The Simultaneous Equations Model with Generalized Autoregressive Conditional Heteroskedasticity: The SEM-GARCH Model

International Finance Discussion Paper Number 322.

May 1988

THE SIMULTANEOUS EQUATIONS MODEL WITH GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY: THE SEM-GARCH MODEL

Richard Harmon

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to

unpublished material) should be cleared with the author or authors.

ABSTRACT

In this paper I generalize the standard simultaneous equations model by allowing the innovations of the structural equations to exhibit Generalized Autoregressive Conditional Heteroskedasticity (GARCH). I refer to this new specification as the SEM-GARCH model. I develop two estimation strategies: LIM-GARCH, a limited information estimator, and FIM-GARCH, a full informaticn estimator. I show that these estimators are consistent and asymptotically normal. Following Weiss (1986) I show that when the errors in the SEM-GARCH process are incorrectly assumed to be conditionally normal the likelihood function is still maximized at the true parameters, given certain regularity conditions. This results in the asymptotic

variance-covariance matrix being: more complex than the usual inverse of the

information matrix.

THE SIMULTANEOUS EQUATIONS MODEL WITH GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY: THE SEM-GARCH MODEL

* Richard Harmon

In this paper I generalize the standard simultaneous equations model (SEM) by allowing the innovations of the structural equations to exhibit Generalized Autoregressive Conditional Heteroskedasticity (GARCH). A GARCH(p,q) process is a process whose conditional variance at time t is a function of the information available at time t-l. Specifically, it is a function of variances of past innovations and of the squared realizations of past innovations. Unconditionally, the current innovation reverts to the standard specification: white noise with fixed variances over time. I refer to this new specification as the SEM-GARCH model. I develop two estimation strategies: LIM-GARCH, a limited information estimator, and FIM-GARCH, a full information estimator. I show that these estimators are consistent and asymptotically normal.

The outline of the paper is as follows. Section I presents the unrestricted and diagonal SEM-GARCH(p,q) models. Sections II and III derive the LIM-GARCH estimator and its asymptotic properties. Sections IV

and V develop the FIM-GARCH estimator and its asymptotic properties.

*I am grateful to James Albrecht, Dale W. Henderson, and especially Dan Westbrook for their patience and guidance throughout the course of research embodied in this paper. Helpful suggestions are also appreciated from Frank Diebold, Neil Ericsson, David Howard, Ralph Tryon and the participants of a seminar at the Federal Reserve Board. This work was completed while I was an intern in the International Finance Division of the Federal Reserve Board. This paper represents the views of the author and should not be interpreted as reflecting the view of the Board of Governors of the Federal Reserve System. Unfortunately, any remaining errors are my responsibility.

Finally in Section VI the Information Matrix Test of White (1982) is used to ascertain the correct form of the variance-covariance matrix and test for misspecification. This is followed by some concluding remarks. I. THE SEM-GARCH(p,q) MODEL

The Autoregressive Conditional Heteroskedastic model (ARCH) developed by Engle (1982) and its generalization, the GARCH model, developed by Bollerslev (1986) specifies the conditional variance of the current innovation as a function of the available information set; specifically,

the conditional variance is a function of the squared realizations of past

innovations and of variances of past innovations. In the GARCH(‘p,q) model, the conditional variance, denoted ne, has the following specification q P (1.1) he = vw + ae - + & he : t ° i-t-i : i t-i i=l i=l

Kraft and Engle (1982) extend the ARCH model into a multivariate time series framework. Extensions with respect to multivariate GARCH models are given by Bollerslev, Engle, and Wooldridge (1985). The SEM-'SARCH model, to be specified below, generalizes the standard SEM by allowing the innovations of the structural equations to exhibit GARCH processes. This type of specification has many potential applications. For example, in models of foreign exchange rate determination it can be used to model the joint determination of the foreign exchange rate with domestic and foreign

interest rates.

The standard SEM consists of M linear equations: (1.2) yo + XB ~ €& t=$1,...,T,

where T is a M x M matrix of coefficients of current endogenous variables,

B is a K x M matrix of coefficients of predetermined variables, ee is

row vector consisting of M innovations at time t, and Y, and x. are row

a

vectors consisting of observations on M endogenous variables and K

predetermined variables, respectively, at time t. It is assumed that the

vector of innovations, ¢€ is given by

t’

(1.3) e.7 N(O,z) ,

where = is nonsingular. The likelihood function for the sample Y Y

porte conditional on X is given by

(1.4) LOYy yes Vp hR) = (2m M2 | |Taf 7/7 ii -1 exp|- 3 = (v0 + X,A)= (YF + x,p)'| i-1

Leaving aside the variance-covariance matrix 2, which has M(M+1)/2 distinct elements, the likelihood function defined by (1.4) has a + MK parameters. Clearly, without a priori restrictions onTI, B, or X, none of

the parameters of the structural model are identified.

For now I restrict my attention to a model whose predetermined variables are exogenous; dynamic linear SEM remain a topic for future research. Following Kraft and Engle's multivariate ARCH specification,

the SEM-GARCH(p,q) model is given by

(1.5) YF + X86 = «€, (1.6) Ce.[T,3) ~- NOH.) , where Hae Maae oe ime (1.7) H = H,. H H

t 21,t §22,t °"" 2M,t

Hae Hye t ee Ha

= W+ Ee @ cea ]ey[ty @ 4 tose + [H ® cealal te ® co

e

+ [1 @ b_4)>, [* ®@ | ticee + [1 @ be PPa| ®@ h._,| ,

where fey and hey are M-element row vectors consisting of ith lagged innovations and ith lagged conditional standard deviations. Y, and x. are row vectors consisting of M endogenous variables and K _ exogenous variables, respectively. Hist (i,j=-1,...,M) is a scalar and represents the conditional variance of the ith equation when i=j and the conditional

covariance between the ith and jth equations when i»j. The primary

restriction imposed on H. is that it must be’ bounded = and positive-definite. C and D are Mw x m2 symmetric matrices with M x M symmetric blocks of Ci, and Diy? respectively. W, a M x M scalar matrix, nests the hypothesis of homoskedastic ey within the SEM-GARCH specification and allows the conditional variance-covariance matrix to exist when all C and D are zero.

While I explicitly assume that the conditional distribution of the innovations is normal, in many circumstances this may not be the case. If the conditional probability model is non-normal, then the estimators proposecl here are quasi-maximum likelihood estimators (QMLE). White (1982) notes that the asymptotic variance-covariance matrix of the QMLE no longer equals the inverse of Fisher's Information matrix, but can be consistently estimated by a more complex form. For expository purposes I will continue to assume that the conditional probability model is correctly specified and explore the consequences of it being incorrectly

specified when the asymptotic properties of the estimates are derived.

The following assumptions will be made throughout:

= Q , where Q is a finite and nonsingular matrix,

(4) . pain

is,

where = is a bounded and positive-definite unconditional variance-covariance matrix.!

