How does method of moments estimation compare to MLE?

Question

AcadiFi · Accepted Answer

Method of moments (MoM) equates sample moments to population moments and solves for parameters. If your model has k parameters, you match k moments.

Jasmin Okafor at Accra Fixed Income estimates a two-parameter gamma distribution for loss severities. Gamma has mean = alpha/beta and variance = alpha/beta^2. Sample mean = 4.72, sample variance = 12.85.

Solving:
- alpha/beta = 4.72
- alpha/beta^2 = 12.85
- Dividing: beta = 4.72/12.85 = 0.367
- Then alpha = 4.72 * 0.367 = 1.732

No optimizer needed. Contrast with MLE which requires numerical maximization of a gamma log-likelihood over (alpha, beta).

MoM vs MLE tradeoffs:

| Property | MoM | MLE |
|---|---|---|
| Computation | Simple, closed-form | Numerical, may fail |
| Efficiency | Lower (higher variance) | Asymptotically efficient |
| Requires density | No | Yes |
| Robust to misspec | More | Less |
| Small sample | Unbiased often | Biased but lower variance |

When MoM wins:
1. Density is hard to specify but moments are known
2. Need quick estimates for starting values in MLE
3. Distribution has heavy tails where MLE is unstable
4. Teaching or quick back-of-envelope work

When MLE wins:
1. Full distribution is needed for VaR / expected shortfall
2. Sample is small and you need efficiency
3. Model has many parameters

Generalized Method of Moments (GMM) extends MoM to handle MORE moments than parameters, weighting them optimally. GMM is the workhorse for macroeconomic and asset pricing models where likelihood is intractable.

How does method of moments estimation compare to MLE?

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