Continuous-Updating GMM

Instead of taking the weight matrix as given when evaluating the criterion function, a contiuous-updating (also known as continuously-updated, or CU) GMM estimator updates the weight matrix simultaneously when the parameter vector $\theta$ is altered. Specifically, the criterion function

\[Q(\theta) = \left[\frac{1}{N}\sum_{i=1}^N \mathbf{g}_i(\theta)\right]'\mathbf{W}(\theta)\left[\frac{1}{N}\sum_{i=1}^N \mathbf{g}_i(\theta)\right]\]

now takes $\mathbf{W}$ as a function of $\theta$, which typically is the inverse of the variance-covariance estimator. Starting from a guess for the initial value of $\theta$, the CUGMM estimator solves an optimization problem with $Q(\theta)$ as the objective function.

Implementations for both nonlinear and linear moment conditions are provided via estimator types CUGMM and LinearCUGMM. Clearly, the CUGMM estimator is relevant only when the number of moment conditions exceeds the number of parameters.

Example: IV Estimation with CUGMM

We revisit the GMM IV example with CUGMM:

vce = ClusterVCE(data, :idcode, 6, 8)
eq = (:ln_wage, (:tenure=>[:union, :wks_work, :msp], :age, :age2, :birth_yr, :grade))
r = fit(LinearCUGMM, Hybrid, vce, data, eq)

NonlinearGMM with 8 moments and 6 parameters over 18625 observations:
  Continuously updated GMM estimator:
    Jstat = 12.68        Pr(>J) = 0.0018
  Cluster-robust covariance estimator: idcode
────────────────────────────────────────────────────────────────────────────────
              Estimate   Std. Error      z  Pr(>|z|)     Lower 95%     Upper 95%
────────────────────────────────────────────────────────────────────────────────
tenure     0.100188     0.00379273   26.42    <1e-99   0.0927543     0.107621
age        0.0169596    0.00672394    2.52    0.0117   0.00378087    0.0301382
age2      -0.000519433  0.000111552  -4.66    <1e-05  -0.000738072  -0.000300795
birth_yr  -0.00867626   0.00220182   -3.94    <1e-04  -0.0129917    -0.00436078
grade      0.0714924    0.00300753   23.77    <1e-99   0.0655977     0.0773871
cons       0.864279     0.162432      5.32    <1e-06   0.545918      1.18264
────────────────────────────────────────────────────────────────────────────────

Above, we have used the Hybrid solver from NonlinearSystems.jl. The moment conditions are specified in the same way as before for Linear GMM.

Nonlinear CUGMM estimation is supported with a syntax similar to that for IteratedGMM via CUGMM.