Generalized Spectral Estimation

Home , Whittle likelihood

Generalized Sp ectral Estimation

Jeremy Berkowitz

June 12, 1996

Abstract

This pap er provides a framework for estimating parameters in a wide class

of dynamic rational exp ectations mo dels. The framework recognizes that dy-

namic RE mo dels are often meant to match the data only in limited ways.

In particular, interest may fo cus on a subset of frequencies. Thus, this pap er

designs a frequency domain version of GMM. The estimator has several advan-

tages over traditional GMM. Aside from allowing band-restricted estimation, it

do es not require making arbitrary instrumentorweighting matrix choices. The

general estimation framework also includes least squares, maximum likeliho o d

and band restricted maximum likeliho o d as sp ecial cases.

Key Words: Estimation, Frequency Domain, Missp eci cation

JEL Classi cation: C13, C22

Federal Reserve Board, Mail Stop 61A, Washington, D.C. 20551. Telephone: (202) 736-5581.

Email: [email protected]. I would like to thank Frank Dieb old, Lee Ohanian, Valentina Corradi

and Jin Hahn for helpful suggestions. The view presented are solely those of the author and do not

necessarily represent those of the Federal Reserve Board or its sta .

1 Intro duction

This pap er develops frequency domain techniques for estimating dynamic rational

exp ectations mo dels. This approach allows, mo dels to be estimated and tested over

a subset of frequencies, such as business cycle frequencies, seasonal frequencies, or

long horizons. The techniques describ ed in this work are also particularly useful in

allowing researchers to deal squarely with high frequency measurement error.

It is natural for researchers interested in avoiding high frequency noise or in match-

ing particular cyclical b ehavior to carry out estimation and evaluation of such mo dels

in the frequency domain. The frequency domain provides an orthogonalization of the

uctuations in the observed data. Engle (1974) intro duced band sp ectral regression as

a means to assess the relationship b etween economic variables at sp eci c frequencies.

In that work, the criterion b eing minimized was restricted to linear least squares.

Generalized sp ectral estimation (GSE) allows for a much wider class of minimiza-

tion criteria than was previously p ossible. The GSE framework includes a new class

of estimators which I will call 'whitening estimators', as well as least squares, band

sp ectrum regression, and 'Whittle likeliho o d' estimation (which is asymptotically

maximum likeliho o d).

This pap er builds on Dieb old, Ohanian and Berkowitz (1995), who prop ose tech-

niques for estimating and evaluating dynamic rational exp ectations mo dels in the

frequency domain. Their framework allows for parameter estimation and mo del as-

sessment inavery general setting. Parameters are estimated by minimizing distance

between sp ectra of observed data and mo del-generated data. Distance may b e de ned

by the user and may fo cus on any relevant subset of frequencies. A shortcoming of

this framework is that in all but the simplest cases, the mo del must b e approximated

and simulated in order to carry out the estimation.

Since, in general, analytic solutions are not available for nonlinear dynamic equi-

librium mo dels, one must cho ose an approximate solution metho d. Furthermore, in

order to pro ceed with estimation, the approximation to the mo del must be carried

out at each parameter con guration. That means the mo del must b e simulated hun-

dreds if not thousands of times. Solution metho ds which can be chosen arbitrarily

'close' to the true mo del (such as discretizing the parameter space) are generally pre-

cluded b ecause of the extreme computational intensity asso ciated with simulating the

mo del only once. For estimation, then, faster and less accurate solution metho ds are

required. To the extent that the approximate solution di ers from the true mo del,

the Dieb old, Ohanian and Berkowitz (1995) estimated parameters will di er from the

parameters which minimize loss. It is dicult to make general statements regarding

For a discussion of band-restricted maximization of the Whittle likeliho o d function see Engle

(1980), Dieb old, Ohanian, and Berkowitz (1995). 2

this sort of approximation error. However, Taylor and Uhlig (1990), in a compari-

son of 14 approximation metho ds applied to the sto chastic growth mo del, concluded

that "the simulated sample paths generated by the di erent solution metho ds have

signi cantly di erent prop erties."

