test with missing standard errors - stata

Question

How can I conduct a hypothesis test in Stata when my predictor perfectly predicts my dependent variable?
I would like to run the same regression over many subsets of my data. For each regression, I would then like to test the hypothesis that beta_1 = 1/2. However, for some subsets, I have perfect collinearity, and Stata is not able to calculate standard errors.
For example, in the below case,
sysuse auto, clear
gen value = 2*foreign*(price<6165)
gen value2 = 2*foreign*(price>6165)
gen id = 1 + (price<6165)
I get the output
. reg foreign value value2 weight length, noconstant
Source | SS df MS Number of obs = 74
-------------+------------------------------ F( 4, 70) = .
Model | 22 4 5.5 Prob > F = .
Residual | 0 70 0 R-squared = 1.0000
-------------+------------------------------ Adj R-squared = 1.0000
Total | 22 74 .297297297 Root MSE = 0
------------------------------------------------------------------------------
foreign | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
value | .5 . . . . .
value2 | .5 . . . . .
weight | 3.54e-19 . . . . .
length | -6.31e-18 . . . . .
------------------------------------------------------------------------------
and
. test value = .5
( 1) value = .5
F( 1, 70) = .
Prob > F = .
In the actual data, there is usually more variation. So I can identify the cases where the predictor does a very good job of predicting the DV--but I miss those cases where prediction is perfect. Is there a way to conduct a hypothesis test that catches these cases?
EDIT:
The end goal would be to classify observations within subsets based on the hypothesis test. If I cannot reject the hypothesis at the 95% confidence level, I classify the observation as type 1. Below, both groups would be classified as type 1, though I only want the second group.
gen type = .
for values 1/2 {
quietly: reg foreign value value2 weight length if id = `i', noconstant
test value = .5
replace type = 1 if r(p)>.05
}

Answer 1

There is no way to do this out of the box that I'm aware of. Of course you could program it yourself to get an approximation of the p-value in these cases. The standard error is missing here because the relationship between x and y is perfectly collinear. There is no noise in the model, nothing deviates.
Interestingly enough though, the standard error of the estimate is useless in this case anyway. test performs a Wald test for beta_i = exp against beta_i != exp, not a t-test.
The Wald test uses the variance-covariance matrix from the regression. To see this yourself, refer to the Methods and formulas section here and run the following code:
(also, if you remove the -1 from gen mpg2 = and run, you will see the issue)
sysuse auto, clear
gen mpg2 = mpg * 2.5 - 1
qui reg mpg2 mpg, nocons
* collect matrices to calculate Wald statistic
mat b = e(b) // Vector of Coefficients
mat V = e(V) // Var-Cov matrix
mat R = (1) // for use in Rb-r. This does not == [0,1] because of
the use of the noconstant option in regress
mat r = (2.5) // Value you want to test for equality
mat W = (R*b-r)'*inv(R*V*R')*(R*b-r)
// This is where it breaks for you, because with perfect collinearity, V == 0
reg mpg2 mpg, nocons
test mpg = 2.5
sca F = r(F)
sca list F
mat list W
Now, as #Brendan Cox suggested, you might be able to simply use the missing value returned in r(p) to condition your replace command. Depending on exactly how you are using it. A word of caution on this, however, is that when the relationship between some x and y is such that y = 2x, and you want to test x = 5 vs test x = 2, you will want to be very careful about the interpretation of missing p-values - In both cases they are classified as type == 1, where the test x = 2 command should not result in that outcome.
Another work-around would be to simply set p = 0 in these cases, since the variance estimate will asymptotically approach 0 as the linear relationship becomes near perfect, and thus the Wald statistic will approach infinity (driving p down, all else equal).
A final yet more complicated work-around in this case could be to calculate the F-statistic manually using the formula in the manual, and setting V to some arbitrary, yet infinitesimally small number. I've included code to do this below, but it is quite a bit more involved than simply issuing the test command, and in truth only an approximation of the actual p-value from the F distribution.
clear *
sysuse auto
gen i = ceil(_n/5)
qui sum i
gen mpg2 = mpg * 2 if i <= 5 // Get different estimation results
replace mpg2 = mpg * 10 if i > 5 // over different subsets of data
gen type = .
local N = _N // use for d.f. calculation later
local iMax = r(max) // use to iterate loop
forvalues i = 1/`iMax' {
qui reg mpg2 mpg if i == `i', nocons
mat b`i' = e(b) // collect returned results for Wald stat
mat V`i' = e(V)
sca cov`i' = V`i'[1,1]
mat R`i' = (1)
mat r`i' = (2) // Value you wish to test against
if (cov`i' == 0) { // set V to be very small if Variance = 0 & calculate Wald
mat V`i' = 1.0e-14
}
mat W`i' = (R`i'*b`i'-r`i')'*inv(R`i'*V`i'*R`i'')*(R`i'*b`i'-r`i')
sca W`i' = W`i'[1,1] // collect Wald statistic into scalar
sca p`i' = Ftail(1,`N'-2, W`i') // pull p-value from F dist
if p`i' > .05 {
replace type = 1 if i == `i'
}
}
Also note that this workaround will become slightly more involved if you want to test multiple coefficients.
I'm not sure if I advise these approaches without issuing a word of caution considering you are in a very real sense "making up" variance estimates, but without a variance estimate you wont be able to test the coefficients at all.