Crude Oil and Natural Gas

Econbrowser has the explanations and the equations.

"Let Δo_t denote the monthly percent change in in oil prices (technically, the change in the natural logarithm) and z_t the percentage gap in cost (technically, z_t = ln(o_t/6g_t)). If you use a regression to try to predict oil prices from their own lagged values and the lagged oil-gas cost gap, a positive gap such as we have at the moment does tend to tug down future oil prices slightly, though the coefficient is not statistically significant. Here are the regression coefficients, with standard errors in parentheses:

On the other hand, the cost gap does seem to help significantly to predict where natural gas prices might go. With the gap currently at z_t = 1.13, the historical regression below might lead you to expect natural gas prices to climb by 10% a month (0.103 x 1.13 = 0.116) until the gap is closed.

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"The Relationship Between Crude Oil and Natural Gas Prices"

43 pages added to the pile!

Natural Gas Prices are here

Crude Prices are here and they moved up strongly while nat gas are lagging.

Summary from above paper p. 40f:
"Economic theory suggests that there is a relation between natural gas and oil prices, but the influence of an increase in oil prices may conflict in its effects on natural gas supply, and therefore, prices. Production of natural gas may increase as a co-product of oil, or may decrease as a result of higher-cost productive resources. While the net effect of an increase in oil prices on natural gas supply may be ambiguous, the effect on natural gas demand is clear, resulting in a positive relation between oil and natural gas prices. Given the relative inelasticity of natural gas supply in the short term owing to factors such as a 12-18 month lag in the production response to drilling changes, it appears that the effect of oil prices on natural gas demand is dominant in the short run.
...
The analysis supports the presence of a cointegrating relationship between the crude oil and natural gas price time series, providing significant statistical evidence that WTI crude oil and Henry Hub natural gas prices have a long-run cointegrating relationship. A key finding of the analysis is that natural gas and crude oil prices historically have had a stable relationship, despite periods where they may have appeared to decouple.
...
The estimation of the model resulted in identifying evidence of a stable relationship between natural gas and crude oil prices. The statistical evidence also supported the a priori expectation that while oil prices may influence the natural gas price, the impact of natural gas prices on the oil price is negligible. With crude oil prices weakly exogenous to natural gas prices, the short-run response of the natural gas price to contemporaneous changes in the oil price was found to be statistically significant."