Econ 299 Chapter3c - 3.9 Properties of the OLS Estimator...

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1 3.9 Properties of the OLS Estimator There exist a variety of methods to estimate  the coefficients of our model (b 1  and b 2 ) Why use Ordinary Least Squares (OLS)? 1) OLS minimizes the sum of squared errors,  creating a line that fits best with the  observations 2) With certain assumptions, OLS exhibits  beneficial statistical properties.  In particular,  OLS is BLUE.
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2 3.9.1 The OLS Estimator As seen before, our model is: Y i  = b + b 2 X i  + є i Where b1 and b2 are not observed.  OLS estimates  these parameters through: X b Y b X X Y Y X X b i i i 2 1 2 2 ˆ ˆ ) ( ) )( ( ˆ - = - - - =
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3 3.9 The OLS Estimator These OLS estimates create a straight line going  through the “middle” of the estimates: Studying and Marks 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 Marks Studying
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4 3.9.2 Fitted or Predicted Values From the above we see that often the actual data  points lie above or below the estimated line. Points on the line give us ESTIMATED y values for  each given x. The predicted or fitted y values are found using our x  data and our estimated b’s: i i X b b Y 2 1 ˆ ˆ + =
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5 3.9.2 Estimators Example Ols Estimation Qhat = 110/6 –(5/6)P For Price =4, Qhat = 110/6-(5/6)4 = (110-24)/6 = 14.3 Price 4 3 3 6 Pbar = 4 Quantity 10 15 20 15 Qbar=15
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6 3.9.3 Estimating Errors or Residuals The estimated y values (yhat) are rarely equal to their  actual values (y).  The difference is the error term: Y Y E i i ˆ - =
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7 3.9.3 Estimators Example Ols Estimation Qhat = 110/6 –(5/6)P For Price =4, Qhat =  14.3 Ehat = Y – Yhat = 10-14.3 = -4.3 Price 4 3 3 6 Pbar = 4 Quantity 10 15 20 15 Qbar=15
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8 3.9.4 Rationale for OLS Estimators Reason 1: OLS minimizes the sum of the squared errors,  providing the best fit line between the  observed data. Section 4 will use calculus to show that b1hat and  b2hat minimize squared errors.
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9 3.9.4 Rationale for OLS Estimators Reason 2: Under certain error term assumptions, OLS  estimators have good statistical properties. Of all linear and unbiased estimators, the OLS  estimators have the smallest variance.
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10 3.9.5 Statistical Properties of OLS In our model: Y, the dependent variable, is made up of two  components: a) b + b 2 X i  – a non-random component that  indicates the effect of X on Y.  In this course,  X is non-random. b) Є i  – a random error term representing other  influences on Y.
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11 3.9.5 Statistical Properties of OLS Error Assumptions: a) E(є i ) = 0; we expect no error; we assume the  model is complete b) Var(є i ) =  σ 2 ; the error term has a constant  variance c) Cov(є i , є j ) = 0; error terms from two different  observations are uncorrelated.  If the last error  was positive, the next error need not be  negative.
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This note was uploaded on 03/14/2009 for the course ECON ECON 299 taught by Professor Priemaza during the Spring '08 term at University of Alberta.

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Econ 299 Chapter3c - 3.9 Properties of the OLS Estimator...

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