Generalized method of moments (GMM)

Recall: $x_t = m_t^T \phi + r_t $ with the previous definition of the symbols. The residuals are $x_t- m_t^T\phi$ and $m_t$ has no correlation:

\[\begin{aligned} \mathbb{E}[(x_t-m_t^T\phi)m_t] &= \mathbb{E}[\mathbb{E}[(x_t-m_t^T\phi)m_t|\mathcal{F}_{t-1}]]\\ &=\mathbb{E}[m_t \mathbb{E}[\sigma_t\epsilon_t|\mathcal{F}_{t-1}]] \\ &=\mathbb{E}[m_t\sigma_t\mathbb{E}[\epsilon_t|\mathcal{F}_{t-1}]] = 0 \end{aligned}\]

The correlation of the errors of the second moments with the series:

\[\begin{aligned} \mathbb{E}[(r_t^2-\sigma_t^2)z_t(\phi)] &= \mathbb{E}[\mathbb{E}[(r_t^2-\sigma_t^2)z_t(\phi)|\mathcal{F}_{t-1}]] \\ &= \mathbb{E} [\sigma_t^2 z_t(\phi) \mathbb{E}[\epsilon_t^2-1]]\\ &= 0 \end{aligned}\]

Thus, we have two constraints. The GMM method replaces the above expectation constraints by the samples.

\[\begin{aligned} &\frac{1}{T}\sum_{t=1}^T (x_t - m_t^T\phi) m_t = 0 \\ &\frac{1}{T}\sum_{t=1}^T \left((x_t-m_t^T\phi)^2 - z_t(\phi)^T \alpha\right)z_t(\phi) = 0 \end{aligned}\]

Methods of moments:

\[\min_{\alpha,\phi} \left\| \frac{1}{T}\sum_{t=1}^T (x_t-m_t^T\phi) m_t \right\|^2 + \left\| \frac{1}{T}\sum_{t=1}^T \left((x_t-m_t^T\phi)^2- z_t(\phi)^T\alpha \right) z_t(\phi)\right\|^2\]

There are some correlations between the first part and the second part and we thus diagonlaize the quadratic form for the two parts. and we get

\[\min_{\phi,\alpha} g(\phi,\alpha)^T \hat{S}_T^{-1} g(\phi,\alpha)\]

where $S_T$ is the square root matrix of the estimated covariance of the two equations from the methods of moments.


\[g(\phi,\alpha) = \begin{pmatrix} \frac{1}{T}\sum_{t=1}^T(x_t-m_t^T\phi)m_t \\ \frac{1}{T}\sum_{t=1}^T((x_t-m_t^T\phi)^2-z_t(\phi)^T\alpha)z_t(\phi) \end{pmatrix}\]

$g(\phi,\alpha)$ is a $k+p+2$ vector


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