Multivariate Exponential Smoothing: prediction, signal ...

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Abstract. This paper provides analytical formulae that allow expressing the reduced form parameters of the multivariate exponential smoothing model as ...
Multivariate Exponential Smoothing: prediction, signal extraction and control Giacomo SBRANA∗

Abstract This paper provides analytical formulae that allow expressing the reduced form parameters of the multivariate exponential smoothing model as algebraic functions of the structural form parameters. The results are also valid in the context of common trends when cointegration among trends arises. The analytical results shed light on the crucial role of the signal-noise matrix ratio in filtering and signal extraction. Moreover, the results provides an easy and fast estimation method based on sample covariances. An empirical illustration modeling stock market assets prices shows the importance of this method when large-scale models are employed.

Keywords: Multivariate Exponential Smoothing Model, Prediction, Signal extraction, VARMA, Common trends.

1

Introduction

Exponential smoothing models are widely used in forecasting time series. The generalization of these models in the multivariate context was first discussed by Jones(1966). This paper provides algebraic results that explain how the structural form parameters determines the prediction and signal extraction processes. More specifically, it is shown that the reduced form parameters of the multivariate exponential smoothing model are analytical functions of the shocks belonging to the structural process. These results represent the multivariate generalization of the well known result of the univariate case. In ∗

´ Bureau d’Economie Th´eorique et Appliqu´ee (BETA), Universit´e de Strasbourg, 61 Avenue de la Fˆ oret Noire, 67085 STRASBOURG Cedex, France. E-mail address: [email protected]

1

fact, as shown in Harvey (1989), the reduced form parameters of the univariate exponential smoothing (i.e. random walk plus noise model) are exact functions of the so called “signal-noise ratio” (i.e. q-ratio). In this paper it is shown that in the multivariate case, the reduced form parameters are analytical function of the “signal-noise Q matrix ratio”. These results are valid also in the context of common trends. In fact, when cointegration arises, the reduced form parameters can be still implied analytically. This paper discusses some important consequences of these algebraic linkages. In fact, they shed light on the interpretation of standard filtering and signal extraction formulae in the multivariate context. In particular, it is shown the crucial role played by the eigenvalues belonging to the signal-noise Q matrix for prediction and signal extraction. Moreover, the results provides an easy and fast estimation method (of moments) of the reduced form parameters. In fact, it is shown that these parameters are exact functions of the sample covariance matrices. The use of this estimation method gains importance when modeling large-scale systems. It is in fact well known that the Maximum Likelihood estimator is computationally demanding for high dimensional systems. An empirical exercise focusing on stock market prices data is shown to help clarify this issue. Using a system of eight variables it is shown that both Maximum Likelihood and the suggested method of moments achieve similar results, but the computational costs of using the Maximum Likelihood estimation are high.

2

Results

Consider the standard multivariate exponential smoothing model: yt = μ t +  t μt = μt−1 + ηt

2

(1)

Where yt is a vector of N variables and t = 1, · · · , T . In addition, the shocks have the following properties:  cov

t ηt



 =

Σ 0 0 Ση

 (2)

Where Σ , Ση and 0 are N ×N matrices. Therefore the reduced form is a VARIMA(0,1,1):

Δyt = (I − L)yt = ηt + (I − L)t = (I + ΘL)ξt

with

E(ξt ξt ) = Σξ

(3)

