Introduction Model Empirical Applications Conclusion State-Varying Factor Models of Large Dimensions Markus Pelger and Ruoxuan Xiong Stanford University European Econometric Society Meeting August 27, 2018 Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 1/24
Introduction Model Empirical Applications Conclusion Introduction Motivation Conventional large-dimensional latent factor model assumes the exposures to factors (factor loadings) are constant over time Observation: Asset prices’ exposures to the market (and other risk factors) are time-varying Example: Term-structure factor exposure is different in recessions and booms. Figure: PCA Factor Loadings for Treasuries in Boom and Recession (a) Level Factor (b) Slope Factor (c) Curvature Factor Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 2/24
Introduction Model Empirical Applications Conclusion Introduction This paper Research Question: Find latent factors and loadings that are state-dependent. 1 Test if factor model is state-dependent. 2 Key elements of estimator Statistical factors instead of pre-specified (and potentially 1 miss-specified) factors Uses information from large panel data sets: Many cross-section 2 units with many time observations Factor structure can be time-varying as a general non-linear 3 function of the state process Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 3/24
Introduction Model Empirical Applications Conclusion Introduction Contribution of this paper Contribution Theoretical PCA estimator combined with kernel projection for factors, state-varying factor loadings and common components Asymptotic inferential theory for estimators for N , T → ∞ : consistency normal distribution and standard errors Test for state-dependency of latent factor model Generalized correlation test statistic detects for which states model changes Non-standard superconsistency Empirical State-dependency of factor loadings in US Treasury securities Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 4/24
Introduction Model Empirical Applications Conclusion Literature Literature (partial list) Large-dimensional factor models with constant loadings Bai (2003): Distribution theory Fan et al. (2013): Sparse matrices in factor modeling Large-dimensional factor models with time-varying loadings Su and Wang (2017): Local time-window Pelger (2018), A¨ ıt-Sahalia and Xiu (2017): High-frequency Fan et al. (2016): Projected PCA Large-dimensional factor models with structural breaks Stock and Watson (2009): Inconsistency Breitung and Eickmeier (2011), Chen et al. (2014): Detection Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 5/24
Introduction Model Empirical Applications Conclusion Model The Model State-varying factor model X it is the observed data for the i -th cross-section unit at time t State variable S t at time t X it = Λ i ( S t ) F t + e it i = 1 , · · · N , t = 1 , · · · T r × 1 ���� 1 × r idiosyncratic ���� � �� � factors loadings N cross-section units (large), time horizon T (large) r systematic factors (fixed) Factors F , loadings Λ( S t ), idiosyncratic components e are unknown Data X and state process S t observed Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 6/24
Introduction Model Empirical Applications Conclusion Model The Model Examples (with one factor) equivalent to multi-factor representation Loadings linear in state: Λ i ( S t ) = Λ i , 1 + Λ i , 2 S t X it = Λ i , 1 F t +Λ i , 2 ( S t F t ) + e it ���� � �� � F t , 1 F t , 2 Loadings nonlinear in discrete state: Λ i ( S t ) = g i ( S t ), S t ∈ { s 1 , s 2 } X it = g i ( s 1 ) ✶ { S t = s 1 } F t + g i ( s 2 ) ✶ { S t = s 2 } F t + e it � �� � � �� � � �� � � �� � Λ i , 1 Λ i , 2 F t , 1 F t , 2 Our model Loadings nonlinear in non-discrete state: Λ i ( S t ) = g i ( S t ) with continuous distribution function for S t ⇒ Cumbersome/No multi-factor representation Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 7/24
Introduction Model Empirical Applications Conclusion Assumptions The Model: Main Assumptions Approximate state-varying factor model Systematic factors explain a large portion of the variance Idiosyncratic risk is nonsystematic: Weak time-series and cross-sectional correlation State: recurrent (infinite observations around the state to condition on) with continuous stationary PDF Factor Loadings: deterministic functions of the state and the functions are Lipschitz continuous (observations in the nearby state are useful) ∃ C , � Λ i ( s + ∆ s ) − Λ i ( s ) � ≤ C | ∆ s | Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 8/24
