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Abstract
In the paper, the autoregressive moving average model for matrix time series (MARMA) is investigated. The properties of the MARMA model are investigated by using the conditional least square estimation, the conditional maximum likelihood estimation, the projection theorem in Hilbert space and the decomposition technique of time series, which include necessary and sufficient conditions for stationarity and invertibility, model parameter estimation, model testing and model forecasting.
1. Introduction
Matrix time series is a time series whose cross-sectional data are matrices, which can be found in a variety of fields such as economics, business, ecology, psychology, meteorology, biology and fMRI (Samadi, Citation2014). For example, consider two stocks, and
, as potential investment products, whose prices and volumes are selected as two analysis factors. Denote the price and volume of stock
at time t by
and
, k = 1, 2, and then a
-dimensional matrix time series can be constructed as follows:
Matrix time series has attracted a few scholars' attention and research at the beginning of the century. Walden and Serroukh (Citation2002) studied the construction of matrix-valued filters for multi-resolution analysis of matrix time series. Samadi (Citation2014) brought forward and investigated a p-order autoregressive model for matrix time series, which is essentially a VAR(p) model in matrix form. D. Wang et al. (Citation2019) proposed a novel factor model
where
and
are matrix time series. Chen et al. (Citation2021) first proposed one-order autoregressive model for matrix time series in the bilinear form, denoted by MAR(1),
(1)
(1) and investigated its stationarity, causality, method of parameter estimation, and asymptotics of statistic. Wu and Hua (Citation2022) independently proposed the p-order autoregressive model for matrix time series in the bilinear form, denoted by MAR(p),
(2)
(2) and presented parameter estimation, model identification criterion and model checking. For more literature studies on matrix time series, one can refer to H. Wang and West (Citation2009), Zhou et al. (Citation2018), Getmanov et al. (Citation2021) and their references.
It is widely known that the autoregressive moving average model of time series (ARMA) plays a very important role in the theory and the application of one-dimensional time series, and we will show later that a bilinear model has its unique advantages for matrix time series. In the paper, autoregressive moving average models for matrix time series (MARMA) are first proposed and investigated. Necessary and sufficient conditions for stationarity of MARMA are provided, and parameter estimations are also considered by the conditional least squares method and the conditional maximum likelihood estimation method. At last, an example is presented to show the applications of the MARMA model.
2. Preliminaries
Let be a probability space with a σ-filtration
in which the second moment of each variable exists, and
.
Definition 2.1
For any given positive integers m and n, an -dimensional matrix time series refers to
(3)
(3) where
is a one-dimensional time series on a probability space
for any
and
.
Definition 2.2
Let be an
-dimensional matrix time series defined by (Equation3
(3)
(3) ), and then its mean function follows as
(4)
(4) Additionally, its autocovariance function follows as
(5)
(5) where
,
and
;
, and
is the vectorization of
by columns, that is,
(6)
(6)
Stationarity and matrix white noise play a very important role on time series analysis. Thus, we will introduce the concept of stationary matrix time series and matrix white noise in the following.
Definition 2.3
Let be a matrix time series defined by (Equation3
(3)
(3) ) and
be the vectorization of
defined by (Equation6
(6)
(6) ). Then
is a stationary matrix time series if and only if
is stationary.
Definition 2.4
For any given positive integers m and n, denote an -dimensional matrix time series
, and then ε is called an
-dimensional matrix white noise, if it satisfies the following conditions.
Its mean function
for all
, where
is the
-dimensional zero matrix.
Its autocovariance function
defined by Definition 2.3 satisfies that
where
is the
-dimensional zero matrix, and
(7)
(7) is the
-dimensional diagonal matrix with diagonal entries
,
, …,
,
,
, …,
,
.
For any matrix white noise , if its vectorization by columns,
, is Gaussian, then
is called a matrix Gaussian white noise.
Property 2.1
For any -dimensional matrix time series
, it is an
-dimensional matrix white noise if and only if
is an mn-dimensional vector white noise, where
.
The proof of Property 2.1 is not difficult, so we omit it.
