Search

Summary We present three non-parametric trispectrum tests that can establish whether the spectral decomposition of kurtosis of high frequency financial asset price time series is consistent with the assumptions of Gaussianity, linearity and time reversiblility. The detection of non-linear and time irreversible probabilistic structure has important implications for the choice and implementation of a range of models of the evolution of asset prices, including Black–Scholes–Merton option pricing model, auto-regressive conditional heteroscedastic or generalized auto-regressive conditional heteroscedastic and stochastic volatility models. We apply the tests to a selection of high frequency Australian stocks.

We perform non-linearity tests using daily data for leading currencies that include the Australian dollar, British pound, Brazilian real, Canadian dollar, euro, Japanese yen, Mexican peso, and the Swiss franc to resolve the issue of whether these currencies are driven by fundamentals or exogenous shocks to the global economy. In particular, we use a new method of testing for linear and nonlinear lead/lag relationships between time series, introduced by Brooks and Hinich (J Empir Finance 20:385–404, 1999), based on the concepts of cross-correlation and cross-bicorrelation. Our evidence points to a relatively rare episodic nonlinearity within and across foreign exchange rates. We also test the validity of specifying ARCH-type error structures for foreign exchange rates. In doing so, we estimate Bollerslev’s (J Econom 31:307–327, 1986) generalized ARCH (GARCH) model and Nelson’s (1988) exponential GARCH (EGARCH) model, using a variety of error densities [including the normal, the Student-t distribution, and the Generalized Error Distribution (GED)] and a comprehensive set of diagnostic checks. We apply the Brooks and Hinich (1999) nonlinearity test to the standardized residuals of the optimal GARCH/EGARCH model for each exchange rate series and show that the nonlinearity in the exchange rates is not due to ARCH-type effects. This result has important implications for the interpretation of the recent voluminous literature which attempts to model financial asset returns using this family of models.

This paper investigates the duration dependence of the US stock market cycles. A new classification method for bull and bear market regimes based on the crossing of the market index and its moving average is proposed. We show evidence of duration dependence in whole cycles. The half cycles, however, are found to be duration independent. More importantly, we find that the degree of duration dependence of the US stock market cycles has dropped after the launch of the NASDAQ index.

Using a modification of the Hinich, J Time Ser Anal 3(3):169–176, (1982) bispectrum test for nonlinearity and Gaussianity, the residuals of the Tiao and Box, J Am Stat Assoc 76:802–816, (1981) constrained and unconstrained VAR models for the gas furnace data reject the assumption of Gaussianity and linearity over a grid of bandwidths for estimating the bispectrum. These findings call into question the specification of the linear VAR and VARMA models assumed by Tiao and Box, J Am Stat Assoc 76:802–816, (1981). Utilizing the alternative Hinich J Nonparametr Stat 6:205–221, (1996) nonlinearity test, the residuals of the VAR model were shown to exhibit episodic nonlinearity. The sensitivity of the findings to outliers is investigated by estimating and testing the residuals of L1 and MINIMAX models from 1–6 lags. Building on the linear dynamic specification, a multivariate adaptive regression splines (MARS) model is estimated, using two software implementations, and shown to remove the nonlinearity in the residuals. Leverage plots were used to illustrate the “cost” of imposing a linearity assumption. Out-of-sample forecasting tests from 1–6 periods ahead found that using the sum-of-squared errors criteria, the MARS model out performed ACE, GAM and projection pursuit models.

In this article, we show how tests of nonlinear serial dependence can be applied to high-frequency time series data that exhibit high volatility, strong mean reversion, and leptokurtotis. Portmanteau correlation, bicorrelation, and tricorrelation tests are used to detect nonlinear serial dependence in the data. Trimming is used to control for the presence of outliers in the data. The data that are employed are 161,786 half-hourly spot electricity price observations recorded over nearly a decade in the wholesale electricity market in New South Wales, Australia. Strong evidence of nonlinear serial dependence is found and its implications for time series modeling are discussed.

