International Journal of Forecasting 16 (2000) 117–119
www.elsevier.com / locate / ijforecast
Modeling variables of different frequencies
Tilak Abeysinghe*
Department of Economics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
Abstract
The transformation introduced in Abeysinghe (1998: International Journal of Forecasting 14, 505-513) to model dynamic
regressions with variables of different frequencies creates an autocorrelation problem when applied to flow variables. This
exercise shows that the magnitude of the autocorrelation is rather small and offers a solution to the problem.
 2000 Elsevier Science B.V. All rights reserved.
Keywords: Flow variables; Autocorrelation; IV estimator
1. Introduction
2. Flow dependent variable
The transformation proposed by Abeysinghe
(1998) to regress a low-frequency variable on
high-frequency variables works exactly for
stock dependent variables. This transformation
leads to an autocorrelation problem in flow
variables. The model fitted to flow variables by
Abeysinghe passes through the autocorrelation
tests because the autocorrelation present in the
residuals is too small to be detected by standard
tests. In this note we present a more natural
transformation that can be used for flow dependent variables and address the issue of autocorrelation and examine the problems associated
with estimation.
The basic model considered in Abeysinghe
(1998) is
y t 5 a0 1 b0 x t 1 l y t 2t 1 u t ,
t 5 t, 2t, 3t, . . . ,T,
where t is a fraction of the time interval at
which y is observed. If y is quarterly and x is
monthly then t 51 / 3. u t is assumed to be a
white noise process with zero mean and constant variance. When y t in (1) is a flow variable,
the recorded values of y are the period aggregates. By aggregating (1) over the range of
r 5 0, 1, . . . , 1 /t 2 1 we get
12t
Yt 5 a0 ]]
1 b0 Xt 1 lYt 2t 1 Ut ,
2t 2
t 5 t, 2t, . . . , T
*Tel.: 11-65-874-6116; fax: 11-65-775-2646.
E-mail address: tilakabey@nus.edu.sg (T. Abeysinghe)
(1)
(2)
where the upper case letters indicate the (moving) aggregates:
0169-2070 / 00 / $ – see front matter  2000 Elsevier Science B.V. All rights reserved.
PII: S0169-2070( 99 )00028-X
T. Abeysinghe / International Journal of Forecasting 16 (2000) 117 – 119
118
Oy ,
X 5 Ox , Y
5 Oy
and
U 5 Ou .
Yt 5
Ol ,
i
t2rt
t2rt
t
j
di 5
j50
t2t
and
t2(r11 )t
l j 11 ,
j5i 21 / t
S
D
1 1
1
if i 5 ], ] 1 1, . . . , 2 ] 2 1 .
t t
t
Note that Yt , Xt , and Ut constitute non-overlapping sums at integer lags whereas Yt2t involves the sums of overlapping periods. Model
(2) cannot be estimated because at integer lags
Yt2t is not observed. The transformation
Abeysinghe (1998) introduced to convert the
fractional lag (t 2 t ) to integer lag (t 2 1)
involves lagging (2) by rt and multiplying by
l r and summing over the range of r. This will
yield
From (4) we can see that the first 1 /t terms
of u (i 5 0, 1, . . . , 1 /t 2 1) belong to the
current period t (t 5 1, 2, . . . , T ) and the rest of
the (1 /t 2 1) terms belongs to the period (t 2
1). It can easily be verified that Vt is an MA(1)
process. To compute the first order autocorrelation coefficient r, let d 95[d0 , d1 , . . . , d2(1 / t 2 1 ) ],
d 91 5 [d0 , d1 , . . . , d1 / t 2 2 ], and d 92 5 [d1 / t ,
d1 / t 11 , . . . , d2(1 / t 21) ] be column vectors. Then it
can easily be verified that
2
9
E(Vt ) 5 0, Var(Vt ) 5 s 2ud 9d, E(VV
t t21 ) 5 s u d 1 d2
Yt 5 a 1 b0 Zt 1 l 1 / t Yt 21 1Vt ,
t 5 1, 2, . . . ,T,
(3)
with
E(VV
t t21 )
r 5 ]]].
