-12-

for maximizing the probabilities of intervals around actual realizations
(see (1)). Therefore, predictors should satisfy condition (2). If predictors
satisfying the criterion of highest concentration do not generally exist,
then we should satisfy condition (2)(minimum mean square error) as closely
as possible.

How do we nearly satisfy condition (2)? As explained by Chipman
(1976, pp. 612-613), the conditional expectation of the random variable
Thee given y will have the minimum mean square error about Wete within a wide
class of functions of y. If we restrict this class to linear functions or if
we assume that Whe and y* are jointly normally distributed random variables
satisfying equations (6) and (4) respectively, then the conditional expectation
of whe given y is the same as the predictor (8) with 7 replaced by 7 (see
Chipman (1976, pp. 603-604)). Thus, the conditional expectation of hte given
y nearly satisfies condition (2), and it can be evaluated fairly accurately,
provided model (6) and assumptions (5), (7a)-(7c) are true.? Another difficulty
is that if we believe a priori that model (6) and assumptions (5), (7a)-(7c)
are true, then the conditional mean of whe given y involves 1, w and o2y
which are usually unknown. The result (12) shows that any procedure cof
estimating these unknown quantities that attempts to reduce the variance of

an estimator of 1 does not necessarily lead to better predictors of Yay. .
‘Ss

Minimum Mean Square Error Predictors

An appropriate way to overcome this difficulty is to estimate models

using a smaller mean square error criterion. We can consider two cases:

9 Unfortunately, it is not possible to establish the truth of any logically
valid model (see Swamy, Conway and von zur Muehlen (1985)).


-3-

even reduce, forecast accuracy. Note that this is a theoretical result about
the forecasting properties of “rolling” regressions and it simply states that
there is no reason to believe that “rolling” improves forecast accuracy. The
logic of this argument extends to more general cases, where two or more
predictors are compared for forecast accuracy using the same sample.

Section 2 introduces and briefly discusses two forecast criteria
that compare forecasts with realizations rather than the means of predictive
distribut:ions. We then discuss conventional estimators satisfying certain
sampling properties in Section 3. Operationally, some versions of these
estimators may not have finite variances. Using the realization-based crite-
rion, Section 4 shows that the best predictor must have minimum mean square
error, and that there is no reason why the method of sequential estimation

achieves this. Conclusions are presented in Section 5.

2. Forecasting Criteria Based on Realizations

Suppose that we are interested in predicting the value that would
be taken by the random variable yt in a future period T+s, where T is the
terminal period of the currently available sample observations on yte4

We now define two criteria for comparing predictors.

Criterion of Highest Concentration

The criterion of highest concentration compares predictions with actual

realizations and is defined as follows: an operational predictor Vn of the
‘Ss

4 We distinguish a random variable from its value by an asterisk. For
example, y, is the value taken by the random variable y, in period t.


-18-

International Finance Discussion Papers

IFDP
NUMBER TITLES AUTHOR(s)
1986

287 Should Fixed Coefficients be Reestimated P.A.V.B. Swamy
Every Period for Extrapolation? Garry J. Schinasi

286 An Empirical Analysis of Policy Hali J. Edison
Coordination in the U.S., Japan and Ralph Tryor,
Europe

285, Comovements in Aggregate and Relative B. Dianne Fauls
Prices: Some Evidence on Neutrality

284 Labor Market Rigidities and Unemployment: Michael K. Gavin
the Case of Severance Costs

283 A Framework for Analyzing the Process Allen B. Frankel
of Financial Innovation Catherine L. Mann

282 Is the ECU an Optimal Currency Basket? Hali J. Edison

281 Are Foreign Exchange Forecasts Rational? Kathryn M. Dominguez
New Evidence from Survey Data

280 Taxation of Capital Gains on Foreign Garry J. Schinasi
Exchange Transactions and the Non-neutrality
of Changes in Anticipated Inflation

279 The Prospect of a Depreciating Dollar Jacques Melitz
and Possible Tension Inside the EMS

278 The Stock Market and Exchange Rate Dynamics Michael K. Gavin

277 Can Debtor Countries Service Their Debts? Jaime Marquez
Income and Price Elasticities for Exports Caryl McNeilly
of Developing Countries

276 Post-simulation Analysis of Monte Carlo Neil R. Ericsson
Experiments: Interpreting Pesaran's (1974)
Study of Non-nested Hypothesis Test Statistics

275 A Method for Solving Systems of First Order Robert A. Johnson

Linear Homogeneous Differential Equations
When the Elements of the Forcing Vector
are Modelled as Step Functions

Please address requests for copies to International Finance Discussion
Papers, Division of International Finance, Stop 24, Board of Governors of
the Federal Reserve System, Washington, D.C. 20551.


