## ⓘ Dynamic unobserved effects model

A dynamic unobserved effects model is a statistical model used in econometrics. It is characterized by the influence of previous values of the dependent variable on its present value, and by the presence of unobservable explanatory variables.

The term" dynamic” here means the dependence of the dependent variable on its past history; this is usually used to model the" state dependence” in economics. For instance, for a person who cannot find a job this year, it will be harder to find a job next year because her present lack of a job will be a negative signal for the potential employers." Unobserved effects” means that one or some of the explanatory variables are unobservable: for example, consumption choice of one flavor of ice cream over another is a function of personal preference, but preference is unobservable.

## 1. Formulation

A typical dynamic unobserved effects model is represented as:

P = G z it δ + ρ y i,t-1 + c i

where c i is an unobservable explanatory variable, z it are explanatory variables which are exogenous conditional on the c i, and G∙ is a cumulative distribution function.

## 2. Estimates of parameters

In this type of model, economists have a special interest in ρ, which is used to characterize the state dependence. For example, y i,t can be a womans choice whether to work or not, z it includes the i -th individuals age, education level, number of children, and other factors. c i can be some individual specific characteristic which cannot be observed by economists. It is a reasonable conjecture that ones labor choice in period t should depend on his or her choice in period t − 1 due to habit formation or other reasons. This dependence is characterized by parameter ρ.

There are several MLE-based approaches to estimate δ and ρ consistently. The simplest way is to treat y i,0 as non-stochastic and assume c i is independent with z i. Then by integrating P against the density of c i, we can obtain the conditional density P. The objective function for the conditional MLE can be represented as: ∑ i = 1 N {\displaystyle \sum _{i=1}^{N}} log).

Treating y i,0 as non-stochastic implicitly assumes the independence of y i,0 on z i. But in most cases in reality, y i,0 depends on c i and c i also depends on z i. An improvement on the approach above is to assume a density of y i,0 conditional on c i, z i and conditional likelihood P can be obtained. By integrating this likelihood against the density of c i conditional on z i, we can obtain the conditional density P. The objective function for the conditional MLE is ∑ i = 1 N {\displaystyle \sum _{i=1}^{N}} log).

Based on the estimates for δ, ρ and the corresponding variance, values of the coefficients can be tested and the average partial effect can be calculated.

Latent variable model |

Doubly stochastic model |

Factor regression model |

First-difference estimator |

Nuisance variable |

Partial least squares regression |

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