Combining State-Dependent Forecasts of Equity Risk Premium

Combining State-Dependent Forecasts of Equity Risk Premium Daniel de Almeida, Ana-Maria Fuertes and Luiz Koodi Hotta Universidad Carlos III de Madrid September 15, 216 Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 1 / 28

Introduction Motivation Equity premium forecasts are inputs for asset pricing, asset allocation and risk management. In-sample evidence that macroeconomic variables explain future risk premia. Some authors remain skeptical regarding predictability: restricted to in-sample evaluations and inconsistent out-of-sample performance. Welch and Goyal (Review of Financial Studies, 28) conclude that several macroeconomic predictors fail to beat the simple historical average (HA) benchmark. Researchers should explore alternative variables and/or more sophisticated models. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 2 / 28

Introduction Figure 1: Cumulative square forecast error for the historical average minus cumulative square forecast error for individual macroeconomic variables. 3 D/Y 3 E/P 3 D/E 2 2 2 1 1 1 1 1 1 2 2 2 3 3 3 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 3 RVOL 3 DFR 3 INFL 2 2 2 1 1 1 1 1 1 2 2 2 3 3 3 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 Dividend yield, earnings-price ratio, dividend payout ratio, equity premium volatility, default return spread and inflation. Inconsistent out-of-sample performance. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 3 / 28

Background literature Introduction Rapach et al. (Review of Financial Studies, 21) - combining forecasts - provide out-sample evidence of predictability. Large body of literature indicating that asset returns follow a process with more than one regime, e.g., Guidolin and Timmermann (Journal of Economic Dynamics and Control, 27), Henkel et al. (Journal of Financial Economics, 211), Zhu and Zhu (Journal of Banking & Finance, 213): Markov-Switching (MS) forecasting strategies. Neely et al. (Management Science, 214) show that technical indicators can predict the equity risk premium and that they are not encompassed by macroeconomic variables by using principal component (PC) analysis. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 4 / 28

Introduction Objective To improve existing equity premium forecasts. To propose parsimonious state-dependent (regime-switching with observable state) predictive regressions. To propose new combining approaches: alternative to equal-weight (EW) combination. To compare our forecast method with existing approaches. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 5 / 28

Introduction Contribution 1 Combination of parsimonious regime-switching models to forecast equity premia out-of sample. 2 Technical variables as proxy of current state of economy. 3 Sparse EW (SPAR) combining method: mean average of forecasts from models whose slope parameters are jointly significant (it excludes of the combination models whose slope parameters are not significant according to a statistic test). * Consistent statistical and economic gains of our methods in relation to univariate regressions and existing combined forecasts. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 6 / 28

Forecasting methodology Traditional predictive variables Equity premia can be explained by macroeconomic variables r t+1 = α i + β i x i,t + ε t+1, t = 1, 2,, T, (1) where r t+1 is the continuously compounded return (including dividends) in excess of the risk-free interest rate and ε t+1 is a zero-mean error. x i is a macroeconomic variable: dividend yield (D/Y), earnings-price ratio (E/P), dividend payout ratio (D/E), equity premium volatility (RVOL), book-to-market ratio (B/M), net equity expansion (NTIS), treasury bill (TBill), long-term rate of returns (LTR), term spread (TMS), default yield spread (DFY), default return spread (DFR), inflation (INFL). 1-month-ahead equity risk premium forecast is ˆr i,t+1 = α i + β i x it. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 7 / 28

Forecasting methodology Technical predictive variables Equity risk premium forecasts can also be predicted by technical variables r t+1 = α i + β i TECH i,t + ε t+1, (2) where TECH i,t is the i-th technical indicator at time t. We consider the 14 TECH of Neely et al. (Management Science, 214): takes values (positive trends) or 1 (negative trends). 1 6 combinations of moving average rule with lengths a and b, MA(a,b): a = 1, 2, 3 and b = 9, 12. 2 The moment rule with level l, denoted by MOM(l), l = 9, 12. 3 6 combinations of the on-balance volume, VOL(a,b), : a = 1, 2, 3 and b = 9, 12. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 8 / 28

