A full ARMA model for counts with bounded support and its application to rainy-days time series

Abstract

Motivated by a large dataset containing time series of weekly number of rainy days collected over two thousand locations across Europe and Russia for the period 2000–2010, we propose a new class of ARMA-like model for time series of bounded counts, which can also handle extra-binomial variation. We abbreviate this model as bvARMA, as it is based upon a novel operation referred to as binomial variation. After having discussed important stochastic properties and proposed a model-fitting approach relying on maximum likelihood estimation, we apply the bvARMA model family to the rainy-days time series. Results show that both bvAR and bvMA models are adequate and exhibit a similar performance. Furthermore, bvARMA results outperform those obtained by fitting ordinary discrete ARMA (NDARMA) models of the same order.

Appendices

Proofs

1.1 Proof of Theorem 2

For $k\ge 1$, by conditioning,

$$\begin{aligned} \gamma&(k)\ =\ Cov[ X_{t}, X_{t-k} ] \\&=\ E\big [ Cov[ X_{t}, X_{t-k} | X_{t-1}, \dots , \epsilon _{t}, \epsilon _{t-1}, \dots ] \big ] + Cov\big [ E[ X_{t} | X_{t-1}, \dots , \epsilon _{t}, \epsilon _{t-1}, \dots ], X_{t-k}\big ]\\&\quad =\ 0 + Cov\Big [ \sum _{i=1}^{{p }} \phi _{i} X_{t-i} + \sum _{j=0}^{{q }} \phi _{-j} \epsilon _{t-j}, X_{t-k}\Big ]\\& =\ \sum _{i=1}^{{p }} \phi _{i} Cov[X_{t-i}, X_{t-k}] + \sum _{j=0}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, X_{t-k}]\\& =\ \sum _{i=1}^{{p }} \phi _{i} \gamma ( | k-i| ) + \sum _{j=k}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, X_{t-k}]. \end{aligned}$$

Let us now look at the cross-covariances $a_{k}:= Cov[X_{t}, \epsilon _{t-k}]$ with $k\in {\mathbb {Z}}$, which satisfies $a_{k}=0$ for $k<0$. At lag 0, we obtain by conditioning (in analogy to above)

$$\begin{aligned} a_{0}\ =\ Cov[X_{t}, \epsilon _{t}] \ =\ \sum _{i=1}^{{p }} \phi _{i} Cov[X_{t-i}, \epsilon _{t}]\ +\ \sum _{j=0}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, \epsilon _{t}] \ =\ \phi _{0} \sigma _{\epsilon }^{2}. \end{aligned}$$

For lags $k\ge 1$,

$$\begin{aligned} a_{k} \ =\ \sum _{i=1}^{{p }} \phi _{i} Cov[ X_{t-i}, \epsilon _{t-k}]\ +\ \sum _{j=0}^{{q }} \phi _{-j} Cov[\epsilon _{t-j}, \epsilon _{t-k}] \ =\ \sum _{i=1}^{{p }} \phi _{i} a_{k-i}\ +\ \phi _{-k}\sigma _{\epsilon }^{2}. \end{aligned}$$

If we now define $b_{k}:=a_{k}/\sigma _{\epsilon }^{2}$ for $k=0,1,\dots , {q }$, we get the recursion stated in Theorem 2.