A alternative parameterization of the SEM-GARCH(p,q) inodel facilitates estimation and highlights the alternative variance-covariance specifications available. Let h, be the M(M+1)/2 column vector whose components are the unique elements in He taken by vectorizing the lower triangle of H.- For expository purposes I will focus on_ the

SEM-GARCH(1,1) model in a two equation system (M=2). Here the conditional

variance-covariance matrix is defined to be

(1.8) H, =

+ [yen |r, [nen].

where Cc) and Dd, are 4 x 4 parameter matrices given by

tan excellent discussion of Central Limit Theory for the non-i.i.d. case is provided in White (1984) Ch.5:

(1.9)

(1.10)

1 %12 : “13 “14 “o1 S22 : S23 S24 Cay S39 + &33 O34 Cc. C,:6,. ¢

Dir P12 P13 Pe Poa Daa: O23 Dee. D31 P32 : 233° P34 Dar Pa2 : Pas Pua

Note that the symmetry of C and D require that Co - Cho: C53 - Chay: Ca =

Cao)

(o =

43 ~ ©34°

Diary Pos = Paar Par 7 P32 and Pus = Pgy- The

alternative parameterization consists of vectorizing the lower triangle of

He»

* denot:ed h,,

(1.11)

h*

Ait Hort Hoot w A A A e2 11 11 “12 413 1,t-1 vor | + | Aor Aca Aas |} €2,e-161,e-2 | 52 Az, Azo 433 | Oe

11 Bia Bas |} Baa, e-2 91 Boo Boz |} Horta. | °

31 332 333 |} Boo,e-4

which in matrix notation can be written as

1.12) ht + A + Bh. (1.12) t 7 M1 tl

where 1-47 vec(es 1&1) and f.47 (ey ea? > 1° This setup will be referred to as the "“unrestricted" SEM-GARCH(1,1) model since no restrictions are imposed on matrices A and B except those required to ensure the positive definiteness of H,- Expansion of (1.11) highlights

why this specification is referred to as "unrestricted",

2 (1.13) yy ge = yt Agate * Are (€2,e-281,e-2? + A ez + B..H + B..H 13°2,t-1 18a. t-1 12491 t-1 + By 3Hoo t-1 (1.14) 4 ~ vw. + Ane +a 21,t 21 21°1,t-1 222, t-1°1,t-L. + Ae + B,.H + B,.H 23°2,t-1 21841, t-1 22o1 t-1 +

Bo 342 t-1 ?

. 2 (1-15) Hoa tg 7 Man + Agfa eed * Age€o t-181 ev + Anne? + BH + BoH 33°2,t-1 31°11,t-1 32°21,t-1 + B33Hoo ta °

As shown in Table 1, the conditional variance-covariance matrix of a two equation SEM-GARCH(1,1) model has 21 free parameters. While there should be no problem estimating the parameters of this small system, the number of parameters rapidly increases as the system gets larger. Table 1 shows that a five equation SEM-GARCH(1,1) model has 465 parameters. Many time se‘ries data sets do not have enough observations for estimation; clearly a more parsimonious specification is required.

One such specification models the conditional variance of each equation as a function of its own lagged squared innovations and lagged conditional variances. Similarly, the conditional covariances can be modelled’ as functions of the lagged cross-innovations and lagged conditional covariances. Imposing this structure on the SEM-GARCH(1,1)

model yields (1.8) with Cc) and D) given by

11 14 0 0 :C¢ 0

(1.16) Cc) - beceeeeeeet es’? seen 0 C35 0 0 Cc 0 0 oC

GARCH

Model

(1,1)

(2,2)

(3,3)

Table 1

THE SEM-GAR‘ MODE

The Conditional Variance-Covariance Matrix

Equations Unrestricted” Diagonal” 2 21 9 3 ee: 18 4 210 30 5 465 45 2 ; 39 4s 3 150 30 4 410 50 5 915 75 2 | 57 21 3 222 42 4 610 70 5 . 1365 105

* Unrestricted Var-Cov: # = [M(M+1) /2]2(p+q) + [M(M+1)/2]

M

**k Diagonal Var-Cov: # = (l+ptq) Zi where M is the

number of equations in the system.

10

Dd, 0 0 Ply 0 oOo :D 0

(47) Dp os Jee. 238 0 Ds: 0 0

Di 9 0 Dyy

where symmetry implies Co3 - Cha: Cha - C35: Do3 - Diy? and Dal - D35-

* Under the more parsimonious parameterization, h, becomes

H

1l,t 1.18 h* H (1.18) t 21,t Hoot A 0 oO 2 “11 | “11 “1,t-1 - | %1] + 0 Ag, 0 £9 e-191,t+1 2 “99 0 0 433 9 t-1 Bi, 9 820 Aya t + 0 Boy 0 Hor t-1 0 90 B33 tt Boot Expanding (1.18) yields (1.19) H.. . = w + Ace . + BLAH 11,¢ 11 11°1,t-1 117q1 e-1 °

11

(1.20) H = w + A

21,t + B,.H

21 2262, t-1°1,t-1 22°21,t-1 ’

(1.21) H =

22,t Yoo + A

+ By 3Ho 1

2 33°2,t-1 Specification (1.18) is referred to as the "diagonal" SEM-GARCH(1,1) model. Table 1 summarizes the parametric requirements of the full and diagonal SEM-GARCH(1,1) model. For the SEM-GARCH(1,1) model in a two equation system the number of parameters is reduced from 21 to 9. For a five equation system the number of parameters is reduced from 465 to only 45.

As stated earlier, H, is required to be bounded and positive

t definite. For the SEM-GARCH(1, 1) model this requires that

(1.22) Ait

and

Ww Ol.

(1-23) 0 Aya Hoo t 7 Bore

Clearly, a sufficient condition for He to be positive definite is having every term in w, A, and B be positive definite. A less stringent sufficient condition is established by setting A=B=0 which from (1.22) and

(1.23) implies

12

| 2 (1.24) 14 > 0, Woo >0 and 11 ¥o9 — Wo >o. Similarly, setting w=B=0 yields 2 (1.25) All >o, Ayo > 0 and Ay y409 - Aoy >0 Alternatively, setting w=A=0 yields 2 (1.26) Bit >o, Boo >0 and Bi 1359 ad Boy >o.

For higher order SEM-GARCH models the parameter constraints to ensure the positive definiteness of He are extremely complicated. In those cases, one might impose penalty functions to ensure that H. is positive

definite.

Il. ‘THE LIM-GARCH ESTIMATOR

Since the reduced form innovations involve linear combinations of all the structural innovations, my focus is on structural form estimation to facilitate the conditional heteroskedasticity specification. I develop two estimation strategies for the SEM-GARCH(p,q) model. The first is a limited information approach that concentrates upon a single equation of the simultaneous equations system while disregarding the parametric restrictions that bind the system as a whole. The second is a full system

estimation that makes efficient use of all available information. The

13

limited information approach is useful when the full model is too complex to be estimated by a full information technique or when one suspects specification errors in equations other than the equation of primary interest.”

As described in Section I, the structural model consist:s of M

equations with a single equation, say the first, given by

(2.1) YI. + X84 7 eee

where Ty By: and «¢ are the first columns of I, #, and e,

1t respectively. The variables are arranged so that the usual identifying

restrictions may be shown by the following partitions:

and

The number of included and excluded endogenous variables in the first

2The LIM-GARCH estimator can be viewed as a particular case o FIM-GARCH where the other M-1 equations are just identified and have non-ARCH innovations... ,

14

* equatiion are denoted by m) and m, (= M - m,); respectively. The numbers

*

of included and excluded exogenous variables are k, and k, (= K - k))-. Hence, a, and b, are column vectors with m and ky elements, respectively. The matrices of endogeneous and exogenous variables are partitioned in

(2.3) to correspond to the partitioning of the coefficient vectors in

(2.2). The usual order and rank conditions for identification of the

first equation are given by

* (Order Condition): k) = m ~ 1 (Rank Condition) : Rank(, A, ) -M-1

where: >, is a R X (MtK) selection matrix composed of zeros and ones, and Ay = (T'y> B')) = (at : 0’ : be : 0’)’ is a (M+K)—element column vector composed of all the parameters of the system.

The limited information approach to estimation of the SEM-GARCH(p,q) model is referred to as the LIM-GARCH estimator. Following the standard derivation of the LIML estimator, [c.f. Koopmans, Rubin, and Leipnik (1950), Dhrymes (1970):328-357 and Schmidt (1976):184-195], I begin with the conditional log-likelihood function of the LIM-GARCH estimator for the

structural equation defined in (2.1):

(2.4) L(a,,.b,.v 0,641.4) = t

(2.

where

(2.

(2

(2.

(2.

(2.

(2.

(2.

