How does maximum likelihood estimation work in finance?

Question

AcadiFi · Accepted Answer

MLE finds parameters that maximize the probability of observing your sample. For a model f(x|theta), the likelihood is L(theta) = product of f(x_i|theta); log-likelihood logL = sum of log f(x_i|theta).

Celestin Abioye at Lagos Volatility Research fits a GARCH(1,1) to 2,600 daily returns of the Nigerian stock index. The model: r_t = mu + epsilon_t, epsilon_t = sigma_t * z_t where z_t ~ N(0,1), and sigma_t^2 = omega + alpha * epsilon_{t-1}^2 + beta * sigma_{t-1}^2.

The log-likelihood (assuming normal innovations):
logL = -0.5 * sum(log(2*pi*sigma_t^2) + (r_t - mu)^2 / sigma_t^2)

Numerical optimization (BFGS, Nelder-Mead) searches over (mu, omega, alpha, beta) maximizing logL. Celestin's fit produces mu=0.0003, omega=0.000002, alpha=0.082, beta=0.896, with persistence alpha+beta=0.978.

```mermaid
flowchart TD
  A[Sample Data] --> B[Specify Density f]
  B --> C[Build Log-Likelihood]
  C --> D[Numerical Optimizer]
  D --> E[Parameter Estimates]
  E --> F[Standard Errors via Hessian]
```

MLE properties under regularity:
1. Consistent — converges to true theta as n grows
2. Asymptotically normal — sqrt(n) * (theta_hat - theta) ~ N(0, I^(-1)) where I is Fisher information
3. Asymptotically efficient — lowest variance among consistent estimators

Pitfalls: local maxima (use multiple starting values), misspecified density (quasi-MLE gives consistent estimates but different standard errors via sandwich formula), and flat likelihoods with weakly identified parameters. For GARCH, always verify alpha+beta<1 for stationarity and check alpha>0 for well-defined volatility.

How does maximum likelihood estimation work in finance?

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