# Least square estimation error, calculus question

I want to estimate $n+m$ parameters with the equation: $$\hat\theta=\left[\frac{1}{N}\sum_{t=1}^N \varphi(t)\varphi^T(t)\right]^{-1}\left[\frac{1}{N}\sum_{t=1}^N\varphi(t)y(t)\right]$$ where $\varphi$ is a vector of regressors and $\theta$ the vector of parameters. Then my book derive an expression for the estimation error: $$\hat\theta-\theta_0 =\left[\frac{1}{N}\sum_{t=1}^N\varphi(t)\varphi^T(t)\right]^{-1} \left[\frac{1}{N}\sum_{t=1}^N\varphi(t)y(t)-\left\{\frac{1}{N}\sum_{t=1}^N\varphi(t)\varphi^T(t)\right\}\theta_0\right]$$ this makes no sense to me! Why the term in curly braces? And why inside the second factor? Please help.

• whoa... I guess I really have some sort of phobia for big formulae. Thanks! – gurghet Jun 18 '11 at 18:11
Let $A=\frac{1}{N}\sum_{t=1}^N \varphi(t)\varphi^T(t)$ and $B=\frac{1}{N}\sum_{t=1}^N\varphi(t)y(t)$
Then by distributivity $A^{-1}[B-A\theta_0]=A^{-1}B-A^{-1}A\theta_0=\hat{\theta}-\theta_0$.