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## Parameter Estimation: The Least Squares Method

Given data for the dependent and independent variables, X and Y, how should we estimate the values for , the model parameters? For this we can use the least squares procedure. That is, estimate by minimizing the total squared differences between observed and predicted values. The difference between the observed and predicted values, often times called the residual, is, in matrix notation, . The squared residual is . Thus,

To minimize Equation 3.13.4, we take its derivative with respect to , set it equal to zero and solve for .

and thus3.14

 (3.13.5)

If we substitute our estimated parameters, , into Equation 3.13.4, we get the following simplification for calculating the squared residual:

 (3.13.6)

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Frank Starmer 2004-05-19
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