Factor Analysis for Volatility - Part II

Transcription

1 Factor Analysis for Volatility - Part II Ross Askanazi and Jacob Warren September 4, 2015 Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

2 Review - Intro Canonical factor model: y t = βf t + e t Induces the convenient decomposition on the covariance matrix: Σ y = βσ f β + Σ e Where Σ e is likely diagonal (or at least sparse) And we can make any of the above three objects time-varying to produce time-varying covariances Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

3 Review - Literature Classical models (Diebold and Nerlove, 1989, many others): Σ y t = βσ f t β + Σ e log(σ f t ) = α f log(σ f t 1) + u f t Simple extensions (Pitt and Shephard, 1999, many others): Σ y t = βσ f t β + Σ e t log(σ f t ) = α f log(σ f t 1) + u f t log(σ e t,i) = α e i log(σ e t 1,i) + u e t,i Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

4 Review - Literature Factor for Volatility (Barigozzi and Hallin (2015), Herskovic, Kelly, et al (2014), growing literature) Σ y t = βσ f t β + Σ e t log(σ f t ) = β f V t + ut f log(σt,i) e = βi e V t + ut,i e V t = β v V t 1 + ut v Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

5 Review - Empirics Fit factor model at high frequency using intraday returns Record realized variance of factor and idiosyncratic error Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

6 Conditional Mean Dynamics If we want to have any hope capturing conditional variance dynamics, we need to be sure of the conditional mean dynamics first. Could the observed comovement between factor and idiosyncratic volatilities be due to omitted conditional mean dynamics? Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

7 Quadratic factor If the true DGP is: Yet the estimated model is: y t = β 1 f t + β 2 (ft 2 f t N(0, σf 2,t ) σ 2 f t ) + e t y t = β 1 f t + ē t Then: Cov(f t, ft E[ β 2 ) 1 ] = β 1 + β 2 = β 1 V[f t ] under symmetry (1) ē t = β 2 (ft 2 σf 2 t ) + e t (2) V t [ē t ] = 2σ 4 f t β 2 β 2 + V t [e t ] (3) Even if V t [e t ] = c, V t [ē t ] will be time-varying and comove with market volatility! Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

8 White s Theorem and Cubic factor White s Theorem: Any nonlinear model can be well-approximated by a time-varying parameter linear model. Although the research is inconclusive, some work (including ours) show that βs vary over time Could the time-variation also be a result of ommited variables? Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

9 Cubic Factor Now let the DGP be: y t = β 1 f t + β 2 (ft 2 σf 2 t ) + β 3 ft 3 + e t and we fit a misspecified linear model with time-varying coefficients: Then: y t = β 1,t f t + ē t, ē t N(0, Σ t ) β 1,t = cov t(y t, f t ) V t [f t ] cov t (ft 3, f t ) = β 1 + β 3 = β 1 + 3β 3 σf 2 V t [f t ] t Time-varying βs with factor structure! This is White s Theorem in action, since the time-varying parameters pick up the nonlinearities. Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

10 Cubic Factor (Cont d) Looking at the residuals: ē t = β 2 (f 2 t σ 2 f t ) + β 3 f 3 t 3β 3 σ 2 f t f t + e t (4) V t [ē t ] = 2σ 4 f t β 2 β 2 + (9σ 4 f,t 3σ6 f,t )β 3β 3 + V t [e t ] (5) And once again, we get a factor structure on volatility, even if V t [e t ] is actually constant! Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

11 Leverage Effects Leverage Effects are the phenomenon that there is general negative correlation between an asset return and its (changes in) volatility The story is that a price decrease results in a more leveraged position, since the value of debt rises relative to that of equity Black (1976) and Christie (1982) Risk Premia Effects are the phenomenon that higher returns should be positively correlated with risk Pindyck (1984) and French, Schwert and Stambaugh (1987) For a given stock return, r t, this corresponds to: ( ) σt+1 log = α + λ 0 r t + ε t+1,0 σ t Where λ 0 would be negative (leverage) or positive (risk premia). Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

12 Leverage Effects Duffee (1995) finds that in fact the above formula is misspecified, and should be re-written as: log(σ t ) = α 1 + λ 1 r t + ε t,1 log(σ t+1 ) = α 2 + λ 2 r t + ε t+1,2 Where ˆλ 1 > ˆλ 2, which creates the perceived leverage effect. He then estimates the model in a factor context and finds: log(σ f,t ) = λ f f t + Φ(L)σ f,t + v m,t log(σ i e,t) = λ i f t + Φ(L)σ i e,tv i,t (6) that λ f < 0 and λ i > 0 Though again, there is no consensus about the sign of the coefficients. Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

13 Leverage Effects Can our omitted variable setup explain this puzzle? In the case of a missing square term, Let λ = Cov(f t, σf 2,t) 0, so Then the error term is: E[ β 1 ] = β 1 + λβ 2 (7) ē t = β 2 (f t σ f,t ) λβ 2 f t + e t (8) V t [ē t ] = g( λ, β 2,... )σf 2,t + others (9) = ḡ( λ, β 2,... )f t + others by eqn 6 (10) Nonzero correlation between idiosyncratic variance and market returns! Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

14 Data Are there actually higher-order dynamics? Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

15 Data Are there actually higher-order dynamics? Fit fourth-order polynomial to daily DOW-10 returns with SPY as factor from Jan Sep 2014 Polynomial Order R 2 F-test ** *** *** ** *** *** - *** *** - *** * - * *** * *** ** *** - *** Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

16 Data Fit fourth-order polynomial factor model intraday to DOW-10 with observed SPY factor. Extract residual volatility: Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

17 Data Fit fourth-order polynomial factor model intraday to DOW-10 with observed SPY factor. Extract residual volatility: Screeplot Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

18 Final Thoughts At the end of the day, the units in idiosyncratic volatility are very small. From a forecasting perspective, are we better off just holding them constant? Regime switching with two regimes? Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17

19 Nonlinear Screeplot Back Ross Askanazi and Jacob Warren Factor Analysis for Volatility - Part II September 4, / 17