ACTEX Learning Flashcards · 2019. 3. 5. · ACTEX is committed to making continuous improvements to our study material. We thus invite you to provide us with a critique of these

SOA Exam SRM

Flashcards

Runhuan Feng PhD FSA CERA Danieumll Linders PhDAmbrose Lo PhD FSA CERA

Learning amp Memorizing Key Topics and Formulas

Spring 2019 Edition

ACTEX Learning

Copyright copy 2019 ACTEX Learning a division of SRBooks Inc

No portion may be reproduced or transmittedin any part or by any meanswithout the permission of the publisher

ISBN 978-1-63588-708-2

Printed in the United States of America

ACTEX is committed to making continuous improvements to our study material We thus invite you to provide us with a critique of these flashcards

Publication ACTEX SOA SRM Flashcards Spring 2019 Edition

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Preface

This set of flashcards is meant to complement the ACTEX Study Manual forSOA Exam SRM (Statistics for Risk Modeling) Fully revised in response tothe May 2019 edition of the SRM study manual these flashcards provide aconcise summary of the SRM exam material in a readable and presentation-oriented format with a view to maximizing retention Important formulasare displayed to facilitate identification and memorization Suggestions aregiven as to which formulas in our opinion must be memorized whichformulas are important but can be easily deduced from other results andwhich formulas are of secondary importance The flashcards are particularlysuitable for last-minute reviewmdashdonrsquot forget to take them with you on yourway to the CBT exam center

It should be noted however that these flashcards add value to but areno substitute for reading the SRM study manual Examples and problemswhich are key to exam success are not included or discussed in these flash-cards We suggest that you first read the manual carefully go over the

i

in-text examples and (most of the) end-of-chapter problems then use theflashcards as a means to review what you have learned and to ensure thatyou have mastered all of the key concepts

As with the SRM study manual we would be extremely grateful if youcould share your comments and suggestions on these flashcards with usand bring to our attention any potential errors Please direct your com-ments and questions to ambrose-louiowaedu The authors will try theirbest to respond to any inquiries as soon as possible and an ongoing list ofupdates will be maintained online at httpssitesgooglecomsite

ambroseloyppublicationsSRMWe wish you the best of luck with your SRM exam

Runhuan FengDaniel Linders

Ambrose LoFebruary 2019

Part I

Regression Models

1

Chapter 1

Simple Linear Regression

3

1ndash4

11 Basics

bull Simple linear regression (SLR) model equation An approxi-mately linear relationship between y and x

y︸︷︷︸response

= β0 + β1x︸︷︷︸regression function

+ ε︸︷︷︸error

where

y is the response variable (aka dependent variable)

x is the explanatory variable (aka predictors features)

β0 (intercept) and β1 (slope) are regression coefficients

ε is the random error term

In the above model we say that y is regressed on x (denoted y sim x)

bull Defining property of SLR There is only one explanatory variablenamely x

ACTEX Learning copy 2019 CHAPTER 1 SIMPLE LINEAR REGRESSION

1ndash5

bull Model assumptions

A1 The yirsquos are realizations of random variables while the xirsquosare nonrandom

A2 ε1 ε2 εn are independent with

E[εi] = 0 and Var(εi) = σ2

for all i = 1 2 n

Almost always further assume that εirsquos are normally dis-tributed ie

εiiidsim N(0 σ2)


1ndash6

12 Model Fitting by Least Squares Method

bull Idea of least squares method Choose β0 and β1 to make the sum ofsquares

SS(β0 β1) =

nsumi=1

[ yi︸︷︷︸obs value

minus ( β0 + β1xi︸︷︷︸candidate fitted value

)]2

the ldquoleastrdquo

bull Least squares estimates (LSEs)

β1 =SxySxx

=

sumni=1(xi minus x)(yi minus y)sumn

i=1(xi minus x)2and β0 = y minus β1x

where

Sxy =sumni=1(xi minus x)(yi minus y) =

sumni=1 xiyi minus nxy

Sxx =sumni=1(xi minus x)2 =

sumni=1 x

2i minus nx2

(Suggestion Remember the formulas for β0 and β1)


1ndash7

bull How can the calculation of LSEs be tested

Case 1 Given the raw data (xi yi)ni=1 with a relatively small n(eg n le 10)

Enter the data into your financial calculator and read the out-put from its statistics functions

Case 2 Given summarized data in the form of various sums eg

nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi

Expand the products in the two sums that appear in β1 anduse the alternative form

β1 =

sumni=1 xiyi minus nxysumni=1 x

2i minus nx2


1ndash8

bull An alternative formula for β1 in terms of sample correlation

β1 = r times sysx

(Warning Not r times sx

sy

)where

sx and sy are the sample standard deviations of x and y

r is the sample correlation coefficient between x and y

bull Application of this formula Slope estimates when regressing y on xand regressing x on y are related via

