The nature of Austrian macroeconomic time series unit root tests

It is still an open question in economic and econometric modelling whether the non-stationarity in a time series is captured by detrending or by differencing. We test thirrteen Austrian macroenconomic time series for difference versus trend stationarity using informal methods and formal procedures developed by Dickey-Fuller and Phillips-Perron. To eliminate the effects of seasonal adjustment on the tests we apply a third procedure to the unadjusted data, recently developed by Hylleberg-Engle-Granger-Yoo. Independent of the seasonal adjustment the empirical results indicate that these series are integrated of order 1.     

Testing for Seasonal Fractional Roots in German Real Output

Abstract. This paper examines the seasonal structure of German real GNP per capita by using a version of Robinson's (1994) tests which is suitable in the context of seasonality. This method has several advantages over alternative approaches when testing for seasonal unit roots. First, unlike standard tests, which are nested in AR alternatives, it is embedded in fractional alternatives. Second, it allows testing at the zero frequency and at each of the seasonal frequencies separately. Third, it makes it possible to test for different orders of integration at each of the frequencies simultaneously. The empirical analysis suggests that the real output series may have a unit root at the zero frequency, and fractional rather than unit roots at the seasonal ones. This is in contrast to the findings reported by Lutkepohl et al. (1999) in their study on German money demand, and shows the importance of modelling the seasonal features of the data in alternative ways.     

Outlier detection and seasonal adjustment

By analysing three macroeconomic time series, namely retail sales, purchases of durables and of cars, we show the consequences of the presence of outliers in the data on the outcome of model-based seasonal adjustment. For all three series, we detect substantial negative effects for the resulting seasonally adjusted figures.In a recent paper,Thury — Wüger (1992) demonstrated that the presence of outliers in economic data has serious negative effects for time series modelling. Poorly estimated ARIMA models with an unsatisfactory forecasting performance are the consequence. Beyond that, we suspect that outliers may also cause problems for seasonal adjustment. Since seasonally adjusted data play a prominent role in applied economic research, it seems worthwhile to investigate this problem more deeply. Analysing the same three series as in the above mentioned paper, namely retail sales, purchases of durables and of cars which, as we know, are severely contaminated by outliers, we try to derive the consequences of the existence of outliers in the data for seasonal adjustment. Where monthly observations of our considered data exist, we also enclose calendar effects in the modelbased seasonal adjustment procedure.

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Zusammenfassung Die Existenz von Ausreißern in ökonomischen Zeitreihen führt zu schlecht spezifizierten Zeitreihenmodellen mit verzerrten Parameterschätzwerten. Verwendet man solche Modelle als Ausgangspunkt für eine auf Modellansatz basierende Saisonbereinigung, so erhält man sehr unverläßliche, mit starken Zufallsschwankungen behaftete Ergebnisse.

     

In Defence of Model-Based Seasonal Adjustment: An Illustration Using Australian Data

It is argued that the X-11 seasonal adjustment procedure suffers from severe drawbacks, and so it should be abandoned in favour of model-based seasonal adjustment. Furthermore, it is argued that Harvey's structural time series model is superior to the conventional seasonal ARIMA models for the purpose of model-based seasonal adjustment. It is shown, with the help of a large number of Australian time series, that the nature of seasonality differs from one series to another, and this is why model selection is crucial for seasonal adjustment. It is further shown that model-based seasonal adjustment could produce results that are significantly different from those obtained by applying the X-11 procedure. Since the X-11 procedure is not based on an explicit model and in view of its other serious drawbacks, it is concluded that the procedure should be abandoned in favour of model-based seasonal adjustment.     

Modelling trigonometric seasonal components for monthly economic time series

The basic structural time series model has been designed for the modelling and forecasting of seasonal economic time series. In this article, we explore a generalization of the basic structural time series model in which the time-varying trigonometric terms associated with different seasonal frequencies have different variances for their disturbances. The contribution of the article is two-fold. The first aim is to investigate the dynamic properties of this frequency-specific Basic Structural Model (BSM). The second aim is to relate the model to a comparable generalized version of the Airline model developed at the US Census Bureau. By adopting a quadratic distance metric based on the restricted reduced form moving-average representation of the models, we conclude that the generalized models have properties that are close to each other compared to their default counterparts. In some settings, the distance between the models is almost zero so that the models can be regarded as observationally equivalent. An extensive empirical study on disaggregated monthly shipment and foreign trade series illustrates the improvements of the frequency-specific extension and investigates the relations between the two classes of models.     

