The last column of Table 2.2 reports the unconditional correlation between the chosen market proxy and sector i for i = 1, . . . ,18, denoted as ρ0i. Correlations between an asset and the overall market play an important role in financial markets.
They are widely used in portfolio and risk management applications. For the sample under consideration, all estimated values of ρ0i are higher than 0.65. This indicates a strong linear association between the respective sectors and the market index over the entire sample. However, given the empirical properties described above, especially with regard to the observed time-variation of volatilities and covariances, it is reasonable to assume that the true degree of correlation is not constant but changing over time.
This is illustrated by considering the following regression relationship between sector i and the market proxy:
Ri,t =αi+βiR0,t+i,t, t= 1, . . . , T. (2.3)
The ordinary least squares (OLS) estimator of the slope coefficient in this regression represents the usual estimator of an asset’s beta in the context of the CAPM, which will be dealt with in more detail in the empirical part of this thesis. Under the assumption that the true value of beta is constant, it is defined as
βi = Cov(R0, Ri)
V ar(R0) , (2.4)
whereCov(R0, Ri) is the unconditional covariance of the return of the market proxy with the return of sector i, and V ar(R0) is the unconditional variance of market returns. Taking the observed stylized facts of volatility clustering and volatility co-movements into account, the numerator and denominator of (2.4) should be replaced by the conditional covariance and conditional variance, respectively, to allow the true beta to be time-varying.
Based on the cumulative sum of squares (CUSUMSQ) of recursive residuals, Figure 2.4 demonstrates the instability of beta at the example of the two sectors Insurance and Food & Beverages. The CUSUMSQ test, as proposed by Brown et al.
(1975), detects instability in the regression coefficients caused by time-varying sec-ond moments if the cumulative sum of squares moves outside two critical lines. For both sectors, the signal line moves outside the lower critical bound. This indicates instability in the relationship between the respective sector and the overall market during the sample period. Similar observations leading to the same conclusion have been made for all sectors (not reported here). The test statistics and the significance lines have been computed using the object-oriented matrix programming language Ox 3.30 by Doornik (2001) together with the package SsfPack 2.3 by Koopman et al. (1999).
1990 1995 2000 2005 −0.2
0.0
Figure 2.4: CUSUMSQ tests with 5% confidence intervals for the excess return series of (a) Insurance and (b) Food & Beverages.
The findings above contradict the assumption of beta constancy and further mo-tivate the scope of this thesis to deal with time-varying sensitivities in financial markets. Given previous evidence in the literature and economic arguments, which
contradict the assumption of beta constancy, the focus throughout this thesis will be on the modeling of change, and not on testing the paradigm of beta stability.
For an overview of alternative parameter stability tests in the context of beta, see, for example, Wells (1996, 2).
With regard to the modeling of conditional relationships in financial markets, the literature distinguishes between two different approaches: time-varying sensitiv-ities can be modeled as linear functions of observable state variables as proposed, for example, by Shanken (1990). However, as it is not clear which instrumental vari-ables should be included, any choice may exclude relevant conditioning information.
An alternative procedure is to rely on advanced time series techniques to model time-varying betas as stochastic, possibly hidden processes. As demonstrated by Leusner et al. (1996) a stochastic process may capture the impact of the complete set of potential determinants of systematic risk, thus avoiding the omitted variables problem. Following this route, throughout this thesis conditionality is dealt with by employing time series models. As it is typically not possible to model all distribu-tional and temporal properties of return series simultaneously, different models are typically used to capture different empirical regularities (cf. Ghysels et al., 1996).
Time-varying relationships can be constructed either directly orindirectly. The direct approach will be implemented by employing a state space framework, where beta can be allowed to emerge either as a continuous process estimated via the Kalman filter, or as adiscrete process in a Markov regime switching framework. Al-ternatively, indirect estimates of conditional sensitivities can be derived by capturing the underlying conditional variance and covariance components by a conditional het-eroskedasticity model. Before applying these different concepts to analyze the time-varying relationship between macroeconomics, fundamentals and pan-European in-dustry portfolios, the theoretical groundwork is made available in the subsequent theoretical part of this thesis.
