financial econometrics notes

7/23/2019 Financial Econometrics Notes

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INTRODUCTORYF INANCIALE CONOMETRICSReview of Econometric Theory

3 C REDITS, 51 HOURS

Jianhua Gang

School of FinanceRenmin University of China

Spring 2013

JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric Theory

SPRING2 01 3 1 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION

REVIEWTOPIC1 : SIMPLER EGRESSION

Readings:

Wooldridge, Ch.2

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric Theory

SPRING 2 01 3 2 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION REGRESSION A NALYSIS

REGRESSIONA NALYSIS

Regression analysis involves the estimation and evaluation of therelationship between a variable of interest (dependent variable,explained variable, regressand) and one or more other variables(independent variables, explanatory variables, regressors).

What is estimation, prediction (forecast), the fitting?


SPRING2 01 3 3 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION CLASSICAL N ORMALS IMPLER EGRESSION M ODEL

CLASSICAL N ORMALS IMPLER EGRESSIONM ODEL

Generalized idea of a random sample ofnindependently andidentically distributed (i.i.d.) observations fromN(,2).

Have sample ofnindependent observationsy1, ...,yn, each ofwhich is normally distributed with variance2,but conditionalmean governed by

E(yi) = +xi, i= 1,..., n.

where,

1 and are termed regression parameters/regression coefficients.2 The termxivaries withi, but is not random (nonstochastic, fixed in

repeated sampling).3 What is sampling?


SPRING 2 01 3 4 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION CLASSICAL N ORMAL S IMPLE R EGRESSION M ODEL

CLASSICAL N ORMALS IMPLER EGRESSION M ODEL

If we regard+xias the equation of a straight line, then

1 the interceptis the mean ofywhenxiequals zero2 the slopeis the change in the mean ofywhenxiincreases by one

unit. (This interpretation of the intercept is not always sensible ineconomic applications.)


SPRING2 01 3 5 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION CLASSICAL N ORMALS IMPLER EGRESSION M ODEL

CLASSICAL N ORMALS IMPLER EGRESSIONM ODEL

Ifui = yi (+xi)denotes the error (or disturbance term), thenwrite simple regression model as:

yi = +xi+ui, ui NID(0, 2), i= 1,..., n, (1)

The assumption that the regressorxisNonstochasticisinappropriate in many applications in economics and it is relaxedlater.

More useful to think of the classical assumption as beingappropriate when we conditional on the values ofx1, ..., xn. Thus,conditional upon the values ofx1, ..., xn, theyiare independentnormal variables with means+xiand common constantvariance2 fori = 1,..., n.


SPRING 2 01 3 6 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION ESTIMATION OFPARAMETERS

ESTIMATION OFPARAMETERS

The following general approaches to estimate,and2 areconsidered: method of moments (MM); ordinary least squares

(OLS); and maximum likelihood estimation (MLE).These slides do not contain full mathematical details.


SPRING2 01 3 7 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION METHOD OFM OMENTS E STIMATION

METHOD OFM OMENTS E STIMATION

Population moments conditions(assumptions provided before as in(1)):

E(ui) = 0,

E(xiui) = 0,E(u2i

2) = 0.

Let the MM estimator ofandbeand, with associatedresidualsui = yi (+xi), i= 1,..., n.


SPRING 2 01 3 8 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION METHOD OFM OMENTS ESTIMATION


Obtain MM: solving the derived equations (replacingE(.)byn1

i

(.), anduibyui), the equations are:iui = i [yi (+xi)] =0,

i

xiui = i

xi[yi (+xi)] =0,

i

u2i 2 = 0It can be proved that under weak conditions, MME are consistentand asymptotically normally distributed.


SPRING2 01 3 9 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION ORDINARY LEAST S QUARES E STIMATION(OLS)

ORDINARYL EASTS QUARES E STIMATION(OLS)

Choose estimatesandto get "best fit" in the sense ofminimizing

S(,) = i

[yi (+xi)]2.

First order conditions (the F.O.C.s) are,

S(,)

= 0

S(,)

= 0


SPRING 2 01 3 1 0 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION ORDINARYL EAST S QUARES E STIMATION (OLS)


Ignoring an irrelevant factor of2, these equations are,

i

[yi (+xi)] = i

ui = 0 (2)

i

xi[yi (+xi)] = i xiui = 0 (3)Equations (2) and (3) are called the normal equations (uiis an OLSresidual).

It is clear that the normal equations imply that the OLS estimatesofandare equal to the corresponding MME previously.


SPRING 2 01 3 1 1 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION ORDINARY LEAST S QUARES E STIMATION(OLS)


The solution ofandwhich minimize the objective functionS(,)are,

=

i

(xi x)(yi y)

i(x

i x)2

= yxwherexdenotes a sample average, e.g. x = n1

i

xi.

We have to postpone discussions of estimation of2 later.


SPRING 2 01 3 1 2 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION MAXIMUM L IKELIHOOD E STIMATION

MAXIMUML IKELIHOODE STIMATION

Becauseyi N(+xi, 2), i= 1, ..., n, so that

f(yi) = (22)1/2 exp{[yi (+xi)]

2/22}, i.

We already assume thatyi, ...,ynare independent, so

f(y1, ...,yn) = i f(yi) = L

The log-likelihood is, therefore,

l(,, 2) = n

2ln(22)

i

[yi (+xi)]2

22 .

The MLE ofandmust minimizei

[yi (+xi)]2 and so

equals OLS. The MLE of2 is2 =n1i

u2i =MMestimate.JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric Theory

SPRING 2 01 3 1 3 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION OLS DECOMPOSITION OFS UM OFS QUARES

OLS DECOMPOSITION OFS UM OFS QUARES

Letyi= (+xi)denote a typical OLS predicted value, then thenormal equation for OLS yield several results.

i

yi = i

(yi+ui) = iyi+iui= iyi

i

yiui = i

(+xi)ui=i

ui+i

xiui= 0

i

y2i = i

(yi+ui)2 =i

y2i +i

u2i + 0

i

(yi n1

i

yi)2 =

i

(yi n1i

yi)2 +i

u2i


SPRING 2 01 3 1 4 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION OLS DECOMPOSITION OFS UM OFS QUARES


i

(yi n1

i

yi)2 =

i

(yi n1i

yi)2 +i

u2ior put this in another way,

Total Sum of Squares (TSS)=Explained Sum of Squares(ESS) +Residual Sum of Squares(RSS)

Note sums of squares are measured about sample averages.


SPRING 2 01 3 1 5 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION GOODNESS OFF IT

GOODNESS OFF IT

Coefficient of determinationR2 is index of goodness of fit of OLSline with

R2 = ESS

TSS

= 1 RSS

TSS

, 0 R2 1.

R2 =r2XY, whererXY= XY(correlation coefficient betweenxandy).


SPRING 2 01 3 1 6 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION SAMPLING P ROPERTIES OFOLS ESTIMATORS

SAMPLING P ROPERTIES OFOLS ESTIMATORS

Best linear unbiased estimator (BLUE) ofand, even whenerrorsuiare not normally distributed.

Consistent and asymptotically efficient (MLE).


SPRING 2 01 3 1 7 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION SAMPLING P ROPERTIES OFOLS ESTIMATORS

SAMPLINGD ISTRIBUTION OFOLE ESTIMATORS

For the classical normal simple regression model,andarejointly normally distributed with

E() = E() = Var() = 2

i

(xi x)2

Var() = 2n

+x2Var()Cov(,) = xVar()


SPRING 2 01 3 1 8 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION SAMPLING P ROPERTIES OFOLS ESTIMATORS

SAMPLING D ISTRIBUTION OFOLE ESTIMATORS

The OLS estimator of the regression parameters can be written as

= +i

wiui

= +

i

ziui

where the nonstochastic termswiandzidepend upon theregressor values, e.g.

zi= (xi x)/j

(xj x)2.


SPRING 2 01 3 1 9 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION ESTIMATION OF SIGMA-SQUARE

ESTIMATION OF SIGMA-SQUARE

It can be shown that, in classical normal simple regression model,

i

u2i =RSS

22(n 2)

is independent ofand.Note(n 2)is the number of observations minus the number ofregression parameters estimated toderive the residualsand is calledthedegree of freedomparameter for the regression.


SPRING 2 01 3 2 0 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION ESTIMATION OF SIGMA-SQUARE


Hence,E(

i

u2i) =

2(n 2)

And so the newly-defined (sample) estimator

s2 =

i

u2in 2

is unbiased. The ML estimator, however,2 = [ (n 2)/n] s2 isbiased (of course when sample size gets relatively small).


SPRING 2 01 3 2 1 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION STATISTICALI NFERENCE

STATISTICALI NFERENCESTOCHASTIC SPECIFICATION OF CLASSICAL MODEL

Study of statistical inference requires the specification of theprobabilistic model fory1, ...,yn.We make the followingassumptions.