Monetary policy and dividend growth in Germany: long-run structural modelling versus bounds testing approach

This study examines the long-run relationship between monetary policy and dividend growth in Germany. For this purpose, cointegration is tested for between both variables in the period 1974 to 2003. However, problems related to spurious regression arise from the mixed order of integration of the series used, from mutual causation between the variables and from the lack of a long-run relationship among the variables of the model. These problems are addressed by applying the bounds testing approach to cointegration in addition to a more standard long-run structural modelling approach. In principle, both procedures are capable of dealing with the controversial issue of the exogeneity of monetary policy vis-à-vis dividend growth. However, the structural modelling approach still leaves a certain degree of uncertainty about the integration properties of the interest rate and the dividend growth. Hence, one feels legitimized to refer to the bounds testing procedure and to conclude that in the longer term short-term rates drive stock returns but not vice versa.     

MODELLING TIME SERIES DATA OF MONETARY AGGREGATES USING I(2) AND I(1) COINTEGRATION ANALYSIS

The objective of this paper is to consider methodology for modelling time series data of monetary aggregates such as monetary base and broad money. A brief review is made with regard to the likelihood‐based cointegration analysis of I(2) (integrated of order 2) data and I(2)‐to‐I(1) transformations. The paper then investigates procedures for econometric modelling of monetary aggregates, which are in general deemed to be I(2) variables analogous to price indices. It is shown that I(2)‐to‐I(1) transformations centering on a money multiplier play an important role in the modelling procedures. Finally, the study presents an empirical illustration of the proposed methodology using monetary aggregate data from Japan.     

实际经济时间序列的计算、季节调整及相关经济含义

本文首先讨论了计算中国实际经济时间序列的不同做法 ,并分析了其对季节调整的影响 ,指出通过同比增长率计算实际变量并进行季节调整是一个可以接受的做法 ,可以得到非常接近真实的季调后序列 ,并且在中国现有数据资源的限制下拥有一些特别的优势。然后本文具体讨论了对几个不同经济变量进行季节调整的方法 ,并给出了一些在经济数据分析与预测中的简单应用。方法的关键是采用regARIMA模型 ,从而可以对工作日变化、放长假、春节因素等作出一个估计和调整。作为一个副产品 ,本文引荐了一个相对较新的季节调整程序 (方法 ) ,TRAMO SEATS ,简单介绍了它的原理和优势 ,希望今后能得到更广泛的应用。     

Comparison of Stochastic and Spline Models for Temperature‐based Derivatives in China

In this paper, we propose modelling the seasonal variation of temperature with a stochastic process to achieve normality of residuals. We conduct a heuristic comparison of the new stochastic seasonal variation model with three established empirical temperature and pricing models: the model of Alaton et al., the continuous autoregressive model and the spline model. The test criteria are residual normality, the Akaike information criterion, relative errors, and stability of price behaviour. The objective of the paper is to find the most suitable model for the application of temperature‐based derivatives in China. Therefore, 30 years of daily average temperature data from 12 cities in mainland China are applied. The results show that the stochastic seasonal variation model dominates the other three models by providing a more precise fitting of the temperature process. Furthermore, the spline model displays inconsistencies when it is applied to Chinese temperature data. This model has the smallest relative errors, but the worst results for normality of residuals.     

Unit and Fractional Roots at the Long Run and the Seasonal Frequencies in Macroeconomic Time Series

In this article, we show that macroeconomic time series may contain unit and fractional roots at both, at zero and at zero and at the seasonal frequencies. The importance of the root at the long run or zero frequency requires in many cases to consider this root at both, separately in an independent polynomial, and also included in the seasonal one. Several Monte Carlo experiments are conducted to examine cases when the root at the zero frequency is not appropriately considered. An empirical application based on the tests of Robinson, Peter M. “Efficient Tests of Nonstationary Hypotheses,” Journal of the American Statistical Association, 89, 1994, pp. 1420–37 is also carried out at the end of the article.The author gratefully acknowledges financial support from the Government of Navarra (“Ayudas de Formación e Investigación y Desarrollo”).     

