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<h1 class="post-title" itemprop="name headline">工具变量和孟德尔随机化原理</h1>
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<p>工具变量(Instrumental-Variable)本身是一个计量经济学的概念，它的出现是为了克服普通最小二乘法中的内生性问题。</p>
<h1 id="内生性问题">内生性问题</h1>
<p>内生性问题的原因主要有以下三种</p>
<ul>
<li>遗漏变量</li>
<li>变量有测量误差</li>
<li>双向因果关系：比如受教育年限影响工资，工资反过来影响受教育年限</li>
</ul>
<h1 id="假设">假设</h1>
<p>孟德尔随机化需要三个核心工具变量假设，即：</p>
<ul>
<li>相关性假设：用作工具变量的遗传变异 (<span
class="math inline">\(Z\)</span> )与暴露 (<span
class="math inline">\(X\)</span>) 有关，即存在 <span
class="math inline">\(Z \to X\)</span></li>
<li>独立性/可交换性假设：遗传变异和目标结果没有共同的原因（即没有干扰因子），即虽然可能存在
<span class="math inline">\(U \to Y\)</span>，但是不存在 <span
class="math inline">\(U \to Z\)</span></li>
<li>排斥性/无水平多效性假设：除了通过暴露( <span
class="math inline">\(X\)</span> )外，遗传变异( <span
class="math inline">\(Z\)</span> )和结果( <span
class="math inline">\(Y\)</span> )之间没有独立的因果途径，不存在 <span
class="math inline">\(Z \to Y\)</span> 不经过 <span
class="math inline">\(X\)</span></li>
</ul>
<h1 id="模型解释">模型解释</h1>
<ul>
<li>用作工具变量的遗传变异( <span class="math inline">\(Z\)</span>
)</li>
<li>暴露( <span class="math inline">\(X\)</span> )</li>
<li>结果( <span class="math inline">\(Y\)</span> )</li>
<li>混杂因素( <span class="math inline">\(U\)</span> )</li>
</ul>
<figure>
<img src="/img/2022-5-9-1.png" alt="孟德尔随机化模型" />
<figcaption aria-hidden="true">孟德尔随机化模型</figcaption>
</figure>
<h1 id="模型使用">模型使用</h1>
<p>使用两阶最小二乘（Two Stage Least
Squares，TSLS）来替代原来的最小二乘法</p>
<p><span class="math display">\[Y = \sum \beta_i X_i +
\delta\]</span></p>
<h2 id="第一步建立自变量x和工具变量的回归模型">第一步：建立自变量<span
class="math inline">\(X\)</span>和工具变量的回归模型</h2>
<p><span class="math display">\[X = \alpha Z + \epsilon\]</span></p>
<p>在这一步中，需要验证，<span class="math inline">\(Cov(Z,
\epsilon)=0\)</span>，也即工具变量和混杂因素无关。
注：我感觉应该是残差吧！</p>
<p>另外还需要考虑上述方程的决定系数或者说是F统计量，来判断<span
class="math inline">\(Z\)</span>和<span
class="math inline">\(X\)</span>的关联强度。通常情况下，决定系数或者F统计量越大，说明Z和X的关联程度越大。</p>
<h2
id="第二步将第一步中拟合的x的估计量带入如下方程">第二步：将第一步中拟合的X的估计量带入如下方程：</h2>
<p><span class="math display">\[Y = \beta \hat{X} + \delta\]</span></p>
<p>其中<span class="math inline">\(\beta\)</span>便是<span
class="math inline">\(X\)</span>对<span
class="math inline">\(Y\)</span>的纯净效应量。</p>
<h1 id="参考资料">参考资料</h1>
<ul>
<li><a
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