ECMT6002 Chap.10 Random Effects and Maximum Likelihood
Random Effects and Maximum Likelihood
Random effects models the unit component as part of a structured composite disturbance and requires it to be uncorrelated with the regressors. Generalised least squares uses partial demeaning rather than removing all between unit information. The chapter then develops maximum likelihood as a general principle: specify a density, form the joint likelihood or log likelihood, differentiate and assess curvature.
Both topics reward close attention to maintained distributional and independence assumptions. In the composite panel error, every observation for a unit shares the same random component while idiosyncratic shocks vary over time. The diagonal covariance therefore contains both variance components and each within-unit off-diagonal element contains only the shared component.
GLS uses this covariance to transform the model and exploit both within and between variation. Feasible GLS first estimates the unknown variance components, so its efficiency claim is tied to correct covariance and orthogonality assumptions. If the unit effect correlates with regressors, random effects is inconsistent even when its standard errors are small. Maximum likelihood begins from a fully specified outcome density.
Independence converts the sample likelihood into a product, and logarithms convert that product into a sum without moving the maximum. The score identifies stationary candidates, while curvature and boundaries determine whether a candidate is a maximum. Under regularity conditions, information also describes asymptotic precision.
A likelihood value is not the probability that a parameter is true; it is the observed-data objective evaluated at that parameter.
What this chapter covers
- 01
Random effects assumptions
- 02
Composite error covariance
- 03
GLS and partial demeaning
- 04
Fixed versus random effects
- 05
Likelihood and log likelihood
- 06
Score, information and curvature
Worked example · free
Maximising a Bernoulli likelihood
- +1Write the likelihood as p cubed times (1 minus p).
- +1Take logs to obtain 3 log p plus log(1 minus p).
- +1Set the score 3 over p minus 1 over (1 minus p) equal to zero.
- +1Solve to obtain p hat = 3 divided by 4 = 0.75, with negative curvature at the solution.
Key terms
- Random effects
- A panel model treating the unit component as random and uncorrelated with included regressors.
- Generalised least squares
- Least squares after a transformation based on a nonidentity disturbance covariance matrix.
- Likelihood
- The joint density of the observed sample viewed as a function of unknown parameters.
- Score
- The first derivative of the log likelihood with respect to a parameter.
Random Effects and Maximum Likelihood FAQ
What is the key random effects restriction?
The time invariant unit effect must be uncorrelated with the regressors under the maintained model. If that restriction fails, random effects combines contaminated between variation with within variation and becomes inconsistent.
Why take logarithms of the likelihood?
Products of densities become sums, which are easier to differentiate and more stable numerically. Because the logarithm is strictly increasing, it does not change the parameter value that maximises the objective.
Does a zero score guarantee a maximum?
No. It identifies a stationary point when differentiability conditions hold. Curvature, boundary cases and a comparison of objective values are needed to determine whether the point is the relevant maximum.
Why can random effects be more efficient?
Under orthogonality between unit effects and regressors, random effects retains credible between-unit information and uses the composite-error covariance in a GLS transformation. Fixed effects discards that between variation. The efficiency advantage disappears as a justification if the orthogonality restriction fails, because a precise estimator centred on the wrong target is not preferable.
Exam move
For panel model choice, write the correlation assumption beside the source of identifying variation. For likelihood problems, follow one fixed chain: density, likelihood, log likelihood, score, solution and curvature. Practise with Bernoulli and normal examples until each line is automatic, then explain which distributional assumptions give the objective its interpretation.
Draw the random-effects covariance matrix for three dates and label why the diagonal and off-diagonal entries differ. Convert the variance components into a within-unit correlation and predict how the GLS transformation changes as the shared component grows. Contrast full demeaning in fixed effects with partial demeaning in random effects, linking the difference to the orthogonality assumption.
For likelihood, write the single-observation contribution before multiplying or summing across the sample. Differentiate the Bernoulli and normal mean log likelihoods and verify the stationary solution against intuition. Check the second derivative or objective values rather than stopping when the score is zero. Separate the likelihood function from a sampling probability statement and from a Bayesian posterior.
When software reports a numerical optimum, inspect convergence, parameter boundaries and identification. Close each derivation with the maintained density assumptions and an interpretation of what information supplies the estimate.
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