- Email: [email protected]
Bound and convergence of the non-constant dynamic linear model
Journal of Statistical Planning and Inference 141 (2011) 602–609
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Short communication
Bound and convergence of the non-constant dynamic linear model$ Wenhui Liao, Yan Liu Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521, PR China
a r t i c l e i n f o
abstract
Article history: Received 24 September 2009 Received in revised form 20 July 2010 Accepted 21 July 2010 Available online 29 July 2010
This paper extends the limiting results of West and Harrison (1997, section 5.5) about the convergence of the variances of time series dynamic linear models (TSDLMs) when both, the variances of the observation and evolution errors of the model, are timevarying with steady limits. Analytical results are derived and an illustrative example is provided. & 2010 Elsevier B.V. All rights reserved.
Keywords: Bayesian statistics Bayesian dynamic model Non-constant dynamic linear model Posterior variance Time series dynamic linear model (TSDLM)
1. Introduction The dynamic linear models (DLMs) are based on the fundamental of Bayesian statistics to model and forecast the time series. Harrison and Stevens (1976) firstly brought forward the conception of dynamic models. As yet, the theory shows strong vitality. The prime reason for DLM is to provide efficient learning processes that will increase understanding and enable wise decisions. The DLMs are true of open systems, the models can incessantly revise parameter according to all kinds of exterior information. The principles, models and methods of DLMs have been developed extensively to cover business, industry and economics, etc. In the last decade, the dynamic linear models (DLMs) have been further developed in Salvador et al. (2003, 2004), Triantafyllopoulos and Pikoulas (2002) and Triantafyllopoulos (2006, 2008), etc. The general normal dynamic linear models (DLMs) are characterized by a set of quadruples {Ft, Gt, Vt, Wt}, for each time t. The constant model means that all components that characterize and distinguish the model from other models are constant over time. Limiting results for constant DLMs have been examined in Harrison (1996). West and Harrison (1997) establishes that for any observable constant DLMs the variances of the posterior state and one-step forecast distributions converge rapidly to constant values. The term ‘observability’ means that the number of the parameters is sufficient for estimation and forecasting. That is the model which is not overparamaterized. For more details one could refer to West and Harrison (1997, Chapter 5). However, the constant DLMs are so restricted to request that all components are constant over time that it is inferior to use them in practice. Limiting results for non-constant DLMs have been developed in the first-order non-constant DLMs (Harrison, 1985) or discount TSDLMs limit result in West and Harrison (1997, Chapter 6).
$ The work was supported by the National Natural Science Foundation of China (Grant ]10971234), and Science Foundation of Guangdong University of Finance (Grant ] 09XJ02-04). Corresponding author. E-mail address: [email protected] (Y. Liu).
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.07.012
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
603
To get more generic results, this paper aims to introduce a class of non-constant DLMs that exhibit a similar limiting value as in the constant case. The paper is organized as follows. Section 2 defines the non-constant DLMs and proves the variances bounding and limiting of the non-constant DLMs. The results are very interesting and useful complementary of the Bayesian Time Series Dynamic Model Theory. Section 3 gives a practical and real example to illustrate the utility of the results. We analyze data from the Shanghai Stock Exchange market. 2. A kind of non-constant DLMs Limiting results for constant DLMs were fully considered in West and Harrison (1997, Chapter 5). However, these authors assume that the variances of the observation and evolution errors of model are constant, which is a very restrictive constraint. In this section we extend their limiting results assuming the more realistic assumption that these variances are time-varying with steady limits. 2.1. Definition of the non-constant DLMs Consider the DLMs given by equations {F, G, Vt, Wt}, the working model is defined by ( vt N½0,Vt , Yt ¼ Fuyt þ vt ,
yt ¼ Gyt1 þ ot , ot Nn ½0,Wt ,
ð2:1Þ
where (a) (b) (c) (d)
F is a known (n r) matrix; G is a known (n n) matrix; Vt is a known (r r) variance matrix; Wt is a known (n n) variance matrix.
Yt is an observation (r 1) variable, yt the (n 1) parameter vector. We assume that the model is observable, that is to say Fa0. Clearly, this is a minimum and reasonable assumption, because otherwise the DLM is reduced to a purely random process and yt has no contribution to the model. At such time t, the available information Dt is defined as Dt ={Yt, Dt 1}. Initially D0 is known and it is assumed that ðy0 jD0 Þ N½m0 ,C0
ð2:2Þ
for some known moments m0, C0. Then, it was established updating equations for the distributions of ðYt jDt1 Þ and ðyt jDi Þi ¼ t1,t in Kalman (1963) or Harrison and Stevens (1976). They are ðyt jDt1 Þ N½at ,Rt , ðYt jDt1 Þ N½ft ,Qt , and ðyt jDt Þ N½mt ,Ct with at =Gmt 1, Rt ¼ GC t1 Gu þ Wt ,ft ¼ Fuat , Qt ¼ FuRt F þVt , mt =at +Atet, Ct ¼ Rt At Qt Aut , At = RtFQt 1, where et = Yt ft is the one-step error and the n vector At is called the adaptive vector. Consider the non-constant observable DLMs {F G Vt Wt}, with observability matrix 0 1 Fu B FuG C B C T ¼B C: @ ^ A FuGn1 Define ! Yt ¼ ðYtn þ 1 Yt Þu,
ftn þ 1 ¼ vtn þ 1 , ^
ftn þ 1 þ i ¼ vtn þ 1 þ i þ
i1 X
FuGj otn þ 1 þ ij
ði ¼ 1,2, . . . ,n1Þ,
j¼0
!