Where L is the backshift operator and I is an N × N identity matrix. Proposition 1. Given (1) and (2), provided that Σ and Ση are both positive definite and defining Q = Ση Σ−1  , the exact linkages between reduced and structural parameters are:   1 Θ = 12 −Q − 2I + (QQ + 4Q) 2 −1   (4) 1 Σ Σξ = − 12 −Q − 2I + (QQ + 4Q) 2 −1 −1 In addition, these expressions hold since the matrix Ση Σ−1  Ση Σ + 4Ση Σ has only

positive eigenvalues, such as it is diagonalizable and therefore it admits the square root. Proof. First of all, the autocovariance functions of the reduced form (3) are: E(Δyt Δyt ) = Γ0 = Σξ + ΘΣξ Θ = Ση + 2Σ  ) = Γ = ΘΣ = −Σ E(Δyt Δyt−1 1  ξ It is relevant noting that Γ1 is a symmetric matrix. In addition, Γ0 can also be expressed as follows: Γ0 = Σξ + Γ1 Σ−1 ξ Γ1 Since Γ1 = ΘΣξ = Σξ Θ . Post-multiplying the previous expression for Γ0 by Σ−1 ξ , we have: −1 −1 −1 Γ0 Σ−1 ξ = I + Γ1 Σξ Γ1 Σξ = I + ΘΘ = Γ0 Γ1 Θ

3

Therefore we have the following quadratic matrix equation: ΘΘ − Γ0 Γ−1 1 Θ+I =0 It can be seen that the previous expression is satisfied whenever: −1 Γ0 Γ−1 1 =Θ+Θ −1 Note that the matrix Γ0 Γ−1 1 = −Ση Σ − 2I is always diagonalizable. In fact, recalling

the following important theorem shown in Horn & Johnson (1985) p. 465: “the product of a positive definite matrix A and an Hermitian matrix B is a diagonalizable matrix, all of whose eigenvalues are real. The matrix AB has the same number of positive, negative and −1 , and, given the assumption zero eigenvalues as B.” Thus, we have that Ση Σ−1  = P DP

of positive definite matrices, all the eigenvalues in D are strictly positive. Therefore, we have that: P (−D − 2I)P −1 = Θ + Θ−1



Θ + Θ−1 = P (G + G−1 )P −1

Where G is the diagonal matrix containing the eigenvalues of Θ. In fact, note that Θ is always diagonalizable. Since, by definition of the autocovariance of order one, Θ = −1 −Σ Σ−1 ξ , and thus, following the theorem of Horn & Johnson, we can express Θ = P GP

. Finally, the equality is satisfied whenever G + G−1 = −D − 2I, that is, post-multiplying both sides by G, whenever GG + (D + 2I)G + I = 0. Given that G and D are diagonal   1 matrices, the eigenvalues of the unknown matrix are G = 2−1 −D − 2I ± (DD + 4D) 2 . √ −di −2± d2i +4di This implies that the i-th element of the diagonal matrix G is gi = . There2 fore, the (usually) non-symmetric solutions are:

Θ=

 1   1 1 1 P (−D − 2I) P −1 ± P D2 + 4D 2 P −1 = −Q − 2I ± (QQ + 4Q) 2 2 2 4

However, the only invertible solution (with all roots of Θ lying outside the unit circle) is the one with positive sign in front of the square root. As a consequence, Σξ = −P G−1 P −1 Σ , implying that −P G−1 P −1 must be a positive definite matrix.

Finally, recalling the theorem as in Horn & Johnson, (4) exists since the matrix −1 −1 Ση Σ−1  Ση Σ + 4Ση Σ is always diagonalizable and it contains only positive eigenvalues.

This implies the existence of the square root matrix and concludes the proof.

The results as in (4) is the multivariate extension of the univariate result as shown in Harvey(1989) p.68. In fact, the matrix Q represents the signal-noise matrix ratio.

The result for Σξ in (4) shows that we have full control of the prediction error covariance matrix of the system given the structural shocks. Furthermore, the expression for Θ in (4) provides an important tool for the interpretation of multivariate filtering and signal extraction. In fact, it is now possible to claim that the eigenvalues of the signal-noise Q matrix ratio play the crucial role in determining how observations should be weighted for prediction and signal extraction. To see this, note that the optimal linear predictor of a future observation in the multivariate exponential smoothing model depends only on Θ. More specifically, the optimal linear predictor of yt+1 given the observations up to time t is: y˜t+1|t = μt+1|t = (I + Θ)(I + ΘL)−1 yt = (I + Θ)



(−Θ)j yt−j

j=0

A proof that (I + Θ)−1 =



j=0 (−Θ)

j

can be found in Abadir & Magnus (2005) p. 249.

