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The Kalman filter formula, given by the linear recursive algorithm, is usually used for estimation of the time-varying parameter model. The filtering formula, introduced by Kalman (1960) and Kalman and Bucy (1961), requires the initial state variable. The obtained state estimates are influenced by the initial value when the initial variance is not too large. To avoid the choice of the initial state variable, in this paper we utilize the diffuse prior for the initial density. Moreover, using the Gibbs sampler, random draws of the state variables given all the data are generated, which implies that random draws are generated from the fixed-interval smoothing densities. Using the EM algorithm, the unknown parameters included in the system are estimated. As an example, we estimate a traditional consumption function for both the U.S. and Japan.

In this paper, we propose two parametric alternatives to the standard GJR-GARCH model of Glosten et al. (1993), based on additive and multiplicative decompositions of the variance. They allow the variance of the model to have a smooth time-varying structure. The suggested parameterizations describe structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. The main focus is on the multiplicative decomposition of the variance into an unconditional and conditional components. Estimation of the multiplicative model is discussed in detail. An empirical application to daily stock returns illustrates the functioning of the model. The results show that the ‘long memory type behaviour’ of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance., COMPETE, QREN, FEDER, Fundação para a Ciência e a Tecnologia (FCT)

This paper estimates a reaction function with forward-looking time-varying parameters for changes in the Brazilian monetary policy under inflation targeting regime. As the policy rule has endogenous regressors, the conventional Kalman filter can t be applied. Thus, a two-step procedure of the type Heckman (1976) is used to estimate the hyperparameters consistent model. The results show that: i) there is strong empirical evidence of endogeneity of the regressors of monetary policy rule, ii) the expected interest rate was above 10% throughout the analysis period to an average of 11%; iii) response the Selic rate to inflation varies considerably throughout the period and has shown a declining trend, iv) the response of interest rates relative to inflation deviation from the target with the principle of Taylor; v) the coefficient of smoothing rate interest has been constant throughout the period; vi) that the BCB had in much of the period analyzed an aversion recession by allowing inflation above target. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Este trabalho estima uma função de reação forward-looking com parâmetros variando no tempo para verificar mudanças na condução da política monetária brasileira sob o regime de metas de inflação. Como a regra de política apresenta regressores endógenos, o filtro de Kalman convencional não pode ser aplicado. Diante disso, um procedimento em dois passos do tipo de Heckman (1976) é utilizado para estimação consistente dos hiperparâmetros do modelo. Os resultados mostram que: i) há forte evidência empírica de endogeneidade dos regressores da regra de política monetária; ii) que a taxa de juros esperada esteve acima de 10% durante todo o período analisado, tendo uma média de 11%; iii) a resposta da taxa Selic à inflação varia consideravelmente ao longo do período e tem mostrado uma tendência decrescente; iv) a resposta da taxa de juros em relação ao desvio da inflação a meta respeita o princípio de Taylor; v) o coeficiente de suavização da taxa de juros foi constante durante todo o período analisado; vi) que o BCB teve em boa parte do período analisado uma aversão recessão, permitindo uma inflação acima da meta.

We examine the evolution of monetary policy rules in a group of inflation targeting countries (Australia, Canada, New Zealand, Sweden and the United Kingdom), applying a moment-based estimator in a time-varying parameter model with endogenous regressors. Using this novel flexible framework, our main findings are threefold. First, monetary policy rules change gradually, pointing to the importance of applying a time-varying estimation framework. Second, the interest rate smoothing parameter is much lower than typically reported by previous time-invariant estimates of policy rules. External factors matter for all countries, although the importance of the exchange rate diminishes after the adoption of inflation targeting. Third, the response of interest rates to inflation is particularly strong during periods when central bankers want to break a record of high inflation, such as in the UK or Australia at the beginning of the 1980s. Contrary to common wisdom, the response becomes less aggressive after the adoption of inflation targeting, suggesting a positive anchoring effect of this regime on inflation expectations. This result is supported by our finding that inflation persistence as well as the policy neutral rate typically decreased after the adoption of inflation targeting.