ft ¼ ðftn þ 1 ft Þu,
and n2 ! ! X i et ¼ G oti Gn1 T 1 ft : i¼0
ð2:3Þ
604
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
From Eqs. (2.1) ! ! Yt ¼ T ytn þ 1 þ ft
ð2:4Þ
and
yt ¼ Gn1 ytn þ 1 þ
n2 X
Gi oti ,
i¼0
!
yt ¼ Gn1 T 1 Yt þ
n2 X
! Gi oti Gn1 T 1 ft ,
i¼0
so that ! !
yt ¼ Gn1 T 1 Yt þ et :
ð2:5Þ
! Thus, et is a linear function of fotn þ 2 , . . . , ot ,vtn þ 1 , . . . ,vt g. 2.2. Bound and convergence of the non-constant DLMs A feature of constant DLMs is that variances often convergence to limiting values rapidly. From West and Harrison (1997, Chapter 5), we know that the variance sequence {Ct} of constant DLMs is bounded above and below with 0 r Ct r S and the limiting variance limt-1 Ct ¼ C exists. To get the same feature of non-constant DLM, we firstly prove that the posterior variance Ct ¼ varðyt jDt Þ is bounded in the working model. Theorem 2.1. In the observable DLMs (2.1), for all t Z 0 if the parameters 0 rVt r V and 0 r Wt rW exist, then, the variance sequence {Ct} is bounded above and below, with 0 r Ct r S: Proof. Using (2.3)–(2.5), we have ! !
yt ¼ Gn1 T 1 Yt þ et and
! ! ! 0 r Ct ¼ V½yt jDt ¼ V½yt jYt , . . . ,Ytn þ 1 ,Dtn r V½yt jYt ¼ V½yt jGn1 T 1 Yt ¼ V½ et r S:
! The inequality V½yt jYt , . . . ,Ytn þ 1 ,Dtn r V½yt jYt can be proved as follows: * ! + V½yt jYt , . . . ,Ytn þ 1 ,Dtn ¼ V½yt jYtn þ 1:t cov
yt
Ytn þ 1:t
*
,Ytn ðVðYtn ÞÞ1 cov
yt Ytn þ 1:t
!
+ ! ,Ytn u rV½yt jYt ,
where Ytn ¼ ðYt , . . . ,Ytn Þu and Ytn þ 1:t ¼ ðYtn þ 1 , . . . ,Yt Þu. ! Furthermore, from Section 2.1, it follows that et is a linear function of fotn þ 2 , . . . , ot ,vtn þ 1 , . . . ,vt g. Hence, and taking into account that for all t Z n, the parameters 0 r V½vt ¼ Vt r V and 0 rV½wt ¼ Wt r W, it follows that a constant matrix S ! exists such that V½ et rS. & Now, we prepare two lemmas for fetching the convergence of the posterior variance Ct ¼ varðyt jDt Þ after proving the bound. Lemma 2.1. For two observable constant DLMs {F G V(1) W(1)} and {F G V(2) W(2)} with the same initial information D0, Ct1 the variance sequence of the DLM {F G V(1) W(1)} and Ct2 is the variance sequence of the DLM {F G V(2) W(2)}, if V ð1Þ o V ð2Þ ,W ð1Þ o W ð2Þ , then Ct1 oCt2 exists for all t 4 0. Proof. (11) t = 1, W ð1Þ o W ð2Þ , and R11 ¼ GC 0 Gu þW ð1Þ oR12 ¼ GC 0 Gu þW ð2Þ so that 1 R1 12 o R11 :
Since V
ð1Þ
oV
ð2Þ
, we get ðV
ð2:6Þ ð1Þ 1
Þ
4 ðV
ð2Þ 1
Þ
:
FðV ð1Þ Þ1 Fu 4FðV ð2Þ Þ1 Fu: From Eqs. (2.6) and (2.7), we have 1 ð1Þ 1 1 ð2Þ 1 ¼ R1 Þ Fu 4 C12 ¼ R1 Þ Fu: C11 11 þ FðV 12 þ FðV
ð2:7Þ
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
605
We can easily get C12 4 C11 : (21) When t = k, assuming Ck1 o Ck2 , we have GC k2 Gu 4 GC k1 Gu, so Rk þ 1,1 ¼ GC k1 Gu þ W ð1Þ oRk þ 1,2 ¼ GC k2 Gu þW ð2Þ , then 1 R1 k þ 1,2 o Rk þ 1,1 :
Thus we have 1 ð1Þ 1 1 ð2Þ 1 Þ Fu 4 Ck1 Þ Fu Ck1 þ 1,1 ¼ Rk þ 1,1 þFðV þ 1,2 ¼ Rk þ 1,2 þ FðV
and Ck þ 1,1 o Ck þ 1,2 : Combing the above discussions, we have proved the Lemma 2.1.
