Establish the order of integration of the futures and the

Question

Price Discovery in Commodity Markets

Context

Commodity futures and cash prices are linked by no-arbitrage conditions, suggesting a long run relationship between price series. The no-arbitrage condition between cash and futures prices can be expressed as in (1), with F_t being the futures price at time t that matures at time T, S_t being the cash price at t, r_t and w_t being continuously compounded risk free interest rate and storage costs over time τ = T - t and y_t being the convenience yield; a utility based rewards that accrues to the holder of inventories.

F_t,T = S_te(r_t + w_t - y_t)τ (1)

Taking logs of (1), with τ = 1 for simplicity:

f_t,t+1 = s_t + r_t + w_t - y_t (2)

Interest rate and storage costs less convenience yield can be summarised as net-carry costs c_t.

f_t,t+1 = s_t + c_t (3)

At maturity τ → 0 so that f_t,t = s_t. Over a futures contract's life cycle (τ > 0), the futures and cash price can deviate by net-carry costs (and other random shocks).

Task

Establish the relationship between a commodity futures price and its underlying physical price in three steps:
(i) Establish the order of integration of the futures and the physical price series.
(ii) Establish the lead-relationship between the futures and its underlying physical price.
(iii) Establish the relationship between the futures and its underlying physical price.

Please use textbooks, lecture notes and additional handouts for guidance. You might also find the references such as provided below useful.