-14-

then used as a sampling distribution with equal mass assigned to
each member. Each model (13) and (14) is then used to simulate data for

714 periods. The simulation model in each case was an autoregressive

approximation to (13) or (14). The filter (1-8)? was approximated by a
truncated version of equation (9). This filter was then convoluted with
(1-43), (1-487), or (1-498) as the situation required. In all
examples to follow the autoregressive approximating polynomial was of
degree 56. Generally, 800 fitted residuals were sampled from the
empirical distribution to start up the simulation model and to obtain 714
simulated observations.

The simulated data, which are denoted ce were then used to
reestimate the model. Parameter estimates resulting from the use of this

wRQ vk ok ok wk

data are denoted Oy d, > $35 and 459° The ‘bootstrap procedure was
repeated 1000 times in all cases presented here. In Tables 1 and 2
columns containing sample estimates are simply labelled “estimate.” The
mean and standard deviation of bootstrapped sampling distributions for
a‘ and a y and % are also provided. Finally, the estimated standard
error for d using the conventional formula under ordinary least squares
and the estimated standard errors for the autoregressive parameters using
the diagonal of the inverse information matrix from a maximum likelihood
estimation procedure (Box and Jenkins 1976, p.227) are presented. These
are called the nominal standard errors and their repeated calculation in
the bootstrap estimation process allows us to calculate their root mean

*
square. It is important to note that the bootstrap error terms, ¢€ , are


~A2-

maturity. This is done with the realization that term structure
approximations at all maturities are possibly affected by the faulty
local. approximations at the end of the yield curve. Researchers who use
such approximations should be aware of these problems.

The approximations, not surprisingly, closely replicate the
data which appear in Roll's Figure 5-1 and the actual average yields by
maturity contained in his Table 5-3. The convenience of the
approximations is that only the few estimated coefficients of the
approximating function need be saved and manipulated to quickly generate
any transformation of the interest rate term structure that is desired.
The data used in this study are actually from the final 714 weeks of the

sample's post Treasury Accord period.


TABLE 4

Excess Variability Test Statistics

Model I Model If
rao ? a8 as
In-Samle Full-Sample
Bootstrap Bootstrap Bootstrap
a Eetimace Mean SD Estimate Mean Estimate Mean SD a Setinate = Mean sD
1 072.072 .006 +060 060.005 +063 063.005 1 063 962 005
2 221.222 024 183 184.018 162 + 184.065 2 169 169 06
3 +438 440.059 2356357 Ob 352.394 .096 3 £323 363 08
4 721.725 «105 579.579 .075 -504 <573 161 4 557 621 avs
5 1.09 1.09159 861.863.1186 +906 = «1.01251 5 +835 +933 22
6 1.52 1,52 238 1191.20 287 1.24 1,40 .365 6 LAS 1,29 232
7 2.00 1.99 346 1,551.55 238 1.62 1.85505 7 1.49 1.69 hb
8 2.55 7,56 481 1.95 1.95 .333 2.03 2.34 .670 8 1.87 2.13 58
9 3. 1608 2.36 2.37 462 248 2,88 861 9 2.26 2.61 7h
10 3.72 365786 2.79 BL «575 2.95 3.46 1.08 10 2.69 3.13 +92
n 4.53 4.596937 3.38 336.877 3.45 4,08 1.31 uw 3.13 3.68 1.12
Model IIT Model IV Model ¥
o2 32 o 82 ot 82
Bootstrap Bootstrap Sootetrap

a Estimate = Mean sD a Estinace = Mean sD a Estimace aa sD
1 +063 +063 +005 1 +063 +063 +005 1 063 062 005
2 187 -170 +04 2 2137 +169 +03 2 “137 -170 .03
3 +320 +366 08 3 +298 +365 07 3 2297 368 .07
4 2552 +629 16 4 +513 $27 213 4 515 630 12
5 +826 +9, 22 5 77h =342 -20 5 33 946 at)
6 aes 1.31 231 6 1.06 1.30 229 6 1.06 1.31 8
7 1,48 171 43 7 1.38 1.71 239 7 1.38 71

8 1.85 2.16 -56 3 1.72 2.35 352 8 1.72 2.16 50
9 2.26 2.55 nm 9 2.08 2.45 -66 9 2.09 2.64 64
10 2.66 3.17 -89 10 2.66 3.15 +82 10 2.47 3.16 -80
u 3.10 3.73 +08 uw 2.86 3.7 1.00 u 2.87 3,72 97


other ARIMA models which impose an incorrect order of differencing on the
time series. We would therefore expect detectably better forecasting
performance from a general integrated model and especially better long-
term forecasting capabilities if the time series truly has 'long memory’.
In this section we estimate several such long memory models and select
some to address the ultimate objective of this paper, a variance bounds
test for the linear rational expectations theory of the term structure.
Forecasting capabilities are an important criterion in the model
selection process. By this means it is hoped the reader will be
convinced that only by using suboptimal long-term linear forecasting
models for interest rates will the rational expectations hypothesis be
rejected.

Identification, estimation, forecasting and hypothesis testing
with general linear integrated models present some special difficulties.
For example, in trying to identify a preliminary time series model,
inspection of the autocorrelation function of a general integrated series
is not as helpful in obtaining preliminary parameter estimates as is the
case when trying to identify an ARMA model when d is assumed to be
integer and known. This is because the autocorrelations are complex
combinations of d and the ARMA parameters and cannot be readily factored
into simple expressions suggestive of the model's structure (Porter-Hudak
1982). This complexity also presents problems for estimation. Granger
and Joyeux (1980) and Porter-Hudak (1982) review estimation theory for
these models, most of which is designed to circumvent the problem of not
being able to write down a manageable likelihood function for a

fractional differencing filter. A line of attack hinted at by Granger


TABLE 2

Autoregressive Parameter Estimates

1
Bootstrap
Model 2SsEatimate © Nominal SE Means SDSS Nominal SE
tS? “5 202 <10 20 205
Iz. .69 “9 203 -09 08
mt. 7 a9 205 lk .09 205
Ww. .73 122 sto -08 96
TS 2 203 A308 06
vr. -
vit. ~~
vitt. -
Model = = Estiimace = Nominal Mem SD_—-3MS Nominal SE
1. =
I. -
IL. -
Ww. --
on
v.58 -.12 8 = .06 06
WIT. 85 “1 05 = 06 205
VIL. 456 -.08 05 “07.06 06
#52
Bootstrap
Model a Estimate Nominal SE Mean.—=—s SD. 34S Nominal SE
TST 125 .07 2h .05 205
Ir 488 2h 05 105 08
Mt. 71 2th 05 wh 05 205
Ww. .73 th 08 mh 08 .06
wT 2th 06 “05 108
vt. .6h 225 205 206 .08
vit. 465 25 205 OS 205
vit. 66 23 05 23.05 205


