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Size: 1918
Comment: Regression
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Size: 1022
Comment: Estimates and residuals
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| Based on [[Econometrics/LinearRegression|regression]], the estimated outcome for observation ''i'' is: | |
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| {{attachment:estimate.svg}} | |
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| == Regression == | And the residual is: |
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| A regression line passes through two points: {{attachment:regression1.svg}} and {{attachment:regression2.svg}} Take the generic equation form of a line: {{attachment:b01.svg}} Insert the first point into this form. {{attachment:b02.svg}} This can be trivially rewritten to solve for ''a'' in terms of ''b'': {{attachment:b03.svg}} Insert the second point into the original form. {{attachment:b04.svg}} Now additionally insert the solution for ''a'' in terms of ''b''. {{attachment:b05.svg}} Expand all terms to produce: {{attachment:b06.svg}} This can now be eliminated into: {{attachment:b07.svg}} Giving a solution for ''b'': {{attachment:b08.svg}} This solution is trivially rewritten as: {{attachment:b09.svg}} Expand the formula for correlation as: {{attachment:b10.svg}} This can now be eliminated into: {{attachment:b11.svg}} Finally, ''b'' can be eloquently written as: {{attachment:b12.svg}} Giving a generic formula for the regression line: {{attachment:b13.svg}} |
{{attachment:residual.svg}} |
Econometrics Notation
Data
The number of observations is n.
The outcome variable is y. For observation i, the outcome value is yi.
The treatment variable is x1. For observation i, the treatment value is x1i.
The control variables are x2 through xk (up to k - 1 control variables). For observation i, a control value might be x2i.
Statistics
The average outcome is:
The variance is:
The standard deviation is:
The covariance between the treatment and outcome is:
The correlation between the treatment and outcome is:
Based on regression, the estimated outcome for observation i is:
And the residual is:
