State-Varying Factor Models of Large Dimensions

Abstract

This article develops an inferential theory for state-varying factor models of large dimensions. Unlike constant factor models, loadings are general functions of some recurrent state process. We develop an estimator for the latent factors and state-varying loadings under a large cross-section and time dimension. Our estimator combines nonparametric methods with principal component analysis. We derive the rate of convergence and limiting normal distribution for the factors, loadings, and common components. In addition, we develop a statistical test for a change in the factor structure in different states. We apply the estimator to the U.S. Treasury yields and S&P500 stock returns. The systematic factor structure in treasury yields differs in times of booms and recessions as well as in periods of high market volatility. State-varying factors based on the VIX capture significantly more variation and pricing information in individual stocks than constant factor models.

KEYWORDS:

• Factor analysis
• Kernel-regression
• Large-dimensional panel data
• Large N and T
• Principal components
• State-varying

Supplementary Materials

The online appendix collects the proofs and additional results that support the main text. The additional theoretical results include a detailed description of special cases and related models and an extension to noisy and misspecified state processes. We also provide an estimator for the number of factors. The additional empirical results consider alternative state processes and discuss the choice of tuning parameters. We also study a portfolio application of our state-varying factors. The extensive simulation section compares the performance relative to alternative latent factor models that allow for time-variation and studies the choice of bandwidth and number of factors with cross-validation arguments. Lastly, we collect the detailed proofs for all the theoretical statements.

Acknowledgments

We thank Yacine Aït-Sahalia, Daniele Bianchi, Frank Diebold, Kay Giesecke, Eric Ghysels, Lisa Goldberg and seminar and conference participants at Stanford, the NBER-NSF Time-Series Conference, the Society for Financial Econometrics, the North American Summer Meeting of the Econometric Society, the European Meeting of the Econometric Society, the Western Mathematical Finance Conference, the SIAM Conference on Financial Mathematics & Engineering and INFORMS for helpful comments.

Notes

1 Extensions of the constant loading model include sparse and interpretable latent factors in Pelger and Xiong (2020), estimation from incomplete datasets in Xiong and Pelger (2020) and including additional moments to estimate weak factors as in Lettau and Pelger (2020a,b).

2 Pelger (2020) provided empirical evidence for time-variation in latent factor models that is related to recessions.

3 Fan et al.’s (2016) model loadings as nonlinear functions of time-varying features of the cross-sectional units. Their estimation approach applies PCA to the data matrix that is projected in the cross-section on the subject-specific covariates. We also apply PCA to a projected data matrix, but our projection is applied in the time dimension.

4 In this article, we consider a scalar state process and leave the extension to multivariate state processes to future research. We expect multivariate state processes to lead to lower convergence rates due to the “curse of dimensionality” inherent in higher dimensional kernel projections. An additional challenge is that many multivariate state processes do not have the recurrence property.

5 Wang et al. (2019) also study a state-dependent latent factor model. Their focus is the estimation of a large-dimensional state-dependent covariance matrix, while we derive the inferential theory for the factors and conditional loadings. Ma et al. (2020) developed a high-dimensional asset pricing test for conditional factor model with time-varying loadings and a large number of test assets.

6 Generalized correlations or canonical correlations have been studied in Anderson (1958), Yuan and Bentler (2000), Bai and Ng (2006), Pelger (2019), and Andreou et al. (2019).

7 Our test differs from the existing tests, such as those of Breitung and Eickmeier (2011), Chen, Dolado, and Gonzalo (2014), Han and Inoue (2015), and Yamamoto and Tanaka (2015), which check the stability of the moments of factor loadings or common factors, but do not take invertible transformations into account. Our test takes a “micro” view to compare loadings in any two states, while Su and Wang (2017) takes a “global” view to test whether loadings change in the whole time dimension. In the context of asset pricing, Ma et al. (2020) focused on testing alpha, which is the first moment, while our test of loadings resembles testing beta, which is related to second moments.

