BAYESIAN DYNAMIC LINEAR MODELS FOR STRATEGIC ASSET ALLOCATION

Transcription

1 BAYESIAN DYNAMIC LINEAR MODELS FOR STRATEGIC ASSET ALLOCATION Carlos M. Carvalho University of Texas at Austin Davide Pettenuzzo Brandeis University March 14, 2017 Jared D. Fisher University of Texas at Austin Abstract We find statistical and economic evidence of the predictability of risk premia for multiple risky assets. We present a unifying model that accounts for parameter uncertainty, stochastic volatility, and time-varying parameters through a Bayesian dynamic linear model framework with variance discounting. The Bayesian dynamic linear model naturally allow for updates the model parameters as new data is obtained. To account for model uncertainty, we average over many potential models of different combinations of predictors and discount factors. For an investor allocating his wealth between a risk-free asset, stocks, and bonds with maturities from 2 to 5 years, we demonstrate an ensemble of features that is preferable to others, including the models assumed by the efficient markets hypothesis. Keywords: time-series, asset allocation, Bayesian econometrics JEL Classifications: McCombs Schools of Business, University of Texas, 2110 Speedway Stop, Austin, TX carlos. carvalho@mccombs.utexas.edu McCombs Schools of Business, University of Texas, 2110 Speedway Stop, Austin, TX jared. fisher@utexas.edu Brandeis University, Sachar International Center, 415 South St, Waltham, MA dpettenu@ brandeis.edu 1

2 1 Introduction There is an extensive literature on excess return predictability in financial markets. Perhaps not surprisingly, a large share of this literature has focused on excess stock returns. 1 Over the years, this earlier literature triggered interests in examining the predictability of excess returns from other asset classes, such as government bonds, currencies, real estate, and commodities, and across different countries. Interestingly, while focusing on a wide range of assets, the various strands of literature that have addressed the question of predictability in financial markets have for the most part focused on a single risky asset. However, most investors portfolios contain multiple risky assets from multiple classes of assets. A few notable exceptions in the literature include Merton (1971), Schroder and Skiadis (1999) and (2003), Campbell, Chan and Viceira (2003), Lynch (2001) and Liu (2007), and Wachter and Warusawitharana (2009). For example, Campbell and Viceira (2003) provide theory addressing how investors should best allocate their wealth into multiple results, but their methods require that the risk premia and volatilities (i.e. the mean and covariance matrix of excess returns) be known. Wachter and Warusawitharana (2009) model the risk premia of both a stock index and the ten-year Treasury bond, checking for predictability. However, their models only use a single predictor at a time, and these papers do not incorporate the optimal features expressed in the large literature on the single asset risk premium. Within the single asset literature, recently Johannes et al. (2014) and Pettenuzzo et al. (2016) have shown that to predict stock and government bond excess return, it is important to work with very rich and flexible model. These models, at a minimum, should account for time-varying volatility and estimation risk. Unfortunately, none of the existing literature on predictability across multiple risky assets has acknowledged these features. In this paper, we fill this gap and present new methods for modeling the joint predictive distribution of excess returns of multiple risky assets while accounting for these needs expressed in the literature on 1 Earlier on, this literature uncovered that some of the time-variation in excess stock returns can be explained by financial ratios, such as the price-dividend or price-earnings ratio. Later on, macroeconomic variables, including the spread between long-term and short-term bond yields, the consumption-wealth ratio and corporate decision variables were also shown to have some predictive power. 2

3 a single risky asset. Using our generalized model, we ask a few questions of the multi-asset risk premia. What do we gain by modeling asset returns jointly, as opposed to independently? Is there predictability across multiple assets? And, for multiple assets, what set of features is ideal? Our work is most similar to Dangl and Halling (2012), Johannes et al. (2014), and Pettenuzzo et al. (2016). Dangl and Halling (2012) model the equity risk premium using a Bayesian dynamic linear model (DLM) that allows the regression coefficient parameters to vary over time (see West and Harrison, 1997). This time-variation is enabled by discounting the variance of past data, thus weighting older data less than more recent data in the time-series. Johannes, Korteweg and Polson (2014) employ what they term is an ensemble of features to address model uncertainty, estimation risk/parameter uncertainty, and time-variation in not just the coefficients, but in the volatilities too. They find that time-varying (stochastic) volatility is more impactful than time-varying coefficients. Both papers average over models to address model uncertainty, and both papers find statistical and economic evidence of predictability when multiple features are employed. Furthermore, both papers models are essentially special cases of ours, as we generalize their models to allow for a union of their employed features, as well as a multivariate response to capture multiple assets returns. The DLM, as used by Dangl and Halling (2012), allows for Bayesian updating of the distribution of returns as more information is learned. However, extending this model from a univariate to a multivariate time-series is nontrivial. We look at two approaches. The Wishart DLM (W-DLM) is the multivariate version of Dangl and Halling (2012) s model, which also comes from Harrison and West (1997, section 16.4). The W-DLM provides the exact distributions at each time point and thus is very computationally efficient compared to models that merely estimate a distribution through sampling. However, the model assumes each asset is predicted with the same covariates, which is likely undesirable, as some predictors may be inappropriate for certain assets and only amount to noise instead of signal. To address this undesirable feature, we also look at the Simultaneous Graphical DLM (SG-DLM) from Gruber and West (2016). The SG-DLM relaxes the constraining assumption of the W-DLM 3

