^{1}

^{*}

^{1}

^{1}

Three methods are considered in this paper: Simple exponential smoothing (SES), Holt-Winters exponential smoothing (HWES) and autoregressive integrated moving average (ARIMA). The best fit model was then used to forecast Zambia’s annual net foreign direct investment (FDI) inflows from 1970 to 2014. Foreign direct investment is foreign capital investment to Zambia. Throughout the paper the methods are illustrated using Zambia’s annual Net FDI inflows. A comparison of the three methods shows that the ARIMA (1, 1, 5) is the best fit model because it has the minimum error. Forecasting results give a gradual increase in annual net FDI inflows of about 44.36% by 2024. Forecasting results plays a vital role to policy makers. Decision making, coming up with good policies and suitable strategic plans, depends on accurate forecasts. Zambian FDI policy makers can use the results obtained in this study and create suitable strategic plans to promote FDI.

Forecasting is key in many fields of science. In this paper, three methods are of interest and they are: SES, HWES and ARIMA. Throughout the paper the methods are illustrated using Zambia’s annual net FDI inflows. The best fit model is used to forecast Zambia’s annual net foreign direct investment (FDI) from 1970 to 2014.

SES method is a simple tool for forecasting time series data. Smoothing implies removing the unwanted noise so that the general path is created. This method is suitable for forecasting data without trend or seasonal pattern. It is basically a recursive computing procedure [

FDI is foreign capital investment to a country. Foreign direct investment to a country results in increasing productivity, reducing unemployment, and increasing the use of technology. The need for FDI came as a result of shortages in domestic funding sources to finance development projects in developing countries. These developing countries realized that it is through FDI that they can achieve economic growth. According to [

Studies by [

The main goal for the Zambia government is to increase and sustain FDI inflows beyond the current levels to highly benefit the country. Zambian FDI inflows are mainly in copper and cobalt extraction, agricultural sector particular in horticulture and floriculture production, and in tourism. Firms or groups of firms from countries like United Kingdom and South Africa have traditionally been the main contributors of foreign direct investment though FDI inflow from other countries drastically increases. The net inflow to other countries is negative indicating outflows that are FDI from those countries (inflows) are less than those from Zambia (outflows). However, the scope of this research is to discuss FDI inflows (Net) to Zambia. An analysis of the FDI flows by source country in 2012 shows that Canada (US $724.3 million), South Africa (US $426.0 million), the Netherlands (US $262.2 million) and the United Kingdom (US $227.2 million), were the major source countries of Zambia’s FDI inflows, accounting for 94.7 per cent of total inflows, collectively. The other source countries are Switzerland (US $166.9 million), China (US $141.9 million), Nigeria (US $94.6 million), Singapore (US $62.0 million), Congo DR (US $28.6 million) and France (US $20.2 million) [

Forecasting results plays a vital role to policy makers. Decision making, coming up with good policies and suitable strategic plans, depends on accurate forecasts [

Simple exponential smoothing method involves smoothing out random fluctuations of time series data. The method is suitable for forecasting data without trend or seasonal pattern. This method gives past data weights known as smoothing constants that decrease exponentially with time. Below is the exponential smoothing model for time series data Χ t is shown below:

X ¯ t = α X t + ( 1 − α ) X ¯ t − 1 (1)

X ⌢ t + h = X ¯ t (2)

Holt-Winters exponential smoothing method is an extension of SES and uses a linear combination of the previous values of a series for generating and modeling future values. It is applies to time series data that has trend. Recent time series recordings are key to forecasting future values of a series. The model for time series data X t is as shown below:

X ¯ t = α X t + ( 1 − α ) ( X ¯ t − 1 + b t − 1 ) 0 < α < 1 (3)

b t = β ( X ¯ t − X ¯ t − 1 ) + ( 1 − β ) b t − 1 0 < β < 1 (4)

where α is the smoothing constant, β is the trend smoothing constants, X t is raw data, X ¯ t is smoothed data and b t is the trend estimates.

The h -step-ahead forecast equation is

X ⌢ t + h = X ¯ t + h b t ( [

Stochastic models attributed to Box-Jenkins known as the ARIMA have been found to be more efficient and reliable even for short term forecasting. Further, stochastic models are distribution-free as no assumptions are required about the data [

AR model: X ⌢ t = ∑ i = 1 p ϑ i X t − i + ε t (6)

MA model: X ⌢ t = ∑ i = 1 q φ i ε t − i , (7)

ARMA model: X ⌢ t = ∑ i = 1 p ϑ i Χ t − i + ε t + ∑ i = 1 q φ i ε t − i (8)

where ϑ t is the autoregressive parameter at time t, ε t is the error term at time t and φ t is the moving-average parameter at time t [

The error measures are used to compare how well models fit the time series. According to [

The SES, HWES and ARIMA models are used to forecasting Zambia’s annual Net Foreign direct investment (FDI) inflows from 1970 to 2014. R is a widely used statistical software package for statistical analysis. It was used to come up with SES, HWES and ARIMA models. R contains built-in functions that allow the user to determine model parameters spontaneously; the only requirement in this software is the time series data to be analysed. Using R, the SES model indicates that the parameter α = 0.73 is the best parameter value. The equation for this model thus takes the form

X ¯ t = 0.73 X t + 0.27 X ¯ t − 1 (9)

The HWES model indicate that the parameters α = 0.31 and β = 0.41 , giving us the following equations:

X ¯ t = 0.31 X t + 0.69 ( X ¯ t − 1 + b t − 1 ) (10)

b t = 0.41 ( X ¯ t − X ¯ t − 1 ) + 0.59 b t − 1 (11)

For ARIMA model, the procedure is achieved by considering the following steps: identification, model selection, parameter estimation and diagnostic check [

Step 1: ARIMA model identification:

Time plot is the first step of ARIMA model identification of time series. A time plot of the FDI is plotted in

Step 2: Model selection

The ACF and PACF plots for d = 1 in

Step 3: Model fitting and Parameter estimation.

R output (version 0.99.903) for estimated parameter and p-value:

Arima (1, 1, 5))

Coefficients:

ar1 ma1 ma2 ma3 ma4 ma5

0.8300 −1.3240 0.4039 0.5289 −0.8271 0.6342

sigma^2 estimated as 27972: log likelihood = −290.67, aic = 595.33

The parameters found to be significance at 5% in

Criteria | Formula | Criteria | Formula |
---|---|---|---|

MPE | RMSE | ||

MAE | MAPE | ||

MASE |

TENTATIVE MODEL | ARIMA (1,1,5) | ARIMA (5,1,1) | ARIMA (1,1,3) | ARIMA (5,1,0) | ARIMA (0,1,2) | ARIMA (3,1,0) | ARIMA (1,1,0) | ARIMA (0,1,5) |
---|---|---|---|---|---|---|---|---|

AIC | 595.33 | 598.89 | 599.69 | 599.76 | 600.25 | 600.29 | 600.69 | 600.74 |

TENTATIVE MODEL | ARIMA (0,1,3) | ARIMA (4,1,0) | ARIMA (0,1,1) | ARIMA (0,1,4