Regression: statistically significant difference between transformed coefficients

Hi All,

I know it is straightforward enough to test for a statistically significant difference between two coefficients... but I have kind of a funny case that has gotten me rather confused (and a push in the right direction would be great). I am running 4 regressions. Let's say they are

y = a1 + a2x1 + additional things

y = b1 + b2x1 + additional things

y = c1 + c2x1 + additional things

y = d1 + d2x1 + additional things

Where "additional things" is different between each regression. I am interested in the value F1 = exp(a2)/exp(b2) and F2 = exp(c2)/exp(d2). I want to test whether or not there is a statistically significant difference between F1 and F2, but this has thrown me for quite a loop because I am using both ratios and exponentials. Can anybody give me any insight in this respect? I would really appreciate it!

Nick

Re: Regression: statistically significant difference between transformed coefficients

Hey salohcin.

Since exp(a-b)=exp(a)/exp(b) you will need the distribution of a-b and then the distribution of exp(a-b).

If a2, b2, c2, and d2 are approximately normal then exp(a-b) will be approximately log-normal. If you have covariance between parameters then the covariance matrix of variables (i.e. covariance of each pair of variables) needs to be taken into account. You can estimate co-variance of two variables using standard techniques.

Note that if the estimators you are using are maximum likelihood ones, then you can use results regarding estimators of functions of parameters as well as asymptotic results regarding normal approximations.

Re: Regression: statistically significant difference between transformed coefficients

Hi Chrio,

Thanks so much for your response. This is very helpful. I was hoping to ask a quick follow up if that is okay. I was realizing today I could do something much simpler. I could just check whether (a2-b2) - (c2-d2) is statistically different than zero using a plain old t-test. If this is not statistically different than zero, the exp(a2-b2) - exp(c2-d2) necessarily would not be statistically different than zero. Does this seem reasonable to you or am I fundamentally missing something? Thanks so much,

Nick

Re: Regression: statistically significant difference between transformed coefficients

You can do that, but you need to take into account the nature of the exponential function.

Because of the non-linear nature of the exponential function, testing the non-exponential hypothesis will not capture the non-linearity which means that the variance will either be too small or too large depending on the nature of the transformation.

I'd strongly recommend to look at the test statistic for your original hypothesis involving the exponential form if that is what you need to test.

As an example, if our linear confidence interval was [1,2] then the exponentiated interval would be [2.71828,7.38906] which is a lot wider than the normal linear interval.

Are you aware of how to get a test statistic for a function of a parameter given that the estimator is a maximum likelihood estimator (MLE)?

Re: Regression: statistically significant difference between transformed coefficients

Thanks so much for your response! I am estimating all of this using OLS actually. I will go about this comparing my original specification (in exponentials). Thanks again