Lecture4 - Lecture 4 Ordinary Least Square(OLS Econ 444...

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Unformatted text preview: Lecture 4 : Ordinary Least Square(OLS) Econ 444, Fall 2010 Jan 21, 2010 The Estimated Regression Consider the regression equation, Y i = β + β 1 X i + i i = 1 , 2 ...., N (1) The problem is in real life we only observe a sample and hence the population parameters β and β 1 are unknown. The regression coefficients β and β 1 need to be estimated from a data set (or a sample). We denote the estimators by ˆ β and ˆ β 1 , respectively.Then, ˆ Y i = ˆ β + ˆ β 1 X i (2) is called the estimated (or sample) regression . ˆ Y i is called the estimated or fitted value and ˆ β and ˆ β 1 are called estimated regression coefficients . Estimating a regression using OLS In this class we discuss how to estimate the unknown parameters β and β 1 in the regression equation, Y i = β + β 1 X i + i (3) This is equivalent to fitting a straight line to the data. Visually inspecting the data and fitting a line is not a good idea as it is too arbitrary and imprecise. We choose β and β 1 in such a way that the residual e i = Y i- ˆ Y i = Y i- ( ˆ β + ˆ β 1 X i ) (4) is as "small" as possible under a certain criterion. There are many criteria that one could use but for this course we will use the the Ordinary Least Squares method. The estimators of β and β 1 based on this method are called OLS estimators are denoted by ˆ β and ˆ β 1 respectively. Algebraically, the least squares principle tells us to choose ˆ β and ˆ β 1 , so that we minimize residual sum of squares (RSS) N X i = 1 e 2 i = N X i = 1 ( Y i- ( ˆ β + ˆ β 1 X i ) 2 (5) by choosing ˆ β and ˆ β 1 appropriately. How Does OLS Work?...
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Lecture4 - Lecture 4 Ordinary Least Square(OLS Econ 444...

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