Multivariate Unobserved Components Research Articles

Multivariate Trend‐Cycle‐Seasonal Decompositions with Correlated Innovations*

AbstractMultivariate analysis can help to focus on important phenomena, including trend and cyclical movements, but any economic information in seasonality is typically ignored. The present paper aims to more fully exploit time series information through a multivariate unobserved component model for quarterly data that exhibits seasonality together with cross‐variable component correlations. We show that economic restrictions, including common trends, common cycles and common seasonals can aid identification. The approach is illustrated using Italian GDP and consumption data.

Quantifying the uncertainty of long-term macroeconomic projections

We propose a practical approach to measuring the uncertainty of long-term economic projections. The presented method quantifies the uncertainty of economic variables by using simulations from a multivariate unobserved components model in which variables are formulated as sums of stationary and nonstationary components. The method captures the correlations between both the stationary and nonstationary components of the variables and offers a seamless analysis of short- and long-term uncertainty. Experiments on artificial data demonstrate that, despite its simplicity, the method performs fairly well compared with alternative methods in terms of long-term predictive accuracy and coverage.

The Global Business Cycle within the New Commodities and the Financial Cycle: An Empirical Evidence Based on a Multivariate Unobserved Components Model (UCM)

The Global Business Cycle within the New Commodities and the Financial Cycle: An Empirical Evidence Based on a Multivariate Unobserved Components Model (UCM)

Separating aggregate discouraged and added worker effects: the case of a former transition country

Research background: We analyse the added worker effect (AWE) and the discouraged worker effect (DWE) from an aggregate perspective. The first effect refers to an increase in labour force participation in response to a decrease in the wage rate. The second effect refers to the decision by workers who have been unsuccessful in their job search to leave the labour market or to decrease their labour force participation. For our analysis, we use the case of Poland, a country with a persistently low labour force participation rate.
 Purpose of the article: While previous studies focused on the net of the two effects, we aim to analyse the two effects both separately and simultaneously. We propose a new approach for analysing the two effects. We generalise and model them as resulting from different shocks: (i) the AWE as the result of a negative wage income shock, and (ii) the DWE as the result of a positive job search time shock. The underlying assumption is that both shocks have at least a transitory effect on the labour force participation rate. However, we also track the potential long-lasting effects of these shocks, and we analyse the reactions of gender and age groups to them. While this approach demonstrates the robustness of our results, it also provides the range of the sensitivity, as it shows that there are large differences in the magnitude of the AWE and the DWE for different labour market cohorts.
 Methods: We use the multivariate unobserved component model to extract the AWE and the DWE, and we then use VAR models, applying sign and exclusion restrictions to model the underlying shocks. We use quarterly data for Poland in 1995?2019. Most of these data come from the Labour Force Survey, while the rest come from Statistics Poland.
 Findings & value added: In contrast to previous literature, which analysed only the net effect of the two effects, we model the AWE and the DWE separately. Contrary to the findings of previous research, our approach seems to confirm that both effects are simultaneously present in the labour market, and both effects influence the labour force participation rate. Thus, we find that both effects are significant. Specifically, we show that the AWE is stronger, but transitory; while the DWE is weaker, but long-lasting.

Estimating a neutral real interest rate in Russia during inflation targeting

The article aims to estimate a short-term neutral real interest rate (NRI) in Russia after transition to inflation targeting. The article provides a review of theoretical factors, which demonstrates the difference between short-term and long-term neutral real interest rate. The following analysis of estimates of NRI in different countries shows that its decline in developed and developing countries is due to various factors. Further, the article provides a comparative analysis of advantages and disadvantages of three main approaches to NRI estimation: using financial market indicators, statistical methods and multilevel theoretical models. Based on multivariate unobserved components model (UCM) for the Russian economy, the article concludes that the short-term NRI decreased from 5 to 1% in 2016—2020, whereas monetary policy was predominantly neutral.