Modelling seasonality in innovation diffusion

The ability to forecast new product growth is especially important for innovative firms that compete in the marketplace. Today many new products exhibit very strong seasonal behaviour, which may deserve specific modelling, both for producing better forecasts in the short term and for better explaining special market dynamics and related managerial decisions. By considering seasonality as a deterministic component to be estimated jointly with the trend through Nonlinear Least Squares methods, we have developed two extensions of the Guseo–Guidolin model that are able to simultaneously describe trend and seasonality. Such models are based on two different but equally reasonable approaches: in one case we consider a simple additive decomposition of a time series and design a model in which seasonality is directly added to the trend and jointly estimated with it; in the other we design a more complex structure, mimicking that of a Generalized Bass model and embed two separate seasonal perturbations within the dynamic market potential and the corresponding adoption process. The different characteristics of two products, a pharmaceutical drug and an IT device, make it possible to appreciate empirically various modelling options and performances. Both models are quite simple to implement and to interpret from a managerial point of view.     

Cyclical common factors in cointegrated systems

When working with vectors of time series which fluctuate regularly we may possibly want to consider the presence of common factors characterized by cyclical or seasonal behavior as well as trend. For example, Deaton89 provides a hint of a theoretical model where cointegration at the annual frequency may exist between consumption and income in addition to the usual secular cointegration. It is well known that a non-cyclical system cointegrated at frequency zero has a common trend (CT) representation Stock-Watson: 88. In this paper we show that a time series vector that is cointegrated at one or several frequencies simultaneously (e.g. seasonal data) has a common factors (CF) representation which belongs to a class of common factor models that encompasses many cointegrating situations found in the literature. We study these issues and extend the method proposed by Gonzalo-Granger: 95 to the estimation and testing of common factors which may combine trend as well as cyclical or seasonal characteristics. Two illustrative applications are also provided. JEL Classification: C10, C32, C50 Javier Fernández-Macho: Financial support from research group grant 9/UPV00038.321-13503/2001 of UPV/EHU is gratefully acknowledged by both authors and from research project BEC2003-02028 of Ministerio de Ciencia y Tecnología by the first author. We are also indebted to two anonymous referees for their helpful comments.     

Testing for Seasonal Stability in Unemployment Series: International Evidence

Recently, it has been shown that seasonal and business cycles are related and a similar economic mechanism is at work in producing both types of cycles (Miron 1996). Thus, an analysis of seasonal fluctuations sheds light on the nature of the business cycle. This paper uses the classical test developed by Hylleberg et al. and the LM-type tests proposed by Canova and Hansen (1995) to investigate seasonal behavior in the unemployment series of Australia, Canada, Japan, New Zealand, the US and a number of OECD countries. The main findings are that the Australian, Austrian and Canadian series are non-stationary at all seasonal frequencies, French, Japan, the NZ and the UK series are stationary at all seasonal frequencies and the USA series is stationary only at the annual frequency. The test results for other series are mixed, suggesting that further analysis is required to reach a definite conclusion. The series, except for France, Japan, New Zealand and the UK, appear to possess unstable seasonal patterns, indicating changing business cycle conditions.     

Seasonality, Cointegration, and Forecasting UK Residential Energy Demand

Much of the short‐run movement in energy demand in the UK is seasonal, and the contribution of long‐run factors to short‐run forecasts is slight. Nevertheless, using a variety of techniques, including a recently developed estimation procedure that is applicable irrespective of the orders of integration of the data, we obtain a long‐run income elasticity of demand of about one third, and we are unable to reject a zero price elasticity. An econometric model is shown to provide superior short‐run forecasts to well‐known seasonal time series models ex post , but is inferior to Box‐Jenkins SARMA models when the determinants themselves have to be forecast. However, the relatively short data sample and small number of forecasts suggest caution in generalising these results.     