A Minimum Variance Predictor

It has been shown by Goldberger (1962) that the minimum variance

linear “unbiased” predictor of y is6
T+s

Yous 7 “het + wv7l(y® - xm) (8)

o
if 7 = (xtvrlxyalxryclys is the minimum variance linear unbiased estimator of

1/7

It is also shown by Goldberger (1962) that

m-y* 2 (9)

EG -y* )2« ER Tts

Tts Tts

with the equality holding when w = 0.

A difficulty with the minimum variance predictor (8) is that w and o2v
are usually unknown. The inequality (9) may be reversed if Yt+s represents
an operational predictor of the form

Xpygt t wv liye - xa) , (10)
oO

based on some sample estimates of wy 02, and V, and 7 represents an operational
estimator

(xtVl xy “lxty-lys ql)

6 A predictor y is said to be “unbiased” if E y =Ey* ,
T+s Tts Tts

7 The minimum variance linear unbiased predictor of an element of a vector
variable following a vector autoregressive (VAR) model also has the seme
form. For example, if equation (4) is an autoregressive model imbedded in a
VAR model, then Ds includes the lagged values of Yong? w represents

the contemporaneous covariances between the dependent variable of equation (4)
and the other dependent variables included in the VAR model, o“V represents
the contemporaneous covariance matrix of these other variables, and y* and X
consist of the current and lagged values of the other variables respectively.


-5-

With these assumptions, Peddada (1985) has proved that if

(Gi) EL (y .< ELL, for i = 1,2; 3,
tras? Tos 1946? Tops? FOF (a)
(41) ey = Ee, = EL,(¥ - EL,(Y.
i i 1046? Yois? 1% n46? Toes)
< -2.67 for i = 1,2; and (3b)
(4441) E(eq-ey)5 < 9! for i = 1,2; j =1,2,..., (3e)

where

e, =L(y_ >
i 1 Fn 4, y, )

»- LY,
Tts itis’ "THs

then is closer to y than ¥ in the PN sense.
Tts Tts Tts

3. Fixed Coefficients Approaches

With these desirable properties of predictors in hand, we next
discuss conventional predictors and their statistical properties. The first
step in any statistical method of generating predictions is to formulate a
statistical model about the possible data generating process. It is usually
postulated that the observations on ye are generated by the following reduced
form equation with fixed coefficients:

y* = Xn + e%, (4)
where

y* = the (Txl) vector of variables ve t = 1,2,...T3

X = the (TxK) matrix of sample observations on K "fixed" exogenous
variables;

w = the (Kxl) vector of reduced-form coefficients; and

e* = the (Txl) vector of reduced-form disturbances.


-4-

actual value y is better than any other operational predictor ¥ Lf
Tts T+s

the probabilities satisfy the condition

Pr(¢ ~rAV<
tes “1S rts

< +o) > Pr¢ --\< yO < + A») qd)
aes) “2? 2 OTS ATS yg S Toye? %2

Tt:
for all possible values of ] and Az in a chosen interval (0, A) and
for all possible realizations vn .

+s
A necessary condition that (1) be satisfied for all A is

eo. 2 y - 2 2
Oey Yrs SEG, y fy (2)

Ss +s Tts

that is, the mean square error of Votes about the actual realization Yous isa
minimum (see Rao (1973, p. 315)). This condition indicates, for example, when
a predictor does not satisfy the so-called criterion of highest concentration
in (1).

