Sources of error in economic time series

This paper classifies and measures the major sources of error, uncertainty or noise in economic data, regarding such data as observations from stationary and non-stationary time series. Uncertainties due to seasonal adjustment, sampling, and transitory variation are studied, both as observable error (eventually removed from preliminary figures when they are revised) and as unobservable error (imbedded in both preliminary and final data). Correlations between different error sources are derived. The results are illustrated with U.S. money supply series.     

Experience with using the Box-Cox transformation when forecasting economic time series

Experience using twenty-one actual economic series suggests that using the Box-Cox transform does not consistently produce superior forecasts. The procedure used was to consider transformations $\text{x}{}^{\mathrm{(\lambda )}}\text{=}\text{(x}{}^{\mathrm{\lambda }}\text{?1)}\text{λ}$, where λ is chosen by maximum likelihood, a linear ARIMA model fitted to x^(λ) and forecasts produced, and finally forecasts constructed for the original series. A main problem found was that no value of λ appeared to produce normally distributed data and so the maximum likelihood procedure was inappropriate.     

Bayes model averaging of cyclical decompositions in economic time series

A flexible decomposition of a time series into stochastic cycles under possible non‐stationarity is specified, providing both a useful data analysis tool and a very wide model class. A Bayes procedure using Markov Chain Monte Carlo (MCMC) is introduced with a model averaging approach which explicitly deals with the uncertainty on the appropriate number of cycles. The convergence of the MCMC method is substantially accelerated through a convenient reparametrization based on a hierarchical structure of variances in a state space model. The model and corresponding inferential procedure are applied to simulated data and to cyclical economic time series like US industrial production and unemployment. We derive the implied posterior distributions of model parameters and some relevant functions thereof, shedding light on several key features of economic time series. Copyright © 2006 John Wiley & Sons, Ltd.     

Trends and cycles in economic time series: A Bayesian approach

Trends and cyclical components in economic time series are modeled in a Bayesian framework. This enables prior notions about the duration of cycles to be used, while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted. The posterior distributions of such underlying cycles can be very informative for policy makers, particularly with regard to the size and direction of the output gap and potential turning points. From the technical point of view a contribution is made in investigating the most appropriate prior distributions for the parameters in the cyclical components and in developing Markov chain Monte Carlo methods for both univariate and multivariate models. Applications to US macroeconomic series are presented.     

Conditional Markov chain and its application in economic time series analysis

Motivated by the great moderation in major US macroeconomic time series, we formulate the regime switching problem through a conditional Markov chain. We model the long‐run volatility change as a recurrent structure change, while short‐run changes in the mean growth rate as regime switches. Both structure and regime are unobserved. The structure is assumed to be Markovian. Conditioning on the structure, the regime is also Markovian, whose transition matrix is structure‐dependent. This formulation imposes interpretable restrictions on the Hamilton Markov switching model. Empirical studies show that this restricted model well identifies both short‐run regime switches and long‐run structure changes in the US macroeconomic data. Copyright © 2010 John Wiley & Sons, Ltd.     

Size and power of tests of stationarity in highly autocorrelated time series

Tests of stationarity are routinely applied to highly autocorrelated time series. Following Kwiatkowski et al. (J. Econom. 54 (1992) 159), standard stationarity tests employ a rescaling by an estimator of the long-run variance of the (potentially) stationary series. This paper analytically investigates the size and power properties of such tests when the series are strongly autocorrelated in a local-to-unity asymptotic framework. It is shown that the behavior of the tests strongly depends on the long-run variance estimator employed, but is in general highly undesirable. Either the tests fail to control size even for strongly mean reverting series, or they are inconsistent against an integrated process and discriminate only poorly between stationary and integrated processes compared to optimal statistics.     

Urbanisation,fiscal decentralisation and economic development: A time series evidence

Several empirical tests using Multiple Regression Analyses were conducted on several hypotheses using time series data obtained from the federal and state governments. The results of our analyses establish that the degree of fiscal decentralisation is dependent on intergovernmental transfers and states income per capita. However intergovernmental transfers were not dependent on expenditure decentralisationper se. The degree of urbanisation was found to be inversely related to fiscal decentralisation. The variable measuring the degree of openess was found not statistically significant as an explanatory variable for fiscal decentralisation. However, the share of agriculture was found significant in revenue decentralisation but loses its importance in expenditure decentralisation.The policy implications of the study are that: (i) There is need for the states to develop plans to increase their per capita income, improve their tax collection system, introduce new tax bases and reduce the high degree of free ridership in public goods and services exhibited by the urban population. (ii) The Federal Government should provide specific grants to state governments for urban development, because of the high per capita cost of public goods and services. (iii) The existing Revenue Allocation Act should be reviewed to reflect state government efforts in generating their own revenue from internal sources.     