8 See Diebold, Piazzesi, and Rudebusch (2005), Diebold and Li (2006), Cochrane and Piazzesi (2005), and Cochrane (2009).

9 Breitung and Eickmeier (2011) developed three test statistics for structural breaks. Chen, Dolado, and Gonzalo (2014) study the detection of large breaks in loadings through a two-stage procedure. Han and Inoue’s (2015) test for structural breaks by studying the stability in the second moments. Yamamoto and Tanaka (2015) generalized Breitung and Eickmeier’s (2011) test. Cheng, Liao, and Schorfheide (2016) proposed a test where both the factor loadings and the number of factors change simultaneously. Baltagi, Kao, and Wang (2020) and Bai, Han, and Shi (2020) estimated structural breaks with pseudo factors, Ma and Su (2018) proposed a three-step approach with local estimates and Barigozzi, Cho, and Fryzlewicz (2018) estimated structural breaks with wavelets. Chen, Wang, and Wu (2020) study the detection and inference of change points when factors are nonstationary and loadings are nonparametric functions of characteristics.

10 Compared with the convergence rate of the estimated loadings in the constant loading model, our convergence rate is slower by a polynomial factor while the convergence rate of the structural break models is slower by a logarithmic factor (e.g., Bai, Han, and Shi 2020; Chen, Wang, and Wu 2020).

11 In contrast to other time-varying factor models, for example, Bates et al. (2013), Cheng, Liao, and Schorfheide (2016), and Su and Wang (2017), our model directly incorporates the driving forces for the changes in loadings. Park et al. (2009) study a similar semiparametric factor model but require cross-sectional variation in the loadings to come from observable covariates, this means they estimate a function $\Lambda (C_{it})$ of observable covariates C_it, where $\Lambda (.)\in {\mathbb{R}}^r$ is the same function for all i. They apply B-splines to estimate the unknown loading function and estimate the factors with a Newton–Raphson algorithm. Our approach is based on a simple-to-implement PCA method, which allows us to derive an inferential theory.

12 Many common kernels satisfy this assumption: (i) Gaussian kernel $K(u)=\frac{1}{\sqrt{2\pi }}\text{exp}\hspace{0.17em}(-\frac{u^2}{2})$. (ii) Uniform kernel $K(u)=\frac{1}{2}1(|u|\le 1)$. (iii) Epanechnikov kernel $k(u)=\frac{3}{4}(1-u^2)1(|u|\le 1)$. (iv) Biweight kernel ($k(u)=\frac{15}{16}{(1-u^2)}^{2}1(|u|\le 1)$). (v) Triweight kernel ($k(u)=\frac{35}{32}{(1-u^2)}^{3}1(|u|\le 1)$)). (vi) Many higher order kernels obtained by multiplying them by a higher order polynomial in u ${}^2$.

13 We can relax the stationarity assumption of S_t, if S_t is smooth and we have a growing number of observations in a Section 3 upweights observations in a local window.

14 Su and Wang (2017) provided a “global” test for the constancy of factor loadings over time. Similar arguments could be applied to our framework. The proof would go through with some modification about controlling the bias from using data in other states. In a similar spirit, Kong (2018) provided a global test in a high-frequency setup.