FOOTNOTES

* International Finance Division, Board of Governors of the Federal
Reserve System and The Pennsylvania State University. This paper
represents the views of the author and should not be interpreted as
reflecting the views of the Board of Governors of the Federal Reserve
System or other members of its staff. This paper builds upon material in
the author's dissertation (Shea 1982) and thereby benefitted from
comments from Charles R. Nelson. A number of my colleagues at the Board
have patiently discussed with me material covered in this paper. Under
no circumstances would any blame attach to them for my own errors. These
helpful discussants were Ed Green, Richard Freeman, and Jeffrey
Marquardt. Conversations with Susan Porter-Hudak and John F. Geweke were
particularly helpful in solving or in suggesting solutions to some of the
technical problems encountered. David Laughton provided valuable
programming assistance at several moments of crisis. Margaret Gray and

Kathy Krasney ably typed the manuscript.

1 Tests performed by Singleton (1980) were repeated using the data from
the Appendix. The results were much the same as those reported by
Singleton, so the objection that the data are especially favorable
towards the rational expectations hypothesis cannot be raised on the

point that the hypothesis might be rejected using his test procedures.


Shiller and Grossman studies, Marsh and Merton (1984) have also
questioned the assumption of whether dividend flows are adequately
described as stationary stochastic processes.

If interest rate time series are nonstationary, would there be
interesting implications for how excess volatility tests are applied to
equilibrium notions of the term structure? This paper answers that in
the affirmative and introduces a particular class of discrete linear
nonstationary stochastic processes which are better models of short-term
interest rates than either the random walk or the discrete stationary
linear models. These models are the general integrated models
investigated by Granger and Joyeux (1980), Geweke and Porter-Hudak (1983)
and Mandelbrot (1972) and are otherwise known as fractional difference or
long memory time series models. In the remainder of this paper there are
developed a number of general linear integrated models for short-term
interest rates, which are estimated and then compared to alternatives.
The variation of short-term rates relative to that of holding period
yields for longer term bonds under the linear rational expectations
hypothesis is examined. It is shown that, joint with the estimated
general integrated model for short-term rates, the linear rational
expectations hypothesis cannot be rejected using a variance bounds test.
The controversial bootstrap methodology is used to construct confidence

intervals for hypothesis testing.


-26-

combination of d and other parameters of a long memory model. Taking
equation (13) as an example, we can write down the first four p

coefficients as

Yo = 1,
T(d + 1) -
*T@ray 7 74> bp
T(d + 2)
vo = Fara) ~ (4 - 4)

asd 4 oa - 62, and

T 3. 2
¥3 oat oo + [a?ea+o,a-¢7] 4

(a+ 2)¢a +1d , [a?ea+9, 4-47] 7

If equation (21) is to be used as the basis of a test of the rational
expectations hypothesis when interest rates follow a long memory process,
sampling distributions for oan will have to be constructed and we will
have to be concerned with the joint distributional properties of d and
the $ coeffcients. Moreover, one has to be able to evaluate the moments
of 6's sampling distribution, This distribution will, in general be a
function of the factorial and product moments of the jointly distributed
d and ¢ parameters. Even if these parameters were joint normally
distributed and model estimation was based upon that fact, the evaluation

of sampling distributions for s is difficult. As useful as it was to


-15-

independently sampled from the common distribution of fitted residuals,

€. The sampled e, however, are used only to simulate the bootstrapped

data, oa Sample nominal standard errors are formed by using € and the

bootstrap standard errors are formed by using e, the fitted errors from
each bootstrap replication.

Models I through V correspond to equation (13) and models VI
through VIII correspond to equation (14). In Table 1 estimated o in
most all cases is very close to its theoretical value of 12/6 = 1.645 and
to the mean of its bootstrapped sampling distribution. Estimated d also
seems to be well within its bootstrapped sampling distribution. In each
of the simulated worlds represented by these models, nominal standard
errors are close to their ‘true’ bootstrapped standard deviations and,
moreover, are probably reliable guides for hypothesis testing judging
from the similarity between estimated d's bootstrapped standard
deviations and the root mean square nominal standard errors. This
answers in part the question raised by Geweke and Porter-Hudak (1983) as
to whether it is sufficient to use conventional formulas for calculating
standard errors or should the known characteristics of the theoretical
distribution of v = log { I(A)/£(A) } be used instead.

Table 1 would seem to suggest that a good case can be made for
d lying between .5 and 1. It was found in several experiments that
estimated d was inversely related to a until a was between .70 and .75,
at which point values of estimated d were stable and apparently different
from either .5 or 1. There are instances in Table 1 in which estimated d

or the mean of its bootstrapped sampling distribution are not too far


TABLE

3

Long Memory Model Forecast Accuracy

Estimated SE of ¥

Root Mean Square Forecast Error

Ta-Sample Out—of-Sample
Bootstrap Bootstrap Boot strap
Model Estimate Mean SD Zstimace Mean SD imate Mean SD
e) @ a () «@) @) (9) (0)
i -60 65.10 67 “65.07 a 65.10
In “37 62.99 68 152.06 a 63.09
UL 237 62.09 68 $2.07 ol 8209
Iv. 55 62.08 68 81.06 ot 61.09
v. 55 62.08 68 61.06 at 69.09
vw a +66 1b 75 7742 36 BL .20
vit. 61 62 1 27h 70.10 Me 66.19
VEIT. 252 52.10 72 56.06 32 56.09
AR(56) 21h 85.09 57 86.16
Ix. -61 59.08 7h 70.10 i 1216
x. 45 44.03 nm 62.06 -36 62.09
XI. 2 .72 08 1 96.21 40 1.02.32

¢ Horizon — 26 Weeks
Estimated SE of Forecast Root Mean Square Forecast Error

Ta-Sampie Out-of-Sample
Bootstrap Bootstrap
Modei Zatimace Mean SD Estimate Mean «SDs istimate Mean. «SD
aM @ GQ © G) (6) 7) (a) a) Go)
I. 7h 8300.17 97 81.12 -40 8117
IL -70 77 AS 7 wT AL 41 76 1S
IIT. 59 78 14 97 78 LL Al SAS
Iv. -66 17813 97 73.10 42 73 1S
+86 +78 13 “97 s72 20 4 +73
vr. 81 82.23 1.16 Liz 425 33 1.20.40
vit. 275 78 aes 99.20 51 1.05.32
VIII, 62 62.16 1.06 moat mrs 72.16
AR(56) 1.27 1.54.21 +96 1.6025
IX. -76 7313 1.13 98.20 -50 1.03.31
x 23h 54.06 95 -70 .09 48 70 AB
XI. 95 95.06 1.22 1.56.46 +62 1.72.67