Therefore, using the solution as above, the analytical optimal predictor can be expresses 5

such as: ∞ 

y˜t+1|t = μt+1|t = P G∗ P −1 −P Gj P −1 yt−j j=0

where G∗ is a diagonal matrix and its i-th element is gi∗ =



(5)

d2i +4di −di . 2

Then, it can be seen that the more close to zero are the di ’s (i.e. the more close to -1 are the roots of Θ), the more past observations are used in (5) when forecasting the future. On the other hand, the higher the di ’s the less past observations are used in forecasting the future. The case in which one or more di ’s are equal to zero (i.e. when some roots of Θ lie on the unit circle) is discussed in the cointegration section. Similar consequences hold for signal extraction. In fact, the multivariate WienerKolmogorov filter (WK) can be expressed as follows: W (L) = Γμ (L)Γy (L)−1

(6)

Where Γμ (L) and Γy (L) are the autocovariance generating functions of, respectively, μ and y. As shown for example in Proietti(2007), the above matrix can be expressed as a function of the reduced form parameters, performing two-sided exponential smoothing, such that: −1 W (L) = (I + Θ)Σξ (I + Θ )(I + Θ L−1 )−1 Σ−1 ξ (I + ΘL)

(7)

Therefore the autocovariances generating functions as in (6) can be derived analytically such that: 

Γμ (L) = −P G∗ G−1 P −1 Σ P −1 G∗ P 

(8)

and 

−1 Γy (L)−1 = (I + P −1 GP  L−1 )−1 − Σ−1 (I + P GP −1 L)−1  P GP

6

(9)

Therefore, considering (8) and (9) and bearing in mind that gi =

−di −2+

√ 2

d2i +4di

, it

can be clearly seen that when the eigenvalues of the signal-noise Q matrix ratio are close to zero we have that more past and future observations are used when signal extraction is carried out. On the other hand, when the eigenvalues are high the reverse is true and the closest observations receive a bigger weight in the signal extraction. It is worth noting that if we relax the assumption that E(t ηt ) = 0, then (4) does not hold unless E(t ηt ) is a symmetric matrix. In the univariate case the solution always holds as already shown in Harvey & Koopman(2000).

2.1

Commuting matrices

A special case arises when the covariance matrices (Σ and Ση ) commute. In this case the signal-noise Q-matrix ratio as well as Θ are symmetric matrices. Therefore, under this assumption, it is relevant noting that Σ and Ση share the same eigenvectors. As a consequence, it is interesting remarking the evident similarities with the univariate result as shown in Harvey(1989) p.68. To see this, define Σ = P Λ P −1 and Ση = P Λη P −1 with eigenvalues respectively as Λ = diag{λ1 , · · · , λN } and Λη = diag{λη1 , · · · , ληN }. Thus −1 has eigenvalues such that d = the signal-noise matrix Q = P DP −1 = P Λη Λ−1 i  P

In addition the reduced form parameters are:  2 −2 1 −1 2 ]P −1 Θ = P 12 [−Λη Λ−1  − 2I + Λη Λ + 4Λη Λ  2 −2 1 −1 2 ]−1 2Λ P −1 Σξ = −P [−Λη Λ−1   − 2I + Λη Λ + 4Λη Λ

λi ληi .

