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Forecasting Asset Returns in Realistic Environments David E. Rapach Saint Louis University CFA Montreal Asset Management Forum October 8, 2015 David E. Rapach Forecasting Asset Returns Introduction Forecasting asset returns is fascinating

  1. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  2. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  3. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  4. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  5. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  6. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  7. Forecasting challenges & strategies Stating the obvious I We don’t know The Model of asset returns Actual DGP is highly complex & constantly evolving Substantial model uncertainty & instability Hedgehog approach inadvisable Need to incorporate info from many predictors But also need to avoid overfitting Effective strategies from recent literature Forecast combination & diffusion indices Stating the obvious II I’m a fox David E. Rapach Forecasting Asset Returns

  8. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  9. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  10. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  11. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  12. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  13. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  14. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  15. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  16. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  17. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  18. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  19. Stylized example: equity risk premium ERP: S&P 500 return minus risk-free return Huge literature on predicting ERP (Rapach & Zhou 2013) Predictive regression forecast: ˆ α t + ˆ R t + 1 = ˆ β t x t Incorporate info from x t to forecast R t + 1 Prevailing mean benchmark forecast: ¯ R t + 1 Assumes no return predictability (RW with drift) Stringent out-of-sample benchmark (Goyal & Welch 2008) Out-of-sample R 2 (Campbell & Thompson 2008) Proportional ↓ in MSFE for PR vis-à-vis PM Sample period: 1926:01–2014:12 Forecast evaluation period: 1960:01–2014:12 David E. Rapach Forecasting Asset Returns

  20. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  21. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  22. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  23. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  24. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  25. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  26. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  27. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  28. Highly plausible predictors Dividend yield (Fama & French 1988) T-bill yield (Ang & Bekaert 2007) Term spread (Campbell 1987) Default spread (Fama & French 1989) Inflation (Nelson 1976) Output gap (Cooper & Priestly 2009) Moving-average signal (Neely, Rapach, Tu, & Zhou 2014) Momentum signal (Neely, Rapach, Tu, & Zhou 2014) David E. Rapach Forecasting Asset Returns

  29. Individual monthly PR forecasts Dividend yield T-bill yield 2 2 1 1 0 0 -1 -1 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 Term spread Default spread 2 2 1 1 0 0 -1 -1 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 David E. Rapach Forecasting Asset Returns

  30. Individual monthly PR forecasts Inflation Output gap 2 2 1 1 0 0 -1 -1 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 MA signal Momentum signal 2 2 1 1 0 0 -1 -1 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 David E. Rapach Forecasting Asset Returns

  31. R 2 OS stats (%) Semi- Predictor Monthly Quarterly annual Annual Dividend yield − 0.03 − 2.27 1 . 11 − 1.87 T-bill yield − 0.10 − 0.47 − 1.86 − 2.59 Term spread 0 . 25 0 . 45 0.40 2 . 24 Default spread 0.30 1 . 11 1 . 29 3 . 28 Inflation 0.03 0.63 1 . 07 1 . 76 Output gap 0.17 0.45 0 . 88 1.20 MA signal 0 . 41 − 0.16 − 1.05 − 1.66 Momentum signal − 0.09 − 0.46 − 1.12 − 0.30 David E. Rapach Forecasting Asset Returns

  32. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  33. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  34. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  35. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  36. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  37. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  38. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  39. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  40. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  41. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  42. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  43. Drawbacks to individual PR forecasts Individual PR forecasts perform inconsistently/unevenly We can’t identify best predictor a priori Relying on individual PR forecast is too risky Too ‘hedgehogy’ We need to get foxy Incorporate info from multiple plausible predictors Forecast diversification (Timmermann 2006) Bad strategy: include all predictors in one regression Kitchen sink PR: ˆ α t + ˆ β 1 , t x 1 , t + · · · + ˆ R t + 1 = ˆ β K , t x K , t R 2 OS stats: − 0 . 95 % , − 6 . 22 % , − 5 . 25 % , − 9 . 64 % In-sample overfitting (avoid overfitting like the plague) David E. Rapach Forecasting Asset Returns

  44. Monthly kitchen sink PR forecast 2 1 0 -1 1960 1970 1980 1990 2000 2010 David E. Rapach Forecasting Asset Returns

  45. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  46. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  47. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  48. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  49. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  50. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  51. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  52. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  53. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  54. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  55. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  56. Good foxy strategies Incorporate info from all predictors without overfitting Forecast combination (Rapach, Strauss, & Zhou 2010) Take simple average of individual PR forecasts Avoid overfitting by shrinking forecast to PM Shrink to ‘outside view’ R 2 OS stats: 0 . 77 % , 2 . 17 % , 2 . 52 % , 6 . 08 % Diffusion index (Ludvigson & Ng 2007) Extract first few principal components from all predictors PCs serve as predictors for PR forecast Filter noise in individual predictors ⇒ more reliable signal R 2 OS stats: 0 . 60 % , 1 . 62 % , 2 . 66 % , 6 . 97 % David E. Rapach Forecasting Asset Returns

  57. Monthly combination forecast 2 1 0 -1 1960 1970 1980 1990 2000 2010 David E. Rapach Forecasting Asset Returns

  58. Monthly diffusion index forecast 2 1 0 -1 1960 1970 1980 1990 2000 2010 David E. Rapach Forecasting Asset Returns

  59. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  60. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  61. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  62. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  63. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  64. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  65. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  66. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  67. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  68. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  69. Dynamic asset allocation MV investor allocates monthly across stocks & bills Allocation to risky stocks: w t = ( 1 /γ )(ˆ σ 2 R t + 1 / ˆ t + 1 ) Realistic portfolio constraint: − 0 . 5 ≤ w t ≤ 1 . 5 σ 2 Certainty equivalent return (CER): ˆ µ p − 0 . 5 γ ˆ p σ 2 µ p ( ˆ ˆ p ): portfolio mean (variance) over evaluation period Annualized CER gain vis-à-vis PM for γ = 5 Combination forecast: 1.79% Diffusion index forecast: 2.46% Return predictability is economically valuable NB: assuming ‘small’ investor David E. Rapach Forecasting Asset Returns

  70. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  71. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  72. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  73. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  74. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  75. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  76. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  77. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  78. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  79. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  80. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  81. Additional elements Combination weights (Rapach, Strauss, & Zhou 2010) Place more weight on particular forecasts NB: typically best to hew close to equal weighting Economic restrictions ⇒ ↓ estimation uncertainty Sign restrictions (Campbell & Thompson 2008) Valuation ratios (Ferreira & Santa-Clara 2011) Bayesian (Pettenuzzo, Timmermann, & Valkanov 2014) Useful even if not literally true (bias-efficiency tradeoff) Regime switching (Ang & Timmermann 2012) Markov switching (Henkel, Martin, & Nardari 2011) Time-varying parameters (Dangl & Halling 2012) David E. Rapach Forecasting Asset Returns

  82. Dynamic asset allocation: two recent studies MKT components (Kong, Rapach, Strauss, & Zhou 2011) Allocate across 25 Fama-French size/value portfolios Combination forecasts of component portfolio returns Based on 39 predictor variables Combination forecasts valuable inputs for DAA US stocks/bonds/bills (Almadi, Suri, & Rapach 2014) Active portfolio management vis-à-vis 60/35/5 benchmark Diffusion index forecasts of stock/bond/bill returns Extracted from fundamental, macro, and technical variables Diffusion index forecasts valuable inputs for DAA Largest gains typically realized during contractions/crises David E. Rapach Forecasting Asset Returns

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