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Let us now add another variable Zto the model so that the equation becomes 012ttttYXZuβββ= +++. However, suppose that the Zvariable just the double of Xvariable: 2ttZX=. Can we obtain an estimate for 2βusing OLS estimation? The answer is nosince there is a perfect linear relationship between Xand Z(i.e.,2ttZX=). 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: 012ttttYXZuβββ= +++0122ttttYXXuβββ= +++012(2)tttYXuβββ=+++*01tttYXuββ= ++Hence, using the data of three observations (T=3) for Xand Y: X1=4, Y1=15; X2=8, Y2=23 and X3=12, Y3=28, the equations are written as: *01415ββ+=*01823ββ+=*011228ββ+=As in the previous case, this is the T>k+1situation. There is no perfect fit(data points are noton the line), and hence the relationship between Yand Xis notdeterministic. The estimatesfor 0βand *1βcan be obtained using Ordinary Least Squares (OLS) estimation. Suppose that we have obtained the estimate for *1βas *1ˆ5β=. Using this estimate, can we get the estimates of 1βand 2βusing the relationship *1122βββ=+? The answer is nosince 12ˆˆ52ββ=+is an ill-posed problem: there are two unknowns (1ˆβand 2ˆβ) but we have only one equation, there is an infinity of solutions to this equation. It is said that 1βand 2βare 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