Now sufficient conditions will be given for a criterion weaker than

(1) to be satisfied. This weaker criterion is based on the concept of Pitman's

nearness (PN) which is defined as follows:

Pitman's Nearness (PN)

A predictor y is nearer to the value y than another predictor ¥.
Tts TH T+

8
if Pr[L(y >y )<LY yy -)) > 1/2, where L(V >» Y_ ) represents a
Tts Tts Tts Tts Tts Tt+s

loss in predictin, by y_. We may consider two standard loss functions
P s Yors y Yrs y >

namely,

iy daly - and Ly(y =(y - 2.
Woes? "rs Yas Ynes| 206? Yas * Sng Yous)

s


IFDP

NUMBER

27

273

272

271

270

269

268

267

266

265

264

263

-19-

International Finance Discussion Papers

TITLES

International Comparisons of Fiscal Policy:
The OECD and the IMF Measures of Fiscal
Impulse

An Analysis of the Welfare Implications of
Alternative Exchange Rate Regimes: An
Intertemporal Model with an Application

1985
(partial listing)

Expected Fiscal Policy and the Recession
of 1982

Elections and Macroeconomic Policy Cycles

Assertion Without Empirical Basis: An
Econometric Appraisal of Monetary Trends
in... the United Kingdom by Milton Friedman
and Anna J. Schwartz

Canadian Financial Markets:
Proposal for Reform

Was It Real? The Exchange Rate Interest
Differential Relation, 1973-1984

The U.K. Sector of the Federal Reserve's
Multicountry Model: The Effects of
Monetary and Fiscal Policies

Optimal Currency Basket in a World of
Generalized Floating: An Application to
the Nordic Countries

Money Demand in Open Economies:
Substitution Model for Venezuela

A Currency

Comparing Costs of Note Issuance Facilities
and Eurocredits

Some Implications of the President's Tax
Proposals for U.S. Banks with Claims on
Developing Countries

Monetary Stabilization Policy in an Open
Economy

Anticipatory Capital Flows and the Behaviour
of the Dollar

The Government's

AUTHOR(s)

Garry Schinasi

Andrew Feltenstein
David Lebow
Anne Sibert.

William H. Branson
Arminio Fraga
Robert A. Johnson

Kenneth Rogoff
Anne Sibert

David F. Hendry
Neil R. Ericsson

Garry J. Schinasi
Richard Meese
Kenneth Rogoff

Hali J. Edison

Hali J. Edison
Erling Vardal

Jaime Mar quez
Rodney H. Mills

Allen B. Frankel

Marcus H. Miller

Arnold Kling


-7-

for prediction is the assumption that the actual value of the regressand for
period T+s will be a drawing from a distribution with mean gt and
variance 02.

The second step in any prediction method is to estimate m. The

vector TM can be estimated by one or more of the following procedures:

Estimation Procedures

(1) The least squares procedure;

(2) The generalized least Squares procedure based on an estimated
error covariance matrix;

(3) A fully or partially restricted reduced form procedure that
fully or partially accounts for the connection between 1 and the
coefficients of a structural model;

(4) A Bayes procedure;

(5) Bayes-like procedures; and

(6) Robust procedures.

Let 7 denote any one of these estimators.

The final step is to use an estimated m in predicting the value Jos’
Even though any or all of the estimation procedures (1)-(6) can be considered
for sequential estimation, to save some computations we may, if possible,
want to first find the most efficient of all the above six estimators and then
use it in a sequential estimation. Furthermore, all the criteria of estimation

used by the above estimation procedures may not be compatible with the criterion

(1) or Pitman's nearness, as we show in the following section.


International Finance Discussion Papers
Number 287

August 1986

SHOULD FIXED COEFFICIENTS BE REESTIMATED
EVERY PERIOD FOR EXTRAPOLATION?

by

P.A.V.B. Swamy and Garry J. Schinasi

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 that the writer has had access to unpublished material)
should be cleared with the author or authors.


-10-

Asymptotically Efficient Predictors

If the estimation period is sufficiently long, asymptotic theory
may apply and predictors (10) and (8) may have the same limiting dist:ributions.
This can occur even when 7 represents a partially or fully restricted reduced-
form estimator. It is possible that the asymptotic variance of the predictor
(10) is smaller if # represents a robust estimator than if 7 represents the
estimator (11) or a restricted reduced-form estimator. The comparisons of
asymptotic variances are relevant for moderate sample sizes if the estimators
of TM are convergent in the rth mean. Such estimators are developed in Swamy,
Mehta and Iyengar (1983).

