Identification of Generalized Dynamic Factor Models from mixed-frequency data

In this chapter, we deal with linear dynamic factor models and related topics, such as dynamic principal component analysis (dynamic PCA). The main motivation for the use of such models is the so-called “curse of dimensionality” plaguing modeling of high-dimensional time series by “ordinary” multivariate AR or ARMA models. For instance, consider an AR system for, a say, 20-dimensional time series. Then each of the coefficient matrices contains 400 “free” parameters, if no additional restrictions on the parameter space have been imposed, i.e. in such a case the parameter spaces grow with the square of the output dimension n, whereas the data, for given sample size, grow linearly with n. Thus for moderate sample size and large n (as is the case, e.g. in many situations faced in macroeconomics), reliable parameter estimation in “fully parametrized” AR(X) or ARMA(X) models is hardly possible. On the other hand, e.g. in macroeconomics, for analysis and in particular for short-term forecasting, modeling of “comovement” and of “cross-sectional dependencies” between a large number of univariate time series recently has received increasing attention and appropriate tools for modeling of high-dimensional time series have been developed. Correspondingly, during the last 25 years, a substantial literature has emerged, dealing with such models, methods and applications, in particular for factor models in this context. The first section introduces a general framework for linear dynamic factor models. In the second section, we describe dynamic principal component analysis, which is a generalization of the well-known static principal component analysis to the dynamic case. In practical applications often the generalized dynamic factor model is used, which allows for cross-sectionally weakly dependent noise and assumes strong dependence in latent variables. This model class is suited for very high-dimensional time series.

The increasing prevalence of multiple, long time series data fuels the hope for gaining better understanding and more precise prediction of various phenomena in many domains, hence calling for the development of new methodologies in time series analysis. Two features pertinent to the practical usefulness of any time series model are sparsity and causality. A sparse time series model means that only a small subset of model parameters are non-zero, which is usually preferable to its non-sparse counterparts in terms of interpretability, robustness, parsimony and estimation variability. Causality refers to the dependence structure for which only past and present information is used. Causality is an indispensable feature for certain purposes such as prediction, and is usually desirable when modeling real-world phenomena. This dissertation devotes to the development of two time series methodologies that incorporate sparsity and causality. In the first part of the thesis, we introduce the sparse causal dynamic linear regression model. Two approaches are proposed for construction of such models. The first one utilizes the frequency domain solution of a classic two-sided dynamic linear model and augments it with a simple soft-thresholding operator. This approach is computationally efficient and enjoys good performance theoretically and empirically under the assumption that the true dynamic regression relationship is causal and sparse. The second approach constructs a sparse causal dynamic linear model by minimizing certain penalized mean squared prediction error. This approach is computationally intensive but is superior to the first approach as it does not require the causal assumption. It enjoys better empirical performance and opens up meaningful extensions. In the second part of the thesis, we introduce the sparse causal dynamic principal component analysis. This procedure also has two variants. The first one computes the principal component series as a sparse and causal linear filter of the original data. The filter is obtained by minimizing a penalized reconstruction loss functional in the frequency domain. The principal component series so obtained enjoys good performance in terms of summarizing high-dimensional data, estimating factors in dynamic factor models, and prediction. However, it is computationally intensive. The second approach imposes sparsity structure on the innovation representation of the data generating process, which leads to simpler computational algorithms. In addition, this method is of interest in its own right for certain applications in finance and econometrics.

In this contribution we present a structure theory for generalized linear dynamic factor models. Generalized dynamic factor models have been proposed approximately a decade ago for modeling of high dimensional time series where the cross sectional dimension is of the same order of magnitude as the sample size. In these models the classical assumption for factor models, that the noise components are mutually uncorrelated, is relaxed by allowing for weak dependence. Structure theory turns out to be important for estimation and model selection. The results obtained heavily draw from linear system theory.The contribution consists of two main parts. In the first part we deal with “denoising”, i.e. with getting rid of the noise in the observations. In the second part we deal with constructing linear dynamic systems for the latent variables. Here an important result is the generic zerolessness of the transfer function relating the latent variables and the dynamic factors. This allows for modeling the latent variables by (singular) autoregressions which simplifies estimation.

Factor models in high-dimensional time series—A time-domain approach

In this article, we introduce a two-way dynamic factor model (2w-DFM) for high-dimensional matrix-valued time series and study some of the basic theoretical properties in terms of identifiability and estimation accuracy. The proposed model aims to capture separable and low-dimensional effects of row and column attributes and their correlations across rows, columns, and time points. Complementary to other dynamic factor models for high-dimensional data, the 2w-DFM inherits the dimension-reduction feature of factor models but assumes additive row and column factors for easier interpretability. We provide conditions to ensure model identifiability and consider a quasi-likelihood based two-step method for parameter estimation. Under an asymptotic regime where the size of the data matrices as well as the length of the time series increase, we establish that the estimators achieve the optimal rate of convergence and are asymptotically normal. The asymptotic properties are reaffirmed empirically through simulation studies. An application to air quality data in Chinese cities is given to illustrate the merit of the 2w-DFM.

