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 for the formula that Glen_b linked to, 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, which will give you cross-equation covariances between the coefficients. It's hard to know which is better without the details. One way (out of several possible ways) 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
------------------------------------------------------------------------------