Principal Component Regression and its relation to linear regression I am reading Elements of Statistical Learning. On page 79 it is stated that the principal component regression is defined as:
$\hat{\bf{y}}_{(M)}^{\text{pcr}}=\bar{y}{\bf{1}}+\sum_{m=1}^M \hat{\theta}_m{\bf{z}}_m$
where ${\bf{z}}_m={\bf{X}} v_m $ and $\hat{\theta}_m=\langle{\bf{z}}_m,{\bf{y}}_m\rangle/\langle{\bf{z}}_m,{\bf{z}}_m\rangle$.
Further it is stated that the terms of coefficients can be written as $\hat{\beta}^{\text{pcr}}(M) = \sum_{m=1}^M \hat{\theta}_mv_m$, and that if $M=p$, that is, the number of principal components used is equal to the number of predictors, the coefficients of PCR is equal to the coefficients for least squares. 
For linear regression we have ${\bf{\hat{\beta}}}^{\text{ols}}=(X^TX)^{-1}X^TY$, and according to ELS we should have $\hat{\beta}^{\text{pcr}}(p)={\bf{\hat{\beta}}}^{\text{ols}}$ which I have not been able to show. 
Bonus question: Is it possible to express ${\bf{\hat{y}}}_{(M)}^{\text{pcr}}$ by the coefficients of $\hat{\beta}^{\text{ols}}$?
 A: Since the intercept term is $\bar y$, I take it that $X$ is standardized. Define matrices $Z$ and $V$ containing columns $\bf z_m$ and $\bf v_m$, use subscript to define convention where $V,\hat\theta$ contains columns or elements up to $M$
$$
\begin{aligned}
\hat\beta^\textrm{pcr}(M) &= V_M\hat\theta_M \\
\hat\beta^\textrm{pcr}(p) &= V\hat\theta
\end{aligned}
$$
Also,
$$
Z=XV
$$
We start from definition of $\hat\theta$.
$$
\begin{aligned}
\hat\theta &= (Z^TZ)^{-1}Z^TY \\
Z^TZ\hat\theta &= Z^Ty \\
V^TX^TXV\hat\theta &= V^TX^TY \\
X^TXV\hat\theta &= X^TY \\
V\hat\theta &= (X^TX)^{-1}X^TY = \hat\beta^\textrm{ols}
\end{aligned}
$$
Along the way, we have left multiplied the equation by $V$ to eliminate $V^T$. Thus, it is shown that $\hat\beta^\textrm{pcr}(p)=\beta^\textrm{ols}$
For the second question, we split the following
$$
\begin{aligned}
V\hat\theta &= V_M\theta_M+V_{(M)}\theta_{(M)} \\
V_M\theta_M &= V\hat\theta-V_{(M)}\theta_{(M)}
\end{aligned}
$$
where $V_{(M)},\hat\theta_{(M)}$ contains columns and elements after $M$. With this, we have
$$
\begin{aligned}
\hat{\bf y}^\textrm{pcr}_{(M)} &= \bar y {\bf 1}+XV_M\hat\theta_M \\
  &= \bar y {\bf 1}+ XV\hat\theta - XV_{(M)}\hat\theta_{(M)} \\
  &= \bar y {\bf 1}+ X\hat\beta^\textrm{ols}- XV_{(M)}\hat\theta_{(M)}
\end{aligned}
$$
Unfortunately, this is the best expression I could get that involves $\hat\beta^\textrm{ols}$.
