Regression

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Regression

Econ 240A

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Retrospective Week One

• Descriptive statistics• Exploratory Data Analysis

Week Two• Probability• Binomial Distribution

Week Three• Normal Distribution• Interval Estimation, Hypothesis Testing,

Decision Theory

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Week Four

Bivariate Relationships Correlation and Analysis of Variance

4

Outline A cognitive device to help understand the

formulas for estimating the slope and the intercept, as well as the analysis of variance

Table of Analysis of Variance (ANOVA) for regression

F distribution for testing the significance of the regression, i.e. does the independent variable, x, significantly explain the dependent variable, y?

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Outline (Cont.) The Coefficient of Determination, R2, and

the Coefficient of Correlation, r. Estimate of the error variance, 2. Hypothesis tests on the slope, b.

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Part I: A Cognitive Device

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A Cognitive Device: The Conceptual Model

(1) yi = a + b*xi + ei

Take expectations , E: (2) E yi = a + b*E xi +E ei, where

• assume (3) E ei =0

Subtract (2) from (1) to obtain model in deviations:

(4) [yi - E yi ] = b*[xi - E xi ] + ei

Multiply (3) by [xi - E xi ] and take expectations:

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A Cognitive Device: (Cont.) (5) E{[yi - E yi ] [xi - E xi ]} = b*E[xi - E xi ]2 +

E{ei [xi - E xi ] }, where assume

• E{ei [xi - E xi ] }= 0, i.e. e and x are independent

By definition, (6) cov yx = b* var x, i.e. (7) b= cov yx/ var x The corresponding empirical estimate, by the

method of moments:(8) ˆ b [y(i) y ][x(i) x ] [x(i) x ]2

i

i

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A Cognitive Device (Cont.) The empirical counter part to (2)

Square both sides of (4), and take expectations,

(10) E [yi - E yi ]2 = b2*E[xi - E xi ]2 + 2E{ei*[xi - E xi ]}+ E[ei]2

Where (11) E{ei*[xi - E xi ] = 0 , i.e. the explanatory variable x and the error e are assumed to be independent, cov ex = 0

y a ˆ b * x ,so(9) ˆ a y ˆ b *x

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A Cognitive Device (Cont.) From (10) by definition (11) var y = b2 * var x + var e, this is the

partition of the total variance in y into the variance explained by x, b2 * var x , and the unexplained or error variance, var e.

the empirical counterpart to (11) is the total sum of squares equals the explained sum of squares plus the unexplained sum of squares:

ii i

iexixbyiy 2222 )](ˆ[])([ˆ])([)12(

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A Cognitive Device (Cont.) From Eq. 7, substitute for b in Eq. 11:

• Var y = [covyx]2/Var x + Var e

Divide by Var y: 1 = [covyx]2/vary*varx + var e/var y• or 1 = r2 + var e/var y where r is the correlation

coefficient

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Population Model and Sample Model Side by Side

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Conceptual Vs. Fitted Model Conceptual (1) yi = a + b*xi + ei

Take expectations, E (2) Ey = a + b*Ex +

Eei

(3) Where Eei = 0

Subtract (2) from (1) (4)[yi - Ey] = b*[xi -

Ex] + ei

Fitted

Minimize

)(*ˆˆ)(ˆ)( ixbaiyi )(ˆ)(ˆ)( iyiyeii

i

ixbaiyiv 2)](ˆˆ)(ˆ[)(

2])(ˆ[)(i

ieiii

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Conceptual Vs. Fitted (Cont.) Conceptual Multiply (4) by [xi -

Ex] and take expectations, E

E [yi - Ey] [xi -Ex] = b*E [xi -Ex]2 + Eei* [xi

-Ex], (5) where Eei* [xi -Ex]

= 0 (6) cov[y*x] = b*varx (7) b = cov[y*x]/varx

Fitted First order condition

compare (3) & (vi) From (v) the fitted line

goes through the sample means

i

i

ievi

ixbaiyv

0)(ˆ)(

0)](ˆˆ)([)(

xbayvii ˆˆ)(

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Conceptual vs. Fitted (Cont.)

i

i

ixieviii

ixixbaiyvii

0)(*)(ˆ)(

0)]()][(ˆˆ)([)(

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Part II: ANOVA in Regression

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ANOVA

Testing the significance of the regression, i.e. does x significantly explain y?

