# Extra variable in regression model increases variance of parameter

Suppose my true Model is:

$$y = Xb + u \tag 1$$

But I am estimating:

$$y = Xb + Zd + u \tag 2$$

I can get the estimate of $b$ from $(2)$ by using $Mz$ operator as:

$$\hat{b} = (X'MzX)^{-1}(X'MzY)$$

But what I have to do is to show that this estimate is less efficient than original one. It is true because my degrees of freedom are decreasing as I am increasing parameters. But I am unable to show that mathematically.

• Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Nov 27 '16 at 23:12
• @gung done. Thanks any help though? – Dan ish Nov 28 '16 at 1:45
• I am sure you know the variance covariance matrix for $b$ when estimating model (1)? There likewise is an expression for the variance of the coefficients corresponding to $b$ when estimating (2). Can you derive that one as well? – Christoph Hanck Nov 28 '16 at 5:00