As I recall, I saw an estimator which combined a ratio estimator and a product estimator using one independent variable, x, where the same x was used in each part. Here is my concern: A ratio estimator is based on a positive correlation of x with y. A product estimator is based on a negative correlation of x with y. (See page 186 in Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons.) So how can the same x variable be both positively and negatively correlated with y at the same time?
Can anyone explain how this is supposed to work?
Thank you for your comments.
The direction of correlation of individual components in a set of explanatory variables is largely irrelevant to your question. Considering just the one set of explanatory variables that derive from a single underlying variable x, what you have is a way of parameterizing a function of the form
a f1(x) + b f2(x) + ...
etc.where a, b,... are the regression parameters. What matters is the overall set of shapes of functions of this form, and whether one of these shapes matches your data. If it helps, you might consider plotting the function that has been fitted. Alternatively, you might think of looking at the derivative of the function, which will show that the effect of a small change in x in one part of the range differs from that in another.
I was referring to only one independent variable, when there are no others.
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