The generalized sp ectrum estimator will allow for mo del estimation, inference and

evaluation without requiring an approximate solution of the mo del. We accomplish

this by imp osing moment conditions given by the mo del and then minimizing devi-

ations from these conditions in the frequency domain. It is thus very much in the

spirit of generalized metho d of moment and other minimum distance estimators. For

whitening estimators, we imp ose moment conditions on the residuals which require

that the residuals are 'close' to white noise.

Section 2 de nes the generalized sp ectral estimator and presents some sp ecial

cases for illustration. Section 3 illustrates GSE estimation by presenting the results

of a Monte Carlo exp eriment. In the exp eriment, I maintain the realistic assumption

that the true mo del is unknown. Section 4 concludes.

2 Generalized Sp ectral Estimation

The Euler equation implied by a typical rational exp ectations mo del can be written

E (g (y ; ) j )=0; (1)

t 0 t

where g(; ) is a function given by mo del's rst order conditions, y is an Tx1 vector of

observable data, isavector of parameter values, and is the ( algebra de ned

0 t

by) agent's time t information set. g (y ;) is sometimes called the Euler residual.

Equation 1says that the Euler residual has a zero conditional mean. It implies that

for any rx1 instrument x ,inthe agent's time t information set,

" #!

E g (y ; ) = 0: (2)

t 0

Equation 2 is the familiar basis for GMM estimation. The notion here is that economic

agents should be unable to forecast the mo del residual. Each element of x , each

instrument, gives us an additional residual whose mean should be zero.

x is generally taken to be some number of lags of the endogenous variables,

y : However, consideration of lags of g (y ; ) as instruments will lead to some very

t t 0

interesting results. Note that it is always true that all lags of g (y ; ) are in agents'

t 0

time t information set. The motivation for doing this is that equation 1 says that the

Euler residuals form a martingale di erence sequence. Since martingale di erence 3

sequences are also white noise, equation 2 says that the Euler residual g (y ; ) is

t 0

multivariate white noise .

This fact provides the motivation for a class of GSE estimators, which I will call

'whitening estimators.' The reasoning is as follows. The mo del residual by design

should contain only the unexplainable part of the mo del in question. So, if we wish to

estimate ,wewould like to eliminate as much of the predictable dynamics in g (y ; )

0 t 0

as p ossible. These dynamics should b e incorp orated into the mo del. Another way of

saying this, is wewould like to estimate the parameter con guration in suchawayas

to make the residuals as 'close' as p ossible to white noise. We are, in e ect, trying to

whiten the g (y ; ): Some examples of whitening estimators will b e provided b elow.

t 0

Rewriting equation 3, we have that

E (g (y ; ) g(y ; )) = 0; =1;2; ::: (3)

t 0 t 0

For notational simplicity, let u = g (y ; ), so that u is a single Euler residual. Then

t t 0 jt

equation 3 is equivalenttothe statement,

2l T

f (! )= k; ! = ;l =0; ::: :

u l

T 2

The sp ectral density of each u ; is constantover the entire supp ort, [0, ]: Then given

a nite realization of the residuals, fu g ; we can calculate a consistent estimate of

the sp ectrum, such that f (! ) ! k; 8!:

In reality, though, u = g (y ; ); and we do not know : So we never observe

t t 0 0

a realization of the true Euler residualsfu g ; but rather at b est we can observe

fu ()g ; where is chosen somehow by the user.

2.1 A Simple Example

In order to x ideas, consider a simple linear example, an AR(1),

y = y + u :

t 0 t1 t

The y are observed and assume u (0; 1): In GSE notation, u ( ) = g (y ; ) =

t t t 0 t 0

(1 L)y ; where L is the lag op erator. If we do not know , we may lo ok at the

0 t 0

sp ectrum of the residual implied by any arbitrary ; f (! ); to see whether it is

u()

constant across all frequencies. For this arbitrary ; we can generate a length T

residual,

u ( )=(1L)y

t t

By this, I mean that, if g (y ; ) isavector sequence, each elementof g(y ; ) is white noise.

t 0 t 0 4

=(1( + )L)y

0 t

=u y = u u = A(L)u :

t t1 t t1 t

1 L

And so the sp ectrum of this residual u ( ) is,

f (! )= 1 e

u( )

1 e

Then f (! ) 6= k; across frequencies, unless =0 (that is, at = ):

u( ) 0

It is in this sense natural to estimate , by setting the sp ectrum f (! ) 'close'

u( )

to the sp ectrum of white noise. In other words, nd

=arg min C f (! ) ;

GS E

u( )

where the loss function C() is a measure of divergence of the sp ectrum from a at

line.

