Dynamic unobserved effects model

A dynamic unobserved effects model is a statistical model used in econometrics for panel analysis. 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.

Continuous dependent variable

Censored dependent variable

Binary dependent variable

Formulation

A typical dynamic unobserved effects model with a binary dependent variable is represented[1] as:

P(y_{it}=1\mid y_{i,t-1},\dots ,y_{i,0},z_{i},c_{i})=G(z_{it}\delta +\rho 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.

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 woman's choice whether to work or not, z_it includes the i-th individual's age, education level, number of children, and other factors. c_i can be some individual specific characteristic which cannot be observed by economists.[2] It is a reasonable conjecture that one's 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(y_i,t , y_i,t-1 , … , y_i,1 | y_i,0 , z_i , c_i) against the density of c_i, we can obtain the conditional density P(y_i,t , y_i,t-1 , ... , y_i,1 |y_i,0 , z_i). The objective function for the conditional MLE can be represented as: $\sum _{i=1}^{N}$ log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).

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(y_i,t , y_i,t-1 , … , y_t,1,y_i,0 | c_i, z_i) can be obtained. By integrating this likelihood against the density of c_i conditional on z_i, we can obtain the conditional density P(y_i,t , y_i,t-1 , … , y_i,1 , y_i,0 | z_i). The objective function for the conditional MLE[3] is $\sum _{i=1}^{N}$ log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).

Based on the estimates for (δ, ρ) and the corresponding variance, values of the coefficients can be tested[4] and the average partial effect can be calculated.[5]

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gollark: *Narrator: the caller did not handle the result properly*

References

Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 300.
James J. Heckman (1981): Studies in Labor Markets, University of Chicago Press, Chapter Heterogeneity and State Dependence
Greene, W. H. (2003), Econometric Analysis , Prentice Hall , Upper Saddle River, NJ .
Whitney K. Newey, Daniel McFadden, Chapter 36 Large sample estimation and hypothesis testing, In: Robert F. Engle and Daniel L. McFadden, Editor(s), Handbook of Econometrics, Elsevier, 1994, Volume 4, Pages 2111–2245, ISSN 1573-4412, ISBN 9780444887665,
Chamberlain, G. (1980), “Analysis of Covariance with Qualitative Data,” Journal of Econometrics 18, 5–46

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[1] Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 300.

[2] James J. Heckman (1981): Studies in Labor Markets, University of Chicago Press, Chapter Heterogeneity and State Dependence

[3] Greene, W. H. (2003), Econometric Analysis , Prentice Hall , Upper Saddle River, NJ .

[4] Whitney K. Newey, Daniel McFadden, Chapter 36 Large sample estimation and hypothesis testing, In: Robert F. Engle and Daniel L. McFadden, Editor(s), Handbook of Econometrics, Elsevier, 1994, Volume 4, Pages 2111–2245, ISSN 1573-4412, ISBN 9780444887665,

[5] Chamberlain, G. (1980), “Analysis of Covariance with Qualitative Data,” Journal of Econometrics 18, 5–46