On the basis of either the exact finite sample distributior theory
or the asymptotic theory, a universally preferred choice among the estimation

procedures (1)-(6) is not possible.

4. Does Sequential Estimation Necessarily Improve Forecasting Accuracy?

Suppose that one of the estimation procedures (1)-(6) is used in
sequential estimation and the corresponding predictors of Yor are computed,
and further assume that the corresponding predictors possess finite variances.
Suppose also that these variances decrease as the estimation period increases.
What is the correct interpretation of the predictors of Yors obtained in this
sequential estimation procedure?

We can answer this question by using the following standard result

in probability theory. Let Top and Voy. be two operational predictors
Ss

‘Ss
of y with finite means », and uw, finite variances o2 and o, and
Tts 1 2 1 2

distribution functions Fj and F9 respectively. Suppose that both Yn and
+3

¥ are based on the same formula but y uses a bigger sample than
Tts Tts


-9-

or a restricted estimator based on vy, provided the second-order moments of

(10) and @ are finite (see Rao (1967, 1975)).8 In other words, an operational
version of (8) may not have the minimum variance (about ne and may even have
infinite variance. The estimator (11) may also have infinite variances. The
predictors with infinite variances violate condition (2). They may not even
be good in the PN sense.

The same difficulty may arise if 7 represents a partially or fully
restricted reduced-form estimator. The conditions for the finiteness of the
moments of partially restricted reduced-form estimators in the normal case
are given in Swamy and Mehta (1981, 1982), and Swamy, Mehta and Lyengar
(1983). If the predictors of Yous based on the estimation procedure (3) need
satisfy conditions (3a)-(3c), then Swamy and Mehta's modifications, that
guarantee the existence of moments of all orders in small samples, are necessary.
Using these modifications, whether the predictor (10) has a smaller variance
about Ys than the predictor Xe” depends on how far Ex is from m and
on the precision of the estimators w and v-l(1/62),

A result due to Rao (1967) shows that for a matrix Z of maximum
rank such that X'Z = 0, the estimator (11) will have bigger variances than
the least squares estimator (X'X) “lx-y* if and only if X'VZ is sufficiently
close to a null matrix. In this case the predictor Xp yg (XIX) 1K tys may
have a smaller variance than the predictor (10) or the predictor tg
Thus, it is not possible in practice to recognize an operational predictor

with minimum variance in small samples.

8 this comment does not apply if we assume that w=0 because in this case the
second term in (10) will be zero.


It is usually assumed that
E(e*|X) = E(e*) = 0 and E(e*e*'|X) = o2v. (5)

If model (4) holds for the post-sample periods t = Tts, s = 1,2.....,
also, then for period T+s the variable Yo. given the vector of regressors

will be

* ° *
Ytt+s = *ttsTM + ET,” (6)
where

*
YTts = the scalar regressand;

the (1xK) vector of prediction regressors; and

*T+s

ets the scalar prediction disturbance.

Goldberger (1962) points out that for model (4), an appealing set of assumptions

is one which allows the prediction disturbance to be correlated with the

sample disturbances. Therefore, we shall assume

ECetes lets) = E(etys) = 0, (Ja)
E(e*h,5|xp4,) = 0%, and (7b)
Res e* lx, xg) = 4 (Je)

where w is the Txl vector of covariances of the prediction disturbance with

the vector of sample disturbances.” Thus, implicit in the use of model (4)

5 1£ equation (1) represents an autoregressive model, then X consists of the
lagged values of y and w can be equal to 0. Alternatively, the vector w can

be zero if equation (1) represents a regression model with serially uncorrelated
error term.


-2-

of estimating constant coefficients does not necessarily yield accurate
forecasts. A by-product of the demonstration is that a predictor with a
smaller variance need not be better than another predictor with a larger
variance, even though both predictors are based on the same sample.2

The logic of the basic argument of this paper is as follows. If
the objective of estimation is forecast accuracy, then one should prefer
predictions that are close to actual realizations to predictions that are
close to some other quantities (such as the means of predictors). Consequently,
one should select the predictor that has the highest probability of being
close to actual realizations. Although it is difficult to derive estimators
based on this general criterion, a necesary condition for this probability to
be a maximum is for the mean square error of a predictor (i.e., the expected
squared deviation of a predictor from realization) to be a minimum. “Rolling”
(i.e., sequential estimation) may minimize the forecast variance for broad
classes of estimators, but minimizing variance does not necessarily minimize
mean square error.> Hence, for any given estimator, “rolling” may reduce

the variance of a predictor, but it does not necessarily improve, and may

2 The constant coefficient approach is inherently incorrect if some or all
of the regression slopes change over time, sample enlargement notwithstanding.

