Product of bi-dimensional VAR(1) model components. An application to the cost of electricity load prediction errors

Special cases analysis – Case 2

The formula for E ⁢ ( Y ⁢ ( t ) ) given in equation (4.5) follows directly from equation (3.3). Using equation (3.4), we can calculate the ACVF for { Y ⁢ ( t ) } for h = 0 , 1 , … . Indeed, we have

Now, we can calculate the value

for all t ∈ Z and i , j = 0 , 1 , … , m , p = − h , − h + 1 , … . Using the fact that, for each t ∈ Z , the bi-dimensional residual series Z ⁢ ( t ) is a zero-mean vector and, for t ≠ s , Z ⁢ ( t ) is independent of Z ⁢ ( s ) , one obtains the following:

Thus, we have

where the value m Z = E ⁢ [ Z 1 2 ⁢ ( t ) ⁢ Z 2 2 ⁢ ( t ) ] is independent of 𝑡. Therefore, we have

Finally, taking h = 0 , one obtains V ⁢ ar ⁢ ( Y ⁢ ( t ) ) .

Special cases analysis – Case 3

One can show that, in this case, we have

and the eigenvalues of the matrix Φ are equal to ν 1 = ϕ 11 and ν 2 = ϕ 22 . Thus, according to equations (2.5) and (2.6), the following is fulfilled for j = 1 , 2 , … :

while for j = 0 , ϕ 12 ( j ) = 0 . In order to fulfill condition (2.3), we assume that | ϕ 11 | < 1 and | ϕ 22 | < 2 .

Using equation (2.8), one obtains

Thus, we have

Moreover, from equation (2.9), we have

Thus, from the above, we obtain the formula for the expected value of the random variable Y ⁢ ( t ) for each t ∈ Z ; see equation (4.7).

To obtain the explicit formula for ACVF Y ⁢ ( h ) , we will use equation (3.4). For h = 0 , 1 , 2 ⁢ … , we have

Moreover, the value

is given by

where κ Z = E ⁢ [ Z 2 4 ⁢ ( t ) ] < ∞ is independent of 𝑡. Additionally, one can show that

Now, to make the calculations simpler, let us assume that ϕ 11 = 0 and ϕ 22 ≠ 0 . In this case, the matrix Φ (see equation (2.2)) has two different eigenvalues and ϕ 12 ( j ) = ϕ 12 ⁢ ϕ 22 j − 1 , j = 1 , 2 ⁢ … . Clearly, ϕ 11 ( j ) = 0 for j = 1 , 2 , … , and for j = 0 , ϕ 11 j = 1 . We have the following:

Let us first consider the case h = 0 . We have

On the other hand, for h > 0 , we have

Bivariate Gaussian distribution

The bivariate Gaussian distributed random vector ( Z 1 , Z 2 ) has the following PDF [36]:

where ρ ∈ ( − 1 , 1 ) is the correlation coefficient between random variables Z 1 and Z 2 (denoted in the main text as ρ Z ); μ Z , 1 , μ Z , 2 ∈ R are the corresponding expected values, while σ Z , 1 2 , σ Z , 2 2 > 0 are the corresponding variances. When ρ = 0 , the PDF of the random vector ( Z 1 , Z 2 ) is just the product of the PDFs of the Gaussian distributed random variables.

Bivariate Student’s t distribution

The bivariate Student’s t distributed random vector ( Z 1 , Z 2 ) is constructed as follows. Let us assume that ( N 1 , N 2 ) is the bivariate Gaussian vector defined by the PDF in equation (A.1) with expected values equal to zero, unit variances and ρ ∈ ( − 1 , 1 ) being its correlation coefficient. Moreover, let χ 2 be the one-dimensional random variable with chi-square distribution with η > 0 degrees of freedom and assume that ( N 1 , N 2 ) and χ 2 are independent. Then the random vector defined as

has a bivariate Student’s t distribution with 𝜂 degrees of freedom and its PDF is given by (see [6])

The marginal random variables Z 1 and Z 2 have the one-dimensional Student’s t distribution defined by the following PDF [11]:

where Γ ⁢ ( ⋅ ) is the gamma function, i.e. Γ ⁢ ( α ) = ∫ 0 ∞ t α − 1 ⁢ e − t ⁢ d t for 𝛼 such that Re ⁡ ( α ) > 0 . Note that the number of degrees of freedom, 𝜂, is equal for both marginal variables. It is worth highlighting that the correlation ρ Z between the random variables Z 1 and Z 2 is equal to the parameter 𝜌. However, its zero value is not equivalent to the independence of the random variables Z 1 and Z 2 since, in that case, the PDF of a random vector ( Z 1 , Z 2 ) (see equation (A.2)) is not a product of the PDFs of the marginal distributions; see equation (A.3). Hence, if Z 1 and Z 2 are independent, the PDF of the random vector is given by

where η Z , 1 > 0 , η Z , 2 > 0 are the degrees of freedom parameters of Z 1 and Z 2 , respectively.

The Student’s t distribution defined in (A.3) has zero mean and variance equal to σ Z , 1 2 = η η − 2 . It can be generalized to the Student’s t location-scale distribution by applying the following transformation Z ( μ , λ ) := μ + λ ⁢ Z , where 𝑍 is Student’s t distributed. It yields a three parameter ( μ , λ , η ) distribution, with 𝜇 being the shift parameter, λ > 0 the scale parameter and η > 0 the degrees of freedom. The variance of the Student’s t location-scale random variable is equal to σ Z ( μ , λ ) 2 = λ 2 ⁢ η η − 2 .

Bivariate normal inverse Gaussian distribution

A bivariate normal inverse Gaussian distribution is constructed as a variance-mean mixture of a two-dimensional Gaussian random vector with a univariate inverse Gaussian distributed mixing variable [29]. The density of a bivariate NIG variable Z = ( Z 1 , Z 2 ) is given explicitly by (see [3])

where

K d ⁢ ( x ) is the modified Bessel function of the second kind with index 𝑑 and δ > 0 , α 2 > β T ⁢ Σ ⁢ β , μ ∈ R 2 , β ∈ R 2 , Σ ∈ R 2 × R 2 are the parameters of the distribution. Parameter 𝛼 controls the shape of the density, 𝛿 is the scale parameter, 𝜷 is the skewness parameter, 𝝁 is the translation parameter and the Σ matrix describes degree of correlations between the components of 𝐙. For more details on the NIG distribution, see e.g. [7, 29, 3, 34].