SPRING 2 01 3 2 2 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION STATISTICALI NFERENCE

STATISTICAL I NFERENCESTOCHASTIC SPECIFICATION OF CLASSICAL MODEL

A1 There exist observation invariant parametersandsuch thatE(yi) = +xii;

A2 The regressorxis nonrandom and satisfiesSxx=n

1

(xi x)2> 0

forn > 1. For the purpose of asymptotic theory, it is conventionalto assume 0 < lim n1S < ;

A3 Letui= yi E(yi),common variance (homoskedasticity)var(ui) =

2 i. If theuido not have the same variance, haveheteroskedasticity.

A4 Letui= yi E(yi),uncorrelated disturbances soE(uiuj) = 0 ifi=j.If have time series data and assumption is false then say haveautocorrelation (or serial correlation).

A5 Letui= yi E(yi),normally distributed distanbances (so that A4implies independence).


SPRING 2 01 3 2 3 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION SAMPLING DISTRIBUTIONS FOR INFERENCE

SAMPLING DISTRIBUTIONS FOR INFERENCE

andareN(, var())andN(, var()), respectively, so thatz() = ( )/var() N(0, 1)z() = ()/var() N(0, 1)

RSS= u2i 22(n 2)independently ofand, soRSS2 2(n 2)independently ofz()andz(), so

t() = z() RSS

(n2)2

t(n 2)

t() = z() RSS

(n2)2

t(n 2)


SPRING 2 01 3 2 4 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION SAMPLING DISTRIBUTIONS FOR INFERENCE


RSS(n2) =s

2 so that, for example,

t() = z()s2

2

= ()

var(

)

s2

2

t(n 2)

in which, the denominator

var()(s22

) = ( 2

SXX)(

s2

2) =

s2

SXX

is the estimator ofvar()and the square root of this quantity iscalled the estimated standard error, denoted by

SE() = var()(s22

) =

s2

SXX

var()(when n big)


SPRING 2 01 3 2 5 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION SAMPLING DISTRIBUTIONS FOR INFERENCE


Hence,

t() = SE() t(n 2)

Similar fort(),t() = ( )

SE() t(n 2),


SPRING 2 01 3 2 6 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION CONFIDENCE I NTERVALS(C.I.S)

CONFIDENCEI NTERVALS(C.I.S)

Letd1be such that

prob(d1 t(n 2) d1) = (1 )

Then the(1 ) 100 per cent confidence intervals (C.I.) for

andare given by, d1SE() d1SE()respectively.


SPRING 2 01 3 2 7 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION HYPOTHESIS T ESTING : T S TATISTIC

HYPOTHESIS T ESTING: TS TATISTIC

Consider the null hypothesis that restricts one of the regressionparameters, e.g.H0 : = 0, where0is some specified constant.

For whatever value of,

t(

) =

(

)

SE()t(n 2),

and so ifH0is true,

t0() = (0)SE() t(n 2).

t0()is termed as thetest statistic.JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric Theory

SPRING 2 01 3 2 8 / 1 10


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REVIEWTOPIC1: SIMPLER EGRESSION HYPOTHESIS T ESTING : T S TATISTIC

HYPOTHESIS T ESTING: TS TATISTIC

The critical/rejection region depends upon the nature of thealternative hypothesis and the prespecified significance level,denoted by.

1 H1 : = 0rejectH0if|t0()| > d1,whereprob(t(n 2) > d1) = /2

2 H+1 : > 0rejectH0ift0() > d2,whereprob(t(n 2) > d2) =

3 H1 : < 0rejectH0ift0() < d2,where

prob(t(n 2) < d2) = 4 Just replacebyandbyin the above to obtain test procedures

for(the intercept).

JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 2 9 / 1 10

REVIEWT OPIC1: SIMPLE R EGRESSION RELAXING THEA SSUMPTION OFF IXEDR EGRESSORS

RELAXING THEASSUMPTION OFF IXEDR EGRESSORS

Suppose thatx, likey, is a r.v.. Consider the results above that cannow be regarded as being derived, conditional upon the valuesx1, ..., xn.

1 E(

|x1,..., xn) = ,E(

|x1,..., xn) = andE(s

2|x1,..., xn) = 2.These

expectations do not depend upon thexvaluesand so OLSestimators are unconditionally unbiased. Similar remarks apply toprobability limits;

2 var(|x1,..., xn),var(|x1,..., xn)andcov(,|xx1,..., xn),as givenabove,do depend on the xvalues, and sodo not correspond tounconditional characteristics.

3 Fortunately, 2 does not pose major problems for inference. The

variables( ) /SE()and()/SE()are, givenxvalues,both distributed ast(n 2), still. This distribution does not dependonxvalues, but just on the values of(n 2). Hence thet testsand confidence intervals described above are unconditinally valid.

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 3 0 / 1 10

REVIEWTOPIC1: SIMPLER EGRESSION RELAXING THEA SSUMPTION OFF IXED R EGRESSORS

RELAXING THEA SSUMPTION OFF IXEDR EGRESSORS

It is, however, important to note,

1 It has been assumed that the errorsu1,..., un NID(0,2)whether

or not we condition on the xvalues,i.e. the regressor values and

error terms are statistically independent.2 Assumptions in 1 can be weakened but we cannot expect to getresults that are exact, i.e. valid for finite sample sizes, and oftenhave to resort to asymptotically valid results in practical situations.


REVIEWT OPIC1: SIMPLE R EGRESSION PRESENTATION OFR ESULTS( EARNINGS ON SCHOOLING)

PRESENTATION OFR ESULTS( EARNINGS ONSCHOOLING)



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REVIEWTOPIC1: SIMPLER EGRESSION PREDICTION

PREDICTION

Suppose wish to make predictions for periodf,f > n(the samplesize), withxfknown and assuming the data generation process for

yis unchanged so that,

yf =+xf+ uf, ufN(0,2

).

Prediction ofE(yf): use the predictoryf =+xf, where the OLSestimators use the data for i= 1,..., n. This predictor is BLUE forE(yf) = +xf.The predictoryfis a linear combination of the OLS estimators andso is normally distributed.The variance ofyfcan be estimated, and confidence intervals andtests of hypotheses are feasible.


REVIEWT OPIC1: SIMPLE R EGRESSION PREDICTION

PREDICTION

Suppose wish to make predictions for periodf,f > n(the samplesize), withxfknown and assuming the data generation process for

yis unchanged so that,

yf=+xf+ uf, ufN(0,

2

).

Prediction ofyf : use same predictor which implies a forecast error

of(yfyf) = uf ( )+ xf, which has zeroexpectation, given OLS unbiased and E(uf) = 0.The forecast error is normally distributed, being a linearcombination of three normal variates, and has a variance that can

be estimated. Confidence intervals and tests of hypotheses, e.g.H0 : E(yfyf) = 0,are feasible.


REVIEWTOPIC2: MULTIPLER EGRESSION

REVIEWTOPIC2 : MULTIPLER EGRESSION

READING

Wooldridge, Ch.3, 4


REVIEWTOPIC2: MULTIPLER EGRESSION CLASSICAL M ULTIPLER GRESSION M ODEL

CLASSICAL M ULTIPLER GRESSION M ODEL

Have sample ofnindependent observationsy1, ...,yn, each ofwhich is normally distributed with variance2, but means varyaccording to

E(yi) = +1x1i+ ... +kxki= +j

jxji, i= 1,..., n.

andjare parameters/coefficients.

Regressors xjivary withi, butnonrandom (nonstochastic, i.e. fixedin repeated sampling).can be regarded as an intercept with= E(yi), given allxji = 0.

Slopesjcan often be regarded as partial derivatives:j= E(yi)xji

.

Note: Regressor might be discrete or a nonlinear function of someother regressor; so that interpretations vary.



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REVIEWTOPIC2: MULTIPLER EGRESSION CLASSICAL M ULTIPLER GRESSION M ODEL

THE C LASSICAL M ULTIPLER GRESSION M ODEL

Ifui= yi (+j

jxji)denotes the error or disturbance term,

then write classical normal multiple regression model as:

yi = +j

jxji+ui, ui NID(0, 2), i= 1,..., n,

whereNIDstands for, normally and independently distributed.


REVIEWTOPIC2: MULTIPLER EGRESSION STOCHASTIC S PECIFICATION OFC LASSICAL M ODEL

STOCHASTICS PECIFICATION OFC LASSICAL M ODEL

The following assumptions are made in the classical normalregression model:

A1 There exist observation invariant parametersandj,j= 1, ..., k

such thatE(y

i) = +

j

jx

jii;

A2 The regressorxjiare nonrandom and satisfy

n

1

(xji xj)2> 0, xj = n

1i

xji

wheren > 1 andj= 1,..., k. For the purpose of asymptotic theory,

assume 0 < limnn1

n

1

(xji xj)2< for allj.




The following assumptions are made in the classical normalregression model:

A3 Also need to assume that no regressor is just a linear combinationof the other regressors and the intercept term.

A4 Common variance (homoskedasticity)var(ui) =2 i. If theuido

not have the same variance, have heteroskedasticity.A5 Uncorrelated disturbances soE(uiuj) = 0 ifi=j.If have time series

data and assumption is false then say have autocorrelation/serialcorrelation.

A6 Normally distributed distanbances (so that A5 impliesindependence).




