Much research on modelling the behaviour of stock return or volatility assumes that parameters of models are constant over time This assumption is violated in the real world as we have already known A conventional estimation method, the Kalman Filter technique, which permits variations in parameters and has many applications in economics and finance particularly in empirical work In fact, this method essentially is equivalent to Bayesian Forecasting From a paragragh quoted in West and Harrison’s
book, “Bayesian Forecasting and/dynamic Models” :
"At about the same time in 1969. it became clear that some of the mathematical models were similar to those used in engineering control It is now well-known that, in normal Dynamic Linear Models with known variances, Ihc recurrence relationships for sequential updating of posterior distributions arc essentially equivalent to the Kalman Filter equations, based on the early work of Kalman (I960), (1963) in engineering control, using a minimum variance approach It was clearly not. as many people appear to believe, that Bayesian Forecasting is Founded upon Kalman Filtering (sec Harrison
and Stevens, 1976a. and discussion, and reply to the discussion by Davis of West. Harrison and Migon. 19X5) To say that "Bayesian Forecasting is Kalman Filtering is akin to saying that statistical inference is regressions1" (1989. pi 5)
we know that Bayesian Forecasting is broader and more flexible than Kalman Filtering
In this thesis, one of the estimation methods, the Discount Weighted Estimation (DWE) method based on Bayesian adjustment, has been chosen to estimate and to smooth changing parameters (Simple examples of applying the Discount Weighted Estimation Filtering method can be found in Harrison and Johnston (1984), Ameen and Harrison (1983), (1984), or West and Harrison (1989)) Ameen and Harrison (1984) based Discount Weighted Estimation upon the discount concept that the information content of an observation decays with its age The DWE allows different model components to have different discount weights as there might be numerous characteristics in a system Explicitly, it generalises the Exponentially Weighted Regression (EWR) which simply depends upon one discount factor The DWE technique enables us to estimate and forecast random processes which evolve linearly and are observed subject to noise The method produces recurrence relationships for the sequential updating of the regression parameters and of the variance, and is used not only to forecast but also to smooth the time series of parameters The advantages of using this method are that (1) distinct model components (if there are k independent variables, there will be k discount weights for k parameters) are allowed to have different associated discount weights (for predicting future data, the most recent data are thought more important than the older data), (2) the error term does not need to be homoskedastic, (3) the availability of a large quantity of historical data
is not required, and (4) inverting a correlation matrix is not involved (problems can arise when the independent variables are correlated) Moreover, in empirical work,
practitioners have difficulty in specifying a system variance matrix W But the use of
discount factors, S , in DWE overcomes the major disadvantages of having to give the
values of W , since a discount factor converts its component posterior precision C, ,
at time l - 1 to a prior precision Rt = — , for time / (see Appendix A)
S
The detailed derivations of the Discount Weighted Estimation (DWE) method appear in Appendix A DWE can be applied in two ways: (i) filtering, and (ii) combined filtering and smoothing I first discuss filtering and then combined filtering and smoothing
Assuming that there are k parameters in a Dynamic Linear model that need to
be estimated, and given the information available at time t-l, I), ,, the kxl vector of
filtered beta prior estimates is exactly the same as the kxl vector of posterior betas
/•;[/?,\
d, ,] = h\p, , 11) , , ] = t\p, , ID, 3 ] + a, , (y,, - f; , /•:[/?,, |
d, 2 ])
= l\P i i|M 2]+
where I), , represents the information available at time t-l It includes both the
previous information set /), , and the observation Y, , ,
e, , is the scalar one-step ahead forecast error It is the difference between the
observed value of Y, , and the expected value of Y, ,,
coefficients ( /•.'[/?, , 11), 2 ] ) upon Ï, ,, and 0 < a, , < 1, at , is any one of
the elements in vector A, ,,
Yt , is the dependent scalar variable at time t-1,
/•] is the kxl vector of independent variables at time t-1.
and A'[/?, ,|/>, 2] is the kxl vector of regression parameters at time t-1 given the information available at time t- 1
If the weight. A, ,, is close to zero then A'fP, ,|D, ,] * E|/J , ,|D, 21 and none
of the changes in the time series will be captured by the predictor of Y, ,. That is to
the latest values of the observation series Yt , 7
Another alternative which takes the information available after time t into account is called the smoothed time series For ,v> l, the .v-step smoothed beta estimates is obtained from
where A is the kxk matrix of discount factors (S k ) for the variances of k beta
estimates,
AA is the product of the kxk matrix of A ,
7 Simple examples of applying the Discount Weighted Estimation Filtering method can be referred to Harrison and Johnson (1984). or West and Harrison (1989).
say, the larger the value of A, ,, the more sensitive is the predictor /■] , /',Jy3, ,|A 2] ‘o
/?, | l),.t is the kxl vector of beta estimates at time t given the information available for s time points after time t
Given the information available for s-1 time points after time t+1, this smoothing procedure consists of taking a weighted average of the posterior beta
estimates at time t, p,\l), , and the smoothed beta estimates at time t+1, \/),.s . It
depends on the discount factors which control the magnitude of variance of beta estimates That is, the larger the discount factors, the smaller the magnitude of the changes in the variance Therefore, the smoothing procedure puts more weights on future beta estimates On the other hand, the smaller the discount factors, the larger the magnitude o f the changes in the variance Therefore, the smoothing procedure puts more weight on current beta estimates