On modeling panels of time series*

This paper reviews research issues in modeling panels of time series. Examples of this type of data are annually observed macroeconomic indicators for all countries in the world, daily returns on the individual stocks listed in the S&P500, and the sales records of all items in a retail store. A panel of time series concerns the case where the cross‐sectional dimension and the time dimension are large. Often, there is no a priori reason to select a few series or to aggregate the series over the cross‐sectional dimension. The use of, for example, a vector autoregression or other types of multivariate models then becomes cumbersome. Panel models and associated estimation techniques are more useful. Due to the large time dimension, one should however incorporate the time‐series features. And, the models should not have too many parameters to facilitate interpretation. This paper discusses representation, estimation and inference of relevant models and discusses recently proposed modeling approaches that explicitly aim to meet these requirements. The paper concludes with some reflections on the usefulness of large data sets. These concern sample selection issues and the notion that more detail also requires more complex models.     

Experience with using the Box-Cox transformation when forecasting economic time series: A comment

In a recent study Nelson and Granger (1979) reported on their experience with using the Box-Cox transformation to forecast twenty-one actual economic time series and four simulated series. The objective of this brief note is to offer some qualifications to the results of Nelson and Granger, and to offer a few alternative interpretations.     

The power of four tests of autocorrelation in the linear regression model

The power of each of four tests of first-order autocorrelation in the linear regression model is determined for a simple and multiple regression model whose parameters are presumed to be known. The tests are: Durbin-Watson bounds test, a test based on Theil's best linear unbiased scalar estimator, a test devised by Abrahamse, Koerts and Louter, and an exact test devised by Durbin.For positive values of the coefficient of autocorrelation the Durbin-Watson bounds test is generally better than the tests based on the estimator proposed by Abrahamse, Koerts and Louter, the best linear unbiased scalar estimator, and the Durbin exact test. For negative values of the coefficient of autocorrelation, the pattern of results is mixed for all four test procedures. A byproduct of these experiments is the demonstrated feasibility of enumerating the distribution of the Durbin-Watson test statistic for any regression matrix and thus eliminating the region of indeterminacy from the Durbin-Watson test procedure.     

On the distribution of the sample autocorrelation coefficients

Sample autocorrelation coefficients are widely used to test the randomness of a time series. Despite its unsatisfactory performance, the asymptotic normal distribution is often used to approximate the distribution of the sample autocorrelation coefficients. This is mainly due to the lack of an efficient approach in obtaining the exact distribution of sample autocorrelation coefficients. In this paper, we provide an efficient algorithm for evaluating the exact distribution of the sample autocorrelation coefficients. Under the multivariate elliptical distribution assumption, the exact distribution as well as exact moments and joint moments of sample autocorrelation coefficients are presented. In addition, the exact mean and variance of various autocorrelation-based tests are provided. Actual size properties of the Box–Pierce and Ljung–Box tests are investigated, and they are shown to be poor when the number of lags is moderately large relative to the sample size. Using the exact mean and variance of the Box–Pierce test statistic, we propose an adjusted Box–Pierce test that has a far superior size property than the traditional Box–Pierce and Ljung–Box tests.     

Goodness-of-fit tests for Poisson count time series based on the Stein–Chen identity

To test the null hypothesis of a Poisson marginal distribution, test statistics based on the Stein–Chen identity are proposed. For a wide class of Poisson count time series, the asymptotic distribution of different types of Stein–Chen statistics is derived, also if multiple statistics are jointly applied. The performance of the tests is analyzed with simulations, as well as the question which Stein–Chen functions should be used for which alternative. Illustrative data examples are presented, and possible extensions of the novel Stein–Chen approach are discussed as well.     

On residuals and their autocorrelations in fitted time series models

Summary When discrete autoregressive-moving average time series are fitted by least squares, both the residuals and their autocorrelations are for large n representable as singular linear transformations of the true errors (or white noise) and their autocomlations, respectively, and the matrices of these transformations arc both of the form I-X(X'X) ^-1X, where the rank of X is the number of parameters estimated. However, the large-sample properties of these two sets of statistics are fundamentally different, a phenomenon which is of considerable importance for the use of the residual autocorrelations in performing tests of fit of these models.     

