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This paper derives the limit of the Kalman filter and smoother as the inverse of the process noise covariance tends to zero (the zero informational limit) in the case that there is direct feedthrough (of full column rank) of the process noise input to the measurements. The filter in the limit is closely related to the filter proposed by Gillijns and De Moor who used a constrained optimisation problem to estimate an unknown input. We derive a second form of the limit filter which takes the form of a standard Kalman filter without unknown inputs. This form is used to derive necessary and sufficient conditions for convergence and stability of the filter. These consist of a controllability condition and a minimum phase condition. The filter and smoother are applied to an automotive example to estimate an unknown road profile. The example illustrates the usefulness of the stability and convergence conditions to inform the choice of a suitable set of sensors.

Assessing potential output and the output gap is essential for policy-making and fiscal surveillance. The European Commission proposes a production function methodology that involves the estimation of two classes of Gaussian state space models. This paper presents the R package RGAP which features a flexible modeling framework for the appropriate bivariate unobserved component models and offers frequentist as well as Bayesian estimation techniques. Additional functionalities include direct access to the AMECO database and automated model selection procedures. Multiple illustrative examples outline data preparation, model specification, and estimation processes using RGAP., KOF Working Papers, 503

This paper is concerned with the estimation of unknown drift functions of stochastic differential equations (SDEs) from observations of their sample paths. We propose to formulate this as a non-parametric Gaussian process regression problem and use an Itô–Taylor expansion for approximating the SDE. To address the computational complexity problem of Gaussian process regression, we cast the model in an equivalent state-space representation, such that (non-linear) Kalman filters and smoothers can be used. The benefit of these methods is that computational complexity scales linearly with respect to the number of measurements and hence the method remains tractable also with large amounts of data. The overall complexity of the proposed method is O(N logN), where N is the number of measurements, due to the requirement of sorting the input data. We evaluate the performance of the proposed method using simulated data as well as with realdata applications to sunspot activity and electromyography.

We provide a simulation smoother to a exible state-space model with lagged states and lagged dependent variables. Qian (2014) has introduced this state-space model and proposes a fast Kalman filter with time-varying state dimension in the presence of missing observations in the data. In this paper, we derive the corresponding Kalman smoother moments and propose an efficient simulation smoother, which relies on mean corrections for unconditional vectors. When applied to a factor model, the proposed simulation smoother for the states is efficient compared to other state-space models without lagged states and/or lagged dependent variables in terms of computing time.

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a hybrid Markov Chain Monte Carlo sampling algorithm. Candidate draws for the unobserved volatilities are obtained by applying the Kalman filter and smoother to a linearization of a state-space representation of the model. The method is illustrated using the Heston (1993) stochastic volatility model applied to Australian News Corporation spot and option price data. Alternative models nested in the Heston framework are ranked via Bayes Factors and via fit, predictive and hedging performance.

This paper proposes a method to estimate the parameters of price adjustment model based on the Kalman filter and smoother. From the estimated parameters, we decompose the observed stock return volatility to the value return volatility and the market microstructure noise volatility. We argue that this method may be considered an attractive alternative to the method of observed return's moments. We use data on the eighty stocks of the TOPIX 100 to illustrate the method. The results indicate that the method seems to work well. An empirical analysis provides evidence that the adjustment coefficient represents partial price adjustment to new intrinsic value. We also find that the value return volatility is higher than the market microstructure noise volatility.

This paper proposes a method to estimate the parameters of price adjustment model based on the Kalman filter and smoother. From the estimated parameters, we decompose the observed stock return volatility to the value return volatility and the market microstructure noise volatility. We argue that this method may be considered an attractive alternative to the method of observed return's moments. We use data on the eighty stocks of the TOPIX 100 to illustrate the method. The results indicate that the method seems to work well. An empirical analysis provides evidence that the adjustment coefficient represents partial price adjustment to new intrinsic value. We also find that the value return volatility is higher than the market microstructure noise volatility.