F1, n -2 = EMS/UMS

Distributed with the F distribution with 1 degree of freedom in the numerator and n-2 degrees of freedom in the denominator

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Table of Analysis of Variance (ANOVA)

S o u rc e o fV a r ia t io n

S u m o fS q u a re s

D e g re e s o fF re e d o m

M e a nS q u a re

E x p la in e d ,E S S

b 2 [ x ( i ) x ] 2

i 1 ˆ b 2 [ x ( i ) x ] 2

i

E rro r ,U S S

[ ˆ e ( i ) ]2

i n - 2 [ ˆ e ( i ) ]2

i /n -2

T o ta l , T S S [ y ( i ) y ] 2

i n - 1 [ y ( i ) y ] 2

i /n -1

F1,n -2 = Explained Mean Square / Error Mean Square

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Example from Lab Four

Linear Trend Model for UC Budget

UC Budget, General Fund Component, Millions of Nominal $

$2670.529

y = 81.613x + 19.497

R2 = 0.933

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

Fiscal Year

Mil

lio

ns

$

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.965901681R Square 0.932966058Adjusted R Square 0.931050802Standard Error 240.1544701Observations 37

ANOVAdf SS MS F Significance F

Regression 1 28094446.39 28094446 487.12355 3.992E-22Residual 35 2018595.933 57674.17Total 36 30113042.32

Coefficients Standard Error t Stat P-value Lower 95%Intercept 101.1096814 77.38814601 1.306527 0.1998941 -55.99679935X Variable 1 81.61255073 3.697748649 22.07088 3.992E-22 74.10571271

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 101.1096814 190.19031862 182.7222321 146.5777679

Time index, t = 0 for 1968-69, t=1 for 1969-70 etc

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Example from Lab Four

Exponential trend model for UC Budget UCBud(t) =exp[a+b*t+e(t)] taking the logarithms of both sides ln UCBud(t) = a + b*t +e(t)

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UC Budget

y = 0.3623e0.0654x

R2 = 0.9131

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 5 10 15 20 25 30 35 40

Fiscal year

$ B

illio

ns

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SUMMARY OUTPUT



Regression 1 18.06574216 18.06574 367.8458 3.77305E-20Residual 35 1.718929586 0.049112Total 36 19.78467174

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -0.94975318 0.071413248 -13.2994 3.01E-15 -1.094729955 -0.80478X Variable 1 0.06544472 0.003412257 19.17931 3.77E-20 0.058517462 0.072372

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 -0.94975318 -0.2836484392 -0.88430846 -0.2264776343 -0.81886374 -0.2720780474 -0.75341901 -0.2682322335 -0.68797429 -0.267317175


Exp(-0.950) = 0.387

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Part III: The F Distribution

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The F Distribution

The density function of the F distribution:

1 and 2 are the numerator and denominator degrees of freedom.

0FF

1

F

22

22

22

)F(f2

2

1

22

2

2

1

21

21

21

11

0FF

1

F

22

22

22

)F(f2

2

1

22

2

2

1

21

21

21

11

!!

!

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0

0.002

0.004

0.006

0.008

0.01

0 1 2 3 4 5

This density function generates a rich family of distributions, depending on the values of 1 and 2

The F Distribution

1 = 5, 2 = 10

1 = 50, 2 = 10

00.0010.0020.0030.0040.0050.0060.0070.008

0 1 2 3 4 5

1 = 5, 2 = 10

1 = 5, 2 = 1

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Determining Values of F

The values of the F variable can be found in the F table, Table 6(a) in Appendix B for a type I error of 5%, or Excel .

The entries in the table are the values of the F variable of the right hand tail probability (A), for which P(F1,2>FA) = A.

