1 Econ 240A Power 7

2 This 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(    f ? Sample Sample Statistic: Sample Statistic Pop.

5 The Sample Variance, s 2 Is distributed with n-1 degrees of freedom (text, 12.2 “inference about a population variance) (text, pp , 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. l expected value of the mean: 0.5 l 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:

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

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

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13 The UC Budget

14 The UC Budget §The part of the UC Budget funded by the state from the general fund

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

18 UC General Fund Expenditures, Appendix p , General fund actual, $2,901,257, , estimated $2,175,205, , estimated $2,806,267,000

19 UC General Fund Expenditures, Appendix p. 46

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28 How to Forecast the UC Budget? §Linear Trendline?

29 Trend Models

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32 Slope: increase of Million $ per year Governor’s Proposed Increase Million $

33 Linear Regression Trend Models §A good fit over the years of the data sample may not give a good forecast

34 How to Forecast the UC Budget? §Linear trendline? §Exponential trendline ?

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36 Trend Models

37 An Application

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39 Time Series Trend Analysis §Two Steps l Select a trend model l Fit the trend model Graphically algebraically

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

41 Linear Trend Model Fitted to UC Budget UCBUDB(t) = *t, R 2 = 0.943

42 Time Series Models §Linear l UCBUD(t) = a + b*t + e(t) l where the estimate of a is the intercept: $ Billion in l where the estimate of b is the slope: $ billion/yr l 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

43 Exponential Trend Model Fitted to UC Budget

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45 lnUCBudB(t) = a-hat + b-hat*time lnUCBudB(t) = *time Exp( ) = B ( ) intercept

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47 Time Series Models §Exponential l UCBUD(t) = UCBUD(68-69)*e b*t e u(t) l UCBUD(t) = UCBUD(68-69)*e b*t + u(t) l where the estimate of UCBUD(68-69) is the estimated budget for l where the estimate of b is the exponential rate of growth

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

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

50 Naïve Forecasts §Average §forecast next year to be the same as this year

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52 UC Budget Forecasts for * 1.068x$2,806,207,000; exponential trendline forecast ~$4.5 B Actual:$2,806,207,000 in Governor’s Budget Summary for 05-06

53 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)

54 Application of Bivariate Plot §O-Ring Failure §Plot zeros (no failure) and the ones (failure) versus launch temperature for the 24 launches prior to Challenger

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56 Linear Approximation to Backward Sigmoid

57 Ordinary Least Squares

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

59 §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

60 Minimize the Sum of Squared Errors

61 How to Find a-hat and b-hat? §Methodology l grid search l differential calculus l likelihood function

62 Grid Search, a-hat=0, b-hat=80

63 Grid Search a-hat b-hat Find the point where the sum of squared errors is minimum

64 Differential Calculus §Take the derivative of the sum of squared errors with respect to a-hat and with respect to b-hat and set to zero. §Divide by -2*n §or

65 Least Squares Fitted Parameters §So, the regression line goes through the sample means. §Take the other derivative: §divide by -2

66 Ordinary Least Squares(OLS) §Two linear equations in two unknowns, solve for b-hat and a-hat.

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68 O-Ring Failure Versus launch temperature

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