&
Lemma 2.2. Consider the subset of DLMs defined by {F G V+ x W+ xI} with jxj om, and m is small enough such that {V+ x W+xI} is positive defined matrix and I is the n n identity matrix, for any given x, let the variance sequence be {Ct(x)}, then Ct(x) is bounded, continuous with x, and monotonic in x. The limiting variance limx-0 Ct ðxÞ ¼ Ct ð0Þ exists for all t 4 0. Proof. The character of bound and monotonic is easy to understand from Lemma 2.1 and Theorem 2.1. 8jx1 j,jx2 j om, if x1 ox2 , we have V þ x1 o V þ x2 oV þm and W þ x1 I oW þ x2 I o W þmI, so we have Ct ðx1 Þ o Ct ðx2 Þ oCt ðmÞ r S. Now, we begin to prove the continuous. (11) When t =1, we have 1 limC1 ðxÞ ¼ lim½R1 A1 Q1 Au1 ¼ lim½GC 0 Gu þW þ xIR1 FQ 1 1 Q1 Q1 FuRu1 x-0 x-0 x-0 ðGC 0 Gu þ W þxIÞFFuðGC 0 Gu þ W þ xIÞ ðGC 0 Gu þ WÞFFuðGC 0 Gu þ WÞ ¼ GC 0 Gu þW ¼ C1 ð0Þ: ¼ lim GC 0 Gu þ W þ xI FuGC 0 GuF þ FuðW þ xIÞF þðV þ xÞ FuGC 0 GuF þFuWF þ V x-0
(21) When t =k, assume limx-0 Ck ðxÞ ¼ Ck ð0Þ, then 1 limCk þ 1 ðxÞ ¼ lim½Rk þ 1 Ak þ 1 Qk þ 1 Auk þ 1 ¼ lim½GC k ðxÞGu þ W þ xIRk þ 1 FQ 1 k þ 1 Qk þ 1 Qk þ 1 FuRuk þ 1 x-0 x-0 x-0 ðGC k ðxÞGu þ W þ xIÞFFuðGC k ðxÞGu þ W þxIÞ ¼ lim GC k ðxÞGu þ W þ xI x-0 FuGC k ðxÞGuF þ FuðW þ xIÞF þ ðV þxÞ ðGC k ð0ÞGu þ WÞFFuðGC k ð0ÞGu þ WÞ ¼ GC k ð0ÞGu þW ¼ Ck þ 1 ð0Þ: FuGC k ð0ÞGuF þ FuWF þ V
So we have limCt ðxÞ ¼ Ct ð0Þ x-0
ðjxjo mÞ for all t 4 0:
&
Using the results of Lemmas 2.1 and 2.2 we can prove the following theorem. Theorem 2.2. In the observable DLMs (2.1), for all t Zn, if limt-1 Vt ¼ V and limt-1 Wt ¼ W, the limiting variance lim Ct ¼ C
t-1
exists and depends on the initial information D0. Proof. Define Ct is the variance of the DLM {F G Vt Wt}; Ct ð1=nÞ is the variance of the DLM fF G Vð1=nÞI Wð1=nÞIg; Ct ð1=nÞ is the variance of the DLM fF G V þð1=nÞI W þ ð1=nÞIg.
606
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
From Lemmas 2.1 and 2.2 it is followed that, for any given t 4 0 and for n sufficiently large 1 1 1 1 r Ct ð0Þ r Ct , Ct r Ct r Ct : Ct n n n n
ð2:8Þ
Furthermore West and Harrison (1997, p. 161) proved that there exist C 40 such that limt-1 Ct ð0Þ ¼ C. Thus, from Lemma 2.2 it follows that 1 ¼ lim Ct ð0Þ ¼ C, lim lim Ct t-1n-1 t-1 n lim lim Ct
t-1n-1
1 ¼ lim Ct ð0Þ ¼ C: t-1 n
Hence, from (2.8) we can easily conclude lim Ct ¼ C:
t-1
Thus, we have proved Theorem 2.2.
&
Based on Theorem 2.2, the following results are obvious. Corollary 2.1. lim Rt ¼ R ¼ GCGu þ W,
t-1
lim Qt ¼ Q ¼ FRFu þV,
t-1
lim At ¼ A ¼ RF=Q :
t-1
3. Example: the Shanghai Stock Exchange data 3.1. Description of the data The Shanghai Stock Exchange (SSE) was founded on November 26th 1990 and began operation on December 19th of the same year. After several years’ operation, the SSE has become the most preeminent stock market in Mainland China in terms of number of listed companies, number of shares listed, total market value, tradable market value, securities turnover in value, stock turnover in value and the T bond turnover in value. In this paper we concentrate on the price of SSE Composite Index. Constituents for SSE Composite Index are all listed stocks (A shares and B shares) at Shanghai Stock Exchange. The base day for SSE Composite Index is December 19, 1990. The base period is the total market capitalization of all stocks of that day. The base value is 100. The index was launched on July 15, 1991. The data are collected for every trading week from June 24, 2005 to December 12, 2009, and are plotted in Fig. 1. There are N =226 trading weeks. The data have been obtained from the SSE web site: http://www.sse.com.cn/sseportal/ps/zhs/home.html. 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 0
50
100
150
200
250
Fig. 1. SSE Composite Index data, Constituents for SSE Composite Index are all listed stocks (A shares and B shares) at Shanghai Stock Exchange.