This article shows that the Hinich (1982) bispectrum test for gaussianity and the Hinich and Rothman (1998) test for time reversibility can be used to falsify the null hypothesis that an autoregressive conditionally heteroskedastic model (ARCH) or its generalization (GARCH) generates nonlinear behavior in the variance of an observed time series. The term “falsify” means that the null hypothesis can be rejected with a given size using a nonparametric test based on the bispectrum where the data is trimmed to control the sizes. Rejecting the null hypothesis implies that the ARCH or GARCH model that is estimated from the data is not a complete statistical description of the dependence structure in the variance of the process.

The objective of this paper is to re-examine the weak-form efficiency of 10 Asian emerging stock markets. Using a battery of nonlinearity tests, the statistical results reveal that all the returns series still contain predictable nonlinearities even after removing linear serial correlation from the data. The next stage of sub-sample analysis using the Hinich [Hinich, M., 1996. Testing for dependence in the input to a linear time series model. Journal of Nonparametric Statistics 6, 205–221] bicorrelation test shows that the 10 Asian series follow a pure noise process for long periods of time, only to be interspersed with brief periods of strong nonlinear dependence. The exploratory investigation found that the cross-country differences in nonlinear departure from market efficiency can be explained by market size and trading activity, while the transient burst of nonlinear periods in each individual market can be attributed largely to the occurrence of economic and political events.

The fractional integrated process has been widely studied since the seminal work of Granger and Joyeux (1980). However, most of the tests for fractional integration are time-domain tests. This article develops a Cramer–von Mises type frequency-domain test for fractional integration. Our test is shown to have high power against a wide variety of alternatives. The test is applied to U.S. real disposable income and the quarterly real GDP data of the G7 countries. It is found that most of the series are fractionally integrated at the 5% level. The result is in line with Hinich and Chong (2007).

This article uses daily observations for the Canadian dollar–U.S. dollar nominal exchange rate over the recent flexible exchange rate period and a new statistical technique, recently developed by Hinich, to detect major political and economic events that have affected the exchange rate.

This paper uses a new method of testing for linear and nonlinear lead/lag relationships between time series, introduced by Brooks and Hinich [Brooks, C. and Hinich, M. J. Cross-Correlations and Cross-Bicorrelations in Sterling Exchange Rates. Journal of Empirical Finance 20 (1999), 385–404.], on Alberta's natural gas and power markets. The test, based on the concepts of cross-correlation and cross-bicorrelation, is used after pre-whitening of the data to test for the existence of residual nonlinearity as well as the episodic nature of the nonlinearity. Our evidence points to a relatively rare episodic nonlinearity within and across the two series, having important implications for forecasting these series.

We demonstrate that the process of generating smooth transitions can be viewed as a natural result of the filtering operations implied in the generation of discrete-time series observations from the sampling of data from an underlying continuous time process that has undergone a process of structural change. In order to focus discussion, we utilize the problem of estimating the location of abrupt shifts in some simple time series models. This approach will permit us to address salient issues relating to distortions induced by the inherent aggregation associated with discrete-time sampling of continuous time processes experiencing structural change. We also address the issue of how time irreversible structures may be generated within the smooth transition processes.

This letter applies the Hinich portmanteau bicorrelation test jointly with the windowed testing procedure to detect nonlinear behaviour in the rate of returns series for seven Latin American stock market indices. Our results suggest that the nonlinear serial dependencies are episodic in nature. All the stock returns series are characterized by few brief periods of highly significant nonlinearity, followed by long time periods in which the returns follow a pure noise process. Our findings help explain why there are difficulties in forecasting asset returns.

