Stefan Grzesiak

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 6 – Nicolaus Copernicus University – Toruń – 2004 
  
 
 
 
 
Stefan Grzesiak 
Szczecin University 
 
 
 
Kalman Filters and Specification Errors  
of Hyper-Structure  
 
 
  The article refers to the use of Kalman filters in analyses of econometric 
models with time-varying parameters. These models were considered earlier in 
Grzesiak (1995), (1997), (1999) and also in other papers by the same author. In 
the above-mentioned articles it was stated that a hyper-structure of the linear 
stochastic and dynamic system presented below: 
 
   (1)  , ,..., 2 , 1      , n t x y t t
T
t t = + = ε α
  , ,..., 2 , 1      , 1 n t H Gu F t t t t = + + = − η α α  (2) 
 
consisting of transition matrix F, reaction matrix G, innovation  matrix H as 
well as covariance matrix Q for ηt and variance σ
2 is known
1. 
  However in the empirical analysis of time-varying structural parameters, the 
hyper-structure is not known and has to be determined from the available data. 
Such a situation is defined as the identification of a parameter process. It can be 
formally divided into three stages: first the type of parameter process has to be 
defined – whether it is stationary or non-stationary. Then the structure of the 
parameter process has to be determined. Initially we do not know, for example, 
what order of auto-regression fits in with the particular components of the pa-
rameters vector, which means that both the dimension and structure of transition 
matrix F are not known. And finally the unknown parameters of matrixes F, G, 
H and Q as well as variance σ
2 have to be estimated. 
  Significant is the fact that, in the identification of a parameter process, the 
order of the particular steps or stages is unimportant. We assume that to deter-
mine the type and structure of a parameter process, alternative models of pa-
rameters have to be estimated, which makes the empirical analysis very impor-
                                                      
1 Definition of matrix Q was presented in Grzesiak (1995), p. 53. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Stefan Grzesiak  76
tant. Therefore, we will now assess the unknown parameters of the hyper-
structure, assuming that the structure of the parameter process is known, e.g. we 
will consider the third step of the identification problem first. 
  In our opinion, econometric literature referring to this subject is hardly con-
vincing. In empirical analyses, the authors used and recommended the “random 
walk” model (see: for example Athans (1974), Brännäs (1981), Haas (1983), 
Otter (1978) and Szeto (1973)): 
 
  , ,..., 2 , 1      , 1 n t t t t = + = − η β β  (3) 
  . t t β α =  (4) 
 
  It means that empirical research was limited to a non-stationary parameter 
model with F = H = I and G = 0. In such a situation only covariance matrix Q 
and variance σ
2 must be defined. Additionally, it is assumed that matrix Q is 
diagonal. It is a very restrictive assumption because, in the case of “random 
walk” models, it means that there is no correlation between variances of the 
particular components of the parameters vector βt. 
  However, the same causes of changes of parameters exist in many economic 
relations. This has led to the realisation – when doing research on consumption 
– that changes of consumption behaviour are matched with opposing changes in 
attitudes towards saving. Yet it cannot be properly reflected in a consumption 
function described by the “random walk” parameter model with the diagonal 
matrix Q. 
  In the widely discussed approach to the determination of diagonal covari-
ance matrix Q, the main focus is on heuristic presentation, such as ad hoc speci-
fications, or on a presentation based on a sensitivity analysis
2. The fact that  
a sensitivity analysis can require a relatively large expenditure of computational 
time is not taken into consideration. It particularly refers to the situation when 
errors covariance matrix Σ0|0 of the initial value and variance σ
2 are determined 
by means of a sensitivity analysis. 
  Such a “sampling approach” has found wide application. Yet it can only be 
justified if the Kalman filter turns out to be very resistant to the specification 
errors of the hyper-structure. What are the results of a resistance analysis, both 
theoretical in character and based on simulation studies? 
  The most important results in the theoretical analysis of errors can be found 
in the work by A. Jazwinski
3, who presented how to describe the influence of 
specification errors  , Q and σ 0 | 0 ∑
2 on errors covariance matrixes  1 | − ∑ t t  and  .  t t| ∑
 
 
Theorem 1. 
                                                      
2 See: Alhans (1974), Haas (1983) Otter (1978) for example. 
3 See: Jazwinski (1970), p. 244. 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Kalman Filters and Specification Errors of Hyper-Structure  77
 Let  Qc and   be specified for the covariance matrix Q and variance  . 
Let   be the specification of the error covariance matrix 
2
c σ
2 σ
C
0 | 0 ∑ 0 | 0 ∑ . Let  1 | − ∑ t t  and 
 be error covariance matrixes calculated on the basis of real values of Q, 
 and 
t t| ∑
2 σ 0 | 0 ∑ . Because Q ≤ Qc and  0 | 0 ∑  ≤   occurs, therefore for errors co-
variance matrixes 
C
0 | 0 ∑
C
t t 1 | − ∑  and   calculated together with Q
C
t t| ∑ c,   and  : 
2
c σ
C
0 | 0 ∑
  