Bayesian estimation of the long-run trend of the US economy

The main purpose of this paper is to scrutinize the long-run trend of the US real GDP during the post-war period. In the empirical analysis, we introduce multivariate unobserved components models that accommodate time-varying volatility bounded by an economically reasonable range. After accounting for the cointegration relationship among the real GDP, consumption, and investment, we find the following. (i) There was an abrupt and significant downturn in the long-run growth rate in 2007. (ii) The annualized long-run growth rate of the real GDP declined to approximately 1.69%, decreasing from a peak of nearly 3.21% during the 2000s. (iii) The bounded volatility assumption enables a trend-cycle decomposition of the US real GDP that matches the NBER’s recession dates.

Multivariate decompositions and seasonal gender employment

Multivariate analysis can help to focus on economic phenomena, including trend and cyclical movements. To allow for potential correlation with seasonality, the present paper studies a three component multivariate unobserved component model, focusing on the case of quarterly data and showing that economic restrictions, including common trends and common cycles, can ensure identification. Applied to seasonal aggregate gender employment in Australia, a bivariate male/female model with a common cycle is preferred to both univariate correlated component and bivariate uncorrelated component specifications. This model evidences distinct gender-based seasonal patterns with seasonality declining over time for females and increasing for males.

A Multivariate Unobserved Components Model to Estimate Potential Output in the Euro Area: A Production Function Based Approach

This paper builds an unobserved components model that combines a multivariate filter approach with a Cobb-Douglas production function. This combination allows potential output estimates to incorporate more economic structure than the traditional production function approach, while retaining the ability to conduct growth accounting exercises. The model is a backward-looking state space model estimated with Bayesian methods employing the Kalman filter to jointly decompose six key observable variables (real GDP, unemployment rate, labour force participation rate, hours worked per person, a measure of core inflation and wage inflation) into trend and cyclical components. To do so, it relies on several reduced form relationships across the cyclical components, such as a wage and a price Phillips curve and an Okun's law type relationship, while it also assumes common trends for a few variables and allows for hysteresis effects. The model is estimated on aggregate euro area data with Bayesian methods. The paper finds that the resulting output gap estimates have good revision properties and reasonable forecasting performance in particular in terms of GDP and core inflation vis-a-vis a set of benchmarks.

Approximate state space modelling of unobserved fractional components

We propose convenient inferential methods for potentially nonstationary multivariate unobserved components models with fractional integration and cointegration. Based on finite-order ARMA approximations in the state space representation, maximum likelihood estimation can make use of the EM algorithm and related techniques. The approximation outperforms the frequently used autoregressive or moving average truncation, both in terms of computational costs and with respect to approximation quality. Monte Carlo simulations reveal good estimation properties of the proposed methods for processes of different complexity and dimension.

Multivariate unobserved component model for an oil-exporting economy: the case of Russia

ABSTRACT This paper presents an unobserved component model for real GDP, real household consumption, and real investment of an oil-exporting economy. The model decomposes domestic variables’ dynamics into permanent and transitory components, accounting for dependence on oil prices in the short and long-run, as well as for the common long-run economic growth and the common cyclical behaviour. Estimated on the Russian macroeconomic variables, the model exhibits a strong dependence on oil prices.

Temporal Disaggregation of Overlapping Noisy Quarterly Data: Estimation of Monthly Output from Uk Value-added Tax Data

SummaryThe paper derives monthly estimates of business sector output in the UK from rolling quarterly value-added tax based turnover data. The administrative nature of the value-added tax data implies that their use could ultimately yield a more precise and granular picture of output across the economy. However, they show two particular features which complicate their exploitation: they are overlapping and subject to substantial noise. This motivates our choice of a multivariate unobserved components model for filtering and disaggregating temporally the aggregate figures. After illustrating our method by using one industry as a case-study, we estimate monthly seasonally adjusted gross output figures for the 75 industries for which the data are available. Our results show material differences from the existing output profile.

