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

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.

Appendix

1.1 Gibbs sampling estimation of the predictive model

To conduct estimation, we rely on Gibbs sampling combined with data augmentation. In other words, we augment the parameter space to include any latent states, and sample from the augmented posterior by sequentially sampling from the conditional posteriors. We refer the reader to Koop (2003) for an introduction to Gibbs sampling. Specifically, we divide the augmented posterior of (3.1)–(3.4), $p\left( h_{1:T}^{\eta },h_{1:T}^{X},\zeta _{1:T}^{\eta },\zeta _{1:T}^{X},\psi \mid y_{1:T},X_{1:T}\right)$, where $\psi$ denotes the vector containing the model parameters into the following blocks: The log-volatilities, $h_{1:T}^{\eta }$ and $h_{1:T}^{X}$, conditional mean parameters, $\gamma =\left( \mu ,\phi ,\alpha ,\beta ,\varphi \right) ^{'}$, conditional volatility parameters, $\gamma ^{\eta }$ and $\gamma ^{X}$, where $\gamma ^{\eta }=\left( \mu ^{\eta },\phi ^{\eta },\omega ^{2\eta }\right) ^{'}$, and the mixture component indicators, $\zeta _{1:T}^{\eta }$ and $\zeta _{1:T}^{X}$, which we use to generate $h_{1:T}^{\eta }$ and $h_{1:T}^{X}$ as following Kim et al. (1998). More precisely, posterior draws from $p\left( h_{1:T}^{\eta },h_{1:T}^{X},\zeta _{1:T}^{\eta },\zeta _{1:T}^{X},\psi \mid y_{1:T},X_{1:T}\right)$ are obtained by cycling through the following steps:

• $\gamma \sim p\left( \gamma \mid h_{1:T}^{\eta },h_{1:T}^{X},y_{1:T},X_{1:T}\right)$,

• $h_{1:T}^{\eta }\sim p\left( h_{1:T}^{\eta }\mid \zeta _{1:T}^{y},\gamma ^{\eta },\gamma ,y_{1:T},X_{1:T}\right)$,

• $h_{1:T}^{X}\sim p\left( h_{1:T}^{X}\mid \zeta _{1:T}^{X},\gamma ^{X},\gamma ,X_{1:T}\right)$,

• $\zeta _{1:T}^{\eta }\sim p\left( \zeta _{1:T}^{\eta }\mid h_{1:T}^{y},\gamma ^{\eta },\gamma ,y_{1:T},X_{1:T}\right)$,

• $\zeta _{1:T}^{X}\sim p\left( \zeta _{1:T}^{X}\mid h_{1:T}^{X},\gamma ^{X},\gamma ,X_{1:T}\right)$,

• $\gamma ^{\eta }\sim p\left( \gamma ^{\eta }\mid h_{1:T}^{\eta }\right)$,

• $\gamma ^{X}\sim p\left( \gamma ^{X}\mid h_{1:T}^{X}\right)$.

In order to generate $\gamma \sim p\left( \gamma \mid h_{1:T}^{\eta },h_{1:T}^{X},y_{1:T},x_{1:T}\right)$ in step one, we use simple Gibbs sampling techniques for linear regression models, see Koop (2003) among many other. More precisely, we start by generating $\mu$ and $\phi$. Thereafter, we follow Amihud and Hurvich (2004), and construct the corrected estimator of $\phi$ as follows:

$$\begin{aligned} {\tilde{\phi }}= & {} \phi +\left( 1+3\phi \right) /T+3\left( 1+3\phi \right) /T^{2} \end{aligned}$$

(A.1)

