E-mail address:stephen}[email protected] (D.S.G. Pollock).
Trend estimation and de-trending via rational
square-wave
"
lters
D.S.G. Pollock
!,"
!Queen Mary and Westxeld College, University of London, London, UK
"GREQAM: Groupement de Recherche en Economie Quantitative d+Aix-Marseille, France
Received 13 November 1997; received in revised form 1 December 1999; accepted 13 March 2000
Abstract
This paper gives an account of some techniques of linear"ltering which can be used for extracting trends from economic time series of limited duration and for generating de-trended series. A family of rational square-wave "lters is described which enable designated frequency ranges to be selected or rejected. Their use is advocated in prefer-ence to other "lters which are commonly used in quantitative economic analysis. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C22
Keywords: Signal extraction; Linear"ltering; Frequency-domain analysis; Trend estimation
1. Introduction
Recently, there has been a growing interest amongst economists in the techniques for detrending nonstationary times series and for representing their underlying trends. Much of this interest has been associated with the resurgence of business cycle analysis in the "eld of macroeconomics} see, for example, Hartley et al. (1998).
The technique which has been used predominantly in trend estimation and detrending is that of linear"ltering; and there are some unresolved problems. This paper proposes to tackle these problems by adopting some of the perspect-ives of electrical and audio-acoustic signal processing. The endeavour to adapt the signal processing techniques to the characteristics of economic data series, which are liable to be strongly trended and of a strictly limited duration, leads to some innovations which are presented in this paper.
Underlying the use of "ltering techniques in business cycle analysis is the notion that an economic time series can be represented as the sum of a set of statistically independent components each of which has its own characteristic spectral properties. If the frequency ranges of the components are completely disjoint, then it is possible to achieve a de"nitive separation of the time series into its components. If the frequency ranges of the components overlap, then it is still possible to achieve a tentative separation in which the various components take shares of the cyclical elements of the time series.
The trend of a time series is the component which comprises its noncyclical elements together with the cyclical elements of lowest frequency. If the trend is a well-de"ned entity, then it should be possible to specify a cut-o!frequency which delimits the range of its cyclical elements. The ideal"lter for isolating such a trend would possess a passband, which admits to the estimated trend all elements of frequencies less than the cut-o! value, and a stopband, which impedes all elements of frequencies in excess of that value.
On the other hand, if the frequency range of the trend overlaps substantially with the ranges of the adjacent components, then one might be inclined to adopt
a "lter which shows only a gradual transition from the passband to the
stopband. In that case, the nominal cut-o! point of the "lter becomes the mid-point of the transition band and the rate of transition becomes a feature of
the"lter which needs to be adjusted in view of the extent of the overlap.
The Hodrick}Prescott (H}P)"lter}see Cogley and Nason (1995), King and Rebelo (1993) and Harvey and Jaeger (1993)}which has been used widely by business-cycle analysts, possesses only a single adjustable parameter which a!ects both the rate of transition and the location of the nominal cut-o!point. This so-called smoothing parameter is liable to be"xed by rule of thumb or by convention. The H}P "lter is closely related to the Reinsch (1976) smoothing spline which is used extensively in industrial design.
Whereas the H}P"lter is an excellent device for representing the broad trend of a time series, it often fails in the more exacting task of generating a detrended series. In particular, it sometimes allows powerful low-frequency components to pass through into the detrended series when they ought to be impeded by the
"lter. This de"ciency is the basis of the oft-repeated, albeit inaccurate, claim that
other identi"able components of the economic time series } see Harvey and Todd (1983), Hillmer and Tiao (1982) and Koopman et al. (1995), for examples. Such an approach avoids the use of a rule of thumb in determining the characteristics of the"lter, but it imposes other features which may be undesir-able. In particular, the model-based approach postulates a trend which is the product of an integrated moving-average process of a low order which has no
"rm upper limit to its frequency range.
In this paper, our primary objective is to design a"lter with a well-de"ned cut-o!point and with a rapid transition. The evidence of periodogram analysis suggest that, in many econometric time series, the trend component is clearly segregated from the remaining components; and this indicates a need for a"lter which can e!ect a clear separation.
It transpires that the "lter in question, which ful"ls our design objectives, is an instance of the digital Butterworth "lter which is familiar to electrical engineers}see, for example, Roberts and Mullis (1987). This"lter is commonly regarded as the digital translation of an analogue design. In this light, it is interesting to discover that the"lter arises independently in the digital domain in the pursuit of some simple design objectives.
