# Generalized Empirical Likelihood-Based Focused Information Criterion and Model Averaging

## Abstract

**:**

## 1. Introduction

## 2. Local Misspecification Framework

## 3. Focused Information Criterion

**Assumption 3.1**

- 1.
- $\Theta \subset {\mathbb{R}}^{p}$, $\Gamma \subset {\mathbb{R}}^{q}$, and $\mathcal{T}\subset {\mathbb{R}}^{l}$ are compact.
- 2.
- $m(y,\theta ,\gamma )$ is continuous in $\theta \in \Theta $ and $\gamma \in \Gamma $ for almost every y.
- 3.
- ${sup}_{\theta \in \Theta ,\gamma \in \Gamma ,\tau \in \mathcal{T}}\left|{\widehat{Q}}_{n}(\theta ,\gamma ,\tau )-{Q}_{n}(\theta ,\gamma ,\tau )\right|\stackrel{p}{\to}0$ under the sequence of ${f}_{n}\left(y\right)$.
- 4.
- $\left|{Q}_{n}(\theta ,\gamma ,\tau )-Q(\theta ,\gamma ,\tau )\right|\to 0$ as $n\to \infty $ for all $\theta \in \Theta $, $\gamma \in \Gamma $, and $\tau \in \mathcal{T}$.
- 5.
- $E\left[m({y}_{i},\theta ,\gamma )m{({y}_{i},\theta ,\gamma )}^{\prime}\right]$ is nonsingular for all $\theta \in \Theta $ and $\gamma \in \Gamma $.
- 6.
- $({\theta}_{0},{\gamma}_{0})$ is the unique solution to $E\left[m({y}_{i},\theta ,\gamma )\right]=0$ and $({\theta}_{0},{\gamma}_{0})\in \mathit{int}(\Theta \times \Gamma )$.
- 7.
- $\rho \left(v\right)$ is twice continuously differentiable in a neighborhood of zero.
- 8.
- $E\left[{m}_{\theta i}\right]$ and $E\left[{m}_{\gamma i}\right]$ are of full rank.
- 9.
- ${sup}_{n}{E}_{n}[\parallel {m}_{i}{\parallel}^{2+\alpha}]<\infty $ for some $\alpha >0$.
- 10.
- $m(y,\theta ,\gamma )$ is continuously differentiable in $\theta $ and $\gamma $ in a neighborhood, $\mathcal{N}$, of $({\theta}_{0},{\gamma}_{0})$.
- 11.
- ${sup}_{\theta ,\gamma \in \mathcal{N}}\left|{n}^{-1}{\sum}_{i=1}^{n}\frac{\partial m({y}_{i},\theta ,\gamma )}{\partial {\theta}^{\prime}}-{E}_{n}\left[\frac{\partial m({y}_{i},\theta ,\gamma )}{\partial {\theta}^{\prime}}\right]\right|\stackrel{p}{\to}0$ and ${sup}_{\theta ,\gamma \in \mathcal{N}}\left|{n}^{-1}{\sum}_{i=1}^{n}\frac{\partial m({y}_{i},\theta ,\gamma )}{\partial {\gamma}^{\prime}}-{E}_{n}\left[\frac{\partial m({y}_{i},\theta ,\gamma )}{\partial {\gamma}^{\prime}}\right]\right|\stackrel{p}{\to}0$ under the sequence of ${f}_{n}\left(y\right)$.
- 12.
- $\parallel {E}_{n}\left[{m}_{\theta i}\right]-E\left[{m}_{\theta i}\right]\parallel \to 0$ and $\parallel {E}_{n}\left[{m}_{\gamma i}\right]-E\left[{m}_{\gamma i}\right]\parallel \to 0$ as $n\to \infty $.
- 13.
- $\parallel {E}_{n}\left[{m}_{i}{m}_{i}^{\prime}\right]-E\left[{m}_{i}{m}_{i}^{\prime}\right]\parallel \to 0$ as $n\to \infty $.

**Lemma 3.1**Suppose Assumption 3.1 holds. Then, under the sequence of ${f}_{n}\left(y\right)$, we have:

**Theorem 3.1**Suppose Assumption 3.1 holds. Then, under the sequence of ${f}_{n}\left(y\right)$, we have:

## 4. Model Averaging

## 5. Example

## 6. Monte Carlo Study

**Table 1.**Estimation results; DGP, data generating process; AIC, Akaike information criterion; BIC, Bayesian information criterion; FIC, focused information criterion.

