Testing for common trends in non-stationary large datasets

M. Barigozzi, L. Trapani
Journal of Business & Economic Statistics, 2021

We propose a testing based procedure to determine the number of common trends in a large non-stationary dataset. Our procedure is based on a factor representation, where we determine whether there are (and how many) common factors (i) with linear trends, and (ii) with stochastic trends. Cointegration among the factors is also permit- ted. Our analysis is based on the fact that those largest eigenvalues of a suitably scaled covariance matrix of the data corresponding to the common factor part diverge, as the dimension N of the dataset diverges, whilst the others stay bounded. Therefore, we propose a class of randomised test statistics for the null that the p-th largest eigenvalue diverges, based directly on the estimated eigenvalue. The tests only requires minimal assumptions on the data generating process. Monte Carlo evidence shows that our procedure has very good finite sample properties, clearly dominating competing ap- proaches when no common trends are present. We illustrate our methodology through an application to US bond yields with different maturities observed over the last 30 years.

Consistent estimation of high-dimensional factor models when the factor number is over-estimated

M. Barigozzi, H. Cho
Electronic Journal of Statistics, 2020

A high-dimensional r-factor model for an n-dimensional vector time series is characterised by the presence of a large eigengap (increasing with n) between the r-th and the (r+1)-th largest eigenvalues of the covariance matrix. Consequently, Principal Component (PC) analysis is the most widely-adopted estimation method for factor models and its consistency, when r is correctly estimated, is well-established in the literature. However, popular factor number estimators often suffer from the lack of an obvious eigengap in empirical eigenvalues and tend to over-estimate r due, for example, to the existence of non-pervasive factors a↵ecting only a subset of the variables. We show that the error in the PC estimator resulting from the over-estimation of r is non-negligible, which in turn leads to the violation of the conditions required for factor-based large covariance estimation. To remedy this, we propose new estimators of the factor model based on scaling the entries of the sample eigenvectors. We show both theoretically and numerically that the proposed estimators successfully control for the over-estimation error, and investigate their performance when applied to risk minimisation of a portfolio of financial time series.

Large-dimensional dynamic factor models: estimation of impulse-response functions with I(1) cointegrated factors

M. Barigozzi, M. Lippi, M. Luciani
Journal of Econometrics, 2021

We study a large-dimensional Dynamic Factor Model where: (i) the vector of factors Ft is I(1) and driven by a number of shocks that is smaller than the dimension of Ft; and, (ii) the idiosyncratic components are either I(1) or I(0). Under (i), the factors Ft are cointegrated and can be modeled as a Vector Error Correction Model (VECM). Under (i) and (ii), we provide consistent estimators, as both the cross-sectional size n and the time dimension T go to infinity, for the factors, the loadings, the shocks, the coefficients of the VECM and therefore the Impulse-Response Functions (IRF) of the observed variables to the shocks. Furthermore: possible deterministic linear trends are fully accounted for, and the case of an unrestricted VAR in the levels Ft, instead of a VECM, is also studied. The finite-sample properties the proposed estimators are explored by means of a MonteCarlo exercise. Finally, we revisit two distinct and widely studied empirical applications. By correctly modeling the long-run dynamics of the factors, our results partly overturn those obtained by recent literature. Specifically, we find that: (i) oil price shocks have just a temporary effect on US real activity; and, (ii) in response to a positive news shock, the economy first experiences a significant boom, and then a milder recession.

Time-varying general dynamic factor models and the measurement of financial connectedness

M. Barigozzi, M. Hallin, S. Soccorsi, R. von Sachs
Journal of Econometrics, 2020

We propose a new time-varying Generalized Dynamic Factor Model for high-dimensional, locally stationary time series. Estimation is based on dynamic principal component analysis jointly with singular VAR estimation, and extends to the locally stationary case the one-sided estimation method proposed by Forni et al. (2017) for stationary data. We prove consistency of our estimators of time-varying impulse response functions as both the sample size T and the dimension n of the time series grow to infinity. This approach is used in an empirical application in order to construct a time-varying measure of financial connectedness for a large panel of adjusted intra-day log ranges of stocks. We show that large increases in long-run connectedness are associated with the main financial turmoils. Moreover, we provide evidence of a significant heterogeneity in the dynamic responses to common shocks in time and over didifferent scales, as well as across industrial sectors.

Sequential testing for structural stability in approximate factor models

M. Barigozzi and L. Trapani
Stochastic Processes and their Applications, 2020

We develop a monitoring procedure to detect changes in a large approximate factor model. Letting r be the number of common factors, we base our statistics on the fact that the (r+1)-th eigenvalue of the sample covariance matrix is bounded under the null of no change, whereas it becomes spiked under changes. Given that sample eigenvalues cannot be estimated consistently under the null, we randomise the test statistic, obtaining a sequence of i.i.d statistics, which are used for the monitoring scheme. Numerical evidence shows a very small probability of false detections, and tight detection times of change-points.

