**4. ARIMA model**

ARIMA models are the most general class of models for forecasting a time series that can be stationary via transformations, such as differencing and lagging. Determine the order(s) of difference required to stabilize the series before determining the best ARIMA model for it.

#### **4.1 Box: Jenkins modeling method**

The Box–Jenkins model is one of the classes of models to choose from the systematic approach to identify the correct model form. There are statistical tests for the validity of the model. Using this test, it is possible to identify the best appropriate Box–Jenkins model for fitting the data on the milk production at Andassa dairy farm.

### **5. Result and discussion**

From the average daily milk production table, the minimum and maximum records of milk production at Andassa dairy farm are **715.0** and 2295.0, respectively, and the mean for the average daily milk production is 1416.9 (**Table 1**).

Test of randomness by using the different sign test.

Test of hypothesis.

H0: Data daily recorded in the average milk production is random. H1: רH0.

Test statistics.

$$Z\_{cal} = \frac{w - E(w)}{\sqrt{Var(w)}}$$

where E wð Þ¼ **<sup>1</sup> <sup>2</sup>** ð Þ N � 1

$$\text{var}\,\left(\mathbf{w}\right) = \frac{\mathbf{1}}{\mathbf{12}}\left(\mathbf{N} + \mathbf{1}\right)$$

From this, we have the number of points increase W = 81



**Table 1.** *Average daily milk production.*

$$\sqrt{Var(w)} = \sqrt{\mathbf{15}} = \mathbf{3.87}$$

Therefore: *Zcal* <sup>¼</sup> <sup>81</sup>�<sup>89</sup> <sup>3</sup>*:*<sup>87</sup> ¼ �2*:*0672*:*

Note: Level of significance used in this case is α = 0.05.

Test rule:

We reject the null hypothesis since <sup>∣</sup>*Zcal*<sup>∣</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>0672</sup> <sup>&</sup>gt;*<sup>Z</sup> <sup>α</sup>* <sup>2</sup> ¼ 1*:*96. Thus, at five percent (5%) level of significance, the data on the average daily milk production is not random. This means that there is a systematic pattern. Meaning, the dissemination of the data is not balanced and it must be transformed into random form by using different methods.