-17-

procedures for the autoregressive models. We could abandon the
nonparametric bootstrap in favor of a parametric bootstrap experiment
instead. In particular, this would mean that bootstrap error terms, e,
will be sampled form a normal distribution having the estimated sample
standard error; this is consistent with the maximum likelihood procedure
used in estimation of the autoregressive coefficients. From the limited
experiments carried out here, however, it is fairly clear that sampling
error may have played a role in the appearance of nonexistent low-order
autoregressive parameters in these models. On the other hand, a seasonal
autoregressive parameter and an order of differencing between .5 and 1
are tenable components of a general linear integrated model for The
Evidence of the forecasting abilities of long memory models for
Tr is presented in Table 3. Forecast statistics are displayed for the
models I through VIII. Other models are also-presented as benchmarks
with which we can compare the long memory models. The first benchmark
model is a straightforward autoregression of order 56. Since forecasting
experiments were performed by constructing autoregressive projecting
filters with 56 lags for all models, it is important to show that a long
autoregression, which might naively be estimated without account being
taken of the data's nonstationary character, does not have forecasting
capabilties comparable to autoregressive filters of similar size
constructed from long memory models. This long autoregression, by
necessity of its length and the large sample sizes employed in this

study, was estimated by Whittle's (1983) algorithm for the recursive


-27-

use the Monte Carlo methods to simulate distributions in the previous
section, it is essential in this section to use the same methods to
simulate the distributions of sample 6's.

Using the models of interest rates presented in the previous
section, the v coefficients in the infinite moving average representation
can be calculated to give estimated ane for any value of n. From the

Treasury Bill yield curve data an unbiased estimate of the variance of

ee They is obtained by taking the noncentral second moments of
HD Th for any particular n. In Table 4 sample and bootstrap

(n)_

: 2 : .
estimates of BCH, Toy) are presented. The in-sample estimates refer
to estimates using the 514 observations which figured in model

estimation. The so-called full-sample examples use the full complement

(n)

of 713 available observations of r 1 and Hq.

t- . For each model I through

VIII, X, and XI a panel of estimated and bootstrap simulated ons (the
latter being based upon estimated WV) are presented.

The results are easily enough stated. For all models except X
the hypothesis that holding period yields are not too variable to agree
with the rational expectations theory and a long memory characterization
of interest rates is not rejected. Only in the case in which d is fixed
and known to be .5 is a long memory model (X) close to describing a
stationary interest rates series. In that case the mean and standard
deviation of bootstrapped ote? are small enough to reject equation (21)
as a tenable hypothesis. If we are inclined to assume that interest:
rates are mean stationary stochastic variables and especially if we
assume that d equals zero rather than some other value, it is rather easy

to bias tests toward the rejection of the rational expectations


-13-

In the course of the bootstrap experiments, several notable
features of the model came to light. When d is fixed, the estimated
standard errors for $5, % and yg at first made it appear that these
were statistically significant parts of the model, but the bootstrapped
standard errors seemed to demonstrate that their presence was illusory.
Moreover, for a less than -67 or thereabouts, did not seem to be
significant whereas for a greater than .67, $5, did not appear to be
significant. Tables 1 and 2 therefore display the results of estimation

of the models
(1 = 4B) = 45B°00- Bor, = & a3)
1 52 t t
for values of a between .67 and .75 and
_ 3 _ 52. 7,_ pd. =
(1 = $5B°) C1 ~ $598 dC Br, = g (14)

for values of a between .64 and .66. Using (11) as a regression equation
and letting v = log { I(A)/£0A) }, Table 1 presents estimated d and
estimated a and their associated statistics for each of the two
respective ARMA models for selected values of a. Table 2 similarly
displays results for estimated o> $3» and 59°

The bootstrap starts by saving the fitted residuals. From the
original 714 post Treasury Accord observations of Treasury Bill yield
curves 200 observations have been saved for the out-of-sample forecasting

experiments to follow. The empirical distribution of fitted residuals is


LOG SPECTRUM OF ONE-WEEK TREASURY BILL YIELDS

-14

ZCDAQMVH OO

0.0 0.1 0.2 0.3 0.4 0.5
FREQUENCY

Figure 1


~24-

period t+k is just the appropriate linear combination of expected shocks
(e's) which are zero for periods subsequent to t-m and are known for all
periods prior to and up through t-m.

The rational expectations equilibrium between a long-term rate
and the short-term rate works through the expected equality of a long-
term interest rate with an average of short-term rates expected to

prevail over the life of the bond. That is, if R

) is the yield to

maturity of a bond with n periods left to redemption,
n-l

yTM
-im tg yet gg
ko FOR F429 5x0

-]  &

v

(n)
Re (17)

jtk ore

The one-period yield on a long-term bond can be calculated as the return
on the purchase of a bond and its sale in the succeeding period. The
yield realized in period t from holding and selling a bond in period t
when it has n periods left to maturity is therefore
(n) (n+1) (n)
Ae (n+l) Reo) nko. (18)
The yield to maturity in period t-l from a bond with n+l periods to

maturity is

n-l ©

1
E,_j(r Ye fF fF hie . . (19)
0 t-l* t-l+k ntl =1 j=l jtk t-j

n
gant) 2 1
t-1 ntl k=

By combining the right hand sides of equations (17) and (19) we can

obtain an expression for a holding period yield in terms of the ¥


-22-

a” strongly suggest that d is significantly greater than .5 and probably
less than 1. There is some question about how to properly model the high
frequency components of the series, however. This is not too surprising
in light of the experience of Geweke and Porter-Hudak (1983) in
simulation experiments in which the chances of properly recognizing high
frequency components of a long memory model were reasonable using only
samples of fairly large size. The forecasting performances of models I
through VIII compare quite favorably to those of the AR(56) model, other
stationary ARMA models and an ARMA model with a unit root in its
solution. There is some evidence favoring models I through V on the
basis of their relative forecasting performance. In closing this section
it will be important to remember that long-term forecasting ability seems
to be strongly influenced by the level of a and, judging from the
bootstrap standard deviations in Table 3, the level of d also strongly
influences the simulated variability of forecasts. Forecast accuracy and
variability play crucial roles in the tests conducted in the next

section.
IV. Linear Rational Expectations and Variance Bound Tests

This section will look at one notion of variance bounds
implicit in the term structure of interest rates, the variance bounds
relation between holding period yields and short-term interest rates.
The stochastic properties of holding period yields from long-term bonds
and short-term interest rates are dramatically different and have been
the basis of rejections of the linear rational expectations hypothesis of

the term structure by Shiller and Singleton. Wide swings in holding


reliability with which long-term forecasts can be made and will certainly
alter the expression of theoretical variance bounds on yields from long-

term bonds.

