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Let us now add another variable z to the model so

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Let us now add another variable Z to the model so that the equation becomes 0 1 2 t t t t Y X Z u β β β = + + + . However, suppose that the Z variable just the double of X variable: 2 t t Z X = . Can we obtain an estimate for 2 β using OLS estimation? The answer is no since there is a perfect linear relationship between X and Z (i.e., 2 t t Z X = ). 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: 0 1 2 t t t t Y X Z u β β β = + + + 0 1 2 2 t t t t Y X X u β β β = + + + 0 1 2 ( 2 ) t t t Y X u β β β = + + + * 0 1 t t t Y X u β β = + + 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: * 0 1 4 15 β β + = * 0 1 8 23 β β + = * 0 1 12 28 β β + = 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 β and * 1 β can be obtained using Ordinary Least Squares (OLS) estimation. Suppose that we have obtained the estimate for * 1 β as * 1 ˆ 5 β = . Using this estimate, can we get the estimates of 1 β and 2 β using the relationship * 1 1 2 2 β β β = + ? The answer is no since 1 2 ˆ ˆ 5 2 β β = + is an ill-posed problem : there are two unknowns ( 1 ˆ β and 2 ˆ β ) but we have only one equation, there is an infinity of solutions to this equation. It is said that 1 β and 2 β 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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