8/12/2019 Poe 4 Formulas

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The Rules of Summation

n

i1xi x1 x2 xn

n

i1 a nan

i1axi a

n

i1xi

n

i1xi yi

n

i1xi

n

i1yi

n

i1axi byi a

n

i1xi b

n

i1yi

n

i1a bxi na b

n

i1xi

x n

i1xin x1 x2 xn

n

n

i1xi x 0

2

i1

3

j1fxi;yj 2

i1 fxi;y1 fxi;y2 fxi;y3 fx1;y1 fx1;y2 fx1;y3

fx2;y1 fx2;y2 fx2;y3

Expected Values & Variances

EX x1fx1 x2fx2 xnfxn

n

i1xifxi

xx fx

E gX x

gxfx

E g1X g2X x

g1x g2x fx

x

g1x

f

x

x

g2x

f

x

E g1X E g2X E(c)cE(cX)cE(X)E(a cX)a cE(X)var(X)s2 E[X E(X)]2 E(X2) [E(X)]2var(a cX) E[(a cX) E(a cX)]2 c2var(X)

Marginal and Conditional Distributions

fx y

fx;y for each valueXcan take

fy x

fx;y for each valueYcan take

fxjy

P X

xjY

y

fx;yfy

If X and Y are independent random variables, then

f(x,y) f(x)f(y) for each and every pair of valuesx and y. The converse is also true.

IfXand Yare independent random variables, then the

conditional probability density function ofXgiven that

Y y is fxjy fx;yfy

fxfyfy fx

foreach andevery pair of valuesx andy. Theconverse is

also true.

Expectations, Variances & Covariances

covX; Y EXEXYEY

xy

x EX y EY fx;y

r covX;YffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarXvarY

pE(c1X c2Y) c1E(X)c2E(Y)E(X Y)E(X)E(Y)var(aX bY cZ) a2var(X) b2var(Y) c2var(Z) 2abcov(X,Y) 2accov(X,Z) 2bccov(Y,Z)IfX,Y, andZare independent, or uncorrelated, random

variables, then the covariance terms are zero and:

varaX bY cZ a2varX b2varY c2varZ

Normal Probabilities

IfX N(m,s2), then ZX ms

N0; 1IfX N(m,s2) and a is a constant, then

PX a P Z a ms

If XNm;s2 and a and b are constants; then

Pa X b P ams

Z b ms

Assumptions of the Simple Linear Regression

Model

SR1 The value ofy, for each value ofx, isyb1b2x e

SR2 The average value of the random error e is

E(e)0 since we assumethat E(y) b1b2xSR3 The variance of the random errore is var(e)

s2 var(y)SR4 The covariance between any pair of random

errors,ei and ejis cov(ei, ej)cov(yi, yj)0SR5 The variablex is not random and must take at

least two different values.

SR6 (optional ) The values ofe are normally dis-

tributedabout their mean e N(0,s2)

Least Squares Estimation

Ifb1and b2are the least squares estimates, then

yi b1 b2xiei yi ^yi yi b1 b2xi

The Normal Equations

Nb1 Sxib2 SyiSxib1 Sx2ib2 Sxiyi

Least Squares Estimators

b2 Sxi xyi yS xi x2

b1 y b2x

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Elasticity

h percentage change in ypercentage change in x

Dy=yDx=x

DyDx

xy

h DEy=EyDx=x DEyDx xEy b2 xEyLeast Squares Expressions Useful for Theory

b2 b2 Swiei

wi xi xSxi x2

Swi 0; Swixi 1; Sw2i 1=Sxi x2

Properties of the Least Squares Estimators

varb1 s2 Sx2i

NSxi x2" #

varb2 s2

Sxi x2

covb1; b2 s2

x

Sxi x2" #

Gauss-Markov Theorem: Under the assumptions

SR1SR5 of the linear regression model the estimators

b1and b2have the smallest variance of all linear and

unbiased estimators ofb1 and b2. They are the Best

Linear Unbiased Estimators (BLUE) ofb1 and b2.

If we make the normality assumption, assumption

SR6, about the error term, then the least squares esti-

mators are normally distributed.

b1 N b1; s2 x2i

NSxi x2 !

; b2 N b2; s2

Sxi x2 !

Estimated Error Variance

s2 Se2i

N 2

Estimator Standard Errors

seb1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibvarb1q

; seb2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibvarb2q

t-distribution

If assumptionsSR1SR6of the simple linear regression

model hold, then

t bk bksebk tN2; k 1; 2

Interval Estimates

P[b2 tcse(b2) b2 b2tcse(b2)] 1 aHypothesis Testing

Components of Hypothesis Tests

1. A null hypothesis,H02. An alternative hypothesis, H13. A test statistic

4. A rejection region

5. A conclusion

If the null hypothesis H0: b2 c is true, then

t b2 cseb2 tN2

Rejection rule for a two-tail test: If the value of the

test statistic falls in the rejection region, either tail of

the t-distribution, then we reject the null hypothesis

and accept the alternative.

Type I error: The null hypothesis istrueand we decidetorejectit.

TypeII error:The null hypothesisisfalse andwe decide

notto reject it.

p-value rejection rule:When thep-value of a hypoth-

esis test issmallerthan the chosen value ofa, then the

test procedure leads to rejectionof the null hypothesis.