4 and has a more flexible structure. This is done by updating separate, univariate models for each asset with contemporary dependency on the other assets, so that the predictive distributions from each of the separate models can be combined hen joint inference is needed. However, this approach requires a larger computational burden as sampling and optimization routines are needed. To evaluate these models and explore the benefit of the aforementioned features in multiple asset prediction, we test the models out of sample. Employing the equity predictors from Goyal and Welch (2008) updated to 2014, as well as predictors for and data on U.S. treasury bonds from Gargano, Pettenuzzo, and Timmermann (2016), we look at a Bayesian investor with power utility who holds a portfolio containing two risky assets, a stock index fund and treasury bonds, as well as a risk-free asset. The bonds maturities range from 2-5 years. As the investor would not know a priori which combinations of predictors and variance discount factors are optimal, we average across models of different predictors, volatility discount factors and regression coefficient variance discount factors. As Dangl and Halling (2012) said, If we want to analyze, for example, the empirical support for models including a specific predictive variable or having a certain degree of time-variation, we simply average across all models with this specific characteristic. Model averages are either equal weighted or weighted by past performance in utility or log score. This allows us to specify whether specific features are included, or not, yet let the data determine the strength of the discount factor and/or which predictors to focus on. We look at three different measures of empirical support. Predictive accuracy is measured by mean-squared prediction error. Statistical fit is measured by log score. Economic significance is measured by certainty equivalent returns (CER). Given the mean vector and covariance matrix from the risky assets multivariate next-period excess return predictive distribution, the investor allocates her wealth according to Campbell and Viceira (2003) s result for optimal investment given power utility. Then, upon seeing the actual returns, she revises her beliefs and holdings according to Bayesian updating. We thusly weight CER s the most in our evaluation, just as our investor would. From this empirical examination, we find large gains in both the economic and the 4

5 statistical performance of models of multiple risky assets when employing an appropriate ensemble of features. The full ensemble typically improves annual certainty equivalent returns by basis points over the classical benchmark historic mean model. In regards to predictability, we submit that there is meaningful information in the covariates, when the correct modeling practice is used. The best models for maturities 3-5 require predictors. Also, to our last question, yes, there is some-degree of time-variation in the parameters of the distribution of excess returns, as they clearly contribute to the superior performance of the full ensemble. Employing stochastic volatility brings drastic improvements to statistical fit. Also, Time-varying parameters usually yields marginal improvements both statistically and economically. For equity returns, we find little to no benefit in mean squared error when including a predictor, and thus we do not disagree with Goyal and Welch s (2008) conclusion. Also, for bond excess returns under this standard linear regression, we see results similar to Gargano, Pettenuzzo, Timmermann (2016), namely significance for some of their predictive covariates. Removing time variation in the regression coefficient and volatility estimates still presents encouraging results. Reducing the features to only averaging over models of different predictors, annualized certainty equivalent returns improve basis points over the benchmark, supporting the conclusions of Avramov (2002), among others. 2 Predictability of Multiple Risk Premia 2.1 Guidance from the Literature Few sources address the joint prediction of multiple risk premia. As a well known source on the asset allocation problem, Cambell and Viceira (2002) provide an theoretical exposition of the implications that different utility functions and horizons have on portfolio allocation choices. They do not, however, describe in detail how to estimate the moments of the distribution of excess returns and the consequences of different methods of estimation. They do fit multiple assets with vector auto-regression (VAR), and find that stocks have over double the Sharpe ratio as bonds. One thing we have learned from the risk premium prediction literature is the importance of parameter uncertainty, which is not a feature of the standard VAR. A Bayesian 5

6 implementation would more appropriately account for parameter uncertainty. Wachter and Warusawitharana (2009) model the risk premia of both a value-weighted stock index and the ten-year Treasury bond. They consider as predictors both the dividendprice ratio and the yield spread, separately. Their aim is to point to evaluate the effect of different prior beliefs on the existence of predictability, namely prior beliefs about the R 2 of the predictive regression. Checking a handful of different priors, ranging from the no-predictability prior (efficient markets/expectations hypothesis) to a diffuse prior, they find that a moderate skepticism (but not complete acceptance or dismissal) of predictability performs best out-ofsample. They build these prior beliefs onto system and evolution equations of their predictors and assets, only using a single predictor at a time, in a historical window regression. Their model is not a filter, so they must resort to estimating the model parameters via MCMC at every time step, a computational burden. However, their model does not employ several of the important feature identified in the single risk premium prediction literature. Without allowing coefficients and voltatilities to vary through time, it is likely the estimators are not flexible enough to capture the truth. Also, their model must use the same predictor(s) for stocks and bonds, which may fail to detect true signal and actually add more noise. Ideally, you can forecast the equity premium with the dividend-price ratio and the debt premium with the yield spread. In this paper, we present two models of risk premia, both of which employ the features found important in the research on singular risk premium prediction. 2.2 Literature on Predicting a Single Risk Premium Stocks Historically, the typical model assumption in the literature is a linear relationship between a chosen predictor and the excess return, with constant volatility: Y i,t+1 = β 0 + X tβ 1 + ɛ t+1, V ar(ɛ) = σ 2, (1) where Y i,t+1 is the log excess return of asset i when bought in month t and sold in month t + 1, X t is a vector of information about the asset and the economy available at time t. Most tests were performed in-sample, not out-of-sample (OOS), and comparisons are made against 6

