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Econometrics

Econometrics. Chapter 7 Properties of OLS Estimators. Properties of Least Squares Estimators - Chapter 7. Unbiased Consistent Efficient Assumptions of the Classic Regression Model (Gauss-Markov) Homoskedasticity Serial Correlation Exogenous/Endogenous Linearity (Parameters/Variables).

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Econometrics

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  1. Econometrics Chapter 7 Properties of OLS Estimators

  2. Properties of Least Squares Estimators - Chapter 7 • Unbiased • Consistent • Efficient • Assumptions of the Classic Regression Model (Gauss-Markov) • Homoskedasticity • Serial Correlation • Exogenous/Endogenous • Linearity (Parameters/Variables)

  3. Properties of OLS Estimators – continuedDefinitions/Desirable Characteristics of Estimators • Unbiased – An estimator’s expected value equals the true value of the parameter it estimates: • Consistent – As the data sample gets larger and larger, the difference between the Estimated Value and the True Value gets smaller and smaller • Efficient – There is no other unbiased estimator with a lower variance than

  4. Properties of OLS Estimators – continuedThe Classical Regression Model: Gauss-Markov Assumptions • A1:The true data generating process is seen by our model; • Yi = β0 + β1Xi + εi • A2:X does not take the same value for all observations • A3:Given the values for Xi, the variance of the error term is the same for each observation • A4:The error terms are Not correlated with each other • A5:The error terms are Not correlated with the value of X; the mean value of ε is 0 no matter what the value of X is

  5. Properties of OLS Estimators – continuedThe Classical Regression Model: Gauss-Markov Assumptions - Implications • A1:Model Matches Process (no way to confirm this – just build best model we can) • A2:Guarantees that Sum of Squared difference between observed and mean values of X is NOT equal to 0 • A3-A5:These are TESTABLE and can change the way we estimate the model if they are false

  6. Properties of OLS Estimators – continuedThe Classical Regression Model: Gauss-Markov Assumptions - Implications • A3:Error Terms have a common variance (Homoskedasticity not Heteroskedasticity) • A4:Error Terms are uncorrelated and do not influence each other (Error Terms are Not Serially Correlated) • A5:Knowing the value of X does not tell you anything about the value of ε (RHS Variables must not be Endogenous – must be Exogenous)

  7. Properties of OLS Estimators – continuedThe Classical Regression Model: Gauss-Markov Assumptions • Homoskedasticity: Each error term has the same variance → each data point is likely to lie just as far from the true model as any other • No Serial Correlation: Correlation of εi and εj equals 0 • Exogenous: Values determined outside the model (Independent) • Endogenous: Values determined within the model ( Y depends on X) • Linearity: WRT variables or Parameters or both

  8. Properties of OLS Estimators – continuedThe Classical Regression Model: Gauss-Markov Assumptions • Gauss-Markov Theorem: The estimators we get from OLS are efficient, that is; of all linear estimators, OLS provides ones with the lowest variance • A6: The Error Term is Normally Distributed; then OLS estimators have the Lowest Variance of other possible Unbiased Estimators • BLUE and BUE: A1-A5 → BLUE; add A6 and OLS estimators are BUE

  9. Properties of OLS Estimators – continuedSummary – Chapter 7 • An Estimator is Unbiased if its expected value is the True Value • An Estimator is Consistent if the chance of the Estimated Value being within any given distance of the True Value rises as the sample gets larger • An Estimator is Efficient within a class of unbiased estimators if it has a lower variance than other estimators in Its class

  10. Properties of OLS Estimators – continuedSummary – Chapter 7 • A1-A5 imply that Least Squares Estimators are Unbiased, Consistent, and More Efficient than any other Linear, Unbiased Estimator (BLUE) • A6 (Error Term is Normally Distributed) implies that OLS Estimator is BUE (linear or non-linear

  11. Properties of OLS Estimators – continuedSummary – Chapter 7 • The SER (Standard Error of the Regression) is an Unbiased and Consistent Estimator of the Variance of the Error Terms • The larger the SER, the larger the variances of the OLS parameters; but more observations imply smaller variances

  12. Properties of OLS Estimators – continuedSummary – Chapter 7 • The OLS Estimators are Normally Distributed if the Error Terms are Normally Distributed – They are approximately Normally Distributed, even if the Error Terms are not • We can use the OLS Estimates and their Standard Deviations to calculate Confidence Intervals for the True Values of β0 and β1 and to Test Hypothesis about those True Values

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