The"lter can also be derived by applying the Wiener}Kolmogorov theory of
signal extraction to a speci"c statistical model. (See Kolmogorov (1941) and Wiener (1950) for the original expositions of the theory.) We shall use a" nite-sample version of Wiener}Kolmogorov approach in adapting the"lter to cope with samples of limited duration. Our method of coping with the problem of nonstationarity will also be given a statistical justi"cation.
2. Rational square-wave5lters
The"lters which we shall consider in this paper are of the bidirectional variety
which are applied by passing forwards and backward through the data series. The same"lter is used in both directions. Thus, ify(t)"My
t; t"0,$1,$2,2N represents a discrete-time signal, then the two "ltering operations can be described by the equations
c(¸)q(t)"d(¸)y(t), (1a)
c(F)x(t)"d(F)q(t), (1b)
whereinq(t) is the intermediate output from the forwards pass of the"lter and x(t) is the "nal output which is generated in the backward pass. Here c(¸) andd(¸) stand for polynomials of the lag operator¸, whilstc(F) andd(F) have the inverse forwards-shift operatorF"¸~1in place of¸.
The"lter must ful"l the condition of stability which requires the roots of the
For convenience, the two "lters can be combined to form a symmetric two-sided rational"lter
t(¸)"d(F)d(¸)
c(F)c(¸), (2)
but, of necessity, this must be applied in the manner indicated by (1a) and (1b). The e!ect of a linear"lter is to modify a signal by altering the amplitudes of its cyclical components and by advancing and delaying them in time. These
modi"cations are described respectively as the gain e!ect and the phase e!ect;
and they are referred to jointly as the frequency response of the"lter.
The frequency response of the "lter is obtained by replacing the ¸ by the complex argumentz. The resulting complex-valued function can be expressed in polar form as
t(z)"Dt(z)Dexp[i ArgMt(z)N], (3)
where Dt(z)D"Jt(z)t(z~1) is the complex modulus of t(z), and
ArgMt(z)N"ArctanMt*.(z)/t3%(z)Nis the argument oft(z). The gain of the"lter, which is a function of u3[0,n], is obtained by setting z"e~*uin Dt(z)D. The phase lag of the"lter, which is also regarded as a function ofu, is obtained by settingz"e~*uin ArgMt(z)N.
The bidirectional rational "lter of Eq. (2) ful"ls the condition that t(z)"t(z~1). This is the condition of the so-called phase neutrality, and it implies thatt(z)"Dt(z)Dand that Arg(z)"0. In e!ect, the real-time phase lag which is induced by the forwards pass of the"lter represented by Eq. (1a) is exactly o!set by a reverse-time phase lag induced by the backwards pass of (1b). In the terminology of digital signal processing, an ideal frequency-selective
"lter is a phase-neutral square-wave "lter for which the gain is unity over
a certain range of frequencies, described as the passband, and zero over the remaining frequencies, which constitute the stopband. In a lowpass"ltertL, the passband covers a frequency interval [0,u
#) ranging from zero to a cut-o!point.
In the complementary highpass"ltertH, it is the stopband which stands on this interval. Thus,
An ideal"lter t(z) ful"lling one or other of these conditions together with the condition of phase neutrality constitutes a positive semi-de"nite idempotent function. The function is idempotent in the sense thatt(z)"t2(z). It is positive semi-de"nite in the sense that
where o(z) is any polynomial or power series in z. Here, the "nal equality depends upon the assumption of phase neutrality which, in combination with the condition of idempotency, implies that t(z)"t2(z)"t(z)t(z~1). When the locus of this contour integral is the unit circle, the expression stands for the integral of the gain of the composite"lter o(u)t(u). Observe that an equality holds on the LHS wheno(z)"1!t(z). In that case,o(u) represents an ideal
"lter which is the complement oft(u) and which nulli"es the output of the latter
which is nonzero only over its passband.
The objective in constructing a practical frequency-selective"lter, is to"nd a rational function, embodying a limited number of coe$cients, whose fre-quency response is a good approximation to the square wave. In fact, the idealised conditions of (4) are impossible to ful"l in practice. In particular, the stability condition a!ectingc(z), which implies thatt(z) must be a
positive-de"nite function, precludes the ful"lment of the conditions. Moreover,
improve-ments in the accuracy of the approximation, which are bound to be purchased at the cost of increasing the number of"lter coe$cients, are liable to exacerbate the problems of stability.