Forecasting methodology State-dependent prediction Our forecasting strategy consists on modeling the equity premium by a regime-switching regressive model with observable state r t+1 = α i + β 1i x i,t + β 2i S t x i,t + ε t+1, (3) where S t is a dummy state-variable: takes values or 1. Our approach: information from technical indicators as candidates for S t. Advantages: parsimonious, strongly positive correlated with the NBER-cycle, can identify price trends, easily interpretable. We report two alternatives for S t : the moving average MA(2,12) and the agreement variable A 1 such that A 1 = 1 if 14 i=1 TECH i,t 1, and A 1 = if 14 i=1 TECH i,t < 1. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 9 / 28

Forecasting methodology Comparison to alternative methods PC analysis of Neely et al. (Management Science, 214). Combination of individual MS models of the form r t+1 = α i,st + β i,st x i,t + ε i,st, (4) where s t is the regime at time t, following a Markov chain with transition probabilities between regimes at times t and t + 1 given by ( ) p11 1 p Π = 11, (5) 1 p 22 p 22 where p ij are transition probabilities from regime j to i, i, j = 1, 2. * We consider MS model in (4) with switching in the intercept, slope and/or variance of errors (7 cases in total). * The MS models are estimate by the maximum likelihood estimator with the unobserved transition matrix Π being estimated by the Hamilton s Filter. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 1 / 28

Forecasting methodology Figure 2: Recession months for the MA(2,12) over the full sample: Dec/195 to Dec/214. 1.9.8.7 MA(2,12).6.5.4.3.2.1 196 197 198 199 2 21 1 means recession and means expansion. Shaded areas indicate NBER-dated recession months. More than 8% of agreement the NBER business cycle and less then 7.5% of transitions. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 11 / 28

Combining forecasts Forecasting methodology Combined forecast is a weighted average of N individual forecasts N ˆr t+1 C = ω i,tˆr i,t+1, (6) where ˆr i,t+1 is the i-th individual forecast of r t+1, with the corresponding loading or weight ω i,t. EW combining forecasts is the most popular: ω i,t = 1/N. i=1 Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 12 / 28

Combining forecasts Forecasting methodology Elliot et al. (Journal of Econometrics, 213) combine forecasts from all linear regression models with a fixed number of predictors, k. For k = 2, e.g., total of ( N 2) models with two variables r t+1 = α i + β i x i,t + β j x j,t + ε t+1, (7) such that i j. Combining one-step ahead forecast: average of all the ( ) N 2 forecasts. Here we consider k = 1, 2, 3. For all k, we compare combining forecasts of one-state and state-dependent models. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 13 / 28

Forecasting methodology Sparse combining forecasts Sparse EW combination (SPAR): mean average of forecasts from models whose slope parameters are jointly significant. Hard thresholding: statistical test to determine if the i-th model is significant without regard for the other predictors being considered (Bai and Ng, Journal of Econometrics, 28). Kostakis et al. (Review of Financial Studies, 215) test: robust to regressors degree of persistence and possible to multiple regression models (k > 1). The SPAR forecast is i E ˆr i,t+1, if E, ˆr t+1 SPAR M = T t=1 r t, if E =, T where E = {i; predictive variable(s) (is)are (jointly) significant at 1% level; i = 1,, ( N k) } and M is the cardinality of E. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 14 / 28 (8)

Out-of-sample forecast evaluation Statistical measures of forecast accuracy The out-of-sample R 2 OOS is R 2 OOS = 1 H h=1 (ˆr T+h r T+h ) 2 H h=1 ( r T+h r T+h ) 2, (9) where h = 1,, H are out-of-sample months, ˆr T+h is 1-ahead forecast r T+h = T+h 1 t=1 r t T+h 1 is the HA forecast. R 2 OOS > ˆr T+h is better than HA. R 2 OOS ˆr T+h is not better than HA. Separately for expansionary (EXP) and recessionary (REC) months: NBER dating. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 15 / 28

