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Unformatted text preview: FILTERING MACROECONOMIC DATA
Wiener–Kolmogorov Filtering of Stationary Sequences
The classical theory of linear ﬁltering was formulated independently by
Norbert Wiener (1941) and Andrei Nikolaevich Kolmogorov (1941) during
the Second World War. They were both considering the problem of how
to target radarassisted antiaircraft guns on incoming enemy aircraft.
The theory has found widespread application in analog and digital signal
processing and in telecommunications in general. Also, it has provided a
basic technique for the enhancement of recorded music.
The classical theory assumes that the data sequences are generated by
stationary stochastic processes and that these are of suﬃcient length to
justify the assumption that they constitute doublyinﬁnite sequences.
For econometrics, the theory must to be adapted to cater to short trended
sequences. Then, Wiener–Kolmogorov ﬁlters can used to extract trends
from economic data sequences and for generating seasonally adjusted data.
1 D.S.G. POLLOCK: Filtering Macroeconomic Data
Consider a vector y with a signal component ξ and a noise component η :
(1) y = ξ + η. These components are assumed to be independently normally distributed
with zero means and with positivedeﬁnite dispersion matrices. Then,
E (ξ ) = 0,
(2) D(ξ ) = Ωξ , E (η ) = 0, D(η ) = Ωη , and C (ξ, η ) = 0.
A consequence of the independence of ξ and η is that
(3) D(y ) = Ωξ + Ωη and C (ξ, y ) = D(ξ ) = Ωξ . The signal component is estimated by a linear transformation x = Ψx y
of the data vector that suppresses the noise component. Usually, the
signal comprises lowfrequency elements and the noise comprises elements
of higher frequencies.
2 D.S.G. POLLOCK: Filtering Macroeconomic Data
The Minimum MeanSquared Error Estimator
The principle of linear minimum meansquared error estimation indicates
that the error ξ − x in representing ξ by x should be uncorrelated with
the data in y : (4) 0 = C (ξ − x, y ) = C (ξ, y ) − C (x, y )
= C (ξ, y ) − Ψx C (y, y )
= Ωξ − Ψx (Ωξ + Ωη ). This indicates that the estimate is
(5) x = Ψx y = Ωξ (Ωξ + Ωη )−1 y. The corresponding estimate of the noise component η is
(6) h = Ψh y = Ωη (Ωξ + Ωη )−1 y. It will be observed that Ψξ + Ψη = I and, therefore, that x + h = y .
3 D.S.G. POLLOCK: Filtering Macroeconomic Data
Conditional Expectations
In deriving the estimator, we might have used the formula for conditional
expectations. In the case of two linearly related scalar random variables
ξ and y , the conditional expectation of ξ given y is
(7) E (ξ y ) = E (ξ ) + C (ξ, y )
{y − E (y )}.
V (y ) In the case of two vector quantities, this becomes
(8) E (ξ y ) = E (ξ ) + C (ξ, y )D−1 (y ){y − E (y )}. By setting
C (ξ, y ) = Ωξ and D(y ) = Ωξ + Ωη , as in (3), and by setting E (ξ ) = E (y ) = 0, we get the expression that is
to be found under (5):
x = Ωξ (Ωξ + Ωη )−1 y. 4 D.S.G. POLLOCK: Filtering Macroeconomic Data
The Diﬀerence Operator and Polynomial Regression
The lag operator L, which is commonly deﬁned in respect of a doublyinﬁnite sequence x(t) = {xt ; t = 0 ± 1, ±2, . . .}, has the eﬀect that Lx(t) =
x(t − 1).
The (backwards) diﬀerence operator ∇ = 1 − L has the eﬀect that ∇x(t) =
x(t) − x(t − 1). It serves to reduce a constant function to zero and to reduce
a linear function to a constant. The secondorder or twofold diﬀerence
operator
∇2 = 1 − 2L + L2
is eﬀective in reducing a linear function to zero.
A diﬀerence operator ∇d of order d is commonly employed in the context of an ARIMA(p, d, q ) model to reduce the data to stationarity. Then,
the diﬀerenced data can be modelled by an ARMA(p, q ) process. In such
circumstances, the diﬀerence operator takes the form of a matrix transformation. 5 D.S.G. POLLOCK: Filtering Macroeconomic Data 6 4 2 0
0 π/4 π/2 3π/4 π Figure 1. The squared gain of the diﬀerence operator, which has a zero at zero
frequency, and the squared gain of the summation operator, which is unbounded
at zero frequency. 6 D.S.G. POLLOCK: Filtering Macroeconomic Data
The Matrix Diﬀerence Operator
The matrix analogue of the secondorder diﬀerence operator in the case
of T = 5 , for example, is given by (9) ∇2 =
5 Q∗
Q 1
0 −2 1 = 1 −2 0
1
0
0 0
0
1
−2
1 0
0 0 0. 10
−2 1
0
0 The ﬁrst two rows, which do not produce true diﬀerences, are liable to be
discarded.
The diﬀerence operator nulliﬁes data elements at zero frequency and it
severely attenuates those at the adjacent frequencies. This is a disadvantage when the low frequency elements are of primary interest. Another
way of detrending the data is to ﬁt a polynomial trend by leastsquares
regression and to take the residual sequence as the detrended data.
