There are three data series in the Data01 data frame: RIGS, WTI, and CPIU.

RIGS is the total rig count for oil and gas wells in the lower 48 states.

WTI is the spot price of West Texas Intermediate crude oil, the marker price for crude oil in the United States.

CPIU is the consumer price index for all urban consumers (seasonally adjusted).

Rigs are the large apparatus you find in oil and gas fields used in the exploration for crude oil and natural gas. They are very expensive and are usually leased on a daily or weekly basis.

In this first homework assignment:

1. Start by regressing RIGS on WTI and CPIU

Suggested code:

• rigs.model <- lm(RIGS~WTI+CPIU, data=Data01)
  1. What sign would you expect for WTI? Why?

1. What is the interpretation of the WTI coefficient?
2. Is the WTI coefficient significant at the .05 level? At the .01 level? Please refer to either the t-statistics or the p-values.

1. Is the regression significant? Would you use a t test or an F test to answer this question? Why?

1. Graph the residual with Date on the x-axis

Suggested code:

• rigs.model$model$Date <- Data01$Date
  • plot(x=rigs.model$model$Date, y=rigs.model$model$residuals, ylab='OLS residuals', xlab='Date')

1. Do the residuals seem to have a pattern? Do positive residuals tend to follow positive residuals and do negative residuals tend to follow negative residuals?
2. Calculate the correlation between the error term and the lagged error term. Suggested code is

• cor(rigs.model$model$residuals[-1], dplyr::lag(rigs.model$model$residuals)[-1])
  1. Does this correlation suggest a problem with the regression? (Is there a violation of OLS assumptions? If so, what is the violation?)

1. Graph the squared residual vs Date.

Suggested code:

• plot(x=rigs.model$model$Date, y=rigs.model$model$residuals^2, ylab='Squared OLS residuals', xlab='Date', pch=16)
  1. Does the variance appear to be constant across the sample or does it vary?

1. Does this pattern suggest a problem with the regression? Is there a violation of OLS assumptions? If so, what is the violation and what are its implications? Please explain with reference to either t-statistics or p-values.
2. Perform forecasts and confidence intervals of forecasts.

1. Using the regression from #1, what is your forecast for rig count in January and February of 2023 and what is the 95% confidence interval of that forecast?
2. RIGS, WTI and CPIU for January 2023 and February 2023 are:

Date	RIGS	WTI	CPIU
Jan-23	772	78.15	300.5
Feb-23	758	76.86	301.2

What is your opinion of the regression in terms of forecasting accuracy?

1. One might hypothesize that, if crude oil prices get high enough, consumers might eventually substitute out of oil products and into alternative fuels and technologies such as biofuels and electric cars. At some high oil price, then, rig counts may start to decline. We can test this hypothesis using the equation

• rigs.model_2 <- lm(RIGS~WTI+I(WTI^2)+CPIU, data=Data01)

where we now have both