7 the historic mean model Y i,t+1 = β 0 + ɛ t+1, V ar(ɛ) = σ 2 (2) which reflects the efficient markets hypothesis that there is no predictability. Welch and Goyal (2008) show that popular factors that had previously shown predictive prowess in-sample do not do well OOS, nor do they do well in-sample or OOS if the time period of the sample is expanded to the present. They show that any predictive power in aggregate stems from the oil shock of Cooper and Gulen (2006) find that the OOS tests are rather sensitive, and that much of the predictability findings is consistent with data snooping. However, this is also addressed by Campbell and Thompson (2008) who impose theory based constraints to find small, but significant, predictive improvement. Clearly we need to be more careful with our modeling than simple linear regression assumptions Bonds Naturally, others have explored predictability in bond returns. Fama and Bliss (1987) examine U.S. Treasuries with maturities up to five years. They find that 1-year expected excess returns on these bonds vary through time, and that 1-year forward rates track some of this variation. This predictor is referred to as FB. Seeking to extend their work, Cochrane and Piazzesi (2005) run regressions and find that a linear combination of 1 to 5 year forward rates, CP, predict excess returns on bonds, for maturities of 2-5 years. This combination is tentshaped, with larger, positive weight given to the 3-year forward rate and negative weight to the 1 and 5-year forward rates. The tent-shape is more peaked the greater the maturity of the bond being forecasted. However, Thornton and Valente (2012) examine both of these predictors, while also accounting for parameter uncertainty through an informative prior and stochastic volatility through a rolling window method. They find that neither CP or FB lead to economic benefits for an investor over the no-predictability benchmark out-of-sample. Predictability is also found in, and perhaps stronger with, predictors other than forward rates. Ludvigson and Ng (2009) explore the affect of cyclical macroeconomic activity. Constructing factors from macroeconomic fundamentals using principal components analysis, they show that the factors are needed to show the cyclical nature of bond risk premia and have predictive power independent of financial variables such as CP. Their main 7

8 factor includes five components, which we ll call LN. Gargano, Pettenuzzo and Timmermann (2016) forecast out-of-sample bond excess returns using combinations of FB, CP, and LN. They account for stochastic volatility, time-variation in the regression coefficients, and model uncertainty by averaging over multiple regressions (2 k ) with those variables. Using several model averaging methods (equal weighted, BMA and a pooled method), they find that these Bayesian models outperform not only their OLS equivalents, but the expectation hypothesis/efficient market benchmark Time-variation There is evidence that regression coefficients β are not constant over time, or, in other words, that the relationship between a predictor and excess returns changes over time. A straightforward way to allow the regression coefficients/parameters to vary over time is to break up the timeline and fit the model repetitively over the sections. A common variant of this is the rolling window regression, where only a set number of years are included in the regression at a time (e.g. Thorton and Valente, 2012). An alternative approach to this is structural breaks, where models incorporate discrete breaks are fit to different sections of time (see Pastor and Stambaugh, 2001; Kim, Morley, and Nelson, 2005; Viceira, 1997; Paye and Timmermann, 2006). But, as noted by Lettau and Van Nieuwerburgh (2008) models incorporating structural breaks are hard to use in real time and hence have poor out-of-sample performance. Pettenuzzo and Timmermann (2011) present a methodology that allows for future structural breaks as well, permitting both the coefficients and the volatilities to vary across breaks. Using a linear model for a limited number of time steps, they allow the parameters (means, volatilities and covariances) to be different for different time periods. By exploring Bayesian modeling averaging (BMA) over different numbers of breaks, and different risk aversion coefficients, they find that ignoring potential future breaks in the model negatively impacts investors returns. There are models, however, that do not require firm breaks in the timeline. Dangl and Halling (2012) use a forward-filtering model that allows regression coefficients to vary across time, known as time-varying parameters (TVP). This is through Bayesian dynamic linear models. This model adjusts the classic linear regression model setup in equation 1, as used in 8

9 the literature, to permit the coefficients β to adapt to new trends, through the discounting of past information. They note that the degree of time-variance in the parameters is undecided, and propose a grid of values. They fit models using the discount factors from this grid and then average over them and different predictors from Welch and Goyal (2008) using BMA. They show, that the ideal amount of time-variance in the β parameters is increasing over time (see Dangl and Halling, 2012, figure 5). They conclude that OOS predictability exists when time-varying coefficients (β) are employed, as there are improvements in MSPE and utility gains over the no-predictability benchmark. Johannes, Korteweg, and Polson (2013) find predictable returns using a Bayesian model via particle filters. They incorporate time variance in both the regression coefficient and the volatility, but only use two variations of dividend yield as the predictors. They find that stochastic volatility (SV) is more crucial than TVP. The significant economic impact (as determined by Sharpe ratios and certainty equivalent returns) requires that an ensemble of features are accounted for, namely predictable expected returns, time-varying volatility, and parameter uncertainty Model Selection and Averaging Model uncertainty is most frequently solved by averaging over many different models. These averages are usually weighted either by statistical or economic performance, but can be equally-weighted. This issue is first addressed by Avramov (2002). As it is well documented that individual regression models of returns underperform compared to the historical mean model, he suggests averaging over univariate models with different single predictors, using Bayesian model averaging. This brings in-sample and out-of-sample predictability, and it is also shown to be superior than simply choosing a single model using model selection criteria, such as R 2, AIC, etc. BMA has continued to be a solution to model uncertainty, as seen in Pettenuzzo and Timmermann (2011), Dangl and Halling (2012). While BMA may be the single most popular choice, there are other methods for attacking model uncertainty. Rapach, Strauss, and Zhou (2010) average across equity premium prediction models to incorporate information from several variables and reduce volatility, which yields statistically and economically significant OOS gains compared to the 9