This paper applies the Becker-Murphy (1988) theory of rational addiction to the case of carbonated soft drinks, using a time-varying parameter model and scanner data from 46 U.S. cities. Empirical results provide strong evidence that carbonated soft drinks are rationally addictive, thus opening the door to taxation and regulation. Taking rational addition into account, estimated demand elasticities are much lower than previous estimates using scanner data.

Material from this paper has been presented at the International Symposium on Econometric Theory and Applications, Xiamen, April 2006; 5th Annual International Conference Forecasting Financial Markets and Economic Decision-making, Lodz, May 2006; 13th International Conference on Forecasting Financial Markets, Marseille, May-June 2006; 26th International Symposium on Forecasting, Santander, June 2006; Workshop Volatility day, Stockholm, November 2006; Nordic Econometric Meeting, Tartu, May 2007; Symposium on "Long Memory", Aarhus, June-July 2007; LACEA-LAMES, Bogotá, October 2007; and at the seminars at Banca d’Italia, Rome, European University Institute, Florence, Humboldt University, Berlin, University of Minho, Braga, Stockholm School of Economics, Leonard N. Stern School of Business at New York University, and University of Vilnius. We would like to thank the articipants at these occasions for their comments, and Stefan Lundbergh, Mika Meitz, Anders Rahbek and Esther Ruiz for useful discussions and suggestions. The responsibility for any errors and hortcomings in this article remains ours., In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the conditional variance to have a smooth time-varying structure of either additive or multiplicative type. The suggested parameterizations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. A modelling strategy for these new time-varying parameter GARCH models is developed. It relies on a sequence of Lagrange multiplier tests, and the adequacy of the estimated models is investigated by Lagrange multiplier type misspecification tests. Finite-sample properties of these procedures and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance., This research has been supported by the Danish National Research Foundation. NIPE – Núcleo de Investigação em Políticas Económicas – is supported by the Portuguese Foundation for Science and Technology through the Programa Operacional Ciência e Inovação 2010 (POCI 2010) of the III Quadro Comunitário de Apoio (QCA III), which is financed by FEDER and Portuguese funds.

This paper is an empirical research of a monetary policy rule for a small open economy model, taking Switzerland as a case-study. A time-varying parameter model of a monetary policy reaction function is proposed to integrate various trade-offs to be made about various macroeconomic variables -- inflation, the output gap and the real exchange rate gap. The Kalman filter estimations of the time-varying parameters shows how rational economic agents combine past and new information to make new expectations about the state variables. The uncertainty created by the time-varying parameter model, and estimated by the conditional forecast error and conditional variance, is decomposed into two components, the uncertainty related to the time-varying parameters and the uncertainty related to the purely monetary shock. Most of the monetary shock uncertainty comes from the time-varying parameters and not from the pure monetary shock. The Lucas and Friedman hypotheses about the impact of uncertainty on output are revisited, using a conditional variance to test them. Both hypothesis are confirmed, using the one-step ahead conditional variance of the monetary shock. An inverse relation between the magnitude of the response on output to the nominal shock and the variance of this shock is found, as Lucas had predicted. Moreover, there is a direct negative impact of uncertainty which reduces output in the long-term.

This article considers constructing monetary aggregates in the presence of financial market innovations and changes in the relationship between individual assets and output. We propose two procedures for constructing a monetary aggregate with the objective of providing a reliable monetary leading indicator of nominal GDP. In the first, subaggregates discretely switch in and out; in the second, the aggregate's growth is a time-varying weighted average of the growth of the subaggregates, where the weights follow a multivariate random walk. These procedures are used to examine augmenting M2 with stock and/or bond mutual funds. The alternative aggregates are broadly similar to M2, but during 1992–1993 they outperform M2., Economics