Enders 1995 ARCH

Applied Econometric Time Series

WALTER ENDERS Iowa State University

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Library of Congress Cataloging-in-Publication Data

Enders. Walter. 1948- Applied economehic time series / Walter Enders. - 1st ed.

p. cm. - Wiley series in probability and mathematical statistics) Includes bibliographical references. ISBN 0-47 1-0394 1- 1 I. Econometrics. 2. Time-series analysis. I. Title.

XI. Series. HB139.ES5 1995 33U.01'5 195-dc20 94-27849

CIP

Rinted in the United States of America

Chapter 3

MODELING ECONOMIC TlME SERIES: TRENDS AND VOLATILITY

- 0 - c r - , ~ -c m&c; t Ce3 h. I ' f~Jw kd S&b M a n y economic time series do not have a constant mean and most exhibit phases of relative tranquility followed by periods of high volatility. Much of the current econometric research is concerned with extending the Box-Jenkins methodology to

7 analyze this type of time-series behavior. The a m s of this chapter are to:

1. Examine the so-called st&&&.& concerning the properties of economic time- - senes data. Casual inspection of GNP, financial aggregates, interest and exchange rates suggests they do not have a constant mean and variance. A stochastic variable with a constant variance is called homoskedastic as opposed to

mav be cons though the v-e during some'eriods is u n u s ~ a 5 ~ heteroskedastic.' For series exhibiting volatility, the unconditional variance,

large. You will learn how to use the tools developed in Chapter 2 to,model s a -. conditional heteroskedasticity.

2. Formalize simvle -1s of variables with a time-dewdent m ean. Certainly, the i mean value of GNP, various price indices, and the money supply have been increasing over time. The - displayed by these variables may contain deterministic andlor s t o c ~ o m p o n e n t s . Learning about the properties of the two types of trends is important. It makes a great deal of difference if a series is estimated and forecasted under the hypothesis of a deterministic versus stochastic trend. \

3. Illustrate the difference between stochastic and deterministic trends by consider- ing the modem view of the business cycle. A methodology that can be used to decompose a series into its temporary and permanent components is presented.

1. ECONOMIC TlME SERIES: THE STYLIZED FACTS

Figures 3.1 through 3.8 illustrate - the behavior of some of the more important vari- * - ables encountered in m a c r o e c o n o ~ s . Casual inspection does have its per- ils and formal testing is necessary to substantiate any first impressions. However,

136 Modeling Economic Time Series: Trends and Volatility

Figure 3.1 U.S. GNP (1985 prices). 5

the strong visual pattern is that these series are not; the sample means do not appear to be constant andlor there is the strong appearance of h~oskedas t ic - 1 T W e can chara-e the kev features of the various series with these "stvliier

@t of the series contain a clear trend. Real GNP and its subcomponents and the supplies of short-term financia~struments exhibit a decidedly upward trend. For some series (interest, and inflation rates), the positive trend is inter- rupted by a marked decline, followed by a resumption of the positive growth. Nevertheless, it is hard to maintain that these series do have a time-invariant mean. As such, they are not stationary.

Figure 3.2 Investment and government consumption (1985 prices). 1000 0

- Government Consumption 800 4 Investment

Economic Time Series: The Stylized Facts 137

Figure 3.3 Checkable deposits and money market instruments. 800

- Checkable deposits - Money market instruments

Some series seem to m e r . The poundldollar exchange rate shows no par- -3 - ticular tendency to increase or decrease. The pound seems to go through sus- tained periods of appreciation and then depreciation with no tendency to revert to a long-run mean. This type of "random walk" behavior is typical of nonsta- tionary series. c*2$ oh

I ,3 Any shock to a series displays a high degree of pmistece. Notice that the + - Federal Funds Rate experienced a violently upward surge in 1973 and remained

I at the higher level for nearly 2 years. In the same way, U.K. industrial production plummeted in the late 1970s, not returning to its previous level until the mid-1980s.

1 /&The volatility - of many series is ot nt over time. During the 1970s,

U.S. producer prices fluctuated w i e p a r e d with the 1960s and 1980s. Real investment grew smoothly throughout most of the 1960s, but became

1 highly variable in the 1970s also. Such series are called conditionally het- ! eroskedastic if the unconditional @r long-run) variance is constant but there are -

periods in which the variance is relatively hich.

I with other series. Large shocks to U.S. in- I to be a s i e l y to those in the- a.

Short- and lona-term interest rates track each other quite closely. The presence - of such comovements should nLt be too su*. We might e E t that the un- derlying ec+c forces affecting U.S. industry also affect industry internation- - ally.


Figure 3.4 U.S. money supply: M2.

ti) 'C 0

E 2 - 0 .- - - .- c

1 - -

Please be aware that "eyeballing" the data is cot a substitute for f o r m a l l [ for the presence of conditional h sticity or non ry behavior. Although most of the v a d g u r e s u < r s s t a t i o n q , the

i issue will n g L d m s be so o w . Fortunately, it is possible to modify the tools developed in the last chapter to help in the identification and estimation of such series. The remainder of this chapter considers the issue of conditional heteroskedasticity and presents simple models of ding variables. Formal tests-for the Fes- -

Figure 3.5 Short- and long-term U.S. interest rates. 20%

046% 1960 1966 1972 1978 1984 1990

--- Federal funds - 10-year bond

ARCH Processes 139

Figure 3.6 U.S. price indices (percent change).