Proceeding to the expression of the general integrated model,

we let
vB) = ¢7¢Byecsyqa - B)~4, (2)

where $(B) and 0(B) are finite-order polynomials in the backshift
operat:or B and constitute the short-term ARMA model of the process, so

that
$(B)(1 - By4 r, 7 (BYE, « (3)

This is the complete linear general integrated model.

We now adopt the notation used by Geweke and Porter—Hudak
(1983) to express the model in the frequency domain in which some of the
model's features are more comprehensible and where the estimation of d

takes place. Letting
u, = 4(B)0 (B)r
t t?

the spectral density of re is

-24
F(t) £04) = (o7/2n)|1 - & 1 £O), (4)


Il. A General Integrated Model of Short-Term Interest Rates

I assume that short-term interest rates can be represented by a

discrete linear stochastic model of the form

r= ve, ’ qa)

where T, is a short-term rate, e. is an i.i.d. normal random variable and
~(B) is a polynominal (possibly of infinite order) in the backshift
operator B which operates on the subscript t. Equation (1) is the moving
average form of the model and the coefficients of (B) will be called the
moving average coefficients. Equation (1) can serve as a model for a
nonstationary interest rate time series and I shall assume that a
differencing filter of the form q-s)?, when applied to The will render
the interest rate series stationary. The stationary component of the
time series can then be modelled as an ARMA process.

Economists are used to assuming some prior knowledge of d, the
order of the differencing filter (1-B)4, At most, the assumed value of d
has been tested against an alternative which is usually 0 or 1 when the
possibility of a unit root in the difference equation representation of a
time series arises. In any case, we are mostly used to thinking of ARMA
parameter estimates as conditional upon known integer values of d.

The general integrated model, however, allows for noninteger
values of d and thus gives the modeller some more flexibility in the
representation of the low frequency components of a time series. In
testing for excess volatility this is obviously important since improper

allowance for low frequency components in a time series will affect the


oT’ () a €L° IT’ SL" ey" L9°T 06°T 99° “IIIA
Il’ IT £8" IT’ 98° Lv” OL *T 8L°T s9° “IIA
an Il’ ss° ar’ 18° Ly TL'T 98°T 99° ‘IA
80° 80° 8L° 80° ol €e* OLT €L°T SL° “A
80° 80° LL’ 80° ol’ 9€° TL°T 99°T €L° “AI
60° 60° gL" 60° cL" 8e° OL’T tL°1 TL" ‘111
60° 60° 8L° () el" 17° 69°T 9L°T 69° “Il
or’ or’ 18° IT’ 9L° cy" OL°T °T 149° ‘I
GS TBUIWON SHY as uBay as TeUTWON 97 eu ISA as uray o7eUT ISA Ls T?POW
deijsj00g de13sj00g
P %o
z

(suotjeor[dea de1js300q QOOT UIT ‘yIS=L) UoTssersoy YSPNY-19I10g-9yoMay

T dldvl


= i . _1)dt1
y dy - 7d (d- Dy y+... + C1) Vena * &- (8)

For fractional d, however, the autoregressive representation is of

infinite order and is expressed

o k
y,zad 5 —thk-d BY te, (9)

t klT( -aThk+t ff

T is the gamma or generalized factorial function. So, direct application
of a fractional differencing filter will in principle entail the use of a
long approximating autoregressive filter which may be prohibitively
consumptive of scarce data. The approximation itself may also be less
than satisfactory since the autoregressive coefficients get small only at
long lags, depending on the value of d (Granger and Joyeux 1980; Geweke
and Porter-Hudak 1983); this is a manifestation of the potential long
memory characteristics of these models.

We shall now take a preliminary look at the data to see if
traces of a general linear integrated model can be found there. The data
described in the Appendix are a long time series of continuous term
structure observations stretching from 1951 through 1964. Compared to
what was to follow, this was a quiescent period for interest rates, so if
a noastationary model is not rejected against a stationary model, that
will be telling evidence for the general nonstationary character of
interest rates. A glance at Figure 1 suggests that the yields on

Treasury Bills with one week to maturity have the 'typical’ spectral


-18-

calculation of autoregression coefficients. The other benchmark models,

numbers IX through XI, have the structure
ye. . 2 3 4 _ 52 _ pd es
CL - OB 8 4B $B $598 °)(1 - B) Tre » (5)

Models IX, X, and XI have values of d which respectively are estimated to
be .34 (a was set at .65 just as in model VII) and are constrained to
have the values of .5 and .1. The autoregressive parameters were
estimated in the usual way conditional upon d. These final three models
are meant to show how a long memory version of a model (IX) compares with
versions of the same model displaying two extremes (models X and XI) in
nonstationary behavior.

The two panels in Table 3 pertain to the two forecasting
horizons of 13 and 26 weeks. For each model the estimated standard error
of forecast is presented. By this we mean the standard errors of
forecast based upon the model parameters and the standard error of

estimate. For example, the sample estimate in column (2) of Table 3 is

~ nag

o. sqrt) = ¥

€ . i

i=0

The standard deviation of the empirical distribution of & is 5, . nis
the forecast horizon of either 13 or 26 weeks. Each moving average
paraneter, vi» is constructed from d, o> $0, 35 $,and $59 as
warranted. The standard error of forecast in the bootstrap experiments

is


International Finance Discussion Papers

Number 258

July 1985

LONG MEMORY MODELS OF INTEREST RATES,
THE TERM STRUCTURE, AND VARIANCE BOUNDS TESTS

by

Gary S. Shea

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 materials)
should be cleared with the author or authors.


SPECTRUM OF DIFFERENCED ONE-WEEK YIELDS

0.2504
:
j

0.2254
|

0.2004
0.1754
j

0.1504

0.1254

ZCWAOMUNM

0.1004
0.0754
0.0504

3

0.0254

0. 000-4!

0.0 : 0.1 0.2 0.3 0.4 0.5
FREQUENCY

Figure 2


shape described by Granger. For interest rate levels the spectrum is
very large and perhaps explosive which greatly obscures other features of
the spectrum at low frequencies due to leakage. On the other hand, the
spectrum of the differenced series in Figure 2 clearly goes to zero at
the lowest frequencies instead of flattening which is generally
symptomatic of overdifferencing. Reflecting on his studies after more
than a decade, Granger (1981) was willing to treat this as an additional
feature of an economic time series having the ‘typical’ spectral shape.
Finally, in Figure 3 is the estimated autocorrelation function out to 104
lags. The autocorrelations die out very slowly as we would expect for a
time series with ‘long memory.'

