ECON301_Handout_09_1213_02

# ˆ and ˆ 2 1 ˆ and 1 ˆ 3 ˆ var and ˆ var 4 1 ˆ

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ˆ and * 0 ˆ 2. 1 ˆ and * 1 ˆ 3. 0 ˆ var( ) and * 0 ˆ var( ) 4. 1 ˆ var( ) and * 1 ˆ var( ) 5. 2 ˆ and * 2 ˆ 6. 2 XY R and * * 2 X Y R From least-squares theory, note and show the following results:

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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 12 1. * 0 1 0 ˆ ˆ w 2. * 1 1 1 2 ˆ ˆ w w   3. *2 2 2 1 ˆ ˆ w 4. * 2 0 1 0 ˆ ˆ var( ) var( ) w 5. 2 * 1 1 1 2 ˆ ˆ var( ) var( ) w w 6. * * 2 2 XY X Y R R One can draw some conclusions from the findings above. For example, if 1 w = 2 w , that is, the scaling factors are identical, the slope coefficient and its standard error remain unaffected in going from the ( t Y , t X ) to the ( * t Y , * t X ) scale, which should be intuitively clear. However, the intercept and its standard error are both multiplied by 1 w . 3. Functional Forms of Regression Models In particular, we discuss the following regression models: 1. The Double-log (loglinear) model 2. Semi-log models 3. Reciprocal models
ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 13 A. The Double-log (Loglinear) Model (Constant Elasticity Model) Consider the following model, known as the exponential regression model: 1 0 t u t t Y X e which may also be expressed as follows: 0 1 ln ln ln t t t Y X u or more simply, * 0 1 t t t W Z u where * 0 0 ln , ln t t W Y and ln t t Z X . Note that this model is linear both in the parameters and variables, and hence can be estimated by OLS regression. Because of this linearity, such models are called log-log , double-log , or loglinear models 3 . 3 Do not confuse the expression of loglinear with log-lin . See: Erlat, H. (1997), Introduction to Econometrics , Chapter 2: The Simple Linear Regression Model, p.78. However, Ramanathan uses loglinear expression for log-lin models; this is not compatible what we use in this course. We follow the terminology of Erlat (1997, p.78) and Gujarati (2004, p.175) and hence use the term loglinear to denote double-log models.

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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 14 Here, note that 1 ln ln t t d Y d X . On the other hand, we can write that ln ln t t t t t XY t t t t t dY d Y Y dY X Y d X dX dX X . Hence, in this model the slope coefficient is elasticity (the X elasticity of Y, or elasticity of Y with respect to X). Examples: 1. Cobb Douglas Production Function ( 1 0 t u t t Q L e ) 2. Constant Elasticity Demand Function ( 1 0 t u t t Q P e ) Two special features of the log-linear model may be noted: 1. The model assumes that the elasticity coefficient between Y and X, 1 , remains constant throughout, hence the alternative name constant elasticity model .
ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR

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