GMM assigns a quadratic cost to violations of average moments. Generalized empirical likelihood instead introduces a multiplier vector that tilts the empirical distribution. For moments g_i(\theta) and a concave generating function \rho, the profiled criterion can be written
Trex implements empirical likelihood \rho(v)=\log(1-v), exponential tilting \rho(v)=1-e^v, and a quadratic/CUE-style form \rho(v)=-v-v^2/2. Additive constants do not affect the minimizer. The inner optimization, its domain and the outer parameter fit are distinct numerical problems.
Under valid, well-identified moments, these methods share first-order efficient GMM behavior. Their finite-sample and misspecification properties can differ. Changing the tilt is not a remedy for a false identifying restriction.
An overidentified mean
Suppose both E[Y-\theta]=0 and E[(Y-\theta)^2-1]=0. The second restriction assumes a known variance of one, so there are two moments for one parameter. In this simulation that assumption is true by construction.
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import numpy as npimport pandas as pdfrom trex.gmm import GELEstimator, rho_el, rho_exponential, rho_cuerng = np.random.default_rng(817)y = rng.normal(loc=.7, scale=1, size=500)def moments(data, theta): r = data - theta[0]return np.column_stack([r, r*r-1])rows = []for name, rho in [("EL", rho_el), ("ET", rho_exponential), ("CUE-style", rho_cue)]: fit = GELEstimator(moments, rho=rho).fit(y, startval=np.array([y.mean()])) rows.append({"method": name, "estimate": fit.est[0], "SE": fit.se[0],"J statistic": fit.J_stat, "J p-value": fit.J_pvalue})pd.DataFrame(rows).round(5)
method
estimate
SE
J statistic
J p-value
0
EL
0.70776
0.04641
1.69012
0.19359
1
ET
0.70760
0.04641
1.59936
0.20599
2
CUE-style
0.70733
0.04641
1.50600
0.21975
The code uses NumPy/SciPy, not GPU optimization. The differences in the resulting estimates come from empirical weighting, not different input samples. A large overidentification p-value does not prove both restrictions true or rule out weak identification.
Domain and covariance
For empirical likelihood, each \lambda'g_i must stay below one. Clipping arguments into this domain would change the optimization problem; Trex instead uses a feasibility-preserving line search. Exponential tilting can be numerically difficult for extreme moments. Failure to find a finite feasible optimum is reported rather than replaced by a fabricated estimate.
Let G be the Jacobian of average moments and \Omega=E[g_i g_i']. The implemented correctly specified first-order covariance is
It has parameter dimension p\times p, not moment dimension q\times q. Trex stores it as Sigma, with se its marginal standard errors. est holds coefficients and lam_hat the final multiplier vector. This calculation is not misspecification-robust GEL inference, nor necessarily the same finite-sample reweighted covariance returned by another package.
In just-identified problems, a root of the sample moments often makes the methods coincide; overidentification creates scope for their empirical weighting to matter. In applied work, compare moment definitions and normalization before attributing a discrepancy to an optimizer.