Subtracting t-distributions I have two linear regression parameters of interest, b1 and b2. Both parameters are from a linear model built from 14 datapoints and having 7 model parameters, including an intercept.
Interest lies in testing the null hypothesis b2 - b1 = 0 versus its two-sided alternative.
I have two questions concerning this test:
First, according to Slutsky's lemma, the subtraction of model parameters b1 and b2 converges to the subtraction of the distributions of b2 and b1 (t-distributions with n-p=7 degrees of freedom). However, I am afraid I am not allowed to work asymptotically since I have such a low number of data points, or can I?
Second, how can I calculate a subtraction of these two t-distributions?
 A: I once tried an approximational approach in a similar problem which worked quite well. The idea is to start with the normal distribution where arithmetics with distributions is simple. You doubt in this, and you're right since for small samples (small degrees of freedom) the normal distribution has considerably lighter tails than the $t$-distribution. Thus your critical values would be too small and your test becomes liberal.
I got rid of it the following way: The heavier tails of the $t_k$-distribution correspond (among others) to larger variance compared to the normal distribution: $\frac{k}{k-2}$ instead of $1$ like in case of the normal distribution. So I took the quantiles from the normal distribution but inflated them by $\sqrt{\frac{k}{k-2}}$. This way I accounted for the higher variance if the sample size is finite.
I encourage you to try some simulations if it works in your case. This approach is very easy but yielded quite fair type-I-errors in the small sample cases I simulated. (It was not directly a regression setting but I also had two   dependent $t$-distributed test statistics I wanted to transform linearly.)
I didn't find something similar in the literature. Perhaps it's too straightforward to write it down.
A: I think you are overcomplicating! You have a parameter $b=(b_0,b_1,b_2,\dotsc)$ and want to test the null hypothesis $b_1=b_2$. So define a contrast vector $c=(0,-1,1,0,\dotsc)$ and test the null
$$
   c^T\hat{b}=0
$$
which is just standard linear regression theory.  No need for asymptotics or other approximations.