(10)

Clearly in the commuting case, the roots of both shocks covariance matrices play the crucial role in the interpretation of filtering and signal extraction. In fact, the smaller the eigenvalues of Ση , as well as the higher the eigenvalues of Σ , the more close to -1 are the roots of Θ and therefore the more past observations are discounted in forecasting the future. In signal extraction, given the two-sided smoothing, the smaller the eigenvalues of 7

Ση , as well as the higher the eigenvalues of Σ , the more past and future observations are discounted. Note that, despite that the commuting assumption is a special case, it represents a (much) less restrictive case than the homogeneous-system case as discussed for example in Enns et al.(1982), Harvey(1986) and Harvey & Fernandez(1990). In fact, the homogeneous −1 = qI, that is when Σ = qΣ where q is a scalar. case arises when P Λη Λ−1 η   P

2.2

Common trends

Another important case is when cointegration arises. Consider now the system as in (1) but assume that the matrix Ση is positive semi-definite with rank K < N . Therefore the model contains K common trends (or common levels), such that we can reparametrize it as follows: yt = Ψμ†t + t μ†t = μ†t−1 + ηt†

(11)

where ηt† is a K × 1 vector, Ψ is a N × K matrix of (correlated) factor loadings and Σ†η is a matrix of full rank K. Without loss of generality, the matrix of factor loadings can be reparametrized in a more convenient form such as:

IK Ψ= Ψ The model may be recast in the original multivariate exponential smoothing form (1) with: ⎛

⎞ 0 0 Σ1 Σ12 ⎜ Σ12 Σ2 ⎟ 0 0 t ⎟ =⎜ cov † †  ⎠ ⎝ ηt Ση Ψ 0 0 Ση  0 0 ΨΣ†η ΨΣ†η Ψ Note that the rank of the partitioned matrix Ση is equal to K since: 







† rank(Ση ) = rank(Σ†η ) + rank(ΨΣ†η Ψ − ΨΣ†η Σ†−1 η Ση Ψ ) = K

8

(12)

Therefore Ση has N − K eigenvalues equal to zero while the remaining K eigenvalues are positive. In addition, the reduced form is a VARIMA(0,1,1):

Δyt = (I + Θ† L)ξt

with

E(ξt ξt ) = Σ†ξ

(13)

Thus the following proposition holds: Proposition 2. Given (1) and defining Q† = Ση Σ−1 with shocks as in (12), the exact  linkages between reduced and structural parameters are:   1  Θ† = 12 −Q† − 2I + Q† Q† + 4Q† 2   1 −1  Σ Σ†ξ = − 12 −Q† − 2I + Q† Q† + 4Q† 2

(14)

−1 −1 Moreover Ση Σ−1  Ση Σ + 4Ση Σ has either positive eigenvalues or eigenvalues equal

to zero, such as it is diagonalizable and therefore it admits the square root. Proof. The results follows directly using the proof for Proposition 1. In addition, since Ση is a positive semi-definite Hermitian matrix and Σ is positive definite, recalling the −1 theorem as in Horn and Johnson, it follows that (14) exists since the matrix Ση Σ−1  Ση Σ +

4Ση Σ−1 has K positive eigenvalues and N − K eigenvalues equal to zero, therefore it is  diagonalizable. This implies the existence of the square root matrix. As discussed in Harvey(1989) (chapter 8) the reduced form of the system with common trends is not a strictly invertible VARIMA(0,1,1) process. This is because some roots of Θ† lie on the unit circle. Therefore, another relevant consequence of (4) is the following proposition: Proposition 3. Given (1) with shocks as in (12), the matrix Θ† has K roots lying outside the unit circle and N − K roots lying on the unit circle (being equal to -1). 9



Proof. Recall that Θ† = P † G† P †−1 where gi† =

 † † d†2 +4d −d −2 i i i 2

is the i-th diagonal ele-

† † †−1 . ment of G† and d†i is the i-th diagonal element of the matrix D† since Ση Σ−1  =P D P

It then follows that K elements of d†i are positive while the remaining N − K elements are equal to zero. As a consequence, the gi† will be equal to -1 when its corresponding d†i = 0 When the cointegration arises both prediction and multivariate Wiener-Kolmogorov filter formulae are valid. Yet, given the non strict invertibility of the reduced form, the matrix (−Θ† )j does not collapses to zero as j runs to infinity. In other words, when rank(Ση ) < N , the vector autoregression expression as in (5) contains infinite lags with coefficients that do not decay exponentially. In fact, as j → ∞ we have that N − K roots of the matrix (G† )j remain equal to one while the other K collapse to zero.