Trend Fundamentals and Exchange Rate Dynamics

We estimate a multivariate unobserved components stochastic volatility model to explain the dynamics of a panel of six exchange rates against the US Dollar. The empirical model is based on the assumption that both countries' monetary policy strategies may be well described by Taylor rules with a time-varying inflation target, a time-varying natural rate of unemployment, and interest rate smoothing. The estimates closely track major movements along with important time series properties of real and nominal exchange rates across all currencies considered. The model generally outperforms a benchmark model that does not account for changes in trend inflation and trend unemployment. (authors' abstract)

Trend Fundamentals and Exchange Rate Dynamics

We estimate a multivariate unobserved components stochastic volatility model to explain the dynamics of a panel of six exchange rates against the US dollar. The empirical model is based on the assumption that two countries’ monetary policy strategies may be well described by Taylor rules with a time‐varying inflation target, a time‐varying natural rate of unemployment, and interest rate smoothing. Compared to the existing literature, our model simultaneously provides estimates of the latent components included in a typical Taylor rule specification and the model‐based real exchange rate. Our estimates closely track major movements along with important time series properties of real and nominal exchange rates across all currencies considered, outperforming a benchmark model that does not account for changes in trend inflation and trend unemployment. More precisely, the proposed approach improves on competing models in tracking the actual evolution of the real exchange rate in terms of simple correlations while it appreciably improves on simpler competitors in terms of matching the persistence of the real exchange rate.

Real-Time Macroeconomic Forecasting With a Heteroscedastic Inversion Copula

There is a growing interest in allowing for asymmetry in the density forecasts of macroeconomic variables. In multivariate time series, this can be achieved with a copula model, where both serial and cross-sectional dependence is captured by a copula function, and the margins are nonparametric. Yet most existing copulas cannot capture heteroscedasticity well, which is a feature of many economic and financial time series. To do so, we propose a new copula created by the inversion of a multivariate unobserved component stochastic volatility model, and show how to estimate it using Bayesian methods. We fit the copula model to real-time data on five quarterly U.S. economic and financial variables. The copula model captures heteroscedasticity, dependence in the level, time-variation in higher moments, bounds on variables and other features. Over the window 1975Q1–2016Q2, the real-time density forecasts of all the macroeconomic variables exhibit time-varying asymmetry. In particular, forecasts of GDP growth have increased negative skew during recessions. The point and density forecasts from the copula model are competitive with those from benchmark models—particularly for inflation, a short-term interest rate and current quarter GDP growth. Supplementary materials for this article are available online.

Unemployment or credit: Which one holds the potential? Results for a small open economy with a low degree of financialization

This paper investigates the relative importance of unemployment versus credit in determining the potential level of real activity for a small open economy with a low degree of financialization. We use a multivariate unobserved component model (MUC) to derive the potential output and the associated output gap for the Lithuanian economy. The model is estimated via Bayesian methods and the time paths of unobserved variables are extracted via the Kalman filter. The inclusion of unemployment in the MUC model substantially improves the estimates of the output gap in real time. Adding information about credit further emphasizes the overheating of the economy in the pre-crisis period, both in real time and ex post. Including credit preserves the conclusions regarding turning points. We uncover a strong negative correlation between the model-implied unemployment gap (without accounting for credit) and real credit growth. Data revisions do not appear to be the primary source of revisions of output gap estimates.

Real-Time Macroeconomic Forecasting with a Heteroskedastic Inversion Copula

There is a growing interest in allowing for asymmetry in the density forecasts of macroeconomic variables. In multivariate time series, this can be achieved with a copula model, where both serial and cross-sectional dependence is captured by a copula function, and the margins are nonparametric. Yet most existing copulas cannot capture heteroskedasticity well, which is a feature of many economic and financial time series. To do so, we propose a new copula created by the inversion of a multivariate unobserved component stochastic volatility model, and show how to estimate it using Bayesian methods. We fit the copula model to real-time data on five quarterly U.S. economic and financial variables. The copula model captures heteroskedasticity, dependence in the level, time-variation in higher moments, bounds on variables and other features. Over the window 1975Q1 -- 2016Q2, the real-time density forecasts of all the macroeconomic variables exhibit time-varying asymmetry. In particular, forecasts of GDP growth have increased negative skew during recessions. The point and density forecasts from the copula model are competitive with those from benchmark models -- particularly for inflation, a short term interest rate and current quarter GDP growth.