This particular time series of interest rates, not unlike many
others, appears to be nonstationary. Its nonstationary character is
somewhat shy of that possessed by a random walk, however. In the next
section we estimate a fractional difference filter and accompanying ARMA

model to deal with the data's peculiar character,

Ill. Selecting a General Linear Integrated Model

The previous discussion raises several expectations for an
estimated fractional difference model for short-term interest rates. We
would first naturally expect to get a reliable estimate of the order of
the differencing filter and to be able to ascertain whether it is
statistically different from 0, 1 or some other value. Second, a general
integrated model is supposed to preserve the low frequency components of

a time series and to use them to greater effect for prediction than do


Bibliography-2

Marsh, T. and Merton, R. "Dividend Variability and Variance Bounds Tests
for the Rationality of Stock Market Prices." Discussion paper
presented at the National Bureau of Economic Research Program
on Economic Fuctuations Research Meeting, October 1984.

Michener, R.W. "Variance Bounds in a Simple Model of Asset Pricing."
Journal of Political Economy 90 (February 1982): 166-75.

Nelson, C.R. and Plosser, C.I. "Trends and Random Walks in Macroeconomic
Time Series: Some Evidence and Implications." Journal of
Monetary Economics 10 (September 1982): 139-162.

Peters, S.C. and Freedman, D.A. "Using the Bootstrap to Evaluate
Forecasting Equations." Technical Report No. 20, Department of
Statistics, University of California, Berkely, October 1984.

Porter-Hudak, S. "Long-Term Memory Modelling-A Simplified Spectral
Approach." Unpublished Ph.D. dissertation, University of
Wisconsin, 1982.

Roll, R. The Behavior of Interest Rates, New York: Basic Books, 1971.

Schenker, N. "Qualms about Bootstrap Confidence Intervals." Journal of
the American Statistical Association 80 (June 1985): 360-361.

Shea, G.S. "Pitfalls in Smoothing Interest Rate Term Structure Data:

Equilibrium Models and Spline Approximations." Journal of
Financial and Quantitative Analysis 19 (September 1984): 253-
269.

Shea, G.S. "The Japanese Term Structure of Interest Rates," Unpublished
Ph.D. Dissertation, University of Washington, 1982.

Shiller, R.J. "The Volatility of Long-Term Interest Rates and
Expectations Models of the Term Structure," Journal of
Political Economy 87 (December 1979): 1190-1219.

Shiller, R.J. and Grossman, S.J. "The Determinants of the Variability of
Stock Market Prices." American Economic Review - AEA Papers
and Proceedings 71 (May 1981): 222-227.

Singleton, K.J. "Expectations Models of the Term Structure and Implied
Variance Bounds," Journal of Political Economy 88
(December 1980): 1159-76.

Whittle, P. Prediction and Regulation by Linear Least-Square Methods.
Second Edition Revised. Minneapolis: University of Minnesota
Press, 1983.



ONE-WEEK YIELD AND HOLDING PERIOD YIELD SERIES
(HOLDING PERIOD YIELD FROM A TWELVE-WEEK BILL) PERCENT

| | l | | | ] | | | | PT 24
= ONE-WEEK YIELD

pene ssesaeesees = HOLDING PERIOD YIELD >

— + 16

Z po of gy Ee 78
e e

saseenaee:

' ' Me : N s H
: N " H
: : : : ‘
: : : : :
: : 3 :
= : : 3 : —_
‘ : : :
: : H :
: : :
: : :
: H
: :
: :
: H
: H
: :
:
;

| oe | ! | | | | | | | | 16
1952 1954 1956 1958 1960 1962 1964

Figure 4


-23-

period yields are to be expected on the basis of the capital gains or
losses that can be incurred over a very short holding period, but many
investigators have considered the spectacular capital gains’ and losses
sometimes observed (especially when interest rates are at low levels)
still too great to be in accord with the rational expectations
hypothesis. For example, Figure 4 overlays the time series of one-week
yields to maturity and one-week holding period yields for Treasury Bills
with 12 weeks to maturity. Although the holding period yields seem to
deviate from the short-term rates by many hundreds of basis points at
certain times, is the deviation too great to agree with the expectations
hypothesis?

To answer this we will construct the theoretical variance of
holding period yields under the hypothesized expectation mechanism.
Recall the infinite moving average representation of the short-term

interest rate series,

This linear mechanism can be used to characterize a projection of t into

future periods and using that projection as a model of market

expectations we obtain

)= EF yp es. (16)

E is the expectations operator and equation (16) simply states that the

expectation held in period t-m of the interest rate level to prevail in


where FQ) is the spectral density of ue Alternatively, £(A:d) may be

expressed (Granger and Joyeux 1980) as

o 2 -2d
() [2 G - cosa) ] , (5)

which clearly gets very large as 4 > 0. On the other hand, the spectral

density of the differenced series is

2 2

EN" casa) = GE) [20 - cosa)

lie pa-a

(6)
which goes to zero as } > 0 and d # 1. Thus, for a time series having
spectral density (4) with fractional d, the econometrician runs the risk
of losing low frequency components of the series if he approaches the
problem of achieving stationarity in the conventional manner by taking an

integer-order difference of the series. Only for |a| < .5 will the time

o

series be stationary. That is, for |d| > -5, = y2(B) is unbounded
k=0

(Granger and Joyeux 1980) and for those values of d we must be able to
prefilter the data prior to model estimation.

Applying a differencing filter of fractional order is not as
straightforward as with integer-order prefilters. To see this consider
the expansion of ca-s)4 when d is integer. The autoregressive

representation of a time series, Yer such that

a - By? ye is (7)


-16-

from 1 judging by the standard errors, but these instances are associated
with models of detectably inferior forecasting performance, as will be
shown. In no case, on the other hand, does it appear that estimated or
simulated d is close to .5. Experiments with different prefilters,
values of a and ARMA model specifications yielded more or less the same
results; sample and bootstrapped estimates of d were always well bounded
away from .5. Therefore, this particular series of short-term interest
rates, ri, is of the nonstationary type even though it may not be
possible to reject the hypothesis that d is less than 1.

