Multivariate Structural Time Series Model Research Articles

Bayesian Modeling of Labor Earnings in Construction

Labor costs constitute a significant portion of construction costs. Reliable forecasts provide insight into the movements of labor costs and are critical to the success of projects. Past studies have primarily focused on forecasting construction cost indexes or material costs. Only a few studies have concentrated on forecasting construction labor costs. This study presents a multivariate Bayesian structural time series (MBSTS) model to characterize the future values of construction labor’s average hourly earnings (AHE) using a set of candidate predictors. The methodologies commonly used by past studies do not adequately address the uncertainties associated with the modeling process. In contrast, MBSTS recognizes the uncertainty in its modeling process, which enables practitioners to quantify and account for future labor cost risks in decisions. Furthermore, the MBSTS method results in a transparent model, helping analysts investigate the rationality of the parameters. This article trains MBSTS models under four different data subset lengths (i.e., 150, 144, 138, and 132 months) to study the consistency of the explanatory variables and their corresponding coefficients. The analysis results indicate that the gross domestic product (GDP), housing starts (HS), number of building permits (BP), Construction Cost Index (CCI), Dow Jones Industrial Average (DJI), and Standard and Poor’s 500 index (SPI) are the most frequently used predictors in the regression component of the MBSTS models. The results indicate an inverse relationship between AHE from one side and HS and BP from the other. Likewise, a direct relationship exists between AHE and GDP, CCI, DJI, and SPI. The MBSTS model performed well on the validation subset in the midrange prediction intervals (i.e., 12- and 18-month periods). However, it was outperformed by conventional time series models [i.e., seasonal autoregressive integrated moving average (SARIMA)] when used for short-term forecasting. The proposed framework can be applied to facilitate monetary resource allocation in projects.

Estimating the effectiveness of permanent price reductions for competing products using multivariate Bayesian structural time series models

The Florence branch of an Italian supermarket chain recently implemented a strategy that permanently lowered the price of numerous store brands in several product categories. To quantify the impact of such a policy change, researchers often use synthetic control methods for estimating causal effects when a subset of units receive a single persistent treatment and the rest are unaffected by the change. In our applications, however, competitor brands not assigned to treatment are likely impacted by the intervention because of substitution effects; more broadly, this type of interference occurs whenever the treatment assignment of one unit affects the outcome of another. This paper extends the synthetic control methods to accommodate partial interference, allowing interference within predefined groups but not between them. Focusing on a class of causal estimands that capture the effect both on the treated and control units, we develop a multivariate Bayesian structural time series model for generating synthetic controls that would have occurred in the absence of an intervention, enabling us to estimate our novel effects. In a simulation study we explore our Bayesian procedures’ empirical properties and show that it achieves good frequentists coverage, even when the model is misspecified. We use our new methodology to make causal statements about the impact on sales of the affected store brands and their direct competitors. Our proposed approach is implemented in the CausalMBSTS R package.

Reliability-Based Assessment of Potential Risk for Lane-Changing Maneuvers

Despite the urgent need for continuous risk assessments during autonomous driving, achieving reliable assessment results is still challenging because of the unpredictable behaviors of adjacent human drivers and the resulting complexity. Such complexity increases particularly during lane changes because several vehicles need to interact with other vehicles. Therefore, this paper proposes a new framework to analyze lane-changing risk on freeways considering the forecastability in adjacent vehicles. Virtual lane-change scenarios are constructed based on historical maneuvers in adjacent vehicles, and the risk of potential lane change is evaluated through the safety evaluation result of the scenario. Adjacent vehicles’ future maneuvers are predicted using a multivariate Bayesian structural time series model, and the forecastability is estimated as the standard error of the predicted values. The failure probability of those lane-changing scenarios is obtained through the first-order reliability method, assuming that failure occurred when any time-to-collision value for adjacent vehicles was less than a threshold at the end of the lane change. This study tested two scenarios with three levels of uncertainty to show the effect of uncertainty on the level of risk. The results showed that the reduced uncertainty allowed a clearer distinction between risky situations. The proposed framework differentiates itself from existing methods by estimating higher risk in an adjacent vehicle’s more significant uncertainties. It is expected that the outcome of this study will be valuable in developing reliable lane-change strategies in autonomous driving.

A study of financial cycles and the macroeconomy in Taiwan

This paper studies the characteristics of financial cycles (credit and house prices) and their interactions with business cycles in Taiwan. We employ multivariate structural time series model to estimate trend and cyclical components in real bank credit, real house prices, and real GDP. We find that financial cycles are roughly twice the length of the business cycles, and house price cycles lead both credit and business cycles. Nevertheless, the estimated length of business and financial cycles in Taiwan is much shorter than those in industrialized economies. We then use machine learning to evaluate the importance of a macroeconomic variable that predicts downturns of financial cycles, by conducting both in-sample fitting and out-of-sample forecasting. Those macrovariables selected by machine learning reflects Taiwan’s close linkage in trades and financial interdependence with other countries such as China and spillover effects from the Fed’s monetary policy.