This paper proposes a new framework to analyze and estimate structural fiscal balances. Stochastic trends are properly incorporated, and the numerical solution of the DSGE model serves as part of the Kalman smoother to extract structural fiscal balances. For the UK, a setting of an integrated random walk for the underlying stochastic trends fits the date best. The response of nominal fiscal revenue to the technology shock is small. The shocks to foreign demand and to foreign goods price both have positive effects on fiscal revenue. An expansionary monetary policy shock has a great positive short-run impact on fiscal revenue, but the influence is not persistent because of the open-economy characteristic of the UK. An expansion in government spending can also increase fiscal revenue, but the effect is not persistent as well due to the domestic and external crowd-out effects. A contractionary fiscal policy (cutting government expenditure or increasing the lump-sum tax temporarily), rather than an expasionary one, will benefit economic recovery and also improve fiscal stance. Compared to a temporary increase of the lump-sum tax, cutting government spending is relatively more effective and it alleviates the two kinds of crowd-out effects.

Summary Continuous monitoring of the evolution of the economy is fundamental for the decisions of public and private decision makers. The paper proposes EUROMIND, which is a new monthly indicator of the euro area economic conditions, based on tracking real gross domestic product monthly, relying on information provided in the Eurostat Euro-IND database. EUROMIND has several original economic and statistical features. First, it considers both the output and the expenditure sides of the economy, as it provides a monthly estimate of the value added of the six branches of economic activity and of the main gross domestic product components by type of expenditure (final consumption, gross capital formation and net exports), and combines the estimates with optimal weights reflecting their relative precision. Second, the indicator is based on information at both the monthly and the quarterly level, modelled with a dynamic factor specification cast in state space form. Third, since estimation of the multivariate dynamic factor model with mixed frequency data can be numerically complex, computational efficiency is achieved by implementing univariate filtering and smoothing procedures. Finally, special attention is paid to chain linking and its implications, via a multistep procedure that exploits the additivity of the volume measures expressed at the prices of the previous year.

The paper is concerned with a class of trend cycle filters, encompassing popular ones, such as the Hodrick-Prescott filter, that are derived using the Wiener-Kolmogorov signal extraction theory under maintained models that prove unrealistic in applied time series analysis. As the maintained model is misspecified, inference about the unobserved components, and in particular their first two conditional moments, given the observations, are not delivered by the Kalman filter and smoother or the Wiener-Kolmogorov filter for the maintained model. The paper proposes a model based framework according to which the same class of filters is adapted to the particular time series under investigation; via a suitable decomposition of the innovation process, it is shown that any linear time series with ARIMA representation can be broken down into orthogonal trend and cycle components, for which the class of filters is optimal. Finite sample inferences are provided by the Kalman filter and smoother for the relevant state space representation of the decomposition. In this framework it is possible to discuss two aspects of the reliability of the signals’ estimates: the mean square error of the final estimates and the extent of the revisions. The paper discusses and illustrates how the uncertainty is related to features of the series and the design parameters of the filter, the role of smoothness priors, and the fundamental trade-off between the uncertainty and the magnitude of the revisions as new observations become available.

A continuous monitoring of the evolution of the economy is fundamental for the decisions of public and private decision makers. This paper proposes a new monthly indicator of the euro area real Gross Domestic Product (GDP), with several original features. First, it considers both the output side (six branches of the NACE classification) and the expenditure side (the main GDP components) and combines the two estimates with optimal weights reflecting their relative precision. Second, the indicator is based on information at both the monthly and quarterly level, modelled with a dynamic factor specification cast in state-space form. Third, since estimation of the multivariate dynamic factor model can be numerically complex, computational efficiency is achieved by implementing univariate filtering and smoothing procedures. Finally, special attention is paid to chain-linking and its implications, via a multistep procedure that exploits the additivity of the volume measures expressed at the prices of the previous year.

A continuous monitoring of the evolution of the economy is fundamental for the decisions of public and private decision makers. This paper proposes a new monthly indicator of the euro area real Gross Domestic Product (GDP), with several original features. First, it considers both the output side (six branches of the NACE classification) and the expenditure side (the main GDP components) and combines the two estimates with optimal weights reflecting their relative precision. Second, the indicator is based on information at both the monthly and quarterly level, modelled with a dynamic factor specification cast in state-space form. Third, since estimation of the multivariate dynamic factor model can be numerically complex, computational efficiency is achieved by implementing univariate filtering and smoothing procedures. Finally, special attention is paid to chain-linking and its implications, via a multistep procedure that exploits the additivity of the volume measures expressed at the prices of the previous year.