Note that I

X.. The innovations are assumed to be conditionally normal distributed

t

with the conditional variance, hi following the GARCH(p,q) specification

5)

6)

7)

8)

9)

10)

11)

12)

denotes information available at time t-l, including Y. and

of Bollerslev (1986).

1

2 1 ' 1 plog(h; .) + plog(a W

OW 11, to?

1 ~2 3] 1%. + Xb Ea. FR ePPae |

= Pee)

' -l,, (1 — K,(KEX,) XD)

m4 1 - =| 1n(2r) +1 | + Al 1- 1n|w, | | ;

(YePLY) ) ~ NCO, bh?) t-1 y 1t’% ’ a 2 P42 w + La,e + = 6,h , ° j=l i1,t-i i-1 i1,t-i Yuet. + Xi ,>,

P. is a symmetric and idempotent projection matrix,

16

We is the second moment matrix of residuals of the least squares estimate of the reduced form of the entire system and Wait is the sub-matrix of We pertaining to the included endogenous variables, Y,,.

It should be noted that the standard LIML and 2SLS estimators are consistent due to the conditions (i) — (iii) provided in Section I. The primary benefit of the LIM-GARCH estimator, as with the ARCH and GARCH models, is in terms of efficiency.

Since the conditional log-likelihood function L depends on _ the parameters ao» bo» vw» @ and § in a nonlinear fashion, maximization of L(a,,b,,w,,0,5|I, 1) requires an iterative technique. Maximum Likelihood (ML) estimates of the parameters ao: b5: wo @ and § are derived from the first order conditions of (2.4). These derivatives, given in the Appendix, have a complex recursive structure making it extremely difficult to derive compact analytic expressions. While analytic expressions are in general the desired path to pursue, they are very inflexible to changes in specification and computationally extremely burdensome. Therefore, I rely

on numerical derivatives in the actual estimation procedure.

III. ASYMPTOTIC PROPERTIES OF THE LIM-GARCH ESTIMATOR

This section explores the behavior of the LIM-GARCH estimator in large samples .> In the previous sections conditional normality was explicity assumed. But now, following White (1982) and Weiss (1986), this

assumption is relaxed. As with most QMLE, the likelihood function is

;

“This section is based on Weiss (1986) who derives the asymptotic properties of an extended ARCH model in the context of a dynamic linear regression model with moving average errors.

17

derived as though the innovations are, in fact, conditionally normal. In Theorem I it is shown that in the limit the conditional log-likelihood function is maximized at the true parameters even though the assumption of normality may not be valid. Theorems II and III verify the consistency and asymptotic normality properties of the LIM-GARCH’ estimates, respectively.

As described in Section II, the LIM-GARCH specification involves maximizing the conditional log-likelihood function given by equation (2.4). Throughout this section let #@ to be an s-element column vector (s = m, +k, +p+qt1) containing all the parameters of the 1IM-GARCH

specification, that is,

(3.1) gp’ = (m', v') = (ag, bo, wy, a’, 6")

where @ is partitioned such that m is the parameter vector corresponding to the structural equation under investigation, and v is the parameter vector of the conditional variance specification, given by equation (2.11). Furthermore, I assume # € =, where = is a compact sutspace of Euclidean space, and let o, represent the true parameter vector.

First, a set of lemmas is required to provide the foundation for the ensuing theorems. Lemmas I and II require that certain matrices of

partial derivatives be well defined and positive definite.

18

LEMMA T: For all @ € B, there exists a constant M <o , not depending on @ such that de de (3.2) E _it it < M, dm dm’ and

de de (3.3) Det EI—2= —1t] 5 0. aman!

Proof: See Appendix.

An equivalent requirement for the conditional variance hi, is given

by Lemma II.

LEMMA IT:

Assume that the fourth moment of €1t exists and is bounded.

Then for all 6 € 3, there exists a constant My <o@ , not depending

on. @, such that

de de (3.4) e}—* —#) < mw ov ov’

19

de de

(3.5) Det E|—LE —1t] 5 oo. ava’

Proof: See Appendix.

Lemmas I and II imply that the negative expected value of the matrix of second derivatives of the conditional log-likelihood function is

positive definite:

LEMMA III: Under the same conditions as Lemma II, there exists a constant

M, < © not depending on @ such that

T 22 (3.6) A = —g/9L | M, a0ae'

and (3.7) Det A > 0. Proof: See Appendix.

‘These results provide the basis for the following theorems.

20

THEOREM I:

For the LIM-GARCH specification given by equations (2.4) to

(2.9) and under the same conditions as Lemma II

(3.8) L = lim L(6) (exists a.s. for all 6 € ©)

T-<0

and the lim L(6) is uniquely maximized at O.- Proof: See Appendix.

Taus, in the limit the conditional log-likelihood function is maximized at the true parameters even though the assumption of normality

may not be valid.

THEOREM II: (Consistency) For the LIM-GARCH specification given by equations (2.4) to

(2.9), the maximm likelihood estimate 6 is consistent for 6,

provided o. is interior to =. Proof: See Appendix.

Note that the condition that g, is interior to = ensures that, for T large enough, the first derivatives of L(@) are "well-behaved" at 65:

To consider the asymptotic distribution of the LIM-GARCH estimates, first, define A, and By to be the information matrix in Hessian and outer

product form, respectively.

21

9 (3.9) A, - -E guy. agae’

(3.10) Bo = E qo ob ao 36°

As will be shown, both A, and B) appear in the variance-covariance matrix of the asymptotic distribution of the LIM-GARCH estimates and are

therefore required to be invertible.

THEOREM III: (Asymptotic Normality) For the LIM-GARCH specification under the same conditions as

Theorem I, with the requirement that det (B,) > 0, then

(3.11) po/24 1/2 — 6.) ~ N(O,1)

Furthermore, consistent estimates of A, and B, are given by

2 2 ° ° A T dh a T de de (3.122) A = Qs not Pre Mie + tis yz Et t-1 a0.ao tel 30 a8 and A T of. ae 3.13) B= Try ——& t-1 30 40"

with all derivatives evaluated at 0 = a.

22

Proof: See Appendix.

Thus the asymptotic variance-covariance matrix of @ has the following

general form

(3.14) Asymptotic Var-Cov(#) = ALBA.

since the conditional log-likelihood function may not be correctly specified. When the conditional distribution of eit is normal then A, -

B, and the standard form for the asymptotic variance-covariance matrix is

appropriate, that is

(3.15). Asymptotic Var-Cov(@) = a.

To determine which form of the variance-covariance matrix is

appropriate I rely on White's (1982) information matrix test which is

derived in Section VI.

Iv. ‘THE FIM-GARCH ESTIMATOR

An estimator for the complete SEM-GARCH(p,q) model requires a more complex specification than the LIM-GARCH estimator. The full system estimator will be referred to as the Full Information Model with Generalized Autoregressive Conditional Heteroskedastic error processes, or

the FIM-GARCH estimator. The derivation of the FIM-GARCH estimator will

23

focus on the two equation diagonal SEM-GARCH(1,1) model, defined in Section I, since this is the most tractable model and can easily be generalized.

The alternative parameterization of the diagonal SEM-GARCH(1,1) model, given by (1.18), defines a vector hy consisting of all the unique elements of the conditional variance-covariance matrix. In matrix

notation, this can be written as

* (4.1) ho = wv + An, + Bh.) .

where

2 1,t-1

(4.2) Me. 7 €o,t-1°1,t-1 | ’ € 2 ; 2,t-1

Mya ot 4.3 h” He (4.3) t-l1 7 21,t-1

Hoo t-1

FIM-GARCH estimation of the diagonal SEM-GARCH(1,1) model requires

maximizing the conditional log-likelihood function

24

T (4.4) L-=2 tel where (4.5) 2.v,[t,,) = —Mogcae) + log||r]| — 21og]H_| cre te 208 & 2 Ob

- ame [rr + Xp)‘ (¥,0 + x,prHi."]

where H,I denotes the determinate of H., and [|r] | is the absolute value of the determinate of [. For estimation purposes, H. is constructed from its unique elements, defined by he in equation (1.18).

