GMM Over-Identifying Restrictions
Criterion Function Limiting Distribution
Since
\[\begin{split}\begin{gather}
\sqrt{T}\boldsymbol{g}_{T}(\boldsymbol{\theta}_{0})
\stackrel{d}{\longrightarrow} N(0,S) \\
\implies (\sqrt{T}
\boldsymbol{g}_{T}(
\boldsymbol{\theta}_{0})^{'})S^{-1}(\sqrt{T}\boldsymbol{g}_{T}
(\boldsymbol{\theta}_{0})) = T
\boldsymbol{g}_{T}(\boldsymbol{\theta}_{0})^{'}S^{-1}
\boldsymbol{g}_{T}(\boldsymbol{\theta}_{0})
\overset{d}{\longrightarrow} \chi^{2}(r)
\end{gather}\end{split}\]
where \(\smash{r > k}\) is the number of moment conditions.
Estimated Criterion Function
It turns out that
\[\smash{T
\boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})^{'}\hat{S}^{-1}
\boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})
\overset{d}{\not\to} \chi^{2}(r)}.\]
- This is because \(\smash{k}\) moment conditions will be set to
zero exactly.
Exact Identification
Consider \(\smash{r=k}\). In this case
\[\begin{split}\begin{gather}
\boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}}) = 0 \\
T \boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})^{'}\hat{S}^{-1}
\boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}}) = 0.
\end{gather}\end{split}\]
- \(\smash{r-k}\) of the moment conditions will be non-zero.
Over Identification
In general,
\[\smash{J_{T}(\hat{\boldsymbol{\theta}}) = T
\boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})^{'}\hat{S}^{-1}
\boldsymbol{g}_{T}(\hat{\boldsymbol{\theta}})
\overset{d}{\longrightarrow} \chi^{2}(r-k)}.\]
- To test if our moment conditions are close to zero, we compute
\(\smash{J_{T}(\hat{\boldsymbol{\theta}})}\) and compare with a
\(\smash{\chi^{2}(r-k)}\) distribution.
- If \(\smash{J_{T}(\hat{\boldsymbol{\theta}})}\) is far in the
tail of the \(\smash{\chi^{2}(r-k)}\) distribution, we might
conclude that the model is misspecified.
Asset Pricing with GMM
Suppose an agent derives utility from consumption,
\(\smash{c_{t}}\), and seeks to maximze the discounted sum of
expected utility:
\[\smash{ \sum_{\tau = 0}^{\infty}
\beta^{\tau}E[u(c_{t+\tau})|\Omega_{t}]},\]
where \(\smash{u(c_{t})}\) is the period utility function and
satisfies:
\[\smash{ \frac{\partial u(c_{t})}{\partial c_{t}} > 0 \,\,\, \text{and}
\,\,\, \frac{\partial^{2} u(c_{t})}{\partial c_{t}^{2}} < 0}.\]
Equilibrium Conditions
Suppose that the agent can purchase \(\smash{m}\) assets paying
gross returns \(\smash{(1 + r_{i,t+1})}\) between periods
\(\smash{t}\) and \(\smash{t+1}\), for \(\smash{i =
1,\ldots, m}\).
- The agent’s portfolio must satisfy
\[\smash{u^{'}(c_{t}) = \beta E[(1+r_{i,t+1})u^{'}(c_{t+1}) |
\Omega_{t}] \,\,\, \text{for} \,\,\, i=1,\ldots,m}.\]
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.
- If these conditions didn’t hold, the agent wouldn’t be at an
optimum.
