6.2 Modeling conditional betas
6.2.3 Kalman filter based approaches
6.2.3.4 The generalized random walk model
The Kalman filter models considered so far are based on the assumption of normally distributed errors. As discussed earlier, the assumption of normality is incompatible with the sector series at hand. When the Kalman filter is employed in case of non-normal errors, the Kalman filter still leads to consistent QML estimators as outlined in 3.3.7. However, the obtained estimators are not efficient. As discussed in Chapter 5, the two most important sources of non-Gaussianity are volatility clustering and outliers. As neither of the Kalman filter based approaches above is capable of fully coping with these issues, it might be sensible to remove these influences altogether. For this purpose, a three-stage estimation procedure following Ghysels et al. (1996) is developed:
1. Correct the observations for heteroskedasticity.
2. Cut down the remaining outliers.
3. Apply the Kalman filter based random walk model to the transformed obser-vations.
In order to account for ARCH effects, less weight should be placed on data points with high conditional volatility. In the OLS context, this approach is referred to as generalized least squares (GLS). Following the outline by Davidson and MacKinnon (2004, 7), who give a general introduction to GLS and related topics, the concept of GLS can be summarized as follows: in a standard linear regression model of the form
y=Xβ+, ∼IID(0,Ω), (6.23)
the parameter vector β can only be estimated efficiently by least squares if the residuals are uncorrelated and homoskedastic. Whenever these assumptions are violated, an efficient GLS estimator ofβcan be found by appropriately transforming the regression so that the Gauss-Markov conditions are satisfied. The corresponding transformation depends on Ψ, a quadratic matrix that is usually triangular with
Ω−1 =ΨΨ0. (6.24)
Premultiplication of (6.23) by Ψ0 yields the transformed regression model that can be estimated by OLS to obtain efficient estimates:
Ψ0y =Ψ0Xβ+Ψ0. (6.25)
The GLS estimator forβ is given as
βˆGLS = (X0ΨΨ0X)−1X0ΨΨ0y= (X0Ω−1X)−1X0Ω−1y. (6.26) In case of heteroskedastic but uncorrelated errors, the covariance matrix is diagonal and a GLS estimator can be obtained by means of weighted least squares (WLS).
Each observation is weighted proportionally to the inverse of the nonconstant diag-onal elements of Ω. With wt2 denoting the t-th element of Ω and w−1t denoting the t-th element of Ψ, for a typical observation at time t, the transformed regression model in (6.25) can be written as
w−1t yt =w−1t Xtβ+wt−1t. (6.27) The dependent and independent variables are simply multiplied by wt−1, where the weight observations depend negatively on the variance of the disturbance term. It can be shown that the variance of the disturbances is equal to unity.
As the precise form of the covariance matrix is usually unknown in practice, a consistent estimate of Ω can be employed to get feasible GLS estimators. A common way to correct for heteroskedasticity is to basewt on a filtered or smoothed estimator of the conditional volatility, calculated by one of the methods discussed in Chapter 5. Even though this does not necessarily give the most efficient estimator
in small samples, the feasible GLS estimator is usually more efficient than OLS in the presence of heteroskedasticity (cf. Ghysels et al., 1996).
The concept of WLS can be easily adapted to the random walk model discussed above. For the original state space model of the form
Ri,t =βi,tR0,t+i,t, (6.28)
βi,t+1 =βi,t+ηi,t, (6.29)
the weighted transformation is
Ri,t/wt =βi,tW LS(R0,t/wt) +i,t/wt, (6.30)
βi,t+1W LS =βi,tW LS+ηi,tW LS, (6.31)
with regressand Ri,t/wt and regressor R0,t/wt. Summary statistics are reported in terms of the transformed regressand to ensure orthogonality of the corresponding residuals.
In order to estimate the transformed model in (6.30) and (6.31) consistent es-timates of the weighting factor are needed. In the following, wt is set equal to the filtered conditional standard deviation estimated from an auxiliary heteroskedastic regression model. The behavior of hi,t is modeled by a t-GARCH(1,1) model:
Ri,t =βi∗R0,t+∗i,t, (6.32) ∗i,t =zi,tp
hi,t, (6.33)
hi,t =ωi+γi∗2i,t−1 +δihi,t−1, (6.34)
where zi,t is assumed to be t-distributed. The weighting factor ˆwi,t can now be computed in annualized percentages as
ˆ
wi,t = 100∗p
Ahi,t, (6.35)
where the annualizing factor A = 52 is equal to the number of weekly returns per year (cf. Alexander, 2001, 1.1). For allt= 1, . . . , T, ˆwt depends on the information set Ωt−1, which only contains relevant information through date t−1. Therefore, the transformation based on filtered volatility estimated over the full sample is valid both for in-sample and out-of-sample purposes.
