natural gas spot and future pricesthe law of one

Question

Natural Gas Spot and Future Prices

The law of one price implies that the spot price and the future price of a commodity should not deviate too far away from each other when the net cost of carrying is accounted for. The main driver for such a stable relationship is opportunities of arbitrage. If the two prices become significantly different from each other, market participants take opportunity of arbitrage, driving their levels back to the parity. In other words, arbitrage opportunities prevent the two prices from deviating too far away from each other. Such an equilibrium relationship between the spot and future prices of a commodity is represented by the spot-future parity condition:

F_t = S_te^r_t

where F = future price,
S = spot price, and
r = net cost of carrying.

Taking the log of both sides of (A) and adding a term representing random noises leads to the following regression model.

lnFt = β₁ + β₂lnS_t + β₃r_t + u_t

Note that for spot-future parity to hold, β₁ = 0, β₂ = β₃ = 1.

For the present exercise, we consider the relationship between the spot and future prices of natural gas.

Data (Monthly)

The monthly observations on prices are obtained from the U.S. federal government's Energy Information Administration website, and the data on interest rate (as net cost of carrying) are obtained from the Federal Reserve's website.

The variables are defined as follows:

F_t = future price of natural gas in $US per million Btu¹ for the delivery in the following month,

S_t = spot price in $US per million Btu, and
r_t = one-month bank bill rate (% per year).

Generate lnF and lnS after importing the data into Gretl.

Questions

Use the variables: lnF, lnS and r.

(1) Obtain a time-series plot and correlogram for each variable, and comment on them with regard to the stationarity/non-stationarity property of the variable.

(2) Test, at 10%, for unit root on both the level (Yt) and the first-difference (ΔYt) of each variable (and the second difference, Δ2
Yt, if necessary) to determine the order of integration.

Use the Augmented Dickey-Fuller test with constant (τc), and let the optimum number of lags in the test equation be determined by "modified AIC". In Gretl, leave the (maximum) lag order at the default level (14) and select "test down from the maximum lag order".

A preliminary analysis reveals that the net cost of carrying (r) is not that important in the parity condition, (A). Ignore it for the remaining questions.

Q(3)-Q(6): Consider the model in first differences.

Δ lnFt = β1 + β2Δ lnSt + ut (C)

(3) Estimate (C) by the OLS method. Then, test for heteroscedasticity and fourth-order autocorrelation at the 5% significance level. (Use the Koenker version of the Breusch-Pagan test for heteroscedasticity and the Breusch-Godfrey test for autocorrelation. Use the N×R² statistic in both cases.)

(4) Provide a summary report of the OLS estimation of (C), taking into account the test results in (3).

(5) Assume that the log of the variance function of ut is given by

lnσ_t² =  α₀ + α₁u²_t-1

where ^u_t are OLS residuals. Estimate (C) by the FGLS (WLS) method and report the estimation result.

(6) Estimate (C) by the Cochrane-Orcutt method (with the Paris-Winsten transformation) correcting for first-order autocorrelation. Provide a summary report of the result. (Use Model > Time series > AR(1) in Gretl.)

(7) Using the Engle-Granger residual-based method, test at 10% if the following is a cointegrating relationship.

lnF_t = βlnS_t + u_t (E)

where u_t are random errors.

In Gretl, leave the (maximum) lag order at the default level (12) and select "test down from maximum lag order" so that the optimum number of lags in the test equation is automatically determined from 0-12. Note that (E) does not have a constant term (i.e., intercept).

(8) Discuss the implications of the test results from (3) and (7) for the estimation results reported in (4).

(9) Estimate the following error correction model by the OLS method and provide a summary report of the estimated equation.

Δ lnF_t = β₀ + αu_t-1 + β₁ΔlnF_t-1 + β₂Δ lnS_t-1 + v_t (F)

where ΔY_t = Y_t - Y_t-1 and ^u_t are the OLS residuals from (E). α and β are the regression coefficients.

(10) Obtain a time-series plot and correlogram of the residuals from (9) and comment on them with regard to the violation of classical assumptions. Test for fourth-order autocorrelation using the Breusch-Godfrey (N×R²) procedure.

Download:- data.xlsx