I want to test if the slope of a regression line is -1. I used the t test as such:
t = β-(-1)/SE(β)
where β is the slope coefficient and the SE(β) is the standard error of the slope coefficient.
I defined α as 0.05 and calculated the critical value for t-statistic as 1.96 since the number of samples in my dataset is 10000.
If I understand correctly, if the
|t-statistic| is smaller than the critical vale then we can assume that the hypothesis
H0 : β=-1 cannot be rejected.
When I actually tested this on my dataset, I got the following results:
β = 2002.39
SE(β) = 1564.45
|t-statistic| = 1.28 which is smaller than critical value so Ho is not rejected.
But the β value of 2002 is far away from -1.
Also, there is another dataset which gives the following
SE(β) = 0.39
|t-statistic|= 38.46 which is greater than critical value so Ho is rejected
But one could say the β value of -16 is a lot closer to -1 than 2002 so the t-statistic should be closer to the critical value in the case of -16.
Am I wrong in this assumption ? Is the formula for t-test of β=-1 correct ?
Is there any other way to test if β is equal to -1? I also want a "measure of evidence" statistic that shows how close β is to -1.