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
607
3.2. Statistical analysis We use the model Yt ¼ Fuyt þvt ,
nt N½0,Vt ,
yt ¼ Gyt1 þ ot , ot N½0,Wt , Þ and Yt is the price of the SSE Composite Index at time t {t =1,2,3,y,225,226}. The quadruple {F, G, Vt, Wt} is where yt ¼ ðyyt1 t2 equal to 8 0 19 1 > > > > 20 þ 10ð1Þt 0 < 30 1:15 0 B C= 1 t C : , ,301þ 200ð1Þt , B 1 A> > t @ 1:5 0 1 > > 0 50 þ10ð1Þt ; : t Obviously, the error variances Vt and Wt are convergent with t- þ1. Their values describe the fluctuation of the dynamic parameter vector yt and the observation Yt. In fact, it is quite difficult to get practical number for the quadruple {F, G, Vt, Wt}. These variables are evaluated from our subjective experience to show the convergence of the posterior variance Ct. We think that the number 30 is the rational price-earning ratios in Chinese capital market and the error variance Vt and Wt are evaluated from the historical exchange data. It is reasonable that the yt and Yt ’s fluctuation becomes stable under time after operation time. Figs. 2 and 3 show the convergence of the posterior variance Ct. From Theorem 2.2, we know that the eigenvalues of the posterior variance Ct should be convergent. By computing, we found one of the two eigenvalues of the variance Ct is 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
0
50
100
150
200
250
Fig. 2. Dynamic evolution of the first eigenvalue of Ct.
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1
0
50
100
150
200
Fig. 3. Dynamic evolution of the second eigenvalue of Ct.
250
608
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
Table 1 Dynamic evolution of the DLM’s variances. t 1 10 20 50 112 150 198 215 220 224 226
eigC 1t
eigC 2t
ft
Yt
Vt
eigW 1t
eigW 2t
107.178 551.938 1047.065 2379.371 3959.302 4328.33 4512.631 4542.612 4549.028 4553.868 4556.123
0.112 0.349 0.339 0.333 0.33 0.33 0.329 0.327 0.329 0.329 0.329
1213.5 1314.504 1254.273 1782.688 5900.189 3472.211 2632.496 3039.586 3201.609 3430.892 3436.467
1101.88 1171.86 1090.19 1605.71 5312.18 2868.799 2632.929 2838.838 2995.849 3096.259 3247.319
101 321 311 305 302.786 302.333 302.01 300.07 301.909 301.893 301.885
26.972 18.967 19.492 19.799 19.91 19.933 19.949 20.046 19.954 19.955 19.956
63.028 49.033 49.508 49.801 49.911 49.933 49.95 50.047 49.955 49.955 49.956
7000 price forecast 95% credible intervals
6000
5000
4000
3000
2000
1000
0
50
100
150
200
250
Fig. 4. The forecast price and true price of the SSE Composite Index.
convergent to 0.329, the other one is about 4560. Fig. 2 illustrates the first one and Fig. 3 shows the second. Table 1 shows the values of the error variances Vt and the eigenvalues of the error variances matrices Wt. The variables eigC 1 and eigC 2 are values of the posterior variance Ct ’s eigenvalues. To economize the length, we choose 11 times to explain the convergent process. From the above results we can clearly see a convergent limit of the posterior variance Ct of the dynamic parameter vector of a dynamic linear model (DLM) with non-constant error variances Vt and Wt. It was successfully validated that Ct -C when Vt -V and Wt -W. Fig. 4 shows the forecast effect of the model which we used. From the second half of the time series, we found that the forecast price of The SSE Composite Index is quite accordant to the true price. It proves that the example is right and the DLM succeeded.
4. Conclusion This paper extends the limiting results of West and Harrison (1997, section 5.5) about the convergence of the variances of time series dynamic linear models (TSDLMs) when both, the variances of the observation and evolution errors of the model, are time-varying with steady limits. Now, the similarity results perhaps can be extended when all components {Ft Gt Vt Wt} are time-varying with steady limits. We are taking into account that the components fFt Gt g are timevarying with 0 r Ft r F and 0 rGt r G. It would be more interesting if we can extend the similarity results. References Harrison, P.J., Stevens, C., 1976. Bayesian forecasting (with discussion). Journal of the Royal Statistical Society (Series B) 38, 205–247. Harrison, P.J., 1985. First-order constant dynamic models. Research Report 66, Department of Statistics, University of Warwick. Harrison, P.J., 1996. Convergence and the constant dynamic linear model. Journal of Forecasting 16, 287–292. Kalman, R.E., 1963. New methods in Wiener filtering theory. In: Boganoff, J.L., Kozin, F. (Eds.), Proceeding of the First Symposium in Engineering Applicational of Random Function Theory and Probability. Wiley, New York.