Generalized dynamic factor models and volatilities: consistency, rates, and prediction intervals

M. Barigozzi and M. Hallin
Journal of Econometrics, 2020

Volatilities, in high-dimensional panels of economic time series with a dynamic factor structure on the levels or returns, typically also admit a dynamic factor decomposition. We consider a two-stage dynamic factor model method recovering the common and idiosyncratic components of both levels and log-volatilities. Specifically, in a first estimation step, we extract the common and idiosyncratic shocks for the levels, from which a log-volatility proxy is computed. In a second step, we estimate a dynamic factor model, which is equivalent to a multiplicative factor structure for volatilities, for the log-volatility panel. By exploiting this two-stage factor approach, we build one-step-ahead conditional prediction intervals for large n x T panels of returns. Those intervals are based on empirical quantiles, not on conditional variances; they can be either equal- or unequal- tailed. We provide uniform consistency and consistency rates results for the proposed estimators as both n and T tend to infinity. We study the finite-sample properties of our estimators by means of Monte Carlo simulations. Finally, we apply our methodology to a panel of asset returns belonging to the S&P100 index in order to compute one-step-ahead conditional prediction intervals for the period 2006-2013. A comparison with the componentwise GARCH benchmark (which does not take advantage of cross-sectional information) demon- strates the superiority of our approach, which is genuinely multivariate (and high-dimensional), nonparametric, and model-free.

Cointegration and error correction mechanisms for singular stochastic vectors

M. Barigozzi, M. Lippi, and M. Luciani
Econometrics, 2020

Large-Dimensional Dynamic Factor Models and Dynamic Stochastic General Equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e. vectors of dimension r which are driven by a q-dimensional white noise, with q < r. The present paper studies Cointegration and Error Correction representations for an I(1) singular stochastic vector yt. It is easily seen that yt is necessarily cointegrated with cointegrating rank c ≥ r − q. Our contributions are: (i) we generalize Johansen’s proof of the Granger Representation Theorem to I(1) singular vectors under the assumption that yt has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of yt has a finite-degree polynomial. The relationship between cointegration of the factors and cointegration of the observable variables in a Large-Dimensional Factor Model is also discussed.

NETS: Network estimation for time series

M. Barigozzi and C. Brownlees
Journal of Applied Econometrics, 2019

We model a large panel of time series as a VAR where the autoregressive matrices and the inverse covariance matrix of the system innovations are assumed to be sparse. The system has a network representation in terms of a directed graph representing predictive Granger relations and an undirected graph representing contemporaneous partial correlations. A lasso algorithm called NETS is introduced to estimate the model. We apply the methodology to analyse a panel of volatility measures of ninety bluechips. The model captures an important fraction of total variability, on top of what is explained by volatility factors, and improves out-of-sample forecasting.

Power-law partial correlation network models

M. Barigozzi, C. Brownlees, and G. Lugosi
Electronic Journal of Statistics, 2018

We introduce a class partial correlation network models whose network structure is determined by a random graph. I n particular in this work we focus on a version of the model in which the random graph has a power-law degree distribution. A number of cross-sectional dependence properties of this class of models are derived. The main results we establish is that when the random graph is power-law, the system exhibits a high degree of collinearity. More precisely, the largest eigenvalues of the inverse covariance matrix converge to an affine function of the degrees of the most interconnected vertices in the network. The result implies that the largest eigenvalues of the inverse covariance matrix are approximately power-law distributed, and that, as the system dimension increases, the eigenvalues diverge. As an empirical illustration we analyse a panel of stock returns of a large set of companies listed in the S&P 500 and show that the covariance matrix of returns exhibits empirical features that are consistent with our power-law model.

Simultaneous multiple change-point and factor analysis for high-dimensional time series

M. Barigozzi, H. Cho, and P. Fryzlewicz
Journal of Econometrics, 2018

We propose the first comprehensive treatment of high-dimensional time series factor models with multiple change-points in their second-order structure. We operate under the most flexible definition of piecewise stationarity, and estimate the number and loca- tions of change-points consistently as well as identifying whether they originate in the common or idiosyncratic components. Through the use of wavelets, we transform the problem of change-point detection in the second-order structure of a high-dimensional time series, into the (relatively easier) problem of change-point detection in the means of high-dimensional panel data. Also, our methodology circumvents the difficult issue of the accurate estimation of the true number of factors in the presence of multiple change-points by adopting a screening procedure. We further show that consistent factor analysis is achieved over each segment defined by the change-points estimated by the proposed methodology. In extensive simulation studies, we observe that factor analysis prior to change-point detection improves the detectability of change-points, and identify and describe an interesting ‘spillover’ effect in which substantial breaks in the idiosyncratic components get, naturally enough, identified as change-points in the common compo- nents, which prompts us to regard the corresponding change-points as also acting as a form of ‘factors’. Our methodology is implemented in the R package factorcpt, available from CRAN.

