Signal Extraction Problem Research Articles

Central Bank Forecasts of Liquidity Factors: Quality, Publication and the Control of the Overnight Rate

A simple model of the interaction between central bank liquidity management and the inter-bank overnight rate is suggested which helps understanding the effects of the publication of forecasts of liquidity factors by the European Central Bank adopted in June 2000. The paper argues that the main practical advantage of the publication of these forecasts is that it makes the signal extraction problem with regard to the central bank's operational intentions trivial and hence allows establishing a superior behavioural equilibrium between the central bank and the money market participants. In this equilibrium, the central bank can achieve a better steering of overnight rates than under private autonomous factor forecasts, depending of course also on the quality of liquidity forecasts. It is furthermore shown that the publication of an average of autonomous factors, such as adopted by the ECB, is, at least within the model presented, superior to the separate publication of autonomous factors for each single day.

Forecasting the Forecasts of Others in the Frequency Domain

This paper studies a class of models developed by R. M. Townsend (1983, J. Polit. Econ.91, 546–588) and T. J. Sargent (1991, J. Econom. Dynam. Control15, 245–273). These models feature dynamic signal extraction problems and an infinite regress in expectations. This paper uses frequency domain methods to compute an analytical solution to the fixed point problem posed by the infinite regress in expectations. The advantage of a frequency domain approach vis-à-vis a time domain approach derives from the fact that these models produce equilibria with non-fundamental moving average representations, in which market observations do not reveal the underlying shocks to agents' information sets. As a result, decision rules contain moving average components that are more easily handled in the frequency domain than in the time domain.

On Signal Extraction and Non-Certainty-Equivalence in Optimal Monetary Policy Rules

A standard result in the literature on monetary policy rules is that of certainty equivalence: given the expected values of all the state variables of the economy, policy should be set in a way that is independent of all higher moments of those variables. Some exceptions to this rule have been pointed out by Smets (1998), who restricts policy to respond to only a limited subset of state variables, and by Orphanides (1998), who restricts policy to respond to estimates of the state variables that are biased. In contrast, this paper studies unrestricted, fully optimal policy rules with optimal estimation of state variables. The rules in this framework exhibit certainty equivalence with respect to estimates of an unobserved, possibly complicated, state of the economy X, but are not certainty-equivalent when 1) a signal-extraction problem is involved in the estimation of X, and 2) the optimal rule is expressed as a reduced form that combines policymakers' estimation and policy-setting stages. In general, I show that it is optimal for policymakers to attenuate their reaction coefficient on a variable about which uncertainty has increased, while responding more aggressively to all other variables, about which uncertainty hasn't changed.

Central Bank Liquidity Management and the Signal Extraction Problem on the Money Market / Die Liquiditätssteuerung der Notenbank und das Signal-extraktionsproblem am Geldmarkt

Summary Understanding the factors determining overnight rates is crucial both for central bankers and private market participants, since, assuming the validity of the expectation theory of the term structure of interest rates, expectations with regard to this “monadic” maturity should determine longer term rates, which are deemed to be relevant for the transmission of monetary policy. The note proposes a simple model of the money market within a two-day long reserve maintenance period to derive relationships between the relevant quantities, expectations concerning these quantities for the rest of the reserve maintenance period, and overnight rates. It is argued that a signal extraction problem faced by banks when observing quantities such as their aggregate reserve holdings and allotment amounts of monetary policy operations is at the core of these relationships. The usefulness of the model is illustrated by applying it to the analysis of three alternative liquidity management strategies of a central bank.

Semiparametric approaches to signal extraction problems in economic time series

Nonparametric regression methods have become a very useful tool to extract trend signals in economic time series. However, this approach performs poorly when seasonality is present. To overcome this difficulty, we propose two alternative methods to deal with seasonal effects. In both approaches the trend is specified nonparametrically, but the seasonal component specification is different. First, we propose a partial linear model where the parametric part is a dummy-variable specification for the seasonality. Secondly, we consider the seasonal component to be a smooth function of time and, therefore, the model falls within the class of additive models. We offer efficient algorithms for calculating values of the parameter estimators for each of these approaches and we derive asymptotic properties for the estimators in the partial linear model. Finally, we illustrate these methods when applied to the Spanish industrial production index for energy.

Minimax threshold for denoising complex signals with Waveshrink

For the problem of signal extraction from noisy data, Waveshrink has proven to be a powerful tool, both from an empirical and an asymptotic point of view. Waveshrink is especially efficient at estimating spatially inhomogeneous signals. A key step of the procedure is the selection of the threshold parameter. Donoho and Johnstone (1994) propose a selection of the threshold based on a minimax principle. Their derivation is specifically for real signals and real wavelet transforms. In this paper, we propose to extend the use of Waveshrink to denoising complex signals with complex wavelet transforms. We illustrate the problem of denoising complex signals with an electronic surveillance application.

