**3.1. Data sources and procedures**

Three Sub-Saharan African countries were selected to test how climate change factors and population growth have impacted people movement (rural-urban migration). The Sub-Saharan African countries involved in this study are Democratic Republic of Congo (DRC), Kenya (KEN) and Niger (NER). The data collected was over 56 year period from 1960 to 2015. The climate change proxy variable (rainfall and temperature) and the data were collected on monthly bases from the National Oceanic and Atmospheric Administration (NOAA) of USA, 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 migrations time series data.

groups be instrumental in another Y2

based on (Y2,t

A VAR(p) bivariate model for Yt <sup>=</sup> (Y1t

error of a forecast of Y2,t+s

event that both Y2 and Y1

**4. Results and discussion**

. In the opposite case, Y1 does not Granger cause Y2

, Y2,t−<sup>1</sup> , ….

, Y1,t−<sup>1</sup>

can test p linear restrictions on coefficients. The coefficient matrices of VAR are diagonal in the

causality is rather useful in finance and continues to be extensively used because it shows bidirectional as well as uni-directional causal nature of the time series data. In essence, how and in what manner the other variables contribute to the prediction process of a given variable;

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**

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

**Variable Min Max Mean Range P50 SD Variance** DRC\_MR −115,821 79,940 6138.33 195,761 7663.5 37464.27 1.40E^9 DRC\_MU −47,403 59,624 12386.24 107,027 9040 20682.77 4.28E^8 DRC\_Rain 1208.6 1728.9 1488.00 426.5 1494.6 67.3 4534.5 DRC\_Temp 23.60 25.00 24.06 1.299999 24.00 0.28 0.080 KEN\_MR −115,821 79,940 6138.33 195,761 7663.5 37464.27 1.40E^9 KEN\_MU −47,403 59,624 12386.24 107,027 9040 20682.77 4.28E^8 KEN\_Rain 429.9 1005.7 645.60 527.4 640.8 100.7 10136.1 KEN\_Temp 23.60 25.60 24.59 2 24.50 0.34 0.114 NER\_MR −4708 18,450 5396.98 23,158 5649 5420.30 2.94E^7 NER\_MU −13,695 7510 1782.02 21,205 1656.5 3708.77 1.38E^7 NER\_Rain 106.8 269.1 185.60 162.3 183.2 32.3 1041.5 NER\_Temp 26.20 28.80 27.34 2.199999 27.20 0.49 0.240

, given that all p VAR matrices of coefficients are lower triangular. The Wald statistic

) and ( Y1,t

, Y2t ) ′

that is, which variables or which information is crucial for the prediction process.

presents central tenancy, data variabilities as well as spread of the data.

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

based on on (Y2,t

ity notion only suggests the ability to influence another variable.

, Y2,t−<sup>1</sup> , ….

causes Y2

cause Y1

forecast of Y2,t+s

variable's or variable group's prediction, then Y1

The Effects of Climate Change on Rural-Urban Migration in Sub-Saharan Africa (SSA)—The…

, can be used to identify the failure of Y2

fail to Granger cause each other. It is important to note that Granger

Granger

17

to Granger

if for all s > 0 the mean squared

http://dx.doi.org/10.5772/intechopen.72226

) is the same as the mean squared error of a

, ….). It is worth noting that Granger's causal-

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

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 covariance stationary multivariate time series using VAR models. Suppose that *Yt* <sup>=</sup> (*Y*1,*<sup>t</sup>* ,  *Y*2,*<sup>t</sup>* , …., *Yn*,*<sup>t</sup>*) ′ represents a time series variable vector (*k* × 1) then an autoregressive model of the basic vector, having order *p,* VAR (*p*) becomes:

$$Y\_{\boldsymbol{t}} = \boldsymbol{\mu}\_{1} + \boldsymbol{\alpha}\_{11}Y\_{\boldsymbol{t}-1} + \boldsymbol{\alpha}\_{12}Y\_{\boldsymbol{t}-1} + \dots + \boldsymbol{\alpha}\_{1p}Y\_{\boldsymbol{t}-p} + \boldsymbol{\beta}\_{11}X\_{\boldsymbol{t}-1} + \boldsymbol{\beta}\_{12}X\_{\boldsymbol{t}-1} + \dots + \boldsymbol{\beta}\_{1p}X\_{\boldsymbol{t}-p} + \boldsymbol{u}\_{\boldsymbol{t}} \tag{1}$$

$$X\_{\iota} = \mu\_{2} + \alpha\_{21}Y\_{\iota-1} + \alpha\_{22}Y\_{\iota-1} + \dots + \alpha\_{2p}Y\_{\iota-p} + \beta\_{21}X\_{\iota-1} + \beta\_{22}X\_{\iota-1} + \dots + \beta\_{2p}X\_{\iota-p} + u\_{\iota} \tag{2}$$

OR

$$Y\_t = c + \prod\_1 Y\_{t-1} + \prod\_2 Y\_{t-2} + \dots + \prod\_p Y\_{t-p} + u\_t \tag{3}$$

where *t* = 1, …, *T* and (*<sup>k</sup>* <sup>×</sup> 1) and ∏*<sup>i</sup>* are coefficient matrices, *c* a (*<sup>k</sup>* <sup>×</sup> 1) constant vector and *ut* a (*k* × 1) 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.

#### **3.3. Granger causality test**

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 variable or variable groups be instrumental in another Y2 variable's or variable group's prediction, then Y1 Granger causes Y2 . In the opposite case, Y1 does not Granger cause Y2 if for all s > 0 the mean squared error of a forecast of Y2,t+s based on on (Y2,t , Y2,t−<sup>1</sup> , …. ) is the same as the mean squared error of a forecast of Y2,t+s based on (Y2,t , Y2,t−<sup>1</sup> , …. ) and ( Y1,t , Y1,t−<sup>1</sup> , ….). It is worth noting that Granger's causality notion only suggests the ability to influence another variable.

A VAR(p) bivariate model for Yt <sup>=</sup> (Y1t , Y2t ) ′ , can be used to identify the failure of Y2 to Granger cause Y1 , given that all p VAR matrices of coefficients are lower triangular. The Wald statistic can test p linear restrictions on coefficients. The coefficient matrices of VAR are diagonal in the event that both Y2 and Y1 fail to Granger cause each other. It is important to note that Granger causality is rather useful in finance and continues to be extensively used because it shows bidirectional as well as uni-directional causal nature of the time series data. In essence, how and in what manner the other variables contribute to the prediction process of a given variable; that is, which variables or which information is crucial for the prediction process.
