Optimal Signal Extraction with Correlated Components

Derivation of the BN filter

The original BN filter of Beveridge and Nelson (1981) was applied to nonseasonal time series, yielding a decomposition into permanent and transitory components. This was achieved by supposing the innovations of signal and noise to be identical (rather than orthogonal). We extend this notion of the BN filter to the general scenario outlined in Section 3 (and following) by taking at=bt=ct in eq. [6]. Then the MSE optimal filters have zero error and are given by de-correlating the aggregate process (via the filter φ(z)/θ(z)) followed by re-correlating according to the signal’s pattern, namely by the filter θS(z)/φS(z). Multiplying these filters yields θS(z)φN(z)/θ(z). The formula we provide in Section 3 generalizes this treatment slightly to the case of a filter derived under the assumption that the innovations are fully (positively) correlated (i.e. they may have different variances). This leads to the insertion of the factor σb/σa in the formula for ΨBN.

By definition, this filter will produce optimal extractions under the full positive correlation hypothesis, and of course the signal and noise extractions aggregate back to the original aggregate variables. This defines the filter; of course it may be applied to data that does not satisfy the hypotheses under which the filter was derived. It is in this sense that we may speak of the hysteretic filter as being a convex combination of WK and BN filters – it is an algebraic fact, involving statistical quantities derived under incompatible stochastic assumptions.

We are not aware of prior references to this generalized BN filter, but are not comfortable claiming that our derivation is novel. This sort of idea, and construction, has close precedents in Morley, Nelson, and Zivot (2003) and Proietti (2006).

We also mention an interesting interpretation of the BN filter in the case of a trend plus noise decomposition, namely that it can be interpreted as the optimal concurrent filter in an orthogonal decomposition. Consider the special case of an I(1) data process {Yt} that satisfies (1−B)Yt=θ(B)at, where the polynomial (or causal power series) θ(z) is invertible. Then the BN decomposition can be written

which can be compared to (1−B)Yt=(1−B)St+(1−B)Nt. Thus, using the notation of eq. [6] we have θS(z)=θ(1), a constant, while θN(z)=(θ(z)−θ(1))/(1−z) and φS(z)=1−z and φN(z)=1. Note that a single innovation {at} drives both component processes. The assumed invertibility of θ(z) ensures that θN(z) has a convergent power series expansion. The BN filter can then be expressed as θ(1)/θ(z).

Suppose now that we take the same component model definitions – namely (1−B)St=bt, a white noise of variance θ2(1)σa2, and Nt=θN(B)ct, with {ct} another white noise – but instead of having a common innovation drive the components, we assume they are uncorrelated. The concurrent filter in this case, using the formulas of Bell and Martin (2004), is

which simplifies to θ(1)/θ(z). Here the bracket notation with the plus subscript indicates that in the power series expansion, we only retain the terms corresponding to non-negative powers of z.

Proofs

Proof of Theorem 1. Following the same technique as McElroy (2008), it suffices to show that the error process ε=Sˆ−S is uncorrelated with Y. Using I to denote an identity matrix, the error process is

From Assumption A, it follows that ε is uncorrelated with Y∗. Since Y can be expressed as a linear combination of Y∗ and W, as discussed in McElroy (2008), it suffices to show that ε is uncorrelated with W. To that end, we have

using eq. [4]. This establishes MSE linear optimality. For the error covariance matrix, we obtain the formula by expanding E[εε′] and simplifying the algebra.

It may be instructive to offer a constructive derivation of the formula for F. Recall that U=ΔSS and V=ΔNN; the minimum MSE linear estimator of U given Y is the same as the optimal estimator of U given W (by Assumption A), and hence its formula is

which utilizes eq. [3]. By linearity, the above estimate must be equal to ΔS times the optimal estimate of S given Y. Similarly, we can derive the minimum MSE linear estimator of V given Y via

(This tactic, whereby we directly compute the best linear estimate of S given Y, can be used to derive F as well, but requires considerably more algebra than the approach given here.) Collecting these results, and letting F denote the matrix such that FY is the minimum MSE linear estimator of S given Y, we obtain

The second equation utilizes ΔNF=ΔN−ΔN(I−F) and the fact that I−F is the linear optimal estimator of the noise N given Y. Now knowing algebraically the form of both ΔSF and ΔNF permits us to solve for F, presuming that the intersection of the null spaces of ΔS and ΔN is the zero vector; this follows from the assumption that δS(z) and δN(z) are relatively prime. Then multiply ΔSF by ΔS′ΓU−1(this is not a unique choice, e.g. we could just utilize ΔS′ instead and get an alternative expression for the unique F) and ΔNF by ΔN′ΓV−1 and add the result. Then utilizing eq. [2], this yields

which simplifies to ΔN′ΓV−1ΔN+PΓW−1Δ. Inverting M then yields the formula for F. ⃞