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
609
Salvador, M., Gallizo, J.L., Gargallo, P., 2004. Bayesian inference in a matrix normal dynamic linear model with unknown covariance matrices. Statistics 38, 307–335. Salvador, M., Gallizo, J.L., Gargallo, P., 2003. A dynamic principal components analysis based on multivariate matrix normal dynamic linear models. Journal of Forecasting 22, 457–478. Triantafyllopoulos, K., 2006. Multivariate discount weighted regression and local level models. Computational Statistic and Data Analysis 50, 3702–3720. Triantafyllopoulos, K., 2008. Multivariate stochastic volatility with Bayesian dynamic linear models. Journal of Statistical Planning and Inference 138 (4), 1021–1037. Triantafyllopoulos, K., Pikoulas, J., 2002. Multivariate Bayesian regression applied to the problem of network security models. Journal of Forecasting 21, 579–594. West, M., Harrison, P.J., 1997. Bayesian Forecasting and Dynamic Models.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Short communication
Bound and convergence of the non-constant dynamic linear model$ Wenhui Liao, Yan Liu Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521, PR China
a r t i c l e i n f o
abstract
Article history: Received 24 September 2009 Received in revised form 20 July 2010 Accepted 21 July 2010 Available online 29 July 2010
This paper extends the limiting results of West and Harrison (1997, section 5.5) about the convergence of the variances of time series dynamic linear models (TSDLMs) when both, the variances of the observation and evolution errors of the model, are timevarying with steady limits. Analytical results are derived and an illustrative example is provided. & 2010 Elsevier B.V. All rights reserved.
Keywords: Bayesian statistics Bayesian dynamic model Non-constant dynamic linear model Posterior variance Time series dynamic linear model (TSDLM)
1. Introduction The dynamic linear models (DLMs) are based on the fundamental of Bayesian statistics to model and forecast the time series. Harrison and Stevens (1976) firstly brought forward the conception of dynamic models. As yet, the theory shows strong vitality. The prime reason for DLM is to provide efficient learning processes that will increase understanding and enable wise decisions. The DLMs are true of open systems, the models can incessantly revise parameter according to all kinds of exterior information. The principles, models and methods of DLMs have been developed extensively to cover business, industry and economics, etc. In the last decade, the dynamic linear models (DLMs) have been further developed in Salvador et al. (2003, 2004), Triantafyllopoulos and Pikoulas (2002) and Triantafyllopoulos (2006, 2008), etc. The general normal dynamic linear models (DLMs) are characterized by a set of quadruples {Ft, Gt, Vt, Wt}, for each time t. The constant model means that all components that characterize and distinguish the model from other models are constant over time. Limiting results for constant DLMs have been examined in Harrison (1996). West and Harrison (1997) establishes that for any observable constant DLMs the variances of the posterior state and one-step forecast distributions converge rapidly to constant values. The term ‘observability’ means that the number of the parameters is sufficient for estimation and forecasting. That is the model which is not overparamaterized. For more details one could refer to West and Harrison (1997, Chapter 5). However, the constant DLMs are so restricted to request that all components are constant over time that it is inferior to use them in practice. Limiting results for non-constant DLMs have been developed in the first-order non-constant DLMs (Harrison, 1985) or discount TSDLMs limit result in West and Harrison (1997, Chapter 6).
$ The work was supported by the National Natural Science Foundation of China (Grant ]10971234), and Science Foundation of Guangdong University of Finance (Grant ] 09XJ02-04). Corresponding author. E-mail address: [email protected] (Y. Liu).
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.07.012
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
603
To get more generic results, this paper aims to introduce a class of non-constant DLMs that exhibit a similar limiting value as in the constant case. The paper is organized as follows. Section 2 defines the non-constant DLMs and proves the variances bounding and limiting of the non-constant DLMs. The results are very interesting and useful complementary of the Bayesian Time Series Dynamic Model Theory. Section 3 gives a practical and real example to illustrate the utility of the results. We analyze data from the Shanghai Stock Exchange market. 2. A kind of non-constant DLMs Limiting results for constant DLMs were fully considered in West and Harrison (1997, Chapter 5). However, these authors assume that the variances of the observation and evolution errors of model are constant, which is a very restrictive constraint. In this section we extend their limiting results assuming the more realistic assumption that these variances are time-varying with steady limits. 2.1. Definition of the non-constant DLMs Consider the DLMs given by equations {F, G, Vt, Wt}, the working model is defined by ( vt N½0,Vt , Yt ¼ Fuyt þ vt ,
yt ¼ Gyt1 þ ot , ot Nn ½0,Wt ,
ð2:1Þ
where (a) (b) (c) (d)
F is a known (n r) matrix; G is a known (n n) matrix; Vt is a known (r r) variance matrix; Wt is a known (n n) variance matrix.
Yt is an observation (r 1) variable, yt the (n 1) parameter vector. We assume that the model is observable, that is to say Fa0. Clearly, this is a minimum and reasonable assumption, because otherwise the DLM is reduced to a purely random process and yt has no contribution to the model. At such time t, the available information Dt is defined as Dt ={Yt, Dt 1}. Initially D0 is known and it is assumed that ðy0 jD0 Þ N½m0 ,C0
ð2:2Þ
for some known moments m0, C0. Then, it was established updating equations for the distributions of ðYt jDt1 Þ and ðyt jDi Þi ¼ t1,t in Kalman (1963) or Harrison and Stevens (1976). They are ðyt jDt1 Þ N½at ,Rt , ðYt jDt1 Þ N½ft ,Qt , and ðyt jDt Þ N½mt ,Ct with at =Gmt 1, Rt ¼ GC t1 Gu þ Wt ,ft ¼ Fuat , Qt ¼ FuRt F þVt , mt =at +Atet, Ct ¼ Rt At Qt Aut , At = RtFQt 1, where et = Yt ft is the one-step error and the n vector At is called the adaptive vector. Consider the non-constant observable DLMs {F G Vt Wt}, with observability matrix 0 1 Fu B FuG C B C T ¼B C: @ ^ A FuGn1 Define ! Yt ¼ ðYtn þ 1 Yt Þu,
ftn þ 1 ¼ vtn þ 1 , ^
ftn þ 1 þ i ¼ vtn þ 1 þ i þ
i1 X
FuGj otn þ 1 þ ij
ði ¼ 1,2, . . . ,n1Þ,
j¼0
!