Assumption A2 is often too restrictive for economic applicationsin which some regressors are probably better regarded as random,rather than fixed in repeated sampling.

As in the case of the simple regression model, we can start bythinking about the conditional distribution ofyi, holding the

valuesxji(i= 1, ..., n;j= 1,...k)constant. Having derived resultsfor the conditinal model, we can see which of them will apply tothe unconditional model for y.

For the former model, we have that, given the values of theregressors, the variatesyiare independent with conditionaldistributionsN(+

j

xjij, 2)fori = 1, ..., n.



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REVIEWTOPIC2: MULTIPLER EGRESSION METHOD OFM OMENTS ESTIMATION


Have,E(ui) = 0 andE(xjiui) = 0 forj = 1,..., k.

Therefore, MM estimators, denoted by, can be derived form

i

ui = 0

i xjiui = 0forj= 1,..., k, whereuiis the residualyi (+

j

jxji), i= 1, ..., n.The MM estimate of2 can be derived from

E(u2i 2) = 0,

it is 2 =n1i

u2i.JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 4 1 / 1 10

REVIEWTOPIC2: MULTIPLER EGRESSION ORDINARY LEAST S QUARES E STIMATION

ORDINARYL EASTS QUARES E STIMATION

The OLS estimators are chosen to minimize,

S(,1, ...,k) = i

yi

+

j

jxji

2

The F.O.C.s yields the normal equations,

i

ui = 0

i

xjiui = 0forj= 1,..., k, whereuiis the OLS residual

yi

+j

jxji

, i= 1,..., n.

Hence the OLS estimators are equal to the MM estimators.


REVIEWTOPIC2: MULTIPLER EGRESSION MAXIMUM L IKELIHOOD E STIMATION

MAXIMUML IKELIHOODE STIMATION

Using methods similar to those appropriate in the context of thesimple regression model, it can be shown that the log likelihoodfunctionis given by,

l(,1, ...,k, 2) = (

n

2

) ln(22)S(,1, ...,k)

22

.

The MLE of the regression parameters must minimizeS(,1, ...,k)and soOLSE= MLE.

The MLE of2 is R SSn , whereRSS= i

u2i is the OLS residual sumof squares function.


REVIEWTOPIC2: MULTIPLER EGRESSION SAMPLING P ROPERTIES OFOLS ESTIMATORS

SAMPLINGP ROPERTIES OFOLS ESTIMATORS

Best linear unbiased estimator (BLUE) ofandj,j= 1,..., k,evenwhen errorsuiare not normally distributed.

Consistent and asympototically efficient (MLE).



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REVIEWTOPIC2: MULTIPLER EGRESSION OLS DECOMPOSITION OFS UM OFS QUARES


Letyi =+j

jxjidenote a typical OLS predicted value.Thenormal equation for OLS yield several results,

i

yi = i

(

yi+

ui)=

i

yi+

i

ui=

i

yi

iyiui = i (+jjxji)ui=

i

ui+j

ji

xjiui= 0

i

y2i = i

(yi+ui)2=

i

y2i +i

u2i ,given2i

yiui = 0

i

(yi n1

i

yi)2 =

i

(yi n1i

yi)2 +i

u2iJIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 4 5 / 1 10

REVIEWTOPIC2: MULTIPLER EGRESSION OLS DECOMPOSITION OFS UM OFS QUARES


or put it another way,

Total Sum of Squares (TSS)=Explained Sum of Squares(ESS) +Residual Sum of Squares(RSS)

Note sums of squares are measured about sample averages.


REVIEWTOPIC2: MULTIPLER EGRESSION GOODNESS OFF IT

GOODNESS OFF IT

Coefficient of determinationR2 is index of goodness of fit of OLSline withR2 = ESSTSS =1

RSSTSS , 0 R

2 1.

Some use degree-of-freedom adjustedR2, denoted byR2,and

defined byR

2

=1 {RSS/ (n k 1) / [TSS/ (n 1)]} .Thisindex can be negative.

If add regressors to a model and re-estimate by OLS, R2 cannot

fall (monotonic function on # of parameters), but R2

can.


REVIEWTOPIC2: MULTIPLER EGRESSION EXPRESSIONS FOROLS ESTIMATORS

EXPRESSIONS FOROLS ESTIMATORS

It can be shown that = y j

jxj,with a typical slope estimator given by

j= i xjiyi

i

x2ji ,where xjiis theith residual from the OLS regression of the

jthregressor on the other(k 1)regressors and the intercept term.



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REVIEWTOPIC2: MULTIPLER EGRESSION EXPRESSIONS FOROLS ESTIMATORS

EXPRESSIONS FOROLS ESTIMATORS

It can also be shown that

j= j+ i xjiui

i

x2ji = j+

i xjiuiRSSj ,whereRSSjis the residual sum of squares from the OLS

estimation of the auxiliary regression of thejth regressor on theother(k 1)regressors and the intercept term.


REVIEWTOPIC2: MULTIPLER EGRESSION SAMPLING D ISTRIBUTION OFOLE ESTIMATORS

SAMPLINGD ISTRIBUTION OFOLE ESTIMATORS

For the classical normal multiple regression model,

N(, var(

)).

Since the OLS estimators of the slope parameters can be written asj = j+ixjiui/

ix2ji= j+

ixjiui/RSSjand the disturbances

uiareNID(0, 2), they are all normally distributed with

E(j) = jvar(j) = 2/RSSj,j= 1,..., k.


REVIEWTOPIC2: MULTIPLER EGRESSION ESTIMATION OF SIGMA-SQUARE


It can be shown that, in classical normal simple regression model,

i

u2i =RSS 22(n k 1)independentlyof

and

j,j.

Note that(n k 1)is thenumber of observations minus the

number of regression parameters estimated to derive theresidualsand is calledthe degree of freedomparameter for theregression.

E(i

u2i) = 2(n k 1)and so the estimator s2 = 1(nk1) (i

u2i)isunbiased.

However, the MLE estimator,2 = [ (n k 1)/n] s2 isbiased(ofcourse when sample size is relatively small).


REVIEWTOPIC2: MULTIPLER EGRESSION SAMPLING DISTRIBUTIONS FOR INFERENCE


andjareN(, var())andN(j, var(j)), respectively, so thatz() = ( )/var() N(0, 1)

z(

j) = (

j j)/

var(

j) N(0, 1).

RSS= iu2i 22(n k 1)independently ofandj, soRSS/2 2(n k 1)independently ofz()andz(j), so

t() = z()/[RSS/(n k 1)] /2 t(n k 1)t(j) = z(j)/[RSS/(n k 1)] /2 t(n k 1).



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REVIEWTOPIC2: MULTIPLER EGRESSION SAMPLING DISTRIBUTIONS FOR INFERENCE


We knowRSS/(n k 1) = s2, so that, for example,

t(j) = z(j)/s2/2 = (j j)/var(j)s2/2

var(j)(s2/2) = (2/RSSj)(s2/2) = s2/RSSjwhich is theestimator ofvar(j)and the square root of this quantity is calledthe (estimated) standard error, denoted by SE().Hence,

t(j) = j j/SE(j) t(n k 1)simlarly

t() = ( ) /SE() t(n k 1)JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 5 3 / 1 10

REVIEWTOPIC2: MULTIPLER EGRESSION CONFIDENCE I NTERVALS

CONFIDENCEI NTERVALS

Letd1be such thatprob(d1 t(n k 1) d1) = (1 )

the(1 ) 100 per cent confidence intervals for andjare

given by d1SE()andj d1SE(j), respectively.


REVIEWTOPIC2: MULTIPLER EGRESSION TEST OFH YPOTHESES USING TS TATISTICS

TEST OFH YPOTHESES USING TS TATISTICS

Consider null hypothesis that restricts one of the regressionparameters, e.g.H0 : j= j0(some specified constant),

For whatever value ofj,t(

j) = (

j j)/SE(

j) t(n k 1)

,and so ifH0is truet0(j) = (j j0)/SE(j) t(n k 1).Thent0(j)is the test statistic. The critical/rejection regiondepends upon the nature of the alternative hypothesis and theprespecified significance level, denoted by .


REVIEWTOPIC2: MULTIPLER EGRESSION TEST OFH YPOTHESES USING T S TATISTICS

TEST OFH YPOTHESES USING TS TATISTICS

H1 : j= j0 rejectH0if |t0(j)| > d1,where

prob(t(n k 1) > d1) = /2

H+1 :j > j0 rejectH0ift0(j) > d2,where

prob(t(n k 1) > d2) =

H1 :j < j0 rejectH0ift0(j) < d2,where

prob(t(n k 1) < d2) =

Just replacejby andjbyin the above to obtain test

procedures relevant to testing hypotheses concerning theintercept.



15/115

REVIEWTOPIC2: MULTIPLER EGRESSION F T EST OFS EVERAL L INEAR R ESTRICTIONS

F TEST OFS EVERALL INEARR ESTRICTIONS

EXAMPLE

Suppose that the null hypothesis to be tested is denoted by H0and

consists of several linear restrictions on the parameters of theregression model. ThusH0specifies thevalues of, say,q < (k+ 1)linear combinations of the regression coefficients. For example, withk= 4 andq= 3,H0could consist of the following restrictions:+1 = 0;2 = 1; and4= 0. We now need a joint test ofall therestrictionsofH0,rather than a collection of separate t-tests.


