2. The case $d=2$

In the following lemma, we present a new method for deriving the solutions to the recursive equations involving the MA coefficients.

Lemma 2.1. Let $\mathbf{b}=\{b=b_{k_1,k_2},\,(k_1,k_2)\in\mathbb{Z}^{2++}\}$ , $b_{(0,0)}=1$ , $b_{(k_1,k_2)}=0$ for $k_1<0$ or $k_2<0$ , be a sequence of real numbers. Also assume that $\alpha,\beta,\theta$ and real numbers that are not all zero. Then $\mathbf{b}$ satisfies the system of the recursive equations

(2.1) \begin{equation} b_{k_1,k_2} = \alpha b_{k_1-1,k_2} + \beta b_{k_1,k_2-1} + \theta b_{k_1-1,k_2-1}, \quad (k_1,k_2)\in\mathbb{Z}^{2++}, \end{equation}

if and only if

(2.2) \begin{equation} b_{k_1,k_2}c = c\sum^{\min(k_1,k_2)}_{r=0}{\frac{(k_1+k_2-r)!}{(k_1-r)!(k_2-r)!r!} {\alpha}^{k_1-r}{\beta}^{k_2-r}{\theta}^r}, \quad (k_1,k_2)\in\mathbb{Z}^{2++}. \end{equation}

Figure 1.

A path from (0, 0) to (m, n).

The following lemma gives insights into the MA coefficients of SAR(1) models.

Lemma 2.2. For real numbers ${\alpha}$ , ${\beta}$ , and ${\theta}$ , not all equal to zero,

(2.3) \begin{equation} \frac{1}{1-\alpha Z_1-\beta Z_2-\theta Z_1Z_2} = \sum^{\infty}_{k_1=0}{\sum^{\infty}_{k_2=0}{b_{k_1,k_2}{{Z_1}^{k_1}}{{Z_2}^{k_2}}}} \end{equation}

for non-zero complex numbers $Z_1$ and $Z_2$ if and only if

(2.4) \begin{equation} b_{k_1,k_2}=\alpha b_{k_1-1,k_2} + \beta b_{k_1,k_2-1} + \theta b_{k_1-1,k_2-1} \quad \text{for all } (k_1,k_2)>0, \end{equation}

and $b_{0,0}=1$ and $b_{k_1,k_2}=0$ for $k_1<0$ or $k_2 < 0 $ .

Proof. The result follows by equating the coefficients of $Z_1^{k_1}Z_2^{k_2}$ on both sides of

\begin{align*} & (1-\alpha Z_1-\beta Z_2-\theta Z_1Z_2)\Bigg(\sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{Z_1}^{k_1}{Z_2}^{k_2}}\Bigg) \\ & \qquad = \sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{Z_1}^{k_1}{Z_2}^{k_2}} - \alpha\sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{Z_1}^{k_1+1}{Z_2}^{k_2}} \\ & \qquad\quad - \beta\sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{Z_1}^{k_1}{Z_2}^{k_2+1}} - \theta\sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{Z_1}^{k_1+1}{Z_2}^{k_2+1}} \\ & \qquad = 1 - \sum^{\infty}_{({k_1,k}_2)>(0,0)}{{Z_1}^{k_1}{Z_2}^{k_2}(b_{k_1,k_2} - \alpha{b_{k_1-1,k_2}} - \beta{b_{k_1,k_2-1}} - \theta{b_{k_1-1,k_2-1}})} = 1. \end{align*}

Theorem 2.1. If a SAR(1) model given by

(2.5) \begin{equation} X_{(t_1,t_2)} = \alpha X_{(t_1-1,t_2)} + \beta X_{(t_1,t_2-1)} + \theta X_{(t_1-1,t_2-1)} + {W}_{(t_1,t_2)} \end{equation}

assumes the non-anticipating moving average representation in (1.2), then the MA coefficients are given by (2.2), where ${\alpha}$ , ${\beta}$ , and ${\theta}$ are real numbers not all equal to zero. Conversely, any spatial MA $(\infty)$ process given in (1.2) with the MA coefficients as in (2.2) is SAR(1).