1.2 Derivation of conditional variance

Using (8) for the conditional factorial moments, we obtain

$$\begin{aligned} \begin{array}{l} V[X_{t} | X_{t-1}, \dots , \epsilon _{t-1}, \dots ]\ =\ E[(X_t)_{(2)}\ |\ \ldots ] + E[ X_{t}\ |\ \ldots ] - E[ X_{t}\ |\ \ldots ]^{2}\\ \ =\ \phi _0\cdot E[(\epsilon _t)_{(2)}]\ +\ \left( 1-\tfrac{1}{n}\right) \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}\, X_{t-i}^{2} +\ \left( 1-\tfrac{1}{n}\right) \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\, \epsilon _{t-j}^{2}\\ \quad + \phi _0\cdot \mu \ +\ \mathop {\sum }\nolimits _{i=1}^{{p }}\ \phi _i\, X_{t-i}\ +\ \mathop {\sum }\nolimits _{j=1}^{{q }}\ \phi _{-j}\, \epsilon _{t-j}\\ \quad - \left( \phi _0\cdot \mu \ +\ \mathop {\sum }\nolimits _{i=1}^{{p }}\ \phi _i\, X_{t-i}\ +\ \mathop {\sum }\nolimits _{j=1}^{{q }}\ \phi _{-j}\, \epsilon _{t-j}\right) ^{2}\\ \ =\ \phi _{0} \sigma _{\epsilon }^{2} + \phi _{0} ( 1- \phi _{0}) \mu ^{2} + \left( 1-\tfrac{1}{n}\right) \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}\, X_{t-i}^{2} - \mathop {\sum }\nolimits _{i=1}^{{p }} \mathop {\sum }\nolimits _{k=1}^{{p }} \phi _{i} \phi _{k} X_{t-i}X_{t-k}\\ \quad + (1-2\phi _{0}\mu ) \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i} X_{t-i} + \left( 1- \tfrac{1}{n}\right) \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\, \epsilon _{t-j}^{2}\\ \quad - \mathop {\sum }\nolimits _{j=1}^{{q }} \mathop {\sum }\nolimits _{l=1}^{{q }} \phi _{-j}\phi _{-l} \epsilon _{t-j}\epsilon _{t-l}+ (1-2\phi _{0}\mu ) \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\epsilon _{t-j} - 2\mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}X_{t-i} \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j} \epsilon _{t-j}. \end{array} \end{aligned}$$

So we obtain

$$\begin{aligned} \begin{array}{l} V[X_{t} | X_{t-1}, \dots , \epsilon _{t-1}, \dots ]\\ \ =\ \phi _{0} \sigma _{\epsilon }^{2} + \phi _{0} ( 1- \phi _{0}) \mu ^{2} + (1-\tfrac{1}{n})\, \mathop {\sum }\nolimits _{i=1}^{{p }}\ \phi _{i} X_{t-i}^{2} - \mathop {\sum }\nolimits _{i=1}^{{p }} \mathop {\sum }\nolimits _{k=1}^{{p }} \phi _{i} \phi _{k} X_{t-i}X_{t-k}\\ \quad + (1-\tfrac{1}{n})\, \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j} \epsilon _{t-j}^{2}- \mathop {\sum }\nolimits _{j=1}^{{q }} \mathop {\sum }\nolimits _{l=1}^{{q }} \phi _{-j}\phi _{-l} \epsilon _{t-j}\epsilon _{t-l}\\ \quad + (1-2\phi _{0}\mu ) \left( \mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i} X_{t-i} + \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j}\epsilon _{t-j}\right) - 2\mathop {\sum }\nolimits _{i=1}^{{p }} \phi _{i}X_{t-i} \mathop {\sum }\nolimits _{j=1}^{{q }} \phi _{-j} \epsilon _{t-j}. \end{array} \end{aligned}$$

(12)

Note that the first-order autoregressive case (${p }=1$) gives

$$\begin{aligned} \begin{array}{l} V[X_t\ |\ X_{t-1}]\ =\ E[(X_t)_{(2)}\ |\ X_{t-1}]\ +\ E[X_t\ |\ X_{t-1}]\ -\ E^2[X_t\ |\ X_{t-1}]\\ \ =\ \phi _0\cdot E[(\epsilon _t)_{(2)}]+(1-\phi _0)\,\frac{n-1}{n}\,X_{t-1}^2 \\ \quad +\ \phi _0\cdot \mu +(1-\phi _0)\,X_{t-1} \ -\ \big (\phi _0\cdot \mu +(1-\phi _0)\,X_{t-1}\big )^2\\ \ =\ \phi _0\cdot \sigma _{\epsilon }^2 + \phi _0(1-\phi _0)\,\mu ^2 \\ \quad +\ (1-\phi _0)\,(\phi _0-\frac{1}{n})\,X_{t-1}^2 \ +\ (1-\phi _0)\,(1-2\,\phi _0\, \mu )\,X_{t-1}. \end{array} \end{aligned}$$

