# US Crude Reserve Modeling

Theoretically, petroleum inventory levels are a measure of the balance - or imbalance - between petroleum supply (domestic production and imports) and demand, and thus provide a good market barometer for crude oil price (e.g. spot, futures, etc.) changes.

The US Energy Information Administration (EIA) Petroleum Status Report provides information on the weekly change in petroleum inventories in the U.S., whether produced locally or abroad. This weekly report gives an overview of the level of crude reserves held and produced by the U.S. both domestically and abroad. It is an indicator of current oil prices.

## Why should we care?

The level of inventories helps investors to estimate the prices for petroleum products – (e.g. gasoline, heating oil, diesel, etc.). Just like any other goods and services, prices for petroleum products are determined by supply and demand.

Furthermore, crude oil is an important commodity in the global market and plays a crucial part of the economy, so fluctuations in its price have a direct influence on consumer prices. As such, if low reserves are reported, crude oil prices are most likely to increase and therefore drive up consumer prices as well.

In this paper, we will model the dynamics in the crude oil stockpile (including SPR) level over time, and project a forecast for the next 12 months.

## Background

The general demand for petroleum products is highly seasonal and is greatest during the winter months, when countries in the Northern Hemisphere increase their use of distillated heating oils and residual fuels. Supply of crude oil, including both production and net imports, also shows a similar seasonal variation but with a smaller magnitude.

During the summer months, supply exceeds demand and petroleum inventories normally build; whereas during the winter, demand exceeds supply and inventories are drawn down. As a result, inventories also demonstrate seasonality.

## Analysis

For our sample data, we’ll use the EIA data from the weekly petroleum status report between January 1991 (Post- Gulf War) and August 17th, 2012. The EIA source key (i.e. designation) for this time series is “WCRSTUS1”.

At first glance, the data does not exhibit the 12-month seasonality that we were anticipating, so let’s look closer.

Now, to better distribute the values, we’ll compute the logarithmic transformation of the time series and use it primarily in our analysis.

Using summary statistics, the time series is serially correlated and its probability distribution exhibits fat-tails.

Furthermore, the correlogram analysis (i.e. ACF and PACF plots) uncover an integration issue with a lag-order of one:

Also, look closer to the partial auto-correlation near lag-order 52-55. The leap year phenomenon makes it difficult to pin-point a season length in number of weeks.

To overcome this problem, we’ll use the monthly stock-level and model it instead. The EIA publishes the monthly crude reserve stockpile level, but these levels are published once a month and are generally for the previous three month period.

For monthly time series, we’ll interpolate the reserve level on the last day of each month (similar to the EIA monthly report) using the weekly time series.

Let’s re-run the correlogram analysis on the log-transformed monthly time series:

The correlogram plot suggests a seasonal ARIMA (SARIMA) model with a season length of 12 months. We’ll propose a special case of SARIMA – the Airline Model. The Airline Model can not only capture the seasonality, but it requires only three (3) parameters to estimate, so it is typically a good starting point.

Using the NumXL Airline wizards, we specified a 12-month Airline model and calibrated its parameter values using the log monthly levels.

The calibrated parameters’ values yield a stable model and satisfy the underlying model assumptions. In short, the Airline model is a reasonable candidate model for the monthly log-level process.

### Interpretation:

$$(1-L)(1-L^12)x_t=\mu+(1-\theta L)(1-\Theta L^{12})a_t$$The model is saying that the __year-over-year of the monthly change __($z_t$) is basically a special moving average process of order 13. Let’s go ahead now and use the model for our forecast.

The Airline model’s functions compute the mean and confidence interval of the monthly log-level. To move the forecast back into barrels-unit, we took the exponential transform and adjusted the mean forecast as follows:

$$E[Y_{T+n}]= e^{X_{T+n}+\frac{\sigma_{T+n}^2}{2}} $$Where

- $\sigma_{T+n}$ is the forecast error at time $T+n$

In the graph above, the dotted line represents the Airline mean forecast and the shaded area is the 95% confidence interval.

### EIA Weekly Reports and forecast

To forecast the crude oil reserve level for any arbitrary future date (other than the end of the month), we simply interpolate the value of this date using the monthly forecast. The same can be said for the confidence interval limits.

For the current month where EIA issued one or more of its weekly reports, we use the latest report value and the projected end-of-month forecast value to interpolate the value for an intermediate date.

## Conclusion

In this paper, we analyzed the weekly time series of crude oil reserve levels and demonstrated its key statistical properties: integration and seasonality. Next, using the weekly time series, we interpolated the monthly time series and constructed an Airline (12) model. The calibrated model’s values satisfied all assumptions.

Using Airline(12) we projected an 18 month forecast for the crude reserve level and constructed a 95% confidence interval.