Proof. Let us begin with the representation of the SAR(1) model in (2.5). Then

\begin{equation*} X_{(t_1,t_2)} = \frac{1}{1-\alpha B_1-\beta B_2-\theta B_1B_2}{W}_{(t_1,t_2)}, \end{equation*}

where $B_1$ and $B_2$ are backward shift operators, ${B_1}^i{B_2}^jX_{(t_1,t_2)}=X_{(t_1-i,t_2-j)}$ . We let $b_{k_1,k_2} =\alpha b_{k_1-1,k_2} + \beta b_{k_1,k_2-1} + \theta b_{k_1-1,k_2-1}$ , with $k_1\geq 0$ , $k_2\geq0$ , $b_{0,0}=1$ , and $b_{k_1,k_2}=0$ , for $k_1<0$ or $k_2<0$ . Then, by using Lemma 2.2, we obtain

\begin{equation*} X_{(t_1,t_2)} = \Bigg(\sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{B_1}^{k_1}{B_2}^{k_2}}\Bigg){W}_{(t_1,t_2)} = \sum^{\infty}_{({k_1,k}_2)=(0,0)}{b_{k_1,k_2}{W}_{(t_1-k_1,t_2-k_2)}}. \end{equation*}

The desired result follows using Lemma 2.1.

In the following theorem we provide a sufficient condition for the causality of the SAR(1) process $X_{(t_1,t_2)}$ in (2.5).

Theorem 2.2. Under the assumption that

(2.6) \begin{equation} |\alpha|+|\beta|+|\theta| < 1, \end{equation}

the SAR(1) process in (2.5) is causal, i.e.

(2.7) \begin{equation} \sum_{k_1\geq 0,\,k_2\geq 0}|b_{k_1,k_2}| < \infty. \end{equation}

Proof. Clearly,

(2.8) \begin{equation} \sum_{k_1\geq 0,\,k_2\geq 0}|b_{k_1,k_2}| \leq \sum_{k_1\geq 0,\,k_2\geq 0}\sum^{\min\!(k_1,k_2)}_{r=0}{\frac{(k_1+k_2-r)!}{(k_1-r)!(k_2-r)!r!} {|\alpha|}^{k_1-r}{|\beta|}^{k_2-r}{|\theta|}^r}. \end{equation}

Then, using (2.3) with $\alpha$ , $\beta$ , and $\theta$ replaced by $|\alpha|$ , $|\beta|$ , and $|\theta|$ , the right-hand side of (2.8) is equal to

\begin{equation*} \frac{1}{1- |\alpha| - |\beta| - |\theta|} =\sum_{n=0}^\infty(|\alpha| + |\beta| + |\theta|)^n < \infty \end{equation*}

under (2.6). The proof is complete.

Figure 2.

Plots of $S_{n+10}-S_n$ for $n=80,\ldots,120$ . Left: $\alpha=-0.6$ , $\beta=-0.6$ , $\theta =-0.3$ shows no evidence of causality. Right: $\alpha=-0.4$ , $\beta=-0.4$ , $\theta =-0.3$ shows strong evidence of causality.

Remark 2.2. In [Reference Drapatz5], it is reported from [Reference Fray and Roselle6] that for $d=2$ ,

\begin{equation*} \psi_{n,k} = \sum_{j=0}^n \binom{k}{ j}\binom{n+k-j}{ k}\phi_1^{n-j}\phi_2^{k-j}\phi_3^j, \quad n,k\in\mathbb{N}_0, \end{equation*}

where $\psi_{n,k}$ are the MA coefficients and $\phi_1,\phi_2,\phi_3$ are the AR coefficients. We note that since $\binom{n}{k}=0$ for $k>n$ , $b_{n,k}[\alpha, \beta, \theta]=\psi_{n,k}[\beta, \alpha, \theta]$ .

