Econ 299 Chapter4a - 4 Simple Regression Models Chapter 4...

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4. Simple Regression Models Chapter 4 will expand on concepts introduced in Chapter 3 to cover the following: 1) Estimating parameters using Ordinary Least Squares (OLS) Estimation 2) Hypothesis tests of OLS coefficients 3) Confidence intervals of OLS coefficients 4) Prediction 5) SHAZAM use
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4.1 OLS and Goodness of Fit Reviewing from Chapter 3, we have our model: Y i = b 1 + b 2 X i + є i Where: b 1 and b 2 are unknown (non-random) coefficients X values are non-random The error term, є i , is random with E(є i )=0; no expected error Var(є i )=σ 2 ; constant variance Cov(є i , є j )=0; no covariance between errors
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4.1 OLS and Goodness of Fit Aside from our model, we have a data set containing: N observations of X and Y Actual X and Y values Our data combines with our model and assumptions to estimate our coefficients: i N i i N i i i X b Y b X X Y Y X X b 2 1 1 2 1 2 ˆ ˆ ) ( ) )( ( ˆ - = - - - = = =
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4.1 Predicted and Error Using our estimated coefficients and ACTUAL X values, we obtain ESTIMATED or PREDICTED Y values: Using these predicted values, we can estimate error or the residual: i i X b b Y 2 1 ˆ ˆ ˆ + = i i i Y Y e ˆ ˆ - =
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4.1.1 Deriving OLS OLS is obtained by minimizing the sum of the square errors. This is done using the partial derivative < = - - - = < = - - - = - - = = 2 0 ) ˆ ( 2 ˆ 1 0 ˆ 2 ˆ ˆ : min 2 1 2 2 2 1 1 2 2 1 1 2 ˆ , ˆ 2 1 i i i i i i i i i i N i i b b X X b b Y b e X b b Y b e X b b Y e where e N N N N N N N
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4.1.1 Deriving OLS These can simplify to: < = = - - - - - - - - - - - - - - - - - - - - - - - - - - - - < = = - - a X e OR X X b b Y a e OR X b b Y i i i i i i i i 2 0 0 ) ˆ ( 1 0 0 ˆ 2 1 2 1 1 1 1 1
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4.1.1 Deriving OLS
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