Test sum of coefficient vs lincom

[Diana Yoko]

Please sympathize with me if the question below is trivial. Indeed, I have tried to search the forum but so far failed to clear my mind about that.

My confusion is whether a test for a sum of coefficients can provide a valid conclusion as -lincom-. The below example illustrates my question.

sysuse auto, clear
reg price mpg weight foreign

test weight + foreign = 0

lincom weight + foreign

Your kind guidance is much appreciated.

[Andrew Musau]

lincom reports a t-statistic, whereas test reports an F-statistic. In linear regression, when testing a single coefficient, the relationship between these statistics is \(F=t^2\). Therefore, the results are indeed equivalent in this case.

sysuse auto, clear
reg price mpg weight foreign
test weight + foreign = 0
lincom weight + foreign
display "The t-statistic is `:di %3.2f r(t)' and F-statistic is t^2 = `:di %3.2f r(t)'^2=  `:di %3.2f `=r(t)^2' '"

Res.:

. test weight + foreign = 0

 ( 1)  weight + foreign = 0

       F(  1,    70) =   28.87
            Prob > F =    0.0000

.
. lincom weight + foreign

 ( 1)  weight + foreign = 0

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   3676.525   684.2984     5.37   0.000     2311.735    5041.315
------------------------------------------------------------------------------

.
. display "The t-statistic is `:di %3.2f r(t)' and F-statistic is t^2 = `:di %3.2f r(t)'^2=  `:d
> i %3.2f `=r(t)^2' '"
The t-statistic is 5.37 and F-statistic is t^2 = 5.37^2=  28.87

[Diana Yoko]

Andrew Musau, thank you so much for your detailed guidance.

May I ask whether the same conclusion can always be said if we test the sum of more than two coefficients in more complicated data? I check the code below with 3 coefficients and it does show that F= t^2. However, I still wonder whether it might be different in more complicated situations.

My ultimate concern is whether using -test- for sum coefficients always provides the same conclusion (on reject or fail to reject a null hypothesis) with -lincom-? Your kind further guidance is so much appreciated.

sysuse auto, clear
reg price mpg weight foreign

test mpg + weight + foreign = 0
lincom mpg + weight + foreign

display "The t-statistic is `:di %3.2f r(t)' and F-statistic is t^2 = `:di %3.2f r(t)'^2=  `:di %3.2f `=r(t)^2' '"

. test mpg + weight + foreign = 0

 ( 1)  mpg + weight + foreign = 0

       F(  1,    70) =   27.80
            Prob > F =    0.0000

. lincom mpg + weight + foreign

 ( 1)  mpg + weight + foreign = 0

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   3698.379   701.4449     5.27   0.000     2299.391    5097.366
------------------------------------------------------------------------------

. display "The t-statistic is `:di %3.2f r(t)' and F-statistic is t^2 = `:di %3.2f r(t)'^2=  `:di %3.2f `=r(t)^2' '"
The t-statistic is 5.27 and F-statistic is t^2 = 5.27^2=  27.80

[Andrew Musau]

The test of whether the sum equals zero has one degree of freedom in the numerator, regardless of how many components constitute it, and the \(F-\)distribution simplifies to the square of the corresponding \(t-\)distribution. So yes, that will always be the case.