DGP | |||||

(1) | (2) | (3) | (4) | ||

Full | Bias | -0.104 | -0.109 | - 0.089 | - 0.076 |

Std | 0.544 | 0.533 | 0.509 | 0.489 | |

RMSE | 0.554 | 0.544 | 0.516 | 0.495 | |

Reduced | Bis | -0.279 | -0.057 | -0.148 | -0.048 |

Std | 0.780 | 0.473 | 0.955 | 0.448 | |

RMSE | 0.828 | 0.477 | 0.965 | 0.450 | |

AIC | Bias | -0.113 | -0.099 | -0.101 | -0.079 |

Std | 0.559 | 0.557 | 0.497 | 0.509 | |

RMSE | 0.570 | 0.566 | 0.507 | 0.515 | |

BIC | Bias | -0.136 | -0.088 | -0.104 | -0.073 |

Std | 0.689 | 0.552 | 0.499 | 0.502 | |

RMSE | 0.702 | 0.559 | 0.510 | 0.507 | |

FIC | Bias | -0.139 | -0.095 | -0.112 | -0.076 |

Std | 0.530 | 0.509 | 0.464 | 0.452 | |

RMSE | 0.548 | 0.517 | 0.477 | 0.458 | |

Averaging | Bias | -0.139 | -0.092 | -0.107 | -0.074 |

Std | 0.511 | 0.476 | 0.455 | 0.444 | |

RMSE | 0.529 | 0.484 | 0.468 | 0.450 |

## 7. Conclusions

## Acknowledgments

## References

- R.J. Smith. “Alternative Semi-Parametric Likelihood Approaches to Generalised Method of Moments Estimation.” Econ. J. 107 (1997): 503–519. [Google Scholar]
- W.K. Newey, and R.J. Smith. “Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators.” Econometrica 72 (2004): 219–255. [Google Scholar]
- A.B. Owen. “Empirical Likelihood Ratio Confidence Intervals for a Single Functional.” Biometrika 75 (1988): 237–249. [Google Scholar]
- J. Qin, and J. Lawless. “Empirical Likelihood and General Estimating Equations.” Ann. Stat. 22 (1994): 300–325. [Google Scholar]
- Y. Kitamura, and M. Stutzer. “An Information-Theoretic Alternative to Generalized Method of Moments Estimation.” Econometrica 65 (1997): 861–874. [Google Scholar]
- G.W. Imbens, R.H. Spady, and P. Johnson. “Information Theoretic Approaches to Inference in Moment Condition Models.” Econometrica 66 (1998): 333–357. [Google Scholar]
- G. Claeskens, and N.L. Hjort. “The Focused Information Criterion.” J. Am. Stat. Assoc. 98 (2003): 900–916. [Google Scholar]
- H. Akaike. “Information Theory and an Extension of the Maximum Likelihood Principle.” In Second International Symposium on Information Theory. Edited by B. Petroc and F. Csake. 1973, pp. 267–281. Akademiai Kiado. [Google Scholar]
- G. Schwarz. “Estimating the Dimension of a Model.” Ann. Stat. 6 (1978): 461–464. [Google Scholar]
- B.E. Hansen. “Challenges for Econometric Model Selection.” Economet. Theor. 21 (2005): 60–68. [Google Scholar]
- G. Claeskens, C. Croux, and J.V. Kerckhoven. “Variable Selection for Logistic Regression Using a Prediction-Focused Information Criterion.” Biometrics 62 (2006): 972–979. [Google Scholar]
- N.L. Hjort, and G. Claeskens. “Focused Information Criteria and Model Averaging for the Cox Hazard Regression Model.” J. Am. Stat. Assoc. 101 (2006): 1449–1464. [Google Scholar]
- X. Zhang, and H. Liang. “Focused Information Criterion and Model Averaging for Generalized Additive Partial Linear Models.” Ann. Stat. 39 (2011): 174–200. [Google Scholar]
- D.W. Andrews, and B. Lu. “Consistent Model and Moment Selection Procedures for GMM Estimation with Application to Dynamic Panel Data Models.” J. Econometrics 101 (2001): 123–164. [Google Scholar]
- L.P. Hansen. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica 50 (1982): 1029–1054. [Google Scholar]
- H. Hong, B. Preston, and M. Shum. “Generalized Empirical Likelihood-Based Model Selection Criteria for Moment Condition Models.” Economet. Theor. 19 (2003): 923–943. [Google Scholar]
- N. Sueishi. “Information Criteria for Moment Restriction Models.” Unpublished Manuscript. Kyoto University, 2013. [Google Scholar]