3 In practice, many of these optimal predictors involve unknown parameters and
their operational versions based on some sample estimators of the parameters
may not satisfy all the conditions for achieving minimum variance. This
makes it difficult to recognize an operational predictor with minimum variance
in small samples. Because of this difficulty, attention has been shifted to
the asymptotically optimal predictors which have the smallest asymptotic
variance within the class of asymptotically normal predictors. This does not
eliminate the difficulty because some asymptotically optimal predictors do
not possess finite variances in small samples and hence violate a necessary
condition for maximizing the probability of obtaining forecasts within an
interval around actual realizations.


-13-

normal and non-normal distributions of the errors. For the case in which a
multivariate normal mean under quadratic loss is to be estimated, a paper by
Natarajan and Strawderman (1985) establishes the existence of two-stage sequen-
tial es:imators that are better, both in risk (mean square error) and sample
size, than the usual estimator of a given fixed sample size. When e€ in
equation (1) is normal, the problem of estimating 1 under quadratic loss
becomes identical to the case discussed by Natarajan and Strawderman (1985).
Given any sample size T, we can find a two-stage sequential estimator of 1
truncated at T, with a positive probability of stopping earlier and mean

square error lower than that of an estimator of TM based on all the T observa-
tions. The predictor of Yous based on a Natarajan-Strawderman-like estimator
of TM cana have smaller mean square error than any of the predictors of Yo

+s
based o1 an estimator of TM that uses all the observations available up to T or

Tts-1.

Where normality is not appropriate, the advantages of a median or a
trimmed mean or a robust estimator relative to the mean are well-known (see
Lehmann (1983, pp. 352-388)). The predictors of Yate based on these estimators
of TM may have smaller mean square errors than predictors of y based on

Tts
other estimators of m that use all the sample observations. Thus, there is

no theoretical justification for reestimating models of the type (4) using

all past data prior to each forecast period either in the normal case or in a

nonnormal case.!0

10 If, in fact, the slope coefficients of equation (4) change over time, then it
is incorrect to stack the observations as

' t= [xt ' ' '

ly Ta na! [x apy! a+ [e Eayprreay | .

In other words, the estimators of fixed coefficients are inappropriate when

the slope coefficients vary over time. Appropriate estimators for a time-varying
parameters model are given in Swamy and Tinsley (1980) and two applications

in Resler, Barth, Swamy and Davis (1985) and Swamy, Kennickell and von zur
Muehlen (1986).


-17-

Swamy, P.A.V.B., A.B. Kennickell and P. von zur Muehlen, 1985, Forecasting
tioney demand with econometric models, Special Studies Paper, Federal

Reserve Board, Washington, D.C.


Should Fixed Coefficients be Reestimated Every Period
for Extrapolation?

P.A.V.B. Swamy and Garry J. Schinasi*

1. Introduction

In making forecasts of future variables, some econometricians
reestimate their models using all past data prior to each forecast period.
The process involves fixing the starting date and the initial size of the
sample and enlarging the sample by adding successive observations for reesti-
mation and prediction as new data become available (see, e.g., Fromm and
Klein (1976, p. 9) and Meese and Rogoff (1983, 1985)). It has been suggested
that such a procedure improves forecast accuracy for two reasons: first,
because a larger sample reduces the variances of fixed coefficient estimators;
and second, because sequential estimation or “rolling” captures any variation
in coeffiecients.

The theory underlying the common impression that one should always
use all the available observations in estimation and prediction is more.
ambiguous than may be generally realized.! Without proper theoretical
justification, like an explicit risk minimizing motivation (favoring good
predictions), a procedure that sequentially updates estimates of coefficients
and predictions in a model assumed to have constant coefficients is meaningless.

The primary purpose of the paper is to demonstrate that any sequential method

* Views expressed in this paper are those of the authors and do not reflect
the views of the Board of Governors or the staff of the Federal Reserve System.
The comments by N. Ericsson, J. Marquez, P. von zur Muehlen and P. Tinsley

are greatly appreciated.