In this section, we shall derive a pair of complementary"lters which ful"l the speci"cations of (4) approximately for a cut-o!frequency ofu
#"n/2. Once we
have designed these prototype"lters, we shall be able to apply a transformation which shifts the cut-o!point fromu"n/2 to any other pointu
#3(0,n).
A preliminary step in designing a pair of complementary"lters is to draw up a list of speci"cations which can be ful"lled in practice. We shall be guided by the following conditions:
As we have already noted, a bidirectional rational"lter in the form of (2) ful"ls already the condition (i) of phase neutrality. Given the speci"cation of (2), the condition of complementarity under (ii) implies that the "lters must take the form of
Next, the condition of symmetry under (iii) implies that, when it is re#ected about the axis ofu"n/2, the frequency response of the lowpass"lter becomes the frequency response of the highpass"lter. This implies that the cut-o!point u
# must be located at the mid-point frequency of n/2. The condition requires
thatj"1 and thatc(z)"c(!z), which implies that every root ofc(z)"0 must be a purely imaginary number. The condition also requires that
dL(z)"dH(!z) and dH(z)"dL(!z), (8) which has immediate implications for Eq. (7).
It remains to ful"l the conditions (iv) and (v). Condition (iv) indicates that dH(z) must have a zero atz"1, which is to say that it must incorporate a factor in the form of (1!z)n. Observe that, if the"ltertHis to be used in reducing an ARIMA (p,d,q) process to stationarity, then it is necessary thatn*dwheredis the number of unit roots in the autoregressive operator. Condition (v) indicates thatdL(z) must have a zero atz"!1, which is to say that it must incorporate a factor in the form of (1#z)n. These conditions (iv) and (v) do not preclude the presence of further factors in dL(z) and dH(z). However, the conditions are
ful"lled completely by specifying that
dL(z)"(1#z)n and dH(z)"(1!z)n. (9)
These roots come in reciprocal pairs; and, once they are available, they may be assigned unequivocally to the factorsc(z) andc(z~1). Those roots which lie outside the unit circle belong toc(z) whilst their reciprocals, which lie inside the unit circle, belong toc(z~1).
Consider, therefore, the equation
(1#z)n(1#z~1)n#j(1!z)n(1!z~1)n"0, (12) which is equivalent to the equation
j#
A
i1#zThe roots correspond to a set of 2npoints which are equally spaced around the circumference of a circle of radiusj. The radii which join the points to the centre are separated by angles ofn/n; and the"rst of the radii makes an angle ofn/(2n) with the horizontal real axis.
The inverse of the functions"s(z) is the function
z"s!i
i#s"
(ssH!1)!i(s!sH)
(ssH#1)!i(sH!s), (16)
where the"nal expression, which has a real-valued denominator, comes from multiplying top and bottom of the second expression bysH!i"(i#s)H, where sHdenotes the conjugate of the complex numbers. The elements of this formula are
It is straightforward to determine the value ofjwhich will place the cut-o!of
the "lter at a chosen point u
Fig. 1. The pole}zero diagrams of the lowpass square-wave"lters forn"6 when the cut-o!is at u"n/2 (left) and atu"n/8 (right).
Fig. 2. The frequency responses of the prototype square-wave"lters withn"6 and with a cut-o!at u"n/2.
formula of (10), the lowpass "lter. This gives the following expression for the gain:
tL(e~*u)" 1
1#j(i(1!e~*u)/(1#e~*u))2n
" 1
1#jMtan(u/2)N2n. (18)
At the cut-o! point, the gain must equal 1/2, whence solving the equation
tL(expM!iu
#N)"1/2 givesj"M1/tan(u#/2)N2n.
Fig. 1 shows the disposition in the complex plane of the poles and zeros of the factordL(z~1)/cL(z~1) of the lowpass"ltert(z)Lforn"6 when the cut-o!point is atu"n/2 andn/8. The poles are marked by crosses and the zeros by circles.
Figs. 2 and 3 show the gain of the"lter for these two cases.
Fig. 3. The frequency responses of the square-wave"lters withn"6 and with a cut-o!atu"n/8.
frequency is shifted away from the mid-pointn/2, a similar e!ect ensues whereby the rate of of transition is increased and the poles are brought closer to the perimeter.