This approach has some signi cant advantage relative to Maximum Likeliho o d

and GMM estimation. To continue with the example, ML estimation would pro ceed

by minimizing the distance b etween the mo del data, fy ( )g ; and the observed data,

fy g .

The problem is that in the context of general dynamic equilibrium mo dels the data

generating pro cess (the p olicy function) is not known analytically. The innovation

pro cess, fu ( )g ; must rst be simulated and then the rest of the mo del data must

be generated via an approximate p olicy function. MLE thus requires an additional

layer of approximation.

The advantage of GMM is that one never needs to construct an approximate p olicy

function. Parameters can be estimated using only the observed data. The problem

with GMM is that there is no such thing as the GMM estimator. The researcher is

required to make achoice regarding b oth instruments and the weighting matrix.

GSE, like GMM, can pro ceed without requiring an approximate solution of the

mo del. But unlike GMM, the instruments and weighting are not arbitrarily chosen.

Rather, the user cho oses a frequency band of interest. For example, if the mo del is

designed to explain business cycle uctuations, the user may sp ecify cycles between

2to8 years.

De nition: The generalized sp ectral estimator is de ned as:

=arg min C f (! ) ;

GS E

u( )

! 2$

T 5

where C() may include a wide variety of appropriate loss functions, f (! ) is a

u( )

consistent estimate of the sample sp ectral density matrix (or p ossibly the p erio dogram

matrix ) of u ( )=g(y ;): The summation is taken over the frequencies of interest,

t t

represented by the set $: Wemay appropriately think of the f (! ) as a triangular

u( )

2l T T

array with ! = ;l =0; ::: : The maximum frequencies p er sample size are . This

T 2 2

loss function allows for fo cusing on a subset of frequencies which are judged to be

economically relevant. This is not p ossible in the time domain. To get a b etter feel

for this estimation strategy,wenow describ e some examples of the loss function C():

For what follows, we consider only a univariate u ( ):

2.2 Whitening Estimators

We might sp ecify,

^ ^

=arg min (! ) ! ; (4)

u( )

(! )isthe sp ectral distribution or 'cumulative sp ectrum' of u : We have, where

u( )

in e ect, inverted the Cramer-Von Mises statistic to obtain an estimator . A few

comments are in order here.

1) Rational exp ectations implies that u is white noise, so our moment conditions

are that E F (! ) = !: In words, we have that, in p opulation, the sp ectral

u( )

distribution function of white noise is the 45 line. This estimator minimizes the

L distance between the sample sp ectral distribution and the p opulation sp ectral

distribution of white noise.

2) This estimator is very much like GMM in that it minimizes the squared devi-

ations from moment conditions.

3) There are an in nite numberofvalid functions, C (), with unique minima where

u is white noise. Durlauf (1991), for example, considers a numb er of sp eci c distance

functions that measure deviations of the cumulative p erio dogram in the context of

testing for white noise.

4) We suppress dep endence of the set of ! on the sample size for the remainder

of the pap er for notational simplicity.

A particularly interesting example of a whitening estimator, which I will call

"Sp ectral-GMM", arises when we take C ()as follows,

(BB(z)) dz ; where BB(z) is a Brownian Bridge. The Cramer-Von Mises statistic is actually

0 6

=arg min f ; (5) f (! )

u( ) u ( )

! 2$

where f is the average of the estimated sp ectra (or p erio dograms).

u( )

This estimator displays some interesting prop erties. We will show that it coincides

with a time domain GMM estimator, with the choice of $ corresp onding to the choice

of a GMM weighting matrix. First we need to de ne some notation.

i! j

Set W be the Fourier matrix. It has typical element [W] = [e ], note

y y

that this matrix has the prop erty that W W = I, where W is the conjugate transp ose

of W:

De ne W*, [W*] = [cos (! j )]k (j ), where k(j) is a smo othing window so

mj m

that k(j)=0 for j>B ;m 2 $: So the number of rows of this matrix is equal to the

number of frequencies included in $: Call the number of frequencies included in the

band M, so that W* is an M x T matrix. Let,

2 3

u u

t t1

T 1

6 7

u u

t t2

T 2

^ 6 7

G = (6)

