Econometrics

I just finished up my first of many major Econometrics project, where we were supposed to analyze returns to scale for the electricity industry, looking at the cost of output as compared to the amount of production in billions of killowatts per hour. Above you can see the scatter plot of the residuals of the regression as compared to the natural log of output of killowatts, where the x-axis increases with the size of the firm. Clearly you can see the U shape, which indicates that there is something wrong.

First, we would expect ln(kwh) to be linear representing a steady state of returns to scale regardless of the size of the company. Second, the residual values are clearly correlated to the size of the firm. On the left side, with smaller firms, we have large residuals, indicating that for smaller firms our initial model is probably not a good fit. On the right side of the graph, with the larger firms, we see small residuals, but they are clearly starting to slope upward. Finally, we can look at the correlation between the lnkwh and the residuals, which is practically zero. This makes sense because the positive residuals are counterbalanced by the negative residuals. If there is no correlation between output and the residuals of the regression, yet we see in the graphs that something is wrong, we probably have different returns to scale depending on the size of the company and the amount of output. Even though the industry as a whole has a fairly predictable rate of positive returns to scale individual firms within the industry have different returns to scale depending on the size.
Obviously.