Posts Tagged ‘inference’

Single Linear Combinations of Parameters

March 7, 2010 Leave a comment

Single Linear Combinations of Parameters means we are to test the linear relationship between two parameters in our multiple regression analysis. The simplest case can be

H0: β1 = β2

Or H0:  β1 = 10β2

Our hypothesis can be pretty much anything, as long as β1 and β2 has linear relationship.

Note that we are to test whether or not the effects of the two x variables on y have a linear relationship, NOT the linear relationship between the two x variables on each other (that is the case of perfect multicollinearity).

For example, we are interested in testing

H0: β1 = β2

H1: β1 ≠ β2


Set θ = β1 – β2, then we will have

H0: θ = 0
H1: θ ≠ 0

Set α (if not given, assume it to be .05)

Find critical value: df=n-k-1 (k is the number of x variables), then use the t-table to find critical value.

Calculate test statistic:

(That output above was an example from my class notes.)


Decision: to reject H0 or not (by comparing t0 ­ with the critical value)


If we reject H0 1 = β2), we will conclude that β1 is statistically different from β2 at α level.

If we fail to reject H0 1 = β2), we will conclude that β1 is not statistically different from β2 at α level.


Crazy enough, huh? There is another method that may look easier:


-Set θ = β1 – β2, then β1 = θ + β2

-Substitute β1 in our original model by θ + β2

y= β0 + β1x1 +  β2x2 +  β3x3 + u

y= β0 + (θ + β2)x1+  β2x2 +  β3x3 + u  = β0 + θx1+  β2(x1+x2) + β3x3 + u

Now our 3 variables in the model are x1, x1+x2, x3

-Construct a new variable that is the sum of x1 and x2 (in STATA) by using the command

gen totx12 =  x1 + x2

“totx12” is just the name of the new variable.

-Run the regression of y on x1, totx12, and x3


H0: θ = 0
H1: θ ≠ 0

Now we can look at the t-ratio or p-value of the coefficient on x1 (coefficient on x1 is now θ) then make our decision whether or not to reject H0.