Since the conditional log-likelihood function L depends on the parameters T, 8, w, A, and B.in a highly nonlinear fashion, maximization of L requires iterative techniques. The FIM-GARCH estimator is derived from the first order conditions of (4.4).

As with the LIM-GARCH estimator, the derivatives of He with respect to [ and B are a function of past derivatives of ey and H.. As a result, the analytic derivatives of the FIM-GARCH estimator have a complex recursive structure which are difficult to calculate. Therefore, I will rely on numerical derivatives for actual estimation.

The difficulties involved in deriving the analytical derivatives can be highlighted by examining the partial derivative of the conditional log-likelihood function with respect to the unrestricted structural rm.

parameters, The superscript y denotes a selection operator, as

defined by Hendry (1976), to choose only the unrestricted elements of a

25

matrix. This is necessary since only the derivatives with respect to the

unknown elements are equated to zero.

(4.6) aL _ palogliri] _ [sents [a

ar ary 21 alu| au | jar”

wCe'e) _ ¢, OH

This can be simplified to yield

ar’ 2 ar H H

But the partial derivative of the conditional variance-covariance matrix, H, with respect to [ is also a function of derivatives of lagged residual variances and covariances as well as derivatives of lagged conditional variances and covariances. Similar recursive structures arise with respect to the other parameters of the system.

In estimation, as with the LIM-GARCH estimator, numerical derivatives are used. As explained in Section III, when the model is correctly

specified and the conditional distribution of e¢ is correctly assumed to

1t be normal, then the information matrix can be equivalently expressed in either Hessian or outer product form. In this case, the Berndt, Hall,

Hall, Hausman (BHHH) (1974) algorithm, based on the outer product form of

26

the information matrix, can be used to maximize the conditional log-likelihood functions for the LIM-GARCH and FIM-GARCH estimators. The BHHH method is an iterative method for calculating the optimal parameters,

6. Let §~ denote the parameter estimates after the ith iteration. gitl

is then calculated fron,

T of! @2.J-1T ag! (4.8) gitl _ Reale ae t

te1 aot aot t=1 get

where As is a variable step length chosen to maximize the likelihood function in the given direction.

An alternative algorithm is the Newton-Raphson method which is based on the Hessian form of the information matrix. As mentioned earlier, both methods are equivalent when the conditional log-likelihood function is correctly specified. The asymptotic variance-covariance matrix is given by (3.15) where A, = B.. Alternatively, if the conditional distribution is incorrectly specified to be normal, then under certain regularity

conditions the asymptotic variance-covariance matrix has the form given in

(3.14).

Vv. ASYMPTOTIC PROPERTIES OF THE FIM-GARCH ESTIMATOR

The derivation of the asymptotic properties of the FIM-GARCH estimator closely follows that of the LIM-GARCH estimator. As with the LIM-GARCH estimator, if the conditional distribution is non-normal then the FIM-GARCH estimator is a QMLE. For the FIM-GARCH estimator of the

diagonal SEM-GARCH(p,q) model let @ now be a S-element column vector

27

M (5.1) 6’ = (m’', v’) with S=M+K + (l+ptq)Zi. i=l

where 6, as in Section III, is partitioned such that m is the parameter vector corresponding to the unrestricted parameters of all the structural

equations of the system, that is

(5.2) n= [veccr,a)|* ,

where vec denotes that the matrices are vectorized by column stacking operations and y» denotes a selection operator that chooses only the a priori unrestricted structural parameters. Similarly, v is the parameter

vector of the conditional variance specification, given by equation (4.5), (5.3) v= [reccw.a,B) |" ,

As before, I assume @ € &, where E is a compact subspace of Euclidean space, and let 6, represent the true parameter vector. Theorems IV and V are the FIM-GARCH equivelent to Theorems II and III which verify the

consistency and asymptotic normality properties of the estimator.

THEOREM IV: (Consistency) For the FIM-GARCH specification given by equations (4.1) to Aa

(4.5), the maximum likelihood estimate @ is consistent for 6,

provided §, is interior to E..

Proof: See Appendix.

THEOREM V: (Asymptotic Normality)

28

Assuning that det (B,)>0, the FIM-GARCH specification is distributed

asymptotically normal, (5.4) Bo /2a 1/706 — 6.) ~ N(O,I)

where corsistent estimates of A, and B, are given by

A T dH. dH T 6.5) aA = pts aya yt tert s aya’ t=1- 0606 «6a8’ t=1 and A T @é@2 rip 4 (6.6) B-Ttts —t-—+

t=-1 30 436°

with all derivatives evaluated at 6 = 0.

Proof: See Appendix.

de, de,

a6 308’

Thus the asymptotic variance-covariance matrix of § has the following

general form

. “ -1, .l (5.7) Asymptotic Var-Cov(#) = A, BoA ;

29

since the conditional log-likelihood function may not be correctly specified. When the conditional distribution of ey is truly normal then A, - B, and the standard form for the asymptotic variance-covari.ance

matrix is appropriate, that is

(5.8) Asymptotic Var-Cov(@) = A As with the LIM-GARCH estimator, White’s Information Matrix Test can be

used to determine the correct form for the asymptotic variance-covariance

matrix.

VI. THE INFORMATION MATRIX TEST

A well known test for misspecification associated with maximum likelihood estimation is the information matrix test of White (1982). The test is based on his information matrix equivalence theorem. This theorem essentially says that when the model is correctly specified and the

conditional distribution of 1 is correctly assumed to be normal, then

t

the information matrix can be expressed either in Hessian forn, ACO.) or

in outer product form, B(O,). White shows that under these conditioris (6.1) A(@.) - BC) = 0.

Thus a test for the null hypothesis of conditional normality and

30

correct model specification is based on the difference between consistent a aA

estimates of A) and B,, denoted Ay and By respectively. For the

LIM-GARCH estimator @ is a s-element parameter vector, following White

(1982) and Weiss (1986), I define q [= s(st+l1)/2] vectors d.(8) such that

d (8) has kth element

an, at, 7e, k=1,....4 (6.2) d(6) = |—— + i,j=-1,...,8 a0, 28, 20,80, i<j

The test is based on what White refers to as the "indicators"

(6.3) De) = T B46), tT taf

which are the elements of Ay - B..- Next, define V(@) as (6.4) vV(é) = e[a, (0) aca)" |

V(6) turns out to be the asymptotic variance-covariance matrix of rt/2y, (6). ' yo additional assumptions are required to meet the preconditions for the test to be valid. First, V(#) must be nonsingular. Secondly, for cases when eit is not normally distributed it is necessary

to assume that

31

@o 1t

(6.5) |, (hy)

teal < for all @geé&.

Based on these conditions, White proves that

(6.6) t/?p (9) * N{0,V(9)] and “ a.s. (6.7) Vp(9) > VCO.) where Vip (8) is the estimate of V(O,). Then it follows that the information test statistic, Th is given by (6.8) Ty =- m,.(0) [v.89 D,(9) ~ Xq

To carry out the Information Matrix Test, one computes Ty and compares it to the critical value of the x distribution for a given size of test. If Ty does not exceed this value, then one can not reject the null hypethesis that the model is correctly specified and a may be used as_ the variance-covariance matrix of the LIM-GARCH estimates. This applies

equally well to the FIM-GARCH estimator

VII. CONCLUSION: In this paper I have extended the GARCH(p,q) model of Bollerslev

(1986) into a simultaneous equations framework and derived two estimation

32

strategies: LIM-GARCH and FIM-GARCH. Furthermore, it has been shown that these two estimation strategies have the desireable asymptotic properties, consistency and asymptotic normality, of Maximum Likelihood estimators. Alternative specifications of conditional variances and covariances that are heteroskedastic remain to be explored. My approach has exclusively focused on the ARCH specification originally developed by Engle (1982) and generalized by Bollerslev (1986). Future research will focus on extending the SIIM-GARCH(p,q) model to a dynamic framework by including lagged endogenous variables. In Harmon (1988) I extend the SEM-GARCH model by incorporating the conditional variances and covariances as variables in the structural equations themselves. This is referred to as_ the SEM-GARCH-M model. It is a logical extension of the ARCH(q)—in-Mean model (c.f., Engle, Lilien, and Robins (1987)) and the GARCH(p ,,q)—in-Mean model

(c.f., Bollerslev, Engle, and Wooldridge (1985)).