3. MA coefficients for the general case $d\geq 2$

Let $\{X(\mathbf{t}),\,\mathbf{t}\in \mathbb{Z}^{d}\}$ be a SAR(1) process on $\mathbb{Z}^d$ . We recall from (1.1) and (1.2) that $\beta_{\mathbf{i}}$ , $\mathbf{i}\in \mathbf{I}=\big\{\mathbf{i}=(i_1,\ldots,i_d);\, i_j=0 \text{ or }1;\,\,j=1,\ldots,d;\,1\leq \sum_{j=1}^d i_j \leq d\big\}$ , and $b_{\mathbf{k}}$ , $\mathbf{k}\in \mathbb{Z}^{d++}$ , are the autoregressive parameters and the MA coefficients, respectively. Similar to the case $d=2$ , for the general case $d\geq 2$ the MA coefficients satisfy the recursive equations $b_{\mathbf{k}}=\sum_{\mathbf{i}\in \mathbf{I}}\beta_{\mathbf{i}}b_{\mathbf{k}-\mathbf{i}}$ , $\mathbf{k}\in\mathbb{Z}^{d++}$ , with $b_{\mathbf{k}}=0$ , $\mathbf{k}\in \mathbb{Z}-\mathbb{Z}^{d++}$ , where $\mathbf{I}=\big\{\mathbf{i}=(i_1,\ldots,i_d);\, i_j=0 \text{ or }1;\,\,j=1,\ldots,d;\,1\leq \sum_{j=1}^d i_j \leq d\big\}$ . In the following, we follow the same argument given in the proof of Lemma 2.1 to derive the MA coefficients $b_{\mathbf{k}}$ that are indeed solutions to the recursive equations given above.

The idea goes back to the possible paths and the gain (reward) of each path for a particle with random motion that travels through the lattice points from $\mathbf{0}=(0,\ldots,0)$ to $\mathbf{k}=(k_1,\ldots,k_d)$ in $\mathbb{Z}^{d++}$ step-wise, where only at most one step ahead towards each coordinate and cross coordinate is allowed. Indeed, in one step, $\mathbf{k}=(k_1,\ldots,k_d)$ can be reached from any of the lattice points $\mathbf{k}-\mathbf{i}$ , $\mathbf{i}=(i_1,\ldots,i_d)\in \mathbf{I} $ . In this case the total gain at $\mathbf{k}-\mathbf{i}$ is increased by $\beta_{\mathbf{i}}b_{\mathbf{k}-\mathbf{i}}$ . Clearly the total gain $b_{\mathbf{k}}$ does satisfy the recursive equation given above. Next, we determine the totality of gains incurred by all paths from $\mathbf{0}=(0,\ldots,0) $ to $\mathbf{k}=(k_1,\ldots,k_d)$ that go through the lattice points, as explained above.

The gain of reaching $(k_1,\ldots,k_d)$ from $\mathbf{k}-\mathbf{i} $ is $ \beta_{\mathbf{i}} $ for $\mathbf{i}\in\mathbf{I}$ . The gain of a path is multiplicative: the path total gain is the product of the path step gains. The following procedure provides the number of paths and corresponding gains of reaching $\mathbf{k}=(k_1,\ldots,k_d)$ from $\mathbf{0}=(0,\ldots,0) $ in $\mathbb{Z}^{d++}$ .

We let $\kappa\subset\{1,2,\ldots,d\}$ represent a set of distinct indices. The one-step neighboring lattice points are classified by the $\kappa$ sets.

For $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}^{d++}$ , let $\ell\kappa$ to be non-negative integers subject to

(3.1) \begin{equation} \sum_{\kappa:u\in\kappa}\ell\kappa = k_u, \quad u=1,\ldots,d.\end{equation}

