Degrees of freedom

[prasadhegde1]

Hi David

I believe It is one of the most basic doubt while going through the readings of Gujarati .I would like to know the intuition behind using (n-2) degrees of freedom in t-dist , Is it not n-1 df ? with reference to gujarati pg 179 Ch 7 It has been stated that n-2 degrees of freedom is used .

Is this because of the property of t-dist variance = k/k-2 . I can understand that when a sample variance is used to estimate the population variance 1 df is being lost ,but I cant get the intuition behind n-2 df .Please help

 

[David Harper CFA FRM]

Hi prasad,

Right, tricky, you say "when a sample variance is used to estimate the population variance 1 df is being lost" and that's a good start: in order to estimate the sample variance, we need a sample mean, so we are we are "reusing" one data point (losing one independent observation)...the concept extends...

Chh 7 refers to two-variable regression not to sample mean. Always we refer to estimators! But before estimator = sample mean, now with a two-variable regression, we have three estimators:

slope
intercept
predicted Y

The test for all three of these estimators uses the standard error of regression (SER) as an input into the standard error;
in an analogy to how we need to "consume" 1 sample mean in order to produce a sample variance,
we need to consume slope and intercept in order to produce the SER that gives us a standard error to use for the test
(just as sample variance gives us a standard error for test of sample mean)

In this way, to get a SER that we need for the test will always require k = the number of slope + intercept estimators:
two variable regrssion: df = n -2
3 variable: df = n - 3
k variable: df = n - k
etc
(so we can safely say that k = number of variables and d.f. = n - k but technically k is not the variables but k is the coefficients; ie., the intercept is consumed not the Y variable)

Hope this helps, David

 

[prasadhegde1]

Thanks a ton !

Excellent explanation !! I get your point now i can differentiate liner and multiple regressions better