- , (A-29) where mj, v2 j , and qj, j = 1, 2, ..., 7, are constants specified in Kim et al. (1998) and thus need not be estimated. In turn, (A-29) implies u∗∗ Ä+1
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- , (A-30) 40 However, we modify the algorithm of Primiceri (2005) to reflect the correction to the ordering of steps detailed in Del Negro and Primiceri (2014). where each state has probability Pr (sÄ+1 = j) = qj. (A-31) Draws for the sequence of states st can easily be obtained, noting that each of its elements can be independently drawn from the discrete density defined by Pr sÄ+1 = j| Θ, θt , ht , M0 i, Dt = qjfN r∗∗ Ä+1
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- , we cannot resort to standard Kalman recursions and simulation algorithms such as those in Carter and Kohn (1994) or Durbin and Koopman (2002). To obviate this problem, Kim et al. (1998) employ a data augmentation approach and introduce a new state variable sÄ+1, Ä = 1, .., t−1, turning their focus on drawing from p ht Θ, θt , st, M0 i, Dt instead of p ht Θ, θt , M0 i, Dt . The introduction of the state variable sÄ+1 allows us to rewrite the linear non-Gaussian state space representation in (A-27)-(A-28) as a linear Gaussian state space model, making use of the following approximation, u∗∗ Ä+1 ≈ 7
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- , we follow Primiceri (2005) and employ the algorithm of Kim et al.
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- . (A-41) 3. Draws from p rt+1| θt+1, ht+1, Θ, θt , ht, M0 i, Dt : we have that rt+1| θt+1, ht+1, Θ, θt , ht , M0 i, Dt ∼ N (( + t+1) + (β + βt+1) xt, exp (ht+1)) . (A-42) B Sequential combination In this section, we summarize the prior elicitation and the posterior simulation for the density combination algorithm proposed in Billio et al. (2013), which we extend with a learning mechanism based on the past economic performance of the individual models entering the combination.
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- (1998).40 Define r∗ Ä+1 = rÄ+1 − ( + Ä+1) − (β + βÄ+1) xÄ and note that r∗ Ä+1 is observable conditional on , β, and θt . Next, rewrite (19) as r∗ Ä+1 = exp (hÄ+1) uÄ+1. (A-26) Squaring and taking logs on both sides of (A-26) yields a new state space system that replaces (19)-(21) with r∗∗ Ä+1 = 2hÄ+1 + u∗∗ Ä+1, (A-27) hÄ+1 = λ0 + λ1hÄ + ξÄ+1, (A-28) where r∗∗ Ä+1 = ln h r∗ Ä+1 2 i , and u∗∗ Ä+1 = ln u2 Ä+1 , with u∗∗ Ä independent of ξs for all Ä and s. Since u∗∗ Ä+1 ∼ ln Ç2 1
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- Aastveit, K., F. Ravazzolo, and H. van Dijk (2014). Nowcasting the business cycle in an uncertain enviroment. Norges Bank.
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- All results are based on an evaluation period that extends from January 1947 to December 2007.
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- All results are based on the whole forecast evaluation period, January 1947 -
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- All results are based on the whole forecast evaluation period, January 1947 -
Paper not yet in RePEc: Add citation now
Amisano, G. and R. Giacomini (2007). Comparing density forecasts via weighted likelihood ratio tests. Journal of Business & Economic Statistics 25(2), 177–190.
Andrews, D. W. K. and J. C. Monahan (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60(4), pp. 953–966.
- Avramov, D. (2002). Stock return predictability and model uncertainty. Journal of Financial Economics 64(3), 423 – 458.
Paper not yet in RePEc: Add citation now
Banbura, M., D. Giannone, and L. Reichlin (2010). Large Bayesian vector auto regressions. Journal of Applied Econometrics 25(1), 71–92.
Barberis, N. (2000). Investing for the long run when returns are predictable. The Journal of Finance 55(1), 225 – 264.
- BI Norwegian Business School Centre for Applied Macro - Petroleum economics (CAMP) N-0442 Oslo http://www.bi.no/camp CAMP Working Paper Series ISSN: 1892-2198 CENTRE FOR APPLIED MACRO - AND PETROLEUM ECONOMICS (CAMP) Centre for Applied Macro - and Petroleum economics (CAMP) will bring together economists working on applied macroeconomic issues, with special emphasis on petroleum economics.
Paper not yet in RePEc: Add citation now
Billio, M., R. Casarin, F. Ravazzolo, and H. K. van Dijk (2013). Time-varying combinations of predictive densities using nonlinear filtering. Journal of Econometrics 177(2), 213–232.