33

BIBLIOGRAPHY

Anderson, G.J. "The Structure Of Simultaneous Equations Estimators: A Comment." Journal of Econometrics. Vol. 14 (1980): 271-276.

Basawa, I.V., P.D. Feigin, and C.C. Heyde. "Asymptotic Properties Of Maximum Likelihood Estimators For Stochastic Processes." Sankhya: The Indian Journal of Statistics, Vol. 38, Series A, Part 3 (1976): 259-270.

Berndt, E.K., B.H. Hall, R.E. Hall, and J.A. Hausman. "Estimation and Inference in Nonlinear Strutural Models." Annals of Economic and Social Measurement, Vol. 4 (1974): 653-665.

Bollerslev, Tim. "Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics, Vol. 31 (1986): 307-327.

"A Conditionally Heteroskedastic Time Series Model For Security Price and Rates of Return Data." Discussion Paper #85-32, Department of Economics, University of California, San Diego. (1985).

, R. Engle, and J. Wooldridge. "A Capital Asset Pricing Model With Time Varying Covariances." Discussion Paper #82-4, Department: of Economics, University of California, San Diego. (1982).

Crowder; Martin. "Maximum Likelihood Estimation For Dependent Observations." Journal of the Royal Statistical Society, Series B38, (1976): 45-53.

Dhrymes, P. J. Econometrics: Statistical Foundations and Applications. New York: Springer-Verlag. (1974).

Diebold, Francis X. and M. Nerlove. "ARCH Models of Exchange Rate Fluctuations." Unpublished manuscript, Department of Economics, University of Pennsylvania, (1986).

"Modelling The Persistence Of Conditional Variance: A Comment." Econometric Reviews. Vol.5, No.1 (1986): 51-56.

__. "Temporal Aggregation of ARCH Processes and Distribution of Asset

Returns.," Special Studies Paper #200, Board of Governors of the Federal Reserve System, (1986).

"Testing For Serial Correlation In The Presence Of ARCH." Proceedings of the American Statistical Association, Business and Economic Statistics Section. Washington, D.C.: The American Statistical Society, (1986).

and P. Pauly. "Endogenous Risk In A Portfolio-Balance Rational - Expectations Model Of The Deutschemark-Dollar Rate." Forthcoming, European Economic Review (1988).

34

Domowitz, lan and Craig Hakkio. "Conditional Variance And The Risk Premium In The Foreign Exchange Market." Journal of International Economics,

Vol. 19, (1985): 47-66.

Domowitz, [an and Craig Hakkio. "Testing For Serial Correlation In The Preseence Of Heteroscedasticity With Applications To Exchange Rate

Models." Research Paper #83-11, Federal Reserve Bank of Kansas City. (October, 1983).

Engel, Robert F. “Autoregressive Conditional. Heteroscedasticity With

Estimates Of The Variance Of United Kingdom Inflation." Econometrica, Vol. 50, No. 4 (July, 1982): 987-1007.

"Wald, Likelihood Ratio, And Lagrange Multiplier Tests In Econometrics." Handbook of Econometrics, Volume II. Edited by Z. Griliches and M.D. Intriligator. (1984): 776-826.

. “A General Approach To Lagrange Multiplier Model Diagnostics." Journal of Econometrics, Vol. 20 (1982): 83-104.

and T. Bollerslev. "Modelling The Persistence Of Conditional Variances." Econometric Reviews, Vol.5, No.1 (1986): 1-50.

,C.W.J. Granger and D. Kraft. "Combining Competing Forecasts of

Inflation Using A Bivariate ARCH Model." Journal of Economic Dynamics and Control, Vol. 6 (1984): 151-165.

and D. Kraft. "Multiperiod Forecast Error Variances Of Inflation

Estimated From ARCH Models." In Applied Time Series Analysis of Economic Data, edited by A. Zellner, Bureau of the Census, (1983): 293-202.

, D. Lilien, and R. Robins. "Uncertainty And The Term Structure." Discussion Paper #82-4, Department of Economics, University of California, San Diego. (1982).

, D. Lilien, and R. Robins. "Estimating Time Varying Risk Premia In The Term Structure: The ARCH-M Model." Discussion Paper #85-17, Department of Economics, University of California, San Diego. (1984).

, D. Lilien, and R. Robins. “Estimating Time Varying Risk Premia In The Term Structure: The ARCH-M Model." Econometrica. Vol. 55, No. 2 (1987): 391-407.

Harmon, Richard. “The Simultaneous Equations Model With Generalized Autoregressive Conditional Heteroskedasticity In The Mean: The

SEM-GARCH-M Model. Unpublished Manuscript, Georgetown University. (1988).

Hendry, David F. “The Structure Of Simultaneous Equations Estimators." Journal of Econometrics. Vol. 4 (1976): 51-88.

Koopmans, T.C., H. Rubin and R.B. Leipnik. "Measuring the Equation System of

35

Dynamic Economics." In Ch.2 of Statistical Inference in Dynamic Economic Models, Edited by T.C. Koopmans. New York: Wiley (1950).

Kraft, Dennis and R. Engle. "Autoregressive Conditional Heteroskedasticity In Multiple Time Series Models." Discussion Paper #82-23, Department of Economics, University of California, San Diego. (1982).

Pollock, D.S.G. (1979).

The Algebra of Econometrics. New York: John Wiley & Sons.

"Varieties Of The LIML Estimator."

. Australian Economic Papers. (December, 1983): 499-506. .

Schmidt, P. Econometrics. New York: Marcel Dekker. (1976).

Spanos, Aris. Statistical Foundations Of Econometric Modelling.

Cambridge: Cambridge University Press. 1986. Weiss, Andrew A. "On The Stability Of A Heteroskedastic Process." Journal Of Time Series Analysis. Vol. 7, No. 4, (1985): 303-310. _ “ARMA Models With ARCH Errors." Journal Of Time Series Analysis. Vol. 5, (1984): 129-143.

"ARCH And Bilinear Time-Series Models: Comparison And

Combination." Journal of Business and Economic Statistics, Vol. 4 (1986): 59-70. .

"Asymptotic Theory For ARCH Models: Estimation And Testing." Econometric Theory, Vol. 2, No. 1 (1986): 107-128.

White, Halbert. (1984).

Asymptotic Theory for Econometricians. Academic Press, Inc.

"Maximum Likelihood Estimation Of Misspecified Models." Econometrica, Vol. 50, No. 1 (January, 1982): 1-25.

"Corrigendum." Econometrica, Vol. 51, No. 2 (March, 1983): 513.

Al

APPENDIX

Note: All expectations are conditional on Te unless otherwise stated.

