PhD Econometrics Tutors in New York

Most estimators you met in first year — OLS, IV, maximum likelihood — are the same estimator in different clothes, and the generalised method of moments is what undresses them. You write down what the economics implies about a handful of averages, then pick the parameter that comes closest to making those averages hold. That one idea, due to Hansen (1982), is the spine of modern structural and time-series econometrics — and the topic graduate students bring to our PhD econometrics tutors most. Here it is, from the moment condition up.

1 · Moment conditions: population to sample

A moment condition says some function of the data and parameters has mean zero at the truth:

   

with wᵢ the data on unit i, θ a p-vector, and g returning a q-vector. The economics lives in g — an exogenous instrument z, for instance, gives E[z · u] = 0. You never see the expectation, so you use its sample analogue,

   

Count moments q against parameters p. With q < p the model is under-identified — nothing pins θ down. With q = p it is exactly identified: solve ḡ(θ) = 0 and you hit every moment. With q > p it is over-identified, and that is where GMM earns its keep: no θ zeroes all the moments, so they disagree and you must compromise.

2 · The GMM criterion

Since you cannot solve ḡ(θ) = 0 when over-identified, you get as close as possible. Define the criterion

   

a quadratic form with a symmetric positive-definite weighting matrix W, and set θ̂ = arg min Q(θ). Q is the squared length of ḡ in the metric W, so minimising it drives the moments toward zero — but W decides which moments win when they conflict. Any positive-definite W is consistent; different W give different estimates from the same data. So W is not cosmetic — it fixes both the compromise and the precision.

3 · The optimal weighting matrix

Which W is best? “Best” means smallest asymptotic variance, and the answer is clean — the inverse of the moments’ own covariance:

   

This is the logic of GLS: S⁻¹ divides each moment by its variance, so noisy moments are downweighted and precise ones dominate, and correlated moments are decorrelated too. One catch: S depends on θ. Two-step GMM breaks the loop — estimate with W = I, form Ŝ, re-estimate with W = Ŝ⁻¹; iterating to convergence gives iterated GMM, and both are efficient. When the moments are serially correlated, S needs a heteroskedasticity- and autocorrelation-consistent estimator — its own page.

4 · Asymptotic variance and the Hansen J-test

The estimator is √n-consistent and asymptotically normal. With G = E[∂g/∂θ′] the expected Jacobian, the general sandwich variance is

   

Set W = S⁻¹ and it collapses to (G′S⁻¹G)⁻¹ — less algebra and less variance at once. Over-identification also leaves a testable residual: valid moments cannot all be forced to zero, but they should sit close, no further than noise allows. The Hansen J-statistic measures exactly that:

   

Its degrees of freedom are the over-identifying restrictions, q − p. A large J means the moments disagree by more than noise — at least one is misspecified; exactly-identified models give J = 0, with nothing to test. Field exams test this harder than any other GMM fact.

5 · GMM as the unifying framework

The estimators you already know are special cases of GMM. Let g = x(y − x′β), regressors orthogonal to the error: exactly identified, and GMM returns OLS. Let g = z(y − x′β) with more instruments than regressors and weight efficiently: under homoskedasticity you recover 2SLS. Let g be the score, E[∂ℓ/∂θ] = 0: GMM reproduces maximum likelihood. Same framework, three moment functions — learn it once.

Worked example — two moments for one parameter

You want one parameter θ, the mean of an economic quantity. Two valid moment conditions each say a sample average equals θ, so ḡ(θ) = (m₁ − θ, m₂ − θ)′ — over-identified, two moments for one parameter.

Step 1 — The data. Moment 1 averages m₁ = 10 and is precise, variance σ₁² = 30. Moment 2 averages m₂ = 12 but is noisy, σ₂² = 90. The sample size is n = 90; the moments are uncorrelated, so S = diag(30, 90). Alone they would estimate θ as 10 and as 12 — they conflict.

Step 2 — First step, W = I. Minimise (10 − θ)² + (12 − θ)². The first-order condition gives the simple average, θ̂₍I₎ = 11 — equal weighting trusts the noisy moment as much as the precise.

Step 3 — Efficient step, W = S⁻¹. Minimise (10 − θ)²/30 + (12 − θ)²/90. The weights are 1/30 and 1/90 — three to one. The minimiser is the precision-weighted average, θ̂₍W₎ = (3·10 + 12)/4 = 10.5, pulled toward the precise moment. S⁻¹ has stripped the noisy moment of three-quarters of its say.

Step 4 — The J-test. At θ̂₍W₎ the moments are ḡ = (−0.5, 1.5), so the efficient criterion is 0.25/30 + 2.25/90 = 1/30 and J = 90 · (1/30) = 3.0. The χ²(1) 5% critical value is 3.84. Since 3.0 < 3.84 you do not reject — the gap is within sampling noise at this n.