Treating d as if it were fixed and known can at times give
misleading estimates for standard errors for the ARMA parameters. This
is particularly true for lower order autoregressive parameters which, as
shown in Table 2, generally have estimated standard errors that overstate
their significance. In the simulated bootstrap world the estimated first
order autoregression coefficients are lower than those estimated fiom the
sample and their standard deviations demonstrate there is a considerable
chance they are lower still. This is not quite the case for the third
order autoregressive coefficients although in this instance too it is not

. *
apparent that a correct ARMA representation of r, or T, have any low

t
order autoregressive components. For the seasonal parameter estimates,
however, the nominal standard errors are close to their 'true' values in
the bootstrap experiments and strongly support the inclusion of the
seasonal autoregressive parameter in an ARMA representation of vr, and re

The simulation experiments presented here can be altered, of course, to

further reflect some of the assumptions inherent in the estimation


-28-

hypothesis. For all other long memory models, however, sample estimates
2 aye : : : :
of ECHO - Thy) are well within the bootstrapped sampling distributions

2 : ape
of o28? and sample estimates of a2 are well within the bootstrapped

(n) 2
-r

sampling distributions of ECA, t-D .

Vv. Concluding Remarks

The results in Table 4 suggest that the rational expectations
hypothesis is consistent with our observations of how a set of high
quality term structure data behave. At least we do not find evidence

such as that reported by Singleton (1980) that holding-period yield

variances exceed their implied bounds by a factor of 10!, Confidence

(n)_

intervals around sample ECA,

rep and estimated one suggest that
both are quite close enough to affirm the rational expectations
hypothsis. There is much improvement to be made in the theory of
estimation and forecasting with long memory models and there is some
question (Schenker 1985) as to the reliability of bootstrapped confidence
intervals for complexly constructed test statistics such as those
presented in Table 4. We have tried to overcome some of the qualms about
the bootstrap procedures by using as large sized samples of high quality
data as possible. There are outstanding theoretical questions as to
whether bootstrap techniques are useful for’ bias correction in long
memory models and what effect that would have in aiding the long memory
model selection process, For the data studied here and for the period of
which it is representative, however, it is fairly clear that the mean-
variance stationarity assumptions found in the prevalent excess

volatility literature are unwarranted. How likely is it that the same

assumptions are valid for asset returns in the 1970's and 1980's?


APPENDIX

Data used in this study were collected and described by
Roll (1971, Chapter 5). This high-quality data set is a collection of
bid~ask quotes carefully culled from broker quote sheets and corrected
for calender effects to the greatest extent possible so that quotes are
for Treasury Bills in the brokers' discount market for integer numbers of
weeks to maturity. To construct the longest time series of yield curves
possible only quotes for bills out to 13 weeks to maturity were used.
On some dates quotes for bills with an integer number of weeks to
maturity were not available. For this reason bid-ask quote averages were
smoothed with a constrained cubic B-spline function, a method for
generating an approximation to the term structure described by Shea
(1984). This approximation to the discount curve is constrained so that
it passes through the origin; that is, it is constrained so that the
predicted price of a zero-maturity Treasury Bill is its face value. In
this way it was possible to automatically generate 796 consecutive weekly
discount curves from October 1949 through the last week of December 1964.
The smoothing method is not without its faults. For one, it is
particularly dangerous to use estimates of the term structure implied by
the slope of the approximation at the end of the approximation interval,
in this case, 13 weeks to maturity. There is little that can be done to
correct for the poor yield curve slope approximations that come from
these methods except by undertaking ad hoc constraint experimentation to
manipulate the slopes until they seem reasonable. What has been done

here is to use the term structure estimates out to only 12 weeks to


~20-

column (8) are smaller than figures in column (5). These figures are all
too low to have been sampled from the simulated out-of-sample forecast
error distributions. The primary reason for this is that there is some
marked difference between the variability of the actual data in the 200
post sample observations and that of the other 514 observations of short-
term interest rates. We speculate that since the post sample period
covers most of the time in which Operation Twist was in force, there was
an effort to raise short-term interest rates relative to long-term rates
along with some coincidental smoothing of interest rates resulting from
policy actions. On the other hand, there are some models in Table 3
(notably models VI and VII) in which the estimated out-of-sample root
mean square forecast errors are actually well within their simulated
distributions. All in all, it does appear there may be some preference
for the models with relatively lower values of don the basis of in-
sample and out-of-sample forecasting accuracy for both actual and
simulated data.

The bootstrap methods can also be used to address the question
of which models are likely to yield biased forecasts. We found that the
long memory models with lower d were less likely to yield biased
forecasts. On average, models I through V and model VIII, all of which
generally have d and a* at or below .76, produce forecasts of r very
close to actual we. This was not as prevalent a feature in models VI,
VII, IX, and X. For the less biased forecasting models we would also
expect calculated root mean square forecasting errors to be closer to

their standard errors of forecast based upon parameter estimates. This


stationary linear stochastic processes having finite variance. With that
assumption, rational expectations models of the term structure will imply
particular variance relationships between expected future returns,
certain short-term interest rates, and other returns, such as yields to
maturity or holding period yields. Although most writers have explicitly
admitted that the assumed stationarity of short-term interest rates or
dividend flows is crucial to the working of their tests (a catalog of
such admissions is cited in Flavin's (1983) concluding remarks), there is
only a little dissatisfaction with the assumption yet evident in the
literature. This is unfortunate because, even if the assumption is
valid, there are practical problems in implementing reliable hypothesis
tests. Flavin, for example, shows that it is difficult to reject the
expectations hypothesis using the small samples used by Shiller and
Singleton in their studies. .

In contradiction to the stationarity assumption, there are many
studies which have found interest rate time series to be best
characterized as nonstationary series. Nelson and Plosser (1982) found
interest rates to be perhaps nonstationary as did Brick and Thompson
(1978) and Cargill (1975). Moreover, Granger (1966) in his survey of the
spectral shapes of economic time series found that many typically have
very large power at low frequencies implying that good models for what is
usually called trend are important components in any characterization of
these time series. In concentrating upon interest rates and their
typical spectral shape, Granger and Rees (1968) concluded that low
frequency components were particularly powerful in the spectra of the

time series they examined. In response to the Leroy and Porter and the


-10-

and Joyeux (1980) was extensively developed by Geweke and Porter-Hudak
(1983) and we use it below to estimate our long memory models.
By taking logarithms of both sides of equation (4) and by

adding log £ (0) to both sides Geweke and Porter—Hudak obtained

log £(A) = Log {07 £,(0)/2n} - 4 log {4sin?(r/2)} (10)

+ log {2,00/£,}.

The log periodgram ( log I(A) ) evaluated at the sample's harmonic

frequencies can be added to both sides of (10) to obtain

tog {1(A)} = log {o? £,(0)/2m} - 4 log {4sin2(,/2)} (1)

+ log {£,(A)/£,(0)} + log {1(A)/£(A)}.