Prediction

y0 b1 b2x0 e0; ^y0 b1 b2x0; f y0 y0bvarf s2 1 1

Nx0 x

2

Sxi x2" #

; sef ffiffiffiffiffiffiffiffiffiffiffiffiffibvarfq

A (1a) 100% confidence interval, or predictioninterval, for y0

^y0 tcsefGoodness of Fit

Syi y2 S^yi y2 Se2iSST SSR SSER2 SSR

SST 1 SSE

SST corry;^y2

Log-Linear Model

lny b1b2x e;bln y b1 b2x100 b2 % change iny given a one-unitchangeinx:yn expb1 b2x^yc

exp

b1

b2x

exp

s2=2

Prediction interval:

exp blny tcsefh i

; exp blny tcsefh i

Generalized goodness-of-fit measureR2g corry;yn2

Assumptions of the Multiple RegressionModel

MR1 yib1b2xi2 bKxiK eiMR2 E(yi) b1b2xi2 bKxiK , E(ei)0.MR3 var(yi)var(ei)s2MR4 cov(yi, yj) cov(ei,ej) 0MR5 The values ofxikare not random and are not

exact linear functions of the other explanatory

variables.

MR6 yi Nb1 b2xi2 bKxiK;s2, ei N0;s2

Least Squares Estimates in MR Model

Least squares estimatesb1,b2, . . . ,bKminimize

Sb1, b2, . . . , bK yi b1 b2xi2 bKxiK2

Estimated Error Variance and Estimator

Standard Errors

s2 e2i

N K sebk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibvarbkq

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Hypothesis Tests and Interval Estimates for Single Parameters

Use t-distribution t bk bksebk tNK

t-test for More than One ParameterH0 : b2 cb3 a

When H0 is true t b2 cb3 aseb2 cb3 tNK

seb2 cb3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibvarb2 c2bvarb3 2c bcovb2; b3q

JointF-tests

To testJ joint hypotheses,

F SSER SSEU=JSSEU=N K

To test the overall significance of the model the null and alternative

hypotheses and Fstatistic are

H0 : b2

0; b3

0; : : : ; bK

0

H1: at least one of the bk is nonzero

F SST SSE=K 1SSE=N K

RESET: A Specification Test

yi b1 b2xi2 b3xi3 ei ^yi b1 b2xi2 b3xi3yi b1 b2xi2 b3xi3 g1^y2i ei; H0: g1 0yi b1 b2xi2 b3xi3 g1y2ig2^y3i ei; H0: g1 g2 0Model Selection

AICln(SSE=N) 2K=NSCln(SSE=N) Kln(N)=NCollinearity and Omitted Variables

yi b1 b2xi2 b3xi3 eivarb2 s

2

1 r223 xi2 x22

When x3 is omitted; biasb2 Eb2 b2 b3bcovx2;x3bvarx2

Heteroskedasticity

var(yi)var(ei)s i2General variance function

s2i expa1 a2zi2 aSziSBreusch-Pagan and White Tests for H0:a2a3 aS0

When H0 is true x2 NR2 x2S1

Goldfeld-Quandt test for H0:s2

Ms2

R

versus H1 :s2

M6s2

RWhen H0 is true Fs2M=s2R FNMKM;NRKR

Transformed model for varei s2i s2xiyi=

ffiffiffiffixi

p b1 1=ffiffiffiffi

xip b2 xi=

ffiffiffiffixi

p ei= ffiffiffiffixipEstimating the variance function

lne2i lns2i vi a1 a2zi2 aSziS viGrouped data

varei s2i s2M i 1; 2;. . .;NMs2R i 1; 2;. . .;NR

(

Transformed model for feasible generalized least squares

yi

. ffiffiffiffiffisi

p b1 1

. ffiffiffiffiffisi

p

b2 xi. ffiffiffiffiffisi

p

ei. ffiffiffiffiffisi

p

Regression with Stationary Time Series Variables

Finite distributed lag model

yta b0xt b1xt1 b2xt2 bqxtq vtCorrelogramrk yt yytk y= yt y2

For H0 :rk 0; z ffiffiffiffi

Tp

rk N0; 1LMtest

yt b1 b2xtret1 vt TestH0 :r 0 witht-testet g1 g2xtret1 vt Test usingLM TR2AR(1) error yt b1 b2xt et et ret1 vtNonlinear least squares estimation

yt b11 r b2xt ryt1 b2rxt1 vtARDL(p, q) model

yt d d0xt dlxt1 dqxtq ulyt1

upyt

p

vt

AR(p) forecasting model

yt d ulyt1 u2yt2 upytp vtExponential smoothing ^yt ayt1 1 ayt1Multiplier analysis

d0 d1L d2L2 dqLq 1 u1L u2L2 upLp b0 b1L b2L2

Unit Roots and Cointegration

Unit Root Test for Stationarity: Null hypothesis:

H0 : g 0Dickey-Fuller Test 1 (no constant and no trend):

Dyt gyt1 vtDickey-Fuller Test 2 (with constant but no trend):

Dyt

a

gyt1

vt

Dickey-Fuller Test 3 (with constant and with trend):

Dyt a gyt1 lt vtAugmented Dickey-Fuller Tests:

Dyt a gyt1 m

s1asDyts vt

Test for cointegration

Det get1 vtRandom walk: yt yt1 vtRandom walk with drift: yt ayt1 vtRandom walk model with drift and time trend:

yt a dtyt1 vtPanel Data

Pooled least squares regression

yit b1 b2x2it b3x3it eitCluster robust standard errors cov(eit, e is)c tsFixed effects model

yit b1i b2x2it b3x3it eit b1inotrandomyit yi b2x2it x2i b3x3it x3i eit ei

Random effects model

yitb1i b2x2itb3x3iteit bitb1 ui randomyitayi b11 ab2x2itax2ib3x3itax3ivit

a 1seffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ts2u s2eq

Hausman test

t bFE;k bRE;kbvarbFE;k bvarbRE;kh i1=2