ie_Slide04

ie_Slide04 - Introductory Econometrics ECON2206/ECON3209...

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Introductory Econometrics ECON2206/ECON3209 Slides04 LecturerRachida Ouysse ie_Slides04 RO, School of Economics, UNSW 1
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4. Multiple Regression Model: Inference (Ch4) 4. Multiple Regression Model: Inference • Lecture plan lassical linear model assumptions Classical linear model assumptions – Sampling distribution of OLS estimators under CLM esting hypothesis about one population parameter – Testing hypothesis about one population parameter – p-values onfidence intervals – Confidence intervals – Test hypothesis with CI ie_Slides04 RO, School of Economics, UNSW 2
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4. Multiple Regression Model: Inference (Ch4) • Motivation: y = β 0 + β 1 x 1 +...+ β k x k + u – Goal is to gain knowledge about the population parameters ( β ’s) in the model. – OLS provides the point estimates of the parameters. OLS will get it right on average (being unbiased). – Knowing the mean and variance of is not enough. j ˆ • how to decide if a hypothesis is supported or not? • what we can say about the “true values”? We need the sampling distribution of the OLS estimators to answer these questions. – To simplify, we use a strong assumption here (and will relax it for large-sample cases). ie_Slides04 RO, School of Economics, UNSW 3
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4. Multiple Regression Model: Inference (Ch4) • Normality assumption 6. (MLR6, normality) The disturbance u is independent of all explanatory variables and normally distributed with mean zero and variance σ 2 : u ~ Normal(0, σ 2 ) . This is a very strong assumption. It implies both yg p p MLR4 (ZCM) and MLR5 (homoskedasticity). MLR1-6 together are known as the classical linear model (CLM) assumptions. Under CLM, the OLS produces the minimum variance unbiased estimators. They are the best of unbiased estimators ( not just the est of linear unbiased estimators best of linear unbiased estimators ). ie_Slides04 RO, School of Economics, UNSW 4
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4. Multiple Regression Model: Inference (Ch4) • Normality assumption – CLM implies It is completely characterised by the mean and variance. y | x ~ Normal( β 0 + β 1 x 1 +...+ β k x k , σ 2 ) . LM also implies is normally distributed ˆ CLM also implies is normally distributed. – Whether or not MLR6 is a reasonable assumption j depends on data. Is it reasonable for wage model, given that no wage can be negative? Empirically, it is reasonable for log( wage ) model. – MLR6 is restrictive. But the results here will be useful for large-sample cases (Ch5) without MLR6. ie_Slides04 RO, School of Economics, UNSW 5
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4. Multiple Regression Model: Inference (Ch4) • Sampling distribution of OLS Theorem 4.1 (normal sampling distribution) Under CLM, conditional on independent variables, ormal ˆ 2 where the variance is given in Ch3 (ie_Slides03):   , , Normal ~ j j j . ,..., 1 , ) 1 ( ) ˆ ( 2 2 2 k j R SST Var j j j j – It implies: ˆ . ) ˆ ( ) ˆ ( ), , ( Normal ~ ) ˆ ( j j j j j Var sd sd 1 0 ie_Slides04 RO, School of Economics, UNSW 6
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4. Multiple Regression Model: Inference (Ch4) • Sampling distribution of OLS – In practice, we have to estimate σ 2 and ˆ β Var . ,..., , ) ( ˆ ) ˆ ( ˆ k j R SST ar V j j j 1
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ie_Slide04 - Introductory Econometrics ECON2206/ECON3209...

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