# Forecasting U.S. Aggregate Stock Market Excess Return: Do Functional Data Analysis Add Economic Value?

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## Abstract

**:**

## 1. Introduction

## 2. Excess Return Forecast Models

#### 2.1. Functional Data Methodology and Nonparametric Estimation

#### Estimation Details

#### 2.2. Predictive Regressions

## 3. Data and Results

#### 3.1. Data and Traditional Predictors

- Dividend-price ratio (DP): the difference between the log of a twelve-month moving sum of dividends paid on the S&P500 index and the log of stock prices.
- Dividend yield (DY): the difference between the log of a twelve-month moving sum of dividends and the log of lagged stock prices.
- Earning-price ratio (EP): the log of a twelve-month moving sum of earnings on the S&P 500 index minus the log of stock prices.
- Dividend-payout ratio (DE): DP minus EP.
- Excess stock return volatility (RVOL): the sum of the squared daily returns on the S&P 500 index.
- Book-to-market ratio (BM): book value at the end of the previous year divided by the end-of-month market value of the Dow-Jones Industrial Average index.
- Net equity expansion (NITS): the 12-month moving sum of net equity issues by NYSE-listed stocks divided by the total end-of-year market capitalization of NYSE stocks.
- Treasury bill rate (TBL): three-month Treasury bill interest rate from the secondary market rate.
- Long-term yield (LTY): the long-term government bond yield.
- Term Spread (TMS): LTY minus TBL.
- Default yield spread (DFY): Moody’s BAA- minus AAA-rated corporate bond yields.
- Inflation (INFL): change in the Consumer Price Index (CPI) for all urban consumers (Following common practice in literature, we lag the inflation one extra month due to the delayed release of the CPI index).

#### 3.2. Dataset Used in NP-FDA Estimation

#### 3.3. Forecast Combination

- Equally weighted forecasts or pooled $(\mathrm{POOL}$-$\mathrm{AVG})$: this forecast combination method assigns equal weights to the forecasts of all individual models, i.e., ${w}_{t+1\mid t,m}=1/M$ for $m=1,\dots ,M$. This approach is likely to work well if the forecasting errors of different models have similar variances and are highly correlated, as explained in [30]. Therefore, in many cases, this simple average of forecasts can work well against more sophisticated weighting schemes [26,29].
- Thick Modeling Approach with MSFE $(\mathrm{POOL}$-$\mathrm{DMSFE})$: the second scheme consists of selecting models by means oh thick modeling approach. Following [31], the weight for model m is computed as:$${w}_{t+1\mid t,m}=\frac{{\varphi}_{m,t}^{-1}}{{\displaystyle \sum _{j=1}^{M}}{\varphi}_{m,t}^{-1}},\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{\varphi}_{m,t}=\underset{s=j}{\sum ^{t-1}}{\theta}^{t-1-s}{\left({r}_{s+1}-{\widehat{r}}_{s+1,m}\right)}^{2},$$
- Diffusion Index: this scheme involves the estimation of factors that are subsequently used for forecasting. The idea here is to extract a small number of common factors (often called diffusion indexes) assumed to drive the dynamics associated with a large number of potential return predictors (see, e.g., [9]). The basic intuition behind this approach is to filter out the noise present in the individual predictors, as discussed in [1]. The resulting factorial structure is more parsimoniuos, thus generating a more reliable signal.
- Sum-of-the-Parts Method: the fourth combination scheme is the sum-of-the-parts, which is in line with the ideas that are presented in [4]. The sum-of-the-parts scheme consists of decomposing the return index into three components: the dividend yield, the earnings growth rate, and the growth rate in the price-earnings ratio. Subsequently, each of these components is predicted separately. Ref. [4] show that their sum-of-the-parts forecast scheme significantly outperforms the historical average forecast.

#### 3.4. Forecast Evatuation

#### Out-of-Sample Excess Returns Predictability Results

#### 3.5. Economic-Based Forecast Evaluation

#### 3.6. Robustness Check

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Out of sample forecasting evaluation of S&P500 excess return based on nonparametric functional data analysis (NP-FDA) and regression models.