The paper documents and illustrates state space methods that implement time series disaggregation by regression methods, with dynamics that depend on a single autoregressive parameter. The most popular techniques for the distribution of economic flow variables, such as Chow-Lin, Fernandez and Litterman, are encompassed by this unifying framework. The state space methodology offers the generality that is required to address a variety of inferential issues, such as the role of initial conditions, which are relevant for the properties of the maximum likelihood estimates and for the the derivation of encompassing representations that nest exactly the traditional disaggregation models, and the definition of a suitable set of real time diagnostics on the quality of the disaggregation and revision histories that support model selection. The exact treatment of temporal disaggregation by dynamic regression models, when the latter are formulated in the logarithms, rather than the levels, of an economic variable, is also provided. The properties of the profile and marginal likelihood are investigated and the problems with estimating the Litterman model are illustrated. In the light of the nonstationary nature of the economic time series usually entertained in practice, the suggested strategy is to fit an autoregressive distribute lag model, which, under a reparameterisation and suitable initial conditions, nests both the Chow-Lin and the Fernandez model, thereby incorporating our uncertainty about the presence of cointegration between the aggregated series and the indicators.

The paper derives an algorithm for computing leave-k-out diagnostics for the detection of patches of outliers for stationary and non-stationary state space models with regression effects. The algorithm is based on a reverse run of the Kalman filter on the smoothing errors and is both efficient and easy to implement. An illustration concerning the US index of industrial production for Textiles proves the effectiveness of multiple deletion diagnostics in unmasking clusters of outlying observations.

This paper looks at unobserved components models and examines the implied weighting pat- terns for signal extraction. There are three main themes. The first is the implications of correlated disturbances driving the components, especially those cases in which the correlation is perfect. The second is how setting up models with t- distributed disturbances leads to weighting patterns which are robust to outliers and breaks. The third is a comparison of implied weighting patterns with kernels used in nonparametric trend estimation and equivalent kernels used in spline smoothing. We also examine how weighting patterns are affected by heteroscedasticity and irregular spacing and provide an illustrative example.

The paper derives an algorithm for computing leave-k-out diagnostics for the detection of patches of outliers for stationary and non-stationary state space models with regression effects. The algorithm is based on a reverse run of the Kalman filter on the smoothing errors and is both efficient and easy to implement. An illustration concerning the US index of industrial production for Textiles proves the effectiveness of multiple deletion diagnostics in unmasking clusters of outlying observations.

This paper looks at unobserved components models and examines the implied weighting pat- terns for signal extraction. There are three main themes. The first is the implications of correlated disturbances driving the components, especially those cases in which the correlation is perfect. The second is how setting up models with t- distributed disturbances leads to weighting patterns which are robust to outliers and breaks. The third is a comparison of implied weighting patterns with kernels used in nonparametric trend estimation and equivalent kernels used in spline smoothing. We also examine how weighting patterns are affected by heteroscedasticity and irregular spacing and provide an illustrative example.

This paper looks at unobserved components models and examines the implied weighting pat- terns for signal extraction. There are three main themes. The first is the implications of correlated disturbances driving the components, especially those cases in which the correlation is perfect. The second is how setting up models with t- distributed disturbances leads to weighting patterns which are robust to outliers and breaks. The third is a comparison of implied weighting patterns with kernels used in nonparametric trend estimation and equivalent kernels used in spline smoothing. We also examine how weighting patterns are affected by heteroscedasticity and irregular spacing and provide an illustrative example.

This paper looks at unobserved components models and examines the implied weighting pat- terns for signal extraction. There are three main themes. The first is the implications of correlated disturbances driving the components, especially those cases in which the correlation is perfect. The second is how setting up models with t- distributed disturbances leads to weighting patterns which are robust to outliers and breaks. The third is a comparison of implied weighting patterns with kernels used in nonparametric trend estimation and equivalent kernels used in spline smoothing. We also examine how weighting patterns are affected by heteroscedasticity and irregular spacing and provide an illustrative example.