Normal error regression model

Question

The normal error regression model is assumed to be applicable.

a)When testing $H_0:B_1=5$ vs $H_1:B_1\neq 5$ by means of a general linear test, what is the reduced model? What are the degrees of freedom $df_R$?

b)When testing $H_0:B_0=2,B_1=5$ vs $H_1:$not both $B_0=2,B_1=5$ by means of a general linear test, what is the reduced model? What are the degrees of fredom $df_R$?

The normal error regression model is $$Y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim N(0,\sigma^2)$ and the reduced model is $$Y_i=B_0+\epsilon_i$$

I know there is a difference in the degrees of freedom between the models, since the reduced model is estimated only one parameter, but I do not know essentially what the exercise is wanting.

Answer 1

The reduced model is the restricted model.

In the first question, your restriction is that the slope coefficient equals $5$. You run the regression using this value and note the error sum of squares which you compare with the error sum of squares of the unrestricted model to see if the restriction is too costly, in which case $H_0:b_1=5$ is rejected. The degrees of freedom are clearly $n-1$ since only one parameter is estimated.

Following the same logic, the reduced model in your second question is

$$Y=2+5 x+\varepsilon$$

and you proceed as above to test the joint hypotheses. Now there are two restrictions instead of one since we have also forced the intercept to assume a certain value. Hence the degrees of freedom are $n$, as no parameter is estimated.

Answer 2

You should modify your linear model as below:

If you wish to test a nonzero value, subtract it from the coefficient in the regression output and divide the result by the coefficient's SE. (Use a calculator for this.) Similarly, if you want confidence intervals, use the coefficient plus or minus the product of its s.e. with a t-value for the desired confidence level and 12 degrees of freedom. (Use a calculator for this.) This also works for the intercept using its SE.

http://courses.statistics.com/software/R/R_Ch02.htm

In both your cases, you are performing a linear regression between your data and your hypothesis, so df remains n-2. Same procedure, just that you are regressing to something other than zero.