State Dependence, Unobserved Heterogeneity and Non-Stationarity in Panel Data

1
State Dependence, Unobserved Heterogeneity and
Non-Stationarity in Panel Data

• Session 11

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Importance of Longitudinal and panel data

• they provide scope for extending control for
  variables that have been omitted from the
  analysis, (differencing provides a simple way of
  removing time constant effects (omitted and
  observed) from the analysis.
• Much of human behaviour is influenced by previous
  behaviour and outcomes, that is, there is
  positive feedback' (e.g. the McGinnis (1968)
  axiom of cumulative inertia').

3
Heckman (2001) Nobel Prize speech

• a frequently noted empirical regularity in the
  analysis of unemployment data is that those who
  were unemployed in the past or have worked in the
  past are more likely to be unemployed (or
  working) in the future
• is this due to a causal effect of being
  unemployed (or working) or is it a manifestation
  of a stable trait?

4
SD H and NS

• Inference about feedback effects (SD) are
  particularly prone to bias if the additional
  variation due to omitted variables (stable trait)
  are ignored.
• With dependence upon previous outcome, the
  explanatory variables representing the previous
  outcome will, for structural reasons, normally be
  correlated with omitted explanatory variables (H)
  and therefore always be subject to bias using
  conventional modelling methods.
• The situation is further complicated by changes
  in the scale and relative importance of the
  systematic relationships over time (NS).

5
Initial Conditions

• Most observational schemes for collecting panel
  and other longitudinal data commence with the
  process already under way.
• They will therefore tend to have an informative
  start the initial observed response is typically
  dependent upon pre-sample outcomes and unobserved
  variables.

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Depression Example

• One-year panel study of depression and
  help-seeking behaviour in Los Angeles (Morgan et
  al, 1983).
• Adults were interviewed during the spring and
  summer of 1979 and re-interviewed at 3-monthly
  intervals.
• A respondent was classified as depressed if they
  scored gt16 on a 20-item list of symptoms.

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Depression Example
8
Depression Example

• Depression is difficult to overcome suggesting
  that state dependence might explain at least some
  of the observed temporal dependence, although it
  remains an empirical issue whether true contagion
  extends over three months.
• We might also expect seasonal effects due to the
  weather.
• What is the relative importance of state
  dependence (1st order Markov), non-stationarity
  (seasonal effects) and unobserved heterogeneity

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Likelihood

• subject-specific unobserved effects integrated
  out
• acknowledges the possibility that the multilevel
  effects can depend on the regressors

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Linear predictor of the model

• Also change g(.) to acknowledge 1st Order SD

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Initial Condition

• the data window usually samples an ongoing
  process and the information collected on the
  initial observation rarely contains all of the
  pre-sample response sequence and its determinants
  back to inception.
• Need a model for the initial observed response
• Joint Likelihood with a common random effect

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Conditional Likelihood

• If we omit the 1st term on the RHS of this
  formulation, we have conditioning on the initial
  response.
• The data window interrupts an ongoing process,
  the initial observation will, in part, be
  determined by H and the simplification may induce
  inferential error.

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Conditional Likelihood (1) Naïve Model

• Likelihood simplified by

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Depression Conditional Likelihood (1)
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Depression example Usual Conditional Likelihood

• The coefficient on (s.lag1) is 0.94558 (s.e.
  0.13563), which is highly significant,
• scale parameter is of marginal significance,
  suggesting a nearly homogeneous first order
  model.
• Can we trust this inference?

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Conditional Likelihood (2) random effect
dependent on initial response, Wooldridge

• Rather than we have

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Conditional Likelihood (2)
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Conditional Likelihood (2)

• The coefficients on the constant and the
  time-constant covariates will be confounded.
• For binary initial responses only one parameter
  is needed for y1j, but for the linear model and
  count data, polynomials in y1j, may be needed to
  account more fully for the dependence.

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Depression Conditional Likelihood (1)
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Depression example(2)

• This has the lagged response s.lag1 estimate at
  0.43759E-01 (0.15898), which is not significant,
  while the initial response s1 estimate 1.2873
  (0.19087) and the scale parameter estimate
  0.88018 (0.12553) are highly significant.
• There is also a big improvement in the
  log-likelihood over the model without s1 of
  73.62 for 1 df. This model has no time-constant
  covariates to be confounded by the auxiliary
  model and suggests that depression is a zero
  order process.