Linear Gaussian state space models and the Kalman filter
This chapter introduces the class of linear Gaussian state space models from the classical perspective of maximum likelihood estimation. The first section outlines the basic ideas behind state space modeling. Section 3.2 presents the general state space form of a dynamic system. Section 3.3 develops the Kalman filter and smoother, the basic tools to estimate models in state space form. Section 3.4 describes maximum likelihood estimation procedures for the unknown parameters. Section 3.5 extends the general state space model to allow for the incorporation of explanatory variables with constant as well as time-varying parameters. Model diagnostics and measures to assess the goodness of fit are described in Section 3.6. The chapter finishes with an illustration of how to specify state space models using the software package SsfPack.
3.1 Basic ideas of state space modeling
The state space form of a dynamic system with unobserved components is a very powerful and flexible instrument. A wide range of all linear and many nonlinear time series models can be handled, including regression models with changing coef-ficients, autoregressive integrated moving average (ARIMA) models and unobserved component models. A state space model consists of a state equation and an obser-vation equation. While the state equation formulates the dynamics of the state variables, the observation equation relates the observed variables to the unobserved state vector. The state vector can contain trend, seasonal, cycle and regression components plus an error term. Models that relate the observations over time to different components, which are usually modeled as individual random walks, are referred to as structural time series models. The stochastic behavior of the state variable, its relationship to the data and the covariance structure of the errors de-pend on parameters that are also generally unknown. The state variable and the parameters have to be estimated from the data. Maximum likelihood estimates of
the parameters can be obtained by applying theKalman filter. Named after Kalman (1960, 1963) the Kalman filter is a recursive algorithm that computes estimates for the unobserved components at time t, based on the available information at the same date.
The Kalman filter has originally been applied by engineers and physicists to estimate the state of a noisy system. The classic Kalman filter application is the example of tracking an orbiting satellites whose exact position and speed, which are not directly measurable at any point of time, can be estimated using available data and well established physical laws. A discussion of engineering-type applications of the Kalman filter is provided by Anderson and Moore (1979). In economics and finance, we are regularly confronted with similar situations: either the exact value of the variable of interest is unobservable or the possibly time-varying relationship between two variables is unknown. Nevertheless, the propagation of the Kalman filter among econometricians and applied economists only really began with the introductory works of Harvey (1981) and Meinhold and Singpurwella (1983).
In contrast to the Box-Jenkins methodology, which still plays an important role in teaching and practicing time series analysis, the state space approach allows for a structural analysis of univariate as well as multivariate problems. The different components of a series, such as trend and seasonal terms, and the effects of ex-planatory variables are modeled explicitly. They do not have to be removed prior to the main analysis as is the case in the Box-Jenkins framework. Besides, state space models do not have to be assumed to be homogeneous, which results in a high degree of flexibility. This allows for time-varying regression coefficients, missing ob-servations and calendar adjustments. Transparency is another important feature of structural models as they allow for a visual examination of the single components to check for derivations from expectations; see Durbin and Koopman (2001, 3.5) for a comparison of the state space framework and the Box-Jenkins approach.
Early applications of state space models and the Kalman filter to economics include Fama and Gibbons (1982) who model the unobserved ex-ante real inter-est rate as a state variable that follows an AR(1) process. Clark (1987) uses an unobserved-components model to decompose quarterly real GNP data into the two independent components of a stochastic trend component and a cyclical component.
Another important contribution is the work of Stock and Watson (1991) who define an unobserved variable, which represents the state of the business cycle, to measure the common element of co-movements in various macroeconomic variables. Surveys on the applicability of the state space approach to economics and finance can be found in Hamilton (1994a) and Kim and Nelson (1999).
The state space approach offers attractive features with respect to their general-ity, flexibility and transparency. The lack of publicly available software to estimate these models has been the main reason why only relatively few economic and finance related problems have been analyzed in state space form so far. The subsequent sec-tions aim at providing a presentation of the Gaussian state space model that is as
compact and intuitive as possible, while being as comprehensive as necessary to render the employment of this versatile framework by applied researchers possible.
More detailed treatments of state space models are given by Harvey (1989), Harvey and Shephard (1993) and Hamilton (1994a), among others. An outline with a focus on applications can be found at Kim and Nelson (1999). If not indicated otherwise, Durbin and Koopman (2001, 4–7) serve as standard reference for this chapter.