Null and Alternative hypothesis for multiple linear regression

I have 1 dependent variable and 3 independent variables.

I run multiple regression, and find that the p value for one of the independent variables is higher than 0.05 (95% is my confidence level).

I take that variable out and run it again. Both remaining independent variables have $p$-value less than 0.05 so I conclude I have my model.

Am I correct in thinking that initially, my null hypothesis is

$$H_0= β_1=β_2 = \dots =β_{k-1} = 0$$

and that the alternative hypothesis is

$$H_1=\textrm{At least one } β \neq 0 \textrm{ whilst } p<0.05$$

And that after the first regression, I do not reject, as one variable does not meet my confidence level needs...

So I run it again, and then reject the null as all $p$-values are significant?

Is what I have written accurate?

Edit: Thanks to Bob Jansen for improving this aesthetics of this post.

2 Answers

The hypothesis $H_0: β_1=β_2=\dots =β_{k−1}=0$ is normally tested by the $F$-test for the regression.

You are carrying out 3 independent tests of your coefficients (Do you also have a constant in the regression or is the constant one of your three variables?) If you do three independent tests at a 5% level you have a probability of over 14% of finding one of the coefficients significant at the 5% level even if all coefficients are truly zero (the null hypothesis). This is often ignored but be careful. Even so, If the coefficient is close to significant I would think about the underlying theory before coming to a decision.

If you add dummies you will have a beta for each dummy

• Thanks for your response. I don't have a constant, all of my p-values are very significant (the least is a dummy variable at 0.039). What would my null hypothesis be? My knowledge is that I'm seeking p-values because that'd give me my model. I don't understand the technicalities of it and want to learn it :) – Harry Jan 7 '15 at 22:36
• I think you meant to say 14% of committing a type one error (probability of 0.14 of finding at least one of the coefficient significant when there true value is actually the null hypothesis value) – Kamster Jan 8 '15 at 0:36
• @Kamster Thanks. You are correct and I have amended my answer. – user1483 Jan 21 '15 at 21:26

These are independent variables so the hypothesis applies to each parameter independently.

• +1: Yes, you are right - but the rest of it should be fine – vonjd Jan 2 '15 at 21:18
• sorry, could you clarify? How do I change the equation so it applies to each parameter independently? And also, what is the effect of adding 3 dummy variables. Is it simply 2 more betas? Or do they require their own symbol – Harry Jan 4 '15 at 0:32
• It just means that you have an H_0 and an H_1 for every parameter. – vonjd Jan 4 '15 at 11:33
• Ok I see. Do you know the procedure for dummy variables? Are they just additional beta? Or is it more accurate to refer to them as delta? – Harry Jan 4 '15 at 11:43
• Maybe I have this wrong but isn't it true if you remain your individual significance levels at 0.05 that the probability of type one error (ie the probability that reject null hypothesis when it is actually true; significance level) will be greater than or equal 0.14 – Kamster Jan 8 '15 at 0:43