- J.A. Hoeting, D. Madigan, A.E. Raftery, and C.T. Volinsky. “Bayesian Model Averaging: A Tutorial.” Stat. Sci. 14 (1999): 382–417. [Google Scholar]
- S.T. Buckland, K.P. Burnham, and N.H. Augustin. “Model Selection: An Integral Part of Inference.” Biometrics 53 (1997): 603–618. [Google Scholar]
- K.P. Burnham, and D.R. Anderson. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer, 2002. [Google Scholar]
- N.L. Hjort, and G. Claeskens. “Frequentist Model Average Estimators.” J. Am. Stat. Assoc. 98 (2003): 879–899. [Google Scholar]
- Y. Yang. “Adaptive Regression by Mixing.” J. Am. Stat. Assoc. 96 (2001): 574–588. [Google Scholar]
- G. Leung, and A.R. Barron. “Information Theory and Mixing Least-Squares Regressions.” IEEE T. Inform. Theory 52 (2006): 3396–3410. [Google Scholar]
- A. Goldenshluger. “A Universal Procedure for Aggregating Estimators.” Ann. Stat. 37 (2009): 542–568. [Google Scholar]
- B.E. Hansen. “Least Squares Model Averaging.” Econometrica 75 (2007): 1175–1189. [Google Scholar]
- A.T.K. Wan, X. Zhang, and G. Zou. “Least Squares Model Averaging by Mallows Criterion.” J. Econometrics 156 (2010): 277–283. [Google Scholar]
- B.E. Hansen, and J.S. Racine. “Jackknife Model Averaging.” J. Econometrics 167 (2012): 38–46. [Google Scholar]
- Q. Liu, and R. Okui. “Heteroskedasticity-Robust C
_{p}Model Averaging.” Economet. J., 2013. Forthcoming. [Google Scholar] - F.J. DiTraglia. “Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM.” Unpublished Manuscript. University of Pennsylvania, 2012. [Google Scholar]
- C.A. Liu. “A Plug-In Averaging Estimator for Regressions with Heteroskedastic Errors.” Unpublished Manuscript. National University of Singapore, 2012. [Google Scholar] [Green Version]
- L.F. Martins, and V.J. Gabriel. “Linear Instrumental Variables Model Averaging Estimation.” Comput. Stat. Data An., 2013. Forthcoming. [Google Scholar]
- W.K. Newey. “Generalized Method of Moments Specification Testing.” J. Econometrics 29 (1985): 229–256. [Google Scholar]
- A.R. Hall. “Hypothesis Testing in Models Estimated by Generalized Method of Moments.” In Generalized Method of Moments Estimation. Edited by L. Mátyás. Cambridge University Press, 1999, pp. 75–101. [Google Scholar]
- P.M. Parente, and R.J. Smith. “GEL Methods for Nonsmooth Moment Indicators.” Economet. Theor. 27 (2011): 74–113. [Google Scholar]
- G. Claeskens, and N.L. Hjort. Model Selection and Model Averaging. Cambridge University Press, 2008. [Google Scholar]
- G. Claeskens, and N.L. Hjort. “Minimizing Average Risk in Regression Models.” Economet. Theor. 24 (2008): 493–527. [Google Scholar]

## A. Appendix

^{1.}Although ${y}_{1},\cdots ,{y}_{n}$ is a triangular array, we suppress the additional subscript, n, on y for notational simplicity.^{2.}Simulations were also conducted for different sample sizes. The results are not reported here, because the difference among candidate models is so small for large n that RMSEs are the almost identical for all models.

© 2013 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Sueishi, N.
Generalized Empirical Likelihood-Based Focused Information Criterion and Model Averaging. *Econometrics* **2013**, *1*, 141-156.
https://doi.org/10.3390/econometrics1020141

**AMA Style**

Sueishi N.
Generalized Empirical Likelihood-Based Focused Information Criterion and Model Averaging. *Econometrics*. 2013; 1(2):141-156.
https://doi.org/10.3390/econometrics1020141

**Chicago/Turabian Style**

Sueishi, Naoya.
2013. "Generalized Empirical Likelihood-Based Focused Information Criterion and Model Averaging" *Econometrics* 1, no. 2: 141-156.
https://doi.org/10.3390/econometrics1020141