Predictor | Overall | Expansion | Recession | ||||||
---|---|---|---|---|---|---|---|---|---|

RRMSFE | ${R}_{\mathrm{oos}}^{2}$ | p-Value | RRMSFE | ${R}_{\mathrm{OOS}}^{2}$ | p-Value | RRMSFE | ${R}_{\mathrm{OOS}}^{2}$ | p-Value | |

Individual Models | |||||||||

$\mathrm{NP}$-$\mathrm{FDA}$ | $-0.262$ | $0.374$ | $0.023$ | $-0.151$ | $0.244$ | $0.062$ | $-0.580$ | $0.690$ | $0.002$ |

$log\left(\mathrm{DP}\right)$ | $0.244$ | $0.106$ | $0.101$ | $0.334$ | $0.169$ | $0.025$ | $-0.298$ | $-0.046$ | $0.556$ |

$log\left(\mathrm{DY}\right)$ | $0.366$ | $0.144$ | $0.081$ | $0.690$ | $0.203$ | $0.021$ | $-0.744$ | $0.006$ | $0.450$ |

$log\left(\mathrm{EP}\right)$ | $1.182$ | $-0.535$ | $0.567$ | 1280 | $-0.121$ | $0.297$ | $0.617$ | $-1.518$ | $0.716$ |

$log\left(\mathrm{DE}\right)$ | $1.347$ | $-1.779$ | $0.636$ | $0.469$ | $-0.422$ | $0.084$ | $3.080$ | $-5.003$ | $0.988$ |

$\mathrm{SVAR}$ | $-0.351$ | $0.104$ | $0.096$ | $-0.328$ | $0.183$ | $0.015$ | $-0.736$ | $-0.083$ | $0.629$ |

$\mathrm{BM}$ | $0.746$ | $-0.264$ | $0.448$ | $0.866$ | $0.089$ | $0.196$ | $0.126$ | $-1.103$ | $0.665$ |

$\mathrm{NTIS}$ | $0.172$ | $-0.287$ | $0.243$ | $-0.650$ | $1.249$ | $0.002$ | $1.777$ | $-3.937$ | $0.973$ |

$\mathrm{TBL}$ | $0.152$ | $0.270$ | $0.018$ | $0.243$ | $0.522$ | $0.001$ | $-0.398$ | $-0.328$ | $0.865$ |

$\mathrm{LTY}$ | $0.718$ | $0.275$ | $0.004$ | $1.105$ | $0.455$ | $0.001$ | $-0.538$ | $-0.153$ | $0.738$ |

$\mathrm{LTR}$ | $0.207$ | $0.213$ | $0.124$ | $0.233$ | $0.381$ | $0.094$ | $-0.186$ | $-0.184$ | $0.445$ |

$\mathrm{TMS}$ | $0.169$ | $-0.108$ | $0.489$ | $0.056$ | $0.008$ | $0.340$ | $0.109$ | $-0.382$ | $0.873$ |

$\mathrm{DFY}$ | $0.234$ | $-0.032$ | $0.340$ | $0.023$ | $0.214$ | $0.077$ | $0.406$ | $-0.618$ | $0.953$ |

$\mathrm{DFR}$ | $0.080$ | $-0.225$ | $0.483$ | $-0.041$ | $0.227$ | $0.121$ | $0.038$ | $-1.300$ | $0.821$ |

$\mathrm{INFL}$ | $0.107$ | $0.036$ | $0.331$ | $-0.050$ | $0.270$ | $0.003$ | $0.152$ | $-0.520$ | $0.933$ |

Foreacst Combination | |||||||||

POOL-AVG | $-0.229$ | $0.215$ | $0.105$ | $-0.547$ | $0.559$ | $0.002$ | $-0.062$ | $0.141$ | $0.188$ |

POOL-DMSFE | $-0.283$ | $0.221$ | $0.085$ | $-0.548$ | $0.561$ | $0.002$ | $-0.070$ | $-0.040$ | $0.258$ |

Diffusion index | $-0.365$ | $0.350$ | $0.032$ | $-0.485$ | $0.434$ | $0.025$ | $-0.171$ | $0.111$ | $0.145$ |

Sum-of-the-parts | $-0.497$ | $0.483$ | $0.015$ | $-0.781$ | $1.025$ | $0.005$ | $0.174$ | $-0.286$ | $0.381$ |

**Table 2.**Out of sample forecasting evaluation of Dow Jones Industrial Average (DJIA) excess return based on NP-FDA and regression models.