Indeed, for each $\kappa$ , with $u\in\kappa$ then $\ell\kappa$ is the number of one-step jumps of type $\kappa$ towards the coordinate $k_u$ for reaching $\mathbf{k}=(k_1,\ldots,k_d)$ from $\mathbf{0}=(0,\ldots,0)$ . There are d equations in (3.1). In every equation in that system of equations, exactly $2^{d-1}$ set elements $\kappa$ are involved. Indeed, for a fixed u, $u=1,\ldots,d$ , the number of distinct subsets of $\{1,2,\ldots,d\}$ that contain u is the same as the number of subsets of a set with $d-1$ distinct elements, which is $2^{d-1}$ . We let $\jmath$ stand for the collection of distinct $\kappa$ in the (3.1) equations. We also let $\mathbf{L}$ represent the collection of possible choices for $\ell=\{\ell\kappa,\, \kappa\in\jmath\}$ sets that satisfy the equations in (3.1). Note that since the number $\ell\kappa$ of unknowns in the system of equations in (3.1) is more than the number of equations, the collection of the sets of the possible solutions to (3.1), $\mathbf{L}$ , is well defined. Also, for a set of solutions for $ \{\ell\kappa,\, \kappa\in\jmath\}$ , say $\ell \in \mathbf{L}$ , we let

(3.2) \begin{align} C[\ell] & = \big[\textstyle\sum_{\kappa\in\jmath}\ell\kappa\big]!, \quad \ell \in \mathbf{L}, \\[-10pt] \nonumber\end{align}

(3.3) \begin{align} D[\ell] & = \textstyle\prod_{\kappa\in\jmath}\ell\kappa!, \qquad \ell \in \mathbf{L}. \end{align}

Indeed, this is the multinomial situation. The ratio ${C[\ell]}/{D[\ell]}$ is the number of paths from $\mathbf{0}=(0,\ldots,0)$ to $\mathbf{k}=(k_1,\ldots,k_d) $ with $\ell\kappa$ the number of single steps in the direction $\kappa$ , $\ell\kappa \in \ell$ . The gain of such a path is $\prod_{\kappa\in\jmath}\beta_{{\mathbf{i}_\kappa}}^{\ell\kappa}$ , where $\mathbf{i}_\kappa\in\mathbf{I}$ , $i_j=0$ , $i_j\notin\kappa$ , $i_j=1$ , $i_j\in \kappa$ , $j=1,\ldots,d$ . Therefore, the total of possible gains $b_\mathbf{k}$ is incurred through

(3.4) \begin{equation} b_\mathbf{k}=\sum_{\ell\in\mathbf{L}}\frac{C[\ell]}{D[\ell]} \prod_{\kappa \in \jmath}\beta_{{\mathbf{i}_\kappa}}^{\ell\kappa}, \quad \mathbf{k}\in \mathbb{Z}^{d++},\end{equation}

which satisfies the SAR(1) recursive equations given above.

The following theorem is the version of Theorem 2.1 for $d\geq 2$ .

Proof. We show that the MA coefficients assume the recursive property stated above. The result will follow from the details in deriving (3.4). We observe that

\begin{align*} X_{\mathbf{t}} = \sum_{\mathbf{i}\in\mathbf{I}}\beta_{\mathbf{i}}X_{\mathbf{t}-\mathbf{i}} + W_{\mathbf{t}} = \sum_{\mathbf{i}\in\mathbf{I}}\beta_{\mathbf{i}}\sum_{\mathbf{k}\in\mathbb{Z}^{d++}}b_{\mathbf{k}} W_{\mathbf{t}-\mathbf{i}-\mathbf{k}} + W_{\mathbf{t}} & = \sum_{\mathbf{i}\in\mathbf{I}}\beta_{\mathbf{i}}\sum_{\mathbf{k}\in\mathbb{Z}^{d++}+\mathbf{i}}b_{\mathbf{k}-\mathbf{i}} W_{\mathbf{t}-\mathbf{k}} + W_{\mathbf{t}} \\ & = \sum_{\mathbf{k}\in\mathbb{Z}^{d++}}\Bigg\{\sum_{\mathbf{i}\in\mathbf{I}}\beta_{\mathbf{i}}b_{\mathbf{k}-\mathbf{i}}\Bigg\} W_{\mathbf{t}-\mathbf{k}}. \end{align*}

The proof is complete.

The following theorem provides insights into the SAR(1) processes that admit the MA representation.

