Latent Dynamic Factor Research Articles

CP factor model for dynamic tensors

Abstract Observations in various applications are frequently represented as a time series of multidimensional arrays, called tensor time series, preserving the inherent multidimensional structure. In this paper, we present a factor model approach, in a form similar to tensor CANDECOMP/PARAFAC (CP) decomposition, to the analysis of high-dimensional dynamic tensor time series. As the loading vectors are uniquely defined but not necessarily orthogonal, it is significantly different from the existing tensor factor models based on Tucker-type tensor decomposition. The model structure allows for a set of uncorrelated one-dimensional latent dynamic factor processes, making it much more convenient to study the underlying dynamics of the time series. A new high-order projection estimator is proposed for such a factor model, utilizing the special structure and the idea of the higher order orthogonal iteration procedures commonly used in Tucker-type tensor factor model and general tensor CP decomposition procedures. Theoretical investigation provides statistical error bounds for the proposed methods, which shows the significant advantage of utilizing the special model structure. Simulation study is conducted to further demonstrate the finite sample properties of the estimators. Real data application is used to illustrate the model and its interpretations.

Nowcasting GDP Using Dynamic Factor Model with Unknown Number of Factors and Stochastic Volatility: A Bayesian Approach

Real-time nowcasting is an assessment of current economic conditions from timely released economic series (such as monthly macroeconomic data) before the direct measure (such as quarterly GDP figure) is disseminated. Dynamic factor models (DFMs) are widely used in econometrics to bridge series with different frequencies and achieve a reduction in dimensionality. However, most of the research using DFMs often assumes the number of factors is known. A Bayesian approach is developed to identify the unknown number of factors and estimate the latent dynamic factors of DFMs accurately in a real-time nowcasting framework. The proposed method can deal with unbalanced data, which is typical of a real-time nowcasting analysis. Furthermore, the particle Gibbs with backward simulation algorithm is considered to obtain estimated stochastic volatility (SV) in monthly series efficiently. The validity of the method is demonstrated through simulation studies and an empirical study of nowcasting US’s quarterly GDP growth using monthly data series of several categories in the US market. Both the simulation and empirical studies indicate that the proposed Bayesian approach is a viable option to conduct real-time nowcasting for the US’s quarterly GDP.

On the joint volatility dynamics in international dairy commodity markets*

The present study investigates the price (co)volatility of four dairy commodities—skim milk powder, whole milk powder, butter, and cheddar cheese—in three major dairy markets. It uses a multivariate factor stochastic volatility model for estimating the time‐varying covariance and correlation matrices by imposing a low‐dimensional latent dynamic factor structure. The empirical results support four factors representing the European Union and Oceania dairy sectors as well as the milk powder markets. Factor volatilities and marginal posterior volatilities of each dairy commodity increase after the 2006/07 global (food) crisis, which also coincides with the free trade agreements enacted from 2007 onward and EU and US liberalization policy changes. The model‐implied correlation matrices show increasing dependence during the second half of 2006, throughout the first half of 2007, as well as during 2008 and 2014, which can be attributed to various regional agricultural dairy policies. Furthermore, in‐sample value at risk measures (VaRs and CoVaRs) are provided for each dairy commodity under consideration.

A latent dynamic factor approach to forecasting multivariate stock market volatility

This paper proposes a latent dynamic factor model for high-dimensional realized covariance matrices of stock returns. The approach is based on the matrix logarithm and combines common latent factors driven by HAR processes and idiosyncratic autoregressive dynamics. The model accounts for positive definiteness of covariance matrices without imposing parametric restrictions. Simulated Bayesian parameter estimates are obtained using basic Markov chain Monte Carlo methods. An empirical application to 5-dimensional and 30-dimensional realized covariance matrices shows remarkably good forecasting results, in-sample and out-of-sample.