First: Order Conditions For The LIM-GARCH Estimator: Differentiating the conditional log-likelihood function, given by

equation (2.4), with respect to a, yields

2 T a'W. 6h (A.1) dL = s]— fete _ ; lt da A t=1 a’W a 2h da ° ; 1t

When simplified this can be written as

e! Y (A.2) OL p | - [Sta o 1l,t 7 _ ie it da 0M 11, to 2h),

where

A2

2 2 2 dh q de, p oh, .. (A.3) lt _ a, 1,t-i + 5 6, 1,t-i da i=1 da i=l da 2 a Ps Ob) ei = Za,e! Y + = {x1 il,t-i1,t-i i-1 i da

Similarly, differentiating the conditional log-likelihood function

with respect to b, yields

9 | T e! X oh é! € (A.4) éL ~ sie Lett + ; || ut lt _ 1| - 0 db, A t=1 2h), 2h), ab, hit J b =b ° ° where 2 2 2 dh q de, p dh, (sy HE Ze bet » by She db i=l ~ db - i=l ~ dab °o °o 5 q P ahi t-i - = ae xX. . + = 6 2 im i 1,t-i°1,t-i i-1 i ab

Now, differentiating with respect to the parameters Ww? as, and 5, that comprise the conditional variance, hee yields T dh €1 € (A.6) an - sl-— ; | = lt 1 - 0 dw, A t=1 2h). dw, hj,

A3

but noizice that

2 2 oh, . p @éh (A.7) —it _ F + 5 hes]

isl Ow

°

It is clear from (A.7) that a complex recursive structure exists, which

requires expansions in order to derive an analytic expression for (A.6).

Differentiating with respect to a, yields éL T ah? €1 € (A.8) — = & H i [ste - 0, da, t=1 2hs, da, hie where 2 2 dh p oh, ,., (A.9) ee le at zB §,— ed da J ie aa,

Then, differentiating with respect to 6 yields the following first order

condit:ion

2 . dL T dh e! i € (A.10) — = & [P| [Ass - 0, a6 eo1]2ni las, JL bh),

where in this case

A4

2 2 dh q de, ,_ (amy —4£ = [nt + Bae as. J Gel a5,

The expressions for the first order conditions derived in equations (A.6) through (A.11) clearly show their complex recursive nature. The remaining first order conditions require differentiating L with respect to

the conditional variance, hi.

aL T él sé (A.12) — = & tists - 1 - 0. ony, te1}2hy el hy,

It is not possible to derive general compact analytic expressions due

to the recursive structure of these first order conditions .

Proof of LEMMA I:

Expressions for de), /ém, where m’ = (a; bo): are given by

de 1t (A.13) da, = Ye : de 1t (A.14) “ab, - Xie .

Then for bounded constant vectors A < eo, one has

AS

(a.15) ar——tE LL gez

én

1 , = where Ze [Yee X,,]- Now, one can write

de Oe

(a.16) Efar—2=—2t y} - glarz za

; 1¢71t m om

where 21 it is the cross product matrix and the right hand side is thus a

scalar. Hence there exists a constant M, < © such that

1 de de (A.17) Ejar—tt —4t yg] < M, for all 9 EE om om’ Next, to show that de de (A.18) det E|/—2= —4t] > 0 om om’ is equivalent to showing that de de (A.19) Ejar—4= 28] > 0 for all \ = 0. om Om’

Following Weiss (1986), proof is by contradiction. First, assume there

exists ’ * 0 such that (A.19) equals zero. Then it must true that

de “A.20) 4'—— = 0 a.s. for all t. om

A6

This implies from (A.15) that ANZ), = 0 a.s. for all t. Now let Yue

represent the forecast of Y given information available at time t-l,

lt that is

A

(A.21) oY, = ely, lt.

Then specify Y), to be

(A.22) Y,. = e +

1t 1t lt ’ where ele denotes the true errors. Furthermore, using a mean value expansion one can specify €;, to be de "° 1t (A.23) fie 7 “ae + om _ m,)

* where the derivative is evaluated at m , which lies between m and m2:

Equations (A.22) and (A.23) imply that ele is a function of Ted and Zee

since de, ,/dm' =Z If that is the case then

1t°

° (A.24) elet, I

-1°71¢|

But the LIM-GARCH specification assumes that

(A.25) elee[t1| = 0

and therefore it must be the case that

A7

, ° (A.26) elet,

ey Zre| - 0.

This implies that eit = 0 a.s. for all t, which contradicts the fact that

E|ced,)"| = o > 0 (i.e., the unconditional variance is greater than

zero). Therefore, no such X exists and

de de (A.27) Elar—tt—lt,] 5 9 a.s. dom Om’

as required. Oo Proof of LEMMA IT:

lixpressions for ahs, /ov, where v’ = (w a’, 6’), are given by

(A.28) zh Um= (U1

(A.29) -

(A. 30) > «i

For constant vectors » < ©, write (A.28), (A.29), and (A.30) as

A8

any. (A.31) py >yT 7 BY We where 2 2 oh dh 2 2 1,1-1 __1,t-p (A.32) We - f ; 1 t-1 pees, €1 t-q , a5, peeey a6

Since by assumption e[ced,*] < « then eles, | <M, < © for all 6 € &.,

2 ’ Thus, the first part of the lemma is straightforward. For the second part of Lemma II, apply the same method of proof as in

Lemma I. As before, to show that

. ‘ane ahi (A.33) det Ej-——— —— 0 ov év'

is equivalent to showing that

2 52 (A.34) nfs —it \ > 0 for all \ = 0. ov év'

Assume there exists a A»#0 such that (A.34) equals zero. Then, as in Lemma I, this implies that A'W - 0 a.s. for all t. Using a similar expression to (A.23), which is derived from the mean value theoren, yields

: 2 an expression for e

1t

de de de,

2 o 2 o -. lt lt lt

(A.35) fe 7 (ea) + ere =(v - vi twv- vf — ov avi

(v - v,)

ov

A9

°

This is a quadratic function in fit which yields two solutions,

° .

(A. 36) fie 7 £,(t) or f(t) These solutions are functions of Ty and Xie: But eit having two values ocnditional on Tey and Xie is not permitted. Therefore, no such X exists

and (A.34) holds. oO

Proof of LEMMA IIT: Differentiating the conditional log-likelihood function with respect

to 6 yields

2 2 € de oh € (a.37) 2 - <5 ed ee + : 1t de 4 -0 a0, tel{ hy 1 28, any Lae, JL bp,

i lt lt

where the derivatives with respect to 6, are evaluated at their true values 4,: Second order derivatives can be shown to have the following

general form

Al10

2 2 2,2 -

e710] ere] | Pre 1 | ‘1 oh,

+ Io tt cael h 06. 00 2h h 06.00.

i i j-

it j ree Mae 2 2 2 2 ere Piel ofa r ffre} [Pre] Me rr ed ee 0 ee ny A OO 2 0 ny LL aa a8, nrelhyelL oath 20, 2 2 ~ [| as 4 ante L.a0,tL 28,

Expressions for de, ,/am, and ah, ,/8v, are given by (A.15) and (A.31),

respectively. For ahi ,/am, one has

2 2 dh q de _ p <3éh, ,. (a.39) EE - 2a, Et + 36S om, i=l , om, i=1 om

where my, is the jth element of m. From Lemma II and (A.39) it is easy to

see that ah, ,/29, have bounded second moments.

Since every term in ahs ,/38, and a”hi , /20 00 also appears in he

j 1t

itself, the expressions

2 2,2

oh ah (4.40) 4 and + (4 hi, a6, hi, 00,98,

are uniformily bounded from above. Hence, evaluating equation (A.29) at the true parameter values 6, implies that the first, third, fourth, and

fifth terms are zero, with the sixth term being

All

2 2 2 2 2

(A.41) p| 2 [fae] |2Pre| | Pre] | gf 2 [te] [Pz |, <n

nt In2_}] a¢.|| ae.}} to]. n+ | ae.{| ae.{| 2 ues i j arb 94; j

since E(e* IT,_4) = he. Similarly, the second term of (A.38), evaluated

at Oo» ‘Ls bounded

where the derivatives are evaluated at 6: To show the Det A > 0, I follow the method utilized in Lemmas I and

II. For any A * 0, one can transform A such that

teal teal

2 2 dh dh

2 + 00 ae’

Ae

de de hie 06 a6’

Al2

Both terms on the right hand side are nonnegative. What remains to be shown is that these are both greater than zero. First, partition 1’

= (Az A5) to conform with #9’ = (m’,v’). Now, since de, ,/dv = 0 the second term in (A.44) is equal to

i de de de de (A.45) £E Noe —1t}|4*), |r, | - = Ma —1t}|—*)a ta. hi. L a0 JL ae" ny. L ae JL a0"

From Lemma I this is clearly positive unless do = 0. If ry - 0, then

X

1 * 0 and the first term of (A.44) becomes

24 ray 2 ah, [an (A.46) £E Nem —1t}|—+*), |r ne. | a9 || ae

1t which is positive because of Lemma II. O

teal

2 apa 2 1 fe] Pre E a —] | »y +L a6 SL a

Proof of THEOREM I:

From the ergodic theorem, for any # € &,

(A.47) L(@) = lim Ae

to

_ _il 2} _ 1 ' = B(C,) 5E| 10g eel $e | toga; .,)|

1 2 ,2 - slit | a.s.