Once we compute (A.1), we can construct ${\hat{\varepsilon }}_{1:T}^{X}$. Conditional on ${\hat{\varepsilon }}_{1:T}^{X}$ and the data, we then generate $\alpha$, $\beta$ and $\varphi$ in one block. Likewise, conditional on $\gamma$ and the data, we can generate $h_{1:T}^{\eta }$ and $h_{1:T}^{X}$ from their respective conditional posteriors using Gibbs sampling techniques for linear state-space models, see Carter and Kohn (1994) and Durbin and Koopman (2002) for textbook treatments. For instance, conditional on $\mu$, $\phi$, and $X_{1:T}$, we have that

$$\begin{aligned}&\displaystyle X_{t}-\mu -\phi X_{t-1} = \exp \left( h_{t}^{X}/2\right) \varepsilon _{t}^{X},\quad \varepsilon _{t}^{X}\sim N\left( 0,1\right) , \end{aligned}$$

(A.2)

$$\begin{aligned}&\displaystyle h_{t}^{X}-\mu ^{X} = \phi ^{X}\left( h_{t-1}^{X}-\mu ^{X}\right) +\xi _{t}^{X}\quad \xi _{t}^{X}\sim N\left( 0,\omega ^{2X}\right) . \end{aligned}$$

(A.3)

Evidently, (A.2)–(A.3) is a nonlinear state-space model as the observation equation, (A.2), is nonlinear in the state variable, $h_{t}^{X}$. Therefore, to generate $h_{1:T}^{X}$ from its conditional posterior, we use the auxiliary mixture sampler suggested in Kim et al. (1998). In a nutshell, the idea is to approximate the nonlinear stochastic volatility model using a mixture of linear Gaussian models, which are relatively easier to estimate. More precisely, our goal is to transform (A.2), such that it becomes linear in $h_{1:T}^{X}$ . We can simply do this by squaring both sides of (A.2) and taking the logarithm. This way, the transformed model,

$$\begin{aligned}&\displaystyle \log \left( X_{t}-\mu -\phi X_{t-1}\right) ^{2} = h_{t}^{X}+\log \varepsilon _{t}^{2X}, \end{aligned}$$

(A.4)

$$\begin{aligned}&\displaystyle h_{t}^{X}-\mu ^{X} = \phi ^{X}\left( h_{t-1}^{X}-\mu ^{X}\right) +\xi _{t}^{X}\quad \xi _{t}^{X}\sim N\left( 0,\omega ^{2X}\right) . \end{aligned}$$

(A.5)

is linear in $h_{t}^{X}$. However, the error term in (A.4), $\log \varepsilon _{t}^{2X}$, does not follow a Normal distribution, but a $\log \chi _{1}^{2}$ distribution. Consequently, the mechanism for fitting linear Gaussian state-space models cannot be directly applied to (A.4)–(A.5). To tackle this difficulty, we follow Kim et al. (1998) and approximate $\log \varepsilon _{t}^{2X}$ using a mixture of seven Gaussian densities:

$$\begin{aligned} \log \varepsilon _{t}^{2X}\sim & {} \sum _{i=1}^{7}\varpi _{i}N\left( m_{i}-1.2704,\varsigma _{i}^{2}\right) , \end{aligned}$$

where $\varpi _{i}$ is the probability of the ith mixture component and $\Sigma _{i=1}^{7}\varpi _{i}=1$. The values of $m_{i}$, $\varsigma _{i}^{2}$ and $\varpi _{i}$, $i=1,\ldots ,7$, are all fixed and provided in Kim et al. (1998). We can equivalently write the mixture density in terms of a random indicator variable, $\zeta _{t}^{X}\in \left\{ 1,\ldots ,7\right\}$, (hence, the name of the approach) as follows:

$$\begin{aligned} \log \varepsilon _{t}^{2X}\mid \zeta _{t}^{X}=i\sim & {} N\left( m_{i}-1.2704,\varsigma _{i}^{2}\right) , \end{aligned}$$

(A.6)

$$\begin{aligned} \text {Pr}\left( \zeta _{t}^{X}=i\right)= & {} \varpi _{i},\quad i=1,\ldots ,7. \end{aligned}$$

(A.7)