Predictor | Full Sample | Expansion | Recession | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RRMSFE | ${R}_{\mathrm{oos}}^{2}$ | p-Value | RRMSFE | ${R}_{\mathrm{oos}}^{2}$ | p-Value | RRMSFE | ${R}_{\mathrm{oos}}^{2}$ | p-Value | |||

Individual Models | |||||||||||

$\mathrm{NP}$-$\mathrm{FDA}$ | $-0.287$ | $0.574$ | $0.010$ | $-0.529$ | $1.056$ | $0.001$ | $0.404$ | $0.280$ | $0.089$ | ||

$log\left(\mathrm{DP}\right)$ | $-0.002$ | $0.005$ | $0.382$ | $-0.0640$ | $0.128$ | $0.173$ | $0.168$ | $0.248$ | $0.101$ | ||

$log\left(\mathrm{DY}\right)$ | $-0.058$ | $0.116$ | $0.142$ | $-0.1306$ | $0.262$ | $0.022$ | $0.143$ | $1.110$ | $0.008$ | ||

$log\left(\mathrm{EP}\right)$ | $0.277$ | $-0.555$ | $0.666$ | $0.1092$ | $-0.219$ | $0.433$ | $0.737$ | $-0.617$ | $0.414$ | ||

$log\left(\mathrm{DE}\right)$ | $0.805$ | $-1.618$ | $0.684$ | $0.2395$ | $-0.48$ | $0.143$ | $2.347$ | $-5.453$ | $0.901$ | ||

$\mathrm{SVAR}$ | $-0.066$ | $0.131$ | $0.117$ | $-0.141$ | $0.283$ | $0.015$ | $0.143$ | $1.365$ | $0.119$ | ||

$\mathrm{BM}$ | $0.156$ | $-0.313$ | $0.470$ | $0.008$ | $-0.016$ | $0.271$ | $0.564$ | $-0.116$ | $0.371$ | ||

$\mathrm{NTIS}$ | $0.142$ | $-0.286$ | $0.141$ | $-0.771$ | $1.537$ | $0.000$ | $2.616$ | $-4.782$ | $0.950$ | ||

$\mathrm{TBL}$ | $-0.147$ | $0.295$ | $0.028$ | $-0.305$ | $0.610$ | $0.000$ | $0.287$ | $0.976$ | $0.121$ | ||

$\mathrm{LTY}$ | $-0.147$ | $0.294$ | $0.014$ | $-0.263$ | $0.525$ | $0.000$ | $0.171$ | $0.736$ | $0.099$ | ||

$\mathrm{LTR}$ | $-0.141$ | $0.282$ | $0.089$ | $-0.272$ | $0.543$ | $0.054$ | $0.219$ | $0.863$ | $0.117$ | ||

$\mathrm{TMS}$ | $0.072$ | $-0.145$ | $0.450$ | $-0.014$ | $0.028$ | $0.293$ | $0.308$ | $0.286$ | $0.266$ | ||

$\mathrm{DFY}$ | $0.015$ | $-0.031$ | $0.326$ | $-0.109$ | $0.219$ | $0.087$ | $0.357$ | $-0.836$ | $0.825$ | ||

$\mathrm{DFR}$ | $0.090$ | $-0.182$ | $0.519$ | $-0.172$ | $0.344$ | $0.038$ | $0.810$ | $-0.586$ | $0.533$ | ||

$\mathrm{INFL}$ | $-0.041$ | $0.082$ | $0.204$ | $-0.114$ | $0.228$ | $0.039$ | $0.160$ | $-0.208$ | $0.960$ | ||

Foreacst Combination | |||||||||||

$\mathrm{POOL}$-$\mathrm{AVG}$ | $-0.090$ | $0.181$ | $0.065$ | $-0.262$ | $0.524$ | $0.000$ | $-0.613$ | $1.223$ | $0.007$ | ||

$\mathrm{POOL}$-$\mathrm{DMSFE}$ | $-0.093$ | $0.187$ | $0.058$ | $-0.261$ | $0.523$ | $0.000$ | $-0.657$ | $1.311$ | $0.010$ | ||

$\mathrm{Diffusion}\phantom{\rule{3.33333pt}{0ex}}\mathrm{index}$ | $-0.093$ | $0.187$ | $0.060$ | $-0.270$ | $0.542$ | $0.000$ | $-1.884$ | $3.734$ | $0.000$ | ||

$\mathrm{Sum}\phantom{\rule{3.33333pt}{0ex}}\mathrm{of}\phantom{\rule{3.33333pt}{0ex}}\mathrm{the}\phantom{\rule{3.33333pt}{0ex}}\mathrm{parts}$ | $-0.256$ | $0.513$ | $0.007$ | $-0.797$ | $1.590$ | $0.000$ | $1.968$ | $-3.976$ | $0.999$ |

