Hello All,
This is from Kaplan, quant pg 198. Can somebody explain why the residual vs independent variable plot for
positive correlation is a sine wave and
negative correlation is zig-zag?
Thank you,
P
Positive serial correlation (of error terms) says that if an error term is positive, the next error term in more likely to be positive, and if an error term is negative, the next error term in more likely to be negative. In the sine wave, you can see that this relationship holds true. Note that you have to have some positive and some negative errors about the regression line (the sum of all of the errors has to be zero), so the errors cannot stay positive (they have to go negative at some point) and they cannot stay negative (they have to go positive at some point. Again, the sine wave illustrates this beautifully.
Negative serial correlation (of error terms) says that if an error term is positive, the next error term in more likely to be negative, and if an error term is negative, the next error term in more likely to be positive. Again, you need both positive and negative error terms, so the zig-zag (positive, negative, positive, negative) illustrates this beautifully.
Are these the only situations you can have with positive and negative serial correlation: sine waves and zig-zags? Of course not. But they encapsulate the important features. Beautifully.
For what more could you ask?