1

Econ 240A

Power 7

2

This Week, So Far

Normal DistributionLab Three: Sampling DistributionsInterval Estimation and Hypothesis Testing

3

Outline

Distribution of the sample varianceThe California Budget: Exploratory Data

AnalysisTrend ModelsLinear Regression ModelsOrdinary Least Squares

4

PopulationRandom variable xDistribution f(f ?

Sample

Sample Statistic:

),(~ 2Nx

Sample Statistic

)1/()( 2

1

2

nxxsn

ii

Pop.

5

The Sample Variance, s2

22

1

22

/*)1(

)1/(])([

sn

nxixsn

i

Is distributed 2 with n-1 degrees of

freedom (text, 12.2 “inference about a population variance)(text, pp. 266-270, Chi-Squared distribution)

n

i

n

ii zxxsn

1 1

22222 /)(/)1(

6

TextChi-SquaredDistribution

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TextChi-SquaredTable 5Appendixp. B-10

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Example: Lab Three50 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

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Histogram of 50 sample means

05

1015

20

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.95

Mor

e

Sample Mean

Fre

qu

en

cy

Histogram of 50 Sample Means, Uniform, U(0.5, 1/12)

Average of the 50 sample means: 0.4963

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Histogram of 50 Sample Variances

0

5

10

15

20

0.01

0.03

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.25

Sample variance

Fre

qu

en

cy

Histogram of 50 sample variances, Uniform, U(0.5, 0.0833)

Average sample variance: 0.0832

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Confidence Interval for the first sample variance of 0.07667A 95 % confidence interval

95.0]0562.01161.0[

95.0]01.19/161.8[

95.0]42.71/07667.0*4936.32[

95.0]42.71/*)1(36.32[

95.0]42.7136.32[

2

2

2

22

2

p

p

p

snp

p

Where taking the reciprocal reverses the signs of the inequality

12

13

The UC Budget

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The UC Budget

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

15

16

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Total General Fund ExpendituresAppendix, p.11Schedule 6

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UC General Fund Expenditures, Appendix p. 33

2003-04, General fund actual, $2,901,257,000

2004-05, estimated $2,175,205,000

2005-06, estimated $2,806,267,000

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UC General Fund Expenditures, Appendix p. 46

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21

22

23

24

25

26

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UC Budget in Nominal Millions: 1968-69 through 2007-08

0

500

1000

1500

2000

2500

3000

3500

4000

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

9

80-8

1

82-8

3

84-8

5

86-8

7

88-8

9

90-9

1

92-9

3

94-9

5

96-9

7

98-9

9

00-0

1

02-0

3

04-0

5

06-0

7

Fiscal Year

Mill

ion

s $

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How to Forecast the UC Budget?

Linear Trendline?

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Trend Models

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UC Budget in Nominal Millions: 1968-69 through 2007-08

y = 80.323x + 36.343

R2 = 0.9431

0

500

1000

1500

2000

2500

3000

3500

4000

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

9

80-8

1

82-8

3

84-8

5

86-8

7

88-8

9

90-9

1

92-9

3

94-9

5

96-9

7

98-9

9

00-0

1

02-0

3

04-0

5

06-0

7

Fiscal Year

Mill

ion

s $

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UC Budget In Millions of Nominal Dollars, 1968-69 to 2006-07

y = 80.143x + 38.773

R2 = 0.9385

0

500

1000

1500

2000

2500

3000

3500

4000

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

9

80-8

1

82-8

3

84-8

5

86-8

7

88-8

9

90-9

1

92-9

3

94-9

5

96-9

7

98-9

9

00-0

1

02-0

3

04-0

5

06-0

7

Fiscal Year

Mill

ion

s $

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UC Budget in Nominal Millions: 1968-69 through 2007-08

y = 80.323x + 36.343

R2 = 0.9431

0

500

1000

1500

2000

2500

3000

3500

4000

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

9

80-8

1

82-8

3

84-8

5

86-8

7

88-8

9

90-9

1

92-9

3

94-9

5

96-9

7

98-9

9

00-0

1

02-0

3

04-0

5

06-0

7

Fiscal Year

Mill

ion

s $

Slope: increase of 80.323 Million $ per yearGovernor’s Proposed Increase 186.712 Million $

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Linear Regression Trend Models

A good fit over the years of the data sample may not give a good forecast

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How to Forecast the UC Budget?

Linear trendline?Exponential trendline ?

