ECMT6002 Chap.8 Specification Tests and Prediction
Specification Tests and Prediction
A regression can be estimated perfectly and still answer the wrong question because its functional form, regressor set or disturbance assumptions are misspecified. This chapter organises specification checks as targeted alternatives rather than a ritual battery. It covers added term tests, information criteria, in sample versus out of sample comparison, point prediction and prediction intervals.
The core distinction is between uncertainty about the conditional mean and uncertainty about a new observation. A functional-form test augments the maintained equation with nonlinear terms and asks whether they add systematic explanatory content. Rejection is evidence against the current conditional mean within that alternative family; it does not name the correct replacement.
Theory and data support must guide whether logs, powers, interactions or omitted controls are sensible next candidates. Nested tests compare restrictions inside a larger model, while information criteria trade improved likelihood against a complexity penalty. Predictive comparison should hold the sample and target fixed and use a loss function suited to the application.
Point predictions insert covariates into the estimated conditional mean. A confidence interval around that mean reflects parameter uncertainty, whereas a prediction interval for a new outcome also includes the future disturbance and is therefore wider. Extrapolation adds another risk because functional forms that behave reasonably within observed support can imply implausible curves outside it.
Model assessment should combine diagnostic evidence, interpretability, economic structure and performance rather than treating one test or score as a mechanical winner.
What this chapter covers
- 01
Functional form diagnostics
- 02
Added powers and RESET logic
- 03
Nested and nonnested comparison
- 04
Information criteria
- 05
Point prediction
- 06
Confidence and prediction intervals
Worked example · free
Building a prediction interval
- +1Compute the margin as 1.96 times 2, which equals 3.92.
- +1Subtract the margin from 12 to obtain 8.08.
- +1Add the margin to 12 to obtain 15.92.
Key terms
- Specification error
- A mismatch between the estimated model and the relevant conditional relationship or data process.
- RESET
- An added term test using functions of fitted values to detect certain functional form omissions.
- Information criterion
- A model comparison score balancing fit against a complexity penalty.
- Prediction interval
- An interval designed to cover a future outcome, including irreducible disturbance variation.
Specification Tests and Prediction FAQ
What does rejection of RESET prove?
It indicates that the maintained conditional mean is inadequate against the added function alternatives. It does not identify a unique missing variable or guarantee that one particular nonlinear replacement is correct.
Why is a prediction interval wider than a mean interval?
A mean interval reflects uncertainty in estimating the conditional mean. A prediction interval must additionally allow for the new observation's disturbance, so it contains an extra source of variation.
Can information criteria compare any models?
They are most informative when models use the same response data and comparable likelihood conventions. The direction of preference and penalty definition must be checked for the specific criterion rather than assumed.
What should I do after rejecting a functional form?
Treat rejection as a prompt to redesign the conditional mean, not as identification of one automatic replacement. Inspect residual patterns, theory and data support, then estimate a plausible log, polynomial, interaction or added-variable model on the same sample. Recheck diagnostics and explain the revised marginal effects in terms the research question uses.
Exam move
Create a diagnostic flow from the suspected failure to an explicit alternative and test statistic. When predicting, label whether the target is a conditional mean or a new realised observation before selecting a standard error. Compare models on a common sample, and explain why the chosen loss or information criterion matches the task.
Practise a RESET conclusion in three layers: the null and added terms, the statistical decision, and the limited implication for the maintained form. Follow rejection with two plausible redesigns grounded in the economic relationship, then explain how their marginal effects would be interpreted. For quadratic models, differentiate the entire fitted function and calculate the turning point before describing direction.
For log models, translate coefficients back into percentage language. Build parallel interval calculations for an estimated mean and a new outcome so the extra disturbance variance is explicit. Before forecasting, mark whether the requested covariate point lies inside the sample support. Recompute an information criterion after adding a regressor and identify how much fit improvement is needed to offset its penalty.
Use held-out errors for prediction claims and preserve an untouched evaluation sample. Finish each model comparison with the criterion, the winning model under that criterion, and one reason why a different research objective might rationally prefer another specification.
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