1 Econ 240A Power 7

2 This Week, So Far §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, s 2 Is distributed with n-1 degrees of freedom (text, 12.3 “inference about a population variance) (text, pp , Chi-Squared distribution)

5 Text Chi-Squared Distribution

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

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

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

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

10 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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12 The UC Budget

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

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16 Total General Fund Expenditures Appendix, p.25 Schedule 6

17 UC General Fund Expenditures, Appendix p. 46

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21 p. 94

22 p. 94

23 p. 95

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

26 Trend Models

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28 Forecast increase $84 million

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

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

32 Forecast growth rate: 6.8%/yr

33 Time Series Models §Linear l UCBUD(t) = a + b*t + e(t) l where the estimate of a is the intercept: $ million in l where the estimate of b is the slope: $81.6 million/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

34 intercept slope Error in 01-02

35 Time Series Models §Exponential l UCBUD(t) = UCBUD(68-69)*e b*t e e(t) l UCBUD(t) = UCBUD(68-69)*e b*t + e(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

36 Forecast growth rate: 6.8%/yr 1 year forecast from * = M$ Exponential rate of growth Estimated UCBUD in 68-69

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

38 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 + e(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 + e(t) ] l and the logarithm is the inverse function of the exponential: l ln[UCBUD(t)] = ln[UCBUD(68-69)] + b*t + e(t) l so ln[UCBUD(68-69)] is the intercept “a”

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

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

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

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44 UC Budget Forecasts for * 1.068x$3,038,666,000; exponential trendline forecast ~$4.3 B Actual:$2,670,529,000 in Governor’s Budget Summary

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46 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)

47 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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49 Linear Approximation to Backward Sigmoid

50 Ordinary Least Squares

51 intercept slope Error in 01-02

52 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)

53 §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

54 Minimize the Sum of Squared Errors

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

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

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

58 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

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

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

61 Dependent Variable: UCBUD Method: Least Squares Dependent Variable: UCBUD Sample: Included observations: 36 VariableCoefficientStd. Errort-StatisticProb. C T R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

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63 O-Ring Failure Versus launch temperature