40% %

30% - -

- m a

10% r

-10% 0 1960 1966 1972 1978 1984 1990 --- GNP deflator - Producer prices

I ence of trends (either deterministic and/or stochastic) are contained in the next

I chapter. The issue of comovements must wait until Chapter 6.

1 2. ARCH PROCESSES - n I

'L .II, y 4 > the variance of the disturbance term is as-

I sumed to be constant. However, Figures 3.1 through 3.8 demonstrate that many

I - -

economic time series exhibit periods of unusually large volatility followed by periods of relative tranquility. In such circumstances, the assumption of a constant vari-

( ance (homoskedasticity) is inappropriate. It is easy to imagine instances in which

; 1 you might want to forecast the conditional variance of a series. As an asset holder, you would be. interested in forecasts of the rate of return and its variance over the holding period. The unconditional variance ke., the long-run-- ance) wzuld be unimportant if vou plan to buy the asset at t - + 1.

One approach to forecasting the variance is to explicitly introduce an independent variable that helps to predict the volatility. Consider the simplest case in which

I where y,+, = the variable of interest E,, = a white-noise disturbance term with variance c? x, = an independent variable that can be observed at period 1

&, = x,-, = x,-, = ... = sequence is the familiar white-noise process with a constant , when the realizations of the {x , ] se-


Figure 3.7 Exchange rate indices (currency/dollar). 2.5 I ~ I I I I I I ~ I I I I I I I ~ I I I I I I I / I I

- U.K. OCanada A Japan

I quence are not all equal, the variance of y,+, conditional on the observable value of

I var(y,+, IX,) =$a2 . . Here, the

+ + % al variance of y,+, is dependent on the realized value of x,. Since

you can observe x, at time p e r i o M c a n form the variance of y,+, conditionally -

on the realized value of x,.@the magnitude (xt)' is large (small), the variance of I y,, will be large (small) as well. ~urthermore,@the successive values of {x,) ex-

hibit positive serial correlation (so that a large value of x, tends to be followed by a large value of x,,,), the conditional variance of the {y,) sequence will exhibit posi- hve serial correlation&s well. In this way, the introduction 0-1 sequence can

1 L ' .r explain periods of v ah ity in the {y,} sequence. In practice, you might want to , r modify the basic model by introducing the coefficients a, and a , and estimating the

'11 ' / regression equation in logarithmic form as ;I

11 \ where e, = the error term [formally, e, = ln(~,)] P

I The procedure is simple to implement since the logarithmic transformation re- 1 OLS can be used to estimate a, and a, &ectly.

- is that it a s e s a sp- c-inn variance. Often, you may not have a fm t h m t a L w n for selecting o n e e over other reasonable choices. Was it the oil

I ARCH Processes 141 3

Figure 3.8 Industrial production.

price shocks, a change in the conduct of monetary policy, and/or the breakdown of the Bretton-Woods system that was responsible for the volatile WPI during the 1970s? Moreover, $e t e c h n i q u e n e c e s s i t a t e s a e data such that the resu&ng series ha

. '

e. In the example at hand, the (e ] se- guence is assumed to h-ce If - this assumption is violate- - - other transformation of the data is necessary. +

7

f l ARCH Processes

Instead - of using a m d and/or d-a aosformationsL 9 (1982) shows - that it is possible tosimultane m an and variance o a s&s. As i$-3 to understandi-gy , n& that cm- ditional forecasts e vast1 su nor to unconditional forecasts. To elaborate, y q pose you estim e the stationary ARMA model y, = a, + a,y,, + E, and want to

G c a s t y,,,. e conditional forecast of y,+, is:

'\/ .-

- U.S. 4 U.K. ACanada

I 2 3 EILV,,, - ad(l -aO12) = E[(E,+~ + 016, + ale,-l + a,e t -~ + ... 121

I = &(I - a:)

142 Modeling Economic Time Series: Trends and Volatility I Since 1 4 1 - a:) > 1, the u- a greater variance than the . . -- - I

m t . Thus, c o n d j ~ J o r e c a s t s (since they take into account the known current and past realizations of series) are preferabk.