6 7

:::

4 5

u u

t(T 1)

T (T 1)

be the vector of moment conditions implied by the mo del. Also de ne,

3 2

1 1

:: 1

T T

7 6

:: 1 ::

(7) A =

5 4

1 1

:: 1

T T

V = W A AW :

Now we will show that

^ ^

f (! ) : G V G = f

u( ) u( )

! 2$

0 0 0

0 0

^ ^ ^ ^ ^ ^

By de nition, G V G = G W A AW G =(AW G) (AW G): But this is simply,

2 3

" #

cos(! )k ( )G

X X

6 7

i! 0

:::

cos (! )k ( )G ::: cos (! )e k ( )G A A (8)

4 5

1 M

cos (! )k ( )G

= f ; (9) f (! )

u( ) u( )

! 2$

since multiplication by the matrix A de-means each column. We could equally

well cho ose the smo othing window in W to be k( ) 1 and thus minimize

! 2$

;where I (! ) is the p erio dogram of the residual. I (! )

u( ) u( ) u( )

This "sp ectral-GMM" estimator has the sp ecial prop erty that the moment con-

ditions b eing imp osed are exactly those which corresp ond to requiring that u wn:

This estimator allows for whitening of the residuals and, b ecause it is a GSE, it allows

for band restricted whitening. This estimator also has a particularly simple form.

2.3 Other Frequency Domain Estimators

In this section, we will now present the GSE cost functions which give rise to

the least squares and band restricted least squares estimators, and then (asymptoti-

cally) maximum likeliho o d and band restricted maximum likeliho o d estimators.

First, we can implement least squares bycho osing the trivial cost function, which

yields

=arg min I (! ); (10)

u( )

where I (! )isthe p erio dogram of the residual u : To see that this estimator coin-

u( ) t

cides with least squares, write the nonlinear least squares estimator,

=arg min U U; (11)

0 0 y y

where Uisthe Tx1 vector of residuals u . Now U U = U W WU =(WU) WU

= I (!): (12)

u()

Band sp ectral regression is nested within this estimator. Instead of taking the

summation over T/2 frequencies, we sum only over the frequencies of interest,

=arg min I (! ): (13)

u( )

! 2$

If, for example, we would like to fo cus on uctuations of business cycle frequency we

h i

might restrict $ = : Using quarterly data, this band would isolate cycles of

16; 4

length b etween 2 and 8years.

Now consider Whittle likeliho o d estimation, 8

I (! )

u( )

=arg min

f (! )

u( )

where f (! ) is again, the smo othed sp ectral density of u : Under normality and

u( )

circularity of the residual this estimator can be derived from the familiar time do-

main likeliho o d function (see, for example, Harvey (1989)). Dieb old, Ohanian, and

Berkowitz (1995) advo cate band-restricted maximum likeliho o d, which di ers from

maximum likeliho o d by taking the summation over only those frequencies of foremost

interest,

I (! )

u( )

: (14) =arg min

! 2$

f (! )

u( )

2.4 Consistency

This section delineates sucient conditions for the consistency of the GSE under cor-

rect sp eci cation of the mo del. Although correct sp eci cation is surely an unrealistic

assumption, it is of obvious interest to verify that our estimation pro cedure would b e

asymptotically valid if we did, in fact, know the true mo del.

As ab ove, let =arg min C f (! ) :

u( )

! 2$

Assumption B1: 2 ; a compact subset of R .

Assumption B2: g (;) is Borel measurable for each 2 and g (y ; ) is continuous,

uniformly in y :

Assumption B3: E(g (y ;)) = 0 and g (y ;) has a nite sp ectral density, L -

t t 1

continuous on 2 :

Assumption B4: C() is continuous. For b oth sp ectral-GMM and the CVM es-

timator, this assumption is easily veri ed. In the sp ectral-GMM case, C() is the

quadratic. For the CVM estimator, C() is a comp ound function, a quadratic op er-

ating on a summation.

Assumption B5: There exists a unique such that g(y ; ) = u( ) WN:

0 t 0 0

There must b e a unique parameter vector for which the Euler residual is white noise.

