Forecast Accuracy and Economic Gains from Bayesian Model Averaging Using Time Varying Weight

Hoogerheide, Lennart; Kleijn, Richard; Ravazzolo, Francesco; van Dijk, Herman K.; Verbeek, Marno Working Paper Forecast Accuracy and Economic Gains from Bayesian Model Averaging Using Time Varying Weight Working Paper, No. 2009/10 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Hoogerheide, Lennart; Kleijn, Richard; Ravazzolo, Francesco; van Dijk, Herman K.; Verbeek, Marno (2009) : Forecast Accuracy and Economic Gains from Bayesian Model Averaging Using Time Varying Weight, Working Paper, No. 2009/10, ISBN 978-82-7553-507-6, Norges Bank, Oslo, https://hdl.handle.net/11250/2497634 This Version is available at: https://hdl.handle.net/10419/209926 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no 2009 | 10 Forecast accuracy and economic gains from Bayesian model averaging using time varying weight Lennart Hoogerheide, Richard Kleijn, Francesco Ravazzolo, Herman K. van Dijk and Marno Verbeek Working Paper Research Department Working papers fra Norges Bank, fra 1992/1 til 2009/2 kan bestilles over e-post. [email protected] eller ved henvendelse til: Norges Bank, Abonnementsservice Postboks 1179 Sentrum 0107 Oslo Telefon 22 31 63 83, Telefaks 22 41 31 05 Fra 1999 og fremover er publikasjonene tilgjengelig på www.norges-bank.no Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. 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ISSN 1502-8143 (online) ISBN 978-82-7553-507-6 (online) Forecast Accuracy and Economic Gains from Bayesian Model Averaging using Time Varying Weights Lennart Hoogerheide1Richard Kleijn2Francesco Ravazzolo3 Herman K. van Dijk1Marno Verbeek4 Abstract Several Bayesian model combination schemes, including some novel approaches that simultaneously allow for parameter uncertainty, model uncertainty and robust time varying model weights, are compared in terms of forecast accuracy and economic gains using financial and macroeconomic time series. The results indicate that the proposed time varying model weight schemes outperform other combination schemes in terms of predictive and economic gains. In an empirical application using returns on the S&P 500 index, time varying model weights provide improved forecasts with substantial economic gains in an investment strategy including transaction costs. Another empirical example refers to forecasting US economic growth over the business cycle. It suggests that time varying combination schemes may be very useful in business cycle analysis and forecasting, as these may provide an early indicator for recessions. Key words: forecast combination, Bayesian model averaging, time varying model weights, portfolio optimization, business cycle. 1Econometric and Tinbergen Institutes, Erasmus University Rotterdam, The Netherlands 2PGGM, Zeist, The Netherlands. 3Norges Bank. Correspondence to: Francesco Ravazzolo, Norges Bank, Research Department, Bankplassen 2, 0107 Oslo, Norway. E-mail: francesco.rav[email protected] 4Rotterdam School of Management, Erasmus University Rotterdam, The Netherlands 1 Introduction When an extensive set of forecasts of some future economic event is available, decision makers usually attempt to discover which is the best forecast, then accept this and discard the other forecasts. However, the discarded forecasts may have some independent valuable information and including them in the forecasting process may provide more accurate results. An important explanation is related to the fundamental assumption that in most cases one can not identify a priori the exact true economic process or the forecasting model that generates smaller forecast errors than its competitors. Different models may play a – possibly temporary – complementary role in approximating the data generating process. In these situations, forecast combinations are viewed as a simple and effective way to obtain improvements in forecast accuracy. Since the seminal article of Bates and Granger (1969) several papers have shown that combinations of forecasts can outperform individual forecasts in terms of loss functions. For example, Stock and Watson (2004) find that for predicting output growth in seven countries forecast combinations generally perform better than forecasts based on single models. Marcellino (2004) has extended this analysis to a large European data set with broadly the same conclusion. However, several