1 This impression is also not fully supported by the asymptotic theory, some
simple normal cases apart. For example, Lehmann (1983, pp. 352-388) analyzes
various nonnormal situations showing that a parameter is more efficiently
estimatecl even in large samples by discarding some sample observations than
by using all the observations.


Abstract
Tnis paper demonstrates that forecast accuracy is not

necessarily improved when fixed coefficient models are sequentially
reestimated, and used for prediction, after updating the database with
the latest observation(s). This is at variance with the now popular
method (see Meese and Rogoff (1983, 1985)) of sequentially reestimating
fixed coefficient models for prediction as new data "rolls" in. It is
argued that although "rolling" may minimize the variance of predictions
for some classes of estimators, "rolling" does not necessarily yield
accurate predictions (i.e., predictions that are close to actual data).
Minimizing the mean squared prediction errors is a necessary condition
for maximizing the probability that a given predictor is more accurate
than cther predictors. This minimization need not require, and may even
excluce, the most recent data. A by-product of the demonstration is
that for predictors based on the same sample size, a predictor with

smaller variance need not be more accurate than another predictor with a

larger variance,


-ll-

Fats so that of < 03. Then according to a standard result in

probability theory,

Fily +uy) - FU + U1) > F2¢ + ug) - F2¢ + u2) 12
10a, 1 ey D2ROL 2 2 one 2 (12)

for each possible value Yous implies that ot <x 33. However, the converse
proposition is not true. The inequality ot <x 03 implies that inequality
(12) is true for at least one value Yors but not necessarily for all possible
values of ae (see Rao (1973, p. 96)). Thus, if we have two operational pre-
dictors with finite second-order moments, then it is not necessarily true that
the preclictor with a smaller variance will take values around the actual

value Yoas with a higher probability than the predictor with a larger variance.

These general arguments can now be applied to the specific choice
of two sets of predictions, based on the same predictor using different
samples, We can therefore conclude the following about "rolling:" if we
reestimate a model using all past data prior to each forecast period to
obtain one-step ahead forecasts to reduce the variance, then the one-step
ahead forecasts will not necessarily be closer to the realized values of the
forecasted variable than the multi-step ahead forecasts, even though the
former are based on more observations and hence may have smaller variances
than the latter.

More formally, comparing the inequality (12) with (1) shows that
the criterion of minimum variance ("unbiasedness") prediction only satisfies a
necessary condition for maximizing the probabilities of intervals around the
mean values of predictors (see (12)) while the criterion of minimum mean

square error about actual realizations satisfies a necessary condition


-16-

Rao, C.R., 1973, Linear statistical inference and its applications,
2nd ed. (Wiley, New York).

Rao, C.R., 1975, Simultaneous estimation of parameters in differeat
linear models and applications to Biometric problems, Biometrics 31,
545-554,

Resler, D.H., J.R. Barth, P.A.V.B. Swamy and W.D. Davis, 1985, Detecting
and estimating changing economic relationships: The case of discount
window borrowings, Applied Economics 17, 509-527.

Swamy, P.A.V.B. and J.S. Mehta, 1980, On the existence of moments of
partially restricted reduced-form coefficients, Journal of Econometrics
14, 183-194,

Swamy, P.A.V.B. and J.S. Mehta, 1981, On the existence of moments of
partially restricted reduced-form estimators: A comment, Journal of
Econometrics 17, 389-392.

Swamy, P.A.V.B. and J.S. Mehta, 1983, The existence of moments of ridge-
like k-class and partially restricted reduced-form estimators,
Communications in Statistics 11, 2793-2799.

Swamy, P.A.V.B., J.S. Mehta and N.S. Iyengar, 1983, Convergence of the
moments of the modified K-class estimators, Sankhya B 45, 398-412.
Swamy, P.A.V.B. and P.A. Tinsley, 1980, Linear prediction and estimation

methods for regression models with stationary stochastic coefficients,
Journal of Econometrics 12, 103-142.
Swamy, P.A.V.B., RK. Conway, P. von zur Muehlen, 1985, The foundations

of econometrics--Are there any? (with discussion), Econometric Reviews

4, 1-120.