Step 5 — Perturbation. Suppose moment 2 is measured more crudely, its variance tripling to 270. The weight ratio becomes nine to one, so θ̂₍W₎ = (9·10 + 12)/10 = 10.2 — further toward the precise moment — and J = 90 · (4/300) = 1.2.

Step 6 — Reading it. Both moved as the weighting logic predicts: a less trusted moment 2 lets θ̂ lean harder on moment 1, and the same raw gap of 2 turns less significant once one measurement is known to be noisier. Efficiency is concrete here — the efficient variance (1/30 + 1/90)⁻¹ = 22.5 beats the equally-weighted 30, a standard error of 0.5 against 0.577.

Can you move from the moment condition to the J-test without losing the thread? Deriving why S⁻¹ is the efficient weight — rather than quoting it — is exactly what a qualifying exam rewards. A one-on-one PhD econometrics tutor works GMM from the moment condition through the sandwich variance with you until it is second nature. Book a trial session.

Practice

Q1. Keep only ḡ(θ) = 10 − θ. Solve θ̂, give the minimised criterion and J, and state the degrees of freedom.

Q2. With both moments (σ₁² = 30, σ₂² = 90), a revision moves the second average to m₂ = 9. Recompute θ̂₍I₎ and θ̂₍W₎.

Q3. Take the original data (m₁ = 10, m₂ = 12, variances 30 and 90) but with n = 150. Compute J and test at 5% against χ²(1) = 3.84.

Answers. Q1: exactly identified, so θ̂ = 10 zeroes the single moment, Q(θ̂) = 0 and J = 0 for any weight; q − p = 0, no restriction to test. Q2: θ̂₍I₎ = (10 + 9)/2 = 9.5; θ̂₍W₎ = (3·10 + 9)/4 = 9.75 — with m₂ below m₁, the efficient estimate sits above the average, still leaning toward the precise moment. Q3: Q(θ̂₍W₎) = 1/30 is unchanged, so J = 150 · (1/30) = 5.0 > 3.84 — reject: the discrepancy that passed at n = 90 fails at n = 150, since J scales with n.

Key takeaways

• A moment condition restricts an average, E[g(w, θ₀)] = 0; GMM picks the θ making the sample analogues jointly closest to zero.
• Over-identification is the whole game: with q > p, no θ zeroes every moment, so W in Q = ḡ′Wḡ sets the compromise.
• The optimal weight S⁻¹ downweights noisy and correlated moments and gives the efficient variance (G′S⁻¹G)⁻¹, via two-step or iterated GMM.
• The Hansen J → χ²(q − p) tests the leftover restrictions — zero when exactly identified, large when a moment is misspecified. GMM unifies OLS, IV/2SLS and MLE as special cases.

Why New York PhD students choose our econometrics tutoring

• Comps- and field-exam-ready derivations: the sandwich variance, the efficiency of S⁻¹, and the J-test worked from the moment condition to the χ² result, the way your qualifying exam asks it.
• Matched to your first-year notation: whether your sequence follows Hayashi, Hansen, or your instructor’s own notes, sessions use the identification counting and conventions your problem sets do.
• One-on-one and online: tutors work from your assignments and past qualifying papers over a shared screen, so each session lands on the estimator your exam will test.

FAQ

Q: When should I use GMM instead of maximum likelihood?
A: When you can state moment conditions but do not want to commit to a full distribution. MLE needs the whole likelihood and is efficient if that model is correct; GMM asks only for a few moments, trading efficiency for robustness. With the score as your moments, the two coincide.

Q: What does the Hansen J-test tell you?
A: Whether the over-identifying moments are mutually consistent. A small J means they agree within sampling noise; a large one, against χ²(q − p), means at least one moment is misspecified — though not which.

Q: Why not just set the sample moments to zero?
A: Only exactly-identified models allow it. Over-identified, there are more moment equations than parameters, so no θ solves them all at once; minimising the criterion is how GMM reconciles them.

Q: What is the weighting matrix doing?
A: Setting the metric for “close to zero”. The efficient S⁻¹ divides each moment by its variance and decorrelates them, so precise moments count for more — the GLS intuition.

Q: Does the weighting matrix affect consistency?
A: No. Any positive-definite W is consistent; it changes only efficiency and the finite-sample estimate. S⁻¹ minimises the asymptotic variance.

Work with a PhD econometrics tutor in New York

GMM rewards students who can move between the moment condition, the criterion, and the χ² test without losing the thread — and derive the efficiency result rather than quote it. One-on-one sessions, online, build that fluency on your own problem sets and past qualifying exams. Tell us your programme and your comp date, and we will match you with the right tutor this week.