### Abstract

*L*

_{2}distance between observations from the panel and their model-implied counterparts, augmented by a penalty for the deviation of the extracted factors from the noisy signals of them. When the time dimension is fixed, the limit distribution of the parameter vector is mixed Gaussian with conditional variance depending on the path of the factor realizations. On the other hand, when the time span is large, the convergence rate is faster and the limit distribution is Gaussian with a constant variance. In this case, however, we incur an incidental parameter problem since, at each point in time, we need to recover the concurrent factor realizations. This leads to an asymptotic bias that is absent in the setting with a fixed time span. In either scenario, the limit distribution of the estimates for the factor realizations is mixed Gaussian, but is related to the limiting distribution of the parameter vector only in the scenario with a fixed time horizon. Although the limit behavior is very different for the small versus large time span, we develop a feasible inference theory that applies, without modification, in either case. Hence, the user need not take a stand on the relative size of the time dimension of the panel. Similarly, we propose a time-varying data-driven weighting of the penalty in the objective function, which enhances efficiency by adapting to the relative quality of the signal for the factor realizations.

Original language | English |
---|---|

Journal | Journal of Econometrics |

Volume | 212 |

Issue number | 1 |

Pages (from-to) | 4-25 |

Number of pages | 22 |

ISSN | 0304-4076 |

DOIs | |

Publication status | Published - Sep 2019 |

### Keywords

- Asymptotic bias
- Incidental parameter problem
- Inference
- Large data sets
- Nonlinear factor model
- Options
- Panel data
- Stable convergence
- Stochastic volatility

### Cite this

*Journal of Econometrics*,

*212*(1), 4-25. https://doi.org/10.1016/j.jeconom.2019.04.018

}

*Journal of Econometrics*, vol. 212, no. 1, pp. 4-25. https://doi.org/10.1016/j.jeconom.2019.04.018

**Unified Inference for Nonlinear Factor Models from Panels with Fixed and Large Time Span.** / Andersen, Torben G. ; Fusari, Nicola; Todorov, Viktor; Varneskov, Rasmus T.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Unified Inference for Nonlinear Factor Models from Panels with Fixed and Large Time Span

AU - Andersen, Torben G.

AU - Fusari, Nicola

AU - Todorov, Viktor

AU - Varneskov, Rasmus T.

PY - 2019/9

Y1 - 2019/9

N2 - We provide unifying inference theory for parametric nonlinear factor models based on a panel of noisy observations. The panel has a large cross-section and a time span that may be either small or large. Moreover, we incorporate an additional source of information, provided by noisy observations on some known functions of the factor realizations. The estimation is carried out via penalized least squares, i.e., by minimizing the L2 distance between observations from the panel and their model-implied counterparts, augmented by a penalty for the deviation of the extracted factors from the noisy signals of them. When the time dimension is fixed, the limit distribution of the parameter vector is mixed Gaussian with conditional variance depending on the path of the factor realizations. On the other hand, when the time span is large, the convergence rate is faster and the limit distribution is Gaussian with a constant variance. In this case, however, we incur an incidental parameter problem since, at each point in time, we need to recover the concurrent factor realizations. This leads to an asymptotic bias that is absent in the setting with a fixed time span. In either scenario, the limit distribution of the estimates for the factor realizations is mixed Gaussian, but is related to the limiting distribution of the parameter vector only in the scenario with a fixed time horizon. Although the limit behavior is very different for the small versus large time span, we develop a feasible inference theory that applies, without modification, in either case. Hence, the user need not take a stand on the relative size of the time dimension of the panel. Similarly, we propose a time-varying data-driven weighting of the penalty in the objective function, which enhances efficiency by adapting to the relative quality of the signal for the factor realizations.

AB - We provide unifying inference theory for parametric nonlinear factor models based on a panel of noisy observations. The panel has a large cross-section and a time span that may be either small or large. Moreover, we incorporate an additional source of information, provided by noisy observations on some known functions of the factor realizations. The estimation is carried out via penalized least squares, i.e., by minimizing the L2 distance between observations from the panel and their model-implied counterparts, augmented by a penalty for the deviation of the extracted factors from the noisy signals of them. When the time dimension is fixed, the limit distribution of the parameter vector is mixed Gaussian with conditional variance depending on the path of the factor realizations. On the other hand, when the time span is large, the convergence rate is faster and the limit distribution is Gaussian with a constant variance. In this case, however, we incur an incidental parameter problem since, at each point in time, we need to recover the concurrent factor realizations. This leads to an asymptotic bias that is absent in the setting with a fixed time span. In either scenario, the limit distribution of the estimates for the factor realizations is mixed Gaussian, but is related to the limiting distribution of the parameter vector only in the scenario with a fixed time horizon. Although the limit behavior is very different for the small versus large time span, we develop a feasible inference theory that applies, without modification, in either case. Hence, the user need not take a stand on the relative size of the time dimension of the panel. Similarly, we propose a time-varying data-driven weighting of the penalty in the objective function, which enhances efficiency by adapting to the relative quality of the signal for the factor realizations.

KW - Asymptotic bias

KW - Incidental parameter problem

KW - Inference

KW - Large data sets

KW - Nonlinear factor model

KW - Options

KW - Panel data

KW - Stable convergence

KW - Stochastic volatility

KW - Asymptotic bias

KW - Incidental parameter problem

KW - Inference

KW - Large data sets

KW - Nonlinear factor model

KW - Options

KW - Panel data

KW - Stable convergence

KW - Stochastic volatility

UR - https://sfx-45cbs.hosted.exlibrisgroup.com/45cbs?url_ver=Z39.88-2004&url_ctx_fmt=info:ofi/fmt:kev:mtx:ctx&ctx_enc=info:ofi/enc:UTF-8&ctx_ver=Z39.88-2004&rfr_id=info:sid/sfxit.com:azlist&sfx.ignore_date_threshold=1&rft.object_id=954921387916&rft.object_portfolio_id=&svc.holdings=yes&svc.fulltext=yes

U2 - 10.1016/j.jeconom.2019.04.018

DO - 10.1016/j.jeconom.2019.04.018

M3 - Journal article

VL - 212

SP - 4

EP - 25

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

IS - 1

ER -