(13)

Again from (8) with $r=2$, we obtain the unconditional 2nd factorial moment

$$\begin{aligned} E[(X_t)_{(2)}]\ =\ \phi _0\cdot E[(\epsilon _t)_{(2)}]\ +\ \tfrac{n_{(2)}}{n^2}\,\sum _{i=1}^{{p }}\ \phi _i\, E[X_{t-i}^2]\ +\ \tfrac{n_{(2)}}{n^2}\,\sum _{j=1}^{{q }}\ \phi _{-j}\, E[\epsilon _{t-j}^2]. \end{aligned}$$

Using the stationarity, it follows that

$$\begin{aligned} \left( 1 - (1-\tfrac{1}{n})\, \sum _{i=1}^{{p }}\phi _{i}\right) \, E[X_t^2]\ =\ (1-\phi _0)\mu \ +\ \phi _0\cdot E[\epsilon _t^2]\ +\ (1-\tfrac{1}{n})\,\sum _{j=1}^{{q }}\ \phi _{-j}\, E[\epsilon _{t-j}^2]. \end{aligned}$$

So the unconditional variance $\sigma ^{2}=V[X_t]$ of the stationary process is given by

$$\begin{aligned} \left( 1 - (1-\tfrac{1}{n})\, \sum _{i=1}^{{p }}\phi _{i}\right) \,\sigma ^{2} \ =\ \phi _{0} \sigma _{\epsilon }^{2} + (1-\tfrac{1}{n})\, \sum _{j=1}^{{q }} \phi _{-j} \sigma _{\epsilon }^{2} +(1-\phi _{0})\mu \left( 1-\frac{\mu }{n}\right) . \end{aligned}$$

So Eq. (9) for the BID follows.

1.3 Cohen’s $\kappa$ for bvAR(1) and bvMA(1) models

The computation of Cohen’s $\kappa (h)$ (2) requires knowledge about the bivariate joint distributions with lag h. The bvAR(1) process constitutes a finite Markov chain with the 1-step-ahead transition probabilities $p_{k|l}$ given in Eq. 11; let $\mathbf{P }=(p_{k|l})_{k,l=0,\ldots ,n}$ denote the corresponding transition matrix. Then the h-step-ahead transition probabilities $P(X_t=k\ |\ X_{t-h}=l)$ are the entries of the matrix $\mathbf{P }^h$, and the stationary marginal distribution is given by the solution of the invariance equation $\mathbf{P }\, \varvec{\pi }\ =\ \varvec{\pi }$, i. e., $P(X_t=j)=\pi _j$.

For the bvMA(1) model, the stationary marginal distribution equals

$$\begin{aligned} P(X=x)\ =\ \phi _{0}\, P(\epsilon =x)\ +\ \phi _{-1}\, P({bv _{n}(\epsilon )}=x), \end{aligned}$$

see Example 2. Since $X_t$ and $X_{t-h}$ are independent for lags $h>1$, we then have $\kappa (h)=0$; so it suffices to compute the bivariate probabilities $P(X_t=k, X_{t-1}=l)$ for time lag 1. Using that

$$\begin{aligned} P(X_t=x\ |\ \epsilon _t=u,\epsilon _{t-1}=v)\ =\ \phi _{0}\, \delta _{x,u}\ +\ \phi _{-1}\, P({bv _{n}(v)}=x), \end{aligned}$$