On the stability of euro area money demand and its implications for monetary policy

M. Barigozzi and A. Conti
Oxford Bulletin of Economics and Statistics, 2018

We employ a recent time-varying cointegration test to revisit the usefulness of long-run money demand equations for the ECB, addressing the issue of their instability by means of a model evaluation exercise. Building on the results, we make a twofold contribution. First, we propose a novel stable money demand equation relying on two crucial factors: a speculative motive, represented by domestic and foreign price-earnings ratios, and a precautionary motive, measured by changes in unemployment. Second, we use the model to derive relevant policy implications for the ECB, since excess liquidity looks more useful for forecasting stock market busts than future inflation. Overall, this evidence points to (i) a possible evolution of the monetary pillar in the direction of pursuing financial stability and (ii) the exclusion of a sudden liquidity–driven inflationary burst after the exit from the prolonged period of unconventional monetary measures.

Identification of global and local shocks in international financial markets via general dynamic factor models

M. Barigozzi, M. Hallin, and S. Soccorsi
Journal of Financial Econometrics, 2018

We employ a two-stage general dynamic factor model to analyze co-movements be- tween returns and between volatilities of stocks from the US, European, and Japanese financial markets. We find two common shocks driving the dynamics of volatilities – one global shock and one US-European shock – and four local shocks driving returns, but no global one. Co-movements in returns and volatilities increased considerably in the period 2007-2012 associated with the Great Financial Crisis and the European Sovereign Debt Crisis. We interpret this finding as the sign of a surge, during crises, of interdependencies across markets, as opposed to contagion. Finally, we introduce a new method for structural analysis in general dynamic factor models which is applied to the identification of volatility shocks via natural timing assumptions. The global shock has homogeneous dynamic effects within each individual market but more heterogeneous effects across them, and is useful for predicting aggregate realized volatilities.

Generalized dynamic factor models and volatilities: Estimation and forecasting

M. Barigozzi and M. Hallin
Journal of Econometrics, 2017

In large panels of financial time series with dynamic factor structure on the levels or returns, the volatilities of the common and idiosyncratic components often exhibit strong correlations, indicating that both are exposed to the same market volatility shocks. This suggests, alongside the dynamic factor decomposition of returns, a dynamic factor decomposition of volatilities or volatility proxies. Based on this observation, Barigozzi and Hallin (2015) proposed an entirely non-parametric and model-free two-step general dynamic factor approach which accounts for a joint factor structure of returns and volatilities, and allows for extracting the market volatility shocks. Here, we go one step further, and show how the same two-step approach naturally produces volatility forecasts for the various stocks under study. In an applied exercise, we consider the panel of asset returns of the constituents of the S&P100 index over the period 2000-2009. Numerical results show that the predictors based on our two-step method outperform existing univariate and multivariate GARCH methods, as well as static factor GARCH models, in the pre- diction of daily high–low range—while avoiding the usual problems associated with the curse of dimensionality.

A network analysis of the volatility of high-dimensional financial series

M. Barigozzi and M. Hallin
Journal of the Royal Statistical Society - series C, 2017

Interconnectedness between stocks and firms plays a crucial role in the volatility contagion phenomena that characterise financial crises, and graphs are a natural tool in their analysis. In this paper, we are proposing graphical methods for an analysis of volatil- ity interconnections in the Standard & Poor’s 100 dataset during the period 2000-2013, which contains the 2007-2008 Great Financial Crisis. The challenges are twofold: first, volatilities are not directly observed and have to be extracted from time series of stock returns; second, the observed series, with about 100 stocks, is high-dimensional, and curse of dimensionality problems are to be faced. To overcome this double challenge, we pro- pose a dynamic factor model methodology, decomposing the panel into a factor-driven and an idiosyncratic component modelled as a sparse vector autoregressive model. The inversion of this autoregression, along with suitable identification constraints, produces networks in which, for a given horizon h, the weight associated with edge (i,j) represents the h-step-ahead forecast error variance of variable i accounted for by variable j’s innovations. Then, we show how those graphs yield an assessment of how systemic each firm is. They also demonstrate the prominent role of financial firms as sources of contagion during the 2007-2008 crisis.

