ECON301_Handout_01_1213_02

Instructor dr ozan eruygur e mail oeruygurgmailcom

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Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 14
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ECON 301 (01) - Introduction to Econometrics I March , 2012 METU - Department of Economics To understand the logic, let us we write as follows: 01 2 t t tt Y X Zu ββ β = +++ 2 2 t t Y X Xu = ++ + 2 ( 2) t Y βββ = + * t Y = Hence, using the data of three observations ( T =3) for X and Y : X 1 =4, Y 1 =15; X 2 =8, Y 2 =23 and X 3 =12, Y 3 =28, the equations are written as: * 4 15 += * 8 23 * 12 28 As in the previous case, this is the T > k+1 situation. There is no perfect fit (data points are not on the line), and hence the relationship between Y and X is not deterministic . The estimates for 0 and * 1 can be obtained using Ordinary Least Squares (OLS) estimation. Suppose that we have obtained the estimate for * 1 as * 1 ˆ 5 = . Using this estimate, can we get the estimates of 1 and 2 using the relationship * 11 2 2 = + ? The answer is no since 12 ˆˆ 52 = + is an ill-posed problem : there are two unknowns ( 1 ˆ and 2 ˆ ) but we have only one equation, there is an infinity of solutions to this equation. It is said that 1 and 2 are not identified . Hence, we see that if there is perfect linear relationship between independent variables, then the problem becomes ill-posed , in other words, the problem becomes unsolvable ! This is the situation of perfect (multi)collinearity . Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 15
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ECON 301 (01) - Introduction to Econometrics I March , 2012 METU - Department of Economics Now consider the following simple regression model: 01 t tt Y Xu ββ = ++ If the X t values in a given sample are all same , can we estimate 0 β and 1 using OLS estimation? The answer is no . Let us investigate this issue. Note that t Y = can be equivalently rewritten as follows: t t Y Z =++ where 1 t Z = for all t =1, 2, 3, … T . Moreover, if the X t values in a given sample are all same (a fixed number), then 0 t XX = where 0 X is a fixed number. Hence, it can also be written as 0 X XZ = since 1 t Z = for all t =1, 2, 3, … T . Thus the model reduces to; 0 10 t t Y Z XZ u = 0 () t Y = Denoting * 0 0 X = + , we get: * 0 t Y Zu = Using the data of three observations ( T =3) for Z and Y : Z 1 =1, Y 1 =15; Z 2 =1, Y 2 =23 and Z 3 =1, Y 3 =28, we have: * 0 15 = * 0 23 = * 0 28 = The OLS estimate for * 0 is 22 since in this case * 0 ˆ Y = . Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 16
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ECON 301 (01) - Introduction to Econometrics I March , 2012 METU - Department of Economics Using this estimate, can we get the estimates of 0 β and 1 using the relationship * 0 0 10 X ββ = + ? The answer is no since 0 ˆˆ 22 X = + is an ill-posed problem : there are two unknowns ( 0 ˆ and 1 ˆ ) but we have only one equation, there is an infinity of solutions to this equation: 0 and 1 are not identified . Hence, we see that if the X t values in a given sample are all same, then there is perfect linear relationship between independent variables, hence the problem becomes ill-posed. In other words, the problem becomes again unsolvable !
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Instructor Dr Ozan ERUYGUR e mail oeruygurgmailcom Lecture...

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