Var(Vt )
where
OlX
1 / t 21
Zt 5
O
1 / t 22
di 5
t2rt
t
1
if i 5 0, 1, 2, . . . ,] 2 1,
t
r
OlU
1 / t 21
t 2rt ,
r50
Vt 5
r
t 2rt
In the case of aggregating monthly data to
quarterly data t 5 1 / 3 and
,
r50
r 5 l(1 1 l)2 /(3l 4 1 4l 3 1 5l 2 1 4l 1 3).
and
O
1 2 t 1 / t 21 r
a 5 ]]
a
l.
2t 2 0 r50
To see the autocorrelation in Vt we have to
express Vt in terms of lower case u t . Expanding
over r and after collecting common terms, Vt
can be expressed as
1
2( ]21 )
t
Vt 5
O du
i t 2it
i50
where
For 0 , l , 1, the range most likely for many
economic time series, 0 , r , 4 / 19.
The (non-linear) least squares bias caused by
the MA(1) errors can easily be assessed in the
following simple model that sets a0 5 b0 5 0
and t 5 1 / 3. The resulting model is
3
Yt 5 l Yt21 1Vt , t 5 1, 2, . . . , T, 0 , l , 1,
Vt 5 et 1 uet21
(4)
where et is zero-mean white noise process. Note
that r 5 u /(1 1 u 2 ) and by imposing the invertibility condition we get u 5 (1 2 (1 2 4r 2 )1 / 2 ) /
T. Abeysinghe / International Journal of Forecasting 16 (2000) 117 – 119
2r. By minimizing o V 2t and solving for l we
get
3
1/3
p lim lˆ 5 [ l 1 Cov(Yt 21 ,Vt ) /Var(Yt )]
3
6
3
2
1/3
5[ l 1 u (1 2 l ) /(2ul 1 u 1 1)]
± l.
A plot of the asymptotic autocorrelation bias
( p limlˆ 2 l) against l, 0 , l , 1, shows that
the bias increases first with l and reaches a
maximum of 0.23 when l ¯ 0.14 and then
declines steadily towards zero as l increases.
For larger values of l, the autocorrelation bias
becomes negligibly small.
Although the bias is small, especially for
large values of l, it has to be dealt with
explicitly as it leads to inconsistent LS estimators. In small samples, however, it would be
difficult to detect the presence of autocorrelation
because of its small magnitude. Fortunately, the
MA(1) parameter u is a function of l, though
highly non-linear. What we need is a ‘‘good’’
estimate of l which can then be used to
estimate u, which in turn will be useful in
obtaining improved short-term forecasts. A simple solution is to use an instrumental variable
(IV) estimator using Yt22 as an instrument.
We carried out a limited Monte-Carlo experiment to compare the performance of the LS and
IV estimators in small samples. This experiment
shows that the small-sample bias of both LS and
IV estimators is negligibly small. For l close to
unity the LS bias tends to be smaller than the IV
bias. The IV estimator tends to perform better
when the LS bias is large. To save space we do
not report all the results here. Overall, it seems
advisable to examine both the LS and IV
estimates and compare the results before making a choice.
119
3. Conclusion
Model (3) can easily be generalised as in
Abeysinghe (1998). Although it looks different
from the one that was estimated in Abeysinghe
(1998), the autocorrelation problem is exactly
the same in both cases. The difference lies in
the way the high-frequency variable is handled.
A re-estimation of the Singapore model given in
Abeysinghe (1998) shows that formulation (3)
leads to a more parsimonious model, yet with a
similar underlying distributed lag structure.
Acknowledgements
I would like to thank two anonymous referees
for their valuable comments and working
through my derivations and pointing out some
mistakes. My thanks are also due to Jan G. De
Gooijer, Editor-in-Chief, for further comments
on the revised version. The full paper can be
obtained from the author or be downloaded in
PDF format from http: / / courses.nus.edu.sg /
course / ecstabey / IJF4.PDF.
References
Abeysinghe, T. (1998). Forecasting Singapore’s quarterly
GDP with monthly external trade. International Journal
of Forecasting 14, 505–513.
Biography: Tilak ABEYSINGHE is an Associate Professor in econometrics in the Department of Economics of the
National University of Singapore. He is the Director of the
Econometric Studies Unit, the forecasting and policy
research arm of the Department of Economics.