The portfolio conditions can be rewritten as:
\[\smash{E\left[\left(\beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
-1\right)\bigg|\Omega_{t}\right] = 0 \,\,\, \text{for}
i=1,\ldots,m}.\]
Equilibrium Conditions
Given a vector \(\smash{\boldsymbol{x}_{t} \in \Omega_{t}}\), by
the law of iterated expectations
\[\begin{split}\begin{align}
E\left[\underset{\boldsymbol{h}(\boldsymbol{\theta},
\boldsymbol{y}_{t})}{\underbrace{\left(\beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
-1\right)\boldsymbol{x}_{t}}} \right] & =
E\left[E\left[\left(\beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
-1\right)\boldsymbol{x}_{t} \bigg|\Omega_{t}\right]\right] \\
& E\left[\underset{0}{\underbrace{E\left[\left(\beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})
-1\right) \bigg|\Omega_{t}\right]}} \boldsymbol{x}_{t} \right] = 0,
\end{align}\end{split}\]
for \(\smash{i=1,\ldots,m}\).
Stochastic Discount Factor
Economic theory says that all returns discounted by
\(\smash{\beta \frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}}\) should be
identical:
\[\begin{split}\begin{gather}
E\left[\underset{m_{t,t+1}}{\underbrace{\beta
\frac{u^{'}(c_{t+1})}{u^{'} (c_{t})}(1+r_{i,t+1})}}\right] = 1 \\
\implies E[m_{t,t+1}(1+r_{i,t+1})] = 1.
\end{gather}\end{split}\]
- \(\smash{\beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1}) - 1}\) is a forecast
error and should be uncorrelated with any variable
\(\smash{\boldsymbol{x}_{t} \in \Omega_{t}}\)
Casting as GMM
This problem maps easily into GMM where
\[\begin{split}\begin{gather}
\boldsymbol{y}_{t} =
(r_{1,+1},\ldots,r_{m,t+1},c_{t},c_{t+1},
\boldsymbol{x}_{t}^{'})^{'} \\
\boldsymbol{h}(\boldsymbol{\theta},\boldsymbol{y}_{t}) =
\left[\begin{array}{c} \left(1 - \beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{i,t+1})\right)\boldsymbol{x}_{t}
\\ \vdots \\ \left(1 - \beta
\frac{u^{'}(c_{t+1})}{u^{'}(c_{t})}(1+r_{m,t+1})\right)\boldsymbol{x}_{t}
\\ \end{array} \right] \\
\boldsymbol{g}_{T}(\boldsymbol{\theta}) = \frac{1}{T}
\sum_{t=0}^{T}
\boldsymbol{h}(\boldsymbol{\theta},\boldsymbol{y}_{t}).
\end{gather}\end{split}\]
Weighting Matrix for Asset Problem
Since the forecast errors in
\(\smash{\boldsymbol{h}(\boldsymbol{\theta},\boldsymbol{y}_{t})}\)
are unpredictable, they exhibit no serial correlation.
- Thus, \(\smash{\boldsymbol{h}(\boldsymbol{\theta},
\boldsymbol{y}_{t})}\) exhibits no serial correlation.
This means \(\smash{S}\) can be simply be estimated by
\[\smash{\hat{S}_{T} = \frac{1}{T} \sum_{t=0}^{T}
\boldsymbol{h}(\hat{\boldsymbol{\theta}}, \boldsymbol{y}_{t})
\boldsymbol{h}(\hat{\boldsymbol{\theta}},\boldsymbol{y}_{t})^{'}}.\]
Hansen and Singleton (1982)
Hansen and Singleton (1982) used GMM to estimate parameters of a model
where
\[\begin{split}\smash{u(c_{t}) = \begin{cases} \frac{c_{t}^{1-\gamma}}{1-\gamma} &
\text{for} \,\,\, \gamma > 0 \,\,\, \text{and} \,\,\, \gamma \neq 1
\\ log(c_{t}) & \text{for} \,\,\, \gamma = 1 \\ \end{cases}.}\end{split}\]
- In this case, \(\smash{\boldsymbol{\theta} =
(\beta,\gamma)^{'}}\).
- Since forecast errors are uncorrelated with past returns and
consumption, the lagged values of asset returns and aggregate
consumption in \(\smash{\boldsymbol{x}_{t}}\).