Seasonality and Markov switching in an unobserved component time series model

It is generally acknowledged that the growth rate of output, the seasonal pattern, and the business cycle are best estimated simultaneously. To achieve this, we develop an unobserved component time series model for seasonally unadjusted US GDP. Our model incorporates a Markov switching regime to produce periods of expansion and recession, both of which are characterized by different underlying growth rates. Although both growth rates are time-varying, they are assumed to be cointegrated. The analysis is Bayesian, which fully accounts for all sources of uncertainty. Comparison with results from a similar model for seasonally adjusted data indicates that the seasonal adjustment of the data significantly alters several aspects of the full model. First Version Received: January 2001/Final Version Received: February 2002 Send offprint requests to: Rob Luginbuhl?Correspondence to: Rob Luginbuhl     

Masking of volatility by seasonal adjustment methods

We report that the X-12 ARIMA and TRAMO–SEATS seasonal adjustment methods consistently underestimate the variability of the differenced seasonally adjusted series. We show that underestimation is due to a non-zero estimation error in estimating the seasonal component at each time period, which is the result of the use of low order seasonal filter in X12-ARIMA for estimating the seasonal component. Hence, we propose the use of high order seasonal filter for estimating the seasonal component, which helps reducing the estimation error noticeably, helps amending the underestimation problem, and helps improving the forecasting accuracy of the series. In TRAMO–SEATS, Airline model is found to deliver the best seasonal filter among other ARIMA models.     

Fuzzy Seasonal Time Series for Forecasting the Production Value of the Mechanical Industry in Taiwan

Based on the seasonal time series ARIMA(p,d,q)(P,D,Q)_s model (SARIMA) and fuzzy regression model, we combine the advantages of two methods to propose a procedure of fuzzy seasonal time series and apply this method to forecasting the production value of the mechanical industry in Taiwan. The intention of the article is to provide the enterprises, in this era of diversified management, with a fresh method to conduct short-term prediction for the future in the hope that these enterprises can perform more accurate planning. This method includes interval models with interval parameters and provides the possibility distribution of future value. From the results of practical application to the mechanical industry, it can be shown that this method makes good forecasts. Further, this method makes it possible for decision makers to forecast the possible situations based on fewer observations than the SARIMA model and has the basis of pre-procedure for fuzzy time series.     

Reciprocal relationships and mechanism design

We study an incomplete information game in which players can coordinate their actions by contracting among themselves. We model this relationship as a reciprocal contracting procedure where each player has the ability to make commitments contingent on the other players' commitments. We differ from the rest of the literature on reciprocal contracting by assuming that punishments cannot be enforced in the event that cooperation breaks down. We fully characterize the outcomes that can be supported as perfect Bayesian equilibrium outcomes in such an environment. We use our characterization to show that the set of supportable outcomes with reciprocal contracting is larger than the set of outcomes available in a centralized mechanism design environment in which the mechanism designer is constrained by his inability to enforce punishments against non‐participants. The difference stems from the players' ability in our contracting game to convey partial information about their types at the time they offer contracts. We discuss the implications of our analysis for modelling collusion between multiple agents interacting with the same principal.     

Cost comparison of four revascularisation procedures for the treatment of multivessel coronary artery disease

Objective: An economic evaluation was performed, using modelling techniques, to compare 1-year total costs of four revascularisation procedures in patients with multivessel disease: on-pump coronary artery bypass grafting (CABG); off-pump CABG; percutaneous coronary intervention (PCI) with bare-metal stents (BMS); and PCI with drug-eluting stents (DES).

Methods: Clinical data were derived from four randomised clinical trials comparing CABG versus PCI, as well as from literature reviews. Resource use and unit cost estimates were modelled to reflect current Canadian practice.

Results: This study demonstrated that 1 year after the initial revascularisation, PCI with BMS is the least costly procedure, followed by off-pump CABG, PCI with DES and on-pump CABG. DES became the most costly procedure if 3.5 or more DES were used or if staged PCI was performed.     

Testing fractional integration with monthly data

Seasonal roots can help to explain the seasonal fluctuations in macroeconomic time series. In this paper we concentrate on monthly data and look at different versions of Robinson’s (1994) tests for testing unit roots and other fractionally integrated hypotheses when the root is located at zero and/or at the seasonal frequencies. A Monte Carlo experiment is carried out to check the power of these tests against different fractional alternatives, and an empirical application, using Spanish monthly data for the consumer price index, is also carried out in the article.