Similarly, if the variance of {e,] is not constant, you c m a t e any tendency -- -d move,ments in the variance using -A model. For ex*, let 141 denote the estimated residuals from the model y, = a, +=,-, + c,, so that & conditional variance of y,+, is

V=CY,+, I Y,) = E,[CY,*, - a, - ~ , Y , ) ~ I *r Thus far, we have set Ere:+, equal to d. &xy s w e that the conditional vari- 7

ance is n m t a n t . One simple strategy is to m g e l the conditional variance as an - AR(q) process using the square of the estimated residuals:

where v, = a white-noise process I I

If the values of a , , g, . . . , a, all equal zero, the estimated variance is simply theconstant % O L w s e , the co&ional vzance ofGvolves according to the -s given by (3.1). As such, you c z use (3.1) to foreczt the conditional variance at t + 1 as 1

* \ w o s k e - For this reason, an equation likes) is called an a u t o r e e conditional_ dastic (ARCH) model. There are many possible applications for ARCH

models since the residuals in (3.1) can come from an autoregression, an ARMA

I I

c, = v,- (3.2) +

where v, = white-noise process such that = 1, v, and E,-, are independent of each other, and a, and a, are constants such that a, > 0 and 0 < a, < 1.

model, or a standard regression model. In ac-, the linear specification of (3.1) i s q p f e most convenient. The rea- - I

son is that $e model for I y,) i-

and the conditional variance are best estimated simul- - taneously using maximum likelihood techniques. Instead of the specification given . .

i by (3.1), it is m m t a b l e to specify v, as a m-e disturbance. I

the [c,} sequence. Since v, is white-noise and indepen- , 1 I

that the elements of t h e w -mean e of The proof is straightforward. Take the unconditional ex- I

I 0, it follows that Q ..PO. .u

1

The s m e s t e x a a e from the class of multiplicative conditionally heteroskedastic models proposed by Engle (1982) is 1

I 1

I ARCH Processes 143 1 I

EE, = E[v,(% + a , ~ ~ ~ ) ' ~ ] = Ev,E(% + a , ~ ~ , ) ' " = 0

ad -t+, AC+$ CM (T- tl. Since Ev,v,-, = 0, it also follows that

The derivation of themonditional variance of E , is also straightforward. Square E , and take the unconditional expectation to form

u variance of E, is identical to that$ E,, (i.e.. - ?.

<a =trl/; SC- ' o ( , ~ t t ( "1 ct-h7,

i

EE: = ad(1 - a,)

TSps, the@conditional mean and variance are unaffected by the i error process given b m . S w l y , it is easy to show that the conditional

of E , is equal to zero. Given that v, and E,, are independent and Ev, = 0, the condi- rCc

tional mean of E, is - I i E(E, I E - I . E , - ~ , . . .) = E v a % + a , ~ ~ - , ) ~ / ~ 0 .'(: 0"

! I ~t this point you &be r4 thinking - that the-e,s of the e , \ sequence are ' 1 not affected by (3.2) s~nce the m m is zero, thek constant, and all autoc -

%ancespre zero. However, theL influent. of ( 3 . z t i r e l y on the c o n d a l I I I

1

variance. --- Since $ = =variance of E , conditioned on the past history %E,-, , E,,, - ... is <ad- A T i

bIhg.;t'i0b 9 5 4

& Y d - ,+o y 2 = E ( E : ~ E , - ~ , E , ~ , . . . (3.6) (3 ?)

&*& - q u ~ - j 0 HX

t In (3.6), the conditional variancezf s i s dependent on the realized value of E,,. + I 1 If the realized is l X e 2 e conditional v a r i a n c e i n i l l be l a e as

well. In (3.6) the conditional variance follows a first-order autoreme- ss CL", denoted by ARCH(1). As opposed to a usual autoregression, the coefficients a,, a@

In order t o ensure that the conditional variance is never .- - -

to assume that both cl, and a, a r e m e . After all@% small realization of E,-, will mean that (3.6) is negative.

a sufficiently large realization of E,-, can render a nega- variance. Moreover, to ensure the stability of the au-

toregressive process, it is-su:h that 0 < a, < 1 .

144 Modeling Economic Time Series: Trends m d Volatility I Equations (3.3), (3.4), (3.3, and (3.6) i w e sential fea-c nf any

ARCH process. F an ARCH model, the error structure is such that the conditional &d unconditional means are equal to zero. Moreover, the {E,] sequence is serially uncorrelated since for all s f 0, EE,E, = 0. The key point is that the errors are not independent since they are related through their second moment (recall that correlation is a linear relationship). The conditional variance itself is an autoregressive process resulting in conditionally heteroskedastic errors. When the realized value of E,, is far from zero-so that a , (~, , )~ is relatively large-the variance of E, will tend to be large. As you will see momentarily, the conditional heteroskedasticity in [E,] will result in [y,] being an ARCH process. Thus, the ARCH model is able to

in the [y,] series. different ARCH models. The upper-

shows 100 serially uncorrelated and normally distributed random deviates. From casual inspection, the [v,] se-

Figure 3.9 Simulated ARCH processes.

