Vtyjkyuk

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 :

$y_t=hatbeta_1+hatbeta_2x_2+hatbeta_3x_3$

and this is the output in gretl of my regression

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

$H0: beta_2=beta_3$

$H1: beta_2neqbeta_3$

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

Equivalent Regression:

$y_t=hatbeta_1+(hatbeta_2-hatbeta_3)x_2+hatbeta_3(x_2+x_3)$

so now i can express the equivalent hypothesis sistem:

$H0: beta_2-beta_3=0$

$H1: beta_2-beta_3neq0$

the F-statistics:

$F=frac[ER]RSS-[OR]RSSm*fracT-k[OR]RSS$

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

so now i have an output of gretl about my test

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 $S^2=frac[ER]RSST-k$

$T-k=5-3$

so i solved and obtained

$[ER]RSS=19.36211$

so the F-stat:

$F=frac19.36211-6.251*frac5-36.25=4.1958$

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

gretl seems to use a derivation of $S^2=fracRSST-k+1$ 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.,
   $$
   S^2_epsilon = fracsum_i=1^T(y_i - haty_i)^2T-k,
   $$
   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
   $$
   y = beta_1 + beta_2x_1 + beta_2 x_2 = beta_1 +beta_2(x_1 + x_2)=beta_1+beta_2x^*,
   $$
   namely, you are estimating $beta_2$ which is the coeff. of $x^*$. I.e., you can use the OLS
   $$
   hatbeta_2=fracsum_i=1^T(y_i - bary_n)(x_i - barx_n)sum_i=1^n(x_i - barx_n)^2,
   $$
   where in the restricted model will be both the coeff. of $x_1$ and $x_2$.