# Getting F statistic from multiple t-statistics?

#### derksheng

##### New Member
If I have

Y = a + b1*X1 + b2*X2 + b3*X3 + error,

and I have calculated sample t-statistics for b1 and b2, from this information (along with sample size) can I construct an F statistic testing Ho : b1 = b2 = 0 ?

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#### Jake

I know how to do it in the case where X1, X2, and X3 are all categorical predictors (i.e., if this were a two-way ANOVA where X3 = X1*X2, or a one-way ANOVA on a 4-level factor), but I am not sure how to work it out with continuous predictors... I will think about it a little more though.

#### noetsi

##### Fortran must die
My question is, why would you not look at the software printout (unless this is a homework problem where you can't)? Any software that will print out a t statistic will a f test.

#### derksheng

##### New Member
Because I'm trying to apply Portfolio OLS by Sefcik & Thompson (1986). This is quite esoteric and no package has it automated (and I'm newb at econometrics).

I ask because they say "Inferences about the significance of any individual $$\hat{\gamma}$$ can be drawn based on tstatistics for the individual portfolio regressions and joint tests can be performed with F-statistics on combinations."

Source:
TITLE: An Approach to Statistical Inference in Cross-Sectional Models with Security AbnormalReturns As Dependent Variable
AUTHORS: Sefcik & Thompson
DATABASE: JSTOR
YEAR: 1986.

#### spunky

##### Doesn't actually exist
I know how to do it in the case where X1, X2, and X3 are all categorical predictors
this sounds really interesting! care to share how?

#### Englund

##### TS Contributor
from this information (along with sample size, estimated b1, estimated b2, etc) can I construct an F statistic testing Ho : b1 = b2 = 0 ?
Using the following equation will test what you asked for

where R is the reduced model and C the complete model. In the denominator it is supposed to be n, not N (my bad). g and k are number of parameter estimates in the reduced model and the complete model, respectively. Using STATA; just write

test _b[X1] = _b[X2] = 0

and STATA will give you the result. This statistica follows an F distribution with k-g numerator degrees of freedom and n-k+1 denominator degrees of freedom.

Edit: It is supposed to be a summation sign at the beginning of the denominator as well...

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#### derksheng

##### New Member
Nice Englud, I see what you did there. FYI you might be interested in this forum's $$tags. Thanks for your help. Let me re-phrase my problem. Using S&T's portfolio OLS, I have the following: (i) The SAME coefficient estimates. (ii) DIFFERENT standard errors. Using your method, as well as the SSR/TSS method, will give me an \(R^2$$ and F-statistic the SAME as the OLS model.

This is very problematic for me because I know with some OLS heteroscedasticity robust SEs, the F changes as well ... But I'm not sure how to get my F to change when ONLY my SEs are changing.\)