The following theorem is for the autocorrelation function of SAR(1) processes.

The MA representation leads to formulations for the forecast recipe and the forecast error. Indeed, in forecasting $X_{\mathbf{t}+\mathbf{n}}$ based on $\{X_\mathbf{s},\,\mathbf{s}\leq \mathbf{t}\}$ , $\mathbf{n}\in \mathbb{Z}^{d++}$ , $\hat{X}_{\mathbf{t}+\mathbf{n}}=E[X_{\mathbf{t}+\mathbf{n}}\,||\,X_\mathbf{s},\,\mathbf{s}\leq \mathbf{t}]$ , the notation $\mathbf{s}\leq \mathbf{t}$ means $s_j\leq t_j$ , $j=1,\ldots,d$ . The corresponding forecast error is defined by $e^2[\mathbf{n}]=\mathrm{Var}(X_{\mathbf{t}+\mathbf{n}}-\hat{X}_{\mathbf{t}+\mathbf{n}})$ , $\mathbf{n}\in \mathbb{Z}^{d++}$ .

Theorem 3.4. Under the assumptions of Theorem 3.1,

\begin{equation*} \hat{X}_{\mathbf{t}+\mathbf{n}} = \sum_{\mathbf{k}\in[\mathbf{n},\mathbf{\infty}]}b_{\mathbf{k}}W_{\mathbf{n}-\mathbf{k}}, \qquad e^2[\mathbf{n}] = \bigg(\sum_{\mathbf{k}\in\mathbb{Z}^{d++}-[\mathbf{n},+\mathbf{\infty}]}b_{\mathbf{k}}^2\bigg)\sigma_W^2. \end{equation*}

Proof. Indeed,

\begin{align*} \hat{X}_{\mathbf{t}+\mathbf{n}} & = E[X_{\mathbf{t}+\mathbf{n}}\,||\,X_\mathbf{s},\,\mathbf{s}\leq\mathbf{t}] \\ & = E\bigg[\sum_{\mathbf{k}\in\mathbb{Z}^{d++}}\!b_{\mathbf{k}}W_{\mathbf{t}+\mathbf{n}-\mathbf{k}}\,||\,W_\mathbf{s},\, \mathbf{s}\leq\mathbf{t}\bigg] = E\bigg[\sum_{\mathbf{k}\in\mathbb{Z}^{d++}}\!b_{\mathbf{k}}W_{\mathbf{n}-\mathbf{k}}\,||\,W_\mathbf{s},\, \mathbf{s}\leq\mathbf{0}\bigg] = \! \sum_{\mathbf{k}\in[\mathbf{n},\mathbf{\infty}]}\!\!b_{\mathbf{k}}W_{\mathbf{n}-\mathbf{k}}. \end{align*}

The corresponding forecast error is given by

\begin{equation*} e^2[\mathbf{n}] = \mathrm{Var}\Bigg(\sum_{\mathbf{k}\in\mathbb{Z}^{d++}-[\mathbf{n},+\mathbf{\infty}]}b_{\mathbf{k}} W_{\mathbf{k}}\Bigg) = \Bigg(\sum_{\mathbf{k}\in\mathbb{Z}^{d++}-[\mathbf{n},+\mathbf{\infty}]}b_{\mathbf{k}}^2\Bigg)\sigma_W^2. \end{equation*}

4. Numerical computation and programming

This section details an algorithm and corresponding Mathematica programs for the numerical derivation of the MA coefficients. To clarify the procedure for evaluation of the MA coefficients given in (3.4), we provide the details for the cases of $d=2$ and $d=3$ here.