Recursive and en-bloc approaches to signal extraction

In the literature on unobservable component models , three main statistical instruments have been used for signal extraction: fixed interval smoothing (FIS), which derives from Kalman's seminal work on optimal state-space filter theory in the time domain; Wiener-Kolmogorov-Whittle optimal signal extraction (OSE) theory, which is normally set in the frequency domain and dominates the field of classical statistics; and regularization , which was developed mainly by numerical analysts but is referred to as 'smoothing' in the statistical literature (such as smoothing splines, kernel smoothers and local regression). Although some minor recognition of the interrelationship between these methods can be discerned from the literature, no clear discussion of their equivalence has appeared. This paper exposes clearly the interrelationships between the three methods; highlights important properties of the smoothing filters used in signal extraction; and stresses the advantages of the FIS algorithms as a practical solution to signal extraction and smoothing problems. It also emphasizes the importance of the classical OSE theory as an analytical tool for obtaining a better understanding of the problem of signal extraction.

SELF-FULFILLING PROPHECIES AND THE BUSINESS CYCLE

We demonstrate that multiple stationary rational-expectations equilibria exist in a version of Lucas's island economy. The existence of these equilibria follows from the fact that there is an indeterminate set of monetary equilibria in the two-period overlapping-generations model. We show how to construct stationary rational-expectations equilibria by randomizing over the set of nonstationary monetary equilibria. In some of our equilibria, a positively sloped Phillips curve exists even though our economy contains no signal-extraction problem as in the original Lucas paper. Our equilibria are indexed by beliefs and are examples of the existence of sunspot equilibria in which allocations may differ across states of nature for which preferences, technology, and endowments are identical. Our technique for constructing stationary sunspot equilibria should prove useful in a wide class of models in which an indeterminate stationary equilibrium exists.

New classical and old Austrian economics: Equilibrium Business Cycle Theory in perspective

The recent flourishing of New Classical economics, and especially its Equilibrium Business Cycle Theory (EBCT), has given a fresh hearing to the Old-but still developing-Austrian Business Cycle Theory (ABCT). While the New and the Old differ radically in both substance and methods, they exhibit a certain formal congruency that has captured the attention of both schools. The formal similarities between EBCT and ABCT invites a point-bypoint comparison, but the comparison itself dramatizes differences between the two views in a way that adds to the integrity and plausibility of the Austrian theory. In modern macroeconomic literature, the label EBCT is applied sometimes so broadly as to include New Keynesian as well as New Classical constructions and sometimes so narrowly as to preclude the very developments within the New Classical school that are most closely related to ABCT. So-called Real Business Cycle Theory, in which cyclical movements of macroeconomic variables are characterized by both market clearing and Pareto optimality, is sometimes designated as the only true equilibrium construction. The comparison of New Classical and Old Austrian theories is best facilitated by letting EBCT refer to those theories in which (a) individuals make the best use of the information available to them and (b) an informational deficiency temporarily masks the intementions of the monetary authority. As exposited by Robert Lucas (1981), Robert Barro (1981) and others, EBCT so conceived accounts for business cycles in terms of the actions of market participants confronted with what has come to be known as a signal-extraction problem. Difficulties in

Convergence of the Kalman filter gain for a class of nondetectable signal extraction problems

Convergence of the filter gain is proved for a class of problems with undetectable modes on the unit circle. This situation arises when the signal is an unstable ARMA process corrupted by ARMA noise with the same unstable modes, since modes common to signal and noise are unobservable.

