Individual vs Joint Significance

[Elliott Blain]

I suspect imperfect multicollinearity between two explanatory variables, x and y, and only variable y is individually significant but the results of a Wald test suggest they are jointly significant.

Would it be incorrect to now describe variable x which was individually insignificant, to be significant? Or is it strictly that the effect of variable x is only jointly significant with the effect of y?

[Carlo Lazzaro]

Elliott:
tons of words cannot replace sharing what you typed and what Stata gave you back (as per FAQ). Thanks.
In addition, as far as quasi-extreme multicollinearity is concerned, see https://www.hup.harvard.edu/catalog....=9780674175440 Chapter 23.
Yes, it would be incorrect. Your last statement conflates coefficient statistical significance with joint statistical significance:

use "C:\Program Files\Stata17\ado\base\a\auto.dta"
. regress price trunk mpg

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     10.14
       Model |   141126459         2  70563229.4   Prob > F        =    0.0001
    Residual |   493938937        71  6956886.44   R-squared       =    0.2222
-------------+----------------------------------   Adj R-squared   =    0.2003
       Total |   635065396        73  8699525.97   Root MSE        =    2637.6

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       trunk |   43.55851   88.71884     0.49   0.625    -133.3418    220.4589
         mpg |  -220.1649   65.59262    -3.36   0.001    -350.9529    -89.3769
       _cons |   10254.95   2349.084     4.37   0.000      5571.01    14938.89
------------------------------------------------------------------------------

. test trunk mpg

 ( 1)  trunk = 0
 ( 2)  mpg = 0

       F(  2,    71) =   10.14
            Prob > F =    0.0001

[Richard Williams]

With a joint test, the null hypothesis is typically all X coefficients = 0, and the alternative is that AT LEAST ONE X coefficient DOES NOT equal zero. So, there could be 50 variables in a model, 49 of which have 0 effect, and only one of which has a nonzero effect. The joint test could come up significant but that doesn't mean all 50 are significant.

Incidentally, you should be careful about including junk variables in a model. If those 49 variables were included "just in case" they might be important, it could be that they will cause you to overlook the variables that really are important because the junk variables will inflate the standard errors of the other variables. For a discussion, see

https://www3.nd.edu/~rwilliam/stats2/l41.pdf

Finally suppose you believed that a significant joint test did mean that all vars included in the test have significant effects. You could sneak in some very dubious hypotheses that way, e.g. "The effects of income, education, race, gender, and eye color are jointly significant. Therefore, exactly as I hypothesized, eye color is a significant determinant of presidential vote!"