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First Derivatives

First Derivatives. y. y. y. x. x. x. dy/dx = 0. dy/dx > 0. dy/dx < 0. First Derivatives. y. y. y. y. x. x. x. x. dy/dx < 0. dy/dx > 0. dy/dx < 0. dy/dx > 0. First and Second Derivatives. y. y. x. x. Y is maximized and minimized when dy/dx = 0. Rules of Differentiation.

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First Derivatives

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  1. First Derivatives y y y x x x dy/dx = 0 dy/dx > 0 dy/dx < 0

  2. First Derivatives y y y y x x x x dy/dx < 0 dy/dx > 0 dy/dx < 0 dy/dx > 0

  3. First and Second Derivatives y y x x Y is maximized and minimized when dy/dx = 0

  4. Rules of Differentiation

  5. Summation Operators

  6. Summation Operators

  7. Expectations Operators

  8. Expectation Operators

  9. Expectations Operation If X,Y are independent:

  10. OLS Estimation • In reality: • OLS line: • Error term: • omitted variables • intrinsic randomness • errors in measurement

  11. OLS Estimation • Objectives • Find average Y given X • Test hypothesis • Predict or forecast

  12. OLS Estimation Minimize the sum of the errors squared: Solve simultaneously:

  13. OLS Problem • Given the price and length of the following textbooks, find the relationship between length and price: Book Price Length 1 $11 100 2 $15 225 3 $24 400 4 $20 350

  14. OLS Problem

  15. OLS Problem

  16. OLS Problem • If X=100, Yhat =$10.46

  17. Special Case

  18. Special Case Deviations from the mean

  19. Special Case

  20. Formal Exposition of Model • Model • X is nonstochastic • E(i) = 0 • Var(i) = 2 • E(i,j)=0 for i=j

  21. Formal Exposition of Model

  22. Formal Exposition of Model

  23. Formal Exposition of Model

  24. Formal Exposition of Model

  25. Gauss Markov Theorem The OLS estimators are BLUE: best, linear, unbiased estimators

  26. Gauss Markov Theorem • The variance of bhat varies: • Directly with the variance of e • Inversely with Sxi2 • The variance of ahat varies: • Directly with s2

  27. Book Example

  28. Confidence Intervals • Ho: b=bo • Ha: bbo • Ho: a=ao • Ha: aao

  29. Confidence Intervals • Ho: b=0 • Ha: b0 • Ho: a=0 • Ha: a0

  30. Book Example • Test if length is a significant explanatory variable of price • Test if b differs from zero • Ho: b=0 • Ha: b0

  31. Book Example • Test if the intercept differs significantly from zero • Test if a differs from zero • Ho: a=0 • Ha: a0

  32. Goodness of Fit • The residuals can help to explain how well the regression line fits the points. • Variation (not variance!) of Y can be broken down into the portion explained by the regression equation and the unexplained portion (error term) of the model

  33. Goodness of Fit

  34. Goodness of Fit

  35. Goodness of Fit

  36. Back to Book Example

  37. ANOVA or F-Test • F-distribution is the ratio of two 2 distributions divided by their respective degrees of freedom. • A 2 distribution is the sum of squares of N independently distributed normal random variables. • The basis of the F-test is the idea that the ratio of the explained variation to the unexplained variation should be high if the tested model is a reasonable approximation of the true model.

  38. ANOVA or F-Test • By dividing RSS and ESS by their degrees of freedom, we will convert the variations to variances. • As long as the error terms are normally distributed with a zero mean, the variances will follow a 2distribution.

  39. ANOVA or F-Test • By calculating an F-statistic: • We are testing the hypothesis that: • Ho: b=0 • Ha: b0

  40. ANOVA or F-Test • Since in the bivariate model case, the null hypothesis for the t-test and the F-test are the same, both should give the same accept/reject answer to the null hypothesis • In fact

  41. Back to Book Example

  42. Scaling and the Units of Measurement • Changing the scale of measurement of the dependent variable changes the corresponding scaling of all the regression coefficients (including residuals and standard errors). • Changing the scale of measurement of a single independent variable changes its coefficient and corresponding standard error but all other statistics are the same.

  43. Forecasting • Point estimate • Interval estimate • Standard error • The prediction variance varies directly with s2 • The denominator of the third term is t-1 times the sample variance of X, so as the variance of X increase, the variance of the prediction decreases • The prediction variance all increase the farther the prediction is from the mean of X.

  44. Formal Exposition of Model

  45. Book Example • How much will a book that has 300 pages sell for? • Point Estimate Yhat = 6.29 + .0417*300 = $18.80 • 95% Confidence Interval Estimate

  46. Causality • Direction of causality • Casual or causal relationship • Spurious correlation • Simultaneity • Crime and police officers

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