   (5)  , 1 | 1 | − − ∑ ≥ ∑ t t
C
t t
   (6)  . | | t t
C
t t ∑ ≥ ∑
 
On the strength of the above-obtained results, the following practical approach 
is recommended: the unknown covariance matrixes Q and  0 | 0 ∑ , as well as vari-
ance   are fixed by “conservative”, i.e. sufficiently large specifications of Q
2 σ c, 
 and  . The Kalman filter provides us with suboptimal estimations of   
and   for a linear stochastic, dynamic system. 
C
0 | 0 ∑
2
c σ
C
t t| α
C
t t| ∑
  Depending on the size of the errors covariance matrix it can eventually be 
decided whether specifications of Qc,   and   provide us with “proper” 
estimations of   or whether Q
C
0 | 0 ∑
2
c σ
C
t t| α c,   and   can be replaced by less “con-
servative” specifications. These recommendations incline us to perform a sensi-
tivity analysis when fixing the elements of a hyper-structure. 
C
0 | 0 ∑
2
c σ
  The following question arises: Is this method of approach also suitable for 
econometric applications of the Kalman filters? 
  More explanations for the influence of hyper-structure specification errors 
on the results of the Kalman filter algorithm are provided by simulation studies. 
The results of two of such studies will be presented below. In the first case, the 
research was carried out by A. McWhorter
4 for the Markov parameter model 
where: 
 
  t t t A η β β + = −1  (7) 
 
and the following results were obtained: 
  Estimates of the Kalman filter yt|t of the dependent variable yt appear to be 
very resistant to errors of specification of Q,  , F and initial value a
2 σ 0. How-
ever, when analysing the resistance of estimates of βt|t, it appears to be defi-
nitely resistant to specification errors of the hyper-structure and initial value, 
where specification errors of transition matrix F and initial value a0 have par-
                                                      
4 Se McWhorter, Spivey, Wrobleski (1976), p. 280. 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Stefan Grzesiak  78
ticularly significant consequences. It motivates McWhorter to the following 
recommendations: 
  Because during the empirical analysis of time-varying parameters there is 
no possibility of checking the correctness of the estimation of βt|t – the real 
courses of βt are not known in any simulation study – exercising caution with 
the economic interpretation of the results of estimating βt|t is highly recom-
mended. In other words, according to McWhorter, the result confirms the use-
fulness of the Kalman filter in the forecasting of dependent variables yt, also, 
when ad hoc specifications of unknown models’ parameters were performed. 
  However, there are significant doubts because only the resistance of the 
filtration of yt|t was analysed. The Kalman filter appears to be very stable be-
cause during the “actualisation” of forecasted values generally great errors of 
forecasts (yt – yt|t-1) are also well smoothed. For the estimation of the prognostic 
ability of the filter, only the forecasted estimated value of yt|t-1 is useful, which 
was not considered in the above-mentioned research. 
  The fact that mere goodness of fit of the filtration of yt|t in the time interval 
ex post is not sufficient for the estimation of prognostic ability, is mentioned in 
an other work by McWhorter, Harasimham and Simonds
5. They estimated the 
well-known model I from Klein (1950) again on the quarterly data from the first 
quarter of 1950 until the fourth quarter of 1974, both by the OLS and the 3SLS 
methods. Besides, time-varying parameters were estimated by means of the 
Kalman filters, while parameters’ dynamics was described by means of the 
“random walk” process. 
  The structure of the covariance matrix Q was accepted ad hoc. It turned out 
that the Kalman filter as compared to the OLS and 3SLS methods has a little 
advantage at ex ante forecasts for one quarter. On the other hand, 4-quarterly ex 
ante forecasts obtained by both OLS and 3SLS methods exceed the forecasts of 
time-varying parameters in all equations of the model. 
  Further simulation study was performed by Brännäs and Westlund
6. The 
results of these studies confirm generally the results of previous research. Errors 
of specifications of initial value and transition matrix F have consequences in 
the form of great errors at the estimation of vector of parameters βt. As com-
pared, the Kalman filter appears to be highly resistant to errors of the specifica-
tion of covariance matrix Q and variance  . However, the pattern of Bran-
nes’s and Westlund’s simulation experiment should be criticised at one point. If 
Q and   are real specifications, then Branas and Westlund in their simulation 
study permanently analyse the reaction of the Kalman filter to the   or 
 type specification errors, while the results of changes of the element 
Q ratio were not verified. 
2 σ
2 σ
cQ QC =
2 2 σ σ c C =
                                                      
5 See: McWhorter, Harasimham and Simonds (1977). 
6 See: Brännäs, Westlund (1981). 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Kalman Filters and Specification Errors of Hyper-Structure  79
  The fact that the Kalman filter does not explicitly react to the proportional 
changes of variance and covariance, considered by the above-mentioned au-
thors, can easily be proved by the following theorem. 
Theorem 2 
 Let  ,   and   be specified variance and co-
variance with C > 0 for the Kalman filter. Then for calculated ratios   and 
 of the state vector 
CQ QC =
2 2 σ σ C C = 0 | 0 0 | 0 ∑ = ∑ C
C
c
t t 1 | − α
c
t t| α t α  the following occurs: 
   (8)  , 1 | 1 | − − = t t
c
t t α α
   (9)  . | | t t
c
t t α α =
 