When the poles are close to the perimeter of the unit circle, there can be problems of stability in implementing the"lter. These can lead to the propaga-tion of numerical rounding errors and to the prolongapropaga-tion of the transient e!ects of ill-chosen start-up conditions. Problems of numerical instability can be handled, within limits, by increasing the numerical precision with which the
"lter coe$cients and the processed series are represented. The problems
asso-ciated with the start-up conditions can be largely overcome by adopting the techniques which are described in the following section.
3. Implementing the5lters
The classical signal-extraction"lters are intended to be applied to lengthy data sets which have been generated by stationary processes. The task of adapting them to limited samples from nonstationary processes often causes di$culties and perplexity. The problems arise from not knowing how to supply the initial conditions with which to start a recursive "ltering process. By choosing inappropriate starting values for the forward or the backward pass, one can generate the so-called transient e!ect which is liable, in fact, to a!ect all of the processed values.
One common approach to the problem of the start-up conditions relies upon the ability to extend the sample by forecasting and backcasting. The additional extra-sample values can be used in a run-up to the"ltering process wherein the
"lter is stabilised by providing it with a plausible history, if it is working in the
An alternative approach to the start-up problem is to estimate the requisite initial conditions. Some of the methods which follow this approach have been devised within the context of the Kalman"lter and the associated smoothing algorithms}see Ansley and Kohn (1985) and De Jong (1991), for example.
The approach which we shall adopt in this paper avoids the start-up problem by applying the"lter, in the"rst instance, to a version of the data sequence which has been reduced to stationarity by repeated di!erencing. In this case, neither the di!erenced version of the trend nor the di!erenced version of the residual sequence requires any start-up values other than the zeros which are the unconditional expectations of their elements.
We can proceed to"nd an estimate of the residual sequence by cumulating its di!erenced version. Here we can pro"t from some carefully estimated start-up values to set the process of cumulation in motion; but, in fact, we can avoid
"nding such values explicitly. Once the residual sequence has been estimated,
the estimates of the trend sequence, which is its compliment within the data sequence, can be found by subtraction.
To clarify these matters, let us begin by considering a speci"c model for which the square-wave "lter would represent the optimal device for extracting the trend, given a sample of in"nite length. The model is represented by the equation
y(t)"m(t)#g(t)
"(1#¸)n
(1!¸)dl(t)#(1!¸)n~de(t), (19)
where m(t) is the stochastic trend and g(t) is the residual sequence. These are driven, respectively, byl(t) ande(t) which are statistically independent sequences generated by normal white-noise processes with<Ml(t)N"p2l and<Me(t)N"p2e. Eq. (19) can be rewritten as
(1!¸)dy(t)"(1#¸)nl(t)#(1!¸)ne(t)
are stationary moving-average processes which are noninvertible.
series g(t), the lowpass "lter tL(z) of Eq. (10) will generate the minimum mean-square-error estimate of the sequence f(t) provided that the smoothing parameter has the value ofj"p2e/p2l. More recently, Bell (1984) has established
that the Wiener}Kolmogorov theory applies equally to nonstationary
pro-cesses. Thus, if it were applied to the nonstationary seriesy(t), the lowpass"lter would generate the minimum mean-square-error estimate of the nonstationary sequencem(t).
The recursive"ltering of nonstationary sequences is beset by two problems. On the one hand, there is the above-mentioned di$culty posed by the initial conditions. On the other hand, there is the danger that the unbounded nature of the sequence and the disparity of the values within it will lead to problems of numerical representation. Therefore, in pursuit of an alternative approach, we may consider the equation
m(t)"y(t)!g(t)
"y(t)! i(t)
(1!¸)d (22)
which can be derived in reference to (19) and (21). Here i(t) is a stationary sequence which can be estimated by applying the highpass"ltertH(¸) of (11) to g(t) which is the di!erenced version of the data sequence. Thereafter, an estimate of the stationary sequenceg(t) can be obtained by ad-fold process of cumulation which usesdzeros for its starting values}the zeros being the expected values of d consecutive elements of g(t). Finally, the estimate m(t) can be obtained by a simple subtraction.
Now imagine that, instead of a lengthy sequence of observations which can be treated as if it were in"nite, there are only¹observations of the processy(t) of Eq. (19) which run fromt"0 to¹!1. These are gathered in a vector
y"m#g
"x#h, (23)
wheremis the trend vector andgis the residual vector which is generated by a stationary process with
E(g)"0 and D(g)"p2eR. (24) The estimates of these vectors are denoted byxandhrespectively.