and denoting $p_{\epsilon ,u}:=P(\epsilon _t=u)$, we obtain

$$\begin{aligned} \begin{array}{l} P(X_t=k, X_{t-1}=l)\ =\ \mathop {\sum }\nolimits _{a,b,c} P(X_t=k, X_{t-1}=l,\ \epsilon _t=a,\epsilon _{t-1}=b,\epsilon _{t-2}=c)\\ \quad =\ \mathop {\sum }\nolimits _{a,b,c} P(X_t=k\ |\ \epsilon _t=a,\epsilon _{t-1}=b)\, P(X_{t-1}=l\ |\ \epsilon _{t-1}=b,\epsilon _{t-2}=c)\, p_{\epsilon ,a}p_{\epsilon ,b}p_{\epsilon ,c}\\ \quad =\ \mathop {\sum }\nolimits _{a,b,c} \big (\phi _{0}\, \delta _{k,a}\ +\ \phi _{-1}\, P({bv _{n}(b)}=k)\big )\, \big (\phi _{0}\, \delta _{l,b}\ +\ \phi _{-1}\, P({bv _{n}(c)}=l)\big )\, p_{\epsilon ,a}p_{\epsilon ,b}p_{\epsilon ,c}\\ \quad =\ \mathop {\sum }\nolimits _{b,c} \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(b)}=k)\big )\, \big (\phi _{0}\, \delta _{l,b}\ +\ \phi _{-1}\, P({bv _{n}(c)}=l)\big )\, p_{\epsilon ,b}p_{\epsilon ,c}\\ \quad =\ \mathop {\sum }\nolimits _{b} \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(b)}=k)\big )\, \big (\phi _{0}\, \delta _{l,b}\ +\ \phi _{-1}\, P({bv _{n}(\epsilon )}=l)\big )\, p_{\epsilon ,b}\\ \quad =\ \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(l)}=k)\big )\, \phi _{0}\, p_{\epsilon ,l}\\ \qquad +\ \big (\phi _{0}\, p_{\epsilon ,k}\ +\ \phi _{-1}\, P({bv _{n}(\epsilon )}=k)\big )\, \phi _{-1}\, P({bv _{n}(\epsilon )}=l)\\ \quad =\ \phi _{0}^2\, p_{\epsilon ,k} p_{\epsilon ,l}\ +\ \phi _{0}\phi _{-1}\, p_{\epsilon ,l}\, P({bv _{n}(l)}=k)\\ \qquad +\ \phi _{0}\phi _{-1}\, p_{\epsilon ,k}\, P({bv _{n}(\epsilon )}=l)\ +\ \phi _{-1}^2\, P({bv _{n}(\epsilon )}=k)\, P({bv _{n}(\epsilon )}=l).\\ \end{array} \end{aligned}$$

Simulation results from Sect. 4.2

See Figs. 9, 10, 11, 12, 13, 14, 15, 16 and 17.

Fig. 9

Violin plots of estimated minus true value of $\phi _{-1}$, $\phi _{0}$, $\phi _{1}$, $\pi$ and d, for fitting bvARMA(1, 1) model to DGP: bvARMA(1, 1) with $n=7$, $\phi _{-1}=0.1$, $\phi _{0}=0.7$, $\phi _{1}=0.2$, $\pi =0.5$, $d=1.8$

Fig. 10

DGP as in Fig. 9, but fitting misspecified bvMA(1) model

Fig. 11

DGP as in Fig. 9, but fitting misspecified bvAR(1) model

Fig. 12

Violin plots of estimated minus true value of $\phi _{-1}$, $\phi _{0}$, $\phi _{1}$, $\pi$ and d, for fitting bvMA(1) model to DGP: bvMA(1) with $n=7$, $\phi _{-1}=0.1$, $\phi _{0}=0.9$, $\phi _{1}=0.0$, $\pi =0.5$, $d=1.8$

Fig. 13

DGP as in Fig. 12, but fitting misspecified bvARMA(1, 1) model

Fig. 14

DGP as in Fig. 12, but fitting misspecified bvAR(1) model

Fig. 15

Violin plots of estimated minus true value of $\phi _{-1}$, $\phi _{0}$, $\phi _{1}$, $\pi$ and d, for fitting bvAR(1) model to DGP: bvAR(1) with $n=7$, $\phi _{-1}=0.0$, $\phi _{0}=0.8$, $\phi _{1}=0.2$, $\pi =0.5$, $d=1.8$

Fig. 16

DGP as in Fig. 15, but fitting misspecified bvARMA(1, 1) model

Fig. 17

DGP as in Fig. 15, but fitting misspecified bvMA(1) model