2.1. Discrete-Time Models

Chapter 1 lays some of the groundwork for the book. It characterizes the financial assets that are studied in subsequent chapters by their monetary cash flows and illustrates the cash flows through bar charts. Various approaches to financial theory are reviewed such as the actuarial approach, the absence of arbitrage hypothesis, and market prices as outcome of aggregate asset demand and supply. The introduction finishes with an overview of particular features of financial data sets. Price data may contain transactions prices or quotes; the frequency of financial data sets could range from less than a minute to annual; marginal distributions of financial variables such as asset returns and trading volumes are typically very different from conditional distributions. Because of the specificity of financial products and the complexity of strategies followed by investors, we can expect many relationships that we find in financial data to be nonlinear.

In Chapter 2, Gourieroux and Jasiak study the statistical properties of univariate linear time series models. For readers with a time series background, it is safe to skip this chapter. All other readers will find a very useful and accessible introduction to autoregressive (AR) models. The chapter starts with a detailed derivation of the basic properties of AR(1) processes. Gourieroux and Jasiak continue with a discussion of least squares (LS) and maximum likelihood (ML) estimation of the AR(1) model parameters. At this point some prior knowledge of econometrics comes in handy as the book does not elaborate on the general principles of LS and ML estimation. Unlike in most introductory treatments of the AR(1) process, Gourieroux and Jasiak devote special attention to the frequency at which the AR process is sampled. In particular, the authors show informally by increasing the sampling frequency of the AR process that it tends to a continuous time Ornstein–Uhlenbeck process in the limit, which will be studied in Chapter 11. A condensed discussion of unit roots follows. Gourieroux and Jasiak provide critical values for simple unit root tests without deriving the limit distribution of the test statistics. The remainder of the chapter generalizes the AR(1) model to a higher order autoregressive and moving-average process. References to “financial” econometrics are brief: the relationship between market efficiency and martingales is explored, and autoregressive moving average (ARMA) models are fitted to S&P 500 prices and returns.

Chapter 3 generalizes univariate ARMA models to multivariate vector processes. As before, non–time series econometricians will appreciate the self-contained analysis of the dynamic properties of vector autoregressive (VAR) processes. As an illustration, Gourieroux and Jasiak fit a bivariate VAR to data on S&P 500 returns and trading volumes. Somewhat surprisingly, we find in the VAR chapter a discussion of mean-variance efficient portfolios. What is the connection? Well, suppose you would like to manage your retirement portfolio. Fit a VAR model to historical asset returns, calculate the conditional mean and variance, and use the theory of mean-variance efficiency to compute the optimal portfolio allocation.

Dynamic simultaneous equations models are the subject of Chapter 4. As in most treatments of simultaneous equations models, a demand and supply system serves as example. In this case, of course, it is the demand and supply for financial assets. The authors review concepts such as endogenous versus exogenous variables, structural versus reduced form, and conditions under which coefficients of a dynamic structural model can be consistently estimated by ordinary LS. Instead of exploring estimation methods for simultaneous equations models such as multistage LS or full-information ML, Gourieroux and Jasiak proceed with the derivation of the CAPM as an example of a demand and supply system. In their discussion of tests for the implications of the CAPM, the authors focus on the careful specification of test statistics rather than on empirical results. The remainder of Chapter 4 is devoted to causality. Causality measures are introduced, and the reader is shown how to construct likelihood-ratio tests for the absence of causality. As an empirical illustration, various causality measures are computed for returns and trading volumes of the Alcatel stock.

The literature on integrated and cointegrated processes is immense. Chapter 5 provides a brief introduction to what has emerged in the past decade as one of the most prolific subjects in time series analysis. In addition to discussing integrated processes of order one, Gourieroux and Jasiak also review the basics of fractionally integrated processes. Long-range persistence appears to be a typical characteristic of power transformations of asset returns, which often serve as measures of volatility.

Time-varying volatilities cannot be captured in linear time series models. As the predictability of volatility is a prevalent characteristic of asset return data, most of the empirical literature uses time series models that allow for conditional heteroskedasticity. Unlike the models examined in Chapters 1–5, the models discussed in Chapter 6 are nonlinear. To be empirically successful, these nonlinear time series models have to capture several stylized facts: leptokurtosis, time-varying volatilities, possibly skewness, and asymmetric relationships between conditional mean and variance dynamics. Gourieroux and Jasiak start with a general specification of a nonlinear autoregressive model:

where μ(·) is the conditional mean function and σ²(·) the conditional variance function. In principle, the functions μ(·) and σ(·) can either be parameterized and estimated with a parametric technique, such as ML, or they can be estimated nonparametrically. To explain the nonparametric estimation, the authors insert a two-page introduction to kernel estimation. After the general discussion of nonlinear AR models, Gourieroux and Jasiak study the statistical properties of the most popular nonlinear AR model in financial econometrics, namely, the autoregressive conditionally heteroskedastic (ARCH) model and its offspring. As an alternative to the ARCH family, the authors consider the class of stochastic volatility models in which the conditional variance of the dependent variable evolves according to a latent stochastic process. The remainder of the chapter provides extensions to multivariate processes.

