A dynamic unobserved effects model is a
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
used in
econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
for
panel analysis. It is characterized by the influence of previous values of the
dependent variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
on its present value, and by the presence of unobservable
explanatory variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
s.
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
In a panel data
tobit model, if the outcome
partially depends on the previous outcome history
this tobit model is called "dynamic". For instance, taking a person who finds a job with a high salary this year, it will be easier for her to find a job with a high salary next year because the fact that she has a high-wage job this year will be a very positive signal for the potential employers. The essence of this type of dynamic effect is the state dependence of the outcome. The "unobservable effects" here refers to the factor which partially determines the outcome of individual but cannot be observed in the data. For instance, the ability of a person is very important in job-hunting, but it is not observable for researchers. A typical dynamic unobserved effects tobit model can be represented as
:
:
:
:
In this specific model,
is the dynamic effect part and
is the unobserved effect part whose distribution is determined by the initial outcome of individual ''i'' and some exogenous features of individual ''i.''
Based on this setup, the likelihood function conditional on
can be given as
:
For the initial values
,there are two different ways to treat them in the construction of the likelihood function: treating them as constant, or imposing a distribution on them and calculate out the unconditional likelihood function. But whichever way is chosen to treat the initial values in the likelihood function, we cannot get rid of the integration inside the likelihood function when estimating the model by maximum likelihood estimation (MLE). Expectation Maximum (EM) algorithm is usually a good solution for this computation issue. Based on the consistent point estimates from MLE, Average Partial Effect (APE) can be calculated correspondingly.
Binary dependent variable
Formulation
A typical dynamic unobserved effects model with a
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
dependent variable is represented as:
:
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
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
.
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. 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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
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: ''
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 is ''
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 and the average partial effect can be calculated.
[Chamberlain, G. (1980), “Analysis of Covariance with Qualitative Data,” Journal of Econometrics 18, 5–46]
References
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