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Modelling the Initial Conditions

• use the same random effect in the initial and
  subsequent responses, e.g. Crouchley and Davies
  (2001)
• use a one-factor decomposition for the initial
  and subsequent responses, e.g. Heckman (1981a),
  Stewart (2007)
• use different (but correlated) random effects for
  the initial and subsequent responses
• embed the Wooldridge (2005) approach in joint
  models for the initial and subsequent responses.

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(1) Same random effect common scale parameter
likelihood

• where

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(1) Same random effect common scale parameter
likelihood

• To set this model up in Sabre 5.0 we combine the
  linear predictors by using dummy variables so
  that for

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Depression Same Scale Joint Likelihood (1)
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Depression Joint Likelihood (1)

• The coefficient of r2s.lag1 is 0.70228E-01
  (0.14048) suggesting that there is no state
  dependence in these data, while the scale
  coefficient 1.0372 (0.10552) suggests
  heterogeneity.

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(2) Same random effect but with different scale
parameters

• This model can be derived from a one-factor
  decomposition of the random effects for the
  initial and subsequent observations for its use
  in this context see Heckman (1981a) and Stewart
  (2007).
• The likelihood is like that for the common scale
  parameter model, except that for i1
• For the rest we have

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Stewart (2007) parameterization

• for i1
• for igt1
• As in the common scale and all the joint models

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Depression Different Scale, Joint Likelihood (2)
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(3) Different (but correlated) random effects
Likelihood

• where

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(3) Different (but correlated) random effects

• The scale parameter for the initial response is
  not identified in the presence of r in the
  binary or linear models
• So in these models we hold it at the same value
  as that of the subsequent responses.
• Again

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Depression Different (but correlated) random
effects (3)
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(4) Depression Different (but correlated) random
effects

• Note that the log likelihood is exactly the same
  as for the previous model
• The scale2 parameter from the previous model has
  the same value as the scale parameter of the
  current model.
• The lagged response r2s.lag1 has an estimate of
  0.50313E-01 (0.15945), which is not significant.
• The correlation between the random effects (corr)
  has estimate 0.97089 (0.10093), which is very
  close to 1 suggesting that the common random
  effects, zero order, single scale parameter model
  is to be preferred.

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(4) Embed the Wooldridge (2005) approach in the
joint models

• We can include the initial response in the linear
  predictors of the subsequent responses of any of
  the joint models, but for simplicity we will use
  the single random effect single scale parameter
  model.
• The likelihood for this model is

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(4) Embed the Wooldridge (2005) approach in the
joint models

• Linear Predictors
• For all i

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Depression Wooldridge (2005) approach in the
joint models(4)
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(4) Embed the Wooldridge (2005) approach in the
joint models

• This joint model has both the lagged response
  r2s.lag1 estimate of 0.61490E-01 (0.15683) and
  the baseline/initial response effect r2s.base
  estimate of -0.33544E-01 (0.26899) as being
  non-significant.
• If we estimate a standard zero order model with
  dummy variables for seasons 2, 3, and 4 to all
  the data we get

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O order Model
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0 Order Model

• This model, without any state dependence,
  suggests that the worst seasons are t3 (autumn)
  and t4 (winter).
• This model also has a good fit to the data, the
  log L(Data)-1141.54, so the ChiSq (goodness of
  fit to the data) is 3.119 (10df)

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Other link functions

• State dependence can also occur in Poisson and
  linear models. For a linear model example, see
  Baltagi and Levin (1992) and Baltagi (2005).
  These data concern the demand for cigarettes per
  capita by state for 46 American States.
• We have found first order state dependence in the
  Poisson data of Hall et al (1984), Hall,
  Griliches and Hausman (1986). The data refer to
  the number of patents awarded to 346 firms each
  year from 1975 to 1979.

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Exercises

• Exercise FOL1 Trade Union Membership 1980-1987 of
  young males, Wooldridge (2005)
• Exercise FOL2 Probit model of union membership of
  females, Stewart (2006)
• Exercise FOL3 Binary Response, Female Labour
  Force Participation in the UK