Comparison of BINAR(1) models with bivariate negative binomial innovations and explanatory variables

Abstract

The bivariate integer-valued autoregressive model of order 1 (BINAR(1)) is popular in fitting bivariate time series of counts, and the bivariate negative binomial (BNB) distribution can be chosen as its innovation's distribution, which is more flexible than the traditional bivariate Poisson distribution. It is well known that BNB distributions can be constructed in different ways, and these distributions will be reviewed in this paper. Performances of BINAR(1) models based on these BNB distributions with explanatory variables being included in the survival probability are compared. To estimate unknown parameters, the conditional maximum likelihood method is considered and evaluated by Monte Carlo simulations. Two sales counts are used to compare performances of the above models, and some interesting conclusions are also given.

Keywords:

• Bivariate integer-valued autoregressive model
• bivariate negative binomial distribution
• explanatory variables
• INAR
• time series of counts

Acknowledgments

We are very grateful to the anonymous referee for providing several constructive comments which led to a significant improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

Notes: The table is limited by text width. Therefore, we omit ‘BINAR(1)’ in the names. BPoisson represents the bivariate Poisson distribution discussed in Pedeli and Karlis [2]. BNB-I represents the BNB-I distribution mentioned in Section 2.

Notes: ${\hat{\zeta }}_i$: constant; ${\hat{\gamma }}_i$: prices. ${\hat{\eta }}_i$: snacks, juices, cookies correspond to the last three models, respectively.

Note: The table is limited by text width, so the names of models omit ‘BINAR(1)’.

Note: The table is limited by text width. Therefore, we omit ‘BINAR(1)’ in the names.

Notes: ${\hat{\zeta }}_i$: constant; ${\hat{\gamma }}_i$: prices. ${\hat{\eta }}_i$: events, temperatures correspond to the last two models, respectively. The table is limited by text width, so the names of models omit ‘BINAR(1)’.

Notes: The table is limited by text width, so the names of models omit ‘BINAR(1)’. The BINAR(1) model with the BNB-COM innovation does not suit this data, which has zero observations (see Remark 2.1).

Note: The AIC of the model with the BNB-COM innovation for the second data is a missing value, because this model does not suit this data with zero observations (see Remark 2.1).

1 The computer processor, used for estimations, is Intel(R) Core(TM) i5-10210U CPU 1.60 GHz 2.11 GHz.

2 http://research.chicagobooth.edu/marketing/databases/dominicks.

3 UPC is a universal product code, which can represent one particular product.

4 The product is ‘$CHICLETSTINYPACK$’, which UPC is ‘1254612128’.

5 The product is ‘$TICTACWINTERGREEN$’, which UPC is ‘980000007’.

6 The data can be downloaded on https://www.ncdc.noaa.gov/cag/city/time-series/.

7 The product is ‘$LCCAFECLSCMTLOAF$’, which UPC is ‘1380016695’.

8 The product is ‘$LCLUNCHEXPTERIYAK$’, which UPC is ‘1380017404’.

9 We use the ‘L-BFGS-B’ method of optim function in R.

Additional information

Funding

Su's work is supported by Graduate Innovation Fund of Jilin University [grant number 101832020CX073]. Zhu's work is supported by the National Natural Science Foundation of China [grant numbers 11871027, 11731015] and Cultivation Plan for Excellent Young Scholar Candidates of Jilin University.