For $d=2$ , the SAR(1) model $X_{\mathbf{t}}$ is given by (2.5). Let $\mathbf{k}=(k_1,k_2)\in \mathbb{Z}^{2++}$ . For $k_1$ and $k_2$ in (3.1), the corresponding indices and non-negative integers are $\{\ell\{1\}, \ell\{1,2\}\}$ and $\{\ell\{2\}, \ell\{1,2\}\}$ , subject to $\ell\{1\} + \ell\{1,2\} = k_1$ and $\ell\{2\} + \ell\{1,2\} = k_2$ . Also, $\jmath=\{\{1\},\{2\}, \{1,2\}\}$ . For given $k_1$ and $k_2$ , if we let $r=\ell\{1,2\}$ , then $\ell\{1\}=k_1-r$ and $\ell\{2\}=k_2-r$ ; possible choices for $\ell$ correspond to possible choices for r. Thus, for $d=2$ , (3.2) is read as $C[\ell]=[(k_1-r)+ (k_2-r) + r]!= (k_1 +k_2 -r)!$ and (3.3) becomes $D[\ell]= (k_1-r)! (k_2-r)! r!$ . For given $k_1, k_2$ , every index $\ell\in\mathbf{L}$ is specified by an $r \in \{0,\ldots,\min\{k_1,k_2\}\}$ . Therefore, (3.4) is read as

\begin{equation*} b_{\mathbf{k}} = \sum_{r=0}^{\min\{k_1,k_2\}}\frac{(k_1+k_2-r)!}{(k_1-r)!(k_2-r)!r!} \beta_1^{k_1-r}\beta_2^{k_2-r}\beta_{12}^r,\end{equation*}

giving (2.2): $\beta_1\equiv\beta_{(1,0)}$ , $\beta_2\equiv\beta_{(0,1)}$ , $\beta_{12}\equiv\beta_{(1,1)}$ .

For $d=3$ , the index set $\mathbf{I}$ has $2^d-1=7$ members. There are seven SAR(1) coefficients, denoted by $\beta_1$ , $\beta_2$ , $\beta_3$ , $\beta_{12}$ , $\beta_{13}$ , $\beta_{23}$ , and $\beta_{123}$ . The SAR(1) model is given by

\begin{align*} X_{(t_1,t_2,t_3)} & = \beta_1X_{(t_1-1,t_2,t_3)} + \beta_2X_{(t_1,t_2-1,t_3)} + \beta_3X_{(t_1,t_2,t_3-1)} \\ & \quad + \beta_{12}X_{(t_1-1,t_2-1,t_3)} + \beta_{13}X_{(t_1-1,t_2,t_3-1)} + \beta_{23}X_{(t_1,t_2-1,t_3-1)} \\ & \quad + \beta_{123}X_{(t_1-1,t_2-1,t_3-1)} + {W}_{(t_1,t_2,t_3)}.\end{align*}

For $d=3$ , the $\jmath$ collection is $\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$ . The corresponding simultaneous equations in (3.1) are

\begin{align*} \ell\{1\} + \ell\{1,2\} + \ell\{1,3\} + \ell\{1,2,3\} & = k_1, \\ \ell\{2\} + \ell\{1,2\} + \ell\{2,3\} + \ell\{1,2,3\} & = k_2, \\ \ell\{3\} + \ell\{1,3\} + \ell\{2,3\} + \ell\{1,2,3\} & = k_3.\end{align*}

Again, if we replace $\ell\{1\}$ , $\ell\{2\}$ , and $\ell\{3\}$ by their equivalent terms from these equations, then (3.2) becomes

\begin{align*} C[\ell] & = \big[(k_1 - \ell\{1,2\} - \ell\{1,3\} - \ell\{1,2,3\}) \\ & \qquad + (k_2 - \ell\{1,2\} - \ell\{2,3\} - \ell\{1,2,3\}) \\ & \qquad + (k_3 - \ell\{1,3\} - \ell\{2,3\} - \ell\{1,2,3\}) \\ & \qquad + \ell\{1,2\} + \ell\{1,3\} + \ell\{2,3\} + \ell\{1,2,3\}\big]! \\ & = [k_1 + k_2 + k_3 - \ell\{1,2\} - \ell\{1,3\} - \ell\{2,3\} - 2\ell\{1,2,3\}]!.\end{align*}

