II_REG_k=2_2011[1]-2

II_REG_k=2_2011[1]-2 - II 1 James B. McDonald Brigham Young...

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II 1 James B. McDonald Brigham Young University 9/2011 II. TWO VARIABLE LINEAR REGRESSION MODEL Several applications about the importance of having information about the relationship between economic variables were illustrated in the introduction. This section provides some essential building blocks used in estimating and analyzing "appropriate" functional relationships between two variables. We first consider estimation problems associated with linear relationships . The properties and distribution of the least squares estimators are considered. Diagnostic and test statistics which are important in evaluating the adequacy of the specified model are then discussed. A methodology for forecasting and the determination of confidence intervals associated with the linear model is presented. Finally, some alternative nonlinear functional forms are presented. A. INTRODUCTION Consider the model Y t = β 1 + β 2 X t + ε t with n observations (X 1 ,Y 1 ), . . ., (X n ,Y n ) which are graphically depicted as ε t : true random disturbance or error term (vertical distance from the observation to the line) Random behavior Measurement error (Y) Omitted variables β 1 + β 2 X t : population regression line β 1 and β 2 are unknown
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II 2 Population Regression Function : The observations don't have to lie on the population regression line, but it is usually assumed that E(Y t | X t ) = β 1 + β 2 X t , i.e., the expected value or the "average" value of Y corresponding to any given value of X lies on the population regression line. An important objective of econometrics is to estimate the unknown parameters (β 1 , β 2 ), and thereby estimate the unknown population regression line. This estimated regression line is referred to as the sample regression line. Again, the sample regression line is an estimator of the population regression line. Sample Regression Function: e t (the residual) is the vertical distance from the Y t to the sample regression line, so t t 1 2 t t t ˆˆ ˆ e Y X Y Y , whereas t t 1 2 t YX It is important to recognize that the residual (e t ) is an estimate of the equation error or random disturbance (ε t ) and may have different properties. 12 observed estimated random Y disturbance or regression "residual" line estimated Y for a given X ˆ t t t tt Y X e Ye sample observed error or population Y random regression disturbance line t t t
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II 3 B. THE ESTIMATION PROBLEM (1) Given a sample of (X t ,Y t ): (X 1 ,Y 1 ), . . ., (X n ,Y n ), Y t _____________________________ X t (2) Estimate the intercept (β 1 ) and the slope (β 2 ). The estimates will be denoted by 12 ˆˆ , . Note that each different guess of β 1 and β 2 , i.e., 1 ˆ β and 2 ˆ β , gives a different sample regression line. How should 1 ˆ β and 2 ˆ β be selected? There are many possible approaches to this problem. We now review five possible alternatives and then carefully develop a method known as least squares.
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This note was uploaded on 02/29/2012 for the course ECON 388 taught by Professor Mcdonald,j during the Winter '08 term at BYU.

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II_REG_k=2_2011[1]-2 - II 1 James B. McDonald Brigham Young...

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