In the last decade, most studies on the interaction between the financial sector and economic growth has focused mainly on the banks and the stock market. Recently, growing attention has shifted to the interaction between the non-bank financial intermediaries (NBFIs) such as the insurance companies and economic growth. In this study, we examined the impact of insurance contributions on economic growth in Nigeria over a twenty seven year period, between 1981 and 2008 using a dynamic factor model. The proposed technique described a number of methods designed to analyze a functional relationship between the volume of insurance contribution and economic growth in terms of underlying but unobservable random quantities called factors. The factor loadings indicate which common trend is related to which set of time series. The result obtained through this approach shows that the functional relationship between the volume of insurance contribution and economic growth in Nigeria is a first order vector autoregressive model. Key words: Insurance contribution, dynamic factor model, economic growth.

The remediation processes in psychosocial crisis intervention were modeled focusing on cognitive orientation. Frequent observations and subsequent process modeling constitute a novel approach to process research and reveal process-outcome associations. A sample of 40 inpatients who were assigned to treatment in a crisis intervention unit was monitored in order to study the process of crisis intervention. The process data consisted of patients' self-ratings of the variables mood, tension, and cognitive orientation, which were assessed three times a day throughout hospitalization (M = 22.6 days). Linear time series models (vector autoregression) of the process data were computed to describe the prototypical dynamic patterns of the sample. Additionally, the outcome of crisis intervention was evaluated by pre-post questionnaires. Linear trends were found pointing to an improvement of mood, a reduction of tension, and an increase of outward cognitive orientation. Time series modeling showed that, on average, outward cognitive orientation preceded improved mood. The time series models partially predicted the treatment effect, notably the outcome domain "reduction of social anxiety," yet did not predict the domain of symptom reduction. In conclusion, crisis intervention should focus on having patients increasingly engage in outward cognitive orientation in order to stabilize mood, reduce anxiety, and activate their resources.

Most of the time series models discussed in the previous chapters are linear time series models. Although they remain at the forefront of academic and applied research, it has often been found that simple linear time series models usually leave certain aspects of economic and financial data unexplained. Since economic and financial systems are known to go through both structural and behavioral changes, it is reasonable to assume that different time series models may be required to explain the empirical data at different times. This chapter introduces some popular nonlinear time series models that have been found to be effective at modeling nonlinear behavior in economic and financial time series data.

Inflation is a constant and consistent increase in the general price level in the country, due to which the purchasing power of a national currency unit decreases. In practice, the measures of inflation are various price indices, such as a consumer price index (CPI), producer price index (PPI), or gross domestic product deflator. However, inflation is usually defined as a change in the HCPI over a year. Time series models, linear regression models and a vector autoregression model (VAR) can be used to model and forecast inflation processes. This paper examines Lithuanian consumer price inflation using a modern stationary time series and econometric theory. The vector autoregression model is proposed for inflation modelling. Theoretical aspects of model estimation are reviewed: time series stationarity, model identification, parameter estimation, model usage and forecasts. The stationarity of the HCPI index and exogenous variables are analyzed using the Augmented Dickey–Fuller (ADF) test. A vector autoregression model of Lithuanian inflation processes is investigated and proposed for inflation modelling. The obtained model is used for forecasting purposes and shows a fairly high degree of accuracy of the inflation forecast in the coming 12-month period.

The state space model provides an unifying framework for analyzing time series. For the classical Box–Jenkins analysis, state space methods are useful when estimating the autoregressive and moving average coefficients by exact maximum likelihood and when missing values are present. In the case of unobserved components time series models with interpretable components such as trend, seasonal, cycle and irregular, state space methods provide the tools for estimation, signal extraction and diagnostic checking. Indeed, all linear time series models can be cast into state space including dynamic regression models and cubic spline smoothing models.

Neftci (1984) proposed a nonparametric test procedure for determining whether business cycles are asymmetric in the sense that contractions are steeper than expansions. He found evidence of this type of asymmetry in postwar quarterly U.S. unemployment variables.' This comment identifies a probable error in Neftci's empirical work that reverses the significance of his results for the unemployment rate, and it suggests that this test may have low power and be sensitive to measurement error.

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the "curse of dimensionality" related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). We assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001-29 October 2020.

