Monday, December 23, 2013

A Simple Regression Problem

Here's a regression problem for student readers of this blog.

Suppose that we estimate the following regression model by OLS:

                     yi = α + β xi + εi .

The model has a single regressor, x, and the point estimate of β turns out to be 10.0.

Now consider the "reverse regression", based on exactly the same data:

                    xi = a + b yi + ui .

What can we say about the value of the OLS point estimate of b?
  • It will be 0.1.
  • It will be less than or equal to 0.1.
  • It will be greater than or equal to 0.1.
  • It's impossible to tell from the information supplied.

© 2013, David E. Giles

11 comments:

  1. I remember answer is less, but at this moment, I got beta/b=y'y/x'x, and I do not know if I need to use chaucy inequality to verify it.

    ReplyDelete
    Replies
    1. Beta*b=∑(xy)^2/∑x^2*∑y^2〈1.

      Delete
  2. i think we need to know the units of variables.

    ReplyDelete
  3. Prof, i have one question not strictly related to above.
    I have two variables X and Y. I have annual data as well as quarterly data. i run two regressions for annual data and quarterly data.
    Should we expect quarterly slope coefficients to be one-fourth of annual estimates and monthly slope coefficients one-twelve of annual estimates?
    Thanking you.

    ReplyDelete
  4. SPOILERS AHEAD:




    Use a fact about sample correlation:

    Cor(x,y) = Cov(x,y) / (Var(x)*Var(y))^(0.5) <= 1

    squaring both sides leaves the inequality intact, because we know the correlation to be positive
    (otherwise, beta would be negative):

    Cov(x,y)^2 / (Var(x)*Var(y)) <= 1

    rearranging:

    Cov(x,y)/Var(y) <= Var(x)/Cov(x,y)

    but the right hand side is nothing but 1/beta = 1/10 = 0.1, and the left hand side is equal to b.
    Hence "less than or equal to 0.1" should be the right answer, no?

    ReplyDelete
  5. Less than or equal to 0.1, essentially because the variance of y is b^2 times the variance of x plus the variance of the error term.

    ReplyDelete
  6. You can also write this in terms of the R-squared of the regression (which of course is the same as the R-squared from the reverse regression).

    b_reverse = R^2/beta <= 1/beta

    ReplyDelete
  7. By the "Iron Law of Econometrics," the confluence of various structural/measurement errors that may have befallen us in this regression will bias the OLS estimate of beta toward zero. Because the true value of beta is bounded below by 10, the true value of b in the reverse regression is bounde above by 1/10. Applying the Iron Law to the reverse regression, the OLS estimate of b will be biased downward as well. Therefore b will be less than or equal to 1/10!

    ReplyDelete
  8. I'll take a stab.

    going to the definitions:

    the ols estimate of \beta is cov(x,y) / var(x)

    the ols estimate of b is cov(x,y) / var(y)

    there is really no reason to expect that the variance of y should be related to the variance of x in any particular way so it could be greater, equal or less than.

    however, if you had did something tricky first like standardize the two variables so that things like units don't matter (like Santosh suggested), then the answer would have been a)

    Merry Christmas!

    Andrew

    ReplyDelete
  9. OK, everyone, see my follow-up post here: http://davegiles.blogspot.ca/2013/12/solution-to-regression-problem.html?showComment=1388161295063#c1666199217587229220

    ReplyDelete