- Bold figures indicate all instances in which the CERD is greater than zero. All results are based on the whole forecast evaluation period, January 1947 - December 2010.
Paper not yet in RePEc: Add citation now
Bossaerts, P. and P. Hillion (1999). Implementing statistical criteria to select return forecasting models: what do we learn? Review of Financial Studies 12(2), 405–428.
Boudoukh, J., M. Richardson, and R. F. Whitelaw (2008). The myth of long-horizon predictability.
Boudoukh, J., R. Michaely, M. Richardson, and M. R. Roberts (2007). On the importance of measuring payout yield: Implications for empirical asset pricing. The Journal of Finance 62(2), 877–915.
- Brown, S. J. (1976). Optimal portfolio choice under uncertainty: a Bayesian approach. Ph.D. Dissertation, University of Chicago.
Paper not yet in RePEc: Add citation now
Campbell, J. Y. and S. B. Thompson (2008). Predicting excess stock returns out of sample: Can anything beat the historical average? Review of Financial Studies 21(4), 1509–1531.
- Carter, C. K. and R. Kohn (1994). On gibbs sampling for state space models. Biometrika 81(3), pp. 541–553.
Paper not yet in RePEc: Add citation now
Cenesizoglu, T. and A. Timmermann (2012). Do return prediction models add economic value? Journal of Banking & Finance 36(11), 2974 – 2987.
Clark, T. E. (2011). Real-time density forecasts from bayesian vector autoregressions with stochastic volatility. Journal of Business & Economic Statistics 29(3), 327–341.
- Clark, T. E. and F. Ravazzolo (2015). Macroeconomic forecasting performance under alternative specifications of time-varying volatility. Journal of Applied Econometrics 30(4), 551–575.
Paper not yet in RePEc: Add citation now
Cochrane, J. H. (2008). The dog that did not bark: A defense of return predictability. Review of Financial Studies 21(4), 1533–1575.
- Columns two to five of Table C.4 show CERD results separately for recession and expansion periods, as defined by the NBER indicator. This type of analysis has been proposed by authors such as Rapach et al. (2010) and Henkel et al. (2011). When focusing on the linear models (columns two and four), we find higher economic predictability in recessions than in expansions. This results is consistent with the findings in these studies. For the TVP-SV models (column three and five), the story is however different. There we find the largest economic gains during expansions. This holds true both for the individual models and the various model combinations. This finding is somewhat surprising, since we would expect time-varying models to help when entering recessions; on the other hand, stochastic volatility might reduce the return volatility during long expansionary periods, having important consequences in the resulting asset allocations.
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- D Additional results In this section, we present a number of supplementary tables and charts, including results for a shorter evaluation sample ending in 2007 before the onset of the latest recession, and a graphical summary of the time dynamics of the CER-based DeCo combination weights.
Paper not yet in RePEc: Add citation now
Dangl, T. and M. Halling (2012). Predictive regressions with time-varying coefficients. Journal of Financial Economics 106(1), 157 – 181.
- Del Negro, M. and G. E. Primiceri (2014). Time-varying structural vector autoregressions and monetary policy: A corrigendum. Working Paper, Northwestern University.
Paper not yet in RePEc: Add citation now
Del Negro, M., R. Hasegawa, and F. Schorfheide (2014). Dynamic prediction pools: an investigation of financial frictions and forecasting performance. Penn Institute for Economic Research Working Paper 14-034.
- Diebold, F. (1991). A note on bayesian forecast combination procedures. In A. Westlund and P. Hackl (Eds.), Economic Structural Change: Analysis and Forecasting, pp. 225–232. New York: Springer-Verlag.
Paper not yet in RePEc: Add citation now
Diebold, F. X. and R. S. Mariano (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics 13(3), 253–263.
Durbin, J. and S. J. Koopman (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika 89(3), pp. 603–615.
Elliott, G. and A. Timmermann (2005). Optimal forecast combination under regime switching.
- Figure D.1. Predictor weights for the CER-based DeCo combination scheme Linear 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007 CER-based DeCo weights 0 0.2 0.4 0.6 0.8 1
Paper not yet in RePEc: Add citation now
Frost, P. A. and J. E. Savarino (1986). An empirical Bayes approach to efficient portfolio selection. The Journal of Financial and Quantitative Analysis 21(3), pp. 293–305.
Geweke, J. (2010). Complete and Incomplete Econometric Models. Princeton: Priceton University Press.
Geweke, J. and G. Amisano (2010). Comparing and evaluating bayesian predictive distributions of asset returns. International Journal of Forecasting 26(2), 216 – 230.
Geweke, J. and G. Amisano (2011). Optimal prediction pools. Journal of Econometrics 164(1), 130 – 141.
