Robust Econometric Inference for Stock Return Predictability

Robust Econometric Inference for Stock Return Predictability Alex Kostakis (MBS), Tassos Magdalinos (Southampton) and Michalis Stamatogiannis (Bath) Alex Kostakis, MBS Marie Curie, Konstanz (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 1 / 26

Presentation Outline Motivation and Related Literature This paper and our contributions Econometric methodology, Wald test and small sample properties Predictability tests for US stock returns (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 2 / 26

Motivation Vast literature on stock return predictability via lagged nancial variables (Keim and Stambaugh, 1986, Fama and French, 1988, Campbell and Shiller, 1988) An unsettled debate (see 2008 RFS special issue). Lettau and Ludvigson (2001) vs. Goyal and Welch (2008) Massive implications ("new facts") for modern nance (Cochrane, 1999): EMH and time-varying premia, dynamic (strategic) asset allocation, conditional asset pricing models and conditional alphas Various econometric issues related to inference in predictive regressions (Stambaugh, 1999): 1 Commonly used regressors are highly persistent series uncertainty regarding their order of integration (biased inference) 2 Innovations of predictor s and return s regressions are correlated (endogeneity) (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 3 / 26

Related Studies Campbell and Yogo (2006), Torous et al. (2004), Hjalmarsson (2011). Setup: y t = µ + Ax t 1 + ε t x t = R n x t 1 + u t Local-to-unity assumption for x t, i.e. R n = 1 + c/n, test H 0 : A = 0 t-statistic has non-standard limit distribution + depends on c Problems of existing methods: Inference is speci c to local-to-unity assumption (not valid for less persistent regressors, e.g. R n = 1 + c/n α, α < 1) ĉ OLS inconsistent complicated inference (need to construct Bonferroni bounds for t-statistics based on con dence interval for c) Statistics developed for scalar regressor. Cannot conduct joint inference on A in multiple regression (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 4 / 26

This paper We propose a new econometric methodology, IVX estimation, for predictive regressions (Magdalinos and Phillips, 2012): 1 Robusti es inference wrt the degree of regressors persistence 2 Accommodates multivariate systems of predictive regressions 3 Easy to implement (Wald test, chi-square inference, no need to construct Bonferroni bounds) We apply this methodology to conduct a series of predictability tests for market, size and value portfolios returns Predictability evidence for div yield, ntis, d/p, e/p and b/m ratios+ various combinations in the full sample period (1927-2011) Predictability stronger for small cap and value portfolios returns Evidence almost entirely disappears in the post-1952 period (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 5 / 26

Allowing for an arbitrary degree of persistence We consider the following multivariate system: y t = µ + Ax t 1 + ε t, x t = R n x t 1 + u t, R n = I K + C n α for some α 0, C = diag(c 1,..., c K ), c i 0 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 6 / 26

Allowing for an arbitrary degree of persistence We consider the following multivariate system: y t = µ + Ax t 1 + ε t, x t = R n x t 1 + u t, R n = I K + C n α for some α 0, C = diag(c 1,..., c K ), c i 0 x t unit root if C = 0 or α > 1 x t local-to-unit root if C < 0 and α = 1 x t mildly integrated if C < 0 and α 2 (0, 1) x t stationary if C < 0 and α = 0 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 6 / 26

Allowing for an arbitrary degree of persistence We consider the following multivariate system: y t = µ + Ax t 1 + ε t, x t = R n x t 1 + u t, R n = I K + C n α for some α 0, C = diag(c 1,..., c K ), c i 0 x t unit root if C = 0 or α > 1 x t local-to-unit root if C < 0 and α = 1 x t mildly integrated if C < 0 and α 2 (0, 1) x t stationary if C < 0 and α = 0 Denoting Y t = y t ȳ n, X t = x t x n 1 and E t = ε t ε n : Y t = AX t 1 + E t (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 6 / 26

Innovations covariance structure Model innovations (ε t ): i.i.d. sequence (0, Σ εε ) Regressor innovations (u t ): stationary linear process u t = C j e t j e t iid (0, Σ e ) j=0 Short run covariance matrices for ε t and u t (sample estimates): Σ εε = E ε t ε 0 t, Σεu = E ε t u 0 t, Σuu = E u t u 0 t, δ = Corr (εt, u t ) Long run covariance matrices (HAC estimation): Ω uu = E u t ut 0 h, Ωεu = Σ εu + Λ 0 uε, Λ uε = E u t ε 0 t h h= h=1 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 7 / 26

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 8 / 26

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments Choose β 2 (0, 1), C z < 0 R nz = I K + C z /n β IVX instruments: z t = R nz z t 1 + x t (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 8 / 26

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments Choose β 2 (0, 1), C z < 0 R nz = I K + C z /n β IVX instruments: z t = R nz z t 1 + x t z t behaves as an n β -mildly integrated process when β < α z t x t (n α -mildly integrated) when α < β (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 8 / 26