*White noise process v, E, = Y, .&iG&J

quence appears to fluctuate around

\ Note the moderate increase in tial condition e,, = 0, these realizations of the [ v , ] sequence were used to construct the next 100 values of the {e , ] sequence using equation &2) and* a,, = 1 and a, = 0.8. As illustrated in the upper-right-hand iraph (b), the [e , ) sequence also has - a mean of zew, but !.he variance appears to e x p e r i e n c e e in voolatlhty around t = 50. C 4 ~ ~ f i [ ~ , " ) , y t r 9 ~ t 4 1 y + + , + G+

How does the error structure affect the [y,} sequence? Clearly, if the autoregressive parameter a , is zero, y, is nothing more than e,. Thus, the upper-right-hand grxh can be used to depict the.tirne path of the iv.1 sequence for t6e case of a, = 0. The lower two graphs (c) and (d) show the behavior of the {y , ] sequence for the cases of a , = 0.2 and 0.9, r m e l y . The essential voir&to note is that the ARCH

of the ( Y . ~ S interact with each that the volatilit of { y , } is in-

creasing in a, and a,. The explanation is intuitive. Any unusually \ large (in abso ute sue) shock in v, will be associated with a persistently large variance in the {e ,] sequence; the larger a,, the loder the persistence. Moreover, the greater the autore-

L gressive parameter a , , the more persistent any g tendency for [y , ] to remain away from its-, - '1'0 tormally examine the properties of the { y , ) s e m e conditional mean and variance are given by

I

1 .I and

Since a, and 6:-, cannot be negative, th

nored), the solution for - y, is

Since Ee, = 0 for all t , the unconditional expectation of (3.7) is Ey, = ad(1 - a , ) . The unconditional variance can be obtained in a similar fashion using (3.7). Given that E E , ~ , ~ is zero for all i # 0, the unconditional variance of y, follows directly

146 Modeling Economic Time Series: Trends and Volariliry i i

from (3.7) as

I

From the result that the unconditional variance of E, is constant [i.e., var(~,) = var(e,-]) = var(e,-J = ... = a d ( l - a,)], it follows that

i-~ Clearly, the variance of the (y,] sequence is increasing in m n d the ab- I solute value of a,. Although the algebra can be a bit tedious, the essential point is - that the ARCH error process can be used to model periods of volatility within the I

univariate framework. The ARCH process given by (3.2) has beengxtendedin several interestin- 1 1

Engle's (1982) original contribution considered the entire class of higher-order ARCH(q) processes: I I

In (3.8), all shocks from E,-, to E,-, have a direct effect on e,, so that the conditional variance acts like an autoregressive process of order q. Question 2 at the end of this chapter asks you to demonstrate that the forecasts for arising from (3.1) and (3.8) have precisely the same form.

The GARCH Model , I I

i Bollerslev (1986) extended Engle's original work by developing a technique that allows the conditional variance to be an A$vf~:ocess. Now let the error process be such that

I

Since [v,] is a white-noise process that is independent of past realizations of e,-,, I

the conditional and unconditional mews of e, are equal to zero. By taking the ex- - I

I

I ARCH Processes 147

pected value of r , it is easy to verify that I

! "

The jmvortant point is that the conditional variance of rr is given by E,-,a:= 1 Thus, the conditional~ariance of <isgiven by h, in (3.9).

- sentation that is m u c h estimate. This is particularly true since d l coefficients in (3.9) must be positive. Moreover, to,ensure that the conditional - - variance is finite, all charactenstic roots of lie ~nside the unit circle. Clearly, the more parsimonious model will entail fewer coefficient restrictions.-'

s - The key feature of els is that the conditional variance of the Jistur- 1 1 bancei of the {y , ) se=tes ARMA pmces: Hence, it is to be ex-

pected that the residuals fmm a fitted ARMA model shouId display this characteris-

4 P

E ~ - ~ E : = a,, + C C L ~ E ? - ~ + C P ~ ~ ~ - ~

Equation (3.10) looks very much like an ARMA(q, p) process in the (E:) sequence. If there is conditional heteroskedasticity, the conelogram should be sug- gestive of such a process. The technique to construct the conelogram of the squared residuals is as follows:

STEP 1: Estimate the (y , ) sequence using the "best-fitting" ARMA model (or regression model) and obtain the squares of the fitted errors g:. Also calculate the sample variance of the residuals (h2) defined as

where T= number of residuals

148 Modeling Economic Time Series: Trends ond Volatiliry

, STEP 2: Calculate and plot the sample autocorrelations of the squared residuals as ! I

/ STEP 3: In large samples, the standard deviation of p(i) can be approximated by T-ln. Individual values of p(i) with a value that is significantly different from zero are indicative of GARCH errors. Ljung-Box Q-statistics can be used to test for groups of significant coefficients. As in Chapter 2, the statistic

has an asymptotic x2 distribution with n degrees of freedom if the 63 are uncorrelated. Rejecting the null hypothesis that the 8: are uncorrelated is equivalent to rejecting the null hypothesis of no ARCH or GARCH errors. In practice, you should consider values of n up to T/4.

The more formal Lagrange multiplier test for ARCH disturbances has been proposed by Engle (1982). The methodology involves the following

!

i two steps:4

r STEP 1: Use OLS to estimate the most appropriate AR(n) (or regression) model:

STEPZ: Obtain the squares of the fitted errors 8:. Regress these squared residuals -2 .2 on a constant and on the q lagged values 8,-, , c,, e,,, . . . ,<?,, that is, es-

I timate

If there are no ARCH or GARCH effects, the estimated values of a, through a, should be zero. Hence, this regression will have little explanatory power so that the coefficient of determination (i.e., the usual ~ ~ - s t a - tistic) will be quite low. With a sample of T residuals, under the null hypothesis of no ARCH errors, the test statistic TR2 converges to a X: distribution. If TR2 is sufficiently large, rejection of the null hypothesis

Al ec ill va ur

I E fl;

I E, tic

I C( 1 C( I tt

h

ti

ARCH and GARCH Estimates of Inflation 149

that a, through a, are jointly equal to zero is equivalent to rejecting the null hypothesis of no ARCH errors. On the other hand, if TR2 is sufficiently low, it is possible to conclude that there are no ARCH effects.