On univariate time series methods and simultaneous equation econometric models

Systematic testing of the implications of the structural assumptions for the properties of the final equations and transfer functions associated with a dynamic econometric model, as proposed by Zellner and Palm (1974–1975), proved to be useful in model building. This paper contains several remarks on the use of univariate time series methods to empirically check out the implications of a linear dynamic simultaneous equation model.     

Forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models

Nine macroeconomic variables are forecast in a real-time scenario using a variety of flexible specification, fixed specification, linear, and nonlinear econometric models. All models are allowed to evolve through time, and our analysis focuses on model selection and performance. In the context of real-time forecasts, flexible specification models (including linear autoregressive models with exogenous variables and nonlinear artificial neural networks) appear to offer a useful and viable alternative to less flexible fixed specification linear models for a subset of the economic variables which we examine, particularly at forecast horizons greater than 1-step ahead. We speculate that one reason for this result is that the economy is evolving (rather slowly) over time. This feature cannot easily be captured by fixed specification linear models, however, and manifests itself in the form of evolving coefficient estimates. We also provide additional evidence supporting the claim that models which ‘win’ based on one model selection criterion (say a squared error measure) do not necessarily win when an alternative selection criterion is used (say a confusion rate measure), thus highlighting the importance of the particular cost function which is used by forecasters and ‘end-users’ to evaluate their models. A wide variety of different model selection criteria and statistical tests are used to illustrate our findings.     

Large shocks,small shocks,and economic fluctuations: Outliers in macroeconomic time series

We analyse fifteen post-World War II US macroeconomic time series using a modified outlier identification procedure based on Tsay (1988a). ‘Large shocks’ appear to be present in all the series we examined. Furthermore, there are three basic outlier patterns: (1) outliers seem to be associated with business cycles, (2) outliers are clustered together—both over time and across series, (3) there appears to be a dichotomy between outlier behaviour of real versus nominal series. Also, after controlling for outliers, much of the evidence of non-linearity in many of the time series is eliminated.     

Efficient tests for the presence of a pair of complex conjugate unit roots in real time series

We introduce a framework which allows us to draw a clear parallel between the test for the presence of seasonal unit roots and that for unit root at frequency 0 (or π$\pi$). It relies on the properties of the complex conjugate integrated of order one processes which are implicitly at work in the real time series. In the same framework as that of Phillips and Perron (Biometrica 75 (1988) 335), we derive tests for the presence of a pair of conjugate complex unit roots. The asymptotic distribution we obtain are formally close to those derived by these authors but expressed with complex Wiener processes. We then introduce sequences of near-integrated processes which allow us to study the local-to-unity asymptotic of the above test statistics. We state a result on the weak convergence of the partial sum of complex near-random walks which leads to complex Orstein–Uhlenbeck processes. Drawing on Elliott et al. (Econometrica 64 (1996) 813) we then study the design of point-optimal invariant test procedures and compute their envelope employing local-to-unity asymptotic approximations. This leads us to introduce new feasible and near efficient seasonal unit root tests. Their finite sample properties are investigated and compared with the different test procedures already available (J. Econometrics 44 (1991) 215; 62 (1994) 415; 85 (1998) 269) and those introduced in the first part of the paper.     

Some comparisons of tests for a shift in the slopes of a multivariate linear time series model

Our objective is to find a simple, robust, reasonably powerful test for a shift in one or more of the slopes in a linear time series model at some unknown point of time. Two such tests are ‘Chow's test’ (1960) for a shift at the midpoint of the record and the ‘Farley-Hinich test’ (1970b); both can be performed easily with standard regression programs. In section 2, we compare the asymptotic properties of these tests when the disturbance variance is known. As expected, Chow's test is superior when the true shift is near the middle of the record; with a single, uniformly-distributed explanatory variable, the Farley-Hinich tests dominates over the remaining eighty-four percent of the record. In section 3, we describe the results of some Monte Carlo experiments with a finite sample, which can be summarized as follows. (i) The asymptotic results of section 2 were appropriate for finite sample power comparisons. (ii) The relative performance of the two tests does not depend appreciably on whether the variance is known. (iii) The likelihood ratio test, which is far more costly to perform than the other two tests, does not dominate either Chow's test or the Farley-Hinich test; it has moderately more power at the ends of the record, moderately less in the middle. The conclusion is clear: at low cost (in terms of computer cost and lost power), one can reduce the probability of over- looking a structural shift by routinely performing Chow's test or the Farley-Hinich test.