The theory for estimating d was developed by noting that log {1(A)/£0) }
is independently distributed across the harmonic frequencies and that,
for the first n harmonic frequencies such that n satisfies a function, g,

of sample size T so that lim g(T)/T = 0 and lim (log 1)?/g(T) = 0,
Te To

log {£,0)/£,(0)} is small compared to log {02 £ (0) /2m} in (11). In
these circumstances use of (11) as a regression equation will give
consistent estimates of d (for d < 0) without account being taken of the
other ARMA parameters. “Moreover, the theoretical error's variance for
this regression is known and is 1/6. Simulation studies by Geweke and
Porter-Hudak suggest that these theoretical results extend to cases in

which 0 < d < 1/2.


-12-

varied between .70 and .87. Of course many other trial regressions
were performed, but as will be shown, the combination of results with
respect to parameter significance and the forecast worthiness of the
long memory model suggest the best values for fractional estimated d's
belong to models with d precisely in the range of .70 to .87.

In experiments with the data prefiltered by (i-B)4, -70<d<.87,
several preliminary ARMA models were identified by inspection of the
filtered data's estimated autocorrelation and partial autocorrelation
functions. What emerged from these experiments was a multiplicative,
seasonal autoregressive model of the form

2 3 4 26 52 d
(1 = GB - $yB°~ 64BP- 4B") — by6B°CL - 59B>-)(1-B) Sr, =. (12)

Since d is unknown and has to be estimated, standard errors of
regression and significance of autoregressive parameters in (12) are
difficult to judge. The approach we take in this paper is to simulate
the sampling distributions for d and other parameters using the
nonparametric bootstrapping methods described by Freedman and Peters
(1984) and to study the distributions of forecast errors using the same
methods described by Peters and Freedman (1984). In the next section it
will also be shown that there is no other practical way to get at
confidence intervals for some crucial test statistics without resort to
these Monte Carlo methods. The bootstrapping methods are appropriate for
simulating confidence intervals in ordinary least squares situations such
as in the estimation of d and for models with simple dynamics such as the

autoregressions in equation (12).


Long Memory Models of Interest Rates,
the Term Structure, and Variance Bounds Tests
by

*
Gary S. Shea

I. Introduction

Equilibrium models of the interest rate term structure under
linear rational expectations are rejected by a number of researchers upon
the basis of empirical evidence that long-term interest rates and holding
period yields are too volatile relative to short-term interest rates.
Prominent among those who believe so are Shiller (1979) and Singleton
(1989). Their views echo the original arguments of Culbertson (1957)
that interest rates are not smooth enough to coincide with the averaging
of expected future short-term rates which are equal to long-term interest
rates under the linear rational expectations hypothesis. Without resort
to a term premium or without direct appeal to risk aversion (Michener
1982), it has been difficult to reconcile observed violations of variance
bounds with linear rational expectations models of asset pricing.
Empirical support for these views is also found in papers on excess
volatility in the stock markets by Leroy and Porter (1981) and Shiller
and Grossman (1981).

The empirical methodologies used in these studies have been
questioned, however, and it is becoming clear that a firm empirical
rejection of the rational expectations hypothesis is not yet at hand.

The primary reason for this is the questionable assumption commonly

found in the excess volatiltity literature that asset returns are mean


TABLE 4 (cont'd.)

Model VI Model VIT Model VIIT
of 32 a2 32 o2 32
Bootstrap Bootstrap

a Estimate = Mean sD 2 Estimate Mean sD a Batimace

1 -063 +064 +005 1 063 +063 +005 1 +065 +065 005
2 2219 219 +03 2 2213 213 +03 2 +198 +195 +03
3 450 +450 +09 3 2427 +432 +09 3 37% +369 .08
4 +800 +803 +22 4 2787 762 21 4 -616 = $06 7
5 Lan 1.13 +34 3 1.03 1.06 33 5 +835 +825 25
6 1.47 1.50 +50 5 1.35 1.41 49 6 1.08 1.07 36
7 1.36 1.90 69 7 1.70 1.78 67 7 1.33 1.33 48
8 2.31 2.38 +92 8 2.09 2.22 -90 8 1.62 1.62 163
9 2.79 2.89 1.18 9 2.52 2.70 1.16 9 1.91 1,93 -80
10 3.32 3.46 1.49 10 2.98 3.22 1.86 10 2.23 2.26 +99
uu 3.88 4.07 1.86 u 3.47 3.77 1.80 u 2.56 2.61 1.20

Model x Model XI
o2 62 o2 62
Bootstrap Bootst raj .

a Estimate Mea sD n Estimate Mean sD

1 06% 063 +005 1 +063 062 -005

2 -072 -072 +009 2 -073 +270 -02

3 128 127 -02 3 -611 +603 205

4 232 +230 +03 4 1.06 1.02 -09

5 340 -336 03 5 1.43 1.41 3

5 485 “479 .05 6 1.80 1.77 at

7 +666 +855 06 7 2.26 2.21 th

8 -880 864 .08 8 2.79 2.75 31

9 1.12 1.09 +10 9 3:62 3.37 239

10 1.37 1.36 “13 10 4 4.06 48

u 1.66 L6l - 17 u 4.85 4.78 58


~19-

n

~ ~* ~ . .

o y SQRT] = We where o , is the standard devia~
€ i=0 €

tion of the empirical distribution of estimated bootstrap errors, 2, and

7k ee sd ed
where each ve is properly constructed fromd , 4) o> $3, $y, and 950°
As in the preceding tables, the mean and standard deviations for these
empirical bootstrap sampling distributions are presented.

To contrast with the parameter-based standard error of forecast
estimates, calculated root mean square forecast errors are also presented
in both in-sample and out-of-sample settings. Since all forecast models
use 56-order autoregressive projecting filters, in all instances
presented in Table 3 there are T-56 = 514-56 = 458 in-sample forecasts
and 200 out-of-sample forecasts. The fifth and eighth columns ir
Table 3, which are simply labeled ‘estimate' refer to the sample root
mean square forecast error for the estimated model. The bootstrap
calculations demonstrate how well the models forecast data which are
generated by the estimated model or, in other words, how well an
estimated model forecasts its own pseudo-series, re

The estimated standard error of forecast and the in-sample root
mean square forecast error are both close to their bootstrapped
counterparts in most cases. There possibly are departures from this
pattern for models I through V at the longer forecast horizons where some
root mean square forecast errors are just in the rightmost 5 percent

tail of the bootstrap sampling distribution. With respect to out-of-

sample performance there is the notable feature that all figures in


Abstract

Variance bounds tests of the rational expectations hypothesis
of the interest rate term structure are sensitive to the stochastic
characterization of short-term interest rates used. When a long memory
or fractional difference nonstationary time series model is used in
preference to a mean stationary model, the rational expectations
hypothesis is not rejected. Long memory models of interest rates are
estimated and tested against alternatives. Their forecasting properties
are also examined. Hypothesis tests are based upon bootstrapping (Monte

Carlo) methodologies.


