Ozan eruygur e mail oeruygurgmailcom lecture notes 15

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Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 15 2. Another feature of the model is that although * 0 ˆ and 1 ˆ are unbiased estimates of * 0 and 1 , 0 (the parameter entering the original model) when estimated as * 0 ˆ 0 ˆ e is itself a biased estimator . 4 0 ˆ is not BLUE because, apart from unbiasedness, 0 ˆ is a non linear estimator. However, 0 ˆ will satisfy the large-sample properties and hence 0 ˆ is a consistent and asymptotically efficient estimator. B. The Semi-log Models: Log Lin and Lin Log Model These are called semilog models because only one variable (in this case the regressand) appears in the logarithmic form. For descriptive purposes a model in which the regressand (dependent variable) is logarithmic will be called a log-lin model . A model in which the regressand (dependent variable) is linear but the regressor(s) (independent variable) are logarithmic is called a lin-log model . i. Log-Lin Model Consider the following model: 0 1 t t X u t Y e   4 In most practical problems, however, the intercept term is of secondary importance, and one need not worry about obtaining its unbiased estimate.
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 16 which may also be expressed as follows: 0 1 ln ln ln t t t Y X u or simply, * * 0 1 ln t t t Y X u (1) where * 0 0 ln and * 1 1 ln . Hence, notice that, the estimators 0 ˆ and 1 ˆ are again non-linear transformations of * 0 ˆ and * 1 ˆ (since * 0 ˆ 0 ˆ e and * 1 ˆ 1 ˆ e ). Thus, 0 ˆ and 1 ˆ estimators are not unbiased but consistent and asymptotically efficient. Here, note that * 1 ln t t d Y dX . On the other hand, we can write that t t relative in Y ln absolute in X t t t t t dY d Y Y dX dX . Hence, in this model: * t 1 t relative in Y absolute in X . Therefore, * 1 100 gives the % in dependent variable given 1 unit of in independent variable (See Appendix 1 for interpretation issue).
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 17 ii. Lin- Log Model Consider the following model: 1 0 t t Y u t e X e which may also be expressed as follows: 0 1 ln ln t t t Y X u (2) or simply, * 0 1 t t t Y Z u where * 0 0 ln and ln t t Z X . Hence, notice that, the estimator 0 ˆ estimator is not unbiased (and BLUE) but consistent and asymptotically efficient. Here, note that 1 ln t t dY d X . On the other hand, we can write that t t absolute in Y ln relative in X t t t t t dY dY d X dX X . Hence, in this model: 1 /100 gives a unit in dependent variable given % in independent variable.
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 18
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