3

Moments Estimator

The Maximum Likelihood Estimation of the model as in (1) has been fully discussed and clarified by Fernandez(1990). This is the method generally employed by standard packages such as STAMP 8.2 (Koopman, Harvey, Doornik & Shephard, 2007). The ML estimator has to be preferred, with no doubt, to any potential alternative when the number of dependent variables is relatively small. However, as noted in Harvey and Shephard(1993), the maximisation of the likelihood function for the model as in (1) can be computationally demanding when N is large. In this section we present a potential alternative, easy and fast to implement, based on the sample covariances. Note, in fact, that the results as in (4) can be expressed in terms of moments of the differenced variables of the system, that is:

10

   12 −1 −1 −1 Γ0 Γ1 + Γ0 Γ1 Γ0 Γ1 − 4I Θ=    12 −1 −1 −1 −1 1 Γ1 Σξ = − 2 Γ0 Γ1 + Γ0 Γ1 Γ0 Γ1 − 4I 1 2

(15)

Where Γ0 Γ−1 1 is the inverse of the autocorrelation matrix as defined in Chitturi(1974). Such that (15) represents the multivariate extension of the well known univariate result expressing the moving average parameters as function of the autocorrelation. Thus filtering and signal extraction can be carried out using the sample covariance matrices. In order to impose symmetric matrices the following covariance estimator can be considered such that:  ˆ 1 = − 1 T −1 Δyt Δy  + T −1 Δyt−1 Δy  Γ t t−1 2 ˆ 0 = T −1 Δyt Δyt Γ

(16)

 where T −1 Δyt Δyt−i are the sample covariance matrices. It is important remarking that

the moments estimator as in (16) is the consistent initial estimator described in Fernandez(1990) to initialize the Maximum Likelihood iterations (see the discussion in Section 5 in Fernandez(1990)). The method (15) is fast and very easy to compute and therefore it is particular convenient to be used for modeling high dimensional systems when the ML method might get computationally intractable. This point is fully clarified in the empirical illustration.

3.1

Illustration: New York Stock Exchange

Here an empirical example is presented to better understand the relevant advantages of using (15) when modeling large-scale systems. The aim of the empirical exercise is twofold: first the estimates produced by STAMP (i.e. using the ML estimator) with those using the sample autocovariance matrices are compared. Secondly the computational costs

11

of using the ML estimator is discussed. The multivariate exponential smoothing is employed using the logarithm of price relative to some assets quoted in the New York Stock Exchange (NYSE). More specifically we consider the price (at weekly frequency) of the following eight titles: Exelon Corp.; Texas Instruments Inc.; Johnson & Johnson; Ford Motor Co.; Walgreen Co.; Apache Corp.; Verizon Communications; Caterpillar Inc. The source data is the Yahoo Finance website. The random walk behavior of the series is evident looking at Figure 1. The sample consists of weekly data ranging from the first week of January 1995 to the last week of December 2007, for a total of 677 observations. According to the diagnostics statistics as in Table 1, the exponential smoothing model seems to be appropriate for this exercise. The followˆ and ing matrices report the elements of the estimated reduced form parameters where Θ Σˆξ are the estimates produced by STAMP 8.2 (Koopman, Harvey, Doornik & Shephard, ˜ and Σ˜ξ have been estimated using (15). 2007) while Θ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ˆ Θ=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ˜ =⎜ Θ ⎜ ⎜ ⎜ ⎜ ⎝

