# Standard error of the quotient of two estimates (Wald estimators) using the delta method

I have two coefficients' estimates from a regression, each of which has an estimated standard error.
I would like to know the quotient of these two estimates -- that is, divide one of the estimates by the other. What would be the corresponding standard error?

Would this be a candidate for the Delta Method?

If so, how should the formula be applied? If this is not something that can be computed in a straightforward manner, is there a way to do this in Stata?

In particular, I am interested in the Wald Estimator. In the Delta Method, there is a covariance term typically. Is it zero or non-zero in this case; that is, when can we assume that the two estimates are independent?

• This might be of some use Jun 4, 2013 at 23:48

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
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. Here you have two choices. You can assume it's zero if that makes sense with your model and do the calculation "manually". 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
------------------------------------------------------------------------------

• If you want to skip to the hypothesis test, you can just testnl _b[x1]/_b[x2]=.5 after the regression. Jun 4, 2013 at 23:41
• Thank you!! Is there a way to do something similar if the estimates are based on coefficients from two separate regressions? Jun 5, 2013 at 1:54
• See my response above. Jun 5, 2013 at 16:34
• What if the two regressions are not conducive to SUR? Is there a formula that I could use? Or, must the two equations be conducive to SUR? Jun 6, 2013 at 13:55
• By SUR, I meant seemingly unrelated regressions as you said. I guess I just meant is whether there is a formula with delta method. I am hesistant about SUR because if I am not normally doing my regressions with SUR why would I want to now? Jun 6, 2013 at 14:50

Another method of looking at the ratio is due to Fieller. An excellent post is at: How to compute the confidence interval of the ratio of two normal means