Proof of Theorem 2. Our strategy is to demonstrate that the filter Ψ(z) produces a signal extraction error process that is orthogonal to the aggregate process, implying MSE linear optimality [cf. Bell (1984)]. It is clear from the given formula that δN(z) can be factored out, leaving Ψ(z)=Ω(z)δN(z) with

Similarly, it is easily verified that 1−Ψ(z)=Φ(z)δS(z) with

Then εt=Ψ(B)Yt−St=Ω(B)Vt−Φ(B)Ut. This shows that {εt} is weakly stationary. The aggregate process {Yt} can be written in terms of a linear combination of d initial values summed with a linear function of the differenced process {Wt}– see Bell (1984). Hence by Assumption A, it is sufficient to demonstrate that εt is uncorrelated with Wt+h for any t and h. Now Wt+h=δN(B)Ut+h+δS(B)Vt+h, so that

which is independent of t, but holds for all h. Thus, taking the Fourier transform yields

Then simple algebra, along with the above formulas for Ω(z) and Φ(z), produces ∑h∈ℤzhE[εtWt+h]=0, and hence E[εtWt+h]=0 for all h. A second calculation produces

Again, summing against zh yields

Now plugging in for Ω and Φ gives the stated result. ⃞

Alternative calculation of F

We mentioned in the proof of Theorem 1 that there is more than one expression for the signal extraction matrix F, although numerically all such formulas are equal. A different approach to the problem, based upon results of Bell and Hillmer (1988), can be developed that has some computational advantages. Let S∗ denote the first dS values of the vector S, conceived of as initial values of the process. Likewise, let N∗ denote the first dN values of the vector N, and Y∗ the first d values of the vector Y. As described in Bell and Hillmer (1988), it is always possible to algebraically describe S∗ as a linear function of Y∗, U, and V, and likewise for N∗. We review and develop these relationships below.

We use the notation In to denote an identity matrix of dimension n. We can relate the signal vector S to its initial values S∗ and differenced values U, and similarly for the noise, via the transformations

Let us denote these matrices by ∇S and ∇N, respectively; they are unit lower triangular and invertible, which allows us to express S directly in terms of S∗ and U (and N in terms of N∗ and V). Taking only the first d rows of ∇S−1 yields

This calculation first uses the fact that – because ∇S is unit lower triangular with first dS rows given by [IdS0]– the first dS rows of ∇S−1 are also given by [IdS0]. The matrices AS and BS correspond to the next dN rows of ∇S−1. With a similar notation and decomposition for N, and noting that S+N=Y, we obtain

The d×d matrix that multiplies the signal and noise initial values is denoted by [H1H2] in Bell and Hillmer (1988) and is there proved to be invertible. We will denote it by the symbol Ω. Note that its inversion is inexpensive due to its relatively low dimension of d. It now follows that

These relations between signal and noise initial values are an exact algebraic relation, and a direct implication of the nonstationary signal and noise structure; no stochastic assumptions have been used yet. The relations can be utilized to produce signal extraction estimates as follows. From S=∇S−1[S∗′,U′]′ we deduce that the linear optimal estimate of S can be constructed from estimates of S∗ and U, followed by application of ∇S−1. This latter matrix does not require inversion in general, but rather expresses in matrix notation the notion of recursion. Namely, we can always compute St from dS prior values (in time) together with Ut, i.e. St=−∑j=1dSδjS/δ0SSt−j+Ut. In the proof of Theorem 1, we discussed the optimal linear estimates of U given Y, and estimates of V given Y, which we shall denote by Uˆ and Vˆ. Then the optimal linear estimates of the signal and noise initial values are just

It is easy to check, using Assumption A, that the error S∗−S∗ˆ is orthogonal to Y. The algorithm is to first compute Uˆ and Vˆ via ΔSFY and ΔN(I−F)Y, i.e.

Then the formula for S∗ˆ is utilized, and finally Sˆ is obtained recursively (and Nˆ=Y−Sˆ). Such an algorithm can avoid the inversion of large matrices, excepting the work involved in inverting ΓW; however, this is a Toeplitz matrix, and hence the innovations algorithm of Brockwell and Davis (1991) can be utilized.

Some readers may find it illuminating to derive F directly from Cov(S,Y)Cov(Y,Y)−1, which we now derive, utilizing the above expressions. Let ∇ denote the differencing matrix with upper rows [1d0] and bottom rows Δ. Because W is orthogonal to Y∗, we find that

Moreover, U is orthogonal to Y∗, so that

Therefore, we obtain

Now utilizing the expression of S∗ above, written in terms of Y∗, U, and V, it follows that

As a result,

This is the same formula for F as given above.