實證白皮書
標(biāo)簽（空格分隔）： Econometrics Empirical_Research
簡介
回歸分析
線性回歸模型是實證分析框架的基礎(chǔ)?；诠诺浼俣ǎ€性回歸模型的最小二乘估計量（OLS）是最優(yōu)線性無偏估計量。因此，實證研究中回歸分析的起點都是最小二乘法。
面板數(shù)據(jù)分析
如果模型的內(nèi)生性問題是由于非觀測效應(yīng)所引起，那么面板數(shù)據(jù)提供了消除這一偏誤絕佳的方法。面板數(shù)據(jù)的優(yōu)點如此明顯，因此在實證中，要充分利用面板數(shù)據(jù)的優(yōu)點進(jìn)行分析。
政策評估方法
政策評估的方法在近年來非常流行，這是因為它的分析框架切合了實證研究的本質(zhì)，即識別和估計因果關(guān)系。它的基本分析框架是潛在結(jié)果模型和魯賓因果框，流行的方法有匹配（Matching）和傾向評分模型（PSM）、雙重差分模型（DID）以及斷點回歸模型（RDD）。
線性回歸模型與最小二乘法
軟件實現(xiàn)及案例
stata
R
面板數(shù)據(jù)的固定效應(yīng)和隨機(jī)效應(yīng)方法
軟件實現(xiàn)及案例
stata
R
潛在結(jié)果估計和魯賓因果框架
Since the early 1990s the potential outcome, or Neyman-Rubin Causal Model, approach to these problems has gained substantial acceptance as a framework for analyzing causal problems.
魯賓因果框架（潛在結(jié)果模型）是因果分析的基本框架。
Causal effects are comparison of pairs of potential outcomes for the same unit, e.g. the difference $Y_{i}(\omega^{'}) - Y_{i}(\omega)$.
因果關(guān)系的衡量方式是對于同一單位處理與控制結(jié)果的比較。
We can never directly observe the causal effects, which is what Holland(1986) calls the "fundamental problem of causal inference".
由于無法同時觀測到兩個結(jié)果，所以無法直接得到因果效應(yīng)，這就是因果推斷的基礎(chǔ)問題。
Esimates of causal effects are ultimately based on comparisons of different untis with different levels of the treatment.
因此，對于處理效應(yīng)的估計依賴于對于不同單位觀測值的比較。
術(shù)語與假設(shè)
Potential outcome model(POM)
$$Y_{i} = Y_{0i} + D_{i}(Y_{1i}-Y_{0i})$$
Treatment effect(TE)
$$TE_{i} = Y_{1i} - Y_{0i}$$
Average Treatment Effect(ATE)
$$ATE = E(Y_{1i} - Y_{0i})$$
Average Treatment Effect on the treated(ATET)
$$ATE = E(Y_{1i} - Y_{0i}|D=1)$$
Average Treatment Effect on the untreated(ATENT)
$$ATE = E(Y_{1i} - Y_{0i}|D=0)$$
Unconfoundedness Assumption
$$(Y_{0},Y_{1})\perp D|X$$
隨機(jī)分配下的識別問題
If the sample was drawn at random(i.e., under random assignment), it would be possible to estimate the ATE as the difference between the sample mean of treated and the sample mean of untreated units, which is the well-known "Difference-in-means"(DIM) estimator of classical statistics.
如果是隨機(jī)分配樣本，那么ATE就是處理樣本和控制樣本的均值差，這就是常用的DIM估計量。
We call this the independence assumption(IA) fromally stating that:
$$(Y_{0},Y_{1})\perp D$$
基于IA條件，這個條件意味著樣本分組與潛在結(jié)果沒有直接聯(lián)系。
DIM
$$\hat{DIM} = \frac{1}{N_{1}} \sum_{i=1}^{N_{1}} Y_{1,i} - \frac{1}{N_{0}} \sum_{i=1}^{N_{0}} Y_{0,i}$$
Regression-Adjustment
RA is suitable only when the conditional independence assumption(CIA) holds. RA方法只有在CIA條件成立時才適用。CIA條件寫為
$$(Y_{0},Y_{1})\perp D|X$$
A less restrictive assumption which only limits independence to the mean is required. 這是一個比CIA更弱的CMI條件。It is known as conditional mean independence(or CMI) and implies that:
$$E(Y_{1}|x,D)=E(Y_{1}|x)$$
以及
$$E(Y_{0}|x,D)=E(Y_{0}|x)$$
因此，當(dāng)存在$x$時，可以導(dǎo)出下列結(jié)論，two identification conditions of the unobservable counterfactual mean potential outcomes:
$$E(Y_{0}|x,D=1)=E(Y_{0}|x,D=0)$$
以及
$$E(Y_{1}|x,D=1)=E(Y_{1}|x,D=0)$$
Under CMI, we could see:
$$ATE(x)=E(Y|x,D=1) - E(Y|x,D=0)$$
By simply denoting:
$$m_{1}(x)=E(Y|x,D=1)$$
$$m_{0}(x)=E(Y|x,D=0)$$
We have that:
$$ATE(x)=m_{1}(x) - m_{0}(x)$$
一旦得到$m_{1}(x)$和$m_{0}(x)$的一致估計量，我們就可以通過上述估計causal parameters。
$$\hat{ATE} = \frac{1}{N}\sum_{i=1}^{N}[\hat{m}{1}(x{i}) - \hat{m}{0}(x{i})]$$
以及
$$\hat{ATET} = \frac{1}{N_{1}}\sum_{i=1}^{N}D_{i}[\hat{m}{1}(x{i}) - \hat{m}{0}(x{i})]$$
以及
$$\hat{ATENT} = \frac{1}{N_{0}}\sum_{i=1}^{N}(1-D_{i})[\hat{m}{1}(x{i}) - \hat{m}{0}(x{i})]$$
這就是Regression-adjustment估計量。其中$m_{1}(x)$和$m_{0}(x)$可以通過參數(shù)、半?yún)⒑头菂⒐烙?。Note that the Regression-adjustment approach only uses the potential outcome means to recover ATEs and does not use the propensity score.