Out-of-sample forecast evaluation Statistical measures MSE-adjusted statistic of the Clark and West (Journal of Econometrics, 27). * Extension of the Diebold and Mariano (Journal of Business & Economic Statistics, 1995) and West (Econometrica, 1996) tests. * Allows to compare forecast of nested models. Variation in cumulative square forecast error (CSE) given by CSE t = t ( r T+h r T+h ) 2 (ˆr T+h r T+h ) 2 (1) h=1 * Curves are positively sloped ˆr T+h outperforms the r T+h. * Negatively sloped curve r T+h is better. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 16 / 28

Out-of-sample forecast evaluation Asset allocation exercise Investors who at time t allocate ω t % to stocks and 1-ω t % to risk-free bills. Total wealth at month t + 1 is W t+1 = [(1 ω t )exp(r f t+1 ) + ω texp(r f t+1 + r t+1)]w t, (11) where r t and the risk-free interest rate, r f t+1, are continuously compounded. Portfolio weights for the period t are the solution of ω t = arg max ω t E t [U(W t+1 )], (12) where U(W t+1 ) is the utility function, mean-variance (MV) or constant relative risk aversion (CRRA), and ω t 1.5 to exclude short sales and leverage above 5%. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 17 / 28

Out-of-sample forecast evaluation CER gain Certainty equivalent return (CER): the risk-free rate of return that an investor is willing to accept instead of adopting the given risky portfolio. CER gain ( ): CER of a investor who uses a forecasting model minus CER of a investor that assumes no predictability. Annualized by multiplying it by 12. Interpretation: annualized fee that an investor would be willing to pay to have access to the forecasting model. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 18 / 28

Empirical analysis Description of the data set Equity risk premium: log return of S&P 5 (including dividends) minus log return of risk-free bill. Predictive variables: 12 macroeconomic and 14 technical variables. Period: from December 195 to December 214. We generated out-of-sample forecasts by expanding windows. First in-sample period: from December 195 to December 1965. H = 588 one-step ahead predictions. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 19 / 28

Empirical analysis Figure 1: Cumulative square forecast error for the historical average minus cumulative square forecast error for individual macroeconomic variables. 3 D/Y 3 E/P 3 D/E 2 2 2 1 1 1 1 1 1 2 2 2 3 3 3 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 3 RVOL 3 DFR 3 INFL 2 2 2 1 1 1 1 1 1 2 2 2 3 3 3 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 Dividend yield, earnings-price ratio, dividend payout ratio, equity premium volatility, default return spread and inflation. Inconsistent out-of-sample performance. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 2 / 28

Empirical analysis Figure 3: Cumulative square forecast error for HA minus cumulative square forecast error for 1-regime model combination. EW Combination of single state models SPAR Combination of single state models PC combination of all variables 3 3 3 PC ALL 2 2 2 1 1 1 1 1 1 2 k=1 2 k=1 2 3 k=2 k=3 3 k=2 k=3 3 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 One-state combinations: predictability deteriorates after the second half of 199 s. PC models do not deliver consistent forecasts. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 21 / 28

Empirical analysis Figure 4: Cumulative square forecast error for HA minus cumulative square forecast error for 2-regime model combination. EW Combination of two state models MA(2,12) as state SPAR Combination of two state models MA(2,12) as state 3 3 2 2 1 1 1 1 2 3 k=1 k=2 k=3 2 3 k=1 k=2 k=3 197 1975 198 1985 199 1995 2 25 21 197 1975 198 1985 199 1995 2 25 21 Predominantly upward slopes. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 22 / 28

