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Data organization

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Regression Models Time series Cross-sectional Panel Multi-dimensional panel

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**Errors in Uni-dimensional Data**

In standard time series or cross-sectional data sets, we must adjust for non-independent errors. Serial correlation Errors correlated across time Spatial correlation Errors correlated across cross-sections Heteroskedasticity Error variance changes over time or cross-sections

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Errors in Panel Data Heterogeneous serial correlation Errors correlated across time and differently for different cross-sections. Heterogeneous spatial correlation Errors correlated across cross-sections but differently for different time periods. Heterogeneous heteroskedasticity Error variance changes over time, but does so differently for different cross-sections. Serial-spatial correlation Past errors from one cross-section are correlated with future errors from a different cross-section.

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**Generalized Least Squares**

The error covariance matrix shows the covariances of error terms across different observations.

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**Ordinary Least Squares Assumptions**

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**Ordinary Least Squares (Heteroskedasticity)**

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**Ordinary Least Squares (Serial Correlation)**

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**Two-Dimensional Panel Data: OLS Assumptions**

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**Two-Dimensional Panel Data: OLS Assumptions**

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**Two-Dimensional Panel Data: OLS (homogeneous serial correlation)**

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**Two-Dimensional Panel Data: OLS (heterogeneous serial correlation)**

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**Two-Dimensional Panel Data: OLS (serial-spatial correlation)**

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**OLS vs. Panel Estimation**

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**Fixed versus Random Effects**

Under the random effects assumption, and are treated as stochastic. Under the fixed effects assumption, they are treated as fixed in repeated samples.

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**Random vs. Fixed Effects**

Random Effects Assumption Pro: Estimators are more efficient Con: Estimators are inconsistent if any of the three errors are not IIN(0,σ2) across all dimensions. Fixed Effects Assumption Pro: Estimators are consistent regardless of and . Con: Estimators are less efficient. See Hausman test for endogeneity.

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**Random vs. Fixed Cross-Sectional Effects**

Test statistic = 22

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**Alternatives to Panel Techniques**

Separate Regressions Drawbacks Less efficient estimators due to lost information about cross-sectional error covariance. Remove the ability to restrict parameter values across cross-sections.

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**Alternatives to Panel Techniques**

Pooled Regression Drawbacks Less efficient estimators due to lost information about cross-sectional error covariance. Restricts parameter values to be equal across cross-sections.

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**Alternatives to Panel Techniques**

Pooled Regression with Cross-Sectional Dummies Drawbacks This is the fixed effects panel technique. If the cross-sectional dummies are IIN, then parameter estimates are less efficient than under the random effects panel technique.

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**Procedures to use with panel data**

Generalized least squares (GLS) Generalized method of moments (GMM) OLS with “automated” corrections for serial correlation, etc. is GLS.

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Extra stuff Panel data reveals information that is unattainable with non-panel data.

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**Three-Dimensional Structure of the ASA-NBER Data Set**

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**Shock Occurrence vs. Shock Impact**

These shocks all impact inflation in quarter 9 but occur in different quarters. These shocks all occur in quarter 6 but impact inflation in different quarters.

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**Cross-sectional shocks**

Shock Occurrence vs. Shock Impact Cumulative shocks Cross-sectional shocks Discrete shocks

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**Shock Occurrence vs. Shock Impact**

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