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Trend Models

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An Application

38

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Time Series Trend Analysis

Two Steps Select a trend model Fit the trend model

• Graphically

• algebraically

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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

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

UC Budget in Nominal Billions $: 1968-69 through 2007-08

y = 0.0803x + 0.0363

R2 = 0.9431

0

0.5

1

1.5

2

2.5

3

3.5

4

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

9

80-8

1

82-8

3

84-8

5

86-8

7

88-8

9

90-9

1

92-9

3

94-9

5

96-9

7

98-9

9

00-0

1

02-0

3

04-0

5

06-0

7

Fiscal Year

Bill

ion

s $

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Time Series ModelsLinear

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

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UC Budget in Billions $:1968-69 through 2007-08

y = 0.3838e0.0611x

R2 = 0.9047

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

9

80-8

1

82-8

3

84-8

5

86-8

7

88-8

9

90-9

1

92-9

3

94-9

5

96-9

7

98-9

9

00-0

1

02-0

3

04-0

5

06-0

7

Fiscal Year

Bill

ion

s $

Exponential Trend Model Fitted to UC Budget

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-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 10 20 30 40

TIME

LN

UC

BU

DB

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lnUCBudB(t) = a-hat + b-hat*timelnUCBudB(t) = -0.896566 + 0.0611 *timeExp(-0.896566) = 0.408 B (1968-69) intercept

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-0.4

-0.2

0.0

0.2

0.4

-2

-1

0

1

2

70 75 80 85 90 95 00 05

Residual Actual Fitted

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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

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Linear Regression Time Series Models

Linear: UCBUD(t) = a + b*t + e(t)How do we get a linear form for the

exponential model?

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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”

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Naïve Forecasts

Averageforecast next year to be the same as this

year

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A Naive Budget Forecast

0

500

1000

1500

2000

2500

3000

3500

4000

68-6

9

71-7

2

74-7

5

77-7

8

80-8

1

83-8

4

86-8

7

89-9

0

92-9

3

95-9

6

98-9

9

01-0

2

Fiscal Year

$ M

illi

on

s

UC Budget

Average

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UC Budget Forecasts for 2006-07

Method Increase Forecast

Linear Trend $80.5 M $2,886,707,000

Exponential Trend 6.4% $2,985,047,000*

Same as 2005-06 0 $2,806,207,000

Average 0 $1,603,671,000

* 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

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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 cycleUCBUD(t) = a + b*t + e(t)

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Application of Bivariate Plot

O-Ring FailurePlot zeros (no failure) and the ones (failure)

versus launch temperature for the 24 launches prior to Challenger

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O-Ring Failure (yes=1, No=0) Versus Launch Temperature

y = -0.0367x + 2.8583

R2 = 0.3254

-0.2

0

0.2

0.4

0.6

0.8

1

30 40 50 60 70 80 90

Launch Temperatue

Pro

ba

bili

ty

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O-Ring Failure (yes=1, No=0) Versus Launch Temperature

y = -0.0367x + 2.8583

R2 = 0.3254

-0.2

0

0.2

0.4

0.6

0.8

1

30 40 50 60 70 80 90

Launch Temperatue

Pro

ba

bili

ty

Linear Approximation to Backward Sigmoid

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Ordinary Least Squares

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Criterion for Fitting a LineMinimize 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)

)(ˆ)(ˆ tytye

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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

tbaty *ˆˆ)(ˆ

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Minimize the Sum of Squared Errors

062005

691968

2

2062005

691968

062005

691968

2

]*ˆˆ)([

])(ˆ)([ˆ

tbaty

tytye

61

How to Find a-hat and b-hat?

Methodology grid search differential calculus likelihood function

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

FiscalYear

Timeindex

UCBUD y-hat Error-hat

68-69 0 291.3 0 291.3

69-70 1 329.3 80 249.3

70-71 2 335.9 160 175.9

… … … … …

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Grid Search

a-hat

-

+

+-0

b-hat

Find the point where the sum of squared errors is minimum

64

Differential CalculusTake 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

0]1[]*ˆˆ)([2ˆ/]*ˆˆ)([0605

6968

0605

6968

2

tbatyatbaty

0605

6968

0605

6968

/ˆˆ/)( ntbanty

tbay *ˆˆ

65

Least Squares Fitted Parameters

So, the regression line goes through the sample means.

Take the other derivative:

divide by -2

0605

6968

0605

8968

2 0]][*ˆˆ)([2ˆ/]*ˆˆ)([ ttbatybtbaty

0605

6968

0605

6968

20605

6968

ˆˆ*)( tbtatty

66

Ordinary Least Squares(OLS)

Two linear equations in two unknowns, solve for b-hat and a-hat.

67

68

O-Ring Failure Versus launch temperature

69