Pooled Time Series Cross Sectional Models

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Pooled Time Series Cross Sectional Models

• Dealing with
• Time Dependence

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The Nature of the Problem

• Assume a simple tscs model
• For OLS to be optimal we require that the
  variance-covariance matrix of the residuals looks
  like this

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This Implies That

• The errors within units
• have the same variance as each other (within-unit
  homoscedasticity)
• are unrelated to the other errors in that unit
  (within-unit temporal independence)
• That the errors across units
• have the same variance (between-unit
  homoscedasticity)
• are unrelated to contemporaneous (or lagged)
  errors in other units (spatial independence).
• When (not if) we violate these assumptions we get
  biased estimates of our parameters standard
  errors

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The O Matrix

• The key to modeling tscs data is O. If we know O
  (or have a good estimate of it) we can use GLS to
  estimate our parameters and ses.
• We do not, of course, ever know O and the best we
  can do is come up with an estimate of it and use
  FGLS.
• Key concerns in modeling tscs data is the
  structure of O.

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The Parks Method

• A standard/traditional approach to dealing with
  tscs data was proposed by Parks (1967). He
  assumed that
• the errors follow a temporal AR1 process
• we can assume that the rhos are common across
  units or that they are different ((since we
  assume that the ßs are constant across units why
  not assume that the rhos are as well?))
• This approach to serial correlation is relatively
  standard use consistent estimates of ß (here
  based on OLS) to generate residuals which provide
  us a way to estimate rho. We can then use the
  Prais-Winsten transformation on each unit.

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• With regard to contemporaneous (Spatial)
  correlation things become more difficult because
  we would the contemporaneous correlation of
  errors.

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• Use S to denote the VCV (variance-covariance)
  matrix of errors and recall that

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• We can look more closely at the elements of S

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• The problem is that there are N(N-1)/2 distinct
  off-diagonal contemporaneous correlations that we
  need to estimate with NT observations. This
  means that we are using 2t/N observations to
  estimate each variance. If T is close to N then
  we are using (on average) only 2 observations to
  estimate each variance. This means
• the Parks method cannot be used if TgtN
• in practice the Parks method gives terrible
  results unless T is SIGNIFICANTLY larger than N

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• Beck and Katz (1995) show via Monte Carlo
  simulations that the Parks method has serious
  problems. Their MCs show that the Parks method
  yields standard error that are far (!!) too small
  up to 600 off this leads to very (!!) bad
  inferences.
• They recommend using Parks only if T is very
  large relative to N.
• In stata the Parks method is implemented via
• xtgls

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Panel-Corrected Standard Errors

• Beck and Katz propose an alternative panel
  corrected standard errors. Note, that in order
  to use pcses we must first rid the data of serial
  correlation (using Prais-Winsten, differencing,
  etc).
• Under these conditions OLS gives consistent
  estimates of ß but the standard errors must be
  adjusted.
• The computations are a bit complex see Beck and
  Katz (1995) for details.
• Note PCSEs are for use when there is no temporal
  autocorrelation.

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A New Data Set

• Eichengreen, Barry, and David Leblang, 2003,
  Capital Account Liberalization and Growth Was
  Mr. Mahathir Right? International Journal of
  Finance and Economics 8205-24
• Research question do capital controls enhance or
  inhibit economic growth? Why?
• Do these variables have a constant or
  time-varying impact conditional on the
  international currency regime (gold, interwar,
  bretton woods, post bretton woods)
• Panel of 21 countries over 21 periods (5 year
  periods).
• Dependent variable is growth rate over 5 year
  period.
• Independent variables of interest capital
  controls, financial crises (bank and currency),
  and interaction between crises and controls.
• Control variables endogenous growth variables
  initial levels of gdp, primary and secondary
  education

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STATA Example
14
Temporal Dynamics

• Both Parks and PCSEs deal with (eliminate?)
  serial correlation by using the Prais-Winsten (or
  other) transformation. This ignores the dynamic
  nature of the data and treats it simply(?) as a
  nuisance relegated to the error term.
• Of course, we need to think about the dynamic
  behavior of the variable in question

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Dealing with Dynamics

• Treat the model as static and the temporal
  correlation as a nuisance. This, as we have
  done, means that we deal with serial correlation
  in the residuals and do not directly confront
  dynamics in the model itself.
• Specify a dynamic model whereby autocorrelation
  is a function of a lagged dependent variable
  (LDV)

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• This model has a lagged endogenous variable and
  unit level effects ignore for now whether those
  unit effects are random or fixed.
• What happens if we estimate this via OLS? If we
  leave out the unit effects we get a model that
  has both unit-level and spherical components in
  the error
• If we lag this one period we have problems

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• Why is this a problem?
• Both yit-1 and uit contain ai which means that
  in estimating the previous equation you know that
  one of your right-hand-side variables is
  correlated with the error term. This yields
  biased and inconsistent estimates of ß and F.
  With regard to F the bias will be
• negative (provided that Fgt0)
• increasing in magnitude in F (the degree of
  autoregression)
• With a lagged endogenous variable model simply
  estimating fixed effects for the ais does not
  solve the problem because there is still a
  correlation between yit-1 and the transformed uit

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Solution First Difference Estimator

• Anderson and Hsiao (1981) suggested differencing
  the model to eliminate unit effects
• we can rewrite this using the first difference
  operator
• while this does eliminate the unit effects it
  does not get rid of the correlation between the
  covariates and the errors because yit-1 is still
  correlated with uit-1

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• Solution to this correlation between errors and
  variables is through the use of instrumental
  variables a set of variables (Z) that are
  highly correlated with X but uncorrelated with
  the errors.
• If, for example, there is no serial correlation
  (which we are assuming for now) then it turns out
  that both ?yit-2 and yit-2 are correlated with
  ?yit-1 through yit-2. Both of these are
  uncorrelated with ?uit.
• Anderson and Hsiao suggest
• use ?yit-2 as Z (as an instrument for ?yit-1) and
  
• use yit-2 itself as Z
• In theory this yields consistent estimates for ß
  and F

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But.

• In practice these estimators have problems
• The estimator is problematic when F is close to 1
• The estimator yields biased results when n is
  small
• The estimator yields biased results when t is
  short
• The estimator is inefficient (of course this is
  relative to an alternative estimator)

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Arellano and Bonds Dynamic Estimator

• AB show that if we have errors that are mean zero
  and serially uncorrelated then the differenced
  residuals ?uit are uncorrelated with all yit and
  Xit from t-2 and before.
• this means that we can use all those values as
  instruments for ?yit-1
• Advantages
• we can get good estimates of the parameters under
  a wide range of circumstances.
• as t increases then we have an increasingly large
  number of instruments which will give us better
  estimates.

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• Disadvantages
• the model is like a fixed-effects estimator
  covariates that do not vary much over time are
  dropped
• the first difference model cannot use the first
  two observations because of differencing and
  lagging. Higher order models lose additional
  observations which is a problem if t is small
• Andrecall that the AB estimator requires that
  the errors are serially uncorrelated. There is,
  of course, a test for this. They note that if
  the us are uncorrelated then the differenced us
  will have
• negative first order serial correlation
• no second and higher order serial correlation
  (because the us are now MA(1)

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Stata Examples