−0.0410 −0.0544 0.0238 0.0120 0.0418 0.0703 −0.0139 0.0558

−0.0128 −0.1455 −0.0032 −0.0128 0.0017 0.0300 0.0004 0.0183

−0.0013 −0.0261 −0.1285 −0.0347 −0.0137 −0.0181 0.0235 0.0219

−0.0021 −0.0029 −0.0113 −0.0242 −0.0036 −0.0029 0.0113 0.0199

0.0116 −0.0178 −0.0213 −0.0401 −0.1623 −0.0460 0.0298 −0.0858

⎞ 0.0146 −0.0107 0.0262 0.0552 0.0335 0.0447 ⎟ ⎟ −0.0159 0.0340 0.0132 ⎟ ⎟ −0.0062 0.0286 0.0050 ⎟ ⎟ −0.0212 0.0578 −0.0534 ⎟ ⎟ −0.1406 −0.0262 0.0254 ⎟ ⎟ −0.0218 −0.0880 0.0232 ⎠ 0.0086 0.0319 −0.1290

−0.0408 −0.0388 0.0253 0.0146 0.0432 0.0705 −0.0074 0.0561

−0.0102 −0.1348 −0.0075 −0.0142 −0.0076 0.0301 0.0084 0.0062

0.0018 −0.0434 −0.1107 −0.0421 −0.0175 −0.0154 0.0198 0.0029

−0.0024 −0.0006 −0.0110 −0.0278 −0.0022 −0.0022 0.0106 0.0259

0.0108 −0.0344 −0.0259 −0.0382 −0.1631 −0.0460 0.0180 −0.0715

⎞ 0.0133 −0.0097 0.0241 0.0540 0.0498 0.0214 ⎟ ⎟ 0.0073 ⎟ −0.0160 0.0264 ⎟ −0.0125 0.0224 0.0166 ⎟ ⎟ −0.0243 0.0427 −0.0376 ⎟ ⎟ −0.1432 −0.0379 0.0168 ⎟ ⎟ −0.0278 −0.0780 0.0071 ⎠ −0.0032 0.0105 −0.1041

12

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ −4 Σˆξ = 10 ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ −4 ⎜ ˜ Σξ = 10 ⎜ ⎜ ⎜ ⎜ ⎝

8.4 −0.4 33.7 1.3 0.4 7.8 1.1 6.1 2.2 20.4 1.0 1.8 3.3 2.7 12.9 2.0 3.2 1.3 2.2 0.3 16.7 1.1 2.4 1.6 3.5 1.3 0.1 11.9 1.2 5.1 2.1 6.2 2.2 4.2 1.5 14.6 8.4 −0.4 33.7 1.3 0.3 7.8 1.1 6.1 2.2 20.4 1.0 1.7 3.3 2.7 12.9 2.0 3.2 1.3 2.2 0.3 16.7 1.1 2.5 1.5 3.5 1.2 0.1 11.9 1.2 5.1 2.1 6.3 2.2 4.2 1.4 14.7

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Note that STAMP does not provides the estimation of Θ. However, since STAMP provides the estimation of the covariance matrices of the shocks together with the preˆ = −Σ ˆ Σ ˆ −1 . A diction error covariance matrix (i.e. Σξ ), it is then possible to imply Θ ξ visual inspection of the reduced form parameters estimates reveals that, despite some small discrepancies, the two estimation methods deliver similar results. However, while the implementation of (15) is instantaneous, the ML estimation process carried out by STAMP took about five minutes using an Intel Pentium(R) processor 4 CPU, 3.00 GHZ. For the sake of space and presentation of results, only eight variables were used in the exercise. Yet, using STAMP, an estimation exercise, with for example twenty variables, might take days to deliver the results.

