models
F. Pegoraro, A. Siegel, L. Tiozzo Pezzoli: Specification Analysis of International Treasury Yield Curve Factors
F. Kleibergen, Z. Zhan: Unexplained Factors and their Effects on Second Pass R-Squared’s and t-Tests- Paper, Slides