alternative combination schemes are available and it is not clear which is the best scheme, either in a frequentist or Bayesian framework. For example, Hendry and Clements (2004) and Timmermann (2006) show that simple combinations1often give better performance than more sophisticated approaches. Further, using a frequentist approach, Granger and Ramanathan (1984) propose the use of coefficient regression methods, Hansen (2007) introduces a Mallows’ criterion, which can be minimized to select the empirical model weights, and Terui and Van Dijk (2002) generalize the least squares model weights by reformulating the linear regression model as a state space specification where the weights are assumed to follow a random walk process. Guidolin and Timmermann (2007) propose a different time varying weight combination scheme where weights have regime switching dynamics. Stock and Watson (2004) and Timmermann (2006) use the inverse mean square prediction error (MSPE) over a set of the most recent observations to compute model weights. In a Bayesian framework, Madigan and Raftery (1994) revitalize the concept of Bayesian model averaging (BMA) and apply it in an empirical application dealing with 1Simple combinations are defined as combinations with model weights that do not involve unknown parameters to be estimated; arithmetic averages constitute a simple example. Complex combinations are defined as combinations that rely on estimating weights that depend on the full variance-covariance matrix and, possibly, allow for time varying model weights. 2 Occam’s Window. Recent applications suggest its relevance for macroeconomics (Fern´andez, Ley, and Steel, 2001 and Sala-i-Martin, Doppelhoffer, and Miller, 2004). Strachan and Van Dijk (2008) compute impulse response paths and effects of policy measures using BMA in the context of a large set of vector autoregressive models. Geweke and Whiteman (2006) apply BMA using predictive likelihoods instead of marginal likelihoods. This paper contributes to the research on forecast combinations by investigating several Bayesian combination schemes. We propose three schemes that allow for parameter uncertainty, model uncertainty and time varying model weights simultaneously. These approaches can be considered Bayesian extensions of the combination scheme of Terui and Van Dijk (2002). We provide two empirical illustrations. The results indicate that time varying model weight schemes outperform other averaging schemes in terms of predictive and economic gains. The first empirical example deals with forecasting the returns on the S&P 500 index by combining individual forecasts from four competing models. The first model assumes that a set of financial and macroeconomic variables that are related to the business cycle have explanatory power. The second model is based on the popular market saying “Sell in May and go away”, also known as the “Halloween indicator”, see for example Bouman and Jacobsen (2002). Low predictability of stock market return data is well documented, see for example Marquering and Verbeek (2004) and so is structural instability in this context, see for example Pesaran and Timmermann (2002) and Ravazzolo, Paap, Van Dijk, and Franses (2007). The third and fourth model are (robust) stochastic volatility models. As an investor is particularly interested in the economic value of a forecasting scheme, we test our findings in an active short-term investment exercise, with an investment horizon of one month. The forecast combination schemes with time-varying model weights provide the highest economic gains. The second empirical example refers to forecasting US economic growth over the business cycle, where we consider combinations of forecasts from six well-known time series models: an autoregressive model, two random walk models (with and without drift), an error correction model and two (robust) stochastic volatility models. It suggests that time varying weighting schemes may provide an early indicator for recessions. The contents of this paper are organized as follows. In Section 2 we describe the different forecast combination schemes. In Section 3 we give results from an empirical application to US stock index returns which show that forecast combinations give economic gains. In Section 4 we report results from macroeconomic forecasts using US GDP growth. Section 5 concludes. 