Chap I Statistical Aspects of Regression

Chap I Statistical Aspects of Regression - Statistical...

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Unformatted text preview: Statistical Aspects of Regression IESEG 3 – Econometrics – H. Daher IESEG Econometrics Regression as a Best Fitting Line • • • • We begin with simple regression to understand the We relationship between two variables, X and Y. relationship Example “house price“ : house price versus lot size. Example Regression fits a line through the points in the XY-plot that plot best captures the relationship between house price and lots size. size. Question: What do we mean by “best fitting” line? Question: 2 Regression as a Best Fitting Line Figure 3.1: XY Plot of House Price vs. Lot Size 200000 180000 House Price (Canadian dollars) 160000 140000 120000 100000 80000 60000 40000 20000 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Lot Size (square feet) 3 Distinction between True and Estimated Coefficient • • • • Assume a linear relationship exists between Y and X: o Y = α + βX α = intercept of line β = slope of line Example: Y = house price, X = lot size Y = 34000 + 7X Regression analysis uses data (X and Y) to make a guess or estimate of what α and β are. ˆ ˆ Notation: α and β are the estimates of α and β. 4 Distinction Between Errors and Residuals • True Regression Line: Y = α + βX + e e = Y - α - βX e = error error Estimated Regression Line: Y= +X+u u=Y - - X u = residual residual • 5 Least square intuition • ˆ ˆ How do we choose α and β ? o find “best fitting” line which makes the residuals as small as possible. o What do we mean by “as small as possible”? The one that minimizes the sum of squared residuals. o Hence, we obtain the “ordinary least squares” or OLS estimator. ∑ (Y − Y )( X − X ) i i ˆ β= ∑ ( X − X )2 i and and ˆ ˆ α = Y−β ×X 6 Least square intuition FIGURE 4.1 Y B u2 u3 ˆˆ Y = α + βX u1 A C X 7 Jargon of Regression • • • • • Y = dependent variable. X = explanatory (or independent) variable. α and β are coefficients. coefficients. and are OLS estimates of coefficients “Run a regression of Y on X 8 Interpreting OLS Estimates ˆˆ Y = α+ β X + u ˆ β Interpretation of • βˆ is estimate of the marginal effect of X on Y • Using regression model: dY = βˆ dX • • A measure of how much Y tends to change when you change X. ˆ “If X changes by 1 unit then Y tends to change by β units”, where “units” refers to what the variables are measured in (e.g. $, £, %, hectares, metres, etc.). 9 R2: A Measure of Fit 10 R2: A Measure of Fit 11 Statistical Aspects of Regression 12 A Confidence Interval for β 13 A Confidence Interval for β 14 15 Hypothesis Testing 16 Hypothesis Testing 17 Hypothesis Testing • • • • • • • Q: What do we mean by “big” and “small”? Q: and A: Look at P-value. value. If P-value ≤ .05 then t is “big” and conclude β≠0. value β≠ If P-value >.05 then t is “small” and conclude β=0. value and Useful (but formally incorrect) intuition: Useful P-value measures the probability that β = 0. value .05 = 5% = level of significance Other levels of significance (e.g. 1% or 10%) occasionally Other used used 18 Example: The Regression of Executive Compensation on Profits • • • • t-ratio is 7.227937. Is this big? ratio Yes, the P-value is 5.5×10-10 which is much less than .05. Yes, Hence, we conclude again that β ≠ 0. Hence, “The coefficient on profits is significantly different from The zero.” zero. “Profits have statistically significant explanatory power for Profits executive compensation.” executive “The hypothesis that β = 0 can be rejected at the 5% The significance level.” 19 Testing on R2: The F-statistic 20 Example: The Regression of Executive Compensation on Profits 21...
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