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# 经济代写|计量经济学代写ECONOMETRICS代考|ECON335 Projection Matrix

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## 经济代写|计量经济学代写ECONOMETRICS代考|Projection Matrix

Define the matrix
$$\boldsymbol{P}=\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}$$
Observe that
$$\boldsymbol{P} \boldsymbol{X}=\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{X}=\boldsymbol{X} .$$
This is a property of a projection matrix. More generally, for any matrix $Z$ which can be written as $\boldsymbol{Z}=\boldsymbol{X} \boldsymbol{\Gamma}$ for some matrix $\boldsymbol{\Gamma}$ (we say that $\boldsymbol{Z}$ lies in the range space of $\boldsymbol{X}$ ), then
$$\boldsymbol{P} \boldsymbol{Z}=\boldsymbol{P} \boldsymbol{X} \boldsymbol{\Gamma}=\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{X} \boldsymbol{\Gamma}=\boldsymbol{X} \boldsymbol{\Gamma}=\boldsymbol{Z} .$$
As an important example, if we partition the matrix $\boldsymbol{X}$ into two matrices $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ so that $\boldsymbol{X}=$ $\left[\begin{array}{ll}\boldsymbol{X}_1 & \boldsymbol{X}_2\end{array}\right]$ then $\boldsymbol{P} \boldsymbol{X}_1=\boldsymbol{X}_1$. (See Exercise 3.7.)

The projection matrix $\boldsymbol{P}$ has the algebraic property that it is idempotent: $\boldsymbol{P} \boldsymbol{P}=\boldsymbol{P}$. See Theorem $3.3 .2$ below. For the general properties of projection matrices see Section A.11.
The matrix $\boldsymbol{P}$ creates the fitted values in a least squares regression:
$$\boldsymbol{P} \boldsymbol{Y}=\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}=\boldsymbol{X} \widehat{\beta}=\widehat{\boldsymbol{Y}} \text {. }$$
Because of this property $\boldsymbol{P}$ is also known as the hat matrix.
A special example of a projection matrix occurs when $\boldsymbol{X}=\mathbf{1}_n$ is an $n$-vector of ones. Then
$$\boldsymbol{P}=\mathbf{1}_n\left(\mathbf{1}_n^{\prime} \mathbf{1}_n\right)^{-1} \mathbf{1}_n^{\prime}=\frac{1}{n} \mathbf{1}_n \mathbf{1}_n^{\prime} .$$
Note that in this case
$$\boldsymbol{P} \boldsymbol{Y}=\mathbf{1}_n\left(\mathbf{1}_n^{\prime} \mathbf{1}_n\right)^{-1} \mathbf{1}_n^{\prime} \boldsymbol{Y}=\mathbf{1}_n \bar{Y}$$
creates an $n$-vector whose elements are the sample mean $\bar{Y}$.

## 经济代写|计量经济学代写ECONOMETRICS代考|Annihilator Matrix

Define
$$\boldsymbol{M}=\boldsymbol{I}_n-\boldsymbol{P}=\boldsymbol{I}_n-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}$$
where $\boldsymbol{I}_n$ is the $n \times n$ identity matrix. Note that
$$\boldsymbol{M} \boldsymbol{X}=\left(\boldsymbol{I}_n-\boldsymbol{P}\right) \boldsymbol{X}=\boldsymbol{X}-\boldsymbol{P} \boldsymbol{X}=\boldsymbol{X}-\boldsymbol{X}=0 .$$
Thus $\boldsymbol{M}$ and $\boldsymbol{X}$ are orthogonal. We call $\boldsymbol{M}$ the annihilator matrix due to the property that for any matrix $\boldsymbol{Z}$ in the range space of $\boldsymbol{X}$ then
$$M Z=Z-P Z=0 .$$

For example, $\boldsymbol{M} \boldsymbol{X}_1=0$ for any subcomponent $\boldsymbol{X}_1$ of $\boldsymbol{X}$, and $\boldsymbol{M P}=0$ (see Exercise 3.7).
The annihilator matrix $\boldsymbol{M}$ has similar properties with $\boldsymbol{P}$, including that $\boldsymbol{M}$ is symmetric $\left(\boldsymbol{M}^{\prime}=\boldsymbol{M}\right)$ and idempotent $(\boldsymbol{M} M=\boldsymbol{M})$. It is thus a projection matrix. Similarly to Theorem 3.3.3 we can calculate
$$\operatorname{tr} M=n-k .$$
(See Exercise 3.9.) One implication is that the rank of $\boldsymbol{M}$ is $n-k$.
While $\boldsymbol{P}$ creates fitted values, $\boldsymbol{M}$ creates least squares residuals:
$$\boldsymbol{M} \boldsymbol{Y}=\boldsymbol{Y}-\boldsymbol{P} \boldsymbol{Y}=\boldsymbol{Y}-\boldsymbol{X} \widehat{\boldsymbol{\beta}}=\widehat{\boldsymbol{e}} .$$
As discussed in the previous section, a special example of a projection matrix occurs when $\boldsymbol{X}=\mathbf{1}_n$ is an $n$-vector of ones, so that $\boldsymbol{P}=\mathbf{1}_n\left(\mathbf{1}_n^{\prime} \mathbf{1}_n\right)^{-1} \mathbf{1}_n^{\prime}$. The associated annihilator matrix is
$$\boldsymbol{M}=\boldsymbol{I}_n-\boldsymbol{P}=\boldsymbol{I}_n-\mathbf{1}_n\left(\mathbf{1}_n^{\prime} \mathbf{1}_n\right)^{-1} \mathbf{1}_n^{\prime} .$$
While $\boldsymbol{P}$ creates a vector of sample means, $\boldsymbol{M}$ creates demeaned values:
$$\boldsymbol{M} \boldsymbol{Y}=\boldsymbol{Y}-\mathbf{1}_n \bar{Y}$$

## 经济代写|计量经济学代写ECONOMETRICS代考|投影矩阵

$$\boldsymbol{P}=\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}$$

$$\boldsymbol{P} \boldsymbol{X}=\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{X}=\boldsymbol{X} .$$

## MATLAB代写

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