10 historical average benchmark. They explore several averaging schemes, including equal weighted, selecting the median, trimmed equal weighted (removing the largest and smallest from the average), and past performance as determined by discounted mean squared prediction error. Their best scheme varies by measurement. Pettenuzzo and Ravazzolo (2016) explore equally weighted, BMA, log score-based weights, and their own utility-based weighting scheme. Their new method is unique, in that the model weights themselves have a model, which gives structure to the incorporation of past performance. This decision-based density combination method ( DB-DeCo ) bests the others in cumulative measures: SSE, log-predictive score, log-probability scores, and CER. This holds when using TVP-SV as well. 2.3 Our Approach To model multiple risky assets we must be careful about our modeling assumptions and practices. We define the one month, buy-and-hold excess return as Y i,t+1 = log(r i,t+1 R f,t+1 ), where R i is the return on some asset i and R f is the risk free rate. Ideally, the risk premium/expected excess return E(Y i,t+1 D t ) is a linear combination of predictors chosen specifically for asset i (which can include the intercept-only case for the EH benchmark). As noted by Johannes, Korteweg and Polson (2014), a particle filter is the logical choice for estimating parameters in a model moving forward in time. Their advantage is two-fold. First, it mimics investor behavior, adding present information to past beliefs, and the past beliefs are all summed up in last period s beliefs. Second, it avoids full Markov chain Monte Carlo at each step in time. Instead, the particle filter approach only approximates distributions at each step, and requires only the parameter estimates from time t and the realized return at time t to predict the returns at time t + 1. The Bayesian dynamic linear model used by Dangl and Halling (2012) is a different filter approach that evaluates the exact distributions of model parameters at each time step. We combine important aspects of their models into ours, which we present here. 10

11 2.3.1 Wishart Dynamic Linear Model (W-DLM) The forward filter scheme used by by Dangl and Halling (2012) is found in Harrison and West (1997), who also extend this model to allow for a multivariate response, or in our case, multiple risky assets (see Harrison and West, 1997, Section 16.4). This allows us to mimic the models from Dangl and Halling (2012) in a multivariate setup, yet also discounts the variation of the errors (volatility) as well as the variance of the parameter evolution. It is essentially a forward-filter analog of Wachter and Warusawitharana (2009) s model. Following West and Harrison (1997), except using lower-case characters for vector and upper-case characters for matrices, Y t = x t 1B t + v t v t Σ t N(0, V t Σ t ) B t = B t 1 + Ω t Ω t Σ t N(0, W t, Σ t ) where x t 1 is a vector of predictor variables (including a constant/intercept term), and B t is the matrix of regression coefficients for this multivariate regression. ν t and Ω t are respectively an error vector and error matrix, mutually independent and each independent over time. Ω t follows a matrix-normal distribution (See Dawid, 1981; West and Harrison, 1997). The initial state at t = 0 is represent by the prior distribution (B 0, Σ 0 D 0 ) NW 1 n 0 (M 0, C 0, S 0 ). B 0 and Σ share a joint matrix-normal/inverse Wishart prior distribution, which implies that B 0 = M 0 + Ω 0, Ω 0 N(0, C 0, S 0 ). Note that D t represents all data and parameter values up to time t, including X t. From this initial state, each of the components simply filters forward with simple update equations (see appendix). Thus, this model deterministically gives the posterior distribution at each time step. The major advantage this yields is that a Bayesian model can be fit without Markov chain Monte Carlo and computation time is almost instant. The stochastic volatility and time-varying parameters effects of the model come into play during these update steps, as represented through their discount factors δ v and δ, respectively. The volatilities become stochastic by shrinking the degrees of freedom the error variance matrix carries over from past data, as δ v (0, 1]: Σ t D t 1 W 1 δ vn t 1 (S t 1 ). 11