UC Budget, General Fund Component, Millions of Nominal $

$2670.529

y = 81.613x + 19.497

R2 = 0.933

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

Fiscal Year

Mil

lio

ns

$

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Time index, t = 0 for 1968-69, t=1 for 1969-70 etcSUMMARY OUTPUT





RESIDUAL OUTPUT


F1, 35 = (n-2)*[R2/(1 - R2) =35*(0.933/0.067)= 487

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1 dof

35 dofF1,35 = 4.12

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Part IV: The Pearson Coefficient of Correlation, r The Pearson coefficient of correlation, r, is

(13) r = cov yx/[var x]1/2 [var y]1/2

Estimated counterpart

Comparing (13) to (7) note that (15) r*{[var y]1/2 /[var x]1/2} = b

(14) ˆ r [y(i) y ][x(i) x ] [y(i) i

i y ]2 [x(i) x ]2

i

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A Cognitive Device: (Cont.)

(5) E{[yi - E yi ] [xi - E xi ]} = b*E[xi - E xi ]2 + E{ei [xi - E xi ] }, where assume

• E{ei [xi - E xi ] }= 0, i.e. e and x are independent

By definition, (6) cov yx = b* var x, i.e. (7) b= cov yx/ var x The corresponding empirical estimate:

(8) ˆ b [y(i) y ][x(i) x ] [x(i) x ]2

i

i

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Part IV (Cont.) The coefficient of Determination, R2 For a bivariate regression of y on a single

explanatory variable, x, R2 = r2, i.e. the coefficient of determination equals the square of the Pearson coefficient of correlation

Using (14) to square the estimate of r

(16)[ ˆ r ]2 { [y(i) y ][x(i) x ]}2 [y(i) i

i y ]2 [x( i) x ]2

i

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Part IV (Cont.) Using (8), (16) can be expressed as

And so

In general, including multivariate regression, the estimate of the coefficient of determination, , can be calculated from (21) =1 -USS/TSS .

(19) ˆ r 2 ˆ b 2 * [x(i) x ]2

i [y(i) y ]2

i ESS / TSS

(20)1 ˆ r 2 1 [ESS / TSS} [TSS ESS]/ TSS USS / TSS

ˆ R 2

ˆ R 2

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Part IV (Cont.) For the bivariate regression, the F-test can

be calculated from F1, n-2 = [(n-2)/1][ESS/TSS]/[USS/TSS] F1, n-2 = [(n-2)/1][ESS/USS]=[(n-2)]

For a multivariate regression with k explanatory variables, the F-test can be calculated as Fk, n-2 = [(n-k-1)/k][ESS/USS] Fk, n-2 = [(n-k-1)/k]

ˆ R 2 [1 ˆ R 2 ]

ˆ R 2 [1 ˆ R 2 ]

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Time index, t = 0 for 1968-69, t=1 for 1969-70 etcSUMMARY OUTPUT





RESIDUAL OUTPUT


R2 = 1 – 2,018,596/30,113,042

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Part V:Estimate of the Error Variance

Var ei =

Estimate is unexplained mean square, UMS

Standard error of the regression is

)2(])(ˆ)([)2()](ˆ[ˆ 222 niyiyniei i

ˆ

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17.5767415.240ˆ UMS

SUMMARY OUTPUT





RESIDUAL OUTPUT


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Part VI: Hypothesis Tests on the Slope

Hypotheses, H0 : b=0; HA: b>0 Test statistic:

Set probability for the type I error, say 5% Note: for bivariate regression, the square of

the t-statistic for the null that the slope is zero is the F-statistic

[ ˆ b E( ˆ b )] ˆ ( ˆ b ),where E( ˆ b ) b under theH0

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t = {81.6 - 0]/3.70 = 22.1

t2 = F, i.e. 22.36*22.36 = 500

SUMMARY OUTPUT





RESIDUAL OUTPUT


t2 = F, i.e. 22.1*22.1 = 488

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Part VII: Student’s t-Distribution

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The Student t Distribution

The Student t density function

is the parameter of the student t distribution

E(t) = 0 V(t) =(– 2)

2/)1(2t1

)]!2[()]!1[(

)t(f

2/)1(2t1

)]!2[()]!1[(

)t(f

(for n > 2)(for n > 2)

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The Student t Distribution

0

0.05

0.1

0.15

0.2

-6 -4 -2 0 2 4 6

0

0.05

0.1

0.15

0.2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

= 3

= 10

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Determining Student t Values

The student t distribution is used extensively in statistical inference.