βysimx1 times βxsimy1 = r2 = R2︸︷︷︸see Sect 13


1ndash9

bull Fitted values and residuals Given β0 and β1 we can compute

1 The fitted value (aka predicted value) yi = β0 + β1xi

Mnemonic Obtained from the model equation by

β0 rarr β0 β1 rarr β1 εi rarr 0

Ideally yi should be close to yi

2 The residual ei = yi minus yi Memory alert Not yi minus yi Completely different from εi which is unobservable and whichei serves to approximate


1ndash10

bull Graphical illustration of fitted regression line and definitions offitted value and residual

0

y

x

(xi yi)

(xi yi)

Slope = β1

fitted regression line

y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11

bull Sum-to-zero constraints on residuals

1sumni=1 ei = 0 provided that β0 is included in the model

Meaning The residuals offset one another to produce a zero sumthey are negatively correlated

2sumni=1 xiei = 0

Meaning The residuals and the explanatory variable values areuncorrelated

Mnemonic β0 and β1 satisfy

part

partβ0SS(β0 β1) = minus2

nsumi=1

[

ei︷︸︸︷yi minus (β0 + β1xi)] = 0

part


nsumi=1

xi[yi minus (β0 + β1xi)︸︷︷︸ei

] = 0


1ndash12

13 Assessing Goodness of Fit of the Model

bull Three kinds of sums of squares

Sum of

SquaresAbbrev Def What Does It Measure

Total SS TSS

Variation of

response values

about y

Amount of variability inher-

ent in the response prior to

performing regression

Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line

bull Goodness of fit of the SLRmodel (the lower the better)

bull Amount of variability of

response left unexplained

even after introduction of x

Regression SS Reg SS

Variation

explained by SLR

(or the knowledge

of x)

How effective SLR model is

in explaining the variation in

y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS

bull Coefficient of determination

Definition R2 =Reg SS

TSS= 1minus RSS

TSS Measures the proportion of variation of response (about its mean)

explained by the SLR model

The higher the better

bull Specialized formulas for Reg SS and R2 under SLR

Reg SS = β21Sxx

R2 = r2 = Corr(x y)2 (square of correlation between x and y)


1ndash14

bull ANOVA table

Source Sum of Squares df Mean Square F -value

Regression Reg SS 1 Reg SS1

Error RSS nminus 2 s2 = RSS(nminus 2)

Total TSS nminus 1

Structure

Different sources of variation in y

Some ldquoinformalrdquo rules for counting df

ndash Reg SS has 1 df because of one explanatory variable

ndash RSS has 2 df subtracted from n because of two parameters β0

and β1

Dividing an SS by its df results in a mean square (MS)


1ndash15

bull Mean square error

s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2

s =radics2 is the residual standard deviation or residual standard error

bull F -test

HypothesesH0 β1 = 0︸︷︷︸

iid model

vs Ha β1 6= 0︸︷︷︸SLR model

a test of the significanceusefulness of x in explaining y

F -statistic

F =Reg SS(df of Reg SS)

RSS(df of RSS)=

Reg SS1

RSS(nminus 2)

Behavior of F -statistic

ndash H0 Expected value close to one

ndash Ha Tends to be large


1ndash16

bull F -test (Cont)

Going between F -statistic and R2

F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2

)(Mnemonic Divide both the numerator and denominator of the F -statistic by TSS to get R2)


1ndash17

14 Statistical Inference about β0 and β1

bull Sampling distributions of β0 and β1

Linear combination formulas

β1 =

nsumi=1

wiyi where wi =xi minus xSxx

β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi

(Suggestion Remembering these weights is recommended but notabsolutely essential)


1ndash18

bull Sampling distributions of β0 and β1 (Cont)

Unbiased E[βj ] = βj for j = 0 1

Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx

(Suggestion Remember these two formulas)

Estimated variances With σ2 rarr s2 (MSE)

Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test

Hypotheses H0 βj = d vs Ha

βj 6= d

βj gt d

βj lt d

Important special case d = 0 (ie to test if x is useful)

t-statistic

t(βj) =LSEminus hypothesized value

standard error of LSE=

βj minus dSE(βj)

Null distribution t(βj)H0sim tnminus2

Decision rules and p-value

Ha Decision Rule p-value (t is the observed value of t(βj))

βj 6= d |t(βj)| gt tnminus2α2 P(|tnminus2| gt |t|) = 2P(tnminus2 gt |t|)βj gt d t(βj) gt tnminus2α P(tnminus2 gt t)

βj lt d t(βj) lt minustnminus2α P(tnminus2 lt t)


1ndash20

bull Confidence intervals (CIs) for β0 and β1 General structure is

LSEplusmn t-quantiletimes Standard error = βj plusmn tnminus2α2 times SE(βj)