This study employs the Hinich portmanteau bicorrelation test (Hinich 1996; Hinich and Patterson 1995) as a diagnostic tool to determine the adequacy of Generalised Autoregressive Conditional Heteroscedasticity (GARCH) models for eight Asian stock markets. The bicorrelation test results demonstrate that this type of model cannot provide an adequate characterisation for the underlying process of all the selected Asian stock markets. Further investigation using the windowed test procedure reveals that the violation of the covariance stationarity assumption as required by the GARCH process is due to the presence of transient epochs of dependencies in the data. The inadequacy of GARCH models has strong implications for the pricing of stock index options, portfolios selection, development of optimal hedging techniques and risk management.

A novel method for modeling and forecasting seasonal time series is proposed. Unlike the traditional approach that depends solely on dynamic models, the proposed method combines stochastic dynamic modeling with an analysis filter bank designed to reduce dimensionality and to extract persistent components for reliable long-term forecasting. The filter bank decomposes the time series of interest into seasonal components, and only those components that are highly coherent across the periods are selected for subsequent modeling and forecasting. Experiments show that under suitable conditions, the use of highly coherent components not only reduces the modeling complexity and the required amount of training data but also limits the impact of noise and occasional corruption in the training data and thus provides robust forecasts with reduced variability. Fourier and wavelet filter banks are discussed in detail. Simulated and real-data examples are used to illustrate the method.

This paper proposes and implements a new methodology for forecasting time series, based on bicorrelations and cross-bicorrelations. It is shown that the forecasting technique arises as a natural extension of, and as a complement to, existing univariate and multivariate non-linearity tests. The formulations are essentially modified autoregressive or vector autoregressive models respectively, which can be estimated using ordinary least squares. The techniques are applied to a set of high-frequency exchange rate returns, and their out-of-sample forecasting performance is compared to that of other time series models

This paper revisits the income convergence hypothesis by using the nonlinear unit root test of Kapetanios et al. [Kapetanios, G., Shin, Y. and A. Snell, 2003. Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 112, 359-379.]. Out of the 12 OECD income gaps in which nonlinearity has been detected, two cases of long-run converging and four cases of catching up are found. © 2008 Elsevier B.V. All rights reserved.

We introduce a frequency-domain test of time reversibility, the REVERSE test. It is based on the bispectrum. We analytically establish the asymptotic distribution of the test and also explore its finite-sample properties through Monte-Carlo simulation. Following other researchers who demonstrated that the problem of business-cycle asymmetry can be stated as whether macroeconomic fluctuations are time irreversible, we use the REVERSE test as a frequency-domain test of business-cycle asymmetry. Our empirical results show that time irreversibility is the rule rather than the exception for a representative set of macroeconomic time series for five OECD countries.

Interest has been growing in testing for nonlinearity and chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We apply five tests for nonlinearity or chaos to various monetary aggregate data series. We find that the inferences vary across tests for the same data, and within tests for varying sample sizes and various methods of aggregation of the data. Robustness of inferences in this area of research seems to be low and may account for the controversies surrounding empirical claims of nonlinearity and chaos in economics.

We model correlated voter-candidate issue data within the framework of the Enelow-Hinich spatial model of predictive dimensions. The empirical consequences of this model of the issue data are surprising and allow for an indirect test of the Enelow-Hinich spatial model. The central prediction of the correlated data model we construct, which depends critically on the underlying spatial model, is tested with issue data from the 1980 NES pre-election interview. The test results are highly supportive of the model's predictions. We conclude both that the spatial model of predictive dimensions is empirically supported and that candidate spatial locations estimated by the model are not an artifact of correlated voter-candidate issue data.

Barnett and Chen [4–6] have displayed evidence of chaos in certain monetary aggregates, but the tests have unknown statistical sampling properties. Using monthly growth rates in monetary aggregates, we conduct bispectral tests for nonlinearity. Our tests have known sampling properties, and we find deep nonlinearity in some monetary aggregate series.

This paper tests the relative stability of voters' evaluations of candidates' policy positions over time. Using survey responses rating political figures in the 1980 and 1984 elections, a scaling procedure recovers the estimated positions of candidates and voters in a two-dimensional ideological space. Comparisons of candidate positions along ideological dimensions reveals little movement during that interval. Incumbency is also proposed as a new measurement for nonpolicy candidate characteristics in empirical tests of the spatial model. The results imply that voter perceptions of candidates are remarkably stable over this four year time period.