REVIEWTOPIC2: MULTIPLER EGRESSION F T EST OFS EVERALL INEARR ESTRICTIONS

F TEST OFS EVERAL L INEARR ESTRICTIONS

LetRSS(H0)be thesum of squared residuals obtained under therestrictions ofH0.In the example of the previous note, RSS(H0)isderived by applying OLS to the restricted model:

(yi x2i) = 1(x1i 1) +3x3i+ui.

LetRSS(H1)be theRSSobtained by applying OLS to theunrestricted model. In the previous example,RSS(H1)is derived by

applying OLS toyi= +4

j=1

jxji+ui.


REVIEWTOPIC2: MULTIPLER EGRESSION F T EST OFS EVERAL L INEAR R ESTRICTIONS

F TEST OFS EVERALL INEARR ESTRICTIONS

DEFINITION

Define theF statisticby the following equation

F= [RSS(H0)RSS(H1)]

RSS(H1)

df(H1)

q ,

in whichdf(H1)is the degrees of freedom parameter for theunrestricted model, i.e.df(H1) = (n k 1).

IfH0is true, thenF F(q, df(H1)).

The null hypothesis is regarded as inconsistent with the data if thesample (observed) value of F is significantly large ,i.e. the test isone-sided.


REVIEWTOPIC2: MULTIPLER EGRESSION PREDICTION

PREDICTION

Suppose wish to make predictions for periodf,f > n(nis thesample size), withxjfknown and it being assumed that the datageneration process (DGP) for yisunchangedso that

yf =+j

jxjf+ uf, ufN(0,2).

Prediction ofE(yf): use the predictoryf =+j

jxjf, where theOLS estimators use the data fori= 1,..., n. This predictor is BLUEforE(yf) = +xf.The predictoryfis a linear combination of the OLS estimators andso is normally distributed. The variance ofyfcan be estimated, andconfidence intervals and tests of hypotheses are feasible.



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REVIEWTOPIC2: MULTIPLER EGRESSION PREDICTION

PREDICTION

Suppose wish to make predictions for periodf,f > n(nis thesample size), withxjfknown and it being assumed that the DGPforyisunchangedso that,

yf=+j

jxjf+ uf, ufN(0,2)

Prediction ofyf : use same predictor which implies a forecast errorof

(yfyf) = uf

( ) +j

j j xjf

which has zero expectation, given OLS unbiased andE(uf) = 0.The forecast error is normally distributed, being a linearcombination of normal variates, and has a variance that can beestimated.Confidence intervals and tests of hypotheses, e.g.

H0 : E(yf

yf) = 0,is feasible.


REVIEWTOPIC3: MULTICOLLINEARITY

REVIEWTOPIC3 : MULTICOLLINEARITY

READINGWooldridge, Ch.3.


REVIEWT OPIC3: MULTICOLLINEARITY MULTICOLLINEARITY

MULTICOLLINEARITY

The information content of a sample available for the purpose ofestimating the individual regression parameters depends, in part,upon theintercorrelations between the regressors.

LetR2j denote theR2 statistic from the OLS estimation of the

auxiliary regression of thejth regressor on the other(k 1)

regressors and the intercept term. Since it has been assumed thatno regressor is a linear combination of the other regressors andthe intercept term, it follows thatR2j < 1 for allj.

IfR2j =1 for somej, then say that there is perfect multicollinearity.

IfR2j is close to 1 for somej, then have a high degree of

multicollinearity.


REVIEWTOPIC3: MULTICOLLINEARITY MULTICOLLINEARITY

MULTICOLLINEARITY

It can be proved that,

var(j) = 2/RSSj= 2/

i

(xji xj)2

1R2j

.

Thus, ceteris paribus,high degrees of multicollinearity lead to high

values of sampling variances.Note: imprecise estimators can lead to wide condidence intervalsand weak tests of hypotheses.

However, in practice, we cannot vary R2j with2 and

i

(xji xj)2

held constant. Variances may be small even when there is a highdegree of multicollinearity, or large when the regressor areuncorrelated.



17/115


MULTICOLLINEARITY

Also note that although the multicollinearity is indeed a problem,but nonthelessno assumptions of the classical multipleregression model have been violated.

Therefore, provided multicollinearity is not perfect, then OLS

estimators are BLUE and MLE. Similarly the standard testprocedures are valid and retain optimality properties relative toother tests.

Klein proposes the rule of thumb that multicollinearity is a"problem" if maxjR

2j > R

2.

If trying to consider multicollinearity, it isnot sufficientto lookonly at pairwise correlations between regressors (might be nestedmodels where reside complex relationship or even stochastic).


REVIEWTOPIC3: MULTICOLLINEARITY MULTICOLLINEARITY

MULTICOLLINEARITY

Multicollinearity is a feature of the nonrandom regressor set andso we cannot test for it. Some measures for multicollinearity have

been proposed, but they are open to objection and the R2j statistics

are simple to calculate and interpret.

Models can be reparameterized to make transformed regressoruncorrelated, but the transformed parametersmay have noeconomic interest.

As noted above, multicollinearity can lead to large variances andweak tests, e.g. might have every individual slope estimate beinginsignificant (as indicated by a t-test), but a highly significant Fstatistic for the hypothesis that all slopes equal zero.



MULTICOLLINEARITY

Multicollinearity can also lead to large changes in parameterestimates when there are small changes in the data.

Various "treatments" have been described, e.g. drop somevariables, use first differences, use outside estimates of somecoefficients. These treatments usually introduce new problems,e.g. dropping an insignificant, but relevant, variable will lead to

biased estimator in the amended model.

Real solutionis to get more valid information, so using falserestrictions is not a good strategy. May also have to wait for moredata.


REVIEWT OPIC4: THE M EAN F UNCTION

REVIEWTOPIC4 : THE M EA NF UNCTION

READING

Wooldridge, Ch.3, Ch. 7, Ch. 9.



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REVIEWTOPIC4: THE M EAN F UNCTIONI I I I

FUNCTION-CONSEQUENCES

CONSEQUENCESCASE 1

Have assumed that there exist observation invariant parametersand1, ...,ksuch that the conditional mean is given by

E(yi|xji,j= 1, ..., k) = +j jxji,

wherexjiisith value ofjth regressor.

1. May have included irrelavant regressors, i.e. somejequals zero.OLS

estimators are stillunbiased and consistent, butno longer efficient(they fail to use valid information set that corresponds to somecoefficients being zero).


REVIEWT OPIC4: THE M EAN F UNCTIONI I I I

FUNCTION -CONSEQUENCES

CONSEQUENCESCASE 2

Have assumed that there exist observation invariant parametersand1, ...,ksuch that the conditional mean is given by

E(yi|xji,j= 1,..., k) = +j

jxji,

wherexjiisith value ofjth regressor.

2. May have omitted some relevant regressors:Write the conditional meanfunction asE(yi|xji,j= 1,..., k) = +

j

jxji+E(fi|xji,j= 1,...k.),

wherefistands for an omitted factor. In general, OLS estimators ofregression parametersandjarebiased and inconsistent. The

estimators2 is biased and inconsistent, and the standard t- andF-tests areno longer valid.


REVIEWTOPIC4: THE M EAN F UNCTIONI I I I

FUNCTION-CONSEQUENCES

CONSEQUENCESCASE 3

May use incorrect functional form, e.g. assume

yi = +j

jxji+ui, ui NID(0, 2),

when the true model is a log-log form

log(yi) = +j

jlog(xji) +vi, vi NID(0, 2).

The OLS estimators of the false linear-linear model do notcorrespond to parameters of economic interest.


REVIEWT OPIC4: THE M EAN F UNCTION TEST P ROCEDURES -RESET TEST

TES TP ROCEDURES-RESET TES T

If have strong belief about the omitted factor, can use precise test.For example, if sure that fiis a linear combination ofqvariableszji,can apply F-test ofH0 : 1 = ...= q= 0 in the expanded model

yi = +j

jxji+j

jzji+ui, ui NID(0,2).

If do not have strong belief, then can use "informationparsimonious" RESET test. In this test, fit the null model

yi= +j

jxji+ui, ui NID(0,2),

by OLS to obtain predicted valuesyi, i= 1,..., n.JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 7 2 / 1 10


19/115

REVIEWTOPIC4: THE M EAN F UNCTION TEST P ROCEDURES -RESET TEST

TES TP ROCEDURES-RESET TES T

Then testH0 : 1 = ...= q= 0 in the artificial model,

yi= +j

jxji+j

j(

yi)

j+1 +ui, ui NID(0, 2).

Notes:

1 Noyiterm because this is a linear combination of the intercept termand the regressorsxji;

2 F-test is valid even though added variables are random;3 Choice ofqhas impact on power;4 No rule for determining the best value of q;5 Often use quite small values of q, e.g. 1 or 2;6 Cannot expect RESET to indicate how a model should be re-specified;7 Cannot assume RESET will always have high power.


