Here's an example in Stata of how to create the ratio and test a hypothesis using nlcom:
. webuse regress
. regress y x1 x2 x3
Source | SS df MS Number of obs = 148
-------------+------------------------------ F( 3, 144) = 96.12
Model | 3259.3561 3 1086.45203 Prob > F = 0.0000
Residual | 1627.56282 144 11.3025196 R-squared = 0.6670
-------------+------------------------------ Adj R-squared = 0.6600
Total | 4886.91892 147 33.2443464 Root MSE = 3.3619
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | 1.457113 1.07461 1.36 0.177 -.666934 3.581161
x2 | 2.221682 .8610358 2.58 0.011 .5197797 3.923583
x3 | -.006139 .0005543 -11.08 0.000 -.0072345 -.0050435
_cons | 36.10135 4.382693 8.24 0.000 27.43863 44.76407
------------------------------------------------------------------------------
. nlcom ratio:_b[x1]/_b[x2], post
ratio: _b[x1]/_b[x2]
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ratio | .6558606 .4221027 1.55 0.122 -.1784571 1.490178
------------------------------------------------------------------------------
. test ratio=.5
( 1) ratio = .5
F( 1, 144) = 0.14
Prob > F = 0.7125
There are formulas in the pdf manual under nlcom. A terse explanation can be found in the Stata FAQ on the delta method.
Added in response to the OP's comment below:
If you have two separate regressions, you have all the ingredients from the formula that Glen_b, other than the covariance term. You can assume it's zero if that makes sense with your model, or you can estimate the two equations as a system. It's hard to know for sure without the details. One way to do the latter is with Seemingly Unrelated Regression:
. webuse regress
. sureg (eq1:y x1 x2) (eq2:y x1 x3)
Seemingly unrelated regression
----------------------------------------------------------------------
Equation Obs Parms RMSE "R-sq" chi2 P
----------------------------------------------------------------------
eq1 148 2 4.54006 0.3758 91.48 0.0000
eq2 148 2 3.770546 0.5694 211.94 0.0000
----------------------------------------------------------------------
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
eq1 |
x1 | 7.472932 .98949 7.55 0.000 5.533568 9.412297
x2 | -.4768772 .7799875 -0.61 0.541 -2.005625 1.05187
_cons | -1.374358 2.883296 -0.48 0.634 -7.025514 4.276798
-------------+----------------------------------------------------------------
eq2 |
x1 | 4.338581 .7852935 5.52 0.000 2.799434 5.877728
x3 | -.0026865 .0003774 -7.12 0.000 -.0034261 -.0019468
_cons | 16.32873 3.214735 5.08 0.000 10.02797 22.6295
------------------------------------------------------------------------------
. nlcom ratio:[eq1]_b[x1]/[eq2]_b[x1]
ratio: [eq1]_b[x1]/[eq2]_b[x1]
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ratio | 1.722437 .2773696 6.21 0.000 1.178803 2.266071
------------------------------------------------------------------------------