Al3

if the expectations exist. From Jensen's Inequality it follows that (A.48) logE(X) = E[10g(x) | ’

for all positive random variables X, with equality only when X is a

constant: a.s. This implies that

2 2 (A.49) Loge [he | = z [10g hie |

Since, by definition, e[nt, | < © then from (A.47) it must be true that

£|1ogin; .)| < o. From Lemma I it was shown that E(e.) <o, for all ge

=. Then since

then clearly 2 ,2 | (A. 51) ef t/t | <o, Since Wii _ is the second moment matrix of residuals from the regression

of Yue on Xie the vector of all exogenous variables, it is equal to zero

when evaluated at the true parameter values.

Next, following Weiss (1986), make the following transformations, 2 2 -2,- ° o (2 (A. 52) ete - E [niece + fie 7 61.

—2,.0 2 -2,_. °o (2 - ela et | + efns cere ~ €14) |

Al4

. - fe] Since h — €

2 1t1e it? and E(es, 11.) = 0. From (A.52) it is clear that

2 depends only on information available at time t-l

°

2 ,2 |. 2,2 (A.53) ef ete/he| > z| (ed) os |

This holds’ with equality when fie 7 ee for all t ak.s. Taking expectations of an expression for ce similar to (A.55) and based on Lemma I, yields Oe de 2 °o (2 ' 1t 1t °o (2 (A. 54) eet, | = e| (ed) + (m m,) | om “| m,) = el Cet)

with equality only at m = m,: Thus from (A.54) one can see that fie 7

° ie: for all t a.s., only when m = m,:

1 2 1 , (A.55) L(6) = E(C,) - 32 108 ie | _ se [log(ar¥,, .2,)|

1 2 ,-2 - peel h | ,

1 2 1 ' s E(C,) - 3B [log ni e| - e[togcas¥,, .4,)|

~ Boel 3 | ,

eine e| cet)” ni? | - E|cet.)” hye (hy)? ae]

A15

1 2 1, E(C,) - 38| log ni | — 5 z| Logica; Wi, L? a,)|

- Boefeleei” (hy) ‘I|- Hoe? (hy. i] ; eine ele | > x tel oi" |

s E(C,) -F 38[10g Mel - se z| Logica; a1, to |

tere at ‘I|- prof? (hy) ‘||. E(C,) - se z|togcnt,) *| - 38 |1ogca: Wi, 2 a,)|

- hoe[ei? oto]

IA

BC) ~ 5 3E|Logche “| - telroe[ce 2? (hy 2 ;

L(6))

with equality only at 6 = 9,- Note that

(A. 56) “t fn? (no) i] - ofiefi 2 ey? i] |

only when (hy, 2 - hi, But based on Lemma II, this can only occur at

§=6.. Oo

Al6

Proof of THEOREM II:

This proof follows closely the proofs of Weiss (1986) and Basawa, Feigin, and Heyde (1976). I begin by first applying a Taylor series expansion to the first order derivatives for the conditional

log-likelihood function. This expansion can be written as

2,0 4 3, (as7) Se - By Shiog_yy 4 LabG_4) 00 06 06 36! 86 ° ° ° where |6 — @)| > [6 - al. which requires that @ lie between @, and 6.

The third term can be considered as a remainder term on the Taylor series

expansion. Equation (A.57) can be rewritten, based on (2.4), as

aL T Ee A T a°2, A T ae, (A.58) =x = T-—— + (- 6.) = + (6 - a) = —— 06 t=1 36, oO t=1 36 (48° t=1 46

Basawa, Feigin, and Heyde derive a set of sufficient conditions for

the existence of a consistent root of the log-likelihood equation

oL oe 00

(A.59) = 0.

These sufficient conditions consists of

| T a2 (I) A 5 -t Bo. T tl 36,

Al7

(II) There exists a nonrandom matrix M(6.) > 0 such that for all e>o0O, 1 T a7e, Pri- - = —— 2 M(@.) > l-e T t=1 60 46’ fe] ° for all T > T, Ce). (III) There exists a constant M < © such that

3 ) 4.

86,30 30,

E

< M for all@eEee=.

Then it follows that 9 —4 a.

The method of proof is to show that these three conditions are

satisfied. I begin by focusing on the parameters of the current

endogenous variables a.

Condition (I): From equation (A.1), one has

T a'W e! Y (A.50) gL - =: -E 11,t | _ itis da, A t=1 aoW a1 to 2h), awa,

A18

When this derivative is evaluated at the true ao: the first term is zero

since Wi t is the second moment matrix of residuals from the regression ’

of Yue on Xx: Then, following Weiss (1986), it can be shown that

(A.61) 2a I | - 0 t-1 i da ° since E(e,,]I, ,) = 0 and E(eo 1 )- n2 The ergodic theorem, see it! t-1 1t' t-1 1t° ,

White (1984), then implies that

T aa. , s — 2 o. t=1 da,

Ir

(A.62)

4

Condition (II): By the ergodic theorem, for any constant vector \ 0,

17 a7a, a°a, (A.63) = =A!‘ » — EIA'—— A a.s. T tel da _ da! da da!’ ° ° ° ° = -rA\'A X °

where A, is defined to be

2 (A.64) A, = -E io da da’ ° °

This second order derivative, after some simplification, has the following

Al19

form,

2 2 T W 'W a yey (a.65) LH = ¢ 11,t polite le le

, = ,

da da’ t=1 aoWi7 to (30M11 £30) 2h5, U — te f1e%1t sily , Ly} on? in. 1t-i 1t-i

a €. € T-1 1 {1e%1t i-l ., * “| - 1 oft “iea%1e-t

T-1 a €1 ¢€ i-l_ , 1 j,, _ o_lt 1t * an 5) dette ne beta * 21 2 2 ° 1t 1t T-1 i-1 >a) Y3 € i-y 1 itt te Taking negative expectations of (A.65) yields 2 T YIY (A.66) - >| | -~ ptt > o, da da? t=1 2h), since E(e,,|I,__,) - E(e |Z...) = 0 and E(e2 JI. ,) = he Now lt! t-1 1t-i' t-1 lt’ t-1 1t’ ,

following Weiss (1986), let

A20

(A.67) 0 < 6(A) <

Nie

ana.

ATE Xr da da' ° fe)

for a given \.Then, for all « > 0, there exists T, = T, Ce) such that

< 6 | > l-e

for all T>T,. Thus, condition (II) is verified by defining M(a,) to be

1

L2 ana, ah, 1s yt y - gh d

(A. 68) Pr . T tel da da! da da’ ° ° ° °

1 1 ak, (A.69) M(a,) - -3 E @a Ja’ . ° ° Then LT ah, (A.70) Pr} — = A'——A > A'M(a,)A > l-e T t=1 da da’

for all T > Ty:

Condition (III): The third condition to be verified requires that the third order derivatives, which can be interpreted to be the remainder term of the Taylor series expansion, be bounded in absolute value. The third order derivative is derived by differentiating equation (A.65) with respect to a. It is clear that the only terms that cannot be dealt with as above are those terms which contain third order derivatives of he ar.d fe: Following Weiss (1986), one is able to show that an extension of the methods applied to the first and second derivatives implies that terms

containing a°hi./a are bounded and that terms containing a°e,,/aa> have

A21

bounded second moments. O

The proof of consistency for the remaining parameters follows as

above.