Dynamic factor models have a wide range of applications in econometrics and applied economics. The basic motivation resides in their capability of reducing a large set of time series to only few indicators (factors). If the number of time series is large compared to the available number of observations then most information may be conveyed to the factors. This way low dimension models may be estimated for explaining and forecasting one or more time series of interest. It is desirable that outlier free time series be available for estimation. In practice, outlying observations are likely to arise at unknown dates due, for instance, to external unusual events or gross data entry errors. Several methods for outlier detection in time series are available. Most methods, however, apply to univariate time series while even methods designed for handling the multivariate framework do not include dynamic factor models explicitly. A method for discovering outliers occurrences in a dynamic factor model is introduced that is based on linear transforms of the observed data. Some strategies to separate outliers that add to the model and outliers within the common component are discussed. Applications to simulated and real data sets are presented to check the effectiveness of the proposed method.

Research purpose: The current work introduces a novel method for time series discriminant analysis (DA). Proposing a version for the Bayes classifier employing Dynamic Linear Models, which we denote by BCDLM This article explores the application of DLMs and the Bayes Classifier in time series classification to promote application in sustainability across diverse sectors. Method: This paper presents some computer simulation studies in which we generate four different scenarios corresponding to time series observations from various Dynamic Linear Models (DLMs). In Discriminant Analysis, we investigated strategies for estimating variance in models and compared the performance of the BCDLM with other common classifiers. Such datasets are composed of real-time series (data from SONY AIBO Robot and spectrometry of coffee types) and pseudo-time series (data from Swedish leaves adapted for time series). We also point out that algorithm was used to determine training and test sets in real-world applications. Results: Considering the real-time series examined in this paper, The results obtained indicate that the parametric approach developed represents a promising alternative for this class of DA problems, with observations of time series in a situation that is quite difficult in practice when we have series with large sizes with respect to the number of observations in the classes, even though more thorough studies are required. Conclusions: It concludes that the BCDLM performed comparably to the results of the classifiers 1NN, RDA, NBND and NBK and superior to the methods LDA and QDA. This offers a powerful combination for time series classification, enabling accurate predictions and informed decision-making in areas such as energy consumption, waste management, and resource allocation.

Time series analysis of medium-resolution multispectral satellite imagery is critical to investigate forest disturbance dynamics at the landscape scale. In particular, the spatial, temporal, and radiometric consistency of Landsat time series data provides unprecedented insight into past disturbances that occurred over the last four decades. Several Landsat time series-based algorithms have been developed to automate the detection of forest disturbances. However, automated detection of non-stand-replacing disturbances based on Landsat time series remains a challenging task due to the difficulty of effectively separating them from spectral noise. Here, we present the High-dimensional detection of Landscape Dynamics (HILANDYN) algorithm, which exploits spatial and spectral information provided by Landsat time series to detect forest disturbance dynamics retrospectively. A novel and unsupervised procedure for changepoint detection in high-dimensional time series allows HILANDYN to perform the temporal segmentation of inter-annual time series into linear trends. The algorithm embeds a noise filter to remove spurious changepoints caused by residual spectral noise in the time series. We tested HILANDYN to detect disturbances that occurred in the forests of the European Alps over a period of 39 years, i.e. between 1984 and 2022, and evaluated its accuracy using a validation dataset of 3000 plots randomly located inside and outside the disturbed patches. We compared HILANDYN with the Bayesian Estimator of Abrupt change, Seasonality, and Trend (BEAST), which is a well-established and high-performing time series-based algorithm for changepoint detection. The quantitative results highlighted that the number of bands, i.e. original Landsat bands and spectral indices, included in the high-dimensional time series and the threshold controlling the significance of changepoints strongly influenced the user’s accuracy (UA). Conversely, changes in the combinations of bands primarily affected the producer’s accuracy (PA). HILANDYN achieved an F1 score of 0.801, which increased to 0.833 when we activated the noise filter, allowing the algorithm to balance UA (83.1%) and PA (83.5%). The qualitative results showed that disturbed forest patches detected by HILANDYN were characterized by a high spatio-temporal consistency, regardless of the disturbance severity. Furthermore, our algorithm was able to detect forest patches associated with secondary disturbances, such as salvage logging, that occur in close succession with respect to the primary event. The comparison with BEAST evidenced a similar sensitivity of the algorithms to non-stand-replacing events, as both achieved comparable PA. However, BEAST struggled to balance UA and PA when using a single parameter set, achieving a maximum F1 score of 0.717. Moreover, the computational efficiency of BEAST in processing high-dimensional time series was very limited due to its univariate nature based on the Bayesian approach. The adaptability of HILANDYN to detect a wide range of disturbance severities using a single parameter set and its computational efficiency in handling high-dimensional time series promotes its scalability to large study areas characterized by heterogeneous ecological conditions.