When we investigate the autoregressive moving average model for matrix time series, we may use the Kronecker product, matrix reshape and derivative of matrix. Thus, we introduce them in the following.
Definition 2.5
Graham, Citation2018
Assume matrices and
, and then the
block matrix
is called the Kronecker product of A and C, denoted by
, that is,
Definition 2.6
For any and positive integers p, q satisfying pq = mn, the
-order reshaped matrix of A, denoted by
, is defined by
where
for all
,
and
, where
is the operator of taking the integer part.
Definition 2.7
Graham, Citation2018
Let and
be two matrices, where m, n, p and q are natural numbers. The derivative of matrix F with respect to matrix X is defined by
where the derivative of matrix F with respect to scalar
is defined by
For the derivative of matrix with respect to matrix, its product rule and two common formulas follow as Properties 2.2 and 2.3.
Property 2.2
Graham, Citation2018
For any ,
and
, it follows that
where
is the
-dimensional identity matrix.
Taking and
into Property 2.2, we obtain Corollary 2.1.
Corollary 2.1
For any and
, it follows that
Property 2.3
Graham, Citation2018
For any ,
and invertible
, it follows that
and
Taking and
into Property 2.3, we obtain Corollary 2.2.
Corollary 2.2
For any and invertible
, it follows that
3. Autoregressive moving average model for matrix time series
The autoregressive moving average model for matrix time series is an extension of the vector autoregressive moving average model (VARMA) to matrix time series. However, we cannot build the autoregressive moving average model for matrix time series like the VARMA model as follows:
(8)
(8) The reason is that the form of (Equation8
(8)
(8) ) cannot describe the dependent relation between the different columns of
according to the rule of matrix multiplication. That is, the ℓth column of
will not be affected by the sth column of
as
.
3.1. ![](//:0)
model
In this section, an autoregressive moving average model for matrix time series (MARMA) is first brought forward, whose degradation model, autoregressive model for matrix-valued time series (MAR), is just the model (Equation2(2)
(2) ) proposed by Wu and Hua (Citation2022) and the extension of model (Equation1
(1)
(1) ) proposed by Chen et al. (Citation2021).
Definition 3.1
Let be an
-dimensional matrix time series. If X is stationary and for each
it follows that
(9)
(9) where C is an
-dimensional matrix;
and
are
-dimensional matrices, and
and
are
-dimensional matrices for each
and
, where p and q are two nonnegative integers;
is an
-dimensional matrix white noise satisfying that
is independent with
for all s<t, and then
is said to follow a
-order autoregressive moving average model for matrix time series, denoted by MARMA
.
When q = 0, MARMA model (Equation9
(9)
(9) ) degenerates into the form
(10)
(10) which is a p-order autoregressive model for matrix time series, MAR(p).
When p = 0, MARMA model (Equation9
(9)
(9) ) degenerates into the form
(11)
(11) which is called a q-order moving average model for matrix time series, denoted by MMA(q).
If is an
-dimensional matrix time series defined by (Equation3
(3)
(3) ) and X is stationary, denote
and then it follows from MARMA
model (Equation9
(9)
(9) ) that
(12)
(12) Denote
It yields from (Equation12
(12)
(12) ) and MARMA
model (Equation9
(9)
(9) ) that
(13)
(13) holds for all
, and then
is said to follow a
-order centralized MARMA
model.
Because every MARMA model can be changed into a centralized MARMA
model and they have the same coefficient parameters. Thus, while estimating coefficient parameters of MARMA
model (Equation9
(9)
(9) ) we will mainly study centralized MARMA
model (Equation13
(13)
(13) ).
For any MARMA model (Equation9
(9)
(9) ), and for any
and
,
and
, it follows that
that is, coefficient parameters of MARMA
model (Equation9
(9)
(9) ) are not unique! Thus, we present constraint conditions that
(14)
(14) and
(15)
(15) for all
and
.