This is not restrictive in the context of dynamic mo dels. However, some time series

mo dels such as ARCH mo dels must be handled with care. Consider for example,

A pro cess is said to b e circular if its auto covariance matrix has the form of a circulant. Letting

( ) denote the auto covariance at lag ; a circulant has the prop erty that ( ) = (T ); for

=1; :::T 1: Circularity do es not hold in in nite moving average pro cesses, in general, but even

without circularity Whittle's derivation holds asymptotically. 9

an AR(1) with ARCH innovations. Even under correct sp eci cation, the innovations

are not uniquely white noise for any set of ARCH parameters. The parameters may

be estimated, however, by noting that the squared innovations have a conditional

homoskedastic ARMA representation.

Assumption B6: The loss function , C (); must have the prop erty that

R R

arg min C f (! ) d! = with 0< C f (! ) d! : This says that, given that

u( ) u( )

$ $

the mo del is identi ed (assumption B4), the loss function must be minimized at

the p opulation parameter vector . This is obviously satis ed for a wide variety

of functions, C (): To make this concrete we will check this assumption for the two

Whitening Estimators intro duced ab ove.

For the CVM estimator, this condition is trivially ful lled since

Z Z

F (!) ! d! ; C f (! ) d! =

u() u( )

$ $

which achieves a minimum at F (!)=!; uniquely at :

u()

Next consider the sp ectral-GMM. Wehave f (!) f d! : This is clearly

u()

minimized at any for which f (! ) = k: Together with assumption B4, is the

u( )

unique value for which this is true.

Theorem 1 Under the assumptions B1-B6, ! : Given the regularity condi-

T 0

tions discussed above, the generalized spectrum estimator is consistent for the true

parameter vector.

Pro of. From assumption B4 and the continuity of C(), C f (! ) d! is

g (y ; )

continuous in :

Next, note that for xed ; we can calculate a consistent estimate of the sp ectrum,

^ ^

f (! ) ! f (! ) uniformly in !: The f (! ) are asymptotically indep endent

g (y ; ) g (y ; ) g (y ; )

t t t

^ ^

with cov f (! ); f (! = O (T ); by, for example, Brillinger (1981). This,

i j

g (y ; ) g (y ; )

t t

in turn, implies that,

T C f (! ) ! C f (! ) d! :

g (y ; ) g (y ; )

t t

Now, since is compact, we can write

C f (! ) C f (! ) d! ! 0: (15) sup T

g (y ; ) g (y ; )

t t

Assumption B3 implies that f (! ) is measurable in y and with continuous C();

g (y ; )

C f (! ) is y-measurable. De ne N( ) as an op en neighb orho o d around ; and

g (y ; ) 0 0

N ( ) as its complement: Then N ( ) \ is compact and min C f (! ) d!

0 0 g (y ; )

2N ( )

exists. Further de ne,

Z Z

= min C f (! ) d! C f (! ) d! :

g (y ; ) g (y ; )

t t

2N ( )

C f (! ) <=2;8: Then Let A b e the event C f (! ) d! T

g (y ; ) g (y ; )

t t

by rearranging terms, A implies

C f (! ) d! =2: (16) C f (! ) d! < T

g (y ; ) g (y ; )

t t

T T

A also implies,

C f (! ) < C f (! ) d! =2: (17) T

g (y ; ) g (y ; )

t t

0 0

Now

X X

1 1

^ ^

C f (! ) ; (18) C f (! )

g(y ; ) g (y ; )

t t

by de nition of : Combine equation 16 with 18, so that,

A )

C f (! ) =2: (19) C f (! ) d! < T

g (y ; ) g (y ; )

t t

Now add, 17 and 19, which leaves,

R R

C f (! ) d! < C f (! ) d! : So that when A is true, 2=

g (y ; ) g (y ; ) T T

t t

N ( ) \ ; or 2N( ):

0 T 0

Now, since, lim P (A )=1;by equation 15, we conclude that ! :

T T 0

T !1

Q.E.D.

2.5 Asymptotic Normality

Under two additional assumptions, it is p ossible to show that the Generalized Sp ec-

trum estimator is asymptotically Normally distributed. The asymptotic variance will

dep end not only on the underlying data generating pro cess, but also on the choice of

C ():

Assumption C1: C()istwice continuously di erentiable.