3. ARCH AND GARCH ESTIMATES OF INFLATION g41 I ggv h ARCH and GARCH models have become very popular in that they enabld thz econometrician to estimate the variance of a series at a articular ~ o i n t in time. To . . illustrate the distinction between the conditional variance and the unconditional variance, consider the nature of the wage bargaining process. Clearly, firms and unions need to forecast the inflation rate over the duration of the labor contract. Economic theory suggests that the terms of the wage contract will depend on the inflation forecasts and uncertainty concerning the accuracy of these forecasts. Let E,K,+, denote the conditional expected rate of inflation for t + 1 and o: the conditional variance. If parties to the contract have rational expectations, the terms of the contract will depend on E,lt,+, and ok, as opposed to the unconditional mean or unconditional variance. Similarly, as mentioned above, asset pricing models indicate that the risk premium will depend on the expected return and variance of that return. The relevant risk measure is the risk over the holding period, not the unconditional risk.

The example illustrates a very important point. The rational expectations hypothesis asserts that agents do not waste useful information. In forecasting any time series, rational agents use the conditional distribution, rather than the unconditional distribution, of that series. Hence, any test of the wage bargaining model above that uses the historical variance of the inflation rate would be inconsistent with the no- tion that rational agents make use of all available information (i.e., conditional means and variances). A student of the "economics of uncertainty" can immedi- ately see the importance of ARCH and GARCH models. Theoretical models using variance as a measure of risk (such as mean variance analysis) can be tested using the conditional variance. As such, the growth in the use of ARCWGARCH methods has been nothing short of impressive.

Engle's Model of U.K. Inflation

Although Section 2 focused on the residuals of a pure ARMA model, it is possible to estimate the residuals of a standard multiple-regression model as ARCH or GARCH processes. In fact, Engle's (1982) seminal paper considered the residuals of the simple model of the wagelprice spiral for the U.K over the 1958:II to 197731 period. Let p, denote the log of the U.K. consumer price index and w, the log of the index of nominal wage rates. Thus, the rate of inflation is K , = p, - p,, and the real wage r, = w, - p,. Engle reports that after some experimentation, he chose the following model of the U.K. inflation rate (standard errors appear in parentheses):

150 Modeling Economic Time Series: Trenah and Volatility

where h, = the variance of (E,]

The nature of the model is such that increases in the previous period's real wage increase the current inflation rate. Lagged inflation rates at t - 4 and t - 5 are in- tended to capture seasonal factors. All coefficients have a t-statistic greater than 3.0, and a battery of diagnostic tests did not indicate the presence of serial correlation. The estimated variance was the constant value 8.9E-5. In testing for ARCH errors, the Lagrange multiplier test for ARCH(1) errors was not significant, but the test for an ARCH(4) error process yielded a value of T R ~ equal to 15.2. At the 0.01 significance level, the critical value of x2 with four degrees of freedom is 13.28; hence, Engle concludes that there are ARCH errors.

Engle specified a ARCH(4) process forcing the following declining set of weights on the errors:

The rationale for choosing a two-parameter variance function was to ensure the nonnegativity and stationarity constraints that might not be satisfied using an unrestricted estimating equation. Given this particular set of weights, the necessary and sufficient conditions for the two constraints to be satisfied are a, > 0 and O < a , < 1.

Engle shows that the estimation of the parameters of (3.12) and (3.13) can be considered separately without loss of asymptotic efficiency. One procedure is to estimate (3.12) using OLS and save the residuals. From these residuals, an estimate of the parameters of (3.13) can be constructed, and based on these estimates, new estimates of (3.12) can be obtained. To estimate both with full efficiency, continued it- erations can be checked to determine whether the separate estimates are converg- ing. Now that many statistical software packages contain nonlinear maximum likelihood estimation routines, the current procedure is to simultaneously estimate both equations using the methodology discussed in Section 7 below.

Engle's maximum likelihood estimates of the model are

The estimated values of h, are one-step ahead forecast error variances. All coefficients (except the own lag of the inflation rate) are significant at conventional levels. For a given real wage, the point estimates of (3.14) imply that the inflation rate

ARCH and GARCH Estimates of Inflation 151

is a convergent process. Using the calculated values of the (h, ) sequence, Engle finds that the standard deviation of inflation forecasts more than doubled as the economy moved from the "predictable sixties into the chaotic seventies." The point estimate of 0.955 indicates an extreme amount of persistence.

Bollerslev's Estimates of U.S. Inflation

Bollerslev's (1986) estimate of U.S. inflation provides an interesting comparison of a standard autoregressive time-series model (which assumes a constant variance), model with ARCH errors, and model with GARCH errors. He notes that the ARCH procedure has been useful in modeling different economic phenomena but points out (see pp. 307-308) that

Common to most. . . applications, however, is the introduction of a rather arbitrary linear declining lag structure in the conditional variance equation to take account of the long memory typically found in empiri- cal work, since estimating a totally free lag distribution often will lead to violation of the non-negativity constraints.