Estimating Causal Effects in the Presence of Partial Interference Using Multivariate Bayesian Structural Time Series Models

Researchers regularly use synthetic control methods for estimating causal effects when a sub-set of units receive a single persistent treatment, and the rest are unaffected by the change. In many applications, however, units not assigned to treatment are nevertheless impacted by the intervention because of cross-unit interactions. This paper extends the synthetic control methods to accommodate partial interference, allowing interactions within predefined groups, but not between them. Focusing on a class of causal estimands that capture the effect both on the treated and control units, we develop a multivariate Bayesian structural time series model for generating synthetic controls that would have occurred in the absence of an intervention enabling us to estimate our novel effects. In a simulation study, we explore our Bayesian procedure’s empirical properties and show that it achieves good frequentists coverage even when the model is misspecified. Our work is motivated by an analysis of a marketing campaign’s effectiveness by an Italian supermarket chain that permanently reduced the price of hundreds of store-brand products. We use our new methodology to make causal statements about the impact on sales of the affected store-brands and their direct competitors. Our proposed approach is implemented in the CausalMBSTS R package.

Consistent Multivariate Seasonal Adjustment for Gross Domestic Product and its Breakdown in Expenditures

Abstract Seasonally adjusted series of Gross Domestic Product (GDP) and its breakdown in underlying categories or domains are generally not consistent with each other. Statistical differences between the total GDP and the sum of the underlying domains arise for two reasons. If series are expressed in constant prices, differences arise due to the process of chain linking. These differences increase if, in addition, a univariate seasonal adjustment, with for instance X-13ARIMA-SEATS, is applied to each series separately. In this article, we propose to model the series for total GDP and its breakdown in underlying domains in a multivariate structural time series model, with the restriction that the sum over the different time series components for the domains are equal to the corresponding values for the total GDP. In the proposed procedure, this approach is applied as a pretreatment to remove outliers, level shifts, seasonal breaks and calendar effects, while obeying the aforementioned consistency restrictions. Subsequently, X-13ARIMA-SEATS is used for seasonal adjustment. This reduces inconsistencies remarkably. Remaining inconsistencies due to seasonal adjustment are removed with a benchmarking procedure.

Bayesian Method for Causal Inference in Spatially-Correlated Multivariate Time Series

Measuring the causal impact of an advertising campaign on sales is an essential task for advertising companies. Challenges arise when companies run advertising campaigns in multiple stores which are spatially correlated, and when the sales data have a low signal-to-noise ratio which makes the advertising effects hard to detect. This paper proposes a solution to address both of these challenges. A novel Bayesian method is proposed to detect weaker impacts and a multivariate structural time series model is used to capture the spatial correlation between stores through placing a G-Wishart prior on the precision matrix. The new method is to compare two posterior distributions of a latent variable—one obtained by using the observed data from the test stores and the other one obtained by using the data from their counterfactual potential outcomes. The counterfactual potential outcomes are estimated from the data of synthetic controls, each of which is a linear combination of sales figures at many control stores over the causal period. Control stores are selected using a revised Expectation-Maximization variable selection (EMVS) method. A two-stage algorithm is proposed to estimate the parameters of the model. To prevent the prediction intervals from being explosive, a stationarity constraint is imposed on the local linear trend of the model through a recently proposed prior. The benefit of using this prior is discussed in this paper. A detailed simulation study shows the effectiveness of using our proposed method to detect weaker causal impact. The new method is applied to measure the causal effect of an advertising campaign for a consumer product sold at stores of a large national retail chain.

Multivariate Bayesian structural time series model

This paper deals with inference and prediction for multiple correlated time series, where one also has the choice of using a candidate pool of contemporaneous predictors for each target series. Starting with a structural model for time series, we use Bayesian tools for model fitting, prediction and feature selection, thus extending some recent works along these lines for the univariate case. The Bayesian paradigm in this multivariate setting helps the model avoid overfitting, as well as captures correlations among multiple target time series with various state components. The model provides needed flexibility in selecting a different set of components and available predictors for each target series. The cyclical component in the model can handle large variations in the short term, which may be caused by external shocks. Extensive simulations were run to investigate properties such as estimation accuracy and performance in forecasting. This was followed by an empirical study with one-step-ahead prediction on the max log return of a portfolio of stocks that involve four leading financial institutions. Both the simulation studies and the extensive empirical study confirm that this multivariate model outperforms three other benchmark models, viz. a model that treats each target series as independent, the autoregressive integrated moving average model with regression (ARIMAX), and the multivariate ARIMAX (MARIMAX) model.

