Linear Return Prediction Models

Transcription

1 Linear Return Prediction Models Oxford, July-August 2013 Allan Timmermann 1 1 UC San Diego, CEPR, CREATES Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

2 1 Linear Prediction Model - Goyal and Welch, RFS Approximate log-linearized PV model - Campbell and Shiller, RFS Predictive Regressions - van Binsbergen and Koijen, JF Sum of the parts - Fereira and Santa-Clara, JFE Predictive regressions - Stambaugh, JFE Predictive Systems - Pastor and Stambaugh, JF Cenesizoglu and Timmermann (JBF, 2012) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

3 Predictability of Stock returns: Empirical Findings Macro variables appear to forecast mean stock returns: investment/capital ratio, consumption wealth ratio (CAY) Predictability linked to few episodes (T-bill rate) and can appear elusive Predictability sometimes pick up a time-varying business cycle component (default spread, term spread) Higher risk premium during recessions (price of risk is countercyclical) Return predictability is low when it could be exploited (and the risk be controlled) at high frequency return prediction models have low signal-to-noise ratios Time-series predictability appears to have declined during the 1990s Standard empirical models predict episodes with negative excess returns on stocks Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

4 Fundamental questions Why are returns predictable? Market ineffi ciency Time-varying investment opportunities 1 = E t [m t+1 r t+1 ] E t [r t+1 ] = 1 cov t (r t+1, m t+1 ) E t [m t+1 ] Data-snooping Learning: parameter estimation unceratinty, model instability and model uncertainty Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

5 Goyal-Welch (2008) exercise Estimate univariate prediction models for the equity premium (excess return), r t+1 : r t+1 = β 0 + β 1 x t + ε t+1 Predictors (x t ): Valuation ratios: dividend price ratio, dividend yield, earnings-price ratio, 10-year earnings-price ratio, book-to-market ratio Bond yields: three-month T-bill rate, yield on long term government bonds, term spread, default yield spread, default return spread Equity risk estimates: long term return, stock variance Corporate finance variables: dividend payout ratio, net equity expansion, percent equity issuing Inflation Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

6 Goyal-Welch (2008): empirical results Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

7 Goyal-Welch (2008): Cumulated SSE differences Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

8 Approximate log-linearized PV model (Campbell-Shiller) From the definition of returns R t+1 = P t+1 + D t+1 P t Denote logs in lowercase letters, r t+1 = log(p t+1 + D t+1 ) log(p t ) = log(p t+1 (1 + D t+1 P t+1 )) log(p t ) = log(p t+1 (1 + exp(log(d t+1 ) log(p t+1 )) log(p t ) = p t+1 p t + log(1 + exp(d t+1 p t+1 )) The last term is a nonlinear function of the log dividend-price ratio Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

9 Log-linearized PV model: returns Using a first-order Taylor expansion around d p f (d t+1 p t+1 ) f (d p) + f (d p)(d t+1 p t+1 (d p)) we have r t+1 p t+1 p t + log(1 + exp(d p)) exp(d p) exp(d p) (d t+1 p t+1 (d p)) k + ρp t+1 + (1 ρ)d t+1 p t where ρ = 1, k = log(ρ) (1 ρ) log(1/ρ 1) 1 + exp(d p) ρ is a constant slightly smaller than one Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

10 Log-linearized PV model: prices Approximate return equation can be rearranged to give a recursive equation for log-prices: p t = k + ρp t+1 + (1 ρ)d t+1 r t+1 Iterating forwards under the assumption that lim j ρ j p t+j = 0, p t = k 1 ρ + ρ j [(1 ρ)d t+1+j r t+1+j ] j=0 Ex-post dynamic accounting identity: Holds under the log-linearized approximation and the transversality condition Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

11 Prices, expected dividends and expected returns Taking expectations conditional on current information, gives the ex ante relationship p t = = [ k ] 1 ρ + E t ρ j [(1 ρ)d t+1+j r t+1+j ] j=0 k 1 ρ + (1 ρ)e t ρ j d t+1+j E t ρ j r t+1+j ] j=0 j=0 }{{}}{{} p dt p rt If stock prices are high today, investors must expect either high future dividends low future returns or Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

12 Log-linearized PV model Use price equation to derive expression for the stationary log dividend-price ratio, d t p t : d t p t = [ ] k 1 ρ + E t ρ j [ d t+1+j + r t+1+j ] j=0 Dividend-price ratio is high when future dividend growth is expected to be low and future returns are expected to be high Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