There is no doubt that the lag structure Engle used to model h, in (3.14) is subject to this criticism. Using quarterly data over the 1948.11 to 1983.N period, Bollerslev (1986) calculates the inflation rate (n,) as the logarithmic change in the U.S. GNP deflator. He then estimates the autoregression:

Equation (3.15) seems to have all the properties of a well-estimated time-series model. All coefficients are significant at conventional levels (the standard errors appear in parentheses) and the estimated values of the autoregressive coefficients imply stationarity. Bollerslev reports that the ACF and PACF do not exhibit any significant coefficients at the 5% significance level. However, as is typical of ARCH errors, the ACF and PACF of the squared residuals (i.e., E:) show significant correlations. The Lagrange multiplier tests for ARCH(l), ARCH(4), and ARCH(8) errors are all highly significant.

Bollerslev next estimates the restricted ARCH(8) model originally proposed by Engle and Kraft (1983). By way of comparison to (3.15). he finds

152 Modeling Economic T i m Series: Trends and Volatility

Note that the autoregressive coefficients of (3.15) and (3.16) are similar. The models of the variance, however, are quite different. Equation (3.15) assumes a constant variance, whereas (3.16) assumes the variance (h,) is a geometrically declining weighted average of the variance in the previous eight quarters.'

Hence, the inflation rate predictions of the two models should be similar, but the confidence intervals surrounding the forecasts will differ. Equation (3.15) yields a constant interval of unchanging width. Equation (3.16) yields a confidence interval that expands during periods of inflation volatility and contracts in relatively tran- quil periods.

In order to test for the presence of a first-order GARCH term in the conditional variance, it is possible to estimate the equation:

The finding that P, = 0 would imply an absence of a first-order moving average term in the conditional variance. Given the difficulties of estimating (3.17), Bollerslev (1986) uses the simpler Lagrange multiplier test. Formally, the test involves constructing the residuals of the conditional variance of (3.16). The next step is to regress these residuals on a constant and h,-,; the expression TR' has a x2 distribution with one degree of freedom. Bollerslev finds that TR2 = 4.57; at the 5% significance level, he cannot reject the presence of a first-order GARCH process. He then estimates the following GARCH(1, 1) model:

Diagnostic checks indicate that the ACF and PACF of the squared residuals do not reveal any coefficients exceeding 2T-'". LM tests for the presence of addi- tional lags of E: and for the presence of h,, are not significant at the 5% level.

4. ESTIMATING A GARCH MODEL OF THE WPI: AN EXAMPLE

To obtain a better idea of the actual process of fitting a GARCH model, reconsider the U.S. Wholesale Price Index data used in the last chapter. Recall that the Box-Jenkins approach led us to estimate a model of the U.S. rate of inflation (n,) having the form:

158 Modeling E C O M ~ ~ C Time Series: Trends and Volatility

'ebb All estimated coefficients are significant at conventional levels and have the ap- I"/

J

propriate sign. An increase in the expected price increases broiler output. Increased I uncertainty, as measured by conditional variance, acts to decrease output. This forward-looking rational expectations formulation is at odds with the more traditional cobweb model discussed in Chapter 1. In order to compare the two formulations, Holt and Aradhyula (1990) also consider an adaptive expectations formulation (see Exercise 2 in Chapter 1). Under adaptive expectations, price expectations are formed according to a weighted average of the previous period's price and the previous period's price expectation:

or solving for p: in terms of the (p,) sequence, we obtain 1

Similarly, the adaptive expectations formulation for conditional risk is given by I

where 0 < p < 1 and (p,,+ -P:-,-~)' = the forecast error variance for period t - i. Note that in (3.29), the expected measure of risk as viewed by producers is not

necessarily the actual conditional variance. The estimates of the two models differ concerning the implied long-run elasticities of supply with respect to expected price and conditional variance.' Respectively, the estimated long-nm elasticities of supply with respect to expected price are 0.587 and 0.399 in the rational expectations and adaptive expectations formulations. Similarly, rational and adaptive expectations formulations yield long-run supply elasticities of conditional variance of -0.030 and -0.013, respectively. Not surprisingly, the adaptive expectations model suggests a more sluggish supply response than the forward-looking rational expectations model.

