The regime-switching risk premium in the gold futures market

Abstract

A bivariate Markov-switching model identifies two regimes in the futures-price and risk-premium models. The persistent underlying states have very different implications for spot and risk-premium forecasts. In the “low” state, a positive bias predicts spot price appreciation. The “high” state is associated with lower spot appreciation and higher risk premiums. The regime-switching framework provides a new perspective on the intertemporal role of gold as a hedge or safe-haven asset. The gold spot-price appreciation regime is shown to be correlated with higher inflation rates and the complement regime is associated with high market returns and stock market risk premia. Since the state-space methodology procedure can be employed using only past data, forecasts of the persistent unobserved underlying state of the gold price appreciation regime will be augmented as more data becomes available.

Appendix A: Markov-switching Kalman filter

The parameter estimates from the state-space model are estimated using GAUSS 10 and the optmum package implemented using code derived in part on code which accompanies Kim (1999). Figures and other empirical results are produced using OxConsole 6.1, Doornik (2007). The procedure requires the filter of Kim (1994); the Kalman filter and the modification for collapsing the probability of each state by Hamilton (1989). Adopting notation where $\boldsymbol {\delta }_{t|t-1}^{[i,j]}=\boldsymbol {E}_{t-1}^{[j]}\boldsymbol {\delta }_{t}^{[i]}$, the expectation of δ _t given that the current state is i and the previous state was j, for all i,j, the Kalman filter and updating equations are given in Eq. 20 through Eq. 25:

$$\begin{array}{@{}rcl@{}} {\boldsymbol{\delta}_{t|t-1}}^{[i,j]}&=&\boldsymbol{\mu}^{[i]}+\boldsymbol{T}^{[i]}\boldsymbol{\delta}_{t-1|t-1}^{[j]} \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \boldsymbol{P_{t|t-1}}^{[i,j]}&=&\boldsymbol{T}^{[i]}\boldsymbol{P}_{t-1|t-1}^{[i]}\boldsymbol{T}^{{\prime}[i]}+\boldsymbol{Q}, \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} \boldsymbol{e}_{t|t-1}^{[i,j]}&=&\boldsymbol{Y}_{t}-\boldsymbol{Z}\boldsymbol{\delta}_{t|t-1}^{[i,j]}-\mathbf{X}\boldsymbol{\beta}^{[i]} \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} \boldsymbol{K}_{t|t-1}^{[i,j]}&=&\boldsymbol{Z}\boldsymbol{P}_{t|t-1}^{[i,j]}\boldsymbol{Z}^{\prime}+ \boldsymbol{R}, \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} \boldsymbol{\delta}_{t|t}^{[i,j]}&=& \boldsymbol{\delta}_{t|t-1}^{[i,j]}+\boldsymbol{P}_{t|t-1}^{[i,j]}\boldsymbol{Z}^{\prime} \left( \boldsymbol{K}_{t|t-1}^{[i,j]}\right)^{-1}\boldsymbol{e}_{t|t-1}^{[i,j]}, \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} \boldsymbol{P}_{t|t}^{[i,j]}&=&\boldsymbol{P}_{t|t-1}^{[i,j]}-\boldsymbol{P}_{t|t-1}^{[i,j]}\boldsymbol{Z}^{\prime} \left( \boldsymbol{K}_{t|t-1}^{[i,j]}\right)^{-1}\boldsymbol{Z}\boldsymbol{P}_{t|t-1}^{[i,j]}. \end{array} $$

(25)

Given the information set at time t is represented by 𝜃 _t , Hamilton’s filter for collapsing probabilities is given by Eq. 26 through Eq. 30:

$$\begin{array}{@{}rcl@{}} f(\boldsymbol{Y}_{t}|S_{t-1}\,=\,i,S_{t}\,=\,j,\theta_{t-1})&=& \frac{1}{\sqrt{2\pi}}\left| \boldsymbol{K}_{t|t-1}^{[i,j]}\right|^{\frac{1}{2}} \exp\left(-\frac{1}{2}{\boldsymbol{e}_{t|t-1}^{[i,j]}}^{\prime} \left( \boldsymbol{K}_{t|t-1}^{[i,j]}\right)^{-1}\boldsymbol{e}_{t|t-1}^{[i,j]}\right) , \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} f(\boldsymbol{Y}_{t})&=&pf(\boldsymbol{Y}_{t}|S_{t+1}=0)+(1-p)f(\boldsymbol{Y}_{t}|S_{t+1}=1), \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} \Pr(S_{t+1}=i|\boldsymbol{Y}_{t})&=&\frac{f(\boldsymbol{Y}_{t}|S_{t+1}=i)}{f(\boldsymbol{Y}_{t})}, \end{array} $$

(28)

$$\begin{array}{@{}rcl@{}} \boldsymbol{\delta}_{t|t}&=&\sum\limits_{i=0}^{m}\boldsymbol{\delta}_{t|t}^{[i]}\Pr(S_{t+1}=i|\boldsymbol{Y}_{t})^{-1}, \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} \boldsymbol{P}_{t|t}&=&\sum\limits_{i=0}^{m} \left(\boldsymbol{P}_{t|t}^{[i]}+\left(\boldsymbol{\delta}_{t|t}-\boldsymbol{\delta}_{t|t}^{[i]}\right)^{2} \right) \Pr(S_{t+1}=i|\boldsymbol{Y}_{t})^{-1}, \end{array} $$

(30)

where p is the estimated transition probability and Pr(⋅) represents the probability operator. The model is estimated by maximizing the vector of parameters δ over the log likelihood:

$$ \max_{\delta} (\mathit{l})=\sum\limits_{t=1}^{T} \ln\left(f(\boldsymbol{Y}_{t})\right). $$

(31)