3 2 Forecast combination schemes Bayesian approaches have been widely used to construct forecast combinations, see for example Leamer (1978), Hodges (1987), Draper (1995), Min and Zellner (1993), and Strachan and Van Dijk (2008). In the Bayesian model averaging approach one derives the posterior density for any individual model and combines these to compute a predictive density of the event of interest. The predictive density accounts then for model uncertainty by averaging over the posterior probabilities of individual models. Since the output is a complete density, not only point forecasts but also distribution and quantile forecasts can be easily derived. We discuss four Bayesian forecast combination schemes. The first scheme is a standard approach known as Bayesian model averaging, the other three schemes obtain model weights as parameters to be estimated in linear and nonlinear regressions. 2.1 Scheme 1: Bayesian Model Averaging (BMA) The predictive density of the variable yat time T+ 1, yT+1, given the data up to time T, DT, is computed by averaging over the conditional predictive densities given the individual models with the posterior probabilities of these models as weights: p(yT+1|DT) = n X i=1 p(yT+1|DT, mi)P(mi|DT) (1) where nis the number of individual models; p(yT+1|DT, mi) is the conditional predictive density given DTand model mi;P(mi|DT) is the posterior probability for model mi. The conditional predictive density given DTand model miis defined as: p(yT+1|DT, mi) = Zp(yT+1|DT, mi, θi)p(θi|DT, mi)dθi(2) where p(yT+1|DT, mi, θi) is the conditional predictive density of yT+1 given DT, the model miand parameters θi;p(θi|DT, mi) is the posterior density for parameters θiin model mi. The posterior probability for model mi,P(mi|DT), can be computed in several ways. Madigan and Raftery (1994) define it as: P(mi|DT) = p(y1:T|mi)P(mi) Pn j=1 p(y1:T|mj)P(mj)(3) where y1:T={yt}T t=1;P(mi) is the prior probability for model mi; and p(y1:T|mi) is the marginal likelihood for model migiven by: p(y1:T|mi) = Zp(y1:T|θi, mi)p(θi|mi)dθi(4) 4 with p(θi|mi) the prior density for the parameters θiin model mi. The integral in equation (4) can be evaluated analytically in the case of linear models, but not for more complex forms. Chib (1995), for example, has derived a method to compute the expression also for nonlinear examples. Laplace methods can also be used, see for example Planas, Rossi, and Fiorentini (2008). A comparative study of Monte Carlo methods for marginal likelihood evaluation, among which importance sampling and bridge sampling, is given by Ardia, Hoogerheide, and Van Dijk (2009). Geweke and Whiteman (2006) propose a BMA scheme based on the idea that a model is as good as its predictions. The predictive density of yT+1 conditional on DThas the same form as equation (1), but the posterior probability of model miconditional on DTis now computed as: P(mi|DT) = p(yT|DT−1, mi)P(mi) Pn j=1 p(yT|DT−1, mj)P(mj)(5) where p(yT|DT−1, mi) is the predictive likelihood for model mi, e.g. the density derived by substituting the realized value yTinto the predictive density of yTconditional on DT−1 given model mi. Mitchell and Hall (2005) discuss the relation of the predictive likelihood to the Kullback-Leibler Information Criterion, and consequently to the frequentist combination scheme based on recursive log-score weights, see for example Kascha and Ravazzolo (2008). We apply BMA using (5) with p(yT|DT−1, mi) replaced by its product over T−kobservations p(yk+1|Dk, mi)×...×p(yT|DT−1, mi), where for increasing Twe hold constant the length kof the ‘initial period’ of data Dkthat are only used for deriving posterior distributions.2That is, for forecasts of yT+1 in later periods the predictive likelihoods and model weights are based on an expanding window of data. The densities p(yt|Dt−1, mi) are evaluated as follows. First, parameters θiare simulated from the conditional distribution on Dt−1. Second, draws ytare simulated conditionally on the θidraws and Dt−1. Third, a kernel smoothing technique is used to estimate the density of ytin model miat its realized value. The performance of alternative approaches for computing predictive likelihoods in our time varying model combination schemes is left as a topic for future research. In all models, we specify uninformative proper priors for the parameters θi. The use of predictive likelihoods rather than marginal likelihoods helps us to avoid the inference problems due to the Bartlett paradox. 