F 6. THE ARCH-M MODEL

- and Robins

ean of a sequence to devend on its ow lass of model, called ARCH-M, is particularlv s The basic in- -

The ARCH-M Model 159

sight is that risk-averse agents will require compensation for holding a a asset. Given that asset's riskiness can be measured by the variance of returns, the r isL

-70 will be an increasg&ction of the conditi=ariance of returns. Engle, Lilien, and Robins express this idea by writing the excess return from hold- - ing a risky a s i s

(3.30)

where y, = excess return from holding a long-term asset relative to a one-period treasury bill

p, = risk premium necessary to induce the risk-averse agent to hold the long-term asset rather than the one-period bond

c, = unforecastable shock to the excess return on the long-term asset --- - ------ 1 To explain (3.30), n s t h a t @e expected excess return from holding the long-

I I term asset must be just equal to the risk premium:"

Engle, Lilien, and Robins assume that the risk premium is an increasing function - I of the conditional variance of E,; In other words, the greater - the conditional variance

of returns, the g-r the compe long-term asset. Mathematically,

i mium can be expressed as -

I when h, is the ARCH(q) process:

As a set, Equations (3.30), (3.31), and (3.32) constitute the basic ARCH-M and (3.31), the c o n d i t i o n a l ~ o f y, depends on the condi-

conditional variance is an ARCH(q) process. conditional variance is constant (i.e., if a, = a, =

I ... = a, = 0). the A R C H - M % ~ ~ I degenerates into the more traditional case of a \ constat risk premiumC& - > .

Flgure 3.1 1 illustrates two different ARCH-M processes. The upper-left-hand graph (a) of the figure shows 60 realizations of a simulated white-nolse process de- - noted by N s the temporary increase i volatility during periods 20 to 30. By initializing E,, = 0, the c o w - e w-s-order ~ R C H process:

0 1 + 0.65r:, -zc

16Q Modeling Economic Time Series: Trends and Volatiliiy

As you can see in the upper-right-hand graph (b), the v w t y in { e . ) translates i m in-the conditional v a r i w . m t h a t large positive and negative realizations of e,, result in a large value of h,; it is the square of each { e , ] realization that enters the conditional variance. In the lower left graph (c), the values of

I p and 6 are set equal to 4 and +4, respectively. As such, the y, sequence is constructed as y, = 4 + 4h, + e,. You can clearly see that y, is above its long-run value during the period of volatility. In the simulation, conditional volatility translates itself into increases in the values of { y , ] . In the latter portion of the sample, the volatility of { e , ) diminishes and the values y,, through y , fluctuate around their long-run mean.

The lower-right-hand graph (d) reduces the influence of ARCH-M effects by re- ducing the magnitude of 6 and p (see Exercise 5 at the end of this chapter). Obviously, if 6 = 0, there are no ARCH-M effects at all. As you can see by compar- ing the two lower graphs, y, more closely mimics the e, sequence when the magnitude of 6 is diminished from 6 = 4 to 6 = 1 .I2

As in ARCH or GARCH models, the form of an ARCH-M model can be deter- mined using Lagrange multiptier tests exactly as in (3.1 1). The LM tests are rela-

Figure 3.11 Simulated ARCH-M processes.

White noise process. h,=ao+al(e,-?I 2

The ARCH-M Model 161

tively simple to conduct since they do not require estimation of the full model. The statistic TR2 is asymptotically distributed as x2 with degrees of freedom equal to the number of restrictions.

Implementation

Using quarterly data from 1960:I to 1984:II, Engle, Lilien, and Robins (1987) constructed the excess yield on 6-month treasury bills as follows. Let r, denote the quarterly yield on a 3-month treasury bill held from r to (t + 1). Rolling over all proceeds, at the end of two quirters an individual investing $1 at the beginning of period t will have (1 + r,)(l + r,+,) dollars. In the same fashion, if R, denotes the quarterly yield on a 6-month treasury bill, buying and holding the 6-month bill for the full two quarters will result in (1 + R,) dollars. The excess yield due to holding the 6-month bill is approximately

The results from regressing the excess yield on a constant are (the t-statistic is in parentheses)

The excess yield of 0.142% per quarter that is over four standard deviations from zero. The problem with this estimation method is that the post-1979 period showed markedly higher volatility than the earlier sample period. To test for the presence of ARCH errors, the squared residuals were regressed on a weighted average of past squared residuals as in (3.13). The LM test for the restriction a, = 0 yields a value of T R ~ = 10.1, which has a x2 distribution with one degree of freedom. At the 1% significance level, the critical value of x2 with one degree of freedom is 6.635; hence, there is strong evidence of heteroskedasticity. Thus, there appear to be ARCH errors so that (3.34) is misspecified if individuals demand a risk premium.

The maximum likelihood estimates of the ARCH-M model and associated t-statistics are

The estimated coefficients imply a time-varying risk premium. The estimated parameter of the ARCH equation of l .64 implies that the unconditional variance is in- finite. Although this is somewhat troublesome, the conditional variance is finite. Shocks to e,-i act to increase the conditional variance so that there are periods of tranquility and volatility. During volatile periods, the risk premium rises as risk- averse agents seek assets that are conditionally less risky.

162 Modeling Economic Time Series: Trends and Vohtility

The next section considers some of the mechanics involved in estimating an ARCH-M model. Exercise 8 at the end of this chapter asks you to estimate such a ARCH-M model using simulated data. The questions are designed to guide you through a typical estimation procedure.