In this article, we use half hourly spot electricity prices and load data for the National Electricity Market (NEM) of Australia for the period from December 1998 to August 2007 to test for randomly modulated periodicity. In doing so, we apply signal coherence spectral analysis to the time series of half hourly spot prices and megawatt-hours (MWh) load demand from 7/12/1998 to 31/08/2007 using the FORTRAN 95 program developed by Hinich (2000). We detect relatively steady weekly and daily cycles in load demand but relatively more unstable cycles in prices.

This paper first introduces the trispectrum-based time reversibility test to complement its bispectrum counterpart introduced earlier in extant literature. Using these frequency domain tests, we then examine whether the returns series of major stock market indices in 48 countries are time reversible. The results consistently show that time irreversibility is the rule rather than the exception for stock market indices. Further investigation on the sources of time irreversibility by applying the bispectrum-based Gaussianity and linearity tests on the original returns series reveals that the rejections in most cases are due to nonlinearity. Implications of the findings are discussed in the paper.

In this article, we use half hourly spot electricity prices and load data for the National Electricity Market (NEM) of Australia for the period from December 1998 to February 2008 to test for episodic nonlinearity in the dynamics governing daily and weekly cycles in load and spot price time series data. We apply the portmanteau correlation, bicorrelation and tricorrelation tests introduced in Hinich (1996) to the time series of half hourly spot prices and load demand from 7/12/1998 to 29/02/2008 using a FORTRAN 95 program. We find the presence of significant third and fourth order (non-linear) serial dependence in the weekly load and spot price data in particular, but to a much more marginal extent, in the daily data.

Summary Many recent papers have documented periodicities in returns, return volatility, bid–ask spreads and trading volume, in both equity and foreign exchange markets. We propose and employ a new test for detecting subtle periodicities in time series data based on a signal coherence function. The technique is applied to a set of seven half-hourly exchange rate series. Overall, we find the signal coherence to be maximal at the 8-h and 12-h frequencies. Retaining only the most coherent frequencies for each series, we implement a trading rule that is based on these observed periodicities. Our results demonstrate in all cases except one that, in gross terms, the rules can generate returns that are considerably greater than those of a buy-and-hold strategy, although they cannot retain their profitability net of transactions costs. We conjecture that this methodology could constitute an important tool for financial market researchers which will enable them to detect, quantify and rank the various periodic components in financial data better.

This paper uses hourly electricity prices and MW hour demand for Alberta, Canada over the deregulated period after 1996 to test for randomly modulated periodicity. In doing so, we apply the signal coherence spectral analysis to the time series of hourly spot prices and megawatt-hours (MWh) demand from 1/1/1996 to 12/7/2003 using the FORTRAN 95 program developed by Hinich (2000). We detect relatively steady weekly and daily cycles in demand but very unstable cycles in prices.

The present study proposes the use of nonlinear serial dependencies in the returns series for assessing market responses to new information. Unlike the standard event study methodology, this paper advocates a reserve form that let data analysis to first detect those periods with significant nonlinearity, and then identify whether their occurrences can be associated with major economic or political events. Our proposed research framework involves the application of the Hinich (1996) portmanteau bicorrelation test statistic in a moving time windows setting. Using the Malaysian stock market as a case study, our empirical application is able to detect not only the presence of nonlinear serial dependencies in the KLCI intraday ten-minute returns, but also the time periods when they occurred. The next stage of event-matching managed to identify major events that unsettled the local stock market in all cases except one. In particular, the Russian crisis, the unorthodox capital control measures and the Brazilian crisis are found to be responsible for the short burst of nonlinear behavior. However, the paper noted that factors that shook the market need not be news announcement in the media. It could be due to, among others, rumors, speculations, expectations, or worries prevail in the investment communities.

