Applications of state space models in finance

132 7 A Kalman filter based conditional multifactor pricing model
with mean
and standard error
ˆσ 2 λk =
ˆλk = 1
T
1
T (T − 1)
T�
t=1
ˆλk,t, (7.13)
T�
( ˆ λk,t − ˆ λk) 2 , (7.14)
where w( ˆ λk) is Student-t distributed with T − 1 degrees of freedom (cf. Campbell et al.
1997, ¢ 5.8).
7.2.2.2 Econometric issues
Several problems related to the methodology may arise. The conditional betas in (7.11)
are unknown and have to be estimated in a first step based on a time series regression.
This induces an errors-in-variables problem. Secondly, even if the true betas were known,
the errors of the proposed cross-sectional regression would be likely to be heteroskedastic
and correlated, because the stock returns that are used as dependent variables are usually
correlated for a given date t. As a consequence, the estimated standard errors might be
unreliable (cf. Ferson and Harvey 1991).
The empirical evidence presented by Ferson and Harvey (1999) suggests that the
main results are robust to errors-in-variables. Nevertheless, different ways to explicitly
cope with the issue have been proposed. Shanken (1992) presents a general procedure
that corrects for errors in the estimates of conditional betas by using adjusted standard
errors. Alternatively, better estimates of the conditional betas can be produced by using
returns of portfolios rather than of single stocks (cf. Chen et al. 1986). As this thesis
exclusively deals with industry portfolios, the impact of a potential errors-in-variables
problem can be assumed to be of limited nature. The second problem of heteroskedastic
and correlated errors is especially alarming in small samples where the t-ratios might
be biased. One way to address this is to employ an efficient generalized least squares
estimator where the T estimates are weighted according their reciprocal variances; see
Ferson and Harvey (1999) for details. In the following, the analysis will be conducted
based on ten years of weekly data. The sample is sufficiently large to assume that the
corresponding t-ratios will be unbiased. Therefore, throughout this chapter, the widely
used unadjusted standard errors of the Fama-MacBeth approach will be relied upon.
7.3 The risk factors
The weekly excess returns on the eighteen DJ Stoxx sector indices introduced in ¢ 2.1
constitute the set of dependent variables to be studied. This section describes the risk
factors to be used in the context of the proposed conditional multifactor model.
According to the literature review provided in ¢ 7.1, various fundamental factors, in
particular size and valuation, as well as different proxies for macroeconomic risks, which
can be thought of having an impact on equity returns, have been previously analyzed.
As this thesis focuses on weekly data, the spectrum of available macroeconomic series is
limited. Many factors that are typically employed in empirical research, such as growth
t=1