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This book consists of three parts: Part One is composed of two introductory chapters. The first chapter provides an instrumental varible interpretation of the state space time series algorithm originally proposed by Aoki (1983), and gives an introductory account for incorporating exogenous signals in state space models. The second chapter, by Havenner, gives practical guidance in apply­ ing this algorithm by one of the most experienced practitioners of the method. Havenner begins by summarizing six reasons state space methods are advanta­ geous, and then walks the reader through construction and evaluation of a state space model for four monthly macroeconomic series: industrial production in­ dex, consumer price index, six month commercial paper rate, and money stock (Ml). To single out one of the several important insights in modeling that he shares with the reader, he discusses in Section 2ii the effects of sampling er­ rors and model misspecification on successful modeling efforts. He argues that model misspecification is an important amplifier of the effects of sampling error that may cause symplectic matrices to have complex unit roots, a theoretical impossibility. Correct model specifications increase efficiency of estimators and often eliminate this finite sample problem. This is an important insight into the positive realness of covariance matrices; positivity has been emphasized by system engineers to the exclusion of other methods of reducing sampling error and alleviating what is simply a finite sample problem. The second and third parts collect papers that describe specific applications.



Introduction to State Space Modeling


1. The SSATS Algorithm and Subspace Methods

This chapter summarizes the development of the State Space Aoki Time Series algorithm since its inception in 1983 as an instrumental variables method, and relates it to subspace methods popular in the engineering literature from the viewpoint of orthogonal projections. Except for the numerical method used to calculate orthogonal projections these two classes of methods are conceptually equivalent.2
Masanao Aoki

2. A Guide to State Space Modeling of Multiple Time Series

In teaching time series analysis over the past several years I have become aware of the gap between the often elegant theory of the methods developed and the sometimes crude issues that arise in their application to real problems. A substantial amount of important practical information is relegated to a minor position and is either haphazardly conveyed in long sessions in the computer laboratory, or not at all. This guide is an attempt to organize and formalize that information as it applies to the use of Aoki’s state space time series procedure, employing monthly data on four macroeconomic series as an illustration. Here I am taking the role of the expert witness, and offering opinions where relevant; the judgmental decisions embodied in the discussion reflect my experience, both from my own personal projects and from those of my students.
Arthur Havenner

Applications of State Space Algorithm


3. Evaluating State Space Forecasts of Soybean Complex Prices

A multivariate state space time series model is fitted to monthly data on six series, soybeans, meal, and oil prices in both the spot and futures markets. These are highly volatile series that are difficult to forecast accurately. The usual statistical criteria are applied to evaluate model performance on the 155 months in sample, and on 47 months out of sample. The model’s root mean squared errors and other summary statistics (including graphs of the actual, forecast, and error) indicate that the model fits very well, both in sample and out. However, a small root mean squared error is not enough to guarantee even a profit, let alone a gain in risk-adjusted utility, from using the model forecasts in actual trading. For example, small errors with random signs can make even a relatively accurate model a financial disaster, as can a few ill-timed large errors. It is very difficult to evaluate the usefulness of model forecasts for financial decisionmaking, where both the direction and magnitude of risky strategies combine to produce success or failure for investors whose well-being not only depends on the mean returns, but also the associated variances. We examine a number of nonstandard methods for evaluating forecasts. For example, a nonparametric test is applied to examine the null hypothesis of randomness in the price direction forecasts, with the result that the randomness hypothesis is rejected in general. Even if the direction forecasts are right on average, however, there is no assurance that a risk averse investor would choose to employ the model given the risk levels involved. To investigate this issue, a particular utility function, logarithmic, is applied to the model moments to calculate the expected utility from a naive speculation rule utilizing the model forecasts, and then realized utility is computed for the 47 months out of sample. We find that the model performs well, earning risk-adjusted utility for a hypothetical logarithmic utility investor. In the course of evaluating the forecasts from the soybean complex model, several important issues in financial model evaluation are raised.
Derek Berwald, Arthur Havenner

4. Forecasts of Monthly U.S. Wheat Prices: A Spatial Market Analysis

The value of timely and accurate forecasts can hardly be overstated. Because remuneration for accurate forecasting is so great, an army of economists, traders, and speculators expend considerable effort to gain insight into the future path of asset prices. Theoretically, the efforts of these agents drives markets to efficiency. In a frictionless world, efficiency implies that asset values would tend to follow a random walk. Under this sort of stochastic process, the best predictor of price in period t+1 is the price at time t.
Channing Arndt, Kenneth Foster

5. Managing the Herd: Price Forecasts for California Cattle Production

A state space model of vertically integrated California cattle prices is constructed and tested in and out of sample. The state space model provides more accurate forecasts than a vector autoregressive model or futures market prices. Since the weight gain characteristics of holding cattle over the relevant production periods are well known, the primary uncertainty lies in the prices. Dependable price forecasts permit market participants to optimize their purchases and sales.
Lorraine M. Egan, Arthur M. Havenner

