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# 经济代写|计量经济学代考ECONOMETRICS代考|ECON502 Illustrations of Best Linear Predictor

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## 经济代写|计量经济学代考ECONOMETRICS代考|Illustrations of Best Linear Predictor

We illustrate the best linear predictor (projection) using three log wage equations introduced in earlier sections.

For our first example, we consider a model with the two dummy variables for gender and race similar to Table 2.1. As we learned in Section 2.17, the entries in this table can be equivalently expressed by a

linear CEF. For simplicity, let’s consider the CEF of $\log ($ wage $)$ as a function of Black and female.
$$\mathbb{E}[\log (\text { wage }) \mid \text { Black, female }]=-0.20 \text { Black }-0.24 \text { female }+0.10 \text { Black } \times \text { female }+3.06 \text {. }$$
This is a CEF as the variables are binary and all interactions are included.
Now consider a simpler model omitting the interaction effect. This is the linear projection on the variables Black and female
$$\mathscr{P}[\log (\text { wage }) \mid \text { Black, female }]=-0.15 \text { Black }-0.23 \text { female }+3.06 .$$
What is the difference? The full CEF (2.34) shows that the race gap is differentiated by gender: it is $20 \%$ for Black men (relative to non-Black men) and $10 \%$ for Black women (relative to non-Black women). The projection model (2.35) simplifies this analysis, calculating an average 15\% wage gap for Black wage earners, ignoring the role of gender. Notice that this is despite the fact that gender is included in (2.35).

## 经济代写|计量经济学代考ECONOMETRICS代考|Linear Predictor Error Variance

As in the CEF model, we define the error variance as $\sigma^{2}=\mathbb{E}\left[e^{2}\right]$. Setting $Q_{Y Y}=\mathbb{E}\left[Y^{2}\right]$ and $\boldsymbol{Q}{Y X}=$ $\mathbb{E}\left[Y X^{\prime}\right]$ we can write $\sigma^{2}$ as \begin{aligned} \sigma^{2} &=\mathbb{E}\left[\left(Y-X^{\prime} \beta\right)^{2}\right] \ &=\mathbb{E}\left[Y^{2}\right]-2 \mathbb{E}\left[Y X^{\prime}\right] \beta+\beta^{\prime} \mathbb{E}\left[X X^{\prime}\right] \beta \ &=Q{Y Y}-2 \boldsymbol{Q}{Y X} \boldsymbol{Q}{X X}^{-1} \boldsymbol{Q}{X Y}+\boldsymbol{Q}{Y X} \boldsymbol{Q}{X X}^{-1} \boldsymbol{Q}{X X} \boldsymbol{Q}{X X}^{-1} \boldsymbol{Q}{X Y} \ &=Q_{Y Y}-\boldsymbol{Q}{Y X} \boldsymbol{Q}{X X}^{-1} \boldsymbol{Q}{X Y} \ & \stackrel{\text { def }}{=} Q{Y Y \cdot X} \end{aligned}
One useful feature of this formula is that it shows that $Q_{Y Y \cdot X}=Q_{Y Y}-\boldsymbol{Q}{Y X} \boldsymbol{Q}{X X}^{-1} \boldsymbol{Q}_{X Y}$ equals the variance of the error from the linear projection of $Y$ on $X$.

## 经济代写|计量经济学代考ECONOMETRICS代考|lllustrations of Best Linear Predictor

$\mathbb{E}[\log ($ wage $) \mid$ Black, female $]=-0.20$ Black $-0.24$ female $+0.10$ Black $\times$ female $+3.06$.

$$\mathscr{P}[\log (\text { wage }) \mid \text { Black, female }]=-0.15 \text { Black }-0.23 \text { female }+3.06 \text {. }$$

## 经济代写|计量经济学代考ECONOMETRICS代考|Linear Predictor Error Variance

$\sigma^{2}=\mathbb{E}\left[\left(Y-X^{\prime} \beta\right)^{2}\right] \quad=\mathbb{E}\left[Y^{2}\right]-2 \mathbb{E}\left[Y X^{\prime}\right] \beta+\beta^{\prime} \mathbb{E}\left[X X^{\prime}\right] \beta=Q Y Y-2 \boldsymbol{Q} Y X \boldsymbol{Q} X X^{-1} \boldsymbol{Q} X Y+\boldsymbol{Q} Y X \boldsymbol{Q} X X^{-1} \boldsymbol{Q} X X \boldsymbol{Q} X X^{-}$

$Y$ 上 $X .$

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