Proof of THEOREM III:

Besawa, Feigin, and Heyde (1976) again provide a set of sufficient

conditions for asymptotic normality. These verifiable conditions are

T aA (I) T 1/2 = —t A N(O,B ) for nonrandom B > 0. fe] fe] t=1 06 ° = T a7a. (II) °° +T = - Py “A, for nonrandom A, > 0. t=1 36 46° ; ° °o CLIT) Condition (III) of Theorem II.

The method of proof, as before, is to show that these three conditions are

satisfied.

Condition (I): Following Weiss (1986), assume that B, > 0. Then

from Condition (1) of the proof of Theorem I, it follows that

(A.71) | —t

Weiss shows that if

A22

a2. a (A.72) | —t_t | < @

and

(A.73) os ol ok | - Bo <o@,

then a Martingale central limit theorem is applicable. Furthermore, Weiss shows that when (A.71) is true then (A.72) and (A.73) are equivalent. Thus, all that is required is to show that (A.72) holds.

To show that (A.72) is true I follow the method in Lemma III used to

show A< ©. From (A.37) one has

Z 2 (a7) gf ete] 8 gf. Ste f2fte] , 2 [Pael | fae _ | a6 a6" n2_| 90 an-_| ao || n2

1t 1t 1t

2 2 ord Laas 1 [Pre] ] f1e ~ 72 + 721, 2 + ne a9 ane} a6 || h

2 2 ne 06 jj 30° ne 06 |} 00’

1t 1t

2 2 2 2 42 It rf empe Pel] fre 2 TN te ose h ant | a0 || ae'|| nh

1t 1t

where the derivatives are evaluated at §,- Taking expectations yields

A23

aL, aL de, ] [ae (A..75) | —tt| -& a —it}|—4t) | , a0 a0" ho. | a6 || a0"

which from Lemma III is clearly bounded.

Condition (II): Since by definition

teal

then Lemma III verifies condition II. oO

2 ) A.

(A.76) A, = —E|—— 00 00’ o 0

Proof of THEOREM IV:

This proof is nearly identical to Theorem II. Again, one must verify the three sufficient conditions derived by Basawa, Feigin, and Heyde (1976) and given in the proof of Theorem II. I begin by focusing on the parameters of the endogenous variables, I.

Condition I: From equation (4.8), the first order condition is given

by

T |: OH, | lele e'Y cac77) 8 me se fats + EY] JE - 1] - 4]. ar? tel 2ler*} | H, H, |

t

yg 2 2 Clearly, since E(e,, 11, = 0 and E(e}, 11,1) hoy ;

(a.78) EJS ar’

Then, the ergodic theorem implies that

T dh sr — Bo. t=1 er’

Ie

(A.79)

ey

Condition (II): By the ergodic theorem, for any constant vector A « O,

1 T a°a, ana, (A.80) = A'|—, d —_— E|A'———_ - A" ASA 7)

, ’ t-1 |ar“ar’ rear’

re

where A, is defined to be

| 2 (A.81) ALS = - B|—°4_ {ar’ar®

This second order derivative for the conditional log-likelihood function

is given by

2 T Y'e_|oaH €'e oH oH (A.82) oh = > |-(r'r) 1, etj_t) _ Ljtety] lye —. arvar® —s_ t= H. ar’| 2, a H.. ar*| | ar?

2 t e ¢ 2 H. or’ ar H. H H.. or

Taking negative expectations conditional on information available at time

A25

t-l of (A.82) yields

2 (A.83) —E{|—? 4 ar’ar4

T aH.) [aH Y'y Ial- 2 -creryt + 2] 4] )—t}}—t | 4 1 5 0 t=1 2H, ar’ | | ar’ H

t

As in Theorem II, for a given \, let 6 be bounded by

ana. A! E|————a ar*ar’

Then, for all « > 0, there exists T

(A.84) 0 < 6(A) <

Nik

17 T, (e) such that

17 a°e, ao, => XS A'—— A - E]A’'—— A

(4.85) Pr . : Tt=1 ar*ar’ ar“ar’

< j >l-e

for all. T > T)-

Thus, condition (II) is verified by defining M(T) to be

1 ak, ar“ar* Then 1 7 a°s, (A.87) Pr} = = A'——=— A > AMPA] > 1l-e T t=1 ar“ar4

for all T > T.

Condition (III): See Theorem II, condition III. oO

A26

The proof of consistency for the remaining parameters follows as above.

Proof of THEOREM V:

As with Theorem IV, this proof is nearly identical to its single equation counterpart, Theorem III. The proof requires verifying the three sufficient conditions for asymptotic normality outlines by Basawa, Feigin, and Heyde (1976) and given in the proof of Theorem III. As before, I will focus on the parameters of the endogenous variables, TI.

Condition I: Since it is assumed that B09, then from Condition I

from the proof of Theorem IV, it follows that

(A.90) oe

a = 0

ar

As explained in Theorem III, given the (A.90) is true, all that is

required to verify Condition I is to show that

ak. ae (A.91) os t | < «

re ar’ art

From (A.77) condition (A.91) can be written as

Ok, AR fe) st ee e’'Y.|' (A.92) ns — - —E arly: + |—tyjeet— i] -—- tt ar’ ar# | ar’ H. H

e’e e/y ftft _ 4] _ feet H

wee nee

A27

Yie .e'Y 1) dH e'e _ fosa « sete _ [el [te — |

, He. 4

e’e ‘10H oH e'eé H. a ar’ H. e'Y _9 (rly t i He

where the derivatives are evaluated at the true I. Taking expectations

yields

ar” ar® HH,

Condition II: By definition

aa. (A.94) A, = - E|——; ar’ar®

teal

An argument identical to Lemma III verifies Condition II.

Condition III: Similar to Theorem II, Condition III. O

IFDP

NUMBER

322

321

320

319

318

317

316

315

314

312

311

310

309

International Finance Discussion Papers

TITLES 1988

The Simultaneous Equations Model with Ceneralized Autoregressive Conditional Heteroskedasticity: The SEM-GARCH Model

Adjustment Costs and International Trade Dynamics

The Capital Flight "Problem"

1987

Modeling Investment Income and Other Services in the U.S. International Transactions Accounts

Improving the Forecast Accuracy of Provisional Data: An Application of the Kalman Filter to Retail Sales Estimates

Monte Carlo Methodology and the Finite Sample Properties of Statistics for Testing Nested and Non-Nested Hypotheses

The U.S. External Deficit: Its Causes and Persistence

Debt Conversions: Economic Issues for Heavily Indebted Developing Countries

Exchange Rate Regimes and Macroeconomic Stabilization in a Developing Country

Monetary Policy in Taiwan, China

The Pricing of Forward Exchange Rates Realignment of the Yen-Dollar Exchange Rate: Aspects of the Adjustment Process in Japan

The Effect of Multilateral Trade Clearinghouses on the Demand for

International Reserves

Protection and Retaliation: the Rules of the Game

Changing

I,

AUTHDR(s)

Richard Harmon

Joseph E. Gagnon

David B. Gordon Ross Levine

William Helkie Lois Stekler

B. Dianne Pauls

Neil R. Ericsson

Peter Hooper Catherine L. Mann

Lewis S. Alexander

David H. Howard

Robert F. Emery Ross Levine

Bonnie E. Loopesko Robert E. Johnson

Ellen EF. Meade

Catherine L. Mann

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the

Federal Reserve System, Washington, D.C.

20551.