**4. Results and discussion**

and then converted to annual terms (1901–2015). Due to migration, data availability issues, the annual population data (rural, urban and total populations) was collected from the World Bank. The vital statistics method has been used to indirectly measure the net internal migration (rural-unban) of these countries. The yearly natural growth rate of the population in each country accounted for and through this process the migration data were obtained. The time series data in this model consist of four variables that are Rural Migration (MR), Urban Migration (MU), Rainfall (Rain) and Temperature (Temp). The data were initially prepared in excel spreadsheets and then transferred to Stata. Stata is a comprehensive, integrated statistical software package that is used for data management, data analysis and to produce graphics to visualize the data; Stata was developed in 1985 by Stata Corporation. Hashizume et al. [74] has utilized Stata software for rainfall and temperature time series analyses, similarly Ebrahim et al. [75] has suggested the appropriateness of Stata software for the analyses of

The vector auto-regression (VAR) model is among the most flexible multivariate analysis, well established, and proven models in multivariate time series analysis. Ideally, the model provides a natural extension from the univariate autoregressive models to multivariate models that include deterministic, endogeneous and exogeneous variables. The VAR has a multivariate advantage, for the forecasts developed can be made conditional on the potential future trends in the other given variables. The proposed study will centre on the analysis of covari-

represents a time series variable vector (*k* × 1) then an autoregressive model of the basic vector,

*Yt* = *c* + ∏<sup>1</sup> *Yt*−<sup>1</sup> + ∏<sup>2</sup> *Yt*−<sup>2</sup> + …+∏*<sup>p</sup> Yt*−*<sup>p</sup>* + *ut* (3)

vector process for white noise with an unobservable zero mean, with ∑ covariance matrix. The Stata software was used to determine the VAR lags most appropriate for each country. The lag selection process was performed using VAR diagnostics and tests method that enables to postestimate the VAR lag order of the time series data. VAR (3) has been the best fit in most cases.

A key part of the use of a VAR model is in its use for forecasting. Its structure gives information on the ability of variables or variable groups to forecast other variables. Granger [76]

introduced this intuitive notion of a variable's ability to forecast. Should Y1

*Yt*−*<sup>p</sup>* + *β*<sup>11</sup> *Xt*−<sup>1</sup> + *β*12*Xt*−<sup>1</sup> + …+*β*1*<sup>p</sup>*

*Yt*−*<sup>p</sup>* + *β*21*Xt*−<sup>1</sup> + *β*22*Xt*−<sup>1</sup> + …+*β*2*<sup>p</sup>*

are coefficient matrices, *c* a (*<sup>k</sup>* <sup>×</sup> 1) constant vector and *ut*

,  *Y*2,*<sup>t</sup>*

*Xt*−*<sup>p</sup>* + *ut* (1)

*Xt*−*<sup>p</sup>* + *ut* (2)

variable or variable

, …., *Yn*,*<sup>t</sup>*) ′

a (*k* × 1)

ance stationary multivariate time series using VAR models. Suppose that *Yt* <sup>=</sup> (*Y*1,*<sup>t</sup>*

migrations time series data.

having order *p,* VAR (*p*) becomes:

where *t* = 1, …, *T* and (*<sup>k</sup>* <sup>×</sup> 1) and ∏*<sup>i</sup>*

**3.3. Granger causality test**

OR

*Yt* = *μ*<sup>1</sup> + *α*11*Yt*−<sup>1</sup> + *α*12*Yt*−<sup>1</sup> + …+*α*1*<sup>p</sup>*

*Xt* = *μ*<sup>2</sup> + *α*21*Yt*−<sup>1</sup> + *α*22*Yt*−<sup>1</sup> + …+*α*2*<sup>p</sup>*

**3.2. Multivariate time series analysis: (VAR)**

16 Applications in Water Systems Management and Modeling

Descriptive statistics briefly summarizes the rural and urban migration, rainfall and temperature time series data (1962–2015) of Democratic Republic of Congo, Kenya and Niger. **Table 5** presents central tenancy, data variabilities as well as spread of the data.

The following section presents VAR results of the analysis time series data of MR, MU, Rain and Temp of DRC, KEN and NER to assess how climate change variables Rain and Temp time series may impact rural-urban migration phenomenon of these countries in the future. The first step of the VAR analysis was the lag selection process. The goodness of fit statistics of


**Table 5.** Descriptive statistics summary of MR, MU, rain and temp time series data (1962–2015).

the VAR model gave the R2 values shown in **Table 6**. The test of the model fit gave favourable results. As shown in **Table 6**, most of R2 values are above 0.70.
