Lecture+1+Review+SLR

I i i i i i the error term the unobservable is not

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Unformatted text preview: residual is not the same as the error term, since ˆˆ ˆ u = y − α − β x , whereas u = y − α − β x (i = 1, 2,..., n) . i • i i i i i The error term (the unobservable) is not observed, while residual is observed. ˆˆˆ The sample regression function (the OLS regression line): Y = α + β X . ˆ ˆ Recall residual is defined as ui = yi − yi . It is the difference between the true value yi ˆ and the predicted value yi . Residual is the part that is not explained by the ˆˆ ˆ regression line yi = α + β xi . • • ˆˆˆˆ ˆ So yi = yi + ui = α + β xi + ui (i = 1, 2,..., n) ˆ Note that ui ≠ ui , i.e., residual is not the same as the error term, since ˆˆ ˆ u = y − α − β x , whereas u = y − α − β x (i = 1, 2,..., n) . i • i i i i i The error term (the unobservable) is not observed, while residual is observed. Figure: Scatter plot, the estimated regression line, and residuals Ordinary Least Squares (OLS) ˆ ˆ How do we calculate α and β ? We need to choose a criterion. OLS Criterion Intuitively, OLS fits a line thro...
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