lecture9 - FILTERING MACROECONOMIC DATA Wiener–Kolmogorov...

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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 radar-assisted anti-aircraft 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 doubly-inﬁnite 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 positive-deﬁnite 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 low-frequency elements and the noise comprises elements of higher frequencies. 2 D.S.G. POLLOCK: Filtering Macroeconomic Data The Minimum Mean-Squared Error Estimator The principle of linear minimum mean-squared 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 second-order 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 second-order 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 least-squares 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 least-squares 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 low-frequency 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 zero-valued initial conditions. Therefore, the starting-value 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 zero-valued 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 so-called smoothing parameter. It will be observed that, as λ → ∞, the vector x tends to that of a linear function interpolated into the data by least-squares 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 cut-oﬀ 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.

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