REVIEWT OPIC4: THE M EAN F UNCTION TESTS FORS TABILITY

TESTS FORS TABILITY

Suppose we divide the sample into two subsamples, denoted by1and2.Let1containsn1observations and2containsn2 = n n1observations. The unrestricted model of thealternative hypothesis is then written as,

yi= +j

jxji+ui, ui NID(0,2), if i 1,

andyi =

+j

jxji+ui, ui NID(0, 2), if i 2.

so that changes in regression coefficients are permitted (under theunrestricted model!). Should note the homoskedasticity.


REVIEWTOPIC4: THE M EAN F UNCTION TESTS FORS TABILITY

TESTS FORS TABILITY

The null hypothesis of constant coefficients consists of the(k+ 1)

restrictions of

H0 : = andj =

j,j= 1,..., k

.

Suppose thatns > (k+ 1), s= 1, 2.LetRSSsdenote the residualsum of squares (RSS) for the OLS regression ofyion the interceptterm and thexjiusing only the observations for s, s= 1, 2, and

RSSdenote the residual sum of squares for this OLS regressionusing allnobservations.H0can be tested using the F statistic

F=RSS (RSS1+RSS2)

RSS1+RSS2

n 2k 2

k+ 1

which isF(k+ 1, (n 2k 2))underH0, with large valuesindicating the inconsistency ofH0.


REVIEWT OPIC4: THE M EAN F UNCTION TESTS FORS TABILITY

TESTS FORS TABILITY

If, say,n2 (k+ 1), then usepredictive failure test. Testn2

restrictionsE

yi

+j

jxji

= 0, i 2,wheredenotes anestimator derived using only the observations of1.TheF-statistics is

F= RSS RSS1RSS1

n1 k 1n2

,

which isF(n2, (n1 k 1))when the model is stable.

However, in case ofn2 < k+ 1,then2restrictions being tested maybe satisfied even thoughH0is false.



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REVIEWTOPIC4: THE M EAN F UNCTION TREATMENT

TREATMENT

The only treatment that allows valid inference is the correctspecification of the mean function.


REVIEWTOPIC5: NON-NORMALD ISTURBANCES

REVIEWTOPIC5 : NON -NORMALD ISTURBANCES

READINGWooldridge, Ch.5.


REVIEWT OPIC5: NON-NORMAL D ISTURBANCES NON-NORMAL D ISTURBANCES

NON -NORMALD ISTURBANCES

Now suppose that the regression model is

yi = +j

jxji+ui, i= 1,..., n,

where the disturbances are independently and identicallydistributed (i.i.d.) with zero mean and variance 2 < ,but thecommondistributionisNOT normal.


REVIEWTOPIC5: NON-NORMALD ISTURBANCES CONSEQUENCES

CONSEQUENCES

OLS estimators are still BLUE, but, in general, are NOT normallydistributed. Thereforethe t and F tests are no longer valid infinite samples.

The standard formulae for confidence intervals are alsoinvalidinfinite samples.

Under weak conditions, OLS estimators are consistent and a

Central Limit Theorem can be used to show that they areasymptotically normally distributed, implying that t and F tests oflinear restrictions on regression coefficients areasymptoticallyvalid. The usual confidence intervals are also asymptoticallyvalid.

The prediction error test is, however,not asymptotically valid.

Since MLE maximizes wrong likelihood function, it does notproduce asymptotically efficient estimators.



21/115

REVIEWT OPIC5: NON-NORMAL D ISTURBANCES TEST P ROCEDURES

TES TP ROCEDURES

When theuiareNID(0, 2), the following conditions are satisfied:

E(u3i) = 0; andE(u4i) 3

4 =0.

If a typical OLS residual is denoted byui,then it is natural to lookat tests based upon the sample moments n1

u3i and

n1u3i 34, where2 =n1u2i .Jarque and Bera propose atest of the joint significance of these terms. However, this test isonly asymptotically valid and,in large samples, there is littleneed to assume normalitywhen examining OLS results for thelinear multiple regression model.

Asymptotic theory sometimes provides a poor approximation tothe actual finite sample behaviour of the Jarque-Bera statisticwhen theuiare normal.

The Jarque-Bera test can have low power under some nonnormaldisturbance distributions.


REVIEWTOPIC5: NON-NORMALD ISTURBANCES TREATMENT

TREATMENT

If haveprecise informationabout the form of the disturbance

distribution, then can derive the likelihood function and obtainthe asymptotically efficient MLE. Otherwise, use OLS and relyupon large sample results.


REVIEWTOPIC6: HETEROSKEDASTICITY

REVIEWTOPIC6 : AUTOCORRELATION ANDHETEROSKEDASTICITY

READING

Wooldridge, Ch.8., Ch.12.


REVIEWTOPIC6: HETEROSKEDASTICITY HETEROSKEDASTICITY-INTRODUCTION

HETEROSKEDASTICITY-I NTRODUCTION

Allowvar(ui)to vary withi, so thaty1, ...,ynare independentN(+

j

jxji, 2i)variables, where

2i denotesvar(ui).

Heteroskedasticity is often regarded as associated withcross-section data, grouped data, or random coefficient models,

but can occur in time-series applications (GARCH-family modelsfor instance).



22/115

REVIEWTOPIC6: HETEROSKEDASTICITY CONSEQUENCES OFH ETEROSKEDASTICITY

CONSEQUENCES OFH ETEROSKEDASTICITY

OLS still unbiased and consistent,but no longer efficientineither large or small samples.

OLS not MLE because MLE maximize likelihood under false

assumption that alluihave same variance.j= j+i

xjiui/i

x2ji = j+i

xjiui/RSSj, so thatvar(j) =

i

x2ji2i/

i

x2ji2

=i

x2jiu2i/ RSSj2 which is not equaltoE(s2)/RSSj.Conventional standard errors are, therefore, biased.

The t- and F-tests are, therefore,invalid.


REVIEWTOPIC6: HETEROSKEDASTICITY TESTSFOR H ETEROSKEDASTICITY

TESTS FORH ETEROSKEDASTICITY

Goldfeld-Quandt Test

A finite sample test that requires normality of the distrubances.

The null hypothesis is that the errors are homoskedastic. It isassumed that information is available about the relativemagnitudes of variances under the alternative hypothesis ofheteroskedasticity.

Using this information, reorder the data so that21 22 ...

2n.

Split the sample into three parts containing m, c, and mobservations, withm > (k+ 1)andn= 2m+c. Drop the middleset of c observations.


REVIEWTOPIC6: HETEROSKEDASTICITY TESTSFORH ETEROSKEDASTICITY


Goldfeld-Quandt Test

LetRSS1andRSS2denote the OLS residual sum of squaresfunctions for estimation using the first m and last m observations,

respectively. Under the null hypothesis of homoskedasticity, thestatisticGQ= RSS2/RSS1is distributed asF(m k 1, m k 1)and large values indicate data inconsistency of null hypothesis.

Problems:a) Choice of m and c; b) Need enough information toreorder data according to values of variances.




Lagrange Multiplier/Score Test

Original form suggested by Breusch-Pagan and Godfrey requiresnormal disturbances even for asymptotic validity, and is notrecommended.



23/115



Studentizedd Score Test

Koenkers Studentized Score test is asymptotically robust tononnormality. Estimate model by OLS using all observations andobtain the residualsui, i= 1,..., n. Assume an alternative of theform2i =g

0+

p

1

jzji

,where the precise form ofg(.)need

not be specified.

Apply OLS to the artificial regression model

u2i =0+ p1

jzji+ai, i= 1,..., n,

and obtain the coefficient of determination denoted byR2K.




Studentizedd Score Test

Koenkers test statistic isnR2Kand, under homoskedasticity,nR2Kisasymptotically distributed as2(p)with large values indicatingthe rejection of the null model.

Problems:a) Large sample test; b) need enough information toselect the variablezji incorrectchoice has impact on power.




Whites Direct Test

Whites test can be regarded as a Koenker-type test with the zji

being the nonredundant terms ofxiqandxiqxir,q, r= 1,..., k.Problems: a) Large sample test; b) need enough information toselect the variablezji incorrectchoice has impact on power.




Autoregressive Conditional Heteroskedasticity Tests

ARCH models are widely used - conditional variance dependsupon squared past values of ui.The test for ARCH is aKoenker-type check withzji=u2ij; i= p+ 1, ...,nandj= 1,...,p.



24/115

REVIEWTOPIC6: HETEROSKEDASTICITY TREATMENT OFH ETEROSKEDASTICITY

TREATMENT OFH ETEROSKEDASTICITY

If know variances up to a constant of proportionality, can applyOLS to transformed data to get efficient estimators. Suppose2

i =2w2

i,with thew2

ibeing known, thenvar(u

i/w

i) = 2 i.In

this case, apply OLS to the transformed model(yi/wi) = (1/wi) +

j

j(xji/wi) + (ui/wi), in which the(ui/wi)

areNID(0,2)variates.

Note:the transformed model may not contain an intercept.