Signal extraction for finite nonstationary time series

SUMMARY For a signal plus noise model, where both signal and noise are generated by nonstation- ary ARIMA, autoregressive integrated moving average, models, we use the transformation approach of Ansley & Kohn (1985) to construct an estimate of the signal and obtain its mean squared error given a finite number of observations. A method for efficient computa- tion of the signal estimate and its mean squared error is also presented. Consider observations z(t) generated by the signal plus noise nmodel z(t) = s(t)+ n(t), (1s1) where both signal s(t) and noise n(t) follow nonstationary ARIMA processes. We give a solution to the finite sample signal extraction problem by defining an estimate of the signal s(t) given observations z(1),... , z(N) and finding its mean squared error. The major difficulty in handling the model with nonstationary components is that the distribution of the observations is not defined unless assumptions are made about the initial values of the series. Moreover, it is usually very difficult to formulate reasonable assumptions about the initial values because they affect the correlation structure of the whole series. See, for example, the discussion of Bell (1984, p. 652). Kohn & Ansley (1986) use a transformation approach to overcome the initial value problem in defining the likelihood and obtaining predittors and interpolators for an ARIMA model which may have missing values. This approach does not require any assumptions about the initial values and generalizes the usual differencing approach of Box & Jenkins (1976) for the case where there are no missing observations. In the present paper we apply the transformation approach of Kohn & Ansley (1986) and Ansley & Kohn (1985) to solve the signal extraction problem. We define a linear estimator A( t) of s( t) so that s( t) - SA( t) is invariant to the initial values and, furthermore, is unbiased in the sense that E{s(t) - s(t)} = 0. Moreover, it has minimum mean squared error amongst all such estimators. Our approach allows missing observations and extends to signal extraction for several additive signals when each is an ARIMA process. We also show how to compute the signal estimate and its mean squared error efficiently by writing the model in state space form, placing a diffuse prior on the initial values of the process and then using the modified Kalman filter and the modified smoothing algorithm of Ansley & Kohn (1985) and Kohn & Ansley (1986). Gersch & Kitagawa (1983), Kitagawa & Gersch (1984) and Harvey & Todd (1983) use

ON SOME AMBIGUITIES ASSOCIATED WITH THE FITTING OF ARMA MODELS TO TIME SERIES

Abstract. Examples are presented illustrating some ambiguities associated with the application of ARMA models to problems of signal extraction, multistep‐ahead forecasting, spectrum approximation and linear quadratic control. Except in the signal extraction example, the ambiguities arise either from lack of sufficient autocovariance data to completely determine the process, or, often relatedly, from the approximate nature of the models used.

Signal Extraction for Nonstationary Time Series

The nonstationary signal extraction problem is to estimate $s_t$ given observations on $z_t = s_t + n_t$ (signal plus noise) when either $s_t$ or $n_t$ or both is nonstationary. Homogeneous or explosive nonstationary time series described by models of the form $\delta(B)z_t = w_t$ where $\delta(B)$ has zeroes on or inside the unit circle and $w_t$ is stationary are considered. For certain cases, approximate solutions to the nonstationary signal extraction problem have been given by Hannan (1967), Sobel (1967), and Cleveland and Tiao (1976). The paper gives exact solutions in the forms of expressions for $E(s_t\mid\{z_t\})$ and $\operatorname{Var}(s_t\mid\{z_t\})$ (assuming normality) under two sets of alternative assumptions regarding the generation of $z_t, s_t$, and $n_t$. Extensions to signal extraction with a finite number of observations, to the nonGaussian case, and to the multivariate case are discussed.

Random telluric field and signal extraction

Scattering of telluric currents (large current sheets induced into the crust) by (1) a horizontal sheet of random (weak) resistivity, (2) a semi-infinite medium of random (weak) resistivity, and (3) a random interface separating two mediums of different resistivities has been studied. Simple relations between the spectral density function of the scattered field and that of the scatterer have been established. Scattering patterns showing the variance of the random vector field as a function of azimuth have been constructed for statistically isotropic and anisotropic sheet scatterers. We have found a significant scattering in the direction normal to the normal field. The problem of signal extraction, especially the difficulties one would face, if he were to apply the well-known linear filter theory, have been briefly discussed in light of the information on the statistical nature of the random background noise.

A theory of adaptive filtering

This paper considers the adaptive signal extraction problem for time-discrete data when only very general a priori assumptions regarding the distributions of signal and noise are possible. Specifically, it is assumed that the noise is white, additive, and signal independent with mean zero and unknown variance and that the signal is band-limited. No stationarity assumptions are required. After a procedure is found under these conditions, the mean-square-error is derived asymptotically under narrower conditions-stationary Gaussian data with mean zero. Finally, a method of estimating the error variance from the data (without knowing the signal directly) is found.

On a characterization of processes for which optimal mean-square systems are of specified form

This paper presents the first results in an unconventional approach to the problem of mean-square optimization. Instead of obtaining a representation for the optimal operator for a process, we seek to characterize the class of processes for which the optimal operator is of specified form. If the processes are given, so that the multivariate characteristic functions are known, then our results can be used to tell whether it is possible for the optimal operator to have a specified form. The bulk of the paper pertains to the signal extraction problem where the signal and noise are independent and additive, and it is desired to estimate some function of the signal. Here, with a slight shift in viewpoint, we phrase the characterization problem in the following way: Given, for example, a noise process, determine the class of signal processes for which the optimal extraction system is of specified form. The case where the noise process is Gaussian comes in for special attention.