==============================================================================
GMM Over-Identifying Restrictions
==============================================================================

Criterion Function Limiting Distribution
==============================================================================
Since

.. math::

   \begin{gather}
   \sqrt{T}\boldsymbol{g}_{T}(\boldsymbol{\theta}_{0})
   \stackrel{d}{\longrightarrow} N(0,S) \\
   \implies  (\sqrt{T}
   \boldsymbol{g}_{T}(
   \boldsymbol{\theta}_{0})^{'})S^{-1}(\sqrt{T}\boldsymbol{g}_{T}
   (\boldsymbol{\theta}_{0})) = T
   \boldsymbol{g}_{T}(\boldsymbol{\theta}_{0})^{'}S^{-1}
   \boldsymbol{g}_{T}(\boldsymbol{\theta}_{0})
   \overset{d}{\longrightarrow} \chi^{2}(r)
   \end{gather}

where :math:`\smash{r > k}` is the number of moment conditions.

Estimated Criterion Function
==============================================================================
It turns out that

.. math::

   \smash{T
   \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})^{'}\hat{S}^{-1}
   \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})
   \overset{d}{\not\to} \chi^{2}(r)}.

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- This is because :math:`\smash{k}` moment conditions will be set to
  zero exactly.

Exact Identification
==============================================================================
Consider :math:`\smash{r=k}`. In this case

.. math::

   \begin{gather}
   \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}}) = 0 \\
   T \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})^{'}\hat{S}^{-1}
   \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}}) = 0.
   \end{gather}

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- :math:`\smash{r-k}` of the moment conditions will be non-zero.

Over Identification
==============================================================================
In general,

.. math::

   \smash{J_{T}(\hat{\boldsymbol{\theta}}) = T
   \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})^{'}\hat{S}^{-1}
   \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})
   \overset{d}{\longrightarrow} \chi^{2}(r-k)}.

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- To test if our moment conditions are close to zero, we compute
  :math:`\smash{J_{T}(\hat{\boldsymbol{\theta}})}` and compare with a
  :math:`\smash{\chi^{2}(r-k)}` distribution.

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- If :math:`\smash{J_{T}(\hat{\boldsymbol{\theta}})}` is far in the
  tail of the :math:`\smash{\chi^{2}(r-k)}` distribution, we might
  conclude that the model is misspecified.

Asset Pricing with GMM
==============================================================================
Suppose an agent derives utility from consumption,
:math:`\smash{c_{t}}`, and seeks to maximze the discounted sum of
expected utility:

.. math::

   \smash{ \sum_{\tau = 0}^{\infty}
   \beta^{\tau}E[u(c_{t+\tau})|\Omega_{t}]},

where :math:`\smash{u(c_{t})}` is the period utility function and
satisfies:

.. math::

   \smash{ \frac{\partial u(c_{t})}{\partial c_{t}} > 0 \,\,\, \text{and}
   \,\,\, \frac{\partial^{2} u(c_{t})}{\partial c_{t}^{2}} < 0}.

Equilibrium Conditions
==============================================================================
Suppose that the agent can purchase :math:`\smash{m}` assets paying
gross returns :math:`\smash{(1 + r_{i,t+1})}` between periods
:math:`\smash{t}` and :math:`\smash{t+1}`, for :math:`\smash{i =
1,\ldots, m}`.

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- The agent's portfolio must satisfy

.. math::

   \smash{u^{'}(c_{t}) = \beta E[(1+r_{i,t+1})u^{'}(c_{t+1}) |
   \Omega_{t}] \,\,\, \text{for} \,\,\, i=1,\ldots,m}.

Equilibrium Conditions
==============================================================================
The equilibrium conditions say that marginal utility of consuming an
extra unit today should be equivalent to the expected marginal
consumption gained by purchasing a unit of any asset.

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- If these conditions didn't hold, the agent wouldn't be at an
  optimum.

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The portfolio conditions can be rewritten as:

.. math::

   \smash{E\left[\left(\beta
   \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
   -1\right)\bigg|\Omega_{t}\right] = 0 \,\,\, \text{for}
   i=1,\ldots,m}.

