02 STAT descriptive stats 3-LONG SLIDES(1)

02 STAT descriptive stats 3-LONG SLIDES(1) - DESCRIPTIVE...

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1 DESCRIPTIVE STATISTICS Dr. V.R. Bencivenga Economic Statistics Economics 329 DESCRIPTIVE STATISTICS PART 3 Outline I. Covariance II. Correlation III. More on linear transformations IV. Regression V. Correlation and regression: “regression to the mean” VI. A simple model of “regression to the mean” Appendix: Two proofs (OPTIONAL) Objectives: Methods for describing how the distributions of two variables are related What it means for two variables to be more “tightly” associated How is strength of association affected by “stretching/shrinking” and “shifting” the variables? Estimating a line through a cloud of points Relationship between “correlation” and the “best fitting line” (regression line)
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2 DESCRIPTIVE STATISTICS III. REGRESSION
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3 DESCRIPTIVE STATISTICS Suppose we have data on two variables, Y and X. Y X
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4 DESCRIPTIVE STATISTICS Let’s call this line X ˆ ˆ 1 0 . Y X For now, we won’t specify the DGP, or investigate all we can do with the estimated line once we compute it. We’ re just fitting a curve, to see what the relationship in the data looks like.
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5 DESCRIPTIVE STATISTICS Y X 0 ˆ intercept
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6 DESCRIPTIVE STATISTICS Y X 1 ˆ slope
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7 DESCRIPTIVE STATISTICS Y X X ˆ ˆ 1 0
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8 DESCRIPTIVE STATISTICS Which line? We want the “best - fitting” line! Y X
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9 DESCRIPTIVE STATISTICS W e decide the “best - fitting” line is the one that minimizes the sum of the squared vertical distances from the data points to the fitted line (sum of squared “residuals”) . Y X positive “residual” negative “residual”
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10 DESCRIPTIVE STATISTICS 2 Y X ˆ ˆ 1 0 2 1 0 X ˆ ˆ fitted value of 1 Y 1 1 0 X ˆ ˆ 2 X 1 X i 1 0 X ˆ ˆ = point on the fitted line corresponding to the value i X ) X ˆ ˆ ( Y i 1 0 i = vertical distance from ) X , Y ( i i to the fitted line (“residual”) n 1 i 2 i 1 0 i ) X ˆ ˆ Y ( = sum of squared residuals (Q) fitted value of 2 Y 2 1 0 X ˆ ˆ 2 Y fitted value of 1 Y 1 1 0 X ˆ ˆ 1 Y
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11 DESCRIPTIVE STATISTICS Here are the equations (“formulas” ) for 0 ˆ and 1 ˆ that minimize the sum of squared residuals (Q): 2 X XY 1 S S ˆ X ˆ Y ˆ 1 0 Slope estimate = covariance between X and Y ÷ variance of X The fitted line goes through the “point of means:” X ˆ ˆ Y 1 0
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