The corresponding $D[\ell]$ term in (3.3) is

\begin{align*} D[\ell]] & = (k_1 - \ell\{1,2\} - \ell\{1,3\} - \ell\{1,2,3\})! \\ & \quad \times (k_2 - \ell\{1,2\} - \ell\{2,3\} - \ell\{1,2,3\})! \\ & \quad \times (k_3 - \ell\{1,3\} - \ell\{2,3\} - \ell\{1,2,3\})! \\ & \quad \times \ell\{1,2\}!\ell\{1,3\}!\ell\{2,3\}!\ell\{1,2,3\}!.\end{align*}

Therefore, the spatial MA coefficient $b_{(k_1,k_2,k_3)}$ in (3.4) is given by

(4.1) \begin{equation} b_{(k_1,k_2,k_3)} = \sum_{\ell}\frac{C[\ell]}{D[\ell]}\beta_1^{\nu_1}\beta_2^{\nu_2}\beta_3^{\nu_3} \beta_{12}^{\ell\{1,2\}}\beta_{13}^{\ell\{1,3\}}\beta_{23}^{\ell\{2,3\}}\beta_{123}^{\ell\{1,2,3\}},\end{equation}

with $\nu_1= k_1 - \ell\{1,2\} - \ell\{1,3\} - \ell\{1,2,3\}$ , $\nu_2 = k_2 - \ell\{1,2\} - \ell\{2,3\} - \ell\{1,2,3\}$ , and $\nu_3 = k_3 - \ell\{1,3\} - \ell\{2,3\} - \ell\{1,2,3\}$ . The possible $\ell$ sets are specified by the possible values for $\ell\{1,2\}$ , $\ell\{1,3\}$ , $\ell\{2,3\}$ , and $\ell\{1,2,3\}$ . Indeed,

• • possible values for $\ell\{1,2,3\}$ are $\{0,1,2,\ldots,\min\{k_1,k_2,k_3\}\}$ ,

• • possible values for $\ell\{2,3\}$ are $\{0,1,2,\ldots,\min\{k_2-\ell\{1,2,3\}, k_3-\ell\{1,2,3\}\}$ ,

• • possible values for $\ell\{1,3\}$ are $\{0,1,2,\ldots,\min\{k_1-\ell\{1,2,3\}, k_3-\ell\{1,2,3\}\}$ , and

• • possible values for $\ell\{1,2\}$ are $\{0,1,2,\ldots,\min\{k_1-\ell\{1,2,3\}-\ell\{1,3\}, k_2-\ell\{1,2,3\}-\ell\{2,3\}\}$ .

Therefore, for $(k_1,k_2,k_3)\in \mathbb{Z}^{3++}$ ,

(4.2) \begin{align} b_{(k_1,k_2,k_3)} & = \sum_{r=0}^{\min\{k_1,k_2,k_3\}}\sum_{u=0}^{\min\{k_2-r,k_3-r\}}\sum_{v=0}^{\min\{k_1-r,k_3-r\}} \sum_{w=0}^{\min\{k_1-r-v,k_2-r-u\}} \nonumber \\ & \quad \times \frac{(k_1+k_2+k_3-v-w-u-2r)!}{(k_1-v-w-r)!(k_2-w-u-r)!(k_3-v-u-r)!v!u!w!r!} \nonumber \\ & \quad \times \beta_1^{k_1-v-w-r}\beta_2^{k_2-w-u-r}\beta_3^{k_3-v-u-r} \beta_{12}^{w}\beta_{13}^{v}\beta_{23}^{u}\beta_{123}^{r}. \end{align}

We note that, for general d, the collection $\jmath$ has exactly $2^d -1$ set elements $\kappa$ . There are d equations in (3.1). In every equation in that system of equations, exactly $2^{d-1}$ set elements $\kappa$ are involved. We may consider $\ell\{j\}$ , $j=1,\ldots,d$ , as dependent variables that are specified by the remaining $\ell\kappa$ and $k_j$ , $j=1,\ldots,d$ . More precisely,

(4.3) \begin{equation} \ell\{j\}=k_j-\sum_{\kappa \supset \{\,j\},\;\kappa\neq \{j\}}\ell\kappa, \quad j=1,\ldots,d.\end{equation}

The equations in (3.1) put restrictions on the possible values for the rest of the $\ell\kappa$ . To deduce the correct range for each $\ell\kappa$ , it will be more convenient to move backward, by removing the elements of $\{1,\ldots,d\}$ one by one. We notice that:

• • $\ell\{1,\ldots,d\}\leq k_u$ , $u=1,\ldots,d$ .

