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Econ 240A

Econ 240A. Power 7. Last Week. Normal Distribution Lab Three: Sampling Distributions Interval Estimation and HypothesisTesting. Outline. Distribution of the sample variance The California Budget: Exploratory Data Analysis Trend Models Linear Regression Models Ordinary Least Squares.

adrienne-roderick
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Econ 240A

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  1. Econ 240A Power 7

  2. Last Week • Normal Distribution • Lab Three: Sampling Distributions • Interval Estimation and HypothesisTesting

  3. Outline • Distribution of the sample variance • The California Budget: Exploratory Data Analysis • Trend Models • Linear Regression Models • Ordinary Least Squares

  4. The Sample Variance, s2 Is distributed with n-1 degrees of freedom (text, 12.3 “inference about a population variance) (text, pp. 258-262, Chi-Squared distribution)

  5. Text Chi-Squared Distribution

  6. Text Chi-Squared Table

  7. Example: Lab Three • 50 replications of a sample of size 50 generated by a Uniform random number generator, range zero to one. • expected value of the mean: 0.5 • expected value of the variance: 1/12

  8. Histogram of 50 Sample Means, Uniform, U(0.5, 1/12) Average of the sample means: 0.4963

  9. Histogram of 50 sample variances, Uniform, U(0.5, 0.0833) Average sample variance: 0.08352

  10. Confidence Interval for the first sample variance of 0.07667 • A 95 % confidence interval Where taking the reciprocal reverses the signs of the inequality

  11. The UC Budget

  12. The UC Budget • The part of the UC Budget funded by the state from the general fund

  13. Appendix p. 25

  14. Appendix p. 25

  15. Appendix p. 47

  16. P. 98

  17. P. 98

  18. P. 99

  19. How to Forecast the UC Budget? • Linear Trendline?

  20. Trend Models

  21. Forecast increase $84 million

  22. Linear Regression Trend Models • A good fit over the years of the data sample may not give a good forecast

  23. How to Forecast the UC Budget? • Linear trendline? • Exponential trendline ?

  24. Forecast growth rate: 6.8%/yr

  25. Time Series Models • Linear • UCBUD(t) = a + b*t + e(t) • where the estimate of a is the intercept: $-10.56 million in 68-69 • where the estimate of b is the slope: $84 million/yr • where the estimate of e(t) is the the difference between the UC Budget at time t and the fitted line for that year • Exponential

  26. Error in 01-02 slope intercept

  27. Time Series Models • Exponential • UCBUD(t) = UCBUD(68-69)*eb*tee(t) • UCBUD(t) = UCBUD(68-69)*eb*t + e(t) • where the estimate of UCBUD(68-69) is the estimated budget for 1968-69 • where the estimate of b is the exponential rate of growth

  28. Estimated UCBUD in 68-69 Exponential rate of growth Forecast growth rate: 6.8%/yr 1 year forecast from 2003-04 1.068*3038.666 = 3245.295 M$

  29. Linear Regression Time Series Models • Linear: UCBUD(t) = a + b*t + e(t) • How do we get a linear form for the exponential model?

  30. Time Series Models • Linear transformation of the exponential • take natural logarithms of both sides • ln[UCBUD(t)] = ln[UCBUD(68-69)*eb*t + e(t)] • where the logarithm of a product is the sum of logarithms: • ln[UCBUD(t)] = ln[UCBUD(68-69)]+ln[eb*t + e(t)] • and the logarithm is the inverse function of the exponential: • ln[UCBUD(t)] = ln[UCBUD(68-69)] + b*t + e(t) • so ln[UCBUD(68-69)] is the intercept “a”

  31. 2003-04 1968-69

  32. Exponential rate of growth ln UCBUD at t=0 exp[5.932] = 376.9 observed = $291.3

  33. Estimated UCBUD in 68-69 Exponential rate of growth Forecast growth rate: 6.8%/yr

  34. Naïve Forecasts • Average • forecast next year to be the same as this year

  35. UC Budget Forecasts for 2004-05 * 1.068x$3,038,666,000; exponential trendline forecast ~$4.3 B

  36. Time Series Forecasts • The best forecast may not be a regression forecast • Time Series Concept: time series(t) = trend + cycle + seasonal + noise(random or error) • fitting just the trend ignores the cycle • UCBUD(t) = a + b*t + e(t)

  37. Ordinary Least Squares

  38. Error in 01-02 slope intercept

  39. Criterion for Fitting a Line • Minimize the sum of the absolute value of the errors? • Minimize the sum of the square of the errors • easier to use • error is the difference between the observed value and the fitted value • example UCBUD(observed) - UCBUD(fitted)

  40. The fitted value: • The fitted value is defined in terms of two parameters, a and b (with hats), that are determined from the data observations, such as to minimize the sum of squared errors

  41. Minimize the Sum of Squared Errors

  42. How to Find a-hat and b-hat? • Methodology • grid search • differential calculus • likelihood function

  43. Grid Search, a-hat=0, b-hat=80

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