REVIEWTOPIC6: HETEROSKEDASTICITY TREATMENT OFHETEROSKEDASTICITY


If suspect heteroskedasticity and do not have very preciseinformation about its form, then can use Whitesheteroskedasticity consistent standard errors, denoted by

WSE(

)andWSE(

j),j = 1,..., k. for asymptotically valid

inference after OLS estimation.White shows that, if

WSE(j) = i

x2jiu2i/ RSSj2,then(j j)/WSE(j)is asymptotically distributed asN(0, 1)inpresence of unspecified heteroskedasticity.




Hence, ifd1is such that

prob(d1 N(0, 1) d1) = (1 ),

the(1 ) 100 per cent confidence intervals for andjare

given by, d1WSE()andj d1WSE(j), respectively.


REVIEWTOPIC6: HETEROSKEDASTICITY TREATMENT OFHETEROSKEDASTICITY


Asymptotically valid tests of hypotheses such as

H0 : j = j0

are based upon

tW0 (j) = (j j0)/WSE(j)N(0, 1)

underH0.

Since the procedures areonly asymptotically valid, can replaceN(0, 1)byt(n k 1)and this is often done. Thus can use thefollowing to obtain asymptotically valid tests ofH0 : j= j0.



25/115



H1 : j =j0rejectH0iftW0 (j) > d1, where

prob(t(n k 1) > d1) = /2;

H+1 :j >j0rejectH0iftW0 (j) > d2, whereprob(t(n k 1) > d2) = ;

H1 :j=j0 rejectH0iftW0 (j) < d2, where

prob(t(n k 1) > d2) = ;

Just replacejby andbyin the above to obtain test

procedures relevant to testing hypotheses concerning theintercept.


REVIEWTOPIC6: HETEROSKEDASTICITY AUTOCORRELATION/SERIAL C ORRELATION

AUTOCORRELATION/S ERIALC ORRELATION-INTRODUCTION

HaveytN(+j

jxjt,2),but no longer assume independence,

t= 1,..., n. Ifut= yt (+j

jxjt),then allowE(utus)=0 for

somet=s.Usetsubscript because autocorrelation is oftendiscussed in a time-series framework, but spatial autocorrelationhas been examined.

The regressors are asumed to be nonrandom. (It would bestraightforward to allow for random regressors withxjtindependentofus, for all j, s and t.) This assumption will berelaxed later. In particular, will consider autocorrelation whenregressors include lagged values of the dependent variable.


REVIEWTOPIC6: HETEROSKEDASTICITY CONSEQUENCES OFA UTOCORRELATION

CONSEQUENCES OFA UTOCORRELATION

OLS still unbiased and consistent,but no longer efficientin eitherlarge or small samples.

OLS not MLE because MLE maximizes likelihood under falseassumption that theutare independent.

j= j+t xjtut/t x2

jt = j+t xjtut/RSSj,and, since the xjtut

are not independent,var

t

xjtut= t

var(xjtut)and sovar(j)=2/RSSj.Conventional standard errors are, therefore,

biased.

The t- and F- tests are invalid.


REVIEWTOPIC6: HETEROSKEDASTICITY TESTS FORA UTOCORRELATION

TESTS FORA UTOCORRELATION

In the lectures given this term, it is assumed that the utarecovariance stationary withE(ututg) = (|g|)for all t, with(|0|) = 2. The autocorrelation of orderg, denoted by(g),is thecorrelation betweenutandutg,i.e.E(ututg)/2, with the

sequence(1),(2), ...being called the autocorrelation function orACF. Under the null hypothesis of serial independence,(g) = 0for allg=0. Different tests check the significance of different setsof estimates of autocorrelations.



26/115

REVIEWTOPIC6: HETEROSKEDASTICITY DURBIN WATSONT EST

DURBIN-WATSONT ES T

Basically a test for nonzero values of(1), based upon OLSresiduals. The test statistic is

d=n

2

(ut ut1)2/n

1u2t

which is approx.2(1 r(1))

where,

r(1) =n

2

utut1/ n1

u2t


REVIEWTOPIC6: HETEROSKEDASTICITY DURBINWATSONT EST

DURBIN-WATSONT ES T

LEMMA

Values of d close to 0 (resp. 4) indicate high level of positive (resp. negative)residual first order serial correlation. The distribution of d under nullhypothesis of independent errors depends upon values of regressors, so criticalvalues vary from one case to another.


REVIEWTOPIC6: HETEROSKEDASTICITY DURBIN WATSONT EST

DURBIN-WATSONT ES T

Have tables for combinations of n and k (and for models with andwithout an intercept) giving bounds for the critical values fortestingH0of serial independence againstH1 : (1) > 0.Theseupper and lower bounds, denoted byduanddl, define an interval

that contains the true known critical value. Ifd < dl, reject.Ifd > du, accept.Ifdl d du,the test is inconclusive. For

H1 : (1) < 0,use 4 duand 4 dlas bounds.


REVIEWTOPIC6: HETEROSKEDASTICITY DURBINWATSONT EST

DURBIN-WATSONT ES T

The Durbin-Watson procedure is a useful test against either firstorder autoregressive (AR(1)) modelut = 1ut1+t,or first ordermoving average (MA(1)) modelut = t+1t1,in which thetNID(0,2 ).For reasons to be discussed later in time series, weassume |1| < 1 and |1| 1.

Problems:

Checks for nonzero values of(1)can be insensitive to(g)=0,g=1,e.g.g = 4, when(1) = 0

Test is inconclusive when sample value of d falls betweenbounds-inconclusive region.

Requires errors to be normal and regressors to be fixed, e.g. nolagged dependent variables.



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REVIEWTOPIC6: HETEROSKEDASTICITY LAGRANGE M ULTIPLIER/SCORET ESTS

LAGRANGEM ULTIPLIER/S CORET ESTS

Very flexible asymptotic test based upon OLS results. It isasymptotically valid for models with nonnormal errors andlagged dependent variables in the regressor set.

If null hypothesis of serial independence is to be tested against

autoregressive or moving average model of orderg, then applyasymptotically valid F-test ofH0 : 1 = 2 = ...= g = 0 after OLS

estimation of the modelyt = +k

1

jxjt+g

1

jutj+ut,in whichtheutjare lagged values of the residuals from the OLS estimationofyt= +

k

1

jxjt+ut.For "gaps" in alternative model, omit

selectedjterms. Iftjis not positive, setutj= 0.JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Review of Econometric TheorySPRING 2 01 3 1 05 / 1 10

REVIEWTOPIC6: HETEROSKEDASTICITY ESTIMATION

ESTIMATION

If have precise information about form of autocorrelation, e.g.type (AR or MA) and order (value ofg), can use asymptoticallyefficient MLE or apporoximation.

Model can then be written as yt = +k1

jxjt+ut,with either

ut = 1ut1+ ... +gutg+t, tNID(0, 2),AR(g), or

ut = t+1t1+ ... +gtg,tNID(0,2 ),MA(g). MLE, orapproximations based upon minimizing

t

2t are available in

econometric softwares.


REVIEWTOPIC6: HETEROSKEDASTICITY OTHERP ROBLEMS OFA UTOCORRELATION

RESIDUAL S ERIALC ORRELATION ORG ENUINEDISTURBANCE A UTOCORRELATION?

Significant outcomes of tests designed for autocorrelation can becaused by misspecification of the mean function, e.g. omit

relevant regressors or use wrong functional form. In such cases,re-estimation allowing for autocorrelation is of little value.



RESIDUALS ERIALC ORRELATION ORG ENUINEDISTURBANCE A UTOCORRELATION?

A procedure, called the COMFAC test, has been developed to testthe null hypothesis that the errors of a regression equation aregenerated by an autoregressive process of specified order. TheCOMFAC test uses as its alternative an expanded version of theoriginal regression equation obtained by adding lagged values ofthe dependent variable and the initial set of regressors. Details arenot provided because this test, while asymptotically valid, hasfinite sample properties that cause concern; see Gregory and Veall,Economic Letters, 1986, 22, 203-208. Moreover, the alternativeadopted in the COMFAC procedure may be inadequate and yielda test that rarely detects a false null hypothesis.



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RESIDUAL S ERIALC ORRELATION ORG ENUINEDISTURBANCE A UTOCORRELATION?

Mizon (A simple message for autocorrelation correctors: dont,Journal of Econometrics, 1995,69, 267-288) offers the followingconclusions:

Although it is important to test for autocorrelation, it is rarelyappropriate to "autocorrelation correct" in response to rejecting thenull hypothesis of independent disturbances;and, when re-estimation assuming autoregressive errors imposesinvalid restrictions, inconsistent parameter estimators will result.

The nature of the restrictions to which Mizon refers can beillustrated by considering a simple case in which the model of thenull isyt = xt+ut,withut = 1ut1+t, t NID(0,

2 ), i.e. the

disturbancesutareAR(1).



RESIDUALS ERIALC ORRELATION ORG ENUINEDISTURBANCE A UTOCORRELATION?

Under this null,

yt = xt+1(yt1 xt1) +t,

or equivalently,

yt= xt+1yt1 1xt1+t, t NID(0, 2 ),

in which the coefficient ofxt1is restricted to be minus theproduct of the coefficients ofxtandyt1. Note that this restrictionis not linear.