A Latent Dynamic Factor Approach to Forecasting Multivariate Stock Market Volatility

This paper proposes a latent dynamic factor model for high-dimensional realized covariance matrices of stock returns. The approach is based on the matrix logarithm and allows for flexible dynamic dependence patterns by combining common latent factors driven by HAR processes and idiosyncratic autoregressive dynamics. The model accounts for symmetry and positive definiteness of covariance matrices without imposing parametric restrictions. The elements of the matrix logarithm are shown to constitute approximations to logarithmic variances and correlations. Common factors can therefore be interpreted as risk-factors related to market risk and diversification risk. Simulated Bayesian parameter estimates as well as positive definite covariance forecasts are obtained using basic Markov Chain Monte Carlo (MCMC) methods. An empirical application to 5-dimensional and 30-dimensional realized covariance matrices of daily New York Stock Exchange (NYSE) stock returns shows remarkably good forecasting results, in-sample and out-of-sample.

Multivariate Seasonal Adjustment, Economic Identities, and Seasonal Taxonomy

This article extends the methodology for multivariate seasonal adjustment by exploring the statistical modeling of seasonality jointly across multiple time series, using latent dynamic factor models fitted using maximum likelihood estimation. Signal extraction methods for the series then allow us to calculate a model-based seasonal adjustment. We emphasize several facets of our analysis: (i) we quantify the efficiency gain in multivariate signal extraction versus univariate approaches; (ii) we address the problem of the preservation of economic identities; (iii) we describe a foray into seasonal taxonomy via the device of seasonal co-integration rank. These contributions are developed through two empirical studies of aggregate U.S. retail trade series and U.S. regional housing starts. Our analysis identifies different seasonal subcomponents that are able to capture the transition from prerecession to postrecession seasonal patterns. We also address the topic of indirect seasonal adjustment by analyzing the regional aggregate series. Supplementary materials for this article are available online.

The long-term population dynamics of common wasps in their native and invaded range.

Populations of introduced species are often thought to perform differently, or experience different population dynamics, in their introduced range compared to their native habitat. Differences between habitats in climate, competition or natural enemies may result in populations with varying density dependence and population dynamics. We examined the long-term population dynamics of the invasive common wasp, Vespula vulgaris, in its native range in England and its invaded range in New Zealand. We used 39years of wasp density data from four sites in England, and 23years of data from six sites in New Zealand. Wasp population time series was examined using partial rate correlation functions. Gompertz population models and multivariate autoregressive state-space (MARSS) models were fitted, incorporating climatic variation. Gompertz models successfully explained 59-66% of the variation in wasp abundance between years. Density dependence in wasp populations appeared to act similarly in both the native and invaded range, with wasp abundance in the previous year as the most important variable in predicting intrinsic rate of increase (r). No evidence of cyclic population dynamics was observed. Both the Gompertz and MARSS models highlighted the role of weather conditions in each country as significant predictors of annual wasp abundance. The temporal evolution of wasp populations at all sites was best modelled jointly using a single latent dynamic factor for local trends, with the inclusion of a latent spring weather covariate. That same parsimonious multivariate model structure was optimal in both the native and invaded range. Density dependence is overwhelmingly important in predicting wasp densities and 'wasp years' in both the native and invaded range. Spring weather conditions in both countries have a major influence, probably through their impact on wasp colony initiation and early development. The population dynamics in the native range and invaded range show no evidence of cyclic boom-and-bust dynamics. Invasive species may not exhibit different population dynamics despite considerable variation in abundances throughout their distribution.