ft ¼ ðftn þ 1 ft Þu,
and n2 ! ! X i et ¼ G oti Gn1 T 1 ft : i¼0
ð2:3Þ
604
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
From Eqs. (2.1) ! ! Yt ¼ T ytn þ 1 þ ft
ð2:4Þ
and
yt ¼ Gn1 ytn þ 1 þ
n2 X
Gi oti ,
i¼0
!
yt ¼ Gn1 T 1 Yt þ
n2 X
! Gi oti Gn1 T 1 ft ,
i¼0
so that ! !
yt ¼ Gn1 T 1 Yt þ et :
ð2:5Þ
! Thus, et is a linear function of fotn þ 2 , . . . , ot ,vtn þ 1 , . . . ,vt g. 2.2. Bound and convergence of the non-constant DLMs A feature of constant DLMs is that variances often convergence to limiting values rapidly. From West and Harrison (1997, Chapter 5), we know that the variance sequence {Ct} of constant DLMs is bounded above and below with 0 r Ct r S and the limiting variance limt-1 Ct ¼ C exists. To get the same feature of non-constant DLM, we firstly prove that the posterior variance Ct ¼ varðyt jDt Þ is bounded in the working model. Theorem 2.1. In the observable DLMs (2.1), for all t Z 0 if the parameters 0 rVt r V and 0 r Wt rW exist, then, the variance sequence {Ct} is bounded above and below, with 0 r Ct r S: Proof. Using (2.3)–(2.5), we have ! !
yt ¼ Gn1 T 1 Yt þ et and
! ! ! 0 r Ct ¼ V½yt jDt ¼ V½yt jYt , . . . ,Ytn þ 1 ,Dtn r V½yt jYt ¼ V½yt jGn1 T 1 Yt ¼ V½ et r S:
! The inequality V½yt jYt , . . . ,Ytn þ 1 ,Dtn r V½yt jYt can be proved as follows: * ! + V½yt jYt , . . . ,Ytn þ 1 ,Dtn ¼ V½yt jYtn þ 1:t cov
yt
Ytn þ 1:t
*
,Ytn ðVðYtn ÞÞ1 cov
yt Ytn þ 1:t
!
+ ! ,Ytn u rV½yt jYt ,
where Ytn ¼ ðYt , . . . ,Ytn Þu and Ytn þ 1:t ¼ ðYtn þ 1 , . . . ,Yt Þu. ! Furthermore, from Section 2.1, it follows that et is a linear function of fotn þ 2 , . . . , ot ,vtn þ 1 , . . . ,vt g. Hence, and taking into account that for all t Z n, the parameters 0 r V½vt ¼ Vt r V and 0 rV½wt ¼ Wt r W, it follows that a constant matrix S ! exists such that V½ et rS. & Now, we prepare two lemmas for fetching the convergence of the posterior variance Ct ¼ varðyt jDt Þ after proving the bound. Lemma 2.1. For two observable constant DLMs {F G V(1) W(1)} and {F G V(2) W(2)} with the same initial information D0, Ct1 the variance sequence of the DLM {F G V(1) W(1)} and Ct2 is the variance sequence of the DLM {F G V(2) W(2)}, if V ð1Þ o V ð2Þ ,W ð1Þ o W ð2Þ , then Ct1 oCt2 exists for all t 4 0. Proof. (11) t = 1, W ð1Þ o W ð2Þ , and R11 ¼ GC 0 Gu þW ð1Þ oR12 ¼ GC 0 Gu þW ð2Þ so that 1 R1 12 o R11 :
Since V
ð1Þ
oV
ð2Þ
, we get ðV
ð2:6Þ ð1Þ 1
Þ
4 ðV
ð2Þ 1
Þ
:
FðV ð1Þ Þ1 Fu 4FðV ð2Þ Þ1 Fu: From Eqs. (2.6) and (2.7), we have 1 ð1Þ 1 1 ð2Þ 1 ¼ R1 Þ Fu 4 C12 ¼ R1 Þ Fu: C11 11 þ FðV 12 þ FðV
ð2:7Þ
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
605
We can easily get C12 4 C11 : (21) When t = k, assuming Ck1 o Ck2 , we have GC k2 Gu 4 GC k1 Gu, so Rk þ 1,1 ¼ GC k1 Gu þ W ð1Þ oRk þ 1,2 ¼ GC k2 Gu þW ð2Þ , then 1 R1 k þ 1,2 o Rk þ 1,1 :
Thus we have 1 ð1Þ 1 1 ð2Þ 1 Þ Fu 4 Ck1 Þ Fu Ck1 þ 1,1 ¼ Rk þ 1,1 þFðV þ 1,2 ¼ Rk þ 1,2 þ FðV
and Ck þ 1,1 o Ck þ 1,2 : Combing the above discussions, we have proved the Lemma 2.1.