3.2. Relationship between MARMA model and VARMA model
When the column number of matrix equals one, i.e., n = 1, MARMA
model (Equation9
(9)
(9) ) degenerates into a
-order vector autoregressive moving average model, VARMA
, as follows:
(16)
(16) where
is an m-dimensional vector time series, C is an m-dimensional vector,
and
are
-dimensional matrices for all
and
, and
is a white noise of the m-dimensional vector time series satisfying that
is independent with
for all s<t. Obviously, VARMA model (Equation16
(16)
(16) ) is a special case of MARMA model (Equation9
(9)
(9) ).
On the other hand, for any -dimensional matrix time series
, its vectorization
is an
-dimensional time series, and the
-order vector autoregressive moving average model VARMA
for
follows as
(17)
(17) where
is an
-dimensional vector;
and
are
-dimensional matrices for
and
; and
is an
-dimensional white noise satisfying that
is independent with
for all s<t.
A natural question is why the authors still bring forward MARMA model (Equation9
(9)
(9) ) for
but directly use VARMA
model (Equation17
(17)
(17) ) for
.
In fact, there are two important reasons that the authors propose MARMA model (Equation9
(9)
(9) ) for
. Firstly, MARMA
model (Equation9
(9)
(9) ) for
can reveal the information structure of
very clearly. Secondly, MARMA
model (Equation9
(9)
(9) ) for
can reduce model parameters more greatly than VARMA
model (Equation17
(17)
(17) ) for
. In fact, the parameter number of MARMA
model (Equation9
(9)
(9) ) for
is
. However, the parameter number of VARMA
model (Equation17
(17)
(17) ) for
is
. Generally,
For example, if p = q = 1 and m = n = 10, then
In today's big data era, m and n are often very large, taking m = n = 100 and p = q = 1 as an example, and then
Remark 3.1
MARMA model (Equation9
(9)
(9) ) greatly reduces model parameters compared with VARMA
model (Equation17
(17)
(17) ).
Although it is not a good idea to replace MARMA model (Equation9
(9)
(9) ) with VARMA
model (Equation17
(17)
(17) ), in the following we will show there exists a special VARMA
model equivalent to MARMA
model, which will play a very important role in theoretical analysis of MARMA
model (Equation9
(9)
(9) ).
Theorem 3.1
MARMA model (Equation9
(9)
(9) ) for
is equivalent to VARMA
model (Equation18
(18)
(18) ) for
as follows:
(18)
(18) where
and
represent the vectorization of matrices
and
by columns, and ⊗ is the Kronecker product.
Theorem 3.1 can be proved by the following equivalence relation: for any matrices ,
,
and
, it follows that
The equivalence relation is not difficult to prove, so we omit the proof and that of Theorem 3.1.
3.3. Stationary and invertible conditions for MARMA model
According to Theorem 3.1, any MARMA model (Equation9
(9)
(9) ) can be converted into its corresponding VARMA
model (Equation18
(18)
(18) ). Furthermore, VARMA
model (Equation18
(18)
(18) ) can be rewritten as
(19)
(19) where
(20)
(20)
(21)
(21) and B is the delay operator, i.e.,
holds for all
.
Theorem 3.2
For MARMA model (Equation9
(9)
(9) ), the necessary and sufficient conditions for stationarity are that any root λ of (Equation22
(22)
(22) ) is in the unit circle, where
(22)
(22) The necessary and sufficient conditions for invertibility are that any root λ of (Equation23
(23)
(23) ) is in the unit circle, where
(23)
(23)
The proof of Theorem 3.2 is presented in Appendix 1.
Corollary 3.1
For MAR(p) model (Equation10(10)
(10) ), the necessary and sufficient conditions for stationarity are that any root λ of (Equation22
(22)
(22) ) is in the unit circle.
Remark 3.2
Corollary 3.1 expands Proposition 1 in Chen et al. (Citation2021).
Corollary 3.2
For MMA(q) model (Equation11(11)
(11) ), the necessary and sufficient conditions for invertibility are that any root λ of (Equation23
(23)
(23) ) is in the unit circle.