2We choose k= 12 for our applications involving monthly data. 5 2.2 Combination schemes using estimated regression coefficients as model weights The next three combination schemes estimate the weights wiof the models mi(i= 1, . . . , n) in regression form. We assume that the data ytsatisfy the linear equation yt=w0+ n X i=1 wiyt,i +utut∼N(0, σ2) i.i.d. t= 1,2, . . . , T (6) where yt,i has the predictive density p(yt|Dt−1, mi) of ytgiven Dt−1in model mi. Clear differences with the BMA approach are that a constant term w0is added, and that there is no restriction that all weights must be non-negative and adding to 1.3Therefore, the weights wi(i= 1, . . . , n) can not be interpreted as model probabilities. Define the model weight vector w= (w0, w1, . . . , wn)0. We propose three novel sampling algorithms for simulating model weight vectors wgiven the data y1:Tand the predictive densities p(yt|Dt−1, mi) (t= 1, . . . , T). Scheme 2: Model weights from Ordinary Least Squares in a linear model (LIN) A set of model weight vectors ws(s= 1, . . . , S) is generated by simulating independently Ssets of T×ndraws ys t,i from the predictive densities p(yt|Dt−1, mi) (t= 1, . . . , T;i= 1, . . . , n), and performing an Ordinary Least Squares (OLS) regression in the model yt=w0+ n X i=1 wiys t,i +us tus t∼N(0, σ2)t= 1,2, . . . , T (7) for each simulated set s= 1, . . . , S. It is well-known that in a linear model as (7) the OLS estimator wsis the posterior mean of wunder a flat prior. The generated model weights ws are used to combine draws ys T+1,i (i= 1, . . . , n) from the predictive densities p(yT+1|DT, mi) into ‘combined draws’ ˜ys T+1: ˜ys T+1 =ws 0+ n X i=1 ws iys T+1,i (8) The median of ˜ys T+1 (s= 1, . . . , S) is our point forecast ˆyT+1 for yT+1, where the median is preferred over the mean because it is more robust to extreme draws. This approach can be considered as an extension of the idea of Granger and Ramanathan (1984) to combine point forecasts using weights that minimize a square loss function, to making use of Bayesian 3Granger and Ramanathan (1984) explain that the constant term must be added to avoid biased forecasts. They also conclude that this strategy is often more accurate than using restricted least squares weights. 6 3.2 Empirical Results The analysis for the active investment strategies is implemented for the period from January 1987 until December 2008, involving T∗= 264 one month ahead excess stock return forecasts. The individual models are estimated recursively using an expanding window of observations. The initial 12 predictions for each individual model are used as training period for combination schemes and making the first combined prediction. The investment strategies are implemented for a level of relative risk aversion of γ= 6.6 Before we analyze the performance of the different portfolios, we summarize the statistical accuracy of the excess return forecasts. All the individual models give similar RMSPE statistics in Table 1, for the RSV model just the smallest and for the LI model the highest. The sign ratio is the highest for the SV model, but hardly exceeds 60%, indicating low predictability. Due to this low predictability, small differences in RMSPE may have substantial economic value. We investigate this in the portfolio exercise. The SV model gives the highest Sharpe ratio, realized final utility and comparison fees ∆ among the individual models. The TVW and RTVW combination schemes, however, provide much higher statistics; in particular RTVW outperforms all the other models in terms of Sharpe ratio and realized utility value, and all three ∆’s are positive. Figure 1 can help to explain these findings. Individual models allocate too low weight to the risky asset resulting in low portfolio returns. BMA has a similar problem. The LIN, TVW and RTVW combinations allocate higher weights to the stock asset, but RTVW is the only scheme that drastically reduces this weight in bear market periods as the burst of the internet bubble in 2001-2003 or the recent financial crisis in the second part of 2007 and 2008. Panel C in Table 1 shows evidence that the findings are similar when taking into account the presence of medium transaction costs. The good performance of RTVW as compared to LIN and TVW shows that its robust flexible structure pays off. The higher portfolio weight of stock in bull markets for RTVW, as compared to the individual models and BMA, is due to the ‘shrunk’ predictive density. This ‘shrunk’ excess return distribution is not so much ‘compressed’ that the risky asset’s portfolio weight switches from 0% to 100% when its mean changes from negative to positive values. Rather, the parameter and model uncertainty that are incorporated in this ‘shrunk’ predictive density imply an investment strategy with a smooth, ‘moderate’, yet flexible 6We also implement exercises with γ= 4 and γ= 8. Results are qualitatively similar and available upon request. 