For the Engel Curve assignment, I chose to analyze the relationship between Median Family Income in the United States (MEFAIN) and Total U.S. Demand for motor gasoline (MPGUS) from 1985 to 2013. Median Family Income data is inflation adjusted and pulled from the St. Louis Federal Reserve website. Total motor gasoline consumption is in thousands of barrels sold in the U.S. per day and is pulled from the U.S. Energy Information Administration website. I chose motor gasoline for my independent variable because gasoline is generally considered to be price insensitive, meaning that consumers do not change their consumption of gasoline very much as factors like the price of oil or tax rates change, but I wanted to see if this trend was persistent across various income levels, as gasoline may be considered a luxury good in some circumstances. The regression equation and resulting Engel Curve are shown below:

MPGUS_{i} = b_{0 }+ b_{1} MEFAIN_{i }+ U_{i}

The Engel Curve above shows us that Gasoline is in fact a normal good, since the Engel Curve has a positive slope. SAS Output for this model is shown below.

The SAS output tells us that for every $1 increase in Median Family Income, there is an increase in Total Motor Gasoline Consumption of 200 barrels sold per day in the United States. The p-value associated with Median Family Income is less than 0.001, telling us that our results are statistically significant, and our R-Squared value of 0.7487 tells us that Median Family Income changes explain changes in Total motor gasoline demand well. However, because this is a time-series analysis, we must check our results for the presence of first-order autocorrelation, as autocorrelation is often present in time-series models. To do this, I ran a Durbin-Watson test, the results of which may be found below.

These results show that there is a high degree of first-order positive autocorrelation in our data, meaning that we must now attempt to correct this problem. To do this, I ran PROC AUTOREG, and lagged my independent variable by two periods, as I found that lagging by one period did little to remedy the situation. Below are the SAS output results of the PROC AUTOREG procedure.

As we can see from the above SAS output, the PROC AUTOREG procedure significantly reduced the level of positive first-order autocorrelation in the data, and in fact brings the Durbin-Watson statistic in between the dU and dL values for a regression with one independent variable and 29 observations, meaning that a hypothesis test to determine the presence of autocorrelation on this dataset would now be inconclusive. Lagging our independent variable also results in a significantly lower transformed R-Squared, which now shows us that only 54% of the dependent variable is explained by the model. To do away with the remaining possibility of autocorrelation in the data and improve the R-Squared for this model, I decided to add a variable, the price of a new car (PNCUS) to this equation to reduce the likelihood of omitted variable bias. The data for the PNCUS variable was pulled from the Chicago Transit Authority website and is adjusted for inflation. The regression equation and SAS output are shown below.

MPGUS_{i} = b_{0 }+ b_{1} MEFAIN_{i }+ b_{2} PNCUS_{i} + U_{i}

This output shows us that adding a second independent variable to the model does not help in concluding whether positive first-order autocorrelation is persistent in the data, but does have a positive effect on our R-Squared value, which now shows us that 79% of the dependent variable is explained within our model. It also shows that for every $1 increase in the price of a new car is associated with a decrease of 139.4 barrels of gasoline per day in the United States.

This model tells us that gasoline is in fact a normal good, as when median family income rises, so does demand for gasoline. There may still be issues in our model with autocorrelation, but it is not inconceivable that this autocorrelation is attributable to the fact that a model with many different independent variables is needed to fully explain changes in demand for motor gasoline in the United States. Clearly, our model does not paint the entire picture, but it provides us with a good starting point for analyzing the factors which influence demand for motor gasoline.