Pooled CrossSectional Time Series

1
Pooled Cross-Sectional Time Series

• Often there are occasions where we are interested
  in looking at a number of similar entities over
  time.
• The archetypal PCSTC (or CSTS, or TSCS, or
  whatever!) data set is the 50-state study, over
  multiple years.
• Note that this represents a set of partitioned
  matrices.

2
The Matrix Model

• The multiple regression model for OLS may be
  easily represented in matrix terms.
• Where the Y, X, B and e are all matrices of data,
  coefficients, or residuals
• The Pooled Design is quite similar

3
The OLS Matrix Model

• The matrices in are
  represented by
• Note that we post-multiply X by B since this
  order makes them conformable.

4
The Pooled Matrix Model

• The matrices in Y XB e
  are represented by

5
The standard assumptions of the model

• The standard assumptions of the model must
  continue to hold, but we must elaborate them
  slightly.

6
Problem with OLS

• If no heteroskedasticity or autocorrelation in
  the residuals exists, then OLS is fine
• If either exists, then detecting them and
  correcting for them is problematic- especially
  autocorrelation
• Significant unit differences (as opposed to time
  points) can produce autocorrleation and
  heteroskedasticity.

7
Generalized Least Squares

• If we find that we have autocorrelation, we must
  correct for it.
• To do so we employ Generalized Least Squares

8
What is GLS

• The standard estimate for OLS in matrix
  notation is
• Generalized least squares uses the following
  estimator.
• See Hibbs (1973) for a fairly clear derivation of
  the GLS estimator

9
The Omega Matrix

• The Omega Matrix is a matrix used to adjust the
  data to remove the effects of the
  autocorrelation.
• It looks like

10
Just Autocorrelation

• If the pooled model has only autocorrelation and
  no heteroskedasticity, then Omega looks like

11
Only Heteroskedasticity

• On the other hand, if there is heteroskedasticity,
  but no autocorrelation then Omega looks like

12
How the Omega Matrix Works

• In principle it works much like the
  Cochran-Orcutt or Durbins methods by
  transforming the data.
• i.e.
• Because ? is estimated, instead of the true
  value, we refer to this Aitkens Generalized
  Estimator or Pseudo-GLS

13
Testing for Autocorrelation in a Pooled Design

• There are several ways to test for AC in the
  residuals of a polled design.
• One of the simplest relies on the Central limit
  theorem.
• Use the point estimates of each cross section as
  a sampling distribution.
• Thus we fit identify an ARMA model for the
  residuals for each cross section
• AR(1) Processes are the most common

14
Estimation of a Pooled Design

• Three software packages to chose from
• Microcrunch
• RATS
• Stata