I did an F-test on a very simple linear regression model for test if the two coefficients have the same effect on the dependent variable y:

i will call this below Original Regression :


and this is the output in gretl of my regression

enter image description here

this is the main idea of what i'm testing:

$H0: \beta_2=\beta_3$

$H1: \beta_2\neq\beta_3$

my professor told that for run this type of test i have to rearrange my model in an

Equivalent Regression:


so now i can express the equivalent hypothesis sistem:

$H0: \beta_2-\beta_3=0$

$H1: \beta_2-\beta_3\neq0$

the F-statistics:


$m$ are the restrictions: in this case $m=1$

so now i have an output of gretl about my test

enter image description here

from this point on i tried to get the same result of the test in gretl but i can't get the same result.

i write my try: the only thing i need for run my test id $[ER]RSS$

but the second output of gretl gave me the S.E of Regression, so $S=3.11144$

since i know that the estimate of Variance of regression is $S^2=\frac{[ER]RSS}{T-k}$


so i solved and obtained


so the F-stat:


this is different from gretl output! of $F=7.29383$

gretl seems to use a derivation of $S^2=\frac{RSS}{T-k+1}$ so if you solve for this you will obtain $[ER]RSS=29.043$ and obtain once again the F-stats you will get the same result $F=7.29383$

My questions are:

(1) where am i doing the test wrong?

(2) how do i get by hand the same coefficients of the output in gretl? i mean $\beta2=1.59549=\beta3$?

Thank You


I hope that I understand your questions correctly..

  1. The most frequently used estimator of variance if the unbiased estimator, i.e., $$ S^2_{\epsilon} = \frac{\sum_{i=1}^T(y_i - \hat{y}_i)^2}{T-k}, $$ where $k$ is the number of estimated coefficients including the intercept. I.e., in a model like $y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2$, $k=3$.

  2. If your hypothesis is $H_0: \beta_2 = \beta_3$, then you can re-express the original model $y = \beta_1 + \beta_2 x_1 + \beta_3 x_2 $ as $$ y = \beta_1 + \beta_2x_1 + \beta_2 x_2 = \beta_1 +\beta_2(x_1 + x_2)=\beta_1+\beta_2{x^*}, $$ namely, you are estimating $\beta_2$ which is the coeff. of $x^*$. I.e., you can use the OLS $$ \hat{\beta}_2=\frac{\sum_{i=1}^T(y_i - \bar{y}_n)(x_i - \bar{x}_n)}{\sum_{i=1}^n(x_i - \bar{x}_n)^2}, $$ where in the restricted model will be both the coeff. of $x_1$ and $x_2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.