Relationship between R square and F stat (Econometrics - Basic)

■ Recast between R square and F

Now we can re cast F in terms of R^2. We an formulate the F stat in terms of R square.

We consider:

Yi = b1 + b2X2i + . . . + bkXki + ui

Let's formulate the hypothesis:

Ho: b2 = b3 = . . . = bk = 0

When we test the overall significance and the intercept is not included, our aim is to find out whether Yi is related to the exolanatory variables or not.

Then, 

F = (ESS ÷ k -1) / RSS ÷ n - k)

where, ESS, RSS, k and n have usual meanings .

F = (ESS ÷ k - 1) × (RSS ÷ n - k)

F = [(n - k) ÷ (k - 1)] × [ESS ÷ RSS]

F = [(n - k) ÷ (k - 1)] × [ESS ÷ TSS - ESS]

F = [(n - k) ÷ (k - 1)] × [ESS/TSS ÷ TSS -          ESS/TSS]

F = [(n - k) ÷ (k - 1)] × [R ^ 2 ÷ 1 - R^2]

F = [(R ^ 2) / (k - 1)] ÷ [(1 - R ^ 2) / (n - k)]

F is directly propotional to R ^ 2

If R ^ 2 = 0 then F = 0

If R ^ 2 = 1 then F = Undefined

Higher the value of F - higher will be the R ^ 2 (F is directly proportional to R ^2).

And we also do know that when Fcal > Ftab.

Thus this is the relationship between R ^ 2 and F. Ho is rejected.

Thank you

Aditya Pokhrel

MBA, MA Economics, MPA