A Dynamic Yield Curve Model with Stochastic Volatility and Non-Gaussian Interactions: An Empirical Study of Non-Standard Monetary Policy in the Euro Area

We develop an econometric methodology for the study of the yield curve and its interactions with measures of non-standard monetary policy during possibly turbulent times. The yield curve is modeled by the dynamic Nelson-Siegel model while the monetary policy measurements are modeled as non-Gaussian variables that interact with latent dynamic factors, including the yield factors of level and slope. Yield developments during the financial and sovereign debt crises require the yield curve model to be extended with stochastic volatility and heavy tailed disturbances. We develop a flexible estimation method for the model parameters with a novel implementation of the importance sampling technique. We empirically investigate how the yields in Germany, France, Italy and Spain have been a ected by monetary policy measures of the European Central Bank. We model the euro area interbank lending rate EONIA by a log-normal distribution and the bond market purchases within the ECB's Securities Markets Programme by a Poisson distribution. We nd evidence that the bond market interventions had a direct and temporary e ect on the yield curve lasting up to ten weeks, and nd limited evidence that purchases changed the relationship between the EONIA rate and the term structure factors.

Interpreting the Latent Dynamic Factors by Threshold FAVAR Model

In this paper, we propose a latent threshold FAVAR model. The novelty is the interpretation of factors by observing how frequently factor loadings fall below estimated thresholds and become irrelevant. The results indicate that we are able to relate the factors to specific categories of the data without any prior specification on the data set. Furthermore, the interpretable factors, e.g., real activity factor, unemployment factor and such, are each given shocks along with policy shock to observe the responses of the other factors and individual series. The resulting impulse responses are of the expected sign and magnitude in general. Overall our results yield an intensive layout which factor is associated with different aspects of the economy and how the shocks affect the other factors and variables.

Generalized Autoregressive Score Model with Realized Measures of Volatility

We propose a new observation-driven time-varying parameter framework to model the financial return and realized variance jointly. The latent dynamic factor is updated by the scaled local density score as a function of past daily return and realized variance. The new model shares the advantages of both the GAS model of Creal et al. (2013) and Realized GARCH model of Hansen et al. (2012). It is robust to extreme outliers in observations as the volatility dynamics is related to the heavy-tailedness of innovation density. In the meanwhile, it adapts quickly to drastic volatility changes by incorporating realized measures of volatility based on high frequency data. We apply the model to a number of equity returns and demonstrate its promising performance, even during the 2008 financial crisis.

Modeling US housing prices by spatial dynamic structural equation models

This article proposes a spatial dynamic structural equation model for the analysis of housing prices at the State level in the USA. The study contributes to the existing literature by extending the use of dynamic factor models to the econometric analysis of multivariate lattice data. One of the main advantages of our model formulation is that by modeling the spatial variation via spatially structured factor loadings, we entertain the possibility of identifying similarity “regions” that share common time series components. The factor loadings are modeled as conditionally independent multivariate Gaussian Markov Random Fields, while the common components are modeled by latent dynamic factors. The general model is proposed in a state-space formulation where both stationary and nonstationary autoregressive distributed-lag processes for the latent factors are considered. For the latent factors which exhibit a common trend, and hence are cointegrated, an error correction specification of the (vector) autoregressive distributed-lag process is proposed. Full probabilistic inference for the model parameters is facilitated by adapting standard Markov chain Monte Carlo (MCMC) algorithms for dynamic linear models to our model formulation. The fit of the model is discussed for a data set of 48 States for which we model the relationship between housing prices and the macroeconomy, using State level unemployment and per capita personal income.

Forecasting the US term structure of interest rates using a macroeconomic smooth dynamic factor model

We extend the class of dynamic factor yield curve models in order to include macroeconomic factors. Our work benefits from recent developments in the dynamic factor literature related to the extraction of the common factors from a large panel of macroeconomic series and the estimation of the parameters in the model. We include these factors in a dynamic factor model for the yield curve, in which we model the salient structure of the yield curve by imposing smoothness restrictions on the yield factor loadings via cubic spline functions. We carry out a likelihood-based analysis in which we jointly consider a factor model for the yield curve, a factor model for the macroeconomic series, and their dynamic interactions with the latent dynamic factors. We illustrate the methodology by forecasting the U.S. term structure of interest rates. For this empirical study, we use a monthly time series panel of unsmoothed Fama–Bliss zero yields for treasuries of different maturities between 1970 and 2009, which we combine with a macro panel of 110 series over the same sample period. We show that the relationship between the macroeconomic factors and the yield curve data has an intuitive interpretation, and that there is interdependence between the yield and macroeconomic factors. Finally, we perform an extensive out-of-sample forecasting study. Our main conclusion is that macroeconomic variables can lead to more accurate yield curve forecasts.