References [1] Abadir K.M. & Magnus J.R. (2005). Matrix Algebra. Cambridge: Cambridge University Press. 13

[2] Chitturi R. V. (1974). Distribution of residual autocorrelation in multiple autoregressive schems. Journal of the American Statistical Association, 69, 348, 928-934. [3] Enns P.G., Machak J.A., Spivey W.A. & Wroblesky W.J (1982). Forecasting Applications of an Adaptive Multiple Exponential Smoothing Model. Management Science, 28, 9, 1035-1044. [4] Fernandez J.(1990). Estimation and Testing of a Multivariate Exponential Smoothing Model. Journal of Time Series Analysis, 11, 2, 89-105. [5] Harvey A.C.(1986). Analysis and Generalization of a Multivariate Exponential Smoothing Model. Management Science, 32, 374-380. [6] Harvey A.C. (1989). Forecasting Structural Time Series and the Kalman Filter. Cambridge: Cambridge University Press. [7] Harvey A.C. & Fernandez J. (1990). Seemingly Unrelated Time Series Equations and a Test for Homogeneity. Journal of Business & Economic Statistics, 8, 71-82. [8] Harvey A.C. & Koopman S.J. (2000). Signal extraction and the formulation of unobserved components models. Econometrics Journal, 3,1, 84-107. [9] Harvey A.C. & Shephard N. (1993). Structural Time Series Models. in Maddala G.S., Rao C.R. and Vinod H.D. (eds), Handbook of Statistics, Volume 11, Elsevier, Amsterdam [10] Horn R.A. & Johnson C.R. (1985). Matrix Analysis. Cambridge: Cambridge University Press.

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[11] Jones R.H. (1966). Exponential Smoothing for Multivariate Time Series Journal of the Royal Statistical Society. Series B (Methodological), 28,1, 241-251. [12] Koopman S.J., Harvey A.C., Doornik J.A. & N. Shephard (2007). STAMP 8.2 Structural Time Series Analyzer, Modeller and Predictor. London: Timberlake Consultants Ltd. [13] Proietti T. (2007), Measuring Core Inflation by Multivariate Structural Time Series Models. in Kontoghiorghes E. J. and Gatu C. (eds), Optimisation, Econometric and Financial Analysis, Springer, pp. 205-226

Table 1: Summary statistics

Exelon Texas J&J Ford Walgreen Apache Verizon Caterpillar Log-likelihood Observations

Std. error

Normality

Q(25)

R2

0.03 0.06 0.03 0.05 0.04 0.04 0.03 0.04

77.5 (*) 83.2 (*) 278.8(*) 179.1(*) 82.9 (*) 9.4 (*) 87.2 (*) 111.9 (*)

26.56 24.64 29.24 35.48 35.47 28.66 35.75 29.18

0.99 0.98 0.99 0.99 0.99 0.99 0.97 0.99

17934.2 677

The normality test statistics is the Bowman-Shenton test statistics with the correction of Doornik and Hansen (1994), distributed under the null hypothesis as a χ2(2) . The Q test statistics of residual autocorrelation is the Box-Ljung statistic. The log-likelihood refers to the multivariate model. In the table (*) denotes rejection of the null hypothesis at 5% level.

15

4.5

4.5

EXELON

TEXAS

4.0

4.0

3.5 3.5 3.0 3.0 2.5 2.5 1996

1998

2000

2002

2004

2006

2008

JNJ

1996

1998

2000

2002

2004

2006

2008 FORD

3.5

4.0 3.0 3.5

2.5

2.0

3.0 1996

4.0

1998

2000

2002

2004

2006

2008

WALGREEN

1996

4.5

3.5

4.0

3.0

3.5

1998

2000

2002

2004

2006

2008

2000

2002

2004

2006

2008

2002

2004

2006

2008

APACHE

3.0 2.5 2.5 1996 4.25

1998

2000

2002

2004

2006

2008

1996 4.5

VERIZON

4.00

1998

CATERPILLAR

4.0

3.75

3.5

3.50

16

3.0

1996

1998

2000

2002

2004

2006

2008

1996

1998

2000

Figure 1: Logarithm of assets prices. Sample: 1/1/1995-31/12/2007. Frequency: weekly. Source: Yahoo Finance.