To"nd the"nite sample counterpart of Eq. (20), we need to represent thedth
di!erence operator (1!¸)d"1#d1¸#2#d
p¸d in the form of a matrix. The matrix which "nds the dth di!erences g
y
Premultiplying Eq. (23) by this matrix gives
g"Q@y"f#i
"z#k, (26)
wheref"Q@mandi"Q@gand wherez"Q@xandk"Q@hare the correspond-ing estimates. The"rst and second moments of the vectorfmay be denoted by
E(f)"0 and D(f)"p2
lXL (27)
and those ofiby
E(i)"0 and D(i)"Q@D(g)Q
"p2eQ@RQ"p2eXH, (28) where bothXL and XH are symmetric Toeplitz matrices with 2n#1 nonzero diagonal bands. The generating functions for the coe$cients of these matrices are, respectively, dL(z)dL(z~1) and dH(z)dH(z~1), wheredL(z) and dH(z) are the polynomials de"ned in (9). The generating function for the coe$cients ofRis
M(1!z)(1!z~1)Nn~d.
The estimates are calculated,"rst, by solving the equation
(XL#jX
H)b"g (31)
for the value ofband, thereafter, by"nding
z"X
Lb and k"jXHb. (32)
The solution of Eq. (31) is found via a Cholesky factorisation which sets
XL#jXH"GG@, whereGis a lower-triangular matrix. The systemGG@b"g may be cast in the form ofGp"gand solved forp. ThenG@b"pcan be solved forb.
Observe that the generating function for the matrix GG@is the polynomial
c(z)c(z~1) de"ned in (7). Moreover, the solution via the Cholesky factorisation is
a"nite-sample analogue of a process of bidirectional"ltering which"nds the
sequence b(t)"Mc(F)c(¸)N~1g(t) via the recursive processes c(¸)p(t)"g(t) and
c(F)b(t)"p(t). The equations of (32) correspond, respectively, to the "ltering
operationsz(t)"d
L(F)dL(¸)b(t) andk(t)"dH(F)dH(¸)b(t), each of which can be
realised in a single pass.
Our objective is to recover fromzor fromkan estimatexof the trend vector m. This would be conceived, ordinarily, as a matter of integrating the vector
z d times via a simple recursion which depends upon d initial conditions.
However, we can circumvent the problem of "nding the initial conditions by seeking the solution to the following problem:
Minimise (y!x)@R~1(y!x) subject to Q@x"z. (33)
The problem is addressed by evaluating the Lagrangean function
¸(x,k)"(y!x)@R~1(y!x)#2k@(Q@x!z). (34) By di!erentiating the function with respect toxand setting the result to zero, we obtain the condition
R~1(y!x)!Qk"0. (35)
Premultiplying byQ@Rgives
Q@(y!x)"Q@RQk. (36)
But, from (31) and (32), it follows that
Putting the"nal expression forkinto (35) gives
x"y!jRQb. (39)
This is our solution to the problem of estimating the trend vectorm. Observe that
h"jRQb
"RQ(Q@RQ)~1k (40)
is the estimate of the vector g. Moreover, it is subject to an identityQ@h"k which corresponds to the identityQ@g"iwhich de"nesi.
Note also that, if yis a vector of ¹values of a polynomial of degreed!1 taken at equally spaced values, theng"Q@y"0 andb"0, whencex"y. Thus, a polynomial time trend of degree less thandis una!ected by the"lter. This is indeed the appropriate outcome; and it would not be forthcoming if the startup problem were handled in another way.
It is notable that there is a criterion function which will enable us to derive the equation of the trend estimation"lter in a single step. The function is
¸(x)"(y!x)@R~1(y!x)#jx@QX~1L Q@x, (41) whereinj"p2
e/p2las before. This is minimised by the value speci"ed in (39). The
criterion function becomes intelligible when we allude to the assumptions that y&N(m,p2
eR) and thatQ@m"f&N(0,p2lXL); for then it plainly resembles a sum
of two independent chi-square variates scaled by a factor ofp2e.
4. An application of the rational5lter
We shall illustrate the uses of the rational"lter by applying it to a series of 66
"gures which constitute the quarterly unemployment statistics for Switzerland
from the"rst quarter of 1980 through to the second quarter of 1996. The graph of this series is given in Fig. 4. Our objective is to discover the pattern of the seasonal#uctuations which surround the longer-term trend. A casual inspection of the graph would suggest that the seasonal motions have been in abeyance in the period of rapidly increasing unemployment in the third segment of the series, only to be resumed when unemployment is stabilised at a higher level at the end of the series.
