725 note 2010_s3_1_linear_3_11

725 note 2010_s3_1_linear_3_11 - 1. LINEAR REGRESSION UNDER...

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1 | Linear Regressions under Ideal Conditions (III) 1. LINEAR REGRESSION UNDER IDEAL CONDITIONS (III) What do we learn in this section? [9] Weaker Assumptions. • Does the OLS estimator have good properties under more realistic circumstances?
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2 | Linear Regressions under Ideal Conditions (III) [9] Nonnormal u and Stochastic Regressors (1) Motivation • If the regressors x t are stochastic? • OLS estimator ˆ is unbiased? • Recall how we have shown the unbiasedness of ˆ under (SIC.8): ˆ = 1 () o XX Xu ? 11 ˆ [ ( ) ] ( ) ( ) oo E E X u X E u     .
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3 | Linear Regressions under Ideal Conditions (III) • Unbiasedness of ˆ does not require nonstochastic regressors. It only requires:   1 | ,..., 0 tT Eu x x  , for all t . (*) O r 1 ( | )0 T E uX :        1 1 ˆˆ () ( | ) ( ) | ( | )( ) . XX o Xo X o o E EE X X X X u X EX X X E u X E    
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4 | Linear Regressions under Ideal Conditions (III) • But… what if 1 tt x y ( 1 t yy u )? 11 1 ( | )( | )0 E ux E u y   . • No longer ˆ is an unbiased estimator. An example for models with lagged dependent variables as regressors: 2 2 3 1 t t t y xx y u   . β 2 /(1- β 3 ) = long-run effect of x t 2 . • If the u t are not normally distributed, all t and F tests are wrong. • Can we use them if T is large?
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5 | Linear Regressions under Ideal Conditions (III) Large-Sample Theories 1. Motivation: ˆ T : An estimator from a sample of size T , { x 1 , . .. , x T }. What would be the statistic properties of ˆ T when T is infinitely large? • What do we wish?
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6 | Linear Regressions under Ideal Conditions (III) 2. Main Points: Rough Definition of Consistency • Suppose that the distribution of ˆ T becomes more and more condensed around θ o as T increases. Then, we say that ˆ T is a consistent estimator. And we use the following notation: plim T ˆ T = θ o (or ˆ T p θ o ). [In fact, this definition is more related to the convergence in mean square.] • The law of large numbers (LLN) says that a sample mean T x ( x from a sample size equal to T ) is a consistent estimator of o . What does it mean?
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7 | Linear Regressions under Ideal Conditions (III) • Gauss Exercise: • A population with N (1,9). • 1000 different random samples of T = 10 to compute 10 x . • 1000 different random samples of T = 100 to compute 100 x . • 1000 different random samples of T = 5000 to compute 5000 x . 10 x 100 x
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8 | Linear Regressions under Ideal Conditions (III) • Relation between unbiasedness and consistency: • Biased estimators could be consistent. Example: Suppose that T is unbiased and consistent. ˆ T = T + 1/ T . E ( ˆ T ) = θ o + 1/ T θ o (biased). ˆ lim lim TT T o pp   (consistent).
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725 note 2010_s3_1_linear_3_11 - 1. LINEAR REGRESSION UNDER...

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