Empirical analysis Statistical results of combining forecasts Equal weight combinations Sparse combinations Predictor R 2 OOS (%) R2 OOS (%) ALL EXP REC ALL EXP REC Panel A: One-state predictive variables k = 1 1.34.722 1.696 1.57.736 3.149 k = 2 1.611 1.5 2.91 1.79.961 3.3 k = 3 1.792.947 3.592 1.779.885 3.682 Panel B: Two-state predictive variables - MA(2,12) k = 1 1.451.523 3.424 2.41.657 4.984 k = 2 1.829.442 4.782 1.98.398 5.121 k = 3 1.769.65 5.394 1.759.9 5.482 Panel C: Two-state predictive variables - A 1 k = 1 1.763.868 3.667 1.894.539 4.778 k = 2 2.23.848 5.171 1.878.38 5.65 k = 3 2.145.552 5.535 2.9.459 5.56 Forecasts of combining regressions with 2 or 3 predictors are better than combinations of individual macroeconomic variables. Forecasts of combining 2-states models are better than nested combining forecasts based on 1-state model. The predictability is substantially larger for recessions vis-à-vis expansions. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 23 / 28

Empirical analysis Figure 5: Scatterplot of forecast variances and squared forecast biases in the out-of-sample period. 19.9 B/M 19.8 D/E 19.7 E/P INFL DFR DFY TBL NTIS Forecast variance 19.6 19.5 D/Y MOM(12) VOL(2,12) MA(2,12) LTR RVOL TMS 19.4 EW(1 state) SPAR(1 state) 19.3 EW(2 states) SPAR(2 state) 19.2 19.1.2.4.6.8.1.12.14.16.18.2 Squared forecast bias Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 24 / 28

Empirical analysis Economic evaluation (asset allocation, γ = 5) of combining forecasts Equal weight SPAR MV CRRA MV CRRA Predictor (%) (%) (%) (%) Panel A: One-state predictive variables k = 1 1.65 1.67 2.75 2.45 k = 2 2.7 2.66 3.2 2.63 k = 3 2.91 2.89 2.85 2.66 Panel B: Two-state predictive variables - MA(2,12) k = 1 3.3 3.53 4.36 4.38 k = 2 4.36 4.46 4.5 4.51 k = 3 4.31 4.5 4.36 4.55 Panel C: Two-state predictive variables - A 1 k = 1 3.62 3.84 4.12 4.31 k = 2 4.38 4.5 4.54 4.63 k = 3 4.25 4.38 4.38 4.49 Combination of forecasts from 1-regime models does not beat the MA(2,12). The 2-regime model combinations outperform (larger CER gains) the 1-regime model combinations. Combination of 2-regime models using the SPAR approach further improves the forecast accuracy. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 25 / 28

Empirical analysis Comparison with another methods The R 2 OOS are larger for combining forecasts from regime-switching models with observable states compared to PC forecasts and combining MS models. The asset allocation exercise confirm the better performance of combining forecasts from parsimonious regime-switching models in relation to PC models and all combining forecasts from MS models (mean-variance CRRA or CRRA gains). Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 26 / 28

Empirical analysis Robust analysis Same conclusions for: * Risk aversions γ = 3, 1; * OOS period starting on January 1976 (after oil crisis) or on Jan/2. OOS on January 1976 or on Jan/2: * R 2 OOS of Rapach et al. (Review of Financial Studies, 21) model and its extension of Elliott et al. (Journal of Econometrics, 213) are not significantly 1% level, whereas combining two-state models are in all the cases; * The CER gains when considering combining two-state models rather than combining one-state models are, in average, 25 and 65 basis points, respectively. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 27 / 28

Concluding remarks Main findings The traditional combination of forecasts from one-state model outperforms statistically and economically the HA benchmark. Combination of forecasts from the regime-switching models with observable states have statistical out-of sample gains in relation to other competitors. Results confirmed by asset allocation exercise with two types of investor s preferences (economic gains). Asset allocation exercise suggests notably better performance of SPAR strategy compared to the EW combined forecasts. Almeida, Fuertes and Hotta (UC3M) Predictability of equity premium September 15, 216 28 / 28