**Table 3.**Performance evaluation for an Investor with Mean Variance Utility with $\gamma =5$, 1956:01-2019:12.

Predictor | S&P500 | DJIA | |||||
---|---|---|---|---|---|---|---|

Full Sample | Expansion | Recession | Full Sample | Expansion | Recession | ||

Individual Models | |||||||

$\mathrm{NP}$-$\mathrm{FDA}$ | $0.983$ | $0.691$ | $2.676$ | $0.277$ | $0.390$ | $0.087$ | |

$log\left(\mathrm{DP}\right)$ | $-0.005$ | $-0.025$ | $0.113$ | $-0.205$ | $-0.223$ | $-0.096$ | |

$log\left(\mathrm{DY}\right)$ | $0.052$ | $0.024$ | $0.216$ | $-0.020$ | $-0.023$ | $0.000$ | |

$log\left(\mathrm{EP}\right)$ | $-1031$ | $-0.164$ | $-6.172$ | $-0.866$ | $-0.34$ | $-3.989$ | |

$log\left(\mathrm{DE}\right)$ | $-3.132$ | $-0.617$ | $-1.773$ | $-2.851$ | $-0.842$ | $-1.458$ | |

$\mathrm{SVAR}$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | |

$\mathrm{BM}$ | $-0.666$ | $-0.019$ | $-4.504$ | $-0.624$ | $-0.189$ | $-3.214$ | |

$\mathrm{NTIS}$ | $-1.303$ | $1.156$ | $-1.566$ | $-0.860$ | $1.482$ | $-1.452$ | |

$\mathrm{TBL}$ | $0.137$ | $0.409$ | $-1.471$ | $0.162$ | $0.452$ | $-1.556$ | |

$\mathrm{LTY}$ | $0.239$ | $0.352$ | $-0.435$ | $0.273$ | $0.363$ | $-0.265$ | |

$\mathrm{LTR}$ | $0.066$ | $0.271$ | $-1154$ | $0.045$ | $0.272$ | $-1.308$ | |

$\mathrm{TMS}$ | $-0.446$ | $-0.237$ | $-1.690$ | $-0.565$ | $-0.350$ | $-1.839$ | |

$\mathrm{DFY}$ | $-0.205$ | $0.027$ | $-1.582$ | $-0.309$ | $-0.147$ | $-1.274$ | |

$\mathrm{DFR}$ | $-0.412$ | $0.022$ | $-2.983$ | $-0.490$ | $0.041$ | $-3.634$ | |

$\mathrm{INFL}$ | $-0.107$ | $0.150$ | $-1.633$ | $-0.092$ | $-0.089$ | $-0.111$ | |

Foreacst Combination | |||||||

POOL-AVG | $2.162$ | $2.853$ | $1.388$ | $0.044$ | $0.034$ | $-0.756$ | |

POOL-DMSFE | $2.269$ | $2.536$ | $1.516$ | $0.073$ | $0.034$ | $-0.571$ | |

Diffusion index | $2.360$ | $2.360$ | $3.921$ | $0.104$ | $0.054$ | $0.050$ | |

Sum-of-the-parts | $1.947$ | $3.149$ | $-0.152$ | $-0.012$ | $1.608$ | $-1.282$ |

RRMSFE | ${\mathit{R}}_{\mathbf{oos}}^{2}$ | p-Value | Δ (annual %) | |
---|---|---|---|---|

Full-Sample | $-0.081$ | $0.163$ | $0.067$ | $0.141$ |

Expansion | $-0.238$ | $0.477$ | $0.008$ | $0.471$ |

Recession | $0.070$ | $-0.142$ | $0.715$ | $0.051$ |

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**MDPI and ACS Style**

Caldeira, J.F.; Gupta, R.; Torrent, H.S. Forecasting U.S. Aggregate Stock Market Excess Return: Do Functional Data Analysis Add Economic Value? *Mathematics* **2020**, *8*, 2042.
https://doi.org/10.3390/math8112042

**AMA Style**

Caldeira JF, Gupta R, Torrent HS. Forecasting U.S. Aggregate Stock Market Excess Return: Do Functional Data Analysis Add Economic Value? *Mathematics*. 2020; 8(11):2042.
https://doi.org/10.3390/math8112042

**Chicago/Turabian Style**

Caldeira, João F., Rangan Gupta, and Hudson S. Torrent. 2020. "Forecasting U.S. Aggregate Stock Market Excess Return: Do Functional Data Analysis Add Economic Value?" *Mathematics* 8, no. 11: 2042.
https://doi.org/10.3390/math8112042