3.4. Parameter estimation for MARMA model
In the section, we will present the conditional least square method and the conditional maximum likelihood estimation method for MARMA model (Equation9
(9)
(9) ).
Let be a series of samples of the centralized matrix time series
defined by (Equation3
(3)
(3) ) with
, where
(24)
(24) where the integer N is the sample length.
When the coefficient parameters of MARMA model (Equation9
(9)
(9) ) have been obtained, it follows from (Equation12
(12)
(12) ) that
and then the constant matrix C of MARMA
model (Equation9
(9)
(9) ) can be estimated as follows:
where
Thus, in the following we always assume the samples come from a centralized MARMA
model (Equation9
(9)
(9) ), i.e.,
.
We use VARMA model (Equation19
(19)
(19) ) with
, equivalent to centralized MARMA
model (Equation9
(9)
(9) ), to estimate the coefficient parameters of MARMA
model (Equation9
(9)
(9) ) by the conditional least square method.
It yields from (Equation19(19)
(19) ) with
that
(25)
(25) where
is the inverse operator of
, and
For the sake of briefness, denote
(26)
(26) and
where we stipulate that
(27)
(27) It follows from (Equation26
(26)
(26) ) that
, which means that
(28)
(28) It yields from (Equation28
(28)
(28) ) and (Equation27
(27)
(27) ) that
(29)
(29) where
.
In summary, centralized MARMA model (Equation9
(9)
(9) ), i.e.,
, is equivalent to VARMA
model (Equation30
(30)
(30) ).
(30)
(30) where
,
, are given by (Equation29
(29)
(29) ).
Theorem 3.3
According to the conditional least square method, the parameters of MARMA model (Equation9
(9)
(9) ) satisfy the following matrix differential equations:
where
is given by (Equation29
(29)
(29) ).
The proof of Theorem 3.3 is presented in Appendix 2.
Corollary 3.3
According to the conditional least square method, the parameters of MAR(p) model (Equation10(10)
(10) ) satisfy the following matrix differential equations:
Theorem 3.4
Assume the innovations are Gaussian with the mean and covariance matrix
. According to the conditional maximum likelihood estimation method, the parameters of centralized MARMA
model (Equation9
(9)
(9) ) satisfy the following matrix differential equations:
where
and
is given by (Equation29
(29)
(29) ).
The proof of Theorem 3.4 is presented in Appendix 3.
Corollary 3.4
Assume the innovations are Gaussian with the mean and covariance matrix
. According to the conditional maximum likelihood estimation method, the parameters of centralized MAR(p) model (Equation10
(10)
(10) ) satisfy the following matrix differential equations:
where
.
Remark 3.3
The matrix differential equations in Theorems 3.3 and 3.4 are very complex. Especially, the coefficients in (Equation29
(29)
(29) ),
, are defined by a series of recursions, whose implied parameters are to be estimated. Thus, it is difficult to obtain its closed solution, but its approximate solutions can be obtained by the numerical computation method.
3.5. Hypothesis testing for the MARMA model
Let be a series of samples of the centralized matrix time series
defined by (Equation3
(3)
(3) ) with
and
for all
. Additionally assume
are Gaussian. In the section, we will test whether
follow MARMA
model (Equation9
(9)
(9) ).
The null hypothesis and the alternative hypothesis follow as
When holds, denote
(31)
(31) where
. It follows from Corollary 5.3 (Karl & Simar, Citation2015) that
where
(32)
(32) It follows from Theorem 5.9 (Karl & Simar, Citation2015) that
that is,
Summarize the above deduction and we obtain Theorem 3.5 for the hypothesis testing on MARMA
model (Equation9
(9)
(9) ).
Theorem 3.5
For any given significance level , if
or
, then reject
following MARMA
model (Equation9
(9)
(9) ); otherwise, accept
following MARMA
model (Equation9
(9)
(9) ), where
and
,
,
are given by (Equation32
(32)
(32) ) and (Equation31
(31)
(31) ).