13 Return innovations r t+1 E t [r t+1 ] = E t+1 ρ j d t+1+j E t ρ j d t+1+j j=0 j=0 }{{} η dt+1 ( ) E t+1 ρ j r t+1+j E t ρ j r t+1+j } j=1 {{ j=1 } η rt+1 Increases to expectations of future dividend growth result in a capital gain while increasing expected future returns leads to a capital loss η dt+1 : changes in expected discounted future dividend growth η rt+1 : changes in expected discounted future returns Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

14 Empirical findings VAR restrictions are typically strongly rejected Large unexplained component is present in the log dividend-price ratio Interest rates and consumption are not good at explaining time-varying expected returns Time-varying expected returns explains between one half and two thirds of the variability in the dividend-price ratio Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

15 Predictive Regressions - van Binsbergen and Koijen (2010) Returns, price-dividend ratio, and dividend growth are defined as ( ) Pt+1 + D r t+1 log t+1 P t PD t P t D t ( ) Dt+1 d t+1 log D t Expected returns, µ t = E t [r t+1 ], and expected dividend growth, g t = E t [ d t+1 ], follow AR(1) processes: µ t+1 = δ 0 + δ 1 (µ t µ 0 ) + ε µ t+1 g t+1 = γ 0 + γ 1 (g t g 0 ) + ε g t+1 Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

16 Predictive regressions: dividend growth and returns Realized dividend growth: d t+1 = g t + ε d t+1 Campbell-Shiller log-linearized returns: r t+1 κ + ρ pd t+1 + d t+1 pd t pd t log(pd t ) pd = E [pd t ] κ = log(1 + exp(pd)) ρ pd ρ = exp(pd)/(1 + exp(pd)) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

17 Predictive regressions: log price-dividend ratio Iterating on the model, the log price-dividend ratio is linear in expected returns and expected dividend growth: pd t = A B 1 (µ t δ 0 ) + B 2 (g t γ 0 ) A = κ 1 ρ + γ 0 δ 0 1 ρ B 1 = 1 1, B 1 2 = ρδ 1 1 ργ 1 Distribution of three shocks: i.i.d. with covariance matrix ε g t+1 σ 2 Σ var ε u g σ g µ σ gd t+1 = σ g µ σ 2 ε d µ σ µd t+1 σ gd σ µd σ 2 d Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

18 Predictive regressions Model simplifies to a state space system with one transition (state) equation and two measurement equations: ĝ t+1 = γ 1 ĝ t + ε g t+1 d t+1 = γ 0 + ĝ t + ε d t+1 pd t+1 = (1 δ 1 )A 1 + B 2 (γ 1 δ 1 )ĝ t + δ 1 pd t B 1 ε µ t+1 + B 2ε g t+1 ĝ t = g t γ 0 : de-meaned dividend growth Linear model can be estimated by means of a Kalman filter Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

19 State space model with market-reinvested dividends Reduced form model for market-reinvested dividends earning a return of exp(ε M t+1 ) D M t+1 = D t+1 exp(ε M t+1 ) ρ M = corr(ε M t+1, εr t+1 ) Linear system with two measurement equations and one state equation - can be estimated by a filter dt+1 M = γ 0 + ĝ t + ε d t+1 + εm t+1 εm t pdt+1 M = (1 δ 1 )A + B 2 (γ 1 δ 1 )ĝ t + δ 1 pdt M B 1 ε µ t+1 + B 2ε g t+1 εm t+1 + δ 1ε M t ĝ t+1 = γ 1 ĝ t + ε g t+1 Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

20 van Binsbergen and Koijen (2010): cash/mkt reinvested dividends Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

21 van Binsbergen and Koijen (2010): MLE results Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

22 van Binsbergen and Koijen (2010): Contrast w. OLS Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

23 van Binsbergen and Koijen: main points Treat conditional expected returns and conditional expected dividend growth as latent variables Use filtering to obtain estimates of expected returns and expected dividend growth Filtered series have strong predictive power over future returns and dividend growth Dividend growth and returns contain persistent, predictable components Expected dividend growth is less persistent than expected returns Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

24 Fereira and Santa-Clara (2011): Sum of parts Decompose returns into capital gains plus dividend yield Capital gains can be written as 1 + R t+1 = 1 + CG t+1 + DY t+1 = P t+1 P t + D t+1 P t 1 + CG t+1 = P t+1 P t = P t+1/e t+1 P t /E t E t+1 E t = M t+1 M t E t+1 E t = (1 + GM t+1 )(1 + GE t+1 ) E t+1 : earnings per share at t + 1 M t+1 : price-earnings multiple at t + 1 GE t+1 : earnings growth rate at t + 1 Decompose the dividend yield into DY t+1 = D t+1 P t = D t+1 P t+1 P t+1 P t = DP t+1 (1 + GM t+1 )(1 + GE t+1 ) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