Quantitative Trendspotting

Trendspotting has become an important marketing intelligence tool for identifying and tracking general tendencies in consumer interest and behavior. Currently, trendspotting is done either qualitatively by trend hunters, who comb through everyday life in search of signs indicating major shifts in consumer needs and wants, or quantitatively by analysts, who monitor individual indicators, such as how many times a keyword has been searched, blogged, or tweeted online. In this study, the authors demonstrate how the latter can be improved by uncovering common trajectories hidden behind the coevolution of a large array of indicators. The authors propose a structural dynamic factor-analytic model that can be applied for simultaneously analyzing tens or even hundreds of time series, distilling them into a few key latent dynamic factors that isolate seasonal cyclic movements from nonseasonal, nonstationary trend lines. The authors demonstrate this novel multivariate approach to quantitative trendspotting in one application involving a promising new source of marketing intelligence—online keyword search data from Google Insights for Search—in which they analyze search volume patterns across 38 major makes of light vehicles over an 81-month period to uncover key common trends in consumer vehicle shopping interest.

A Latent Dynamic Factor Approach to Forecasting Multivariate Stock Market Volatility

This paper proposes a latent dynamic factor model for low- as well as high-dimensional realized covariance matrices of stock returns. The approach is based on the matrix logarithm and allows for flexible dynamic dependence patterns by combining common latent factors driven by HAR dynamics and idiosyncratic AR(1) factors. The model accounts for symmetry and positive definiteness of covariance matrices without imposing parametric restrictions. Simulated Bayesian parameter estimates as well as positive definite (co)variance forecasts are obtained using Markov Chain Monte Carlo (MCMC) methods. An empirical application to 5-dimensional and 30-dimensional realized covariance matrices of daily New York Stock Exchange (NYSE) stock returns shows that the model outperforms other approaches of the extant literature both in-sample and out-of-sample.

Forecasting the U.S. Term Structure of Interest Rates Using a Macroeconomic Smooth Dynamic Factor Model

We extend the class of dynamic factor yield curve models for the inclusion of macro-economic factors. We benefit from recent developments in the dynamic factor literature for extracting the common factors from a large panel of macroeconomic series and for estimating the parameters in the model. We include these factors into a dynamic factor model for the yield curve, in which we model the salient structure of the yield curve by imposing smoothness restrictions on the yield factor loadings via cubic spline functions. We carry out a likelihood-based analysis in which we jointly consider a factor model for the yield curve, a factor model for the macroeconomic series, and their dynamic interactions with the latent dynamic factors. We illustrate the methodology by forecasting the U.S. term structure of interest rates. For this empirical study we use a monthly time series panel of unsmoothed Fama-Bliss zero yields for treasuries of different maturities between 1970 and 2009, which we combine with a macro panel of 110 series over the same sample period. We show that the relation between the macroeconomic factors and yield curve data has an intuitive interpretation, and that there is interdependence between the yield and macroeconomic factors. Finally, we perform an extensive out-of-sample forecasting study. Our main conclusion is that macroeconomic variables can lead to more accurate yield curve forecasts.