For covariance matrices of the errors the following occurs: 
 
    (10)  , 1 | 1 | − − ∑ = ∑ t t
C
t t C
   (11)  . | | t t
C
t t C∑ = ∑
 
Proof: The theorem is proved by means of the complete induction. The begin-
ning of the induction for t = 1 results directly from equations (11)–(15) of the 
Kalman filter
7 for   published in Grzesiak (1995). It was assumed 
that the thesis is true for t – 1. It results from the above-mentioned equation 
11). 
0 | 0 0 | 0 ∑ = ∑ C
C
(
   
   (12)  1 | 1 | 1 1 | 1 | − − − − − = + = + = t t t t t t
c
t t
c
t t Gu F Gu F α α α α
 
and from the equation of forecast of covariance (12) the following results: 
 
 
   (13) 
. 1 | 1 1 | 1 1 | 1 1 | − − − − − − − ∑ = + ∑ = + ∑ = ∑ t t
T T
t t C
T C
t t
C
t t C HCQH F C F H HQ F F
For the filter amplification vector the following results: 
 
   (14) 
t t t t
T
t t t t
C t
C
t t
T
t t
C
t t
C
t
K C x C x x C
x x x K
= + ∑ ∑
= + ∑ ∑ =
−
− −
−
− − −
1 2
1 | 1 |
1 2
1 | 1 | 1
) (
) (
σ
σ
 
Thesis   results directly from the formula of the filter state (14). Also 
thesis (5) results from the formula of the covariance filter (15): 
t t
c
t t | | α α =
 
   (15)  t t t t
T
t t t t
C
t t
T
t
C
t
C
t t
C
t t C C x K C x K | 1 | 1 | 1 | 1 | | ∑ = ∑ − ∑ = ∑ − ∑ = ∑ − − − −
  Practically, not only the absolute size of covariance matrix Q, but also its 
structure is unknown. It is doubtful if Q is diagonal or not, and also the ratio of 
size of particular elements of Q is unknown. 
                                                      
7 These equations are presented in Grzesiak (1995), p. 58. 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Stefan Grzesiak  80
  A particular possibility to estimate an unknown covariance matrix Q offers 
the so-called Sage and Hus filter
8 often referred to in literature. It is a modifica-
tion of the Kalman filter. Unknown variance and covariance are accepted as 
additional components of the state vector and co-estimated recursively. How-
ever it leads to a significant increase in the size of the state vector, which makes 
it difficult to apply in economics due to short data series. The weight of the 
problem is shifted into another point, because the filtration algorithm needs the 
initial values of   and   for variance and covariance, respectively. 
2
0 σ 0 Q
  Summing up the discussion on specification errors it should be pointed out 
that such widespread methods as ad hoc specifications and sensitivity analysis 
are of little use for the estimation of an unknown hyper-structure with time-
varying parameters. And that is exactly what is important for the improvement 
of the prognostic quality of the econometric model. It is also particularly sig-
nificant for the economic interpretation of an estimated course of the parameter. 
The course of the estimated values of the parameter βt|t is definitely dependant 
on the choice of the covariance matrix Q. Moreover, if the influence of specifi-
cation errors of the transition matrix F is considered, we can easily imagine that 
without a proper criterion of estimation, practically every “demanded” course of 
the parameter can be estimated according to the “proper” choice of a hyper-
structure. 
 
 
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Brännäs, K. (1981), On the Estimation of Time-Varying Parameters for Forecasting and 
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8 See: Sage, Husa (1969). 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
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A Robustness Analysis of Kalman Filtering for Estimation of Interdependent Systems,

Adaptive Filtering with unknown Prior Statistic,

An Empirical Examination of the Predictive Performance of an Econometric Model with Random Coefficients,

by The Nicolaus Copernicus University Scientific Publishing House Kalman Filters and Specification Errors of

Economic Fluctuations in the United States, 1921–1941,

Estimation of the Volatility of Securities in the Stock Market by Kalman Filtering Techniques,

Linear Filter Methods: An Application to a Stock Production Model,

O wyznaczaniu wartości początkowych algorytmu filtru Kalmana (On Determining the Initial Values for Kalman Filter Algorithm), Przegląd Statystyczny (Statistical Survey),

On the Estimation of Time-Varying Parameters for Forecasting and Control,

Optimal Filtering,

Problem wygładzania w filtracji kalmanowskiej (The Problem of Smoothing in Kalman Filtration), Przegląd Statystyczny (Statistical Survey),

Równania filtru Kalmana w modelowaniu ekonometrycznym (Kalman Filter Equations in Econometric Modelling), Przegląd Statystyczny (Statistical Survey),

Sage,

The Discrete Kalman Filter Applied to Linear Regression Models: Statistical Considerations and an Application,

Zur Verwendung des Kalman-Ansatzes für eine Verbesserung der Prognosegüte ökonometrischer Modelle, Dissertation,

Kalman Filters and Specification Errors of Hyper-Structure

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Kalman Filters and Specification Errors of Hyper-Structure

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Research Papers in Economics