7 D.S.G. POLLOCK: Filtering Macroeconomic Data 11.5
11
10.5
10
0 50 100 150 Figure 2. The quarterly series of the logarithms of consumption in the U.K., for
the years 1955 to 1994, together with a linear trend interpolated by leastsquares
regression. 8 D.S.G. POLLOCK: Filtering Macroeconomic Data 8
6
4
2
0
0 π/4 π/2 3π/4 Figure 3. The periodogram of the trended logarithmic data. 9 π D.S.G. POLLOCK: Filtering Macroeconomic Data 0.3
0.2
0.1
0
0 π/4 π/2 3π/4 π Figure 4. The periodogram of the diﬀerenced logarithmic consumption
data. 10 D.S.G. POLLOCK: Filtering Macroeconomic Data
Polynomial Regression
Using the matrix Q deﬁned above, we can represent the vector of the
ordinates of a linear trend line interpolated through the data sequence as
(10) x = y − Q(Q Q)−1 Q y . The vector of the residuals is
(11) e = Q(Q Q)−1 Q y . Observe that this vector contains exactly the same information as the
diﬀerenced vector g = Q y . However, whereas the lowfrequency structure
of the data in invisible in the periodogram of the latter, it is entirely visible
in the periodogram of the residuals. 11 D.S.G. POLLOCK: Filtering Macroeconomic Data 0.01
0.0075
0.005
0.0025
0
0 π/4 π/2 3π/4 π Figure 5. The periodogram of the residual sequence obtained from the linear
detrending of the logarithmic consumption data. 12 D.S.G. POLLOCK: Filtering Macroeconomic Data
Filters for Short Trended Sequences
Applying Q to the equation y = ξ + η , representing the trended data,
gives
Qy =Qξ+Qη
= δ + κ = g. (12) The vectors of the expectations and the dispersion matrices of the diﬀerenced vectors are
E (δ ) = 0, D(δ ) = Ωδ = Q D(ξ )Q, E (κ) = 0, D(κ) = Ωκ = Q D(η )Q. (13) The diﬃculty of estimating the trended vector ξ = y − η directly is that
some starting values or initial conditions are required in order to deﬁne
the value at time t = 0. However, since η is from a stationary meanzero process, it requires only zerovalued initial conditions. Therefore,
the startingvalue problem can be circumvented by concentrating on the
estimation of η .
13 D.S.G. POLLOCK: Filtering Macroeconomic Data
The conditional expectation of η , given the diﬀerenced data g = Q y , is
provided by the formula
(14) h = E (η g ) = E (η ) + C (η, g )D−1 (g ){g − E (g )}
= C (η, g )D−1 (g )g, where the second equality follows in view of the zerovalued expectations.
Within this expression, there are
(15) D(g ) = Ωδ + Q Ωη Q and C (η, g ) = Ωη Q. Putting these details into (14) gives the following estimate of η :
(16) h = Ωη Q(Ωδ + Q Ωη Q)−1 Q y . Putting this into the equation x = y − h gives
(17) x = y − Ωη Q(Ωδ + Q Ωη Q)−1 Q y . 14 D.S.G. POLLOCK: Filtering Macroeconomic Data
The Leser (H–P) Filter
We now consider two speciﬁc cases of the Wiener–Kolmogorov ﬁlter. First,
there is the Leser or Hodrick–Prescott ﬁlter. This is derived by setting
(18) 2
D(η ) = Ωη = ση I, 2
D(δ ) = Ωδ = σδ I 2
ση
and λ = 2
σδ within (17) to give
(19) x = y − Q(λ−1 I + Q Q)−1 Q y Here, λ is the socalled smoothing parameter. It will be observed that,
as λ → ∞, the vector x tends to that of a linear function interpolated
into the data by leastsquares regression, which is represented by equation
(10):
x = y − Q(Q Q)−1 Q y . 15 D.S.G. POLLOCK: Filtering Macroeconomic Data 1
0.75
0.5
0.25
0
0 π/4 π/2 3π/4 π Figure 6. The gain of the Hodrick–Prescott lowpass ﬁlter with a smoothing
parameter set to 100, 1,600 and 14,400. 16 D.S.G. POLLOCK: Filtering Macroeconomic Data
The Butterworth Filter
The Butterworth ﬁlter that is appropriate to short trended sequences can
be represented by the equation
(20) x = y − λΣQ(M + λQ ΣQ)−1 Q y . Here, the matrices are
(21) Σ = {2IT − (LT + LT )}n−2 and M = {2IT + (LT + LT )}n , where LT is a matrix of order T with units on the ﬁrst subdiagonal; it can
be veriﬁed that
(22) Q ΣQ = {2IT − (LT + LT )}n . 17 D.S.G. POLLOCK: Filtering Macroeconomic Data 1
0.75
0.5
0.25
0
0 π/4 π/2 3π/4 π Figure 7. The squared gain of the lowpass Butterworth ﬁlters of
orders n = 6 and n = 12 with a nominal cutoﬀ point of 2π/3
radians. 18 D.S.G. POLLOCK: Filtering Macroeconomic Data 19 ...
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This note was uploaded on 03/02/2012 for the course EC 3062 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
 Spring '12
 D.S.G.Pollock

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