- Geweke, J. and G. Amisano (2012). Prediction using several macroeconomic models. Technical report, European Central Bank, Frankfurt.
Paper not yet in RePEc: Add citation now
Gneiting, T. and A. E. Raftery (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association 102(477), 359–378.
Gneiting, T. and R. Ranjan (2011). Comparing density forecasts using threshold- and quantileweighted scoring rules. Journal of Business & Economic Statistics 29(3), 411–422.
Granger, C. W. and M. J. Machina (2006). Forecasting and decision theory. In C. G. G. Elliott and A. Timmermann (Eds.), Handbook of Economic Forecasting, Volume 1, pp. 81 – 98. Elsevier.
Groen, J., R. Paap, and F. Ravazzolo (2013). Real-Time Inflation Forecasting in a Changing World. Journal of Business and Economic Stastistics 31(1), 29–44.
Hall, S. G. and J. Mitchell (2007). Combining density forecasts. International Journal of Forecasting 23(1), 1 – 13.
Hendry, D. F. and M. P. Clements (2004). Pooling of forecasts. The Econometrics Journal 7(1), pp. 1–31.
Henkel, S. J., J. S. Martin, and F. Nardari (2011). Time-varying short-horizon predictability. Journal of Financial Economics 99(3), 560 – 580.
- Hoeting, J. A., D. Madigan, A. E. Raftery, and C. T. Volinsky (1999). Bayesian model averaging: A tutorial. Statistical Science 14, 382–417.
Paper not yet in RePEc: Add citation now
Hoogerheide, L., R. Kleijn, R. Ravazzolo, H. K. van Dijk, and M. Verbeek (2010). Forecast Accuracy and Economic Gains from Bayesian Model Averaging using Time Varying Weights. Journal of Forecasting 29(1-2), 251–269.
Huang, D., F. Jiang, J. Tu, and G. Zhou (2015). Investor sentiment aligned: A powerful predictor of stock returns. Review of Financial Studies 28(3), 791–837.
- Johannes, M., A. Korteweg, and N. Polson (2014). Sequential learning, predictive regressions, and optimal portfolio returns. Journal of Finance 69(2), 611–644.
Paper not yet in RePEc: Add citation now
Jore, A. S., J. Mitchell, and S. P. Vahey (2010). Combining forecast densities from VARs with uncertain instabilities. Journal of Applied Econometrics 25(4), 621–634.
Kan, R. and G. Zhou (2007). Optimal portfolio choice with parameter uncertainty. The Journal of Financial and Quantitative Analysis 42(3), pp. 621–656.
Kandel, S. and R. F. Stambaugh (1996). On the predictability of stock returns: An assetallocation perspective. The Journal of Finance 51(2), pp. 385–424.
Kapetanios, G., J. Mitchell, S. Price, and N. Fawcett (2015). Generalised density forecast combinations. Journal of Econometrics 188(1), 150 – 165.
Kim, S., N. Shephard, and S. Chib (1998). Stochastic volatility: Likelihood inference and comparison with arch models. The Review of Economic Studies 65(3), 361–393.
Klein, R. W. and V. S. Bawa (1976). The effect of estimation risk on optimal portfolio choice. Journal of Financial Economics 3(3), 215 – 231.
- Koop, G. (2003). Bayesian Econometrics. John Wiley & Sons, Ltd.
Paper not yet in RePEc: Add citation now
Koop, G. and D. Korobilis (2011). UK macroeconomic forecasting with many predictors: Which models forecast best and when do they do so? Economic Modelling 28(5), 2307 – 2318.
Koop, G. and D. Korobilis (2012). Forecasting inflation using dynamic model averaging. International Economic Review 53(3), 867–886.
Leitch, G. and J. E. Tanner (1991). Economic forecast evaluation: Profits versus the conventional error measures. The American Economic Review 81(3), pp. 580–590.
Lettau, M. and S. Van Nieuwerburgh (2008). Reconciling the return predictability evidence.
- Log(DE) Log(Smooth EP) Log(EP) NTIS LTY Log(DY) Log(NPY) BM Others TVP-SV 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007 CER-based DeCo weights 0 0.2 0.4 0.6 0.8 1
Paper not yet in RePEc: Add citation now
Mitchell, J. and S. G. Hall (2005). Evaluating, comparing and combining density forecasts using the KLIC with an application to the Bank of England and NIESER “fan†charts of inflation.
- Paye, B. S. (2012). The economic value of estimated porfolio rules under general utility specifications. Terry College of Business, University of Georgia Working Paper.
Paper not yet in RePEc: Add citation now
Paye, B. S. and A. Timmermann (2006). Instability of return prediction models. Journal of Empirical Finance 13(3), 274 – 315.