3.6. Forecasting for the MARMA model
Let be an
-dimensional matrix time series defined by (Equation3
(3)
(3) ) following MARMA
model (Equation9
(9)
(9) ), equivalently,
following VARMA
model (Equation18
(18)
(18) ), that is,
(33)
(33) where
is an
-dimensional matrix white noise.
Denote the forecasting for under the condition that
have been known by
, which refers to the ℓth step forecasting. It follows from (Equation33
(33)
(33) ) and the projection theorem in Hilbert space that
(34)
(34) where
It yields from the equivalence relation of MARMA
model (Equation9
(9)
(9) ) and VARMA
model (Equation18
(18)
(18) ) that
(35)
(35) where
In the following, we will study the interval estimation of MARMA
model (Equation9
(9)
(9) ) and assume the innovations are Gaussian. Equivalently,
follows VARMA
model (Equation19
(19)
(19) ), that is,
where
and
are defined by (Equation20
(20)
(20) ) and (Equation21
(21)
(21) ), and
is a vector white noise.
Denote
and then
(36)
(36) where
(37)
(37) with
.
For any , it follows from (Equation36
(36)
(36) ) and the estimation method of
that
(38)
(38) and then
(39)
(39) For any given
, it yields from (Equation39
(39)
(39) ) that the confidence interval of
with confidence level
follows as
where
refers to the vector composed by all main diagonal elements, and
means taking the square roots of every elements. It yields from the equivalence relation of MARMA
model (Equation9
(9)
(9) ) and VARMA
model (Equation19
(19)
(19) ) that the confidence interval of
with confidence level
follows as
In summary, we can obtain the following results.
Theorem 3.6
Assume follows MARMA
model (Equation9
(9)
(9) ).
(1) For any , the ℓ-step point estimation follows as
where
(2) For any
and
, the ℓ-step interval estimation with confidence level
follows as
where
is the
level lower quantile of standard normal distribution,
the reshape function by Definition 2.6,
the vector composed by all main diagonal elements,
takes the square roots of every elements, and
3.7. Supplementary notes for the MARMA model
3.7.1. Model identification for the MARMA model
According to Theorem 3.1, MARMA model (Equation9
(9)
(9) ) is equivalent to VARMA
model (Equation18
(18)
(18) ). Thus, we can use the model identification method for the VARMA model to identify the order of MARMA model, such as
or alternatively,
where N is the length of observation sequence and
is the logarithm likelihood function.
3.7.2. MARIMA model
For any matrix time series defined by (Equation3
(3)
(3) ), the difference operator Δ for matrix time series follows as
(40)
(40) and Δ defined by (Equation40
(40)
(40) ) has the same effect as the difference operator for vector time series. That is, if
is nonstationary, then we can try to eliminate nonstationarity by Δ defined by (Equation40
(40)
(40) ). If there exists a positive integer d such that
is stationary but
is nonstationary, and
follows a MARMA
model (Equation9
(9)
(9) ), then
is called to follow a
-order autoregressive integrated moving average for matrix time series, and denoted by MARIMA
.
4. An application of the MARMA model
In this section, we will try to model the time series of daily closing prices and daily volumes of Haitong Securities Company Limited (Abbreviated as Haitong Securities; Stock code: 600837) and Ping An Insurance (Group) Company of China, Ltd. (Abbreviated as Ping An; Stock code: 601318). The data are downloaded from the China Stock Market & Accounting Research Database (CSMAR), and the time window is from January 2, 2018 to December 31, 2021, which includes 973 records every stock.
For the sake of clarity, we denote the time series by
where
and
are the daily closing price and daily volume of Haitong Securities, and
and
are the daily closing price and daily volume of Ping An.
4.1. Data preprocessing
We first conduct the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test, i.e., ‘kpsstest’ function in the software MATLAB R2020b, to test the stationarity of the daily closing prices and daily volumes of Haitong Securities and Ping An, and the results show that the daily closing prices and daily volumes of Haitong Securities and Ping An are nonstationary.
