ECON206_1011_01_Handout_10

ECON206_1011_01_Handout_10 - ECON 206 METU- Department of...

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ECON 206 December 27, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 1 LECTURE 10 SIMPLE REGRESSION MODEL - II Outline of today’s lecture: I. Mean and Variance of the Dependent Variable Y . ............................................................. 1 II. Ordinary Least Squares (OLS) Estimation. ....................................................................... 2 A. Mean of 1 ˆ β .................................................................................................................... 6 B. Variance of 1 ˆ ............................................................................................................... 9 C. Mean of 0 ± ................................................................................................................. 10 D. Variance of 0 ± ............................................................................................................ 11 E. Covariance of 0 ˆ and 1 ˆ ............................................................................................. 13 I. Mean and Variance of the Dependent Variable Y The dependent variable Y has mean 01 () tt EY X = + and variance 2 22 ( ) t t Var Y E Y E Y E u σ ⎡⎤ ⎣⎦ =− = = 1. Let us show that the mean of Y t is t X + . Æ By definition the mean of Y t is its expected value. Given that t YX u =+ + . Taking the expected values we get ) ) ( ( t t E X u E X Eu ββ + = ++
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ECON 206 December 27, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 2 Given that 0 β and 1 are parameters and by Assumption 2 the values of X t ’s are a set of fixed numbers (in the process of hypothetical sampling) 01 tt EX X ββ ⎡⎤ = ⎣⎦ + + Furthermore, by Assumption 3, ()0 t Eu = . Therefore, ) ( EY X = + 2. Let us now show that the variance of Y t is 2 σ 2 () t Var Y E Y E Y =− 2 t t Var Y E X u X =+ + + 2 Var Y E u = By Assumption 4, the u t ’s are homoscedastic, that is, they have the constant variance 2 2 2 Var Y E u = = II. Ordinary Least Squares (OLS) Estimation The two-variable population regression function is given by t YX u + , but we do not observe it. Hence we estimate it from the sample regression function
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ECON 206 December 27, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 3 01 ˆ ˆˆ ˆ t tt t Y YX u ββ =+ + ±²³²´ . or ˆ ˆ ttt YYu . We can rewrite the sample regression function as ˆ ˆ tttt t uY YY X β =−=− . In other words, the residuals are the differences between the actual and the estimated t Y values. With T observations, we want to choose 0 ˆ and 1 ˆ such that the sum of the residuals is minimized: 11 ˆ ˆ () . TT t Y == =− ∑∑ this turns out not to be a very good rule because some residuals are negative and some are positive (and they would cancel each other), and
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ECON 206 December 27, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 4 all residuals have the same weight (importance) even though some are small and some are large. 9 To overcome these problems, we use the squares of the residuals instead of their own values. Ordinary Least Squares (OLS) criterion: Minimize 2 2 2 01 ˆˆ ˆ ˆ () ( ) tt t t t uY Y Y X ββ =− ∑∑ wrt 1 ˆ β and 2 ˆ The necessary condition for a minimum is that the first derivatives of the function be equal to zero. Partial differentiation yields 2 0 ˆ 2( ) 0 ˆ t u YX = ( 1 ) 2 1 ˆ ) 0 ˆ t t u X = ( 2 ) From (1) 0 −− = 11 . 0
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ECON206_1011_01_Handout_10 - ECON 206 METU- Department of...

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