Detecting nonlinearity in financial time series is a key point when the main interest is to understand the generating process. One of the main tests for testing linearity in time series is the Hinich Bispectrum Nonlinearity Test (HINBIN). Although this test has been succesfully applied to a vast number of time series, further improvement in the size power of the test is possible. A new method that combines the bispectrum and the surrogate method and bootstrap is then presented for detecting nonlinearity, gaussianity and time reversibility. Simulated and real data examples are given to demonstrate the efficacy of the new tests.

A nation's economy is a complex nonlinear dynamical system with links to other national economies. Classical macroeconomic models typically incorporate linear approximations to the nonlinear world and add simple dynamics to capture adjustments over time. Most of these simplifications offer little insight into the nonlinear structure of economic relationships that exist. Nor do they provide useful predictions beyond the short-term predictions of the autoregressive linear models from which they are derived.

This paper proposes two new tests for linear and nonlinear lead/lag relationships between time series based on the concepts of cross-correlations and cross-bicorrelations, respectively. The tests are then applied to a set of Sterling-denominated exchange rates. Our analysis indicates that there existed periods during the post-Bretton Woods era where the temporal relationship between different exchange rates was strong, although these periods have become less frequent over the past 20 years. In particular, our results demonstrate the episodic nature of the nonlinearity, and have implications for the speed of flow of information between financial series. The method generalises recently proposed tests for nonlinearity to the multivariate context.

The central motivation for this paper is to combine a multivariate nonlinearity test with a nonlinear time series forecasting model of the VAR-type which incorporates lags of the cross-products with the other exchange rate in the system using high frequency data. The paper draws together two sets of literature which have previously been considered entirely separately: namely time series tests for nonlinearity and multivariate exchange rate forecasting. The out-of-sample forecasting performance of the new technique using cross-bicorrelations is inferior to that of univariate time series models, in spite of significant in-sample cross-bicorrelation statistics, although it does seem to be able to predict the signs of the returns well in certain cases. A number of explanations for this apparent paradox are proposed.

We examine a method recently proposed by Hinich and Patterson (mimeo, University of Texas at Austin, 1995) for testing the validity of specifying a GARCH error structure for financial time series data in the context of a set of ten daily Sterling exchange rates. The results demonstrate that there are statistical structures present in the data that cannot be captured by a GARCH model, or any of its variants. This result has important implications for the interpretation of the recent voluminous literature which attempts to model financial asset returns using this family of models.

This is the FINAL draft of this paper reporting the results of a long ongoing competition. The paper now is forthcoming in the Journal of Econometrics. This final version replaces the earlier draft that was also in this archive. Interest has been growing in testing for nonlinearity or chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We apply five highly regarded tests for nonlinearity or chaos to ten simulated data series. The data generating mechanisms include linear processes, chaotic recursions, and nonchaotic stochastic processes; and both large and small samples were included in the experiment. The data series were produced in a single blind manner by the competition manager and sent by e-mail, without identifying information, to the experiment participants. Each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test. The results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos. The simulated data and the code for some of the tests entered into this competition also are online in this archive.

Estimates of the parameters of a linear model are usually obtained by the method of ordinary least-squares (OLS), which is sensitive to large values of the additive error term. By dividing the sample into nonoverlapping subsamples and computing the trimmed means of OLS subsample regression coefficients, we obtain a simple, consistent and asymptotically normal initial estimate of the coefficients, which protects the analyst from large values of ∈ i which are often hard to detect using OLS on a model with many regressors. The technique is applied to the calculation of risk parameters in the capital asset pricing model for securities on the N. Y. Stock Exchange.

Using a scaling procedure that directly estimates the key parameters of a new spatial model of elections described previously, we recover the locations of voters and candidates in the three waves of the 1980 Major Panel File. The candidates lie relatively close to the average voter and exhibit little movement over the campaign. Voter locations are more volatile. A unique equilibrium is computed for each wave of the panel, based on a probabilistic model of individual voting decisions. This equilibrium stays relatively close to the Republican candidates, finally settling in a position between the Democrats and Republicans, but closer to Reagan than Carter.