Introduction Model Empirical Applications Conclusion Assumptions The Model Robustness to Noise in State Process Model The observed state process is a noisy approximation of the underlying state process (e.g. omitted state) X it = (Λ i ( S t ) + ε it ) ⊤ F t + e it i = 1 , 2 , · · · , N and t = 1 , 2 , · · · , T or in vector notation = Λ( S t ) + E t + e t = Λ( S t ) F t + ψ t + e t X t F t F t ���� ���� ���� ���� ���� � �� � N × 1 r × 1 N × r r × 1 N × 1 N × r Under weak conditions noise in state process can be treated like idiosyncratic noise. ⇒ All results hold! Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 9/24
Introduction Model Empirical Applications Conclusion Estimation The Model: Intuition Intuition for Estimation Constant loadings: Loadings are principal components of covariance matrix Cov ( X t ) = Λ Cov ( F t )Λ ⊤ + Cov ( e t ) . State-varying loadings: Loadings for S t = s are principal components of covariance matrix conditioned on the state S t = s : Cov ( X t | S t = s ) = Λ( s ) Cov ( F t | S t = s )Λ( s ) ⊤ + Cov ( e t | S t = s ) . ⇒ Intuition: Estimate conditional covariance matrix Cov ( X t | S t = s ) with kernel projection and apply PCA to it. Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 10/24
Introduction Model Empirical Applications Conclusion Estimation The Model: Nonparametric Estimation Obtective function and nonparametric estimation The estimators minimize mean squared error conditioned on state: N T 1 � � F s , ˆ ˆ K s ( S t )( X it − Λ i ( s ) ′ F t ) 2 Λ( s ) = arg min NT ( s ) F s , Λ( s ) i =1 t =1 � S t − s � 2 π exp {− u 2 Kernel function K s ( S t ) = 1 1 h K (e.g. K ( u ) = 2 } ) √ h T ( s ) = � T p t =1 K s ( S t ), T ( s ) − → π ( s ) (stationary density of S t = s ) T Bandwidth parameter h determines local “state window” Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 11/24
Introduction Model Empirical Applications Conclusion Estimation The Model: Nonparametric Estimation Nonparametric estimation Project square root of kernel on the data and factors X s it = K 1 / 2 F s t = K 1 / 2 ( S t ) X it ( S t ) F t s s PCA solves optimization problem N T 1 � � F s , ˆ ˆ ( X s it − Λ i ( s ) ′ F s t ) 2 Λ( s ) = arg min NT ( s ) F s , Λ( s ) t =1 i =1 ⇒ Apply PCA to conditional covariance matrix F s are the eigenvectors corresponding to top k eigenvalues of ˆ 1 NT ( s ) ( X s ) ′ X s estimated conditional covariance matrix Λ( s ) are coefficients from regressing X s on ˆ ˆ F s Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 12/24
Introduction Model Empirical Applications Conclusion Asymptotic Results The Model: Nonparametric Estimation Major challenge: Bias term from using nearby states X s t = Λ( S t ) F s t + e s t = Λ( s ) F s t + e s + (Λ( S t ) − Λ( s )) F s . t t � �� � � �� � ¯ ∆ X s X s t t ∆ X s it = Λ i ( S t ) F s t − Λ i ( s ) F s t = O p ( h ) Kernel bias complicates problem and lowers convergence rates Theorem: Consistency √ √ Assume N , Th → ∞ and δ NT , h h → 0 with δ NT , h = min ( N , Th ): � 2 � � � � T � ˆ δ 2 1 � F s t − ( H s ) T F s � = O p (1) NT , h t =1 t � T � 2 � � � � N 1 � ˆ δ 2 � Λ i ( s ) − ( H s ) − 1 Λ i ( s ) � = O p (1) � NT , h i =1 N for known full rank matrix H s Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 13/24
Introduction Model Empirical Applications Conclusion Asymptotic Results Limiting Distribution of Estimated Factors Theorem (Factors) √ Nh / ( Th ) → 0, Nh → ∞ and Nh 2 → 0. Then Assume √ � � ( S t ) ˆ K − 1 / 2 F s t − ( H s ) ′ F t N s N r ) − 1 ( ˆ F s ) ′ F s 1 � ( V s = √ Λ i ( s ) e it + o p (1) T N i =1 D N (0 , ( V s ) − 1 Q s Γ s t ( Q s ) ′ ( V s ) − 1 ) − → ( F s ) ′ ˆ Rotation matrix H s = Λ( s ) ′ Λ( s ) F s r ) − 1 ( V s N T √ K − 1 / 2 ( S t ) ˆ F s t converges to some rotation of F t at rate N s Efficiency mainly depends on asymptotic distribution of � N 1 i =1 Λ i ( s ) e it √ N Markus Pelger and Ruoxuan Xiong State-Varying Factor Models of Large Dimensions 14/24
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