13 evolvement of the risky asset’s portfolio weight over time. Lettau and Van Nieuwerburgh (2008) find that the uncertainty on the size of steady-state shifts rather than their dates is responsible for the difficulty of forecasting stock returns in real time. The ‘shrunk’ predictive density of the RTVW scheme may be particularly informative on the current and future evolvement of this steady-state, the driving force of return predictability. This may be the explanation for the RTVW scheme’s good results. We intend to analyze its performance in other portfolio management exercises in future research, in order to investigate the robustness of our findings. 4 US real GDP Growth We now perform an empirical analysis on a key macroeconomic series, the U.S. real Gross Domestic Product (GDP) growth. We collected real GDP (seasonally adjusted) figures from the U.S. Department of Commerce, Bureau of Economic Analysis. The left panel of Figure 2 plots the log quarterly GDP level for our sample 1960:Q1 to 2008:Q3 (195 observations) and shows that GDP has followed an upward sloping pattern but with fluctuations around this trend. The quarterly growth rate, ln GDPt−ln GDPt−1, shown in the right panel of Figure 2, underlines these fluctuations with periods of positive changes followed by periods of negative changes, clearly indicating business cycles; for more details we refer to Harvey, Trimbur, and Van Dijk (2007). As in the previous section, we apply various linear and nonlinear models and forecast combinations to assess these models’ suitability in a pseudo-real-time out-of- sample forecasting exercise. In the forecast exercise we use an initial in-sample period from 1960:Q1 to 1979:Q4 to obtain initial parameter estimates and we forecast the GDP growth figure for 1980:Q1. We then expand the estimation sample with the value in 1980:Q1, reestimating the parameters, and we forecast the next value for 1980:Q2. We continue this procedure up to the last value and we end up with a total of 115 forecasts. We apply n= 6 individual time series models to infer and forecast GDP. Four models are linear specifications, two models are time-varying parameter specifications. The first and second model are random walk models, without and with drift (RW and RWD). The third model is the autoregressive (AR) model of order 1. We follow Schotman and Van Dijk (1991) and specify a weakly informative ‘regularization’ prior that helps to prevent problems that could be encountered during the estimation using the Gibbs sampler, if a flat prior were used. The fourth model we apply is an error correction model (ECM). We apply the same 14 model as in De Pooter, Ravazzolo, Segers, and Van Dijk (2008): ∆yt=δ+ (ρ1+ρ2−1)(yt−1−µ−δ(t−1)) −ρ2(∆yt−1−δ) + εt, εt∼N(0, σ2),(27) which can be rewritten as: yt−δt = (1 −ρ1−ρ2)µ+ρ1(yt−1−δ(t−1)) + ρ2(yt−2−δ(t−2))+ εt, εt∼N(0, σ2).(28) The prior that we use is an extension of the prior of Schotman and Van Dijk (1991). The fifth and sixth models are a state-space model (SSM) and its robust extension (RSSM), that are given by the SV and RSV models of section 3. We use the root mean square prediction error (RMSPE) to compare different point forecasts. Table 2 shows that the random walk models perform poorly. For all other models, the test of Clark and West (2007) for equal forecasting quality of nested models rejects the null hypothesis versus the RW model. The AR model is a bit more precise than the ECM. The models with time varying parameters, SSM and RSSM, perform very well. Figure 3 shows that all models with fixed parameters perform poorly when GDP decreases rapidly and substantially as in NBER recessions, and it takes some quarters for models to adjust, in particular in the 2001 recession and the 2008 recession. Time-varying parameter models seem to cope better with