In the following we will consider the logarithmic rates (log rate) of daily closing prices and daily volumes of Haitong Securities and Ping An. Denote
(41)
(41) where
That is,
is the logarithmic rate of daily closing price of Haitong Securities,
the logarithmic rate of daily volume of Haitong Securities,
the logarithmic rate of daily closing price of Ping An and
the logarithmic rate of daily volume of Ping An.
We conduct the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test, i.e., ‘kpsstest’ function in the software MATLAB R2020b, to test the stationarity of the logarithmic rates of daily closing prices and daily volumes of Haitong Securities and Ping An, and the results show that the logarithmic rates of daily closing prices and daily volumes of Haitong Securities and Ping An are stationary.
Additionally, we conduct a Ljung-Box Q test, i.e., ‘lbqtest’ function in the software MATLAB R2020b, to test the pure randomness of the logarithmic rates of daily closing prices and daily volumes of Haitong Securities and Ping An, and the results show that the logarithmic rates of daily closing prices or daily volumes of Haitong Securities and Ping An are not purely random.
In conclusion, for the stocks of Haitong Securities and Ping An, their daily closing prices and daily volumes are nonstationary, but their logarithmic rates of daily closing prices and daily volumes are stationary, and their logarithmic rates of daily closing prices or daily volumes are not purely random.
4.2. Modelling of ![](//:0)
![](//:0)
We use the Bayesian information criterion (BIC) to select the model, and the results show that MARMA(4,0) is the best. Using the conditional least square method and MATLAB R2020b program, we establish MARMA(4,0) model for by (Equation41
(41)
(41) ) as follows:
(42)
(42) where
and
and then the covariance matrix of residuals
follows as
(43)
(43)
4.3. Evaluation on ![](//:0)
![](//:0)
For the sake of saving space, we will not show the model test, model optimization or forecasting of MARMA(4,0) model (Equation42(42)
(42) ), but present a comparison of the MARMA model and ARMA model in this subsection. We first establish ARMA
model for
and
respectively, and obtain their models as follows:
(44)
(44) where the covariance matrix of residuals
follows as
(45)
(45) It follows from (Equation43
(43)
(43) ) and (Equation45
(45)
(45) ) that the residuals of MARMA(4,0) model (Equation42
(42)
(42) ) are almost consistently less than those of ARMA(4,0) model (Equation44
(44)
(44) ).
In practice, we are more concerned about the residual variance, i.e., the variance of every element of residual. Using (Equation43(43)
(43) ) and (Equation45
(45)
(45) ), we compute the relative change of the residual variance of MARMA(4,0) model (Equation42
(42)
(42) ) to the residual variance of ARMA(4,0) model (Equation44
(44)
(44) ) as follows:
That is, MARMA(4,0) model (Equation42
(42)
(42) ) reduces all residual variance relative to ARMA(4,0) model (Equation44
(44)
(44) ). Especially, the relative change of volume's residual variance exceeds
by MARMA(4,0) model (Equation42
(42)
(42) ) relative to ARMA(4,0) model (Equation44
(44)
(44) ), which means the MARMA model could really improve the prediction accuracy.
5. Conclusion
We proposed an autoregressive moving average model for matrix time series (MARMA), which is an extension of the autoregressive model for matrix time series (MAR). Like the MAR model, the MARMA model retains the original matrix structure, and provides a much more parsimonious model, compared with the approach of the vector autoregressive model for vectorizing the matrix into a long vector. Compared with MAR model, MARMA models are capable of modelling the unknown process with the minimum number of parameters.
As for MARMA model, the necessary and sufficient conditions for stationarity and invertibility are established. Parameter estimation methods are investigated for the conditional least square method and the conditional maximum likelihood estimation method. Point forecasting and interval forecasting are presented by using the projection theorem in the Hilbert space and the decomposition technique of time series. Additionally, model identification, model testing and possible extensions are discussed.
There are many directions to extend the scope of the MARMA model. Random environment such as the Markov environment might be imposed on the MARMA model to depict the impact of environmental change. Additionally, sparsity or group sparsity might be imposed on coefficient matrices to reach a further dimension reduction. Furthermore, the idea of MARMA can be applied for yield modelling, volatility modelling, weather forecast modelling and animal migration modelling.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Funding
References
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Appendices
Appendix 1. Proof of Theorem 3.2
In order to obtain stationary conditions and invertible conditions for MARMA model (Equation9
(9)
(9) ), we first give a lemma as follows.
