Any series of observations ordered along a single dimension, such as time, may be thought of as a time series. The emphasis in time series analysis is on studying the dependence among observations at different points in time. What distinguishes time series analysis from general multivariate analysis is precisely the temporal order imposed on the observations. Many economic variables, such as GNP and its components, price indices, sales, and stock returns are observed over time. In addition to being interested in the contemporaneous relationships among such variables, we are often concerned with relationships between their current and past values, that is, relationships over time.

Time Series Analysis

Forecasting is a common data science task that helps organizations with capacity planning, goal setting, and anomaly detection. Despite its importance, there are serious challenges associated with ...

Forecasting at scale

Bayesian statistics is an approach to data analysis based on Bayes’ theorem, where available knowledge about parameters in a statistical model is updated with the information in observed data. The background knowledge is expressed as a prior distribution and combined with observational data in the form of a likelihood function to determine the posterior distribution. The posterior can also be used for making predictions about future events. This Primer describes the stages involved in Bayesian analysis, from specifying the prior and data models to deriving inference, model checking and refinement. We discuss the importance of prior and posterior predictive checking, selecting a proper technique for sampling from a posterior distribution, variational inference and variable selection. Examples of successful applications of Bayesian analysis across various research fields are provided, including in social sciences, ecology, genetics, medicine and more. We propose strategies for reproducibility and reporting standards, outlining an updated WAMBS (when to Worry and how to Avoid the Misuse of Bayesian Statistics) checklist. Finally, we outline the impact of Bayesian analysis on artificial intelligence, a major goal in the next decade. This Primer on Bayesian statistics summarizes the most important aspects of determining prior distributions, likelihood functions and posterior distributions, in addition to discussing different applications of the method across disciplines.

Bayesian statistics and modelling

Monthly data on road casualties in Great Britain are analyzed in order to assess the effect on casualty rates of the seat belt law introduced on January 31, 1983. Such analysis is known technically as intervention analysis. The form of intervention analysis that is used in this paper is based upon structural time series modelling and differs in significant respects from standard intervention analysis based upon ARIMA modelling. The relative merits of the two approaches are compared. Structural modelling intervention techniques are used to estimate the changes in casualty rates for various categories of road users following the introduction of the seat belt law. We first note the high rate of compliance with the seat belt law. By February 1983, the wearing rate had jumped to 90 percent and the rate has remained at approximately 95 percent from March 1983 onwards. There can be no doubt of the success of the law as regards to compliance. In considering the casualty figures we distinguish between those directly affected by the law, namely car drivers and front seat passengers, and those not directly affected by the law, that is car rear seat passengers, pedestrians and cyclists. Taking first numbers killed and seriously injured (KSI), we found a reduction of 23 percent for car drivers and 30 percent for front seat passengers. Thus, for those directly affected by the law, there have been substantial reductions. For rear seat passengers KSI we found a rise of 3 percent, for pedestrians a fall of one-half percent, and for cyclists an increase of 5 percent -- all three of these of these values being statistically insignificant. We conclude that there is no significant evidence of a change in numbers of KSI of those not directly affected. For numbers killed, we found for those directly affected a reduction of 18 percent for car drivers and 25 percent for front seat passengers. However, for those indirectly affected by the law, our model gave an increase of 27 percent for rear seat passengers, 8 percent for pedestrians, and 13 percent for cyclists. The value for rear seat passengers is highly significant and the other two values are on the borderline of significance. We conclude that there was an increase in fatalities of those not directly affected. We are unable to provide a completely satisfactory explanation of the difference between the figures for KSI and killed for rear seat passengers, pedestrians and cyclists. The article contains 13 pages of comments by discussants and 3 pages of authors' response. Language: en

https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2981553

The Effects of Seat Belt Legislation on British Road Casualties: A Case Study in Structural Time Series Modelling

The good, the bad and the ugly on COVID-19 tourism recovery

A univariate structural time series model based on the traditional decomposition into trend, seasonal and irregular components is defined. A number of methods of computing maximum likelihood estimators are then considered. These include direct maximization of various time domain likelihood function. The asymptotic properties of the estimators are given and a comparison between the various methods in terms of computational efficiency and accuracy is made. The methods are then extended to models with explanatory variables. Ktv WORDS Structural time series model Forecasting Kalman filter Stochastic trend Unobserved components model EM algorithm

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Estimation Procedures for Structural Time Series Models

The question of whether an econometric equation should be formulated in levels or differences is one which has generated a good deal of controversy in recent years; see, for example, Hendry and Mizon (1978). Another important issue concerns the detrending of variables, prior to their inclusion in a regression equation, by regressing each one individually on time. Nelson and Kang (I 984) examine the consequences of such detrending when the appropriate procedure is to estimate the equation in first differences. Since including a time trend in a levels regression is equivalent to prior detrending of the variables, the detrending issue can be regarded as another aspect of the levels versus differences debate. The aim of this paper is to clarify these fundamental issues of dynamic specification. Our approach centres on the introduction of a stochastic trend into a regression model. There are often strong a priori reasons for wanting to include such a term. The application considered here concerns the employment-output relationship, and the role of the stochastic trend in this case is to provide a means of accounting for the underlying productivity trend, part of which derives from technical progress. Technical progress has traditionally been modelled by a deterministic time trend, but as noted by Henry (I979), O'Brien (1983), and Mendis and Muellbauer (1983), equations of this kind failed repeatedly to predict employment satisfactorily after the mid-I970s. One response to these difficulties has been to include ad hoc breaks in the trend and dummy variables; see, for example, HM Treasury (i982). However, we believe that a stochastic trend offers an intuitively more appealing way of modelling variables like productivity and technical progress, and offers a way out of the problems caused by constraining them to be deterministic. From the statistical point of view the key device in modelling a stochastic trend is the state space form. This allows the unknown parameters to be estimated via the prediction error decomposition and for predictiorns to be computed by extending the Kalman filter. In addition, a smoothing algorithm can be used to provide an optimal estimator of the trend at each point throughout the sample. The levels model with a deterministic time trend and the first difference model both emerge as special cases. Section I briefly sets out the economic theory underlying the dynamics of the employment-output equation, while Section II describes the statistical treatment of stochastic trends. The techniques described in Section II are applied to

Simon Peters

Papers

Estimation Procedures for Structural Time Series Models

Stochastic trends in Dynamic Regression Models: An application to the Employment Output Equation