this. The BMA and RTVW combination schemes provide even better statistics than the SSM and RSSM models. LIN is the worst averaging scheme; LIN performs similarly to the AR and ECM models. Figure 4 shows that LIN is performing particularly poorly in the 1980’s and 1990’s. Weight estimates for this scheme may be highly inaccurate as the number of individual models is relatively large and instability possibly high. Moreover, Figure 4 indicates that the other averaging schemes react much faster to sharp decreases in GDP. Especially the RTVW scheme may early indicate recessions: before both the 1991 and 2001 crises its point forecast decreases substantially with approximately 0.5%. To sum up, our results suggest that model averaging may be very beneficial in business cycle analysis and forecasting. The combination method must, however, be chosen carefully and it should cope with estimation efficiency and structural instability, in particular if weights are estimated in regression equations. Again, more extensive studies should be performed to investigate the robustness of our findings, for example over different countries and periods. 15 5 Final remarks The empirical applications have indicated, firstly, that averaging strategies can give higher predictive quality than selecting the best model; secondly, that properly specified time varying model weights yield higher forecast accuracy and substantial economic gains compared with other averaging schemes. The presented results lead to multiple directions for future research. As we already mentioned, interesting possibilities for further research are a rigorous analysis of the impact of some assumptions – both on theoretical aspects and practical applications – and an extensive study on the robustness of our findings. Another topic for further research is to compare our results to other time varying weight combination schemes, such as regime switching, see e.g. Guidolin and Timmermann (2007), or schemes that carefully model breaks, see e.g. Ravazzolo, Paap, Van Dijk, and Franses (2007). For the application to portfolio management, a natural extension is the prediction of multivariate returns processes. The proposed combination schemes can also be adapted to the specific prediction of variance, skewness or kurtosis. Acknowledgements This paper is a substantial revision and extension of Ravazzolo, Van Dijk, and Verbeek (2007). We are very grateful to participants of the Conference of the 50-th Anniversary of the Econometric Institute 2006, and the Conference on Computational Economics and Finance, Geneva, 2007, for their helpful comments on earlier versions of the paper. 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Handbook of Economic Forecasting North-Holland. 20 Table 1: Financial application: statistical and economic performance LI HI SV RSV BMA LIN TVW RTVW Panel A: Statisical accuracy RMSPE 4.618 4.478 4.509 4.470 4.500 4.514 4.484 4.485 Sign Ratio 0.527 0.549 0.614 0.598 0.587 0.610 0.602 0.598 Panel B: Active portfolio performances, γ= 6, transaction costs c= 0 Portfolio mean 4.708 4.741 4.812 4.657 4.701 5.177 5.021 5.785 Portfolio st dev 0.794 0.769 1.139 0.614 0.739 4.356 1.332 3.062 Sharpe ratio 0.110 0.156 0.168 0.060 0.108 0.128 0.301 0.380 Realized Utility -51.77 -51.76 -51.75 -51.79 -51.77 -51.73 -51.70 -51.56 ∆s285.5 288.7 295.2 277.9 283.8 304.3 317.1 381.3 ∆m-63.71 -60.49 -54.03 -71.29 -65.42 -44.95 -32.10 32.10 ∆b11.46 14.68 21.14 3.876 9.748 30.22 43.07 107.3 Panel C: Active portfolio performances, γ= 6, transaction costs c=10 bp Portfolio return 4.708 4.740 4.811 4.657 4.700 5.176 5.020 5.784 Portfolio st dev 0.794 0.769 1.139 0.614 0.739 4.355 1.332 3.062 Sharpe ratio 0.110 0.156 0.167 0.060 0.108 0.128 0.300 0.380 Realized Utility -51.77 -51.77 -51.77 -51.79 -51.78 -51.75 -51.71 -51.58 ∆s284.7 287.9 284.5 276.6 279.1 297.1 311.7 373.6 ∆m-64.65 -61.42 -64.80 -72.72 -70.18 -52.18 -37.66 24.31 ∆b10.81 14.04 10.67 2.741 5.289 23.29 37.81 99.77 21 Figure 1: Financial application: portfolio weight of stock (S&P500) 1987M1 1997M1 2007M1 0 10 20 30 40 50 60 70 80 90 100 % LI HI SV RSV 1987M1 1997M1 2007M1 0 10 20 30 40 50 60 70 80 90 100 % BMA LIN TVW RTVW Note: The graphs show the portfolio weight on the risky asset (S&P500) over the out-of-sample period associated to active asset management given by individual models in the left panel and combination schemes in the right panel. 22