-25-

parameters of the short-term interest rate process. From equation (18)

we have

(n) n-l @ n-l @
AM = EOE we rot Woe.
fsa jar Fe thi k=0 jao Ste tj
o n-1
= Z£ pie ,- £ pe (20)
jar FL ti ycg Rt
n-1
= = = 6 6 i
Te + e en vy, Teo + net » where a is

the sum of the moving average coefficients of the model from index 0
through index n-1. Equation (20) points out that i) an expected holding
period yield in the next period is the current short rate and ii) the
expected variability of holding period yields relative to short-term
interest rates is a function of the bond's term to maturity and the
short-term interest rate model's parameters. In other words,

BCR - 2)? = a (21)

An analytic study of 6 = £ w_ as a function of d is
n k=0 k
potentially interesting, but is the proper object for another paper or
for a later version of this one. It is possible to write down succinct

expressions for 6 and 5 squared and their derivatives with respect to d.

In that way an investigator can search for qualitatively changing
behavior in & » say, as d approaches the stationarity boundary of .5.

Here it will suffice to show that oon is, in general, a complex


BIBLIOGRAPHY

Box, G.E.P. and Jenkins, G.M. Time Series Analysis: Forecasting and
Control. Revised Edition. San Francisco: Holden-Day, 1976.

Brick, R. and Thompson, H.E. "Time Series Analysis of Interest Rates:
Some Additional Evidence," Journal of Finance 33 (March 1978):
93-103.

Cargill, T.F. "The Term Structure of Interest Rates: A Test of the
Expectations Hypothesis." Journal of Finance 30 (June 1975):
761-771.

Culbertson, J.M. "The Term Structure of Interest Rates," Quarterly
Journal of Economics 71 (November 1957): 485-517.

Flavin, M.A. "Excess Volatility in the Financial Markets: A Reassessment
of the Empirical Evidence." Journal of Political Economy 91
(December 1983): 929-956.

Freedman, D.A. and Peters, S.C. "Bootstrapping a Regression Equation:
Some Empirical Results." Journal of the American Statistical
Association 79 (march 1984): 97-106.

Geweke, J. and Porter-Hudak, S$. "The Estimation and Application of Long
Memory Time Series Models." Journal of Time Series Analysis 4
(1983): 221-238,

Granger, C.W.J. "Some Properties of Times Series Data and their Use in
Econometric Model Specification." Journal of Econometrics 16
(May 1981): 121-130.

Granger, C.W.J. "The Typical Spectral Shape of an Economic Variable."
Econometrica 34 (January 1966): 150-161.

Granger, C.W.J. and Joyeux, R. "An Introduction to Long-Memory Time
Series Models and Fractional Differencing." Journal of Time
Series Analysis 1 (1980): 15-39.

Granger, C.W.J. and Rees, H.J.B. "Spectral Analysis of the Term
Structure of Interest Rates," Review of Economic Studies 35
(January 1968): 67-76.

Leroy, S.F. and Porter, R.D. "The Present Value Relation: Tests Based
Upon Implied Variance Bounds," Econometrica 49 (May 1981):
555-74.

Mandelbrot, B.B. "Statistical Methodology for Nonperiodic Cycles: From
the Covariance to R/S Analysis." Annals of Economic and Social
Measurement 1 (July 1972): 259-290.



-21-

is indeed the case as can be confirmed by comparing column (3) with
column (6) or column (9), Although models having d and a “generally
greater than .8 appear to have sample and bootstrapped root mean square
forecasting errors that are very similar, they are models which really do
not forecast very well based upon their own assumptions. Again,
comparisons of column (3) with column (6) or (9) demonstrate this.

With respect to the benchmark models, it can immediately be
noted that the long autoregression, denoted AR(56) in Table 3, is
inferior to the long memory models, particularly in its out-of-sample
performance. Not much better is model XI which presumes a true value of
d equal to 1. Finally, the model with the least variable accuracy in
forecasting we would expect to be the model with the smallest and fixed
value of d. In Table 3 this is model X in which d is constrained to be
+5. Model X has forecasting characteristics comparable to the other
fitted long memory models even though there does not seem to be any other
statistical evidence to favor a value for d as low as .5. Nor is there a
case to be made for even lower values of d improving forecasting
performance. In experiments with conventional stationary ARMA models
(d=0) forecasting performance was on a par or worse than the performance
of the AR(56) model. Other stationary long memory models (0<d<.5)
displayed poorer performance than their nonstationary brothers.

In the balance of the estimation and forecasting results there
are several reasons for preferring long memory nonstationary models as
stochastic characterizations of interest rates. We have seen that

hypothesis testing based upon bootstrapped confidence intervals around


AUTOCORRELATIONS OF ONE-—WEEK YIELDS

1.04

NZOKAYSOMARDOOOACyS
oO
wo
1

0.34

T T T T T T T T T T T T

re) 10 20 30 40 50 60 70 80 90 100 110
LAGS

Figure 3


-11-

This method of estimation for d has been applied to the series
of short-term interest rates from the Appendix. We have seen that the
series clearly’ is a candidate for nonstationarity and, from our
differencing experiment, that d probably is between .5 and 1. Prior to
estimation the series was first put through the filter (a-By*> so that
the filtered series has bounded variance. The differencing method was
that suggested by Geweke and Porter-Hudak (1983) and consists of applying
the discrete Fourier transformation to the interest rates and then
applying the difference filter (1 - ety, which is the analog of
qa - B)"> in the frequency domain. The inverse Fourier transformation
will then give back the fractionally differenced time series of interest
rates.

Following a suggestion by Geweke and Porter-Hudak, g(T) was
chosen to be T% where a varied between .5 and .7. The suggested
functional form for g certainly satisfies the necessary conditions for
consistent estimates of d. Values of a between .5 and .7 were found by
the authors to give the most stable estimates and also the more reliable
estimates of d based upon Monte Carlo simlations. Certainly the
specification of g(T) and its parameters are judgmental exercises at the
current stage of development for this estimation theory, but our
experiences with g(T) experimentation seemed to have paralleled those of
Geweke and Porter-Hudak and we have accepted their guidelines as reliable
for the investigations undertaken here.

With a sample size of T = 714 (11) was used as a regression

equation for a between .64 and .75. In these instances estimated d