Business, Housing, and Credit Cycles – The Case of Hungary

This paper studies the characteristics of financial cycles in Hungary. It applies existing methodology from the literature to Hungarian data to estimate a multivariate structural time-series model. The model allows for a joint examination of the behaviour of the Hungarian financial sector and the overall economy, and estimates their cyclical positions. According to the results of the estimation, the financial sector in Hungary seems to experience volatile cycles, which last more than 15 years on average. Moreover, the cyclical position of output seems to show strong comovement with the long financial-sector cycles. Although the data series available for Hungary are relatively short, the results of the estimation are quite credible, since they conform to the existing international evidence and seem robust to even stricter data limitations.

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.

Small Area Estimation with State Space Common Factor Models for Rotating Panels

SummaryMacroeconomic indicators about the labour force, published by national statistical institutes, are predominantly based on rotating panels. Sample sizes of most labour force surveys in combination with the design-based or model-assisted mode of inference obstruct the publication of such indicators on a monthly frequency. Previous research proposed a multivariate structural time series model to obtain more precise model-based estimates by taking advantage of sample information observed in previous periods. In the paper this model is extended to use sample information from other domains or strongly correlated auxiliary series. A relatively parsimonious version of these models is currently used by Statistics Netherlands to produce official monthly figures about the labour force.

Forecasting Longevity Gains for a Population with Short Time Series Using a Structural SUTSE Model: An Application to Brazilian Annuity Plans

In this article, a multivariate structural time series model with common stochastic trends is proposed to forecast longevity gains of a population with a short time series of observed mortality rates, using the information of a related population for which longer mortality time series exist. The state space model proposed here makes use of the seemingly unrelated time series equation and applies the concepts of related series and common trends to construct a proper model to predict the future mortality rates of a population with little available information. This common trends approach works by assuming the two populations’ mortality rates are affected by common factors. Further, we show how this model can be used by insurers and pension funds to forecast mortality rates of policyholders and beneficiaries. We apply the proposed model to Brazilian annuity plans where life expectancies and their temporal evolution are predicted using the forecast longevity gains. Finally, to demonstrate how the model can be used in actuarial practice, the best estimate of the liabilities and the capital based on underwriting risk are estimated by means of Monte Carlo simulation. The idiosyncratic risk effect in the process of calculating an amount of underwriting capital is also illustrated using that simulation.

Multivariate State Space Approach to Variance Reduction in Series with Level and Variance Breaks Due to Survey Redesigns

Summary Statistics Netherlands applies a design-based estimation procedure to produce road transportation figures. Frequent survey redesigns caused discontinuities in these series which obstruct the comparability of figures over time. Reductions in the sample size and changes in the sample design resulted in variance breaks and unacceptably large sampling errors in the recent part of the series. Both problems are addressed and solved simultaneously. Discontinuities and small sample sizes are accounted for by using a multivariate structural time series model that borrows strength over time and space. The paper illustrates an increased precision when we move from univariate models to a multivariate model where the domains are jointly modelled. This increase is especially significant in the most recent period when sample sizes become smaller, with standard errors of the design-based estimator of the target variables being reduced by 40 & #x2013;70 & #x0025; with the model-based approach.

Monthly Employment Indicators of the Euro Area and Larger Member States: Real‐Time Analysis of Indirect Estimates

ABSTRACTThe paper presents a comparative real‐time analysis of alternative indirect estimates relative to monthly euro area employment. In the experiment quarterly employment is temporally disaggregated using monthly unemployment as related series. The strategies under comparison make use of the contribution of sectoral data of the euro area and its six larger member states. The comparison is carried out among univariate temporal disaggregations of the Chow and Lin type and multivariate structural time series models of small and medium size. Specifications in logarithms are also systematically assessed. All multivariate set‐ups, up to 49 series modelled simultaneously, are estimated via the EM algorithm. Main conclusions are that mean revision errors of disaggregated estimates are overall small, a gain is obtained when the model strategy takes into account the information by both sector and member state and that larger multivariate set‐ups perform very well, with several advantages with respect to simpler models.Copyright © 2014 John Wiley & Sons, Ltd.