In Enelow and Hinich (1982, 1984a), the spatial theory of elections is augmented to incorporate nonspatial candidate characteristics in the voter's candidate evaluations. In this extension, nonpolicy characteristics are represented by an additive term independent of the policy portion of the voter's evaluation. In Grofman (1985), an interactive voting model is introduced, in which nonpolicy characteristics affect the voter's evaluation of the candidate only through the candidate's policy positions. An empirical question is raised by these two contrasting approaches: do nonpolicy candidate qualities affect voter choice independently of candidate policies, interactively with candidate policies, or through both means? This paper is designed to answer that question. An extended form of the Enelow-Hinich model is specified with both main and interaction terms. The model is then used to predict voter choice in the 1972, 1976, and 1980 presidential elections. Statistical evidence for the existence of interaction...

This article applies a newly developed statistical technique to time series of daily rates of return of 15 common stocks. The technique involves estimating the bispectrum of the observed time series. The bispectrum is defined as the double Fourier transform of the third-order cumulant function. If the process generating rates of return is linear with independent innovations, then the skewness of the bispectrum will be constant. The article describes a test that can detect nonconstant skewness in the bispectrum. Hence if the test rejects constant skewness, a nonlinear process is implied. As a consequence, the test can distinguish between white noise and purely random noise. The results suggest that daily stock returns are generated by a nonlinear process.

When issues (i.e., dimensions) are voted on one at a time, a voter whose preferences are not separable across issues must forecast the outcome of later issues in order to know how to vote in the present. This is the problem of expectations. In this article, we develop a general theory designed to handle this problem. Assuming that voters are risk averse and maximize expected utility, we demonstrate that a random variable forecast of how later issues will be decided reduces to a point forecast, which is the mean of the multidimensional random variable. We also show that single-peaked preferences are induced on each issue, and consequently there exists an equilibrium across issues.

Median voter results have played a central role in public economic analyses where there is a linearly ordered set of policy alternatives that is to be voted on. In these analyses, the median voter results themselves have been based on voters' preferences being single-peaked with respect to the ordering for this set. In this paper, we derive conditions that are both necessary and sufficient for such single-peakedness to occur whenever we have economic decision-makers. This leads to certain new median voter results. In addition, it also reveals some important limitations on median voter analyses.

This essay ascertains some general conditions for equivalence and nonequivalence among six election objectives: 01, maximizing expected plurality; 02, maximizing proportion of expected vote; 02, maximizing expected vote; 04, maximizing probability that plurality exceeds some level; 05, maximizing probability that proportion of vote exceeds some level; 06, maximizing probability that vote exceeds some level. The major findings are these: (1) 01, 02, and 03 are equivalent if the election is zero-sum-like in expected vote; (2) 01 and 02 are equivalent if competition is strongly symmetric. A necessary condition for this equivalence is also presented for 2-candidate elections: (3) 01 and 04 are equivalent, as are 03 and 06, if the candidate's forecasting error is independent of all strategies; (4) 01 and 04 are equivalent for two-candidate elections, and for n-candidate elections 02 and 05 are equivalent, as are 03 and 06, if the distribution of a candidate's forecasting error is multivariate normal, and if the level of plurality, proportion, or vote to be exceeded is the minimax value of the election game under 01, 02, or 03; (5) findings of equivalence and nonequivalence depend upon the definition of equivalence (findings 1 and 2 rely upon an election with all candidates at equilibrium strategies, while findings 3 and 4 do not); (6) equivalence and nonequivalence among election objectives may be sensitive to the candidate's attitude toward risk, i.e., to the functional form of his utility function in pluraliy, vote proportion, or vote; election objectives depend on information, competitive environment, and constitutional arrangements. Hence, statements of preference for alternative election systems, laws, and reforms perforce entail reasonable theoretical expectations about the way in which these systems, laws, and reforms affect the candidates' campaign objectives, as well as about equivalence and nonequivalence among these objectives.