Assumption C2: E C ( ) < 1 , is an exp onentially distributed random

! ! !

variable. 11

Theorem 2 Under the additional assumptions C1 and C2,

1 1

! N (0;H( ) H( ) ); where H( ) is the Hessian of the GSE T

0 0 0 GS E 0

minimand.

Pro of. De ne Q ( ) = C (f (! ))d! , the quantity b eing minimized. We have

that Q () exists and is continuous. It is established in the pro of of Theorem 1

that

T Q ( ) ! Q( ); uniformly in :

It is thus straightforward to show that

2 2

@ @

T Q () j ! Q( ): (20)

T 0

2 2

@ @

1=2

Next we consider the quantityT Q () j : From the de nition of the GSE,

@ f (! ) @

Q ()= C (f (!)) : (21)

@ @

@ f (! )

^ ^

C (f (! )) We will evaluate this quantity at the p oint , : The f (! );i=

0 i

0 0 0

^ ^

1,.. T/2, form an asymptotically indep endent triangular array, with cov f (! ); f (! ) =

i j

0 0

O (T ): We maynow apply a Central Limit Theorem due to McLeish (1975),

@ f (! )

0 1=2

C (f (! )) T ! N (0; ); (22)

0 0

@f (!)

0 2

= (f C (!)) d! :

0 0

Combining 20 and 22 will b e sucient to prove the theorem by applying the mean

value theorem,

@Q @Q @ Q

T T T

= ( ); 2 (; ): +

T 0 0

@ @ @

" #

@Q @ Q

T T

1=2 1

T =) T ( )= T

T 0

@ @

h i

2 2

@ Q

@ @

= But, plim T Q( ) by 20. Setting H( ) = Q( ); we conclude

2 2 0 0 2 0

@ @ @

that

1 1

T ( ) ! N (0;H( ) H( ) ):

T 0 0 0 12

3 Simulation Exp eriment

In order to illustrate the b ehavior of GSE, this section rep orts the results of a simple

Monte Carlo exp eriment. The exp eriment is designed to mimic some of the charac-

teristics of daily NYSE sto ck returns data. There is a large literature which studies

the e ect of high frequency noise on observed sto ck returns. The measurement er-

ror arises primarily from bid-ask spreads and asynchronous trading (e.g., Blume and

Stambaugh (1983), Roll (1984)). Consider the following data generating pro cess,

x = x + "

t t1 t

y = x + u :

t t t

The researcher can only observey;which includes high frequency measurement error

u : The researcher do es not the data generating pro cess for u : We are interested

t t

in estimating the unknown parameter : We can pro ceed by using standard ARMA

information criteria and estimation to ols to t an ARMA to y . Alternatively,we can

t an AR(1) to y and omit high frequencies by GSE.

Since we are not interested in the high frequency dynamics in u ; we will use

sp ectral-GMM with a low frequency band. Intuitively, the sp ectrum of y will be

the sum of a low frequency comp onent, x and a high frequency comp onent, u : By

t t

ignoring the high frequencies, we can e ectively estimate the sp ectrum of the low

frequency comp onent and thus invert out the p ersistence parameter.

Following Roll (1984) u is mo deled as an MA(1) with variance equal to 10% of

the variance of y : For the purp oses of the exp eriment, " and u are taken to be

t t t

normally distributed. Table 1 rep orts sp ectral-GMM estimates for using the low

frequency half of available sp ectra, ! 2 [0; ]: Row 1 contains OLS estimates of ;

with Monte Carlo standard errors in Row 2. OLS is severely downward biased due

to the measurement error. The poor p erformance of OLS is not surprising, since

the mo del is, in e ect, missp eci ed. Rows 3 and 4 contain estimates which are con-

structed by tting an ARMA mo del to the data using the Hannan and Rissanen

(1982) information criterion to select the lag lengths and estimating the parameters

via Maximum likeliho o d. This pro cedure delivers consistent ^, since the order of

the ARMA is asymptotically correctly selected. Indeed, the bias of these estimates

disapp ears quickly in the sample sizes studied here. Nevertheless, Sp ectral-GMM es-

timates, displayed in Rows 5 and 6, clearly outp erform the standard ARMA approach

in this exp eriment. Comparison of Rows 3 and 5 indicates that the GSE has b oth

smaller bias and less Monte Carlo variation, in every sample size. 13

4 Conclusion

This pap er suggests some new techniques for estimating dynamic rational exp ecta-

tions mo dels which are explicity designed to match only subsets of uctuations in

observed data. For example, the metho dology allow mo dels to be estimated in the

presence of high frequency noise. Generalized Sp ectral estimators may also be of

use in estimating business cycle mo dels or long term growth mo dels b ecause of their

inherent fo cus on subsets of frequencies.