7. MAXIMUM LIKELIHOOD ESTIMATION OF GARCH AND ARCH-M MODELS

1 T

log 9 = -(T/2) ln(2rc) -(T/2) lno2 -(112o2)x(yr -p12 r=1

i

where log 9 = log of the likelihood function

a

The p v in maximum l i k w estimation is to s t h e distributional . . parameters so as to maximize the hkehho~d of drawing the observed sample. In the example at hand, the problem is to maximize log 9 with respect to p and oZ. The first-order conditions are

Many software packages contain built-in routines that estimate GARCH and ARCH-M models such that the researyher simply s~ecifies the order of the process

and

and the c o m t . Even if you have access to an automated routine, it is if3Eoltant to understand the numerical procedures used by your software pack- age. Other packages require user input in the form of a small optimization alg* rithm. This section explains the maximum l i k e w methods required to understand and write a pro

I Suppose that val 4 distribution having a mean .- C_

p and constant vari n theory, the log likelihood function using T independent observations is

Setting these partial derivatives equal to zero and solving for the values of p and o2 that yield the maximum value of log 2 (denoted by andeZ), we get

Maximum Likelihood Estimation of GARCH and ARCH-M Models 163

Thus, with sample d the maximum likelihod in a regression analysis.

In the lassic regres

11, ance is th9t use a sample with T tion of the above:

log S = -(T/2) ln(2x)-(Tl2) lna2

I Maxi&ing d equation with respect to / ---

! 1 and

I Setting these partial derivatives equal to zero and solving for the values of and

a2 that yield the maximum value of log 2 result in the familiar OLS estimates of the variance and p (denoted by 6' and B). Hence,

6' = Z(E,)~/ T

and from (3.31),

b = Q Y , / V ~ , ) ~

I All this should be familiar ground since most econometric texts concerned with

1 i regression analysis discuss maximum likelihood estimation. The point to ernpha-

* C C ua 164 Modeling Economic Time Series: Trendc and Volatility -A size here is that the first-order conditions Calculating the appropriate sums may be forward. Upfortunately, this is not the case in estimating an ARCH-type m o w

6 since the first-order equations a&&. Instead, the solution requires some sort

e search algorithm. The simplest way to illustrate the issue is to introduce a> ARCH(1) error process into the regression model. Continue to assume that r, is generated by the linear equation r, = y, - Now let r, be given by (3.2):

so that the conditional variance of E, is

Although the conditional variance of E, is not constant, the necessary modiflca- tions are clear. Since each realization of r, has the conditional variance h, the appropriate log likelihood function is

T

log S = -(T/2) ln(2x) - (Tl2) In h, - ( 1 / 2 h , ) ~ ( y , - ,=I

where h, = a, + a,<-:_, = a, + a101,1- P-L,)~

Finally, it is possible to combine the above and then to maximize log S with respect to a,, a , , and p. Fortunately, computers are able to select the parameter values that maximize this log likelihood function. In most time-series software packages, the procedure necessary to write such programs is quite simple. For example, RATS uses a typical set of statements to estimate this ARCH(1) model. Consider:I3

NONLIN p a, a, F R M L E = Y - P FRML h = a, + a,*€:-, FRML LIKELIHOOD = -0.5*[log(hr) + (e:lh,)] COMPUTE P = initial guess, a, = initial guess, a, = initial guess MAXIMIZE(RECURS1VE) LIKELIHOOD 2 end

The first statement prepares the program to estimate a nonlinear model. The second statement sets up the formula (FRML) for E,; E, is defined to be y, - The third statement sets up the formula for h, as an ARCH(1) process. The fourth statement is the key to understanding the program. The formula LIKELIHOOD defines the log likelihood for observation t; the program "understands" that it will

Maximum Likelihood Estimation of GARCH and ARCH-M Models 165 6YWl '"' "I -- - ---.--- r - -

gram to maximize LIKELIHOOD from observation 2 (since the initial observation is lost) to the end of the sample.I4

It is possible to estimate more sophisticated models using a comparable procedure. The key to writing a successful program is to correctly specify the error process and variance. To estimate the ARMA[l, (1, 4)]-ARCH(4) model of the inflation rate given by (3.22), lines 3 and 4 of the program would be replaced with:15

F R l M L ~ = x , - a , - a ~ x , , - b , ~ , , - b , ~ , ~ FRML h = a, + a,(0.4eEI + 0.3&, + 0.2~:-, + 0 .1~ :~ )

Here, the first formula statement defines E, as the residual from an ARMA[(l, (1, 4)] process. The second statement constrains the lagged coefficients to exhibit a smooth decay. Similarly, the GARCH(1, 1) version of this same model-see (3.23)-uses the program steps:

FRMLe=x,-a,,-a:x,-, -b,~, , -b4e, FRML h =a, + ale:, + P,h,-,

The program steps for the ARCH-M model of Engle, Lilien, and Robbins (1987) have the form

FRMLe=y-a,-alh FRML h = a, + a,(O.4e:, + 0.3~:~ + 0.2~:~ + O.ley4)

The first statement defines e, as the value of y, less the conditional variance. The second statement defines the conditional variance.

Finally, it is possible to include explanatory variables in the formula for the conditional variance. In the GARCH(1, 1) inflation model, it is possible to write

FRML h = a, + a l c t l + Plhr-l + B~z,

where z, is an explanatory variable for h.

Enders 1995 ARCH

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