The fundamental process of politics is the aggregation of citizens' preferences into a collective—a social—choice. We develop, interpret, and explain non-technically in this expository essay the definitions, assumptions, and theorems of a mathematical model of one aggregative mechanism—the electoral process. This mechanism is conceptualized here as a multidimensional model of spatial competition in which competition consists of candidates affecting turnout and the electorate's perception of each candidate's positions, and in which the social choice is a policy package which the victorious candidate advocates.This approach, inaugurated by Downs's An Economic Theory of Democracy, and falling under the general rubric “spatial models of party competition,” has been scrutinized, criticized, and reformulated. To clarify the accomplishments of this formulation we identify and discuss in section 2 the general democratic problem of ascertaining a social preference. We review critically in section 3 the definitions and assumptions of our model. We consider in sections 4 and 5 the logic of a competitive electoral equilibrium. We assume in section 4 that the electorate's preferences can be summarized and represented by a single function; the analysis in section 5 pertains to competition between two organizational structures or two opposed ideologies (i.e., when two functions are required to summarize and represent the electorate's preference). Finally, we suggest in section 6 a conceptualization of electoral processes which facilitates extending and empirically testing our model.

Previous research into the Gaussianity of ocean acoustical time series has examined univariate marginal densities. In this paper we present research which examines this issue from a time series point of view. Even series which previously passed univariate tests for normality are shown to be non-Gaussian time series. Additionally, these time series are shown to be nonlinear time series, so that such acoustical series must be modeled in a nonlinear fashion.

Developing a marketing information system involves a variety of rather unique statistical problems; partial solutions are available for some of these problems, while no formal solutions are now available for others. For example, such problems face a group which is responsible fur keeping management of a major petroleum company apprised of movements in competitors’ prices for a wide array of products. Several types of data flow in at different intervals — daily, weekly, and monthly. Some data are reported from within the firm, others come from commercial data collection sources and still others from numerous government agencies. The data inputs are subject to various reporting errors and distortions which might be considered random error or “noise”. Prices in these markets may shift discretely rather than drifting up and down, and there may be a substantial time lag between a de facto change in price by one competitor and a public announcement that such a change has taken place. Tracking these price series for evidence of changes in competitive conditions is thus a major function of the information system.

A well-known model of asset returns is the index equation in which R j, t is the observed rate of return on asset j during time period t; R m, t is the return on a market index; β j, t is a parameter which measures the risk of asset j; α j, t is another parameter which has been given several interpretations by different authors; and e j, t is a stochastic disturbance term with zero mean. We discuss techniques for avoiding some of the pitfalls caused by nonstationarity. We explain a technique of robust regression; when the disturbances do not have all the standard spherical Gaussian properties so familiar from econometrics texts. We then describe a methodology for regression with a nonstationary model. It utilizes orthogonal polynomials of time to track the paths taken by the risk coefficient. This is a very robust specification and it has the potential to side-step many of the troubling theoretical and econometric problems of the model. By using the simple model we are not obliged to take sides on the question of which particular theory is “true” because this specification will be an approximation to all of the currently suggested theories (for a particular data sample). It is also reasonable to assert that the specification will approximate currently unknown models that may be developed in the future.

A new model of voter uncertainty about candidate positions is presented in which voters simplify the issue positions of the candidate by representing them as a random variable on an underlying evaluative dimension. It is further assumed that the degree of voter uncertainty depends upon the mean location of this random variable. It is demonstrated that this type of spatially dependent uncertainty results in a shift of each voter's ideal point on the underlying dimension. We discuss two types of shifts, one in which voter ideal points are shifted toward the extremes and the other in which they are shifted toward the center and comment on the consequences of these shifts for two-candidate electoral competition. Finally, we relate our model to earlier work on the subject by Downs (1957) and Shepsle (1972).