A simple Monte Carlo exp eriment studies the ability of the GSE to estimate an

autoregressive parameter, when the data of interest is observed with high frequency

measurement error. It is assumed that the particular form of the measurement error

is unkown to the economist. The results of the exp eriment suggest that the ability

of the GSE to "ignore" high frequencies leads to far more precise estimation than

standard ARMA techniques in nite samples. 14

References

[1] Amemiya, T., Advanced Econometrics, Cambridge: Harvard University Press,

1985.

[2] Berkowitz, J., "An asymptotically maximum likeliho o d estimator in the fre-

quency domain," Mimeo, Department of Economics, UniversityofPennsylvania,

1994.

[3] Blume, M.E. and R.F. Stambaugh, "Biases in Computed Returns: An Applica-

tion to the Size E ect," Journal of Financial Economics, 12(1983), 387-404.

[4] Brillinger, D. R. Time Series: Data Analysis and Theory , New York: Holden

Day, 1981.

[5] Den Haan, W. J. and A. Marcet, "Accuracy in Simulations," Review of Economic

Studies, 61(1994), 3-17.

[6] Dieb old, F. X., L.E. Ohanian, J. Berkowitz, "Dynamic Equilibrium Economies:

AFramework for Comparing Mo dels and Data," NBER Technical Working Pap er

No. 174, 1995.

[7] Durlauf, S. N., "Sp ectral Based Testing of the Martingale Hyp othesis," Journal

of Econometrics, 50(1991), 355-376.

[8] Engle, R. F., "Band Sp ectrum Regression," International Economic Review,

15(1974), 1-11.

[9] Engle, R. F., "Exact Maximum Likeliho o d Metho ds for Dynamic Regressions

and Band Sp ectrum Regressions," International Economic Review, 21(1980),

391-407.

[10] Granger, C.W.J. and P. Newb old, Forecasting Economic Time Series, San Diego:

Academic Press, 1986.

[11] Hannan, E. J. and J. Rissannen, "Recursive Estimation of Mixed Autoregressive-

Moving Average order," Biometrika, 69(1982), 81-94.

[12] Hansen, L. P., "Large Sample Prop erties of Generalized Metho d of Moment

Estimators," Econometrica, 50(1982), 1029-1055.

[13] Roll, Richard "A Simple Implicit Measure of the E ective Bid-Ask Spread in an

Ecient Market," Journal of Finance, 39(1984), 1127-1139. 15

[14] Uhlig, H. and Taylor, J. B., "Solving Non-linear Sto chastic Growth Mo dels: A

Comparison of Alternative Solution Metho ds," Journal of Business and Eco-

nomic Statistics, 8(1990), 1-17. 16

Table 1. Simulation Results

Sample Size 25 50 75 100 150 200 250 1000

OLS .236 .386 .446 .482 .516 .536 .547 .578

(.284) (.212) (.177) (.154) (.128) (.107) (.097) (.047)

ARMA .333 .664 .770 .815 .844 .861 .870 .894

(.461) (.316) (.216) (.166) (.115) (.085) (.069) (.023)

GSE .787 .838 .856 .856 .875 .882 .886 .896

(.233) (.118) (.084) (.069) (.054) (.043) (.038) (.018)

Notes: Alternative estimates of averaged across 2000 Monte Carlo trials. Monte

Carlo standard errors are in parentheses. Rows 1-2 are OLS estimates. For Rows 3-4,

the order of the moving average is selected by Hannan and Rissanen (1982) infor-

mation criterion, the parameters are then estimated by MLE. Rows 5-6 are sp ectral-

GMM estimates with ! 2 [0; ]:

2 17