12 The smaller the value of δ v, the less information is carried over from previous time points. The variability over time of the regression coefficient matrix B t is controlled by δ (0, 1], which augments the left variance matrix of the evolution errors matrix. W t = 1 δ C t 1 δ Again, the smaller the discount factor δ, the less information that is propagated onward, as the elements of the covariance matrix will be larger allowing B t to be further from B t 1. Without discounting (δ = δ v = 1) and with a scalar/length-one Y t, this model behaves just like an OLS implementation of equation 1, but with the effect of the prior. With variance discounting, this model bridges the gap between recursive time-series models (using equally-weighted data from t = 0 to present) and rolling-window models (equally weighting a set number of previous time steps). For example, if we modified the discount factors to take different values over time, then a rolling window model would have δ = (0,..., 0, 1,..., 1) whereas the historic regressions give equal weight to all points in time, δ = (1,..., 1). Discounting the variance of our past parameter estimates means that the older a data point is, the less influence it has in the estimation of next period s parameters. This W-DLM model, like Wachter and Warusawitharana (2009) s model, uses the same predictors for each asset, namely E(Y t+1 D t ) = (x tb t ) T. Using the same predictor(s) for every asset may make sense, and each asset does have its own coefficients, but this is not general and likely not desirable. Mathematically, E(Y t+1 D t ) is a length m vector, and thus any matrix product of predictors x and coefficients β must be the same. The direct alternative, E(Y t+1 D t ) = X t β t, with matrix X t and coefficient vector β t only makes sense if the different assets have different values of some common predictor. While this may be reasonable if comparing different stocks, we likely want to use entirely different predictors for different classes of investments: stocks, bonds, real estate, precious metals, etc. Additionally, the (Inverse) Wishart prior, while conjugate to the multivariate normal for modeling covariance, hence convenient, is notoriously inflexible. Changing our covariance estimation would require complete renovation of the model and recursive process. Thus, we look at an approach that tries to solve this issue, the simultaneous graphical DLM (SG-DLM). 12

13 2.3.2 Simultaneous Graphical Dynamic Linear Models (SG-DLM) Instead of forcing different assets returns into a single, multivariate model, we leave each asset in its own univariate model and combine them only when necessary. Gruber and West (2016) develop this as a new approach they call simultaneous graphical dynamic linear models, SG-DLM. This is still rather similar to the W-DLM, so here we just point out the major differences. This modeling strategy updates a decoupled univariate model for each asset with its own predictive covariates. When the full multivariate model of all assets is needed, the separate models are recoupled together. Once the predictions are made or the updates completed, the models are decoupled again. The recouple step is possible because each individual predictive equation includes other assets simultaneous returns. Hence, the individual univariate models need not be independent, and we can measure that time-varying covariance more flexibly than assuming an inverse-wishart prior. The log excess return for asset j at time t is y jt, with ( βjt γ jt y jt = x T j,t 1β jt + y j,tγ T jt + ν jt ) ( βj,t 1 = γ j,t 1 ν jt N(0, λ 1 jt ) ) + ω jt ω jt N(0, W jt ). Clearly, this differs from standard models by the term y T j,t γ jt, where j implies all of y t but y jt. 2 Thus, the current return of an asset depends on the simultaneous returns of all other assets. Employing this model out-of-sample requires a priori estimates of each of the assets, which we obtain via the multivariate distribution of assets This multivariate distribution of assets is Y t N ( (I Γ t ) 1 ) µ t, Σ t where Σ t is a function of all λ 1 jt and γ jt, µ t is a vector of all x T j,t 1 β jt, and Γ t is a matrix of the γ vectors with diagonal of zeros. As the full details of the model would clutter this paper, please see Gruber and West (2016) for the full exposition. The genius of the SG- DLM algorithm is pulling information from the joint, multivariate distribution to update the individual, univariate equations. Here, we ll suffice it to say that solving this challenge comes 2 Gruber and West (2016) use sp(j) instead of j to refer to the simultaneous parents of asset j, meaning the other assets that asset j has dependencies on, which can be any subset of y j,t. 13

14 from the additional computational burden. Recoupling requires Monte Carlo sampling, and decoupling requires a iterative numerical solution (Newton-Raphson), both of which fly in the face of the biggest draw to the W-DLM, the speed of its filtering. However, this allows the flexibility for individual models that we may want. When needed, we take 2000 samples to approximate this joint multivariate distribution of excess returns. Thus, there can be error therein, which error we call Monte Carlo error, or in other words, the error inherit with estimating values based on the statistic of a Monte Carlo sample. This error can be reduced by increasing the number of samples, in exchange for computation time Model Averaging As we are unaware a priori which predictors β and discount factors δ, δ v will be most advantageous, we fit a model for each point in grid of plausible combinations of predictors and discount factors. Then, we average over many models of different specifications. A model is fit for every combination of predictor, 3 different, equally spaced δ [0.98, 1.0], and 3 different, equally spaced δ v [0.9, 1.0]. Dangl and Halling (2012) use δ {0.96, 0.98, 1.00}, and find no notable changes by doubling the granularity to δ {0.96, 0.97, 0.98, 0.99, 1.00}. As Dangl and Halling (2012) say, If we want to analyze, for example, the empirical support for models including a specific predictive variable or having a certain degree of time-variation, we simply average across all models with this specific characteristic. We name these different models in Table 1. Note that when averages contain models with different levels of a feature, it also includes the model with the feature turned off. For example, the averaged model with all predictors also includes the model with no predictors, and the averaged model with different levels of stochastic volatility contains the model with δ v = 1 for constant volatility. The model averaging is, however, weighted. For each time t, each of the individual models weight is based on its performance up through time t 1. This rewards the high-performing combinations of parameters and discount factors with more weight in the weighted average. Three weighting schemes are considered: equal weighted, utility weighted, and score weighted. 14