Fig. 4. The quarterly"gures on Swiss unemployment from 1980.1 to 1996.2 (upper panel) together with their trend obtained by smoothing the series using a lowpass"lter of ordern"8.
Fig. 5. The gain of the lowpass"lter of ordern"8 with the nominal cut-o!point atu"3n/8.
Fig. 6. The residual sequence from detrending the Swiss unemployment"gures using the rational "lter.
orders which has a more gradual transition than would be tolerated in more exacting circumstances. The e!ects of the choices of the cut-o! point and the
"lter orders can be seen in Fig. 5.
Fig. 7. The periodogram of the residual sequence obtained by detrending the Swiss unemployment "gures using the rational"lter.
Fig. 8. The gain of the Hodrick}Prescott lowpass"lter with a smoothing parameter of 24.
amplitudes of the seasonal #uctuations are clearly related to the level of unemployment. Thus, in times of high employment, there appears to be a wide-spread hoarding of labour which would be subject, at other times, to seasonal unemployment. This is a feature which one would not have detected by inspect-ing the original data series. It also transpires, from Fig. 6, that, far from beinspect-ing in abeyance during the period of rapidly increasing unemployment, the seasonal #uctuations were present and were of a steadily increasing amplitude.
The regularity of the residual series is re#ected in its periodogram which is represented in Fig. 7. Here, the complete absence of any components of a fre-quency below the cut-o! point is a powerful testimony to the e$cacy of the rational"lter. The tall spike centred atn/2 rad, or 903, represents the power of the seasonal#uctuations.
Fig. 9. The residual sequence from detrending the Swiss unemployment"gures using the Hod-rick}Prescott"lter.
Fig. 10. The periodogram of the residual sequence obtained by detrending the unemployment "gures using the Hodrick}Prescott"lter.
shows that it has allowed some powerful low-frequency components to pass through into the residual series.
This example shows that it is sometimes possible to recover detailed struc-tural information which is buried within aggregate time series; and it emphasises the need, in analysing economic time series, to use frequency-selective"lters with well-de"ned cut-o! points.
5. Programs
The computer programs which have been used in compiling this paper are available on a compact disc which is included in a book by the author: Pollock (1999).
References
Bell, W., 1984. Signal extraction for nonstationary time series. The Annals of Statistics 12, 646}664. Burman, J.P., 1980. Seasonal adjustment by signal extraction. Journal of the Royal Statistical
Society Series A 143, 321}337.
Cogley, T., Nason, J.M., 1995. E!ects of the Hodrick}Prescott "lter on trend and di!erence stationary time series, implications for business cycle research. Journal of Economic Dynamics and Control 19, 253}278.
De Jong, P., 1991. The Di!use Kalman Filter. The Annals of Statistics 19, 1073}1083.
Hartley, J.E., Hoover, K.D., Salyer, K.D. (Eds.), 1998. Real Business Cycles: A Reader. Routledge, London.
Harvey, A.C., Jaeger, A., 1993. Detrending, stylised facts and the business cycle. Journal of Applied Econometrics 8, 231}247.
Harvey, A.C., Todd, P.H., 1983. Forecasting economic time series with structural and Box}Jenkins models: a case study. Journal of Business and Economic Forecasting 1, 299}307.
Hillmer, S.C., Tiao, G.C., 1982. An ARIMA-model-based approach to seasonal adjustment. Journal of the American Statistical Association 77, 63}70.
King, R.G., Rebelo, S.G., 1993. Low frequency "ltering and real business cycles. Journal of Economic Dynamics and Control 17, 207}231.
Kolmogorov, A.N., 1941. Interpolation and extrapolation. Bulletin de l'Academie des Sciences de U.S.S.R., Series Mathematics 5, 3}14.
Koopman, S.J., Harvey, A.C., Doornick, J.A., Shephard, N., 1995. STAMP 5.0: Structural Time Series Analyser Modeller and Predictor}The Manual. Chapman & Hall, London.
Pollock, D.S.G., 1999. A Handbook of Time-Series Analysis, Signal Processing and Dynamics. The Academic Press, London.
Reinsch, C.H., 1976. Smoothing by spline functions. Numerische Mathematik 10, 177}183. Roberts, R.A., Mullis, C.T., 1987. Digital Signal Processing. Addison-Wesley, Reading, MA. Wiener, N., 1950. Extrapolation, Interpolation and Smoothing of Stationary Time Series. MIT
Technology Press, Wiley, New York.