Equilibrium Conditions
==============================================================================
Given a vector :math:`\smash{\boldsymbol{x}_{t} \in \Omega_{t}}`, by
the law of iterated expectations

.. math::

   \begin{align}
   E\left[\underset{\boldsymbol{h}(\boldsymbol{\theta},
   \boldsymbol{y}_{t})}{\underbrace{\left(\beta
   \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
   -1\right)\boldsymbol{x}_{t}}} \right] & =
   E\left[E\left[\left(\beta
   \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
   -1\right)\boldsymbol{x}_{t} \bigg|\Omega_{t}\right]\right] \\
   & E\left[\underset{0}{\underbrace{E\left[\left(\beta
   \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
   -1\right) \bigg|\Omega_{t}\right]}} \boldsymbol{x}_{t} \right] = 0,
   \end{align}

for :math:`\smash{i=1,\ldots,m}`.

Stochastic Discount Factor
==============================================================================
Economic theory says that all returns discounted by
:math:`\smash{\beta \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}}` should be
identical:

.. math::

   \begin{gather}
   E\left[\underset{m_{t,t+1}}{\underbrace{\beta
   \frac{u^{'}(c_{t+1})}{u^{'} (c_{t})}(1+r_{i,t+1})}}\right] = 1 \\
   \implies E[m_{t,t+1}(1+r_{i,t+1})] = 1.
   \end{gather}

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- :math:`\smash{\beta
  \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1}) - 1}` is a forecast
  error and should be uncorrelated with any variable
  :math:`\smash{\boldsymbol{x}_{t} \in \Omega_{t}}`

Casting as GMM
==============================================================================
This problem maps easily into GMM where

.. math::

   \begin{gather}
   \boldsymbol{y}_{t} =
   (r_{1,+1},\ldots,r_{m,t+1},c_{t},c_{t+1},
   \boldsymbol{x}_{t}^{'})^{'} \\
   \boldsymbol{h}(\boldsymbol{\theta},\boldsymbol{y}_{t}) =
   \left[\begin{array}{c} \left(1 - \beta
   \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})\right)\boldsymbol{x}_{t}
   \\ \vdots \\ \left(1 - \beta
   \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{m,t+1})\right)\boldsymbol{x}_{t}
   \\ \end{array} \right] \\
   \boldsymbol{g}_{T}(\boldsymbol{\theta}) =  \frac{1}{T}
   \sum_{t=0}^{T}
   \boldsymbol{h}(\boldsymbol{\theta},\boldsymbol{y}_{t}).
   \end{gather}

Weighting Matrix for Asset Problem
==============================================================================
Since the forecast errors in
:math:`\smash{\boldsymbol{h}(\boldsymbol{\theta},\boldsymbol{y}_{t})}`
are unpredictable, they exhibit no serial correlation.

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- Thus, :math:`\smash{\boldsymbol{h}(\boldsymbol{\theta},
  \boldsymbol{y}_{t})}` exhibits no serial correlation.

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This means :math:`\smash{S}` can be simply be estimated by

.. math::

   \smash{\hat{S}_{T} =  \frac{1}{T}  \sum_{t=0}^{T}
   \boldsymbol{h}(\hat{\boldsymbol{\theta}}, \boldsymbol{y}_{t})
   \boldsymbol{h}(\hat{\boldsymbol{\theta}},\boldsymbol{y}_{t})^{'}}.

Hansen and Singleton (1982)
==============================================================================
Hansen and Singleton (1982) used GMM to estimate parameters of a model
where

.. math::

   \smash{u(c_{t}) = \begin{cases} \frac{c_{t}^{1-\gamma}}{1-\gamma} &
   \text{for} \,\,\, \gamma > 0 \,\,\, \text{and} \,\,\, \gamma \neq 1
   \\ log(c_{t}) & \text{for} \,\,\, \gamma = 1 \\ \end{cases}.}

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- In this case, :math:`\smash{\boldsymbol{\theta} =
  (\beta,\gamma)^{'}}`.

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- Since forecast errors are uncorrelated with past returns and
  consumption, the lagged values of asset returns and aggregate
  consumption in :math:`\smash{\boldsymbol{x}_{t}}`.