15 Table 1. The models and their abbreviations. Each model name has a number (1-3) as a suffix appended to it, based on what model averaging technique was used. For example, TS2 uses utility-based weights to average models of different TVP and SV discount factors, but no predictors. Name Description Formula BENCH DLM historic window model y t = β 0 + ɛ, ɛ N(0, Σ) P Predictors, no time dynamics y t = β 0 + β 1 x j,t 1 + ɛ, ɛ N(0, Σ) T Time-varying mean model y t = β 0,t + ɛ, ɛ N(0, Σ) S Stochastic-volatility mean model y t = β 0 + ɛ, ɛ N(0, Σ t ) PT Predictors and TVP y t = β 0,t + β 1,t x j,t 1 + ɛ, ɛ N(0, Σ) PS Predictors and SV y t = β 0 + β 1 x j,t 1 + ɛ, ɛ N(0, Σ t ) TS TVP-SV y t = β 0,t + ɛ, ɛ N(0, Σ t ) PTS Predictors and TVP-SV y t = β 0,t + β 1,t x j,t 1 + ɛ, ɛ N(0, Σ t ) 1 Equal-weighted average 2 Utility-weighted average 3 Score-weighted average The utility weights are calculated as w U i,τ+1 = ( (1 γ) 1 τ τ t=1 U i,t ) 1 1 γ and then normalized to sum to one. The Score weights are also normalized after calculating ( τ ) ( τ ) wi,τ+1 S = ln(score i,t ) min ln(score j,t ). j t=1 We subtract the minimum unnormalized score weight from all of the others in order to insure that all weights are nonnegative before begin normalized. 3 Empirical Evaluation 3.1 Modeling Implementation Data Description Our portfolio consists of a stock index, treasury bonds, and a risk-free asset. For data on equities, we use the updated dataset from Goyal and Welch (2008), available on Amit Goyal s website. Their 3-month T-bill rate is used as our risk-free asset. We use 3 of their predictors as well. The stock index returns are taken from the CRSP (value-weighted) US stock index returns. For bonds, we use returns from Gargano, Pettenuzzo and Timmermann (2016), for 2-5 year maturities. We borrow there predictive covariates as well, namely CP, a combination t=1 15

16 of forward rates from Cochrane and Piazzesi (2005) and LN, a macro factor from Ludvigson and Ng (2009). There are more predictors we could include, e.g. the forward spread from Fama and Bliss (1987), but we chose a subset of those to ease the computational burden for this proof of concept. Based on univariate, single risky asset performances, we chose the best and worst bond predictors, and the best, worst and a mediocre stock predictors; LN, CP, stock variance, dividend payout ratio and the dividend yield, respectively. The data on bonds begins in 1962, and we use the first 120 months to create the prior distribution parameters, from The models are evaluated on the last 360 months of data, over the period. The gap between the training and testing time periods is to avoiding testing the models during the oil shock of 1974 and the bond market experimentation of the early 1980 s Prior D 0 represents all information obtained before prediction has started, which is incorporated into the prior distribution. The first 120 months of data are set aside as the training dataset D 0. The matrix m 0 is the coefficient estimates from an OLS multivariate predictive regression on this training dataset, and S 0 is the sample covariance of matrix of the training data set excess returns Y train. n 0 is set to 10, meaning there is little weight on the prior (prior sample size). The other parameters are set to be noninformative: Q 0 = 1, C 0 = 100I Portfolio Formation We consider a Bayesian investor that has power utility, as is very common in the literature (e.g. Johannes, Korteweg and Polson 2014), and assume a risk aversion coefficient γ = 5. Such an investor prefers the portfolio with weight vector w t = 1 γ ( 1 ˆΣ t ˆµ t + 1 ) 2 diag(ˆσ t ) as noted in Campbell and Viceira (2003, line 2.26, double check?). To save this investor from some extreme preferences, we place restrictions on single asset weights between 200% and 300% of wealth, reflecting some degree of financing constraints. While restrictive, this is more 16

17 liberal than Dangl and Halling (2012), who limit the share invested in the stock to within 0% to 150% of wealth Model and Portfolio Evaluation We evaluate these models using three measures. To measure point prediction accuracy, we use the traditional mean-squared prediction error ( MSE ). MSE τ = 1 τ τ (Y t f t ) 2 t=1 where f t = E(Y t D t 1 ) is the forecast/point estimate of returns for time t. Each model s MSE is given as a ratio to the benchmark historic mean model s MSE. To measure statistical fit, we look at the log score function, averaged across time steps, as it incorporates our estimate of the variance ( ALS ). ALS τ = 1 τ τ t=1 ( log N[Y t f t, ˆΣ ) t ] Here, N is the Gaussian pdf evaluated at the realized Y t with the prediction s mean and covariance. This measure penalizes wrong return predictions based on the variance of the prediction. If the model is highly confident in the prediction yet wrong, it scores very low. If highly confident and correct, it receives a high score. If the model is unconfident in the prediction, and hence has high variance and a flat pdf, then there is little penalty for being wrong but also little bonus for being correct. Lastly, we also use an economic measure to evaluate this econometric model, namely the annualized certainty equivalent returns ( CER ). where [ CER τ = (1 γ) 1 τ τ t=1 U t ] 12 1 γ U t = 1 1 γ wealth1 γ t (3) wealth t = (1 i w i,t )R f,t+1 + i w i,t R i,t+1 = R f,t+1 + i w i,t Y i,t+1. (4) (JKP2014 comment that CEs are a more relevant benchmark than Sharpe ratios given power utility.) 17