Whereas the previous chapters were organized around statistical models, Chapter 7 explores expectations and present-value models. The first section of the chapter reviews predictions and prediction errors, adaptive expectations, and rational expectations. These expectations play an important role in present-value models, which express the price of an asset as expected sum of discounted future dividends. Gourieroux and Jasiak survey several testable implications of the present-value model, for instance, tests of moment restrictions and tests of restrictions on VAR representations of prices and dividends.

In the present-value model, dividends are discounted at the constant risk free rate. However, in an intertemporal stochastic general equilibrium framework, agents typically adjust the fixed discount factor by the state-dependent marginal utility of consumption. Chapter 8 analyzes asset prices in a simple endowment economy and derives the stochastic discount factor that agents use in equilibrium to price assets. Asset returns R_t and the stochastic discount factor M_t have to satisfy the following moment condition:

, where θ is a parameter vector. The moment restriction can be exploited to estimate the parameter θ by the so-called generalized method of moments (GMM), and Gourieroux and Jasiak finish Chapter 8 with a concise six-page introduction to the theory of GMM. No empirical results are mentioned, and I have my doubts that someone without prior exposure to GMM would actually be able to apply the technique successfully based on the exposition in the book. Although the GMM estimation of discount factor models is very fragile in practice, the text does not mention any of the practical difficulties.

Dynamic factor models can be used to reduce the dimensionality of multivariate models with a large cross-sectional dimension. Chapter 9 provides a brief introduction to linear and nonlinear factor models. It explains how unobservable factors can be integrated out with filtering algorithms and discusses the connection between factor models and arbitrage pricing theory.

Chapter 10 covers dynamic qualitative processes. For instance, the movements of a price process could be classified by multiples of the tick size or simply by “up,” “stable,” and “down.” The chapter begins with an analysis of homogenous Markov chains. As an application, a Markov process is fitted to intraday price movements of the Elf-Aquitaine stock. Instead of characterizing dynamic qualitative processes through transition probabilities between states, they can alternatively be represented in terms of durations, that is, the time that passes between transitions. Moreover, qualitative processes can be used to model latent variables in factor models, as in qualitative threshold ARCH models and regime switching models. These topics are covered in the remainder of Chapter 10.

2.2. Continuous Time Models

Continuous time models play a much more important role in financial econometrics than they do in macroeconometrics. Hence, it is appropriate to devote more than 100 pages to this class of models. Continuous time models typically assume that asset prices satisfy a stochastic differential equation (SDE). Although these models are more difficult to estimate with discrete data, they provide a convenient framework for determining the price of derivative assets. Personally, I think that Chapters 11–13 are the strongest part of the book. Although this important subject is mathematically challenging, the authors crafted an accessible yet comprehensive exposition.

Chapter 11 begins with a review of SDEs. Although stochastic calculus at a rigorous level requires extensive knowledge of measure theoretic probability, Gourieroux and Jasiak's treatment avoids the use of measure theory. After introducing Brownian motions, stochastic integrals, and the Itô lemma, the authors derive geometric Brownian motions, the Ornstein–Uhlenbeck process, and the square root process as solutions to SDEs. For SDEs without closed form solutions various approximation methods are discussed. The remainder of Chapter 11 provides an introduction to derivative pricing under both complete and incomplete markets. The starting point is the explicit derivation of risk neutral probabilities in a binomial tree model. Gourieroux and Jasiak generalize the analysis to the continuous time framework and develop the Black–Scholes formula.