Estimation of the monthly unemployment rate for six domains through structural time series modelling with cointegrated trends

National statistical institutes generally apply design-based techniques like the generalized regression estimator to compile official statistics. These techniques, however, have relatively large design variances in the case of small sample sizes. In such cases, model based small area estimation techniques can be considered to improve the precision of the estimates. A multivariate structural time series model is developed and applied to obtain more precise estimates of the Dutch monthly unemployment rate for six domains. The model takes advantage of sample information from preceding time periods through an appropriate time series model and from other domains by modelling the correlation between the trend components of the time series models for the different domains. The trends of the six domains are cointegrated, which allows the use of a more parsimonious common factor model that is based on three common trends. Although the use of common factor models is a well known approach in econometrics, its application in the context of small area estimation is novel. The standard errors of the direct estimates of the monthly unemployment rates are more than halved with this approach.

Growth accounting for the euro area

This article is concerned with the estimation of euro area potential output growth and its decomposition according to the sources of growth. The growth accounting exercise is based on a multivariate structural time series model which combines the decomposition of total output according to the production function approach with price and wage equations that embody Phillips-type relationships linking inflation and nominal wage dynamics to the output gap and cyclical unemployment, respectively. Assuming a Cobb–Douglas technology with constant returns to scale, potential output results from the combination of the trend levels of total factor productivity (TFP) and factor inputs, capital and labour (hours worked), which is decomposed into labour intensity (average hours worked), the employment rate, the participation rate and population of working age. The nominal variables (prices and wages) play an essential role in defining the trend levels of the components of potential output, as the latter should pose no inflationary pressures on prices and wages. The structural model is further extended to allow for the estimation of potential output growth and the decomposition according to the sources of growth at different horizons (long run, medium run and short run); in particular, we propose and evaluate a model-based approach to the extraction of the low-pass component of potential output growth at different cut-off frequencies. The approach has two important advantages: the signal extraction filters have an automatic adaptation property at the boundaries of the sample period, so that the real-time estimates do not suffer from what is often referred to as the ‘end-of-sample bias’. Secondly, it is possible to assess the uncertainty of potential output growth estimates with different degrees of smoothness.

Multivariate Structural Time Series Models with Dual Cycles: Implications for Measurement of Output Gap and Potential Growth

Structural time series models applied to the factor inputs of a production function often lead to small output gaps and consequently to erratic measures of potential growth. We introduce a dual cycle model which is an extension to the multivariate trend plus cycle model with phase shifts a la Runstler. The dual cycle model is a combination of two types of models: the trend plus cycle model and the cyclical trend model, where the cycle appears in the growth rate of a variable. This property enables hysteresis to be taken into account. Hysteresis is likely to show up in unemployment but it can also affect the capital stock due to the existence of long investment cycles. In the proposed model, hysteresis may affect all the factor inputs of the production function and phase shifts are extended to the dual cycles. Genuine measures of potential growth can be computed that are hysteresis-free and less prone to volatility. A complementary measure of the output gap that takes hysteresis into account can be derived.

Forecasting gas component prices with multivariate structural time series models

Predicting gas component prices over different horizons is important for both energy producers and consumers. In this study, we model and predict the joint dynamics of butanes, propane and naphtha traded in the north European market. Our approach is to use multivariate time series with unobservable components. We applied monthly data over a 10-year period, from 1995 to 2006, and tested the predictive power of fitted models using various hold out samples. The in-sample and out-of-sample results indicated that gas component prices follow stochastic processes with trend and autoregressive effects that continuously change over time while the seasonal patterns seem to be stationary. The prediction results were compared with random walk for one-step and multi-step forecasts in each of the out-of sample periods. The results are promising and indicate that our model can be used for short-/medium term forecasting of gas component prices.

Finding market structure by sales count dynamics—Multivariate structural time series models with hierarchical structure for count data—

In this paper, we propose a multivariate time series model for sales count data. Based on the fact that setting an independent Poisson distribution to each brand’s sales produces the Poisson distribution for their total number, characterized as market sales, and then, conditional on market sales, the brand sales follow a multinomial distribution, we first extend this Poisson–multinomial modeling to a dynamic model in terms of a generalized linear model. We further extend the model to contain nesting hierarchical structures in order to apply it to find the market structure in the field of marketing. As an application using point of sales time series in a store, we compare several possible hypotheses on market structure and choose the most plausible structure by using several model selection criteria, including in-sample fit, out-of-sample forecasting errors, and information criterion.

Assessing inflation targeting through intervention analysis

The aim of this paper is to deal with the empirical aspects of the ‘new’ monetary policy framework, known as Inflation Targeting. Applying Intervention Analysis to multivariate Structural Time Series models, which avoids certain biases encountered in the use of conventional regression estimators, new empirical evidence is produced in the case of a number of OECD countries. These results demonstrate that although Inflation Targeting has gone hand-in-hand with low inflation, the strategy was introduced well after inflation had begun its downward trend. But, then, Inflation Targeting ‘locks in’ low inflation rates. The evidence produced in this paper suggests that non-Inflation Targeting central banks have also been successful on this score.