6. Labor Market and Cyclical Fluctuations

The purpose of this paper is to investigate how labor and good market comovements account for aggregate fluctuations in the US economy after the first oil shock. Labor supply and labor demand are approximated by labor force and employment respectively since we do not assume market clearing at cyclical frequencies. Real wages link labor and good markets by affecting labor quantities and aggregate supply of goods. Aggregate demand for goods is expressed in terms of income determination, which is largely dominated by labor income. Finally, a money reaction function is introduced to assess the role of nominal shocks.
Riccardo Fiorito

7. Modeling Cointegrated Processes by a Vector-Valued State Space Algorithm — Evidence on The Impact of Japanese Stock Prices on The Finnish Derivatives Market

In the present study we provide new evidence on the impact of the Japanese stock market on the Finnish derivatives. Initially, we test for the presence of unit roots among the observed input-output processes. Next, cointegration in the system is explicitly tested, to justify the estimated vector-valued state space model. The trend and cyclical components of the endogenous vector are extracted and analyzed. The content of the cyclical component is analyzed by spectral analysis. Finally, the error processes are subjected to some statistical tests and used as signal variables in nonlinear model building.
Ralf Östermark

8. A Method for Identification of Combined Deterministic Stochastic Systems

The paper présentes a numerically stable and general algorithm for identification and realization of a complete dynamic linear state space model, including the system order, for combined deterministic and stochastic systems from time series. A special property of this algorithm is that the innovations covariance matrix and the Markov parameters for the stochastic sub-system are determined directly from a projection of known data matrices, without e.g. recursions of non-linear matrix Riccatti equations. A realization of the Kaiman filter gain matrix is determined from the estimated extended observability matrix and the Markov parameters. Monte Carlo simulations are used to analyze the statistical properties of the algorithm as well as to compare with existing algorithms.
David Di Ruscio

9. Competing Exchange Rate Models: A State Space Model vs Structural and Time Series Alternatives

Exchange rate modeling has been popular among economists interested in the fields of international trade, finance, monetary policy, and time series analysis for the past twenty years, beginning shortly after the current regime of floating rates. Articles have been written to propose structural models [e.g., Bilson (1979), Dornbusch and Fischer (1980), Edwards (1983), Frankel (1979), Kiguel (1987), Mussa (1976), and Stulz (1987)], to propose time series models [Ahking and Miller (1987), Baillie and Bollerslev (1989), and Havenner and Modjtahedi (1988)], to reject previously proposed models [Meese and Rogoff (1983)], and even to examine the underlying statistical properties of the exchange rates that all the others had been modeling [Boothe and Glassman (1987a) and (1987b)]. This paper will take a somewhat different approach by attempting to provide a fair testing ground for several of the previously proposed models along with one new model, a state space model based on results from linear systems theory. An evaluation of the models will be made with a set of nonnested hypothesis tests. While articles have described the empirical failure of some exchange rate models advanced in out-of-sample forecasting [cf. Meese and Rogoff (1983)], no direct tests of competing paradigms have been done giving equal treatment to all competing hypotheses while including both in-sample and out of-sample tests.
Jeffrey H. Dorfman

10. Application of State-Space Models to Ocean Climate Variability in the Northeast Pacific Ocean

State-space statistical models are used to decompose time series of oceanic parameters from the coastal Northeast Pacific Ocean into a non-parametric trend term, a nonstationary seasonal component, and a stationary autoregressive component. Climate scale variations in upwelling, sea surface temperature, and wind stress are examined, and regional differences in the component series on decadal scales are explored.
Roy Mendelssohn, Franklin B. Schwing

Applications of Neural Networks


11. On the Equivalence Between ARMA Models and Simple Recurrent Neural Networks

This paper presents analytical results for a class of linear discrete time recurrent neural networks. The networks are shown to be able to act as autoregressive moving average models. Minimal network sizes for representing ARMA(p,q) models are derived, and analogies between recurrent networks and state space models are pointed out.
Henrik Saxén

12. Forecasting Stock Market Indices with Recurrent Neural Networks

A recurrent neural network is used to forecast the out-of-sample return of a stock market index. The use of an extensive information set and a stochastic minimization algorithm distinguishes this study from prior work. The data set encompasses daily observations from 1970 through 1993, with the following forecast exercise undertaken. For a variety of model sizes, the network task is to approximate the weekly, monthly or quarterly conditional mean return. These forecasts are conditioned on a daily information set containing a number of index-specific and market-wide variables, term structure and corporate bond yields, and calendar variables. Network performance is evaluated by out-of-sample normalized mean-squared error, sample statistics describing the joint distribution of forecasted and actual returns, and a test for market-timing ability. A further performance evaluation concerns the construction of trading portfolios with transaction costs. Finally, bootstrapping techniques are applied to construct surrogate distributions of the out-of-sample statistics. Neural network models are found to perform more than adequately when compared with a benchmark linear model, and are able to generate large risk-adjusted returns over simple buy-and-hold strategies.
Maxwell J. Rhee


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