Theorem

Suppose that \(z_1\ldots,z_n\) are i.i.d random variables, and \(\hat W\stackrel{P}\rightarrow W\). Suppose that \(g(z,\theta)\) is a moment function. Suppose that

  1. (identification) : \(W\) is positive semi-definite, and \(W E(g(Z,\theta))=0\) only if \(\theta=\theta_0\),
  2. (compactness) : \(\theta_0\in\Theta\), where \(\Theta\) is compact,
  3. (continuity) : \(g(z,\theta)\) is continuous
  4. (uniform convergence) : \(E[\sup_{\theta\in\Theta}| g(Z;\theta)|]<\infty\).

Define \[ \hat Q_n(\theta) = - (n^{-1}\sum_{i=1}^n g(z_i,\theta))' \hat W (n^{-1}\sum_{i=1}^n g(z_i,\theta)), \] and let \(\hat\theta = \arg\max_{\theta\in\Theta}\hat Q_n(\theta)\). Define \(Q_0(\theta) = - g_0(\theta)' W g_0(\theta)\), where \(g_0(\theta)= E(g(Z,\theta))\), and \(g_0(\theta_0)=E(g(Z;\theta_0))=0\).

Then, \(\hat\theta\stackrel{P}\rightarrow \theta_0\).




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