# Specific estimates about fuel consumption - Simple linear regression?

I am working on a story problem for a project.

"You work for a small environmental foundation that wants to analyze fuel consumption. Your boss (in the year 2002) has asked you to help her analyze the 2001 data on highway fuel consumption to find out what a change in the tax rate does to fuel consumption. She has provided a datafile (labelled Fuel2001) that has data on the following variables:

Drivers Number of Licensed drivers in the state

FuelC Gasoline sold for road use (1000s of gal.)

Income Per capita personal income (year 2000)

Miles Miles of Federal-aid highway miles in the state

MPC Estimated miles driven per capita

Pop Population age 16 and over

Tax Gasoline state tax rate, cents per gallon

She wants you to produce an analysis that does the best job of helping her argue that higher tax rates lead to lower fuel consumption. But her initial analysis of the data (simply including all of these variables in a naive regression model) did not find that tax rates had a statistically significant effect. Is there anything that can be done to find out if there is a relationship? Write her a short memo on that subject including your analysis, including specific estimates of the relationship."

I'm supposed to present her with the best case possible to support her cause (raising gas tax to reduce fuel consumption), but I should also include the best model, for her information. I have transformed all the skewed variables by taking their natural log. I ran a bunch of different regressions in Stata until I found one in which tax was statistically significant. When I ran the same regression robust tax was no longer statistically significant.

What would you do and show (graphically, including tables or graphs)? What does it mean that the robust regression makes tax insignificant?

I know it would be helpful to have the data, but I can't figure out how to upload it.

Thanks in advance for any input.

So many people, myself included, learned statistics because someone handed them data problems that they didn't know how to handle.

First comment: you've got a lot to learn, and running regressions without knowing what they mean is dangerous -- you may get caught with your you-know-what hanging out.

Second comment: read Mostly Harmless Econometrics, by Angrist and Pishke. It seems like what you need is a fixed effects model, or maybe a regression discontinuity, or anyway some econometrics-style identification strategy that will let you argue about causation rather than just correlation.

Third comment: in stata, reg y x, robust means that the variance-covariance matrix is calculated as $\left(X'X\right)^{-1}X'\epsilon \epsilon' X \left(X'X\right)^{-1}$. Do you know what that means? If you don't, beware, and read up. A good overview is given in section 20.20 of the Users Guide that comes with Stata and the references therein. If you move onto a panel-data style analysis, you'll want to go further, and cluster your standard errors to account for intra-class correlation (look it up).

Last comment: typically this kind of thing is presented in regression tables. The stata "package" estout is helpful.

Last last comment: you say "best" model. That can mean a lot of things. It can mean the model in which you can argue that your regression coefficients represent unbiased causal effects. It could also mean that youv'e got the best-predicting model, according to Akaike's Information criterion or something similar. What do you mean?

I don't really see a way to estimate it very well from the variation in cross-section data. Everything I know about this literature uses panel data (which may still be bad, but at least you can include fixed effects to control for unobserved time-varying consumption factors) or an instrumental variables with panel data (which is better, but may not identify the parameter you care about here because responses to unexpected oil shocks are different from tax increases or they may be weak or not exogenous).

The problem is that price and quantity demanded are jointly determined, which induces a correlation between price and the error terms that will bias estimates towards zero in a naive regression of quantity on price. I guess you can say that any coefficient you estimate with OLS will be biased downward. This may be the problem you run into. Or you might be able to do something simple, find that it agrees with more sophisticated results, and dismiss the objections on those grounds.

Another strategy would be to take some elasticity estimates from the literature, and use those to see what happens. In the short-run, the demand is going to be very inelastic, so you can assume the full burden of the tax will fall on the consumer. Then you can use your data to show how consumption would change.

There's a nice blog summary of the literature and the econometric issues here. In particular, there are links to two accessible papers. Hughes, Knittel, and Sperling find short-run gas price elasticities of -0.034 to -0.077 in the first five years of this century, which means that a price increase of 1% will lower quantity demanded by between 0.034% and 0.077%. You can also try replicating their specification with your cross-section (using logged price and quantity). The Killian and Davis paper points out some issues with the first approach and find somewhat larger effects with the HKS method (-0.19% with state panel data) and much larger estimates using IV (-0.46% with stata panel data). You might use all of them in your simulations, and compare with your own estimates.