The second issue addressed above is related to outliers. For many estimation methods outliers can be captured straightforwardly by employing a t-distribution for the disturbance terms. The introduction oft-distributed errors to a Kalman filter model is not trivial as the resulting likelihood cannot be evaluated analytically. One way of handling non-Gaussian state space models is to employ importance sampling techniques as introduced in the context of SV models in 5.2.3; see Durbin and Koopman (2000, 11) for details. However, in order to limit the level of complexity, in the following outliers should be treated without relying on simulation-based tech-niques. A common way to remove the influence of outliers is to truncate the variables
that enter the model. According to Granger et al. (2000) outlying observations can be identified as those data points that are at least four standard deviations away from the mean of regressand and regressors, respectively. They propose to employ outlier-reduced series where any data point above or below ˆµ±4ˆσ is replaced by
ˆ
µ±4ˆσ; ˆµ and ˆσ are estimated from the original data. In this thesis, more rigid limit lines at three standard deviations are imposed. Any outlying observation in (6.30) is simply set equal to the mean ofyt/wt orxt/wt plus — or minus for negative outliers — three standard deviations. Obviously, unless outliers and the standard deviations are determined simultaneously, outliers will distort the standard devia-tions to be estimated. This leads to inflated sigmas and possibly masks influential observations. However, the objective of the “three sigma”-rule is not to fundamen-tally remove all outliers, but to approximately remove the biggest distractive effects from the data. Therefore, the chosen procedure can be considered being appropriate in the following.
The resulting model will be referred to as the generalized random walk (GRW) model. The estimates of conditional beta series are denoted as ˆβi,tGRW. Of course, other methods for dealing with non-normality are available. For example, in order to take conditional heteroskedasticity explicitly into account, Harvey et al. (1992) proposed a modified Kalman filter estimated by QML, and Kim (1993) developed a state space models with Markov switching heteroskedasticity. An alternative ap-proach to deal with outliers is discussed by Judge et al. (1985, 20) who, instead of truncating the independent and dependent variables directly, truncate the residuals from a robust regression. While those techniques may be more sophisticated, in the following the methodology should be kept as simple and as relevant for prac-tical implementation purposes as possible. Hence, the preference will be on the methodology described above. Even though this approach is simple, its relevance can be tested in a straightforward fashion: if the forecasting accuracy is superior in comparison to the standard RW model, then the proposed modifications can be
−20
−10 0 10 20
(a) Automobiles returns
Rt (%)
1990 1995 2000 2005 −15
−10
−5 0 5 10 15
(b) DJ Stoxx broad returns
Rt (%)
1990 1995 2000 2005
Figure 6.2: Weekly excess log-return series of (a) Automobiles and (b) the broad market.
regarded as being justified. Note that all reported diagnostics refer to the trimmed generalized input variables, while the error measures to be used in the next section to evaluate the forecast performances will be based on the original return series.
This allows for a fair comparison of the different Kalman filter based models.
The described procedure to deal with volatility clusters and outliers shall be illustrated at the example of the Automobiles sector. Figure 6.2 shows the weekly excess log-returns on the sector and the DJ Stoxx
Broad index. It is obvious that the volatility of both series is not constant over time. It can be seen that some peri-ods are characterized by small absolute returns and others by large absolute returns.
Especially in the second half of the period, which is effected by the Asian crisis, the Russian crises and the boom and bust of the new economy, some outlying absolute returns exceed the value of 10%. As outlined above, the influence of heteroskedas-ticity can be removed by means of weighted least squares. The weighting factor wt
can be derived by an auxiliary heteroskedastic regression model. Figure 6.3 displays the auxiliary residuals in Panel (a); Panel (b) shows the weighting factor computed as the annualized conditional t-GARCH volatility estimate.
−10
−5 0 5 10
(a) Auxiliary heteroskedastic residuals
εt*
1990 1995 2000 2005 5
10 15 20 25 30
(b) Annualized filtered conditional volatility
w^t
1990 1995 2000 2005
Figure 6.3: (a) Residuals from the auxiliary heteroskedastic regression model and (b) GLS weighting factor for Automobiles and the overall market.
The estimated conditional volatility confirms the impression of volatility clustering.
It is used to transform the return series according to (6.30) to yield the weighted return series, which is plotted in Figure 6.4. As expected, the WLS transformation removed the heteroskedasticity in the series. What remains are a few outliers defined as those observations outside ˆµ±3ˆσ that can now be capped (floored) according to the “three sigma”-rule. The trimmed generalized series can now be utilized as dependent and independent variables to estimate time-varying GRW beta series.
Figure 6.5 illustrates the difference between the RW and GRW beta series for the Automobiles sector: the proposed procedure to deal with heteroskedasticity and outliers leads to a smoother conditional beta series whose major pattern remains intact.
−1.0
−0.5 0.0 0.5 1.0
(a) GLS returns − Automobiles
Rt (%)
1990 1995 2000 2005
µ+3σ
µ−3σ
−1.0
−0.5 0.0 0.5 1.0
(b) GLS returns − DJ Stoxx broad
Rt (%)
1990 1995 2000 2005
µ+3σ
µ−3σ
Figure 6.4: Weighted weekly excess log-return series of (a) Automobiles and (b) the broad market.
The estimation results for all GRW models are summarized in Table 6.6. The es-timated variance of observation and state disturbances are significant at the 1%
level for all sectors. Although the null of normality can be rejected without excep-tion, the reportedJB-statistics are all significantly lower than for the Kalman filter based models considered above. The null of no autocorrelation can be rejected at the 5% level for eleven sectors. According to the reported LM-tests, the weighted transformation removed the volatility clusters for all sectors except for Personal &
Household Goods. The reported values for R2 and BIC should be interpreted with caution as they only refer to the second step of the proposed GRW approach. This is the major reason for not relying on fit statistics to compare and evaluate the in-sample performances of the various modeling techniques below.
0.0 0.5 1.0 1.5 2.0
conditional betas
1990 1995 2000 2005
RW GRW
Figure 6.5: Conditional random walk and generalized random walk beta estimates for the Automobiles sector.