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Empirical Economics

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18-02-2019

Does the price of crude oil help predict the conditional distribution of aggregate equity return?

Author: Nima Nonejad

Published in: Empirical Economics | Issue 1/2020

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Abstract

Contrary to point predictions that only convey information about the central tendency of the target variable, or the best prediction, density predictions take into account the whole shape of the conditional distribution, which means that they provide a characterization of prediction uncertainty. They can also be used to assess out-of-sample predictive power when specific regions of the conditional distribution are emphasized, such as the center or the left tail. We carry out an out-of-sample density prediction study for monthly returns on the Standard & Poor’s 500 index from 1859m9 through 2017m12 with a stochastic volatility benchmark and alternatives to it that include the West Texas Intermediate price of crude oil. Results suggest that models employing certain nonlinear transformations of the price of crude oil help deliver statistically significant density prediction improvements relative to the benchmark. The biggest payoff occurs when predicting the left tail of the conditional distribution. They also generate the earliest signal of a market downturn around the 2008 financial crisis.

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There is no shortage of tension in the literature when it comes to the choice between in-sample and out-of-sample evaluation. Lo and MacKinlay (1990), Foster et al. (1997) and Rapach and Wohar (2006) among others argue that out-of-sample analysis is relatively more important as in-sample analysis tends to suffer from data mining. Conversely, Inoue and Kilian (2004) argue that in and out-of-sample tests of predictability are equally reliable against data mining under the null hypothesis of no predictability.

This information is also very useful for making economic decisions. For example, the Bank of England publishes its inflation forecast as a probability distribution in the form of a fan chart, see Britton et al. (1998). Likewise, Alessi et al. (2014) explain how using density forecasts, the New York FED produced measures of macroeconomic risk during the 2008 crisis.

The modern era of oil production in the USA begins in 1859 when Edwin Drake succeeds in producing usable quantities of crude oil for commercial purposes from a 69-foot well in Titusville, Pennsylvania. As a result of Drake’s discovery, WTI crude oil price falls from an average of $\$9.60$ per barrel in 1860 to 10 cents per barrel by the end of 1861, see also Narayan and Gupta (2015).

There are two major crude oil markets: WTI and Brent Blend. WTI is a blend of several US domestic oil fields. Brent Blend is a combination of crude oil from fifteen different oil fields in the North Sea. Both are priced in US dollars. However, monthly time-series of the price of Brent crude oil is only available starting in the early 1950s. An alternative measure of the price crude oil is the price paid by US refiners purchasing crude oil. However, this series is only available starting in 1974m1.

As argued in Alquist et al. (2013) this is explained by the specific regulatory structure of the crude oil industry imposed by the Texas Railroad Commission and other US state regulatory agencies. More precisely, each month these entities would forecast the demand for crude oil and then set the allowable production levels to meet the demand. As a result, much of the cyclically endogenous component of oil demand was reflected in shifts in quantities rather than prices.

The CPI series can be downloaded from Robert J. Shiller’s web site: http://www.econ.yale.edu/~shiller/.

Specifically, in Hamilton (2011), page 370 it is stated: “...deflating by a particular number, such as the CPI, introduces a new source of measurement error, which could lead to deterioration in the forecasting performance. In any case, it is again quite possible that there are differences in the functional form of predictions based on nominal prices instead of real prices”.

The variables, $net^{+}$, $net^{-}$, gap , net, large and $large^{+}$ are well-known from seminal studies, such as Hamilton (2003) and Kilian and Vigfusson (2013), where the emphasis is on exploring the impact of the price of crude oil on GDP growth rate.

We also explore the robustness of our results to different choice of the truncation lag for $net^{+}$, $net^{-}$, gap and net. Overall, we observe that results are similar across the different truncation lags.

An obvious issue with (3.1) is that it ignores the possibility of multiple predictors. However, there is a potential issue of multicollinearity if we were to include more predictors, which explains why researchers generally tend to prefer the single predictor model. In fact, as can be found from the large volume of stock return predictability literature, researchers tend to engage in a horse race among all potential predictors, one by one, see for example, Welch and Goyal (2008) and Westerlund and Narayan (2012). In this study, we choose to keep this tradition and use a single predictor in (3.1).

Stambaugh (1999) and Lewellen (2004) show that $\rho \ne 0$ is a source of major complication in terms of OLS estimation of (3.1). Particularly, the OLS bias is given as: $-\varphi \left( 1+3\phi \right) /T$. Hence, while decreasing in T, the bias is increasing in $\varphi $ and $\phi $, respectively.

Using Monte Carlo simulations, Westerlund and Narayan (2012) document that there are notable gains to be made by accounting for heteroskedasticity in predictive regressions, such as (3.1) and (3.2). Particularly, compared to the constant conditional volatility counterpart, they find that accounting for heteroskedasticity reduces the out-of-sample root mean square error.

As stated in Johannes et al. (2014), the marginal and predictive distributions of returns that integrate out the unobserved log-volatilities are a scale mixture of Normals, which is leptokurtotic.

For instance, $N^{-1}\Sigma _{i=1}^{N}\psi ^{\left( i\right) }$ is a simulation consistent estimate of the posterior mean of $\psi $.

Results are robust to different prior hyperparameter values on the model parameters. It is important to note that we have a relatively large sample, and even when we recursively estimate our models in Sect. 5, each estimation window contains 40 years of data (480 monthly observations). Hence, information from the data will tend to dominate information from the prior.

Because variables are standardized prior to estimation, the intercepts in (3.1) and (3.2) are omitted in our in-sample analysis.

Our decision to adopt the rolling window procedure is motivated by studies, such as Stock and Watson (1996) and Swanson (1998), where it is argued that by allowing the data generating process to evolve over time, a rolling moving window approach can account for parameter instability in the data generating process. Furthermore, as illustrated in Clark and McCracken (2012), under a expanding window approach, test statistics, such as Diebold and Mariano (1995) have a nonstandard limiting distribution.

To compute (5.4), we generate K draws of $y_{t+1}$ using Eqs. (3.1)–(3.4) and then compute it using (5.5).

We follow Groen et al. (2013) and rely on the one-sided Diebold and Mariano (1995) test due to the nested structure of our models. The one-sided test is also more intuitive because we are interested in evaluating the predictive power afforded by employing the price of crude oil in one direction, namely if it adds any.

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Title: Does the price of crude oil help predict the conditional distribution of aggregate equity return?
Author: Nima Nonejad
Publication date: 18-02-2019
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 1/2020
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-019-01643-2

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