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments Choose β 2 (0, 1), C z < 0 R nz = I K + C z /n β IVX instruments: z t = R nz z t 1 + x t z t behaves as an n β -mildly integrated process when β < α z t x t (n α -mildly integrated) when α < β For Y = (Y1 0,..., Y n) 0 0 the IVX estimator is Ã n = Y 0 Z X 0 Z 1 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 8 / 26

Mixed Normality Theorem (i) If β < min (α, 1): n 1+β 2 vec Ã n A ) MN 0, Ψ 1 uu 0 Cz V Cz C z Ψ 1 uu Σ εε, as n!, where 8 >< Ω uu + R 1 0 B udbu 0 under unit root Ψ uu = Ω uu + R 1 0 >: J C djc 0 if local-to-unit root Ω uu + V C C if mildly integrated V C = R 0 erc Ω uu e rc dr and V Cz = R 0 erc z Ω uu e rc z dr. (ii) If α 2 (0, β) then n 1+α 2 vec Ã n A ) N 0, VC 1 Σ εε (iii) If α = β > 0, n 1+α 2 vec Ã n A ) N 0, V 1 C 1 V C C 1 (V 0 ) 1 Σ εε, where V = R 0 erc V C e rc z dr (iv) If α = 0 then p nvec Ã n A ) N 0, (Ex 1 x1 0 ) 1 Σ εε (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 9 / 26

IVX Wald test Testing for general linear restrictions: H 0 : Hvec (A) = h rank (H) = r (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 10 / 26

IVX Wald test Testing for general linear restrictions: H 0 : Hvec (A) = h rank (H) = r Wald statistic: W n = HvecÃ n h 0 Q 1 H HvecÃ n h where Ã n is the IVX estimator and h Q H = H Z 0 X i h 1 Im M M = Z 0 Z ˆΣ εε n z n 1 z 0 n 1 ˆΩ FM ˆΩ FM = ˆΣ εε ˆΩ εu ˆΩ 1 uu ˆΩ 0 εu X 0 Z 1 Im i H 0 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 10 / 26

IVX Wald test Testing for general linear restrictions: H 0 : Hvec (A) = h rank (H) = r Wald statistic: W n = HvecÃ n h 0 Q 1 H HvecÃ n h where Ã n is the IVX estimator and h Q H = H Z 0 X i h 1 Im M M = Z 0 Z ˆΣ εε n z n 1 z 0 n 1 ˆΩ FM ˆΩ FM = ˆΣ εε ˆΩ εu ˆΩ 1 uu ˆΩ 0 εu X 0 Z 1 Im i H 0 Theorem For all α 0 W n ) χ 2 (r) (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 10 / 26

Advantages of the proposed approach Inference valid for a large class of persistent regressors with roots R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 11 / 26

Advantages of the proposed approach Inference valid for a large class of persistent regressors with roots R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context Inference also valid for purely stationary regressors (α = 0) when ε t is an uncorrelated sequence (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 11 / 26

Advantages of the proposed approach Inference valid for a large class of persistent regressors with roots R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context Inference also valid for purely stationary regressors (α = 0) when ε t is an uncorrelated sequence Standard (χ 2 ) and direct (no Bonferroni bounds) inference (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 11 / 26

Advantages of the proposed approach Inference valid for a large class of persistent regressors with roots R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context Inference also valid for purely stationary regressors (α = 0) when ε t is an uncorrelated sequence Standard (χ 2 ) and direct (no Bonferroni bounds) inference Joint inference: Various combinations of regressors/ regressands can be tested (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 11 / 26

Nominal size 10%, no autocorrelation in u_t Monte Carlo simulations, IVX Wald vs CY (2006) Q-statistic (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 12 / 26

Nominal size 10%, strong autocorrelation in u_t (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 13 / 26

Power plots, n=250, innovations corr=-0.95 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 14 / 26

Power plots, n=250, innovations corr=-0.75 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 15 / 26

Power plots, n=250, innovations corr=-0.5 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 16 / 26

Data Commonly used set of persistent regressors from Goyal-Welch (2008) CRSP vw excess returns, 10 size+ value portfolios excess returns (French s website). Monthly frequency, Jan 1927- Dec 2011 (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 17 / 26

Univariate regressions, excess market return (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 18 / 26

Size portfolios excess returns (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 19 / 26

Value portfolios excess returns (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 20 / 26

Multivariate regressions, signi cant combinations (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 21 / 26

Subperiod univariate regressions, xmarket returns (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 22 / 26

Subperiod multivariate regressions (post 1952) (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 23 / 26

Inference robust to choice of instrument persistence (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 24 / 26

Inference robust to choice of instrument persistence (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 25 / 26

Conclusions New econometric methodology for stock return predictability Accommodates a very rich class of predictive regressors, allows for joint tests and is easy to implement via a Wald test Empirical application reveals predictive ability of div yield, net equity expansion, d/p, e/p and b/m ratios over excess market returns Small cap and value portfolios returns are more predictable Nevertheless, predictability evidence almost entirely disappears in the post-1952 period Methodology applicable for predictability tests in other asset classes (bond yields, FOREX returns) that use persistent regressors (Alex Kostakis, MBS) Stock Return Predictability Marie Curie, Konstanz 26 / 26