# Model Selection for Explosive Models

## TL;DR

This paper analyzes the effectiveness of information criteria like AIC, BIC, and HQIC in correctly identifying explosive versus unit root models, revealing their strengths and limitations across different scenarios.

## Contribution

It provides a comprehensive theoretical and simulation analysis of how various information criteria perform in model selection for explosive and unit root models, considering different initial conditions and estimators.

## Key findings

- BIC and HQIC consistently select the unit root model when appropriate.
- AIC has a positive probability of correctly identifying the local-to-unit-root model.
- All criteria reliably select the true model for explosive models with certain parameters.

## Abstract

This paper examines the limit properties of information criteria (such as AIC, BIC, HQIC) for distinguishing between the unit root model and the various kinds of explosive models. The explosive models include the local-to-unit-root model, the mildly explosive model and the regular explosive model. Initial conditions with different order of magnitude are considered. Both the OLS estimator and the indirect inference estimator are studied. It is found that BIC and HQIC, but not AIC, consistently select the unit root model when data come from the unit root model. When data come from the local-to-unit-root model, both BIC and HQIC select the wrong model with probability approaching 1 while AIC has a positive probability of selecting the right model in the limit. When data come from the regular explosive model or from the mildly explosive model in the form of $1+n^{\alpha }/n$ with $\alpha \in (0,1)$, all three information criteria consistently select the true model. Indirect inference estimation can increase or decrease the probability for information criteria to select the right model asymptotically relative to OLS, depending on the information criteria and the true model. Simulation results confirm our asymptotic results in finite sample.