Probit estimation - insignificant coefficients with a significant difference

I'm a postgraduate economics student but not of the statistically inclined sort :)

Im running a regression to estimate the likelihood of a patent being litigated. I have two dummies for the patent owner type; let's call them SmallFirm and LargeFirm. The reference category is individuals.

What I'm actually interested in is if the difference between the coefficients on Large and Small is negative, which would suggest in my particular case that small firms face a higher litigation risk.

I have quite a small sample size and find each coefficient to be insignificant at the 10% level. However, when I run an test of equality of coefficients in stata (test LargeFirm=SmallFirm) I get that they are significantly different at the 5 % level.

Dummy     |coeff     |std err  |z-stat |P-value|       CI
SmallFirm |-.1636389 |.5219826 | -0.31 |0.754  | (-1.186706,    .8594281)
LargeFirm |-.7599469 |.533567  | -1.42 |0.154  | (-1.805719,    .2858252)

. test SmallFirm = LargeFirm

( 1)  [Litigated]SmallFirm - [Litigated]LargeFirm = 0

chi2(  1) =    5.88
Prob > chi2 =    0.0153


What can I conclude from this? Can I say that the significant difference does imply that small firms are associated with a higher likelihood of litigation despite the individual dummies being insignificant?

Thanks so much for the help!

Update! I think I got it now! Thanks Andrea: you got me on the right track. The dummies are insignificant precisely b/c my choice of reference group: neither large or small firms are statistically significantly more likely to be litigated - than individuals-.

this is not what I was interested in anyway.

If I run a regression with the SmallFirm dummy and an Individual dummy making LargeFirm the reference group: surely enough I do find a significant coefficient on SmallFirm!

So thanks!

• Just for clarification: what's the category that is used as a reference? Maybe "MediumFirm"? Because if a firm is either Large or Small, then the 2 dummy variables included in your model would be collinear and one of the two parameters would be omitted automatically by Stata, but this is not the case, I see... – boscovich Mar 26 '12 at 11:30
• Plus: all the tests that you've carried out (the two tests on the parameters being different from 0 $H_0:\beta_{SmallFirm}=0$ and $H_0:\beta_{LargeFirm}=0$) and the test on the difference of the two parameters being different from 0 ($H_0=:\beta_{SmallFirm}-\beta_{LargeFirm}=0$), are actually Wald tests. – boscovich Mar 26 '12 at 11:36
• the reference category is individuals, with the sizes being mece for firms :) – Joe Mar 26 '12 at 11:37
• If SmallFirms and LargeFirms are mutually exclusive and collectively exhaustive categories for the variable "size of the firm", then either one or the other must be the reference. Tertium non datur. Apparently, they're not MECE. – boscovich Mar 26 '12 at 12:17
• misunderstanding: categories are individuals, large firms, small firms for the variable "patent owner type" – Joe Mar 26 '12 at 12:25