&
Lemma 2.2. Consider the subset of DLMs defined by {F G V+ x W+ xI} with jxj om, and m is small enough such that {V+ x W+xI} is positive defined matrix and I is the n n identity matrix, for any given x, let the variance sequence be {Ct(x)}, then Ct(x) is bounded, continuous with x, and monotonic in x. The limiting variance limx-0 Ct ðxÞ ¼ Ct ð0Þ exists for all t 4 0. Proof. The character of bound and monotonic is easy to understand from Lemma 2.1 and Theorem 2.1. 8jx1 j,jx2 j om, if x1 ox2 , we have V þ x1 o V þ x2 oV þm and W þ x1 I oW þ x2 I o W þmI, so we have Ct ðx1 Þ o Ct ðx2 Þ oCt ðmÞ r S. Now, we begin to prove the continuous. (11) When t =1, we have 1 limC1 ðxÞ ¼ lim½R1 A1 Q1 Au1 ¼ lim½GC 0 Gu þW þ xIR1 FQ 1 1 Q1 Q1 FuRu1 x-0 x-0 x-0 ðGC 0 Gu þ W þxIÞFFuðGC 0 Gu þ W þ xIÞ ðGC 0 Gu þ WÞFFuðGC 0 Gu þ WÞ ¼ GC 0 Gu þW ¼ C1 ð0Þ: ¼ lim GC 0 Gu þ W þ xI FuGC 0 GuF þ FuðW þ xIÞF þðV þ xÞ FuGC 0 GuF þFuWF þ V x-0
(21) When t =k, assume limx-0 Ck ðxÞ ¼ Ck ð0Þ, then 1 limCk þ 1 ðxÞ ¼ lim½Rk þ 1 Ak þ 1 Qk þ 1 Auk þ 1 ¼ lim½GC k ðxÞGu þ W þ xIRk þ 1 FQ 1 k þ 1 Qk þ 1 Qk þ 1 FuRuk þ 1 x-0 x-0 x-0 ðGC k ðxÞGu þ W þ xIÞFFuðGC k ðxÞGu þ W þxIÞ ¼ lim GC k ðxÞGu þ W þ xI x-0 FuGC k ðxÞGuF þ FuðW þ xIÞF þ ðV þxÞ ðGC k ð0ÞGu þ WÞFFuðGC k ð0ÞGu þ WÞ ¼ GC k ð0ÞGu þW ¼ Ck þ 1 ð0Þ: FuGC k ð0ÞGuF þ FuWF þ V
So we have limCt ðxÞ ¼ Ct ð0Þ x-0
ðjxjo mÞ for all t 4 0:
&
Using the results of Lemmas 2.1 and 2.2 we can prove the following theorem. Theorem 2.2. In the observable DLMs (2.1), for all t Zn, if limt-1 Vt ¼ V and limt-1 Wt ¼ W, the limiting variance lim Ct ¼ C
t-1
exists and depends on the initial information D0. Proof. Define Ct is the variance of the DLM {F G Vt Wt}; Ct ð1=nÞ is the variance of the DLM fF G Vð1=nÞI Wð1=nÞIg; Ct ð1=nÞ is the variance of the DLM fF G V þð1=nÞI W þ ð1=nÞIg.
606
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
From Lemmas 2.1 and 2.2 it is followed that, for any given t 4 0 and for n sufficiently large 1 1 1 1 r Ct ð0Þ r Ct , Ct r Ct r Ct : Ct n n n n
ð2:8Þ
Furthermore West and Harrison (1997, p. 161) proved that there exist C 40 such that limt-1 Ct ð0Þ ¼ C. Thus, from Lemma 2.2 it follows that 1 ¼ lim Ct ð0Þ ¼ C, lim lim Ct t-1n-1 t-1 n lim lim Ct
t-1n-1
1 ¼ lim Ct ð0Þ ¼ C: t-1 n
Hence, from (2.8) we can easily conclude lim Ct ¼ C:
t-1
Thus, we have proved Theorem 2.2.
&
Based on Theorem 2.2, the following results are obvious. Corollary 2.1. lim Rt ¼ R ¼ GCGu þ W,
t-1
lim Qt ¼ Q ¼ FRFu þV,
t-1
lim At ¼ A ¼ RF=Q :
t-1
3. Example: the Shanghai Stock Exchange data 3.1. Description of the data The Shanghai Stock Exchange (SSE) was founded on November 26th 1990 and began operation on December 19th of the same year. After several years’ operation, the SSE has become the most preeminent stock market in Mainland China in terms of number of listed companies, number of shares listed, total market value, tradable market value, securities turnover in value, stock turnover in value and the T bond turnover in value. In this paper we concentrate on the price of SSE Composite Index. Constituents for SSE Composite Index are all listed stocks (A shares and B shares) at Shanghai Stock Exchange. The base day for SSE Composite Index is December 19, 1990. The base period is the total market capitalization of all stocks of that day. The base value is 100. The index was launched on July 15, 1991. The data are collected for every trading week from June 24, 2005 to December 12, 2009, and are plotted in Fig. 1. There are N =226 trading weeks. The data have been obtained from the SSE web site: http://www.sse.com.cn/sseportal/ps/zhs/home.html. 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 0
50
100
150
200
250
Fig. 1. SSE Composite Index data, Constituents for SSE Composite Index are all listed stocks (A shares and B shares) at Shanghai Stock Exchange.