18 3.2 Main Results Predictive Accuracy The standard out-of-sample measure of a model s predictive accuracy is the mean squared predictive error. Tables 2 and 3 report each model s percent change to the benchmark s MSE, for stocks and bonds respectively. In estimating the equity premium for stocks, we see little to no change in MSE across features. For W-DLM specifications including predictors decreases MSE slightly, by around 1.1% usually, unless the TVP feature is included, then there is only a 0.7% decrease in MSE. The typical effect sizes for SG-DLM s are even smaller, and there s actually slight increase when predictors are added, around 0.5% worse than the benchmark. As is commonly assumed, there is faint signal in the stock predictors as to the expected return of the stock index. The debt premium of treasury bonds has much larger effects. While SV yields no effect on the WDLM s, W-DLM s with TVP generally decrease MSE by %. Including predictive covariates decreases MSE by %. However, these benefits do not appear to additive, as the PT models tend to perform slightly worse than the models with just predictors. Based on the large decreases in MSE, there is quite a bit of signal laced in the bond predictors, but adding TVP overfits to noise. The MSE changes from SG-DLM s are more variable, which is likely due to Monte Carlo error. The TVP effect varies widely depending on the bond maturity. The predictors effect also varies by maturity, more than does the W-DLM, from %. In conclusion, predictors are needed to reduce predictive error, regardless of other features included. The challenge with only using the predictive errors is that they only evaluate the point estimates produced by a given model. The variability of the estimates (i.e. volatility estimates) is given by all the models and is used by most investors utility functions (meanvariance, minimum-variance, power utility, to name a few). Thus it would make sense to take into account the variability or quality/confidence of the estimates Economic Evaluation We measure economic impact and success of our these models and features with CER. Table 4 shows CER for all models and bond maturities. The top performing models for each maturity 18

19 Table 2. Percent Changes in MSE of Stocks, by Model and Maturity. Presented here are the percent changes in MSE from the benchmark model. The benchmark is the historic mean model without TVP, SV, or predictors. Bolded are the best three models, in terms of MSE, for a given maturity (including ties). 2-year Maturity 3-year Maturity 4-year Maturity 5-year Maturity Model W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM S S S T T T TS TS TS P P P PS PS PS PT PT PT PTS PTS PTS are marked in bold. Common amongst all the best models is stochastic volatility. Broadly speaking, the SG-DLM s predictions are better for returns, and the best overall model is PTS3 from the SG-DLM, as it is at or near the top for most maturities. The exception to this is with 2-year bonds, where the TS3 bests PTS3 by 45 basis points. As the least risky of our risky assets, 2-year bonds act the least like stocks, which may lead to the lower impact of predictability we see. It is also alarming to note that certain combinations of features do worse than the simple benchmark, particularly at low maturities. The W-DLM-PS3 model, which is a top performer in both ALS and MSE, has one of the worst CER s of any 2-year maturity model. A similar pattern is seen with W-DLM-PTS3. This problem would occur when a model in the averaging set fit well statistically early in the sample period, but later fit poorly due to ill-suited predictors. This is the issue with the W- DLM formulation, as all chosen predictors must each be used for modeling every asset. Fitting these two specifications with the SG-DLM seems to eliminate this problem, suggesting that the SG-DLM s predictions are more accurate for more complicated, score-weighted models. 19

20 Table 3. Percent Changes in MSE of Bonds, by Model and Maturity. Presented here are the percent changes in OOS MSE from the benchmark model. The benchmark is the historic mean model without TVP, SV, or predictors. Bolded are the best three models, in terms of MSE, for a given maturity (including ties). 2-year Maturity 3-year Maturity 4-year Maturity 5-year Maturity Model W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM S S S T T T TS TS TS P P P PS PS PS PT PT PT PTS PTS PTS We could use a different weighting scheme, but the score-weighted models usually provide more basis points of CER, the best MSE s, and the best ALS s Statistical Evaluation We measure statistical fit with ALS, which incorporates the variability of the estimates, unlike MSE. Table 5 shows ALS for all models and bond maturities. It s abundantly clear from the high values of score that stochastic volatility is the most important feature for optimizing statistical fit. Not only are the best ALS values from models that contain SV, all of the models with SV have higher ALS than those without SV. Also, the W-DLM outperforms the SG-DLM here with higher ALS almost across the board. We submit that this is likely due to Monte Carlo error inherit in the estimations of the SG-DLM. In general, adding any feature improves the fit of the predictive distribution, aside from the TVP only models fit via SG-DLM. Again, this may be due to Monte Carlo error. 20