- Pesaran, M. H. and S. Skouras (2007). Decision-Based Methods for Forecast Evaluation, pp. 241–267. Blackwell Publishing Ltd.
Paper not yet in RePEc: Add citation now
Pettenuzzo, D. and A. Timmermann (2011). Predictability of stock returns and asset allocation under structural breaks. Journal of Econometrics 164(1), 60 – 78.
Pettenuzzo, D., A. Timmermann, and R. Valkanov (2014). Forecasting stock returns under economic constraints. Journal of Financial Economics 114(3), 517–553.
Poirier, D. J. (1995). Intermediate Statistics and Econometrics: A Comparative Approach. MIT Press.
Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. The Review of Economic Studies 72(3), 821–852.
- Raftery, A., M. Karny, and P. Ettler (2010). Online prediction under model uncertainty via dynamic model averaging: Application to a cold rolling mill. Technometrics 52, 52–66.
Paper not yet in RePEc: Add citation now
Rapach, D. E., J. K. Strauss, and G. Zhou (2010). Out-of-sample equity premium prediction: Combination forecasts and links to the real economy. Review of Financial Studies 23(2), 821–862.
- Rapach, D. E., J. K. Strauss, J. Tu, and G. Zhou (2015). Industry interdependencies and cross-industry return predictability. Working paper, Olin School of Business at Washington University.
Paper not yet in RePEc: Add citation now
Sentana, E. (2005). Least squares predictions and mean-variance analysis. Journal of Financial Econometrics 3(1), 56–78.
Stambaugh, R. F. (1997). Analyzing investments whose histories differ in length. Journal of Financial Economics 45(3), 285 – 331.
Stock, J. H. and M. W. Watson (1996). Evidence on structural instability in macroeconomic time series relations. Journal of Business & Economic Statistics 14(1), 11–30.
Stock, J. H. and M. W. Watson (2004). Combination forecasts of output growth in a sevencountry data set. Journal of Forecasting 23(6), 405–430.
- SVAR Log(Smooth EP) Log(DY) BM Log(NPY) Log(DE) NTIS Log(EP) Others This figure plots the posterior means of the CER-based DeCo weights for the top individual linear models (top panel) and TVP-SV models (bottom panel) over the out-of-sample period. The individual predictors showed are Log(DP): log dividend price ratio, Log(DY): log dividend yield, Log(EP): log earning price ratio, Log(Smooth EP): log smooth earning price ratio, Log(DE): log dividend-payout ratio, BM: book-to-market ratio, TBL: T-Bill rate, LTY: long-term yield, LTR: long-term return, TMS: term spread, DFY: default yield spread, DFR: default return spread, SVAR: stock variance, NTIS: net equity expansion, INFL: inflation, and Log(NPY): log total net payout yield. The out of sample period starts in January 1947 and ends in December 2010.
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- The last four columns of Table C.4 show CERD results separately for two out-of-sample periods, 1947-1978 and 1979-2010. Welch and Goyal (2008) argue that the predictive ability of many predictor variables deteriorates markedly after the 1973-1975 oil shock, so we are particularly interested in whether the same holds true here. The results of Table C.4 are overall consistent with this pattern, as we observe smaller gains during the second subsample, both for the individual models and the various model combinations. However, the CER-based DeCo CERDs are still fairly large, as high as 87 basis points in the case of linear models, and as high as 167 basis points in the TVP-SV case.
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Timmermann, A. (2006). Forecast combinations. In G. Elliot, C. Granger, and A. Timmermann (Eds.), North-Holland, Volume 1 of Handbook of Economic Forecasting, Chapter 4, pp. 135– 196. Elsevier.
Tu, J. and G. Zhou (2011). Markowitz meets talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics 99(1), 204 – 215.
Waggoner, D. F. and T. Zha (2012). Confronting model misspecification in macroeconomics. Journal of Econometrics 171(2), 167 – 184. Bayesian Models, Methods and Applications.
- We measure statistical significance relative to the prevailing mean model using the Diebold and Mariano (1995) t-tests for equality of the average loss. One star * indicates significance at 10% level; two stars ** significance at 5% level; three stars *** significance at 1% level. Bold figures indicate all instances in which the forecast accuracy measures are greater than zero.
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- We measure statistical significance relative to the prevailing mean model using the Diebold and Mariano (1995) t-tests for equality of the average loss. One star * indicates significance at 10% level; two stars ** significance at 5% level; three stars *** significance at 1% level. Bold figures indicate all instances in which the forecast accuracy metrics are greater than zero.
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Welch, I. and A. Goyal (2008). A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21(4), 1455–1508.