15 In order to study $\overline{\Lambda }(s_1)$ and $\overline{\Lambda }(s_2)$, we need to redefine $H^{s_l}=\frac{{(F^{s_l})}^{\top }{F}^{s_l}}{T(s_l)}\frac{\Lambda {(s_l)}^{\top }\overline{\Lambda }(s_l)}{N}{(V_r^{s_l})}^{-1},H^{s_l}\stackrel{p}{\to }{(Q^{s_l})}^{-1}$, where $Q^{s_l}=V^{s_l}{({{\rm Y}}^{s_l})}^{\top }{\Sigma }_{F|s_l}^{-1/2}$, and $V^{s_l}$ are eigenvalues of ${\Sigma }_{F|s_l}^{1/2}{\Sigma }_{\Lambda (s_l)}{\Sigma }_{F|s_l}^{1/2}$, ${{\rm Y}}^{s_l}$ is the corresponding eigenvector matrix such that ${({{\rm Y}}^{s_l})}^T{{\rm Y}}^{s_l}=I_r$. Under the same assumptions as Theorem 4, the asymptotic distribution of ${\overline{\Lambda }}_i(s_l)$ is $\sqrt{\text{Th}}({\overline{\Lambda }}_i(s_l)-{(H^{s_l})}^{\top }{\Lambda }_i(s_l))\stackrel{d}{\to }N(0,{(V^{s_l})}^{-1}{Q}^{s_l}{\Phi }_i^s{(Q^{s_l})}^T{(V^{s_l})}^{-1})$, where ${\Phi }_i^s$ is the same as the ${\Phi }_i^s$ in Theorem 4. Let ${\lambda }_{li}={\Lambda }_i(s_l)$ and $v_{li}={(H^{s_l})}^{\top }\frac{\sqrt{Th}}{T(s_l)}\left(\frac{1}{N}{\sum }_{k=1}^N{\lambda }_{lk}{\lambda }_{lk}^{\top }\right)({(F^{s_l})}^{\top }{e}_i^{s_l})$, then we have $\sqrt{Th}({\overline{\Lambda }}_i(s_l)-{(H^{s_l})}^{\top }{\Lambda }_i(s_l))=v_{li}+o_p(1)$ under the same assumptions as in Theorem 4.

16 $V^{s_l}$ here is the same as the $V^{s_l}$ in Theorem 3, since the eigenvalues of ${\Sigma }_{\Lambda (s)}^{1/2}{\Sigma }_{F|s}{\Sigma }_{\Lambda (s)}^{1/2}$ are the same as those of ${\Sigma }_{F|s_l}^{1/2}{\Sigma }_{\Lambda (s_l)}{\Sigma }_{F|s_l}^{1/2}$.

17 Here, we denote by $\text{vec}(·)$ the vectorization operator. Inevitably, the matrix ${\Sigma }_{B,B}$ is singular due to the symmetric nature of the covariance and a proper formulation uses vech operators and elimination matrices.

18 Here, we use the following result:

Lemma 1. Let ${\Lambda }_1\in {\mathbb{R}}^{N\times k_1}$ and ${\Lambda }_2\in {\mathbb{R}}^{N\times k_2}$. Assume $N\ge \max (k_1,k_2),\text{rank}({\Lambda }_1)=k_1$ and $\text{rank}({\Lambda }_2)=k_2$, let $k=\min (k_1,k_2)$, then we have $\rho \le k$.

19 Note that our generalized correlation test statistic would also work when the dimensions of the loading spaces change with the state.

20 The squared error of the nonparametric method is $O_p(\max \left(\frac{1}{N},\frac{1}{\text{Th}},h^2\right))$. In order for the results in Section 5 and 6 to hold, we have $Nh\to \infty ,\text{Th}\to \infty ,N{h}^2\to 0$ and ${\text{Th}}^3\to 0$. This gives us a guideline for selecting the bandwidth h in the simulation and empirical studies and suggests range of 0.1 to 0.5. The appendix (supplementary material) collects the results for various bandwidths and shows that our findings are robust to the choice of h.

21 In Figure IA.15 in the appendix (supplementary material), we compare the estimation results of our state-varying factor model with the local time-varying model of Su and Wang (2017) under the same simulation setup. Our state-varying factor model can recover the correct functional form while the local window estimator fails.

22 Although we correct for the bias, the empirical distribution is still slightly shifted to the left. Our bias correction term only takes into account the dominant bias term. We believe that correcting for higher order bias terms can correct the remaining minor bias. Note that the remaining minor bias makes our test statistic more conservative, that is, we are more likely to reject the null hypothesis.

23 See Diebold, Piazzesi, and Rudebusch (2005), Diebold and Li (2006), Cochrane and Piazzesi (2005) and Cochrane (2009).

24 Our test-statistic is not a global test for changes in the factor structure, but aims at comparing two specific states. In order to use our results for a global test the p-values would need to be adjusted to account for multiple hypothesis testing.