Chapter 12 is devoted to the estimation of diffusion models. The difficulty is that the variables of the continuous time models are only observed at discrete points in time. To obtain a likelihood function, the SDE for a dependent variable y_t has to be solved for the conditional distribution of y_t given y_t−1, y_t−2, …. This procedure is called discretization of a continuous time process. For the geometric Brownian motion, the Ornstein–Uhlenbeck process, and the square root process, exact discretizations are derived in the book. For more complicated processes the discretization can only be obtained with numerical techniques. As an alternative to likelihood-based inference, Gourieroux and Jasiak consider method of moments approaches. Moment conditions can be obtained with the infinitesimal generator, which represents the infinitesimal drift of a transformed process. An explicit example of a method of moments objective function is given. To circumvent identification problems that may arise in the context of moment-based estimation, Gourieroux and Jasiak venture into spectral decompositions of infinitesimal generators and discuss sieve and kernel-based estimators for the spectral decomposition. Further insights into the workings of these procedures are provided by a small Monte Carlo study and the estimation of drift and volatility function for the Alcatel stock based on high-frequency data. The remainder of the chapter surveys a variety of additional estimation procedures: methods based on scale and time deformations of a diffusion process, the method of simulated moments, and indirect inference.

In most cases in which data are available on derivative assets, it is also possible to obtain observations on the underlying asset. Suppose the goal is to estimate a diffusion model for the underlying asset. This can either be done directly based on the price series of the underlying asset or indirectly with derivative prices. In the latter case, the relationship between the parameters of the price process of the underlying asset and the derivative prices is given by the appropriate pricing formula. Moreover, the underlying SDE can in principle be estimated jointly based on asset and derivative prices. The econometrics of derivative assets are explored in Chapter 13. The leading example is the Black–Scholes model. Based on derivative prices, it is possible to recover the implied drift and volatility of the underlying geometric Brownian motion. Regardless of the validity of the Black–Scholes model for a particular option, it is possible to compute an implied volatility for the underlying asset that equates the actual option price with the prediction from the Black–Scholes formula. These implied volatilities can be interpreted as a transformation of the option prices. Gourieroux and Jasiak discuss various descriptive techniques to summarize features of empirical implied volatilities. As an alternative to Black–Scholes type formulas, which are based on absence of arbitrage arguments, the derivatives can be priced with stochastic discount factor models such as the ones discussed in Chapter 8. The joint estimation of a SDE with asset and derivative data is explored in the context of the Hull–White model. The remainder of the chapter covers stochastic risk neutral probability measures and Monte Carlo methods to evaluate expectations with respect to risk neutral probabilities.

2.3. Market Microstructure and Risk Management

Whereas the first two parts of the book mainly focus on the properties of econometric models and occasionally provide empirical illustrations, the last three chapters of the book emphasize institutional details and cover three topics in financial econometrics that are the subject of ongoing research.

Chapter 14 begins with an overview of market characteristics: quote-driven markets, order-driven markets, call auctions, continuous matching, limit orders, market orders, bids and asks, and intraday seasonality. Gourieroux and Jasiak continue with a fairly detailed discussion of market microstructure theory including various models of bid-ask spreads and timing of orders. If asset prices are sampled at very high frequencies, microstructure effects influence the price dynamics. For instance, as a result of a bid-ask bounce high-frequency returns may exhibit a significant negative first-order autocorrelation. Intraday data could be recorded in calendar time or transaction time. Gourieroux and Jasiak explore the effect of time deformation on the properties of a stochastic process. The remainder of the chapter studies intratrade durations including AR correlational duration models.

Indices are used to summarize the joint evolution of multiple assets, two well-known examples being the Dow Jones industrial average and the S&P 500. Because a large fraction of the empirical analysis is based on market indexes, Chapter 15 is a very useful addition to the book. It compares Paasche and Laspeyres indexes, reviews the uses of price indexes, and provides an overview over the major stock market indexes. Empirical tests of the CAPM often use broad market indexes as a proxy for the market portfolio. Gourieroux and Jasiak illustrate that the use of the proxy may produce misleading test results if errors-in-variables are not properly accounted for.

Portfolio managers have to be particularly concerned about the probability of large losses in the value of their assets. To some extent, this risk can be measured by the conditional variance of the portfolio. The variance provides a correct assessment of risk only under specific conditions, such as conditional normality of returns. An alternative measure of risk is the value at risk (VaR), which is the loss on a portfolio that will not be exceeded by the end of the period with a prespecified probability. Chapter 16 provides an introduction to heavy-tail distributions and extreme value theory and studies the VaR measure and its estimation. However, the VaR does not provide a coherent measure as it ignores the expected size of the loss if it occurs. Hence, Gourieroux and Jasiak discuss additional measures of risk that are considered coherent. Whereas mean-variance efficient portfolios were studied in Chapter 3, this last chapter also considers portfolios that maximize the expected return subject to a VaR constraint.