 # F test

Why is the F-test one sided when it’s in fact testing for a null hypothesis that the variables are equal to 0?

Text says the f-test is always one sided?

can someone plz explain why (with an example if possible)?

Interesting …I dont think CFAI or schweser has explained this thing , but I think if you look at the purpose of F Test = ratio of two scaled sums of squares that is explained (MSR) / unexplained variance(MSE) . F test identifies the model that best fits population from which the data were sampled . My rationale for F test being one sided after the distributions plotted on a normal curve is simply because the ratio (explained/unexplained) cannot be less than ’ 0 ’ so on a normal curve the the F statistic cannot be on the left tail .

Am I missing something here ??

H0: No variable has an explanatory power

Ha: Atleast one variable explains the regression.

Logically, it looks to me that it is a one-sided test.

It is two-sided. H0 says all coefficients =0, Ha says al least one is *not* zero, so it could be more than zero or less than zero.

Facor Hedge, I just checked page 364. Your answer closely matches what CFA text book says.

rockmania, it is not as trivial as you may think. Here’s what Schweser has to say about this being a one-sided test:

Professors Note: When testing the hypothesis that all the regression coefficients

~ are simultaneously equal to zero~ the F-test is always a one-tailed test, despite

~ the foct that it looks like it should be a two-tailed test because there is an equal

sign in the null hypothesis. This is a common source of confosion among Level

II candidates; make sure you don’t make that mistake on the exam.

Just remember F-test is one-sided and move on…

rockmania, that’s true, but your statement here is not correct:

• H0: No variable has an explanatory power

• Ha: Atleast one variable explains the regression.

• Logically, it looks to me that it is a one-sided test.

Logically, it should be a two-tailed test. This is for everyone’s benefit.

Its easy to just remember that an F test is one tail…i doubt they will ask for an explanation of why…

What i have trouble remembering is why a T-test would be One tail or Two tail?? any help??

it seems u can always argue for either test… anyone have an easy way to distinguish whether a two tailed t-test or a one tail is needed?? thnx

Logically, for it to be a two-tailed test, there has to be a minimum boundary and maximum boundary and when the t-stat falls out of either boundary, it is considered to be statistically significant.

When you are testing whether the combined explanatory power of the indepent variables is zero or not, it is really asking whether it is greater than zero. How can explanatory power be less than zero? It either does or does not. That is why it’s a one-tailed test

Logically, for it to be a two-tailed test, there has to be a minimum boundary and maximum boundary and when the t-stat falls out of either boundary, it is considered to be statistically significant.

When you are testing whether the combined explanatory power of the indepent variables is zero or not, it is really asking whether it is greater than zero. How can explanatory power be less than zero? It either does or does not. That is why it’s a one-tailed test

I won’t go into detail here since you guys don’t really need all the underlying formula but i’ll be kind :

F-Test is based on 2 variables Normally distribution and compare Variances of the 2 distributions.

1. Use corrected sample variance which is an unbiased estimator of the distribution variance _S1_² and _S2_²

2. we know that (n - 1)S1_²/sigma1² ~Khisqauren - _1 and (m - 1)_S2_²/sigma2² ~Khisquarem- 1

so obviously your F test F = S1² /S2² is distributed as [Khisqaure n-1 /(n-1)] / [Khisqaure m-1 /(m-1)]

since this is symetric and can’t the result can’t go under 0, your two-sided test becomes one sided-test when you use the fisher distribution.

So yes this is a 2 sided test, but with a 1 sided distrubtion. good luck.

If H0: x=0, Ha" x NOT= 0

Then to reject the null, you can show that x > 0 or x < 0, these are two cases, so it’s 2-tailed test.

If H0: x<=0, Ha" x > 0

Then to reject the null, you try to show that x > 0, that’s a 1-tailed test.