Application of various count models: Sahiwal demand from Naivasha

Sahiwal bulls have been bred at the National Sahiwal Stud (NSS) in Naivasha since the late 1960s. The breed is credited for its ability to withstand conditions which other introduced cattle breeds find it difficult, especially those in the ASALs. The sahiwal will produce milk with little supplementation and can let down milk without calf on foot. Farmers interested in acquiring this germplasm to upgrade their local cattle do so either through use of AI with semen from CAIS or alternatively purchase live breeding stock directly from the NSS or other breeding farms. Over the years, this demand for the latter has been recorded through written requests to the farm management for bulls. Recently however, the NSS management has raised concern over its inability to service all the requests for breeding stock. A total of 802 letters were isolated from archived records which represented requests for a total of 5,531 animals from the NSS yielding a rough estimate of 6-7 animals per request where majority of the requests (42%) were for 1-2 animals and an additional 20% are composed of requests for between 3-5 animals. Graphical examination of count of requests for breeding stock for 1971-2007 shows a possible decline in these requests, which is at variance with what management is experiencing. We hypothesize that since the mobile phone boom starting in the early 2000s, demand may have been expressed differently rather than in written form. It would also be expected that as milk prices improve, farmers would increase their demand for breeding stock and conversely, as prices for the animals rise, their demand would decline. Rainfall improves pasture availability and we also hypothesize that this way, farmers are encouraged to increase their stock. To explore these issues more systematically, we fit these monthly count data to Poisson Exponentially Weighted Moving Average (PEWMA), Poisson Autoregressive PAR(p) and poisson models with phone use, milk prices and rainfall as explanatory variables. These models are implemented in R and we use data for the period November 1990 to December 2007. In these models, we use real prices and the price of milk is used in place of the price for breeding animals. We do this for two reasons (i) to avoid multicollinearity since there is a high (+0.98) correlation between sahiwal prices and the price of milk and (ii) we believe that since breeding animals are acquired by farmers to upgrade their local cattle to produce more milk, the price of milk provides more information about the decision to invest in a breeding animal. We begin by examining the ACF plot to identify the presence of dynamics in the data. In addition, zero inflation is negligible and the zero inflation versions of these models are not necessary. The chi-square statistic used to compare the PAR(1) and PAR(2) is not large enough to reject the PAR(1) over the latter. Further, results show that phone use and prices led to reduced number written of requests for sahiwal animals while the contribution of rainfall is positive. The PAR(p) short run multipliers for phone use are computed as -0.77 and -0.67 for the PAR(1) and PAR(2) respectively while the long run multipliers are -0.95 and -0.91. We conclude that phone use may have changed the way demand for breeding animals is expressed.