Lemma A.1
For any square matrices , the operator
is invertible if and only if any root λ of (EquationA1
(A1)
(A1) ) satisfies
, where k is a natural number and B is the delay operator.
(A1)
(A1)
Proof.
For the polynomial with k degree and matrix coefficients
it can be factorized into k linear polynomials with the matrix coefficient in the complex field as follows:
where
are determined by
(A2)
(A2) Thus,
(A3)
(A3) For any
, it is easy to prove that
is invertible if and only if
, that is, all roots of
are in the unit circle. It follows from (EquationA3
(A3)
(A3) ) that
is invertible if and only if all
,
, are invertible. Thus,
is invertible if and only if all roots of
are in the unit circle for all
. According to determinant properties,
is invertible if and only if all roots of
(A4)
(A4) are in the unit circle. It yields from (EquationA2
(A2)
(A2) ) that
Thus,
is invertible if and only if all roots of
are in the unit circle.
Proof
Proof of Theorem 3.2
For VARMA model (Equation19
(19)
(19) ),
It follows from the concept of stationarity that the necessary and sufficient conditions of stationarity are that the operator
is invertible. According to Lemma A.1, the operator
is invertible if and only if any root λ of (Equation22
(22)
(22) ) satisfies
. Thus, VARMA
model (Equation19
(19)
(19) ) is stationary if and only if any root λ of (Equation22
(22)
(22) ) satisfies
. Note that VARMA
model (Equation19
(19)
(19) ) is equivalent to MARMA
model (Equation9
(9)
(9) ), so MARMA
model (Equation9
(9)
(9) ) is stationary if and only if any root λ of (Equation22
(22)
(22) ) satisfies
.
The necessary and sufficient conditions for invertibility can be obtained by the similar method to obtain the necessary and sufficient conditions for stationarity, so we omit it.
Appendix 2. Proof of Theorem 3.3
Noting that is an
-dimensional white noise, and the objective function of VARMA
model (Equation30
(30)
(30) ) using the conditional least square method follows as
(A5)
(A5) where we take
for all
.
Lemma A.2
defined by (EquationA5
(A5)
(A5) ) has the minimum value about
,
,
and
for all
and
.
Proof.
It yields from analysing (Equation29(29)
(29) ) that
by (EquationA5
(A5)
(A5) ) is a multivariate polynomial of
,
,
and
for all
and
. And it is obvious that
by (EquationA5
(A5)
(A5) ) is greater than or equal to zero, which means that
by (EquationA5
(A5)
(A5) ) has lower bound. Thus,
by (EquationA5
(A5)
(A5) ) has the minimum value about
,
,
and
for all
and
.
Proof
Proof of Theorem 3.3.
It follows from Lemma A.2 that, according to the conditional least square method, the parameters of MARMA model (Equation9
(9)
(9) ) satisfy the following matrix differential equations:
Using the derivative of scalar by matrix, it yields from Corollary 2.1 that
Appendix 3. Proof of Theorem 3.4
It yields from (Equation30(30)
(30) ) that
(A6)
(A6) For the sake of briefness, we denote
It yields from (EquationA6
(A6)
(A6) ) that
(A7)
(A7) where
is defined by (Equation7
(7)
(7) ).
Let for all
. It follows from (EquationA7
(A7)
(A7) ) that
(A8)
(A8) and
(A9)
(A9) Thus, the maximum likelihood function of
follows as
where
means the probability density function, and we stipulate
equals zero vector or zero matrix as needed. Therefore, the logarithm maximum likelihood function of
follows as
(A10)
(A10) Using the derivative of scalar by matrix, it yields from (EquationA10
(A10)
(A10) ) that
(A11)
(A11) where
. It yields from Corollary 2.2 and Property 2.3 that