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INTRODUCTORYF INANCIALE CONOMETRICSTopic 1 Introduction of Time Series


Jianhua Gang


Spring 2013

JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 1 Introduction of Time SeriesSPRING 2 01 3 1 / 1 8

TOPIC1 I NTRODUCTION OFT IME S ERIES

TOPIC1 INTRODUCTION OFT IMES ERIES

Statistical analysis of data observed over time.

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 1 Introduction of Time SeriesSPRING2 01 3 2 / 1 8

TOPIC1 I NTRODUCTION OFT IME S ERIES TIME S ERIESD ATA

TIMES ERIESD ATA

DEFINITION (T IME S ERIESDATA)

Data observed between two dates, normalized ast= 1 andt= T.Equispaced, i.e. we observeY1, Y2, ..., Yt, Yt+1, ..., YT1, YTandNOintermediate observation is missing.

Ytdepends onYs(if theres any)if and only ifs < tYtdoes notdepends onYsifs > t.Then, the vector{Y1, Y2, ..., Yt, Yt+1, ..., YT1, YT}

is atime series.


TOPIC1 I NTRODUCTION OFT IME S ERIES MOMENTS

MOMENTS

For a generic random variable we can define themean,variance,and for pairs of random variables we can also define covariance,correlationetc. In a time series we define these for eachYt :

DEFINITIONS (MOMENTS OFT IME SERIES)

Mean:E(Yt) = t;Variance: E (Ytt)2= 2tCovariance:E

(Ytt)(Yt+jt+j)

= t(j)

Correlation: t(j)tt+j



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TOPIC1 I NTRODUCTION OFT IME S ERIES OPERATORS

OPERATORS

Lag operator:L

L Yt = Yt1

So,L1

Yt = Yt+1

First Difference operator:

=1 LYt = (1 L)Yt= Yt Yt1Also, 2Yt = (1 L)2Yt= Yt 2Yt1+Yt2


TOPIC1 I NTRODUCTION OFT IME S ERIES STATIONARITY ANDERGODICITY

STATIONARITY ANDE RGODICITY

PROBLEM

Suppose{Y1, Y2, ..., Yt, Yt+1, ..., YT1, YT} is a single realization from a

stochastic process{Yt} .We areinterested in the model that generated

the time series, but we do not know it. How can we make inference, usingone single realization?

SOLUTION

We must use the fact that this is a T-dimensional observation:

1 Restrict heterogeneity over time;

2 Restrict dependence over time.


TOPIC1 I NTRODUCTION OFT IME S ERIES RESTRICT H ETEROGENEITY

RESTRICTH ETEROGENEITY

Assume some properties are common to all the Yts in{Y1, Y2, ..., YT}

.For example,

DEFINITION (C OVARIANCES TATIONARITY)

For time seriesYt {Yt} ,

E(Yt) = ,t

E

(Yt)(Yt+j)

= (j), t

i.e. the first two moments arefiniteanddo not depend on time(spatial equivalent).



RESTRICTH ETEROGENEITY

In this way, we may try to estimate or(j)using the samplecounterparts. "Covariance stationarity" is also known as a "weakstationarity" or simply as "stationarity" (without other references).

For stationary processes, we shorten the notation and introducejfor(j)to indicate the autocovariance.

The plot ofjagainstjis called autocovariance function.



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RESTRICT HETEROGENEITY

An alternative restriction on heterogeneity is:

DEFINITION (S TRICTS TATIONARITY)

For anyj1,...jn, thejoint distributionofYt+j1 , ..., Yt+jn and ofYt++j1 , ..., Yt++jn

is the same for any .

1 The joint distribution only depends on the spatial difference,not on time;

2 Strict and Covariance stationarity do not imply each other.


TOPIC1 I NTRODUCTION OFT IME S ERIES RESTRICT D EPENDENCE OVERT IME

RESTRICTD EPENDENCE OVERT IM E

Givenn , and given the process is stationary, then the samplemoments would estimate the population moments consistently.

One may generalize this argument and allow for somedependence,provided that it is not too much : a sufficient

condition for consistent estimation ofis j=0

|j| < .

DEFINITION

One restriction on the dependence that allows to consistently estimatethe population moments using the sample moments in stationaryprocesses is called Ergodicity.

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 1 Introduction of Time SeriesSPRING 2 01 3 1 0 / 1 8

TOPIC1 I NTRODUCTION OFT IME S ERIES RESTRICT D EPENDENCE OVERT IME

RESTRICTD EPENDENCE

Often we are interested in time series because we want to answerone of the two questions:

1 Forecasting: What value do you expect forYt+1if you observedY1,..., Yt?

2 Impulse response: What is the consequence onYtof a shock thattook place(t j)periods ago?

We first address these questions in the case of stationary processes.

JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 1 Introduction of Time SeriesSPRING2 01 3 1 1 / 1 8

TOPIC1 I NTRODUCTION OFT IME S ERIES FORECASTS BASED ON A L INEAR P ROJECTION

FORECASTS BASED ON AL INEARP ROJECTION

Assume:Ytis stationary;E(Yt) = 0 (ifE(Yt) = =0, thenconsiderYtinstead). Then,

1 Linear forecast ofYt+1usingYtis

Yt+1|t = a(1)1 Yt;2 Linear forecast ofYt+1usingYtandYt1is

Yt+1|t = a(2)1 Yt+a(2)2 Yt1;3 Linaer forecast ofYt+1usingYt,..., Ytm+1is

Yt+1|t,...,tm+1= a(m)1 Yt+a(m)2 Yt1+ ...a(m)m Ytm+1.



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TOPIC1 I NTRODUCTION OFT IME S ERIES FORECASTS BASED ON AL INEARP ROJECTION

FORECAST

Now, which values of((m)1 ,

(m)2 , ...,

(m)m )

characterise a goodlinear projection?

LetXt = (Yt, ..., Ytm+1), = ((m)1 ,

(m)2 , ...,

(m)m )

,thenmustmeetE [(Yt+1

Xt) Xt]= 0 (i.e., the forecast error Yt+1Xtis

not correlated withXt).

Then, givenYt+1 = Yt+1, (Yt+1being single component),

E(Yt+1Xt)

E(XtXt) = 0

= E(XtXt)1 E(XtYt+1)It can be proved thatgives the best linear forecast.


TOPIC1 I NTRODUCTION OFT IME S ERIES WOLD D ECOMPOSITION

WOL DD ECOMPOSITION

Of course, in some cases a non-linear forecast may be better.

However, a linear model is usually easier to use, so it is importantthat any stationary process may be given a linear representation.This can be discussed using the Wold Decomposition.

DEFINITION (WOL DDECOMPOSITION)Anystationary processYtmay be represented in the form

Yt = kt+

j=0

jtj

where

0 = 1,

j=0

2j <


TOPIC1 I NTRODUCTION OFT IME S ERIES WOLD D ECOMPOSITION

WOL DD ECOMPOSITION

andt,the error made in forecastingYton the basis of a linearfunction,

t = YtE(Yt|Yt1,...)is such that, for anyt,E(t) = 0, E(2t) =

2, E(ts) = 0 ift=s.

ktis the linear deterministic component ofYt: it can be predictedarbitrarily well as a linear function of pastYt, i.e.,kt =E(kt|Yt1,...)and it is such thatE(kttj) = 0j.


TOPIC1 I NTRODUCTION OFT IME S ERIES IMPULSE R ESPONSE

IMPULSE R ESPONSE

For a processYtthat admits

Yt = +

j=0

jtj

fortsuch that, for any t,

E(t) = 0, E(

2

t) =

2

,E(ts) = 0, s=t.

notice thatYttj

=j

sojis the effect onYtof a shock that took place (t j)periods

before. A plot ofj(againtst j)is calledimpulse response function.



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TOPIC1 I NTRODUCTION OFT IME S ERIES ACF

AUTOCORRELATIONF UNCTION

DEFINITION (ACF)

For a stationaryYt,define the autocorrelation,

j=j

0

A plot ofj(againstj) is calledautocorrelation function.


TOPIC1 I NTRODUCTION OFT IME S ERIES PACF

PARTIALA UTOCORRELATIONF UNCTION

DEFINITION (PACF)

For a stationaryYtwithE(Yt) = 0, consider its linear projection,

Yt+1|t,...,tm+1= (m)1 Yt+(m)2 Yt1+ ... +(m)m Ytm+1For different values of m,

(1)1 ,

(2)2 , ...,

(m)m are the firstmpartial

autocorrelations, and a plot of(j)

j (against j) is calledpartial

autocorrelation function.



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INTRODUCTORYF INANCIALE CONOMETRICSTopic 2 MGF


Jianhua Gang


Spring 2013

JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 2 MGF SPRING 2 01 3 1 / 1 9

TOPIC2 M OMENT G ENERATINGF UNCTION

TOPIC2 MOMENTG ENERATINGF UNCTION

It is however essential to consider the MGFs in order todepict/solve relevant time series problems.

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 2 MGF SPRING2 01 3 2 / 1 9

TOPIC2 M OMENTGENERATINGF UNCTION PRELIMINARIES

PRELIMINARIES :SAMPLESPACE ANDR ANDOMVARIABLES

Define, x sample space;x random variable

Then a probability density function (pdf)f(x)is a mapping from

xto the set ofRwith the probability that:

Pr {xx}=

xxf(x) = 1;

xx

f(x)dx= 1.


