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Econ 240A

Econ 240A. Power 7. Last Week, So Far. Normal Distribution Lab Three: Sampling Distributions Interval Estimation and Hypothesis Testing. Outline. Distribution of the sample variance The California Budget: Exploratory Data Analysis Trend Models Linear Regression Models

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Econ 240A

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  1. Econ 240A Power 7

  2. Last Week, So Far • Normal Distribution • Lab Three: Sampling Distributions • Interval Estimation and Hypothesis Testing

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

  4. Population Random variable x Distribution f(m, s2) f ? Pop. Sample Sample Statistic Sample Statistic:

  5. The Sample Variance, s2 Is distributed with n-1 degrees of freedom (text, 12.2 “inference about a population variance) (text, pp. 266-270, Chi-Squared distribution)

  6. Text Chi-Squared Distribution

  7. Text Chi-Squared Table 5 Appendix p. B-10

  8. Example: Lab Three • 50 replications of a sample of size 50 generated by a Uniform random number generator, range zero to one, seed =20. • expected value of the mean: 0.5 • expected value of the variance: 1/12

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

  10. Histogram of 50 sample variances, Uniform, U(0.5, 0.0833) Average sample variance: 0.0832

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

  12. The UC Budget

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

  14. Total General Fund Expenditures Appendix, p.11 Schedule 6

  15. UC General Fund Expenditures, Appendix p. 33 2004-05, estimated $2,175,205,000 2003-04, General fund actual, $2,901,257,000 2005-06, estimated $2,806,267,000

  16. UC General Fund Expenditures, Appendix p. 46

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

  18. Trend Models

  19. Slope: increase of 80.323 Million $ per year Governor’s Proposed Increase 186.712 Million $

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

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

  22. Trend Models

  23. An Application

  24. Time Series Trend Analysis • Two Steps • Select a trend model • Fit the trend model • Graphically • algebraically

  25. Trend Models • Linear Trend: y(t) = a + b*t +e(t) • dy(t)/dt = b • Exponential trend: z(t) = exp(c + d*t + u(t)) ln z(t) = c + d*t + u(t) • (1/z)*dz/dt = d

  26. Linear Trend Model Fitted to UC Budget UCBUDB(t) = 0.0363 + 0.0803*t, R2 = 0.943

  27. Time Series Models • Linear • UCBUD(t) = a + b*t + e(t) • where the estimate of a is the intercept: $0.0363 Billion in 68-69 • where the estimate of b is the slope: $0.0803 billion/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

  28. Exponential Trend Model Fitted to UC Budget

  29. lnUCBudB(t) = a-hat + b-hat*time lnUCBudB(t) = -0.896566 + 0.0611 *time Exp(-0.896566) = 0.408 B (1968-69) intercept

  30. Time Series Models • Exponential • UCBUD(t) = UCBUD(68-69)*eb*teu(t) • UCBUD(t) = UCBUD(68-69)*eb*t + u(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

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

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

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

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