Modeling the Business and Financial Cycle in a Multivariate Structural Time Series Model

We consider a multivariate unobserved component time series model to disentangle the short-term and medium-term cycle for the G7 countries and the Netherlands using four key macroeconomic and financial time series. The novel aspect of our approach is that we simultaneously decompose the short-term and medium-term dynamics of these variables by means of a combination of their estimated cycles. Our results show that the cyclical movements of credit volumes and house prices are mostly driven by the medium-term cycle, while the macroeconomic variables are equally driven by the short-term and medium-term cycle. For most countries, the co-movement between the cycles of the financial and macroeconomic variables is mainly present in the medium-term. First, we find strong co-cyclicality between the medium-term cycles of house prices and GDP in all countries we analyzed. Second, the relation between the medium-term cycles of GDP and credit is more complex. We find strong concordance between both cycles in only three countries. However, in three other countries we find ‘indirect’ concordance, i.e. the medium-term cycles of credit and house prices share co-cyclicality, while in turn the medium-term cycles of house prices and GDP share commonality. This outcome might indicate that the house price cycle is – at least partly – driven by the credit cycle. Lastly, the cross-country concordance of both the short-term cycles and the medium-term cycles of GDP, house prices and credit is low. Hence, the bulk of the cyclical movements seem to be driven by domestic rather than global factors.

Business, housing, and credit cycles

SummaryWe use multivariate unobserved components models to estimate trend and cyclical components in gross domestic product (GDP), credit volumes, and house prices for the USA and the five largest European economies. With the exception of Germany, we find large and long cycles in credit and house prices, which are highly correlated with a medium‐term component in GDP cycles. Differences across countries in the length and size of cycles appear to be related to the properties of national housing markets. The precision of pseudo real‐time estimates of credit and house price cycles is roughly comparable to that of GDP cycles.

Trend inflation estimates for Thailand from disaggregated data

This paper constructs a new trend inflation measure for Thailand based on the multivariate unobserved components model with stochastic volatility and outlier adjustments (MUCSVO) of Stock and Watson (2016). Similar to core inflation, the MUCSVO produces an estimate of trend inflation utilizing information in disaggregated data, but also allows for time-varying weights that depend on the volatility, persistence and comovement of the underlying sectoral inflation series. Based on the empirical results, the majority of sectoral weights show significant time-variation in contrast to their relatively stable expenditure shares. Volatile food and energy sectors that are typically excluded from core inflation measures also turn out to help explain approximately 10 percent of MUCSVO trend inflation rate movements. Compared against other benchmark trend inflation measures, we show that the MUCSVO delivers trend estimates that are smoother, more precise, and are able to forecast average inflation over the 1–3 year horizon more accurately both in-sample and out-of-sample, especially since the year 2000.

Tracking Growth and the Business Cycle: A Stochastic Common Cycle Model for the Euro Area

This paper proposes a new model-based method to obtain a coincident indicator for the business cycle. A dynamic factor model with trend components and a common cycle component is considered which can be estimated using standard maximum likelihood methods. The multivariate unobserved components model includes a stationary higher order cycle. Also, higher order trends can be part of the analysis. These generalisations lead to a business cycle that is similar to a band-pass one. Furthermore, cycle shifts for individual time series are incorporated within the model and estimated simultaneously with the remaining parameters. This feature permits the use of leading, coincident and lagging variables to obtain the business cycle coincident indicator without prior analysis of their lead-lag relationship. Besides the business cycle indicator, the model-based approach also allows to get a growth rate indicator. In the empirical analysis for the Euro area, both indicators are obtained based on nine key economic time series including gross domestic product, industrial production, unemployment, confidence indicators and interest rate spread. This analysis contrasts sharply with earlier multivariate approaches. In particular, our more parsimonious approach leads to a growth rate indicator for the Euro area that is similar to the one of EuroCOIN. The latter is based on a more involved approach by any standard and uses hundreds of time series from individual countries belonging to the Euro area.