# Thread: PDF of (nearly) collinear variables

1. ## PDF of (nearly) collinear variables

My question is motivated by a problem with estimation by maximum likelihood. In simple terms, I want to estimate a parameter and have three variables, , and , such that . Using all three variables is pointless and the joint pdf is singular. Now, suppose that instead of I have such that , where is independent from and and its distribution does not depend on . The joint pdf of , and is not singular. Seems to me obvious that does not contain any additional information about , given and . So, I think it must be true that, in terms of the likelihood function, . If I am correct, what would be the best way to prove it? Thanks.

2. ## Re: PDF of (nearly) collinear variables

The question is equivalent to show that the conditional pdf

is independent of the parameter

Consider the conditional CDF

(by independence)

By assumption, the distribution of is independent of , and therefore

is independent of the beta, and similar for its derivative:

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Sam Vimes (07-18-2014)

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