Time-frequency forecast of the equity premium

Abstract

Any time series can be decomposed into cyclical components fluctuating at different frequencies. Accordingly, in this paper, we propose a method to forecast the equity premium which exploits the frequency relationship between the equity premium and several predictor variables. We evaluate a large set of models and find that, by selecting the relevant frequencies for equity premium forecasting purposes, this method significantly improves in a statistical and economic way upon standard time series forecasting methods. This outperformance is robust regardless of the predictor used, the out-of-sample period considered, and the frequency of the data used.

Keywords:

• Time-frequency forecast
• Equity premium
• Multiresolution analysis

JEL classification:

• C58
• G11
• G17

Acknowledgments

This is a revised version of the paper that circulated as “Forecasting the equity risk premium with frequency-decomposed predictors”. The authors thank Amit Goyal for providing data on his webpage; Gene Ambrocio, Christiane Baumeister, Theo Berger (discussant), Fabio Canova, Patrick Crowley, Eleonora Granziera, Mikael Juselius, Petri Jylhä, Manuel M. F. Martins, Francesco Ravazzolo, António Rua, Pedro Duarte Silva, Goufu Zhou as well as the participants at several conferences and seminars for useful comments; and Riccardo Verona for IT support. The views expressed are those of the authors and do not necessarily reflect those of the Bank of Finland.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Crowley (2007) and Aguiar-Conraria and Soares (2014) provide reviews of economic and finance applications of wavelets tools.

2  Besides the methodological contributions using single-variable predictive regression setup cited in section 1, methodological contributions that make use of several predictors to forecast the equity premium include dynamic factor models (Ludvigson and Ng 2007, Kelly and Pruitt 2013, Neely et al. 2014), forecasts combination from different predictors (Rapach et al. 2010, Pettenuzzo and Ravazzolo 2016), regime-switching vector autoregression models (Henkel et al. 2011), the sum-of-the-parts method (Ferreira and Santa-Clara 2011, Faria and Verona 2018), and Bayesian regime-switching combination or quantile combination approach (Zhu and Zhu 2013 and Lima and Meng 2017, respectively).

3 In this section, we limit the description to the basic concepts which are directly useful to understand our empirical analysis. A more detailed analysis of wavelets methods can be found in Percival and Walden (2000) and in appendix 2. We note that, in this paper, we are not following the convention of using a tilde (˜) to denote the maximal overlap discrete wavelet transform.

4 Examples of papers using the MODWT MRA decomposition include Galagedera and Maharaj (2008), Bekiros and Marcellino (2013), Xue et al. (2014), Barunik and Vacha (2015), Berger (2016), and Faria and Verona (2020). While the Haar filter is simple and widely used (see e.g. Manchaldore et al. 2010, Malagon et al. 2015, Bandi et al. 2019 and Lubik et al. 2019), in section 5.3 we show that the results are robust using other wavelet filters.

5 As regards the choice of J, the number of observations dictates the maximum number of frequency bands that can be used. In particular, if $t_0$ is the number of observations in the in-sample period, then J has to satisfy the constraint $J\le {\log }_2{t}_0$.

6 In the MODWT, each wavelet filter at frequency j approximates an ideal high-pass filter with passband $f\in [1/2^{j+1},1/2^j]$, while the smooth component is associated with frequencies $f\in [0,1/2^{j+1}]$. The level j wavelet components are thus associated to fluctuations with periodicity $[2^j,2^{j+1}]$ (months, in our case).

7 The closest approaches in the literature are those suggested by Rua (2011) and Faria and Verona (2018). This is also the spirit of the scale predictability in Bandi et al. (2019), who explore a model where returns and predictors are linear aggregates of components operating over different frequencies, and where predictability is frequency-specific.

8  This setup is akin to the band spectrum regression proposed by Engle (1974), who tests the hypothesis that the same model applies at various frequencies. In the wavelet literature, there are some recent contributions that run scale-by-scale regressions in a way identical or similar to the one adopted in this paper (see e.g. Gallegati et al. 2011, Gallegati and Ramsey 2013, Aguiar-Conraria et al. 2018 and Verona 2020). In principle, it is possible to fit different forecasting models for each frequency components. For instance, we could use non-linear models when forecasting the high-frequency components of the equity premium, or include more lags of the predictor when forecasting the lowest frequency components of the equity premium. We leave this for future research.

9 We thank Christiane Baumeister for this insight.

10  See Huang et al. (2015) for a detailed description of the bootstrap procedure used to compute empirical p-values.

11 Regarding the WAV model, for each predictor and for each sub-sample period, we use the same weights for the frequencies as the ones in the full OOS period (reported in table 4). This is a conservative approach, as we would expect to improve the performance of the WAV models by choosing the optimal weights of different frequencies for each predictor and for each sub-sample period.

12 Dangl and Halling (2012) and Huang et al. (2017) find positive and statistically significant levels of OOS predictability during expansions using time-varying coefficients regression and state-dependent predictive regression models, respectively.

13 The data for the industrial production in the US was downloaded from Federal Reserve Economic Data at http://research.stlouisfed.org/fred2/.

14 The quarterly time series of the IK and the CAY are available from the Goyal and Welch (2008) updated database. These variables, which are briefly explained in appendix 1, have been used as equity premium predictors when using quarterly data (see e.g. Rapach et al. 2010, Lettau and Ludvigson 2001).

Additional information

Funding

Faria gratefully acknowledges financial support from Fundação para a Ciência e Tecnologia through project UIDB/00731/2020.