21 Table 4. Certainty Equivalent Return by Model and Maturity. Presented here are the certainty equivalent returns (CER) for an investor with power utility, using either the W- DLM or SG-DLM predictions to create a portfolio containing the risk-free asset, a stock index, and bonds of a given maturity. The benchmark is the historic mean model without TVP, SV, or predictors. Bolded are the best three models, in terms of CER, for a given maturity. 2-year Maturity 3-year Maturity 4-year Maturity 5-year Maturity Model W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM BENCH S S S T T T TS TS TS P P P PS PS PS PT PT PT PTS PTS PTS Discussion: Economic and Statistical Tradeoff As previously alluded to, the SG-DLM s do not fit statistically as well as W-DLM, by ALS or MSE. This is attributable to the Monte Carlo error in the sampling approaches embedded in the SG-DLM algorithm. However, there is a tradeoff. As aforementioned, the W-DLM produces low CER for what could otherwise be considered the best models (PS3, PTS3), particularly at lower maturities. We see this in Figure 1. Very few equal-weighted models (*1) are visible as the utility weights are almost uniform, thus all the (*2) models cover the equal weighted ones. Visualizing ALS versus CER shows 3 main points, all reiterating what has been said. First, models including stochastic volatility and score-weighting (*S3 models) fit by W-DLM perform very well statistically but rather poorly in terms of CER. Note that fitting via SG-DLM fixes this problem, restoring the otherwise positive correlation between 21

22 Stocks & 2 year Bonds, Subset of X Stocks & 3 year Bonds, Subset of X Annual Certainty Equivalent Return P3 P1 P2 P1 P2 P3 PT3 PT2 PT1 BENCH1 BENCH2 BENCH3 T1 T2 T3 PT3 BENCH1 BENCH2 BENCH3 PT1 PT2 T3 T2 T1 TS1 TS2 TS3 PTS1 PTS2 PTS3 PS1 PS2 PS3 S1 S2 PS2 PS1 PTS2 PTS1 S1 S2 S3 PTS3 PS3 TS3 S3 Annual Certainty Equivalent Return T1T2 P1 P2 P3 P1 P2 P3 PT3 PT1 PT2 BENCH1 BENCH2 BENCH3 PT3 PT1 PT2 T1 T2 BENCH1 BENCH2 BENCH3 T3 T3 PTS3 PTS1 PTS2 TS2 TS1 TS3 PS1 PS2 PS3 S1 S2 PS2 PS1 PTS2 PTS1 TS1 TS2 S1 S2 S3 PTS3 PS3 TS3 S Average Log Score Average Log Score Stocks & 4 year Bonds, Subset of X Stocks & 5 year Bonds, Subset of X Annual Certainty Equivalent Return T1 T2 T3 TS1 TS2 P1 P2 PT3 P3 PT1 PT2 P1 P2 T1 T2 PT3 PT1 PT2 BENCH1 BENCH2 BENCH3 T3 BENCH1 BENCH2 BENCH3 PTS3 PS2 PS1 TS3 PTS2 PTS1 PS3 PS1 PS2S3 PTS2 PTS1 S1 S2 TS1 TS2 S1 S2 TS3 S3 PTS3 PS3 Annual Certainty Equivalent Return PT2 PT1 T2 T1 P1 P2PT3 P3 P3 P1 P2 T3 BENCH1 BENCH2 BENCH3 T1 T2 PT3 BENCH1 BENCH2 BENCH3 PT1 PT2 T3 PTS3 PTS2 PTS1 PS3 PS1 PS2 S1 S2 TS3 TS1 TS2 S3 PS2 PS1 PTS2 PTS1 PTS3 PS3 TS1 TS2 TS3 S1 S2S3 P Predictors T TVP S SV 1 Equal Weighted 2 Utility Weighted 3 Score Weighted o WDLM o SGDLM Average Log Score Average Log Score Figure 1. Comparisons of W-DLM to SG-DLM of multiple assets, across different maturities of bonds. Abbreviated model names from Table 1 are used as the points. W-DLM s are in black, while SG-DLM s are in red. The dashed axes show the benchmark s levels of CER and ALS. CER measures the economic significance of the model while ALS measures the statistical fit. 22

23 Table 5. Average Log Score by Model and Maturity. Presented here are the average log scores (ALS) for all combinations of models and bond maturities. The benchmark is the historic mean model without TVP, SV, or predictors. Bolded are the best models, in terms of ALS, for a given maturity. Best here means within 0.05 of the maximum score for that maturity bond. 2-year Maturity 3-year Maturity 4-year Maturity 5-year Maturity Model W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM W-DLM SG-DLM BENCH S S S T T T TS TS TS P P P PS PS PS PT PT PT PTS PTS PTS ALS and CER across all most all models. Second, stochastic volatility is the requirement for improving statistical fit, as measured by ALS. There is visible white space between all models with versus without stochastic volatility! Yet, the SG-DLM s don t receive quite the same benefit in statistical fit from employing stochastic volatility, though there is clearly still a benefit. Third, the frontier of models along the top and right borders generally include predictors and stochastic volatility. Furthermore, models with either of these features do significantly better than the benchmark, yielding evidence of predictability and that volatility is not constant. To a degree, Figure 1 show there s a tradeoff between statistical fit and economic impact, when it comes to which model approaches the truth via best performance. Given that the application of these models is to bring higher returns to portfolios, it s most reasonable to deem SG-DLM-PTS3 the most fit for use. While not the highest in ALS in our analysis, 23