• • For $j=1,\ldots,d$ , $\ell\{1,\ldots,d\}_j + \ell\{1,\ldots,d\} \leq k_u$ , $u=1,\ldots,d$ , $u\neq j$ , where $\{1,\ldots,d\}_{\,j}=\{1,\ldots,d\}-\{\,j\}$ .

• • For a given $(k_1,\ldots,k_d)$ , in the ith equation $\ell\{i\}$ is the dependent variable whose value is specified by $k_i$ and all $\ell\kappa$ for which $i\in\kappa$ .

• • For $\kappa=\{j_1,\ldots,j_k\}$ , $k=1,\ldots,d$ ,

  (4.4) \begin{equation} \ell\{j_1,\ldots,j_k\} + \sum\ell_\kappa \leq k_{j_u}, \quad u=1,\ldots,k,\,\kappa\supset\{j_1,\ldots,j_k\}. \end{equation}

By taking these facts into consideration, in the following we provide an algorithm for the numerical evaluation of the MA coefficients $b_\mathbf{k}$ , $\mathbf{k}\in \mathbb{Z}^{d++}$ .

4.2. Mathematica programs

We programmed the algorithm given above using Wolfram Mathematica 11.3. The details are given below.

1. Program 1: In the first part of this program, for $d=3$ the algorithm given above for the SAR(1) parameters

   \begin{align*} [\beta_1=0.2,\,\beta_2=0.3,\,\beta_3=0.1,\,\beta_{12}=0.15,\,\beta_{13}=0.05,\,\beta_{23}=0.07,\, \beta_{123}=0.08] \end{align*}
   was programmed to compute $b_{(k_1,k_2,k_3)}$ , giving $b_{(4,2,1)}=0.031\,08$ . The second part of the program gives $b_{(k_1,k_2,k_3)}$ using (4.2). The output is consistent with the output of the first part.

2. Program 2: For the SAR(1) parameters

   \begin{align*} [\beta_1=0.15,\,\beta_2=0.15,\,\beta_3=0.2,\,\beta_{12}=0.05,\,\beta_{13}=0.12,\,\beta_{23}=0.1,\, \beta_{123}=0.17], \end{align*}
   the algorithm was programmed to compute the MA coefficients $b_{\mathbf{k}}$ , $\mathbf{k}\in[\mathbf{0},\mathbf{n}]$ . Then, $\sum b_{\mathbf{k}}^2$ , $\mathbf{k}\in [\mathbf{0},\mathbf{n}]$ , and $\sum|b_{\mathbf{k}}|$ , $\mathbf{k}\in [\mathbf{0},\mathbf{n}]$ , for $ \mathbf{n}=(1,1,1),\ldots,(n,n,n)$ , $n=1,\ldots,40$ were plotted against n to exhibit the forecast error (stationarity) and the causality.

3. Program 3: This program is similar to Program 2. For $d=2$ , the SAR(1) parameters

   \begin{align*}[\beta_1=0.240,\;\beta_2=0.796,\;\beta_{12}=-0.108]\end{align*}
   were the program inputs. The corresponding SAR(1) model was presented in [Reference Basu and Reinsel1] on the data set from a regular grid presented in [Reference Kempton and Howes10] on the yield of barley (in kilograms) from an agricultural uniformity trial experiment at the Plant Breeding Institute, Cambridge, UK.

The plots for Programs 2 and 3 are depicted in Figure 3. Programs 1, 2, and 3 are given in the Appendix.

Figure 3.

Plots for Programs 2 and 3. Left: The cumulative squared MA coefficients, stationarity (forecast error). Right: The cumulative absolute value MA coefficients, causality.