Mixed Measurement Dynamic Factor Models

We propose a latent dynamic factor model framework for mixed-measurement mixed-frequency panel data. Time series observations may come from different families of parametric distributions, may be observed at different frequencies, and exhibit common dynamics and cross sectional dependence due to shared exposure to latent dynamic factors. As the main complication, the likelihood does not exist in closed form for this class of models. We therefore present three different approaches to parameter and factor estimation in this framework. First, assuming a factor structure for location parameters yields a parameter driven model that can be cast into state space form. Parameters and factors estimation is then accomplished by Monte-Carlo maximum likelihood based on importance sampling. Second, we propose a less complex observation driven alternative to the parameter driven original model, for which the likelihood exists in closed form. Finally, parameter and factor estimates can be obtained by Markov Chain Monte Carlo. We use the new mixed-measurement framework for the estimation and forecasting of intertwined credit and recovery risk conditions for US Moody’s-rated firms from 1982 - 2008. The joint model allows us to construct predictive (conditional) loss densities for portfolios of bank loans and corporate bonds in the presence of non-standard sources of credit risk such as systematic frailty effects and systematic recovery risk.

Extracting Market Expectations from Yield Curves Augmented by Money Market Interest Rates: The Case of Japan

This paper attempts to extract market expectations about the Japanese economy and the BOJ's policy stance from the yen yield curves augmented by money market interest rates, during the period from the end of the quantitative easing policy in March 2006. We use (i) the swap yield curves augmented by OIS interest rates (OIS/Swap), and (ii) the JGB yield curve augmented by FB/TB interest rates. First, using the Nelson-Siegel [1987] model, we estimate three latent dynamic factors, which can be interpreted as reflecting market expectations. Second, we investigate the relative importance of price discovery for each factor between OIS/Swap and FBTB/JGB, and find that the former has a more dominant role of price discovery for all factors. Third, we estimate the efficient price for each factor common to both yield curves using a time-series structural model, which enables us to decompose each factor into the efficient price and idiosyncratic factor.

The multi-state latent factor intensity model for credit rating transitions

A new empirical reduced-form model for credit rating transitions is introduced. It is a parametric intensity-based duration model with multiple states and driven by exogenous covariates and latent dynamic factors. The model has a generalized semi-Markov structure designed to accommodate many of the stylized facts of credit rating migrations. Parameter estimation is based on Monte Carlo maximum likelihood methods for which the details are discussed in this paper. A simulation experiment is carried out to show the effectiveness of the estimation procedure. An empirical application is presented for transitions in a 7 grade rating system. The model includes a common dynamic component that can be interpreted as the credit cycle. Asymmetric effects of this cycle across rating grades and additional semi-Markov dynamics are found to be statistically significant. Finally, we investigate whether the common factor model suffices to capture systematic risk in rating transition data by introducing multiple factors in the model.

Output Volatility and Large Output Drops in Emerging Market and Developing Countries

This paper establishes that output volatility and the size of output drops have declined across all countries over the past three decades, but remain considerably higher in developing countries than in industrial countries. The paper employs a Bayesian latent dynamic factor model to decompose output growth into global, regional, and country-specific components. The favorable trends in output volatility and large output drops in developing countries are found to result from lower country-specific volatility and more benign country-specific events. Evidence from cross-section regressions over the 1970-2003 period suggest that discretionary fiscal spending volatility, and terms of trade volatility together with exchange rate flexibility are key determinants of volatility and large output drops.

The Multi-State Latent Factor Intensity Model for Credit Rating Transitions

A new empirical reduced-form model for credit rating transitions is introduced. It is a parametric intensity-based duration model with multiple states and driven by exogenous covariates and latent dynamic factors. The model has a generalized semi-Markov structure designed to accommodate many of the stylized facts of credit rating migrations. Parameter estimation is based on Monte Carlo maximum likelihood methods for which the details are discussed in this paper. A simulation experiment is carried out to show the effectiveness of the estimation procedure. An empirical application is presented for transitions between investment grade, subinvestment grade, and default ratings for U.S. corporates. The model strongly suggests the presence of a common dynamic component that can be interpreted as the credit cycle. We also show that the impact of this credit cycle is asymmetric with respect to downgrade and upgrade probabilities.