Eg β1 plusmn tnminus2α2 times SE(β1) is the CI for β1

Construction requires formulas of SE(β0) and SE(β1)

bull Relationship between F -test and t-test for H0 β1 = 0

Direct connection between test statistics

F = t(β1)2

Importance Connect

information about β1 (captured by t(β1))

with

information about the whole model (captured by F )


1ndash21

15 Prediction

bull Target (random variable)

ylowast = β0 + β1xlowast + εlowast

where xlowast is explanatory variable value of interest

bull Generic setting

response known values of explanatory variables

y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data

ylowast (target) larr xlowast


1ndash22

bull Point predictorylowast = β0 + β1xlowast

(Mnemonic Set β0 rarr β0 β1 rarr β1 and εlowast rarr 0 same trick as fittedvalues)

bull 100(1minusα) prediction interval

point predictorplusmn t-quantiletimes st error of prediction error

= ylowast plusmn tnminus2α2 times SE(ylowast minus ylowast)

= (β0 + β1xlowast)plusmn tnminus2α2

radics2

[1 +

1

n+

(xlowast minus x)2

Sxx

](Suggestion Remember this formula)


1ndash23

bull Remarks on the structure of the prediction interval Two sourcesof uncertainty associated with prediction

Var(ylowast minus ylowast) = s2︸︷︷︸1copy

+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy

1copy Variability of the random error εlowast Reflected in the extra s2

2copy Estimation of the true regression line at xlowast

β0 and β1 are only estimates of β0 and β1 and are subject tosampling fluctuations

Variance of prediction error minimized when xlowast = x and in-creases quadratically as xlowast moves away from x


Copyright copy 2019 ACTEX Learning a division of SRBooks Inc

No portion may be reproduced or transmittedin any part or by any meanswithout the permission of the publisher

ISBN 978-1-63588-708-2

Printed in the United States of America














Preface



i






Part I

Regression Models

1

Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy



















Preface



i






Part I

Regression Models

1

Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy












Preface



i






Part I

Regression Models

1

Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






Preface



i






Part I

Regression Models

1

Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy











Part I

Regression Models

1

Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






Part I

Regression Models

1

Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






Chapter 1


3

1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash4

11 Basics





where








1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash5





for all i = 1 2 n


εiiidsim N(0 σ2)


1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash6



SS(β0 β1) =

nsumi=1



)]2

the ldquoleastrdquo


β1 =SxySxx

=



where




sumni=1 x

2i minus nx2



1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash7





nsumi=1

xi

nsumi=1

yi

nsumi=1

x2i

nsumi=1

y2i

nsumi=1

xiyi


β1 =


2i minus nx2


1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash8


β1 = r times sysx


sy

)where






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash9








1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash10


0

y

x

(xi yi)

(xi yi)

Slope = β1


y = β0 + β1x

yi minus yi = ei

β0

xi


1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash11




2sumni=1 xiei = 0



part


nsumi=1

[


part


nsumi=1


] = 0


1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash12



Sum of


Total SS TSS

Variation of

response values

about y




Residual SS

or

Error SS

RSS

Variation of

response values

about fitted

regression line






Variation

explained by SLR

(or the knowledge

of x)



y


1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash13

bull ANOVA identity

nsumi=1

(yi minus y)2

︸︷︷︸TSS

=

nsumi=1

(yi minus yi)2

︸︷︷︸RSS

+

nsumi=1

(yi minus y)2

︸︷︷︸Reg SS



TSS= 1minus RSS





Reg SS = β21Sxx



1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash14

bull ANOVA table




Total TSS nminus 1

Structure





and β1



1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash15


s2 =RSS

df of RSS=

sumni=1 e

2i

nminus 2


bull F -test


iid model



F -statistic


RSS(df of RSS)=

Reg SS1

RSS(nminus 2)





1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash16

bull F -test (Cont)


F = (nminus 2)

(Reg SSTSS

RSSTSS

)= (nminus 2)

(R2

1minusR2



1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash17




β1 =

nsumi=1


β0 =

nsumi=1

wi0yi where wi0 =1

nminus xwi



1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash18



Variances

Var(β0) = σ2

(1

n+

x2

Sxx

)and Var(β1) =

σ2

Sxx



Var(β0) = s2

(1

n+

x2

Sxx

)and Var(β1) =

s2

Sxx


1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash19

bull t-test


βj 6= d

βj gt d

βj lt d


t-statistic



βj minus dSE(βj)







1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash20







F = t(β1)2

Importance Connect


with



1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash21

15 Prediction






y x

y1 x1

observed

(past) data

y2 x2

yn xn

Unobserved

(future) data



1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash22







radics2

[1 +

1

n+

(xlowast minus x)2

Sxx



1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






1ndash23



+ s2

[1

n+

(xlowast minus x)2

Sxx

]︸︷︷︸

2copy






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ACTEX Learning Flashcards · 2019. 3. 5. · ACTEX is committed to making continuous improvements to our study material. We thus invite you to provide us with a critique of these