Thus, it is important to determine values of tA associated with a given number of degrees of freedom.

We can do this using• t tables , Table 4 Appendix B

• Excel

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Degrees of Freedom1 3.078 6.314 12.706 31.821 63.6572 1.886 2.92 4.303 6.965 9.925. . . . . .. . . . . .

10 1.372 1.812 2.228 2.764 3.169. . . . . .. . . . . .

200 1.286 1.653 1.972 2.345 2.6011.282 1.645 1.96 2.326 2.576

tA

t.100 t.05 t.025 t.01 t.005

A = .05A = .05

-tA

The t distribution issymmetrical around 0

=1.812=-1.812

The table provides the t values (tA) for which P(t > tA) = A

Using the t Tabletttttttt

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Problem 6.32 in TextTable of Joint Probabilities

Manual Calc. Computer Calc.

Quant Ed. 0.23 0.36

Other Ed. 0.11 0.30

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Problem 6.32 The method of instruction in college and

university applied statistics courses is changing. Historically, most courses were taught with an emphasis on manual calculation. The alternative is to employ a computer and a software package to perform the calculations. An analysis of applied statistics courses investigated whether the instructor’s educational background is primarily mathematics (or statistics) or some other field.

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Problem 6.32 A. What is the probability that a randomly

selected applied statistics course instructor whose education was in statistics emphasizes manual calculations?

What proportion of applied statistics courses employ a computer and software?

Are the educational background of the instructor and the way his or her course are taught independent?

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Midterm 2000• .(15 points) The following table shows the results of

regressing the natural logarithm of California General Fund expenditures, in billions of nominal dollars, against year beginning in 1968 and ending in 2000. A plot of actual, estimated and residual values follows.

– .How much of the variance in the dependent variable is explained by trend?

– .What is the meaning of the F statistic in the table? Is it significant?

– .Interpret the estimated slope.

– .If General Fund expenditures was $68.819 billion in California for fiscal year 2000-2001, provide a point estimate for state expenditures for 2001-2002.

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Cont.• A state senator believes that state expenditures

in nominal dollars have grown over time at 7% a year. Is the senator in the ballpark, or is his impression significantly below the estimated rate, using a 5% level of significance?

• If you were an aide to the Senator, how might you criticize this regression?

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T a b l e

D e p e n d e n t V a r i a b l e : L N G E N F N DM e t h o d : L e a s t S q u a r e s

S a m p l e : 1 9 6 8 2 0 0 0I n c l u d e d o b s e r v a t i o n s : 3 3

V a r i a b l e C o e f f i c i e n t S t d . E r r o r t - S t a t i s t i c P r o b .

Y E A R 0 . 0 8 6 9 5 8 0 . 0 0 3 8 9 5 2 2 . 3 2 8 0 4 0 . 0 0 0 0C - 1 6 9 . 4 7 8 7 7 . 7 2 6 9 2 2 - 2 1 . 9 3 3 5 3 0 . 0 0 0 0

R - s q u a r e d 0 . 9 4 1 4 5 9 M e a n d e p e n d e n t v a r 3 . 0 4 6 4 0 4A d j u s t e d R - s q u a r e d 0 . 9 3 9 5 7 0 S . D . d e p e n d e n t v a r 0 . 8 6 6 5 9 4S . E . o f r e g r e s s i o n 0 . 2 1 3 0 3 0 A k a i k e i n f o c r i t e r i o n - 0 . 1 9 6 0 7 6S u m s q u a r e d r e s i d 1 . 4 0 6 8 3 5 S c h w a r z c r i t e r i o n - 0 . 1 0 5 3 7 9L o g l i k e l i h o o d 5 . 2 3 5 2 5 8 F - s t a t i s t i c 4 9 8 . 5 4 1 6D u r b i n - W a t s o n s t a t 0 . 1 1 8 5 7 5 P r o b ( F - s t a t i s t i c ) 0 . 0 0 0 0 0 0

P lo t

-0.4

-0.2

0.0

0.2

0.4

1

2

3

4

5

70 75 80 85 90 95 00

Residual Actual Fitted

Actual, Fitted and Residual Values from the Regressionof the Logarithm of General Fund Expenditures ($B) on Year

Regression

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