Equations (A.4)–(A.5) with (A.6)–(A.7) constitute a linear state-space model,

$$\begin{aligned} v_{t}= & {} H_{t}\theta _{t}+e_{t},\quad e_{t}\sim N\left( 0,R_{t}\right) ,\\ \theta _{t}= & {} {\widetilde{\mu }}+F\theta _{t-1}+a_{t},\quad a_{t}\sim N\left( 0,Q\right) , \end{aligned}$$

where $v_{t}=\log \left( X_{t}-\mu -\phi X_{t-1}\right) ^{2}-\left( m_{t}-1.2704\right)$, $H_{t}=1$, $\theta _{t}=h_{t}^{X}$, ${\widetilde{\mu }}=\mu ^{X}\left( 1-\phi ^{X}\right)$, $F=\phi ^{X}$, $Q=\omega ^{2X}$ and $R_{t}=\varsigma _{t}^{2}.$ Furthermore, $m_{t}$ and $\varsigma _{t}^{2}$ correspond to $m_{i}$ and $\varsigma _{i}^{2}$, $i=1,\ldots ,7$, with the highest probability and are available from the previous Gibbs iteration. Given these quantities, we can simply use a simulation smoother to generate $h_{1:T}^{X}$ from its conditional posterior, see for example, Kim et al. (1998).

To generate the corresponding mixture component indicator series, $\zeta _{1:T}^{X}$, we use that each $\zeta _{t}^{X}$ can be generated independently conditional on $v_{1:T}$ and $h_{1:T}^{X}$. Specifically, $\zeta _{t}^{X}$ is a discrete random variable that follows a seven-point distribution, and it can be easily sampled via the inverse transform method, see Kim et al. (1998) for details. The processes, $h_{1:T}^{\eta }$ and $\zeta _{1:T}^{\eta }$, are generated following the exact same procedure.

The conditional volatility parameters, $\gamma ^{\eta }=\left( \mu ^{\eta },\phi ^{\eta },\omega ^{2\eta }\right) ^{'}$ and $\gamma ^{X}=\left( \, \mu ^{X},\phi ^{X}, \omega ^{2X}\right) ^{'}$, are generated element-by-element conditional on $h_{1:T}^{\eta }$ and $h_{1:T}^{X}$, respectively. The conditional posteriors of $\mu ^{\eta }$$\left( \mu ^{X}\right)$ and $\omega ^{2\eta }$$\left( \omega ^{2X}\right)$ have closed-formed solutions. Lastly, $\phi ^{\eta }$$\left( \phi ^{X}\right)$ is generated using an independence-chain Metropolis-Hastings step, see Kim et al. (1998) and Chan (2013) for more details.

1.2 Convergence diagnostic test

Table 11 reports p values from the convergence diagnostic test suggested in Geweke (1992). The null hypothesis of this test is that the posterior mean of the model parameters based on the first $10\%$ of the sample of retained draws is the same as the posterior mean of the model parameters based on the last $50\%$ of the sample of retained draws. Following Koop (2003), a p value greater than 0.05 for all parameters suggests that convergence of the proposed Gibbs sampler has been achieved.

Table 11 Convergence diagnostic test for (3.1)–(3.4), where $X_{t}$ is log-crude oil price

Table 12 Six-months ahead out-of-sample avCRPS with $w\left( z\right) =1$, i.e., uniform

Table 13 Six-months ahead out-of-sample avCRPS with $w\left( z\right) =\phi \left( z\right)$, i.e., center

Table 14 Six-months ahead out-of-sample avCRPS with $w\left( z\right) =1-\phi \left( z\right) /\phi \left( 0\right)$, i.e., tails

Table 15 Six-months ahead out-of-sample avCRPS with $w\left( z\right) =\Phi \left( z\right)$, i.e., right tail

Table 16 Six-months ahead out-of-sample avCRPS with $w\left( z\right) =1-\Phi \left( z\right)$, i.e., left tail

1.3 Six-months ahead density prediction results

Tables 12, 13, 14, 15 and 16 report density prediction results from our models relative to the stochastic volatility benchmark at the 6-months ahead prediction horizon.