W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
607
3.2. Statistical analysis We use the model Yt ¼ Fuyt þvt ,
nt N½0,Vt ,
yt ¼ Gyt1 þ ot , ot N½0,Wt , Þ and Yt is the price of the SSE Composite Index at time t {t =1,2,3,y,225,226}. The quadruple {F, G, Vt, Wt} is where yt ¼ ðyyt1 t2 equal to 8 0 19 1 > > > > 20 þ 10ð1Þt 0 < 30 1:15 0 B C= 1 t C : , ,301þ 200ð1Þt , B 1 A> > t @ 1:5 0 1 > > 0 50 þ10ð1Þt ; : t Obviously, the error variances Vt and Wt are convergent with t- þ1. Their values describe the fluctuation of the dynamic parameter vector yt and the observation Yt. In fact, it is quite difficult to get practical number for the quadruple {F, G, Vt, Wt}. These variables are evaluated from our subjective experience to show the convergence of the posterior variance Ct. We think that the number 30 is the rational price-earning ratios in Chinese capital market and the error variance Vt and Wt are evaluated from the historical exchange data. It is reasonable that the yt and Yt ’s fluctuation becomes stable under time after operation time. Figs. 2 and 3 show the convergence of the posterior variance Ct. From Theorem 2.2, we know that the eigenvalues of the posterior variance Ct should be convergent. By computing, we found one of the two eigenvalues of the variance Ct is 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
0
50
100
150
200
250
Fig. 2. Dynamic evolution of the first eigenvalue of Ct.
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1
0
50
100
150
200
Fig. 3. Dynamic evolution of the second eigenvalue of Ct.
250
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W. Liao, Y. Liu / Journal of Statistical Planning and Inference 141 (2011) 602–609
Table 1 Dynamic evolution of the DLM’s variances. t 1 10 20 50 112 150 198 215 220 224 226
eigC 1t
eigC 2t
ft
Yt
Vt
eigW 1t
eigW 2t
107.178 551.938 1047.065 2379.371 3959.302 4328.33 4512.631 4542.612 4549.028 4553.868 4556.123
0.112 0.349 0.339 0.333 0.33 0.33 0.329 0.327 0.329 0.329 0.329
1213.5 1314.504 1254.273 1782.688 5900.189 3472.211 2632.496 3039.586 3201.609 3430.892 3436.467
1101.88 1171.86 1090.19 1605.71 5312.18 2868.799 2632.929 2838.838 2995.849 3096.259 3247.319
101 321 311 305 302.786 302.333 302.01 300.07 301.909 301.893 301.885
26.972 18.967 19.492 19.799 19.91 19.933 19.949 20.046 19.954 19.955 19.956
63.028 49.033 49.508 49.801 49.911 49.933 49.95 50.047 49.955 49.955 49.956
7000 price forecast 95% credible intervals
6000
5000
4000
3000
2000
1000
0
50
100
150
200
250
Fig. 4. The forecast price and true price of the SSE Composite Index.
convergent to 0.329, the other one is about 4560. Fig. 2 illustrates the first one and Fig. 3 shows the second. Table 1 shows the values of the error variances Vt and the eigenvalues of the error variances matrices Wt. The variables eigC 1 and eigC 2 are values of the posterior variance Ct ’s eigenvalues. To economize the length, we choose 11 times to explain the convergent process. From the above results we can clearly see a convergent limit of the posterior variance Ct of the dynamic parameter vector of a dynamic linear model (DLM) with non-constant error variances Vt and Wt. It was successfully validated that Ct -C when Vt -V and Wt -W. Fig. 4 shows the forecast effect of the model which we used. From the second half of the time series, we found that the forecast price of The SSE Composite Index is quite accordant to the true price. It proves that the example is right and the DLM succeeded.
4. Conclusion This paper extends the limiting results of West and Harrison (1997, section 5.5) about the convergence of the variances of time series dynamic linear models (TSDLMs) when both, the variances of the observation and evolution errors of the model, are time-varying with steady limits. Now, the similarity results perhaps can be extended when all components {Ft Gt Vt Wt} are time-varying with steady limits. We are taking into account that the components fFt Gt g are timevarying with 0 r Ft r F and 0 rGt r G. It would be more interesting if we can extend the similarity results. References Harrison, P.J., Stevens, C., 1976. Bayesian forecasting (with discussion). Journal of the Royal Statistical Society (Series B) 38, 205–247. Harrison, P.J., 1985. First-order constant dynamic models. Research Report 66, Department of Statistics, University of Warwick. Harrison, P.J., 1996. Convergence and the constant dynamic linear model. Journal of Forecasting 16, 287–292. Kalman, R.E., 1963. New methods in Wiener filtering theory. In: Boganoff, J.L., Kozin, F. (Eds.), Proceeding of the First Symposium in Engineering Applicational of Random Function Theory and Probability. Wiley, New York.
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