Non-Stationary Dynamic Factor Models Research Articles

Permanent-Transitory decomposition of cointegrated time series via dynamic factor models, with an application to commodity prices

Summary We propose a cointegration-based Permanent-Transitory decomposition for nonstationary dynamic factor models (DFMs). Our methodology exploits the cointegration relations among the observable variables and assumes they are driven by a common and an idiosyncratic component. The common component is further split into a long-term nonstationary and a short-term stationary part. A Monte Carlo experiment shows that incorporating the cointegration structure into the DFM leads to a better reconstruction of the space spanned by the factors, compared to the most standard technique of applying a factor model in differenced systems. We apply our procedure to a set of commodity prices to analyse the co-movement among different markets and find that commodity prices move together mostly due to long-term common forces; while the trend for the prices of most primary goods is declining, metals and energy exhibit an upward or at least stable pattern since the 2000s.

Disinflation and Reliability of Underlying Inflation Measures

We estimated a Non-Stationary Dynamic Factor model and used it to generate artificial episodes of disinflation (permanent change in the mean inflation rate). These datasets were used to test the performance of alternative underlying inflation measures. We found that the benchmark underlying inflation measures (based on unobserved trend extraction) are more severely affected by disinflation than the alternative simpler methods (based on exclusion or reweighting approaches). Alternatively, a Non-Stationary Dynamic Factor model may be employed for extraction of the unobserved trend to be used as an underlying inflation measure.

Estimating Non-stationary Common Factors: Implications for Risk Sharing

In this paper, we analyze and compare the finite sample properties of alternative factor extraction procedures in the context of non-stationary Dynamic Factor Models (DFMs). On top of considering procedures already available in the literature, we extend the hybrid method based on the combination of principal components and Kalman filter and smoothing algorithms to non-stationary models. We show that if the idiosyncratic noises are stationary, procedures based on extracting the factors using the non-stationary original series work better than those based on differenced variables. We apply the methodology to the analysis of cross-border risk sharing by fitting non-stationary DFM to aggregate Gross Domestic Product and consumption of a set of 21 industrialized countries from the Organization for Economic Co-operation and Development (OECD). The goal is to check if international risk sharing is a short- or long-run issue.

Do National Account Statistics Underestimate US Real Output Growth?

In this note, we introduce a new estimate of GDO obtained from a Non-Stationary Dynamic Factor model estimated on a large dataset of US macroeconomic indicators.

Common Factors, Trends, and Cycles in Large Datasets

This paper considers a non-stationary dynamic factor model for large datasets to disentangle long-run from short-run co-movements. We first propose a new Quasi Maximum Likelihood estimator of the model based on the Kalman Smoother and the Expectation Maximisation algorithm. The asymptotic properties of the estimator are discussed. Then, we show how to separate trends and cycles in the factors by mean of eigenanalysis of the estimated non-stationary factors. Finally, we employ our methodology on a panel of US quarterly macroeconomic indicators to estimate aggregate real output, or Gross Domestic Output, and the output gap.

A dynamic factor model for the Mexican economy: are common trends useful when predicting economic activity?

In this paper we propose to use the common trends of the Mexican economy in order to predict economic activity one and two steps ahead. We exploit the cointegration properties of the macroeconomic time series, such that, when the series are I(1) and cointegrated, there is a factor representation, where the common factors are the common trends of the macroeconomic variables. Thus, we estimate a large non-stationary dynamic factor model using principal components (PC) as suggested by Bai (J Econom 122(1):137–183, 2004), where the estimated common factors are used in a factor-augmented vector autoregressive model to forecast the Global Index of Economic Activity. Additionally, we estimate the common trends through partial least squares. The results indicate that the common trends are useful to predict Mexican economic activity, and reduce the forecast error with respect to benchmark models, mainly when estimated using PC.

Non-Stationary Dynamic Factor Models for Large Datasets

We study a Large-Dimensional Non-Stationary Dynamic Factor Model where (1) the factors Ft are I(1) and singular, that is Ft has dimension r and is driven by q dynamic shocks with q less than r, (2) the idiosyncratic components are either I(0) or I(1). Under these assumption the factors Ft are cointegrated and modeled by a singular Error Correction Model. We provide conditions for consistent estimation, as both the cross-sectional size n, and the time dimension T, go to infinity, of the factors, the loadings, the shocks, the ECM coefficients and therefore the Impulse Response Functions. Finally, the numerical properties of our estimator are explored by means of a MonteCarlo exercise and of a real-data application, in which we study the effects of monetary policy and supply shocks on the US economy.

The international transmission channels of US supply and demand shocks: Evidence from a non-stationary dynamic factor model for the G7 countries

We employ a multi-country non-stationary dynamic factor model to assess spillover effects and transmission channels of US supply and demand shocks on a variety of macroeconomic variables in individual non-US G7 countries. We find that trade, financial and confidence channels all play a significant role in the international transmission of US shocks. However, the results point to substantial heterogeneities of shock transmission across the individual G7 economies. In particular, we find negative transmission effects for Italy and Japan as the only two G7 countries not well integrated into global value chains. Moreover, the exchange rate responses of Germany, France and Italy turn out to be far less pronounced in comparison to the other G7 economies which we relate to their membership of the euro area and their coordinated monetary policies prior to the establishment of the euro. Whereas we document a close comovement of stock market dynamics across the G7 countries, we find credit and real estate markets to be less synchronized. We do not find the effects and transmission channels to be fundamentally affected by the post-2008 economic environment.

Dynamic Factor Models, Cointegration, and Error Correction Mechanisms

The paper studies Non-Stationary Dynamic Factor Models such that: (1) the factors Ft are I(1) and singular, i.e. Ft has dimension r and is driven by a q-dimensional white noise, the common shocks, with q < r, and (2) the idiosyncratic components are I(1). We show that Ft is driven by r -c permanent shocks, where c is the cointegration rank of Ft, and q - (r - c) < c transitory shocks, thus the same result as in the non-singular case for the permanent shocks but not for the transitory shocks. Our main result is obtained by combining the classic Granger Representation Theorem with recent results by Anderson and Deistler on singular stochastic vectors: if (1 - L)Ft is singular and has rational spectral density then, for generic values of the parameters, Ft has an autoregressive representation with a finite-degree matrix polynomial fulfilling the restrictions of a Vector Error Correction Mechanism with c error terms. This result is the basis for consistent estimation of Non-Stationary Dynamic Factor Models. The relationship between cointegration of the factors and cointegration of the observable variables is also discussed.

Non-Stationary Dynamic Factor Models for Large Datasets

We study a Large-Dimensional Non-Stationary Dynamic Factor Model where (1) the factors Ft are I (1) and singular, that is Ft has dimension r and is driven by q dynamic shocks with q less than r, (2) the idiosyncratic components are either I (0) or I (1). Under these assumption the factors Ft are cointegrated and modeled by a singular Error Correction Model. We provide conditions for consistent estimation, as both the cross-sectional size n, and the time dimension T, go to infinity, of the factors, the loadings, the shocks, the ECM coefficients and therefore the Impulse Response Functions. Finally, the numerical properties of our estimator are explored by means of a MonteCarlo exercise and of a real-data application, in which we study the effects of monetary policy and supply shocks on the US economy.

Non-Stationary Dynamic Factor Models for Large Datasets

We develop the econometric theory for Non-Stationary Dynamic Factor models for large panels of time series, with a particular focus on building estimators of impulse response functions to unexpected macroeconomic shocks. We derive conditions for consistent estimation of the model as both the cross-sectional size, n, and the time dimension, T, go to infinity, and whether or not cointegration is imposed. We also propose a new estimator for the non-stationary common factors, as well as an information criterion to determine the number of common trends. Finally, the numerical properties of our estimator are explored by means of a MonteCarlo exercise and of a real-data application, in which we study the effects of monetary policy and supply shocks on the US economy.

Comovements and Heterogeneity in the Euro Area Analyzed in a Non-Stationary Dynamic Factor Model

This paper establishes stylized facts on comovements and heterogeneity of individual euro-area countries' output and price developments in the past two decades. For this purpose, a non-stationary structural dynamic factor model is fit to a dataset of euro-area macroeconomic variables. The main results are as follows. Common permanent factors are important in explaining individual countries' output and price developments in the euro area. Output and prices are not only hit by permanent common, but also by permanent idiosyncratic shocks. The latter are mainly responsible for cross-country heterogeneity during most of the sample. The asymmetric transmission of common shocks plays a minor role; there is no strong evidence that some common shocks lead to greater cross-country heterogeneity than others.

Comovements and Heterogeneity in the Comovements and Heterogeneity in the Dynamic Factor Model

This paper seeks to assess comovements and heterogeneity in the euro area by fitting a nonstationary dynamic factor model (Bai and Ng, 2004), augmented with a structural factor setup (Forni and Reichlin, 1998), to a large set of euro-area macroeconomic variables observed between 1982 and 2003. This framework allows us to estimate stationary and non-stationary common factors and idiosyncratic components, to identify the structural shocks behind the common factors and assess their transmission to individual EMU countries. Our most important findings are the following. EMU countries share five common trends. However, the source of non-stationarity of individual countries' key macroeconomic variables is not only pervasive. Instead, most countries' output and inflation are also affected by long-lasting idiosyncratic shocks. Unweighted dispersion is primarily due to idiosyncratic shocks rather than the asymmetric spread of common shocks. However, the latter seems to be the main driving force of weighted dispersion of output at the end of the 1980s and the beginning of the 1990s and again from 1999 on and of inflation in the mid-1980s and the mid-1990s. To examine the transmission of common shocks to individual EMU countries in more detail, we identify five structural common shocks, namely two euro-area supply shocks, one euro-area demand shock, one common monetary policy shock and a US shock. We find similar output and inflation responses across countries (with some exceptions), and similarity generally increases with the horizon.

Joint forecasts of Southern European fertility rates with non-stationary dynamic factor models

Fertility forecasting is the weak point of stochastic population forecasts. Changing trends account for large forecasting errors even a few years ahead. On the other hand, fertility trends have been shown to be common to different European countries. This paper explores the possibility of improving forecasts by jointly modelling total fertility rate (TFR) trends within relatively homogeneous clusters of countries. We propose different varieties of non-stationary dynamic factor models applied to Southern European countries. The forecasting performance of the common factor models is compared to alternative univariate and multivariate forecasting methods using data for the period 1950–2000. Joint forecasts show forecasting gains in terms of root mean square error of prediction (RMSE), particularly for longer forecast horizons. This corroborates the convenience of modelling fertility jointly for population forecasting.

Forecasting with nonstationary dynamic factor models

Abstract In this paper we analyze the structure and the forecasting performance of the dynamic factor model. It is shown that the forecasts obtained by the factor model imply shrinkage pooling terms, similar to the ones obtained from hierarchical Bayesian models that have been applied successfully in the econometric literature. Thus, the results obtained in this paper provide an additional justification for these and other types of pooling procedures. The expected decrease in MSE for using a factor model versus univariate ARIMA and shrinkage models are studied for the one factor model. Monte Carlo simulations are presented to illustrate this result. A factor model is also built to forecast GNP of European countries and it is shown that the factor model can provide a substantial improvement in forecasts with respect to both univariate and shrinkage univariate forecasts.