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\layout Title

Notes on Beck & Katz, 
\begin_inset Quotes eld
\end_inset 

What to do (and not to do) with Time-Series Cross-Section Data
\begin_inset Quotes erd
\end_inset 

APSR, 1995
\layout Standard

These notes assume you have seen the companion 
\begin_inset Quotes eld
\end_inset 

Quick notes on OLS, covariance matrices, and GLS to prepare for CX TS analysis
\begin_inset Quotes erd
\end_inset 

.
 Don't try this without that.
\layout Section

Recall GLS.
\layout Standard

OLS assumes the matrix of covariances for the error terms is very 
\begin_inset Quotes eld
\end_inset 

pleasant
\begin_inset Quotes erd
\end_inset 

.
 No autoregression, no cross-unit correlations, no heteroskedasticity.
\layout Standard

The problem with the error terms causes inefficiency in the estimates of
 
\begin_inset Formula $\hat{b}$
\end_inset 

 and we can address those if we know what 
\begin_inset Formula $\Omega$
\end_inset 

 is.
 The solution from GLS ends up being a close parallel to the OLS solution
 for 
\begin_inset Formula $\hat{b}$
\end_inset 

:
\layout Standard


\begin_inset Formula $\hat{b}=(X'\Omega^{-1}X)^{-1}X'\Omega^{-1}y$
\end_inset 


\layout Standard

and the variance of 
\begin_inset Formula $\hat{b}$
\end_inset 

 is:
\layout Standard


\begin_inset Formula $Var(\hat{b})=(X'\Omega^{-1}X)^{-1}$
\end_inset 


\layout Subsection

FGLS
\layout Standard

Here is where the problem starts, from Beck and Katz's point of view.
 We follow the GLS theory down the road and end up at FGLS, which they claim
 introduces more trouble than it saves.
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Parks's procedure, a variant of FGLS (Feasible Generalized Least Squares)
 is a procedure to estimate a model, then use the residuals to estimate
 the variance of e, and then re-estimate with GLS.
 The procedure may be repeated many times, until the estimates of the b's
 converge to a fixed number.
\layout Standard

Park's method is one way of attacking this problem with CSTS data.
\layout Standard

1.
 estimate by OLS
\layout Standard

2.
 Use residuals to estimate AR(1) models.
\layout Standard

3.
 Use AR(1) models to adjust data and estimate again.
 (see standard stats books under corrections for AR(1), such as Prais-Winsten
 estimators).
\layout Standard

4.
 Use residuals to estimate cross-correlation across units (spatial autocorrelati
on)
\layout Standard

5.
 Use results from 4 to fill in more values of 
\begin_inset Formula $\Omega$
\end_inset 

 and estimate model again.
\layout Subsection

Beck and Katz observe:
\layout Standard

The multi-stage Parks method has dangers.
\layout Subsubsection

The spatial autocorrelation part.
\layout Standard

T has to be much larger than N.
 Here we are forced to estimate a matrix that is NxN, because the covariances
 of the error term for each country influencing each other country have
 to be taken into account.
\layout Standard

The estimates from the Parks method don't take into account uncertainty
 of these estimates, but rather just take the estimates and plug them in!
 
\layout Subsubsection

The AR(1) part.
\layout Standard

If we assume 
\begin_inset Formula $\rho_{i}$
\end_inset 

 is the same for all countries, then we are only estimating one coefficient.
 However, if 
\begin_inset Formula $\rho_{i}$
\end_inset 

 varies across countries, then we estimate N different coefficients, and
 the accumulated uncertainty about those estimates can be large.
\layout Standard

Beck and Kats contend the 
\begin_inset Formula $\rho$
\end_inset 

 should be the same for all countries.
\layout Section

Beck and Kats recommend:
\layout Standard

Use OLS, but correct the estimates of the standard errors of the b's.
 If you suspect there is AR in the errors, they say it should be corrected
 first.
 They contend you should assume that the AR coeffficient 
\begin_inset Formula $\rho_{i}$
\end_inset 

 is the same for all countries.
\layout Standard

The 
\begin_inset Quotes eld
\end_inset 

panel corrected standard errors
\begin_inset Quotes erd
\end_inset 

 are a result of OLS analysis.
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Monte Carlo studies by B&K support the claim that their estimators have
 lower variance than the Parks estimates, and that the Parks estimates lead
 to false t statistics because the Variance of b is underestimated.
 It all happens because the spatial autocorrelation part of the Parks process.
\layout Standard

They also have MC estimates to claim that, even if 
\begin_inset Formula $\rho$
\end_inset 

 is not the same across all countries, it doesn't do much harm to assume
 that it is in order to make these calculations.
\the_end