TOPIC2 M OMENT G ENERATINGF UNCTION PRELIMINARIES

PRELIMINARIES :BINOMIALD ISTRIBUTION

Define as,

f(x) = n!

x!(n x)!px(1 p)nx,for x= 1,2,..., n.

The density arises as a sequence of the binomial expansion of:

(a+b)n =n

x=0

n!

x!(n x)! axbnx,

written asxBin(n,p).



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TOPIC2 M OMENTGENERATINGF UNCTION PRELIMINARIES

PRELIMINARIES :POISSON DISTRIBUTION

Define as,

f(x) = ex

x! ,for x= 1,2,..., n.

The density arises from the identity of:

e =

x=0

x

x!

in which = E(x).


TOPIC2 M OMENT G ENERATINGF UNCTION PRELIMINARIES

PRELIMINARIES :NORMALD ISTRIBUTION

Define as,

f(x) =exp

(x)222 22

written asxN,2 , where < x < .


TOPIC2 M OMENTGENERATINGF UNCTION EXPECTATIONS ANDM OMENTS

EXPECTATIONS ANDL OWER-O RDER M OMENTS

The expectation (or the mean, or 1st. moment) of a randomvariable is defined by,

E(x) =

xx

x f(x) discrete

xxx f(x) dx continuous

i.e., it is a weighted average ofxover all possible outcomes.The expectation of a measurable functiong(x)of a r.v.x istherefore defined by:

E {g(x)}=

xxg (x)f(x)

xx

g (x)f(x)dx


TOPIC2 M OMENT G ENERATINGF UNCTION EXPECTATIONS ANDM OMENTS

EXPECTATIONS ANDL OWER-O RDERM OMENTS

From which we obtain as special case the raw moments:

i = E

xi

And the central moments:

i= E

(x )i

Note that2is the variance ofx, i.e.,2 = 2.



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TOPIC2 M OMENTGENERATINGF UNCTION EXPECTATIONS ANDM OMENTS

HIGHER-O RDERM OMENTS

What about the higher-order (central) moments?In definition, the third and the fourth moments measure thefollowing properties:

3 : Skewness of the distribution

4 : Kurtosis of the distribution

For comparative purpose, skewness and kurtosis are usuallymeasured by:

3 = 33

4 = 44


TOPIC2 M OMENT G ENERATINGF UNCTION EXPECTATIONS ANDM OMENTS

CALCULATION OFM OMENTS

It is simple to show that:

g(x) = cE {g(x)}= cE

{c

g(x)

} = c

E

{g(x)

}E {a+b g(x)} = a+bE {g(x)}E {g(x) +h(x)} = E {g(x)}+E {h(x)}

and hence,

2 =E

(x )2

= E

x2[E(x)]2

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 2 MGF SPRING 2 01 3 1 0 / 1 9

TOPIC2 M OMENTGENERATINGF UNCTION MOMENT G ENERATINGF UNCTIONS (MGFS)

MOMENTG ENERATINGF UNCTIONS (MGFS)

Calculating the moments of even simple r.v.s can be difficult.However, consider the following function:

Mx() = E

ex

=

xx

exf(x)

xx

exf(x)dx

Mx() = E

ex

= E

1 + x+

2x2

2! + ...

= E

i=0

(x)i

i!

=

xx

i=0

(x)i

i!

f(x)dx

JIANHUA G ANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 2 MGF SPRING2 01 3 1 1 / 1 9

TOPIC2 M OMENT G ENERATINGF UNCTION MOMENTGENERATINGF UNCTIONS (MGFS)


=

xx

1 + x+

2x2

2! + ...

f(x)dx

= 1 + 1+2

2!

2+3

3!

3+ ... + i

i!

i+ ...

so that,di [Mx()]

di |=0 = i (raw moments)

Hence we call the function Mx()the MGF of x. Note that thisproperty is true in either the discrete or the continuous case.



37/115

TOPIC2 M OMENTGENERATINGF UNCTION MOMENT G ENERATINGF UNCTIONS (MGFS)


It is also easy to see that the MGF satisfies two very importantproperties.

g(x) = ax+bMg(x)() = ebMx(a)g(x) = x1+x2

Mg(x)() = Mx1()

Mx2()

Therefore, (Given thatx1, x2, x3, ..., xnare independent copies ofthe r.v.x.)

Mn1 xi =n

i=1

Mxi() = [Mx()]n

Mn11n xi

=n

i=1

Mxi(1

n) =

Mx(

n)

n


TOPIC2 M OMENT G ENERATINGF UNCTION EXAMPLE OFMG F

ANE XAMPLE

EXAMPLE

Observationsx1throughxnwhich are independent copies from r.v.

xPo().Suppose were interested in the properties (distribution,moments, etc.) of the sample mean:

X= 1

n

n

i=1

Xi


TOPIC2 M OMENTGENERATINGF UNCTION EXAMPLE OFMGF

ANE XAMPLE

PROBLEM

Calculate the MGF of X;

SOLUTION

Mx() = E

ex

=

x=0 exf(x)

=

x=0

ex ex

x! =

x=0

e e

xx!

= e expe

= exp

e 1



ANE XAMPLE

PROBLEM

Calculate the MGF of Sn= nX;

SOLUTION

MSn() =n

i=1

Mxi () = exp e 1

n

= exp

n

e 1

Note that the MGF of Sn is of the same form as that for x,i.e. letting = n

MSn () = exp

e 1

i.e. SnPo(n) = Po().JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL E CONOMETRICS Topic 2 MGF SPRING 2 01 3 1 6 / 1 9


38/115


ANE XAMPLE

PROBLEM

Calculate the MGF of X;

SOLUTION

MX() = M Snn

() = M xin

() =n

i=1

Mxi (

n)

=

Mx(

n)

n=

exp

en 1

n= exp

n

en 1



ANE XAMPLE

PROBLEM

The moments of X.

SOLUTION

E

Xi

=di exp

n(e

n 1)

di |=0

E

X

=d exp

n(e

n 1)

d

|=0=

E

X2

=d2 exp

n(e

n 1)

d2

|=0=2 +n2

X = E

X

2 E X2 =

n(central moments)



ANE XAMPLE

That is we immediately find that

E

X

=

2X

=

n

If we considerXas an estimator for, we refer to these propertiesas unbiasedness, and given the consistency, that is the variancetends to be zero.



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INTRODUCTORYF INANCIALE CONOMETRICS

Topic 3 ARMA Models3 CREDITS , 51 HOURS

Jianhua Gang


Spring 2013

JIANHUA G ANG (RUC) INTRODUCTORY FINANCIAL E CONOMETRICS Topic 3 ARMA ModelsSPRI NG2 01 3 1 / 4 7

TOPIC 3 ARMA M ODELS

TOPIC 3 ARMA MODELS

We said we are interested in the j in the representation:

Yt=+

j=0

jtj

for the impulse response analysis and for forecasting.

However, in general we dont know the j, and we cant hope toestimate an infinite number of parameters, so we have to proposeparsimonious models.

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL ECONOMETRICS Topic 3 ARMA ModelsSPRING 2 01 3 2 / 4 7

TOPIC 3 ARMA MODELS WHITE N OISE P ROCESS

THE S IMPLEST M ODEL: WHITE N OISE

DEFINITION

{t} is white noise if:

E(t) = 0t

E(2

t) = 2

tE(ts) = 0t (t=s)

so, j=0, j=0, and (j)

j =0 j=0.i.e. the process has no memory.

JIANHUA G ANG (RUC) INTRODUCTORY FINANCIAL E CONOMETRICS Topic 3 ARMA ModelsSPRI NG2 01 3 3 / 4 7

TOPIC 3 ARMA M ODELS WHITE NOI SEP ROCE SS

THE S IMPLESTM ODEL: WHITE N OISE

If t is w.n.(0, 2),

tmay be independent, butneeds not be;tmay be strictly stationary, but needs not be;t is covariance stationary;

and ifYt= +

j=0

jtj,thenYt is stationary if

j=0

2j < ;

and ifYt= +

j=0

jtj,thenYtis stationary and ergodic for the

mean if

j=0

j < .

JIANHUA GANG (RUC) INTRODUCTORY F INANCIAL ECONOMETRICS Topic 3 ARMA ModelsSPRING 2 01 3 4 / 4 7


40/115

TOPIC 3 ARMA MODELS MA(1)

MA(1)

Let t w.n.(0, 2), thenYt= +t+ t1is the MA(1).

We can check stationarity noticing that 0 = 1, 1 = ,so

j=0

2j =1+2< .

Otherwise, we can check that the first two moments do not depend on

time.

1 Mean: E(Yt) = 2 Autocovariances:

0 = E[(Yt )2] = E[(t+ t1)

2] = (1+2)2

1 = E[(Yt )(Yt1 )] =2

j2 = 0

3 Autocorrelations: 1 = 1+2

,j2 = 0.

JIANHUA G ANG (RUC) INTRODUCTORY FINANCIAL E CONOMETRICS Topic 3 ARMA ModelsSPRI NG2

financial econometrics notes

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