25 Returns as sum of parts Using these expressions, we can write 1 + R t+1 = (1 + GM t+1 )(1 + GE t+1 )(1 + DP t+1 ) In logs r t+1 = log(1 + R t+1 ) = m t+1 + ge t+1 + dp t+1 Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

26 Returns forecasts from sum of parts Fereira and Santa-Clara forecast the return components separately: Sum of Parts (SoP) method: ˆµ t = ˆµ gm t + ˆµ ge t + ˆµ dp t ˆµ ge t = ḡt 20 : estimated from 20-year MA of growth in earnings per share ˆµ dp t ˆµ gm t = dp t : Dividend price ratio expected to follow a random walk = 0 : no growth in multiples (simplest model) ˆµ t = ˆµ ge t + ˆµ dp t = ḡt 20 + dp t Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

27 Alternative approaches Linear regression for price multiple: ĝm t+1 = ˆα + ˆβx t Multiple reversion: deduce abnormal part in multiple, u t, from regression m t = a + bx t + u t Expect that m t reverts to its historical mean given current value of x t : gm t+1 = c + d( û t ) + v t ˆµ gm t = ĉ + ˆd( û t ) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

28 Ferreira and Santa-Clara (2011): SoP summary stats Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

29 Ferreira and Santa-Clara (2011): SoP forecasts Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

30 Ferreira and Santa-Clara (2011): Forecasting performance Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

31 Ferreira and Santa-Clara (2011): main points SoP method uses a disaggregated approach to forecasting the dividend yield, earnings growth, and earnings multiples Empirically, the approach appears to generate better out-of-sample forecasts than the prevailing mean of Goyal and Welch (2008) and univariate prediction models "In the investment world, we show that there are important gains from timing the market. To the extent that what we are capturing is excessive predictability rather than a time-varying risk premium, the success of our analysis eventually destroys its usefulness. Once enough investors follow our approach to predict returns, they will impact market prices and again make returns unpredictable." (page 535) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

32 Predictive regressions (Stambaugh, 1999) Return regressions y t = α + βx t 1 + u t E (u t x s, x w ) = 0, s < t w Residuals are correlated with past or future values of the regressor x t is assumed to be persistent: x t = θ + ρx t 1 + v t, ρ < 1 (( ) ut Σ = cov, ( ) ) [ σ 2 u t v t = u σ uv v t σ uv σ 2 v ] Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

33 Distributional results, Stambaugh (1999) Distribution and moments of OLS estimator ˆβ can be derived from Proposition 1 in Stambaugh Assuming Gaussian innovations, Marriott and Pope (1954) and Kendall (1954) show that (1 + 3ρ) E [ˆρ ρ] T Bias in ˆρ transmits to a bias in ˆβ. Stambaugh shows that E [ ˆβ β] = σ uv σ 2 v = σ uv σ 2 v E [ˆρ ρ] (1 + 3ρ) T + O(T 2 ) Bias can be substantial: Stambaugh estimates a bias around 0.40 in a regression of returns on the dividend yield from (T = 240) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

34 Stambaugh (1999): Table 1 - finite sample properties of beta estimate Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

35 Stambaugh (1999): main points "When the innovation in a lagged stochastic regressor is correlated with the regression disturbance, the OLS estimator can exhibit finite-sample properties that deviate sharply from those in the standard regression setting." (p. 408) highly relevant to return regressions that use the dividend yield Finite-sample p-values substantially higher than when computed in the traditional way Bayesian inference can yield sharper results, although they depend on the prior, assumptions about the initial observation (fixed or stochastic), and stationarity assumptions for the predictor: "In the period, for example, the p-value equals 0.15, so a classical test would accept the zero-slope hypothesis at conventional significance levels. In contrast, the posterior probability that the slope is less than or equal to zero ranges between 0.01 and 0.05, depending on the specification of the likelihood and prior." (Stambaugh, p. 408) Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

36 Predictive system, Pastor and Stambaugh, JF 2009 Linear regression r t+1 = a + b x t + e t+1 r t+1 : returns x t : predictors e t+1 : innovation (MDS) This model is too simple if the predictors are imperfectly correlated with expected returns, µ t Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

37 Predictive system State space system r t+1 = µ t + u t+1 x t+1 = (I A)E x + Ax t + v t+1 µ t+1 = (1 β)e r + βµ t + w t+1 u t v t N 0 σ 2 u σ uv σ uw 0, σ vu Σ vv σ vw w t 0 σ wu σ wv σ 2 w µ t : expected return u t+1 : unexpected returns Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

38 Conditional expected returns E [r t+1 D t ] = ) E r + (ω s ε U t s + δ s v t s s=0 ε U t = r t E r ω s = m(β m) s δ s = n(β m) s m = (βq + Cov(u, w v)) (Q + Var(u v)) 1 n = (σ wv mσ uv )Σ 1 vv Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

39 Return equation Plugging µ t = E r + i =0 βi w t i into the return equation, r t+h = E r + β i w t+h 1 i + u t+h i =0 = (1 β h 1 )E r + β h 1 h 1 µ t + β h 1 i w t+i + u t+h i =1 Hence Cov(r t, r t h ) = β h 1 ( βσ2 w 1 β 2 + σ uw ) Persistence in µ t induces positive serial correlation in returns. σ uw can be negative so serial correlation in returns can be of either sign Alternatively, the return process can be written as an ARMA(1,1): r t+1 = (1 β)e r + βr t + ε t+1 γε t Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

40 Predictive regressions: perfect predictors Suppose we regress returns directly on the predictors x t : r t+1 = a + b x t + e t+1 Perfect predictors require that there is a b such that w t = b v t A b = βb In univariate models, we must have ρ vw = ±1, and A = β When this holds, m 0, n b, and so E [r t+1 D t ] = E r + b s=0 A s v t s = E r + b (x t E x ) = a + b x t In this case, forecasts from the predictive system and the linear return regression are identical Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

41 Predictive regressions: imperfect predictors With imperfect predictors full history of the predictors matters through a weighted sum of past innovations full history of returns also affect expected returns Diagnostic for imperfect predictors: cov(e t, e t+1 ) = βvar(µ t x t ) + cov(u t, w t b v t ) Var(µ t x t ) = 0 and ω t = b v t only if the predictors are perfect Otherwise, these terms will be nonzero and so lead to autocorrelation in the residuals from the return equation Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

42 Negative correlation between expected, unexpected returns Strong prior of negative correlation between shocks to expected and unexpected returns: Asset prices fall when discount rates rise: ρ uw < 0 Weight on past return, κ s, in forecast is a function of ρ uw E [r t+1 D t ] = t 1 κ s r t s s=0 Low recent returns suggests that (i) lower expected returns since the sample mean has gone down; (ii) higher expected returns since rising return expectations lead to lower realized returns If ρ uω is suffi ciently negative, the "risk premium" effect dominates the "sample mean" effect, so recent returns get a negative weight in estimates of currrently expected returns, while older return data get a positive weight Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

43 Pastor and Stambaugh (2009): effect of past returns Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

44 Pastor and Stambaugh (2009): Effect of correlation priors Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

45 Pastor and Stambaugh (2009): predictive regressions Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

46 Pastor and Stambaugh (2009): return forecasts Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

47 Pastor and Stambaugh (2009): main points Conditional return forecasts will depend on lagged returns and lagged predictors when the predictor variables are imperfect If a large part of the variation in unexpected returns comes from time-varying expected returns, recent returns will get negative weights Tests for serial correlation in predictive return regression indicates the presence of imperfect predictors Diagnostic for imperfect predictor: sign of correlation between residuals of expected, unexpected return equations Predictive system approach can incorporate priors about correlations between shocks to expected and unexpected returns Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

48 Cenesizoglu and Timmermann (JBF, 2012) Estimate a range of time-varying mean and volatility models for monthly returns of the form r t+1 = β 0 + β 1 x t + ε t+1 ε t+1 N(0, σ 2 t+1 ) log(σ 2 t+1 ) = δ 0 + δ 1 x t + δ 2 log(σ 2 t ) + δ 3 ε t σ t + δ 4 ε t σ t Restricted versions of the model impose constant mean (β 1 = 0) and/or constant conditional volatility (δ 1 = δ 2 = δ 3 = δ 4 = 0) Forecasts are used to select the optimal portfolio weights under mean-variance preferences and under power utility Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

49 Cenesizoglu and Timmermann: Empirical findings Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

50 Cenesizoglu and Timmermann: Empirical findings Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

51 Cenesizoglu and Timmermann: Empirical findings Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

52 Cenesizoglu and Timmermann: Empirical findings Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52

53 Cenesizoglu and Timmermann (2012): main points Very common for return prediction models to produce higher out-of-sample mean squared forecast errors than a model assuming a constant equity premium, yet simultaneously add economic value when their forecasts are used to guide portfolio decisions Although there is generally a positive correlation between a return prediction model s out-of-sample statistical performance and its ability to add economic value, the relation tends to be weak and only explains a small part of the cross-sectional variation in different models economic value Underperformance along conventional measures of forecasting performance such as root mean squared forecast errors contain little information on whether return prediction models that allow for a time-varying mean or variance help or hurt investors Timmermann (UCSD) Linear prediction models July 29 - August 2, / 52