Generalized dynamic factor models and volatilities: Recovering the market volatility shocks

M. Barigozzi and M. Hallin
The Econometrics Journal, 2016

Decomposing volatilities into a common market-driven component and an idiosyncratic item-specific one is an important issue in financial econometrics. This, however, requires the statistical analysis of large panels of time series, hence faces the usual challenges associated with high-dimensional data. Factor model methods in such a context are an ideal tool, but they do not readily apply to the analysis of volatilities. Focusing on the reconstruction of the unobserved market shocks and the way they are loaded by the various items (stocks) in the panel, we propose an entirely non-parametric and model-free two-step general dynamic factor approach to the problem, which avoids the usual curse of dimensionality. Applied to the S&P100 asset return dataset, the method provides evidence that a non-negligible proportion of the market-driven volatility of returns originates in the volatilities of the idiosyncratic components of returns.

Identifying the independent sources of consumption variation

M. Barigozzi and A. Moneta
Journal of Applied Econometrics, 2016

By representing a system of budget shares as an approximate factor model we determine its rank, i.e. the number of common functional forms, or factors and we estimate a base of the factor space by means of approximate principal components. We assume that the extracted factors span the same space of basic Engel curves representing the fundamental forces driving consumers' behaviour. We identify these curves by imposing statistical independence and by studying their dependence on total expenditure using local linear regressions. We prove consistency of the estimates. Using data from the U.K. Family Expenditure Survey from 1977 to 2006, we find strong evidence of two common factors and mixed evidence of a third factor. These are identified as decreasing, increasing, and almost constant Engel curves. The household consumption behaviour is therefore driven by two factors respectively related to necessities (e.g. food), luxuries (e.g. vehicles), and in some cases by a third factor related to goods to which is allocated the same percentage of total budget both by rich and poor households (e.g. housing).

Disentangling systematic and idiosyncratic dynamics in panels of volatility measures

M. Barigozzi, C. Brownlees, G. Gallo, and D. Veredas
Journal of Econometrics, 2014

Realized volatilities observed across several assets show a common secular trend and some idiosyncratic pattern which we accommodate by extending the class of Multiplicative Error Models (MEMs). In our model, the common trend is estimated nonparametrically, while the idiosyncratic dynamics are assumed to follow univariate MEMs. Estimation theory based on seminonparametric methods is developed for this class of models for large cross-sections and large time dimensions. The methodology is illustrated using two panels of realized volatility measures between 2001 and 2008: the SPDR Sectoral Indices of the S&P500 and the constituents of the S&P100. Results show that the shape of the common volatility trend captures the overall level of risk in the market and that the idiosyncratic dynamics have a heterogeneous degree of persistence around the trend. Out-of-sample forecasting shows that the proposed methodology improves volatility prediction over several benchmark specifications.

Do euro area countries respond asymmetrically to the common monetary policy?

M. Barigozzi, A. Conti, and M. Luciani
Oxford Bulletin of Economics and Statistics, 2014

We investigate the possible existence of asymmetries among Euro Area countries reactions to the European Central Bank monetary policy. Our analysis is based on a Structural Dynamic Factor model estimated on a large panel of Euro Area quarterly variables. A lthough the introduction of the euro has changed the monetary transmission mechanism in the individual countries towards a more homogeneous response, we find that differences still remain between North and South Europe in terms of prices and unemployment. These results are the consequence of country-specific structures, rather than of European Central Bank policies.

Non-fundamentalness in structural econometric models: A review

L. Alessi, M. Barigozzi, and M. Capasso
International Statistical Review, 2011

Current economic theory typically assumes that all the macroeconomic variables belonging to a given economy are driven by a small number of structural shocks. As recently argued, apart from negligible cases, the structural shocks can be recovered if the information set contains current and past values of a large, potentially infinite, set of macroeconomic variables. However, the usual practice of estimating small size causal Vector AutoRegressions can be extremely misleading as in many cases such models could fully recover the structural shocks only if future values of the few variables considered were observable. In other words, the structural shocks may be non-fundamental with respect to the small dimensional vector used in current macroeconomic practice. By reviewing a recent strand of econometric literature, we show that, as a solution, econometricians should enlarge the space of observations, and thus consider models able to handle very large panels of related time series. Among several alternatives, we review dynamic factor models together with their economic interpretation, and we show how non-fundamentalness is non-generic in this framework. Finally, using a factor model, we provide new empirical evidence on the effect of technology shocks on labour productivity and hours worked.

Improved penalization for determining the number of factors in approximate factor models

L. Alessi, M. Barigozzi, and M. Capasso
Statistics and Probability Letters, 2010

The procedure proposed by Bai and Ng (2002) for identifying the number of factors in static factor models is revisited. In order to improve its performance, we introduce a tuning multiplicative constant in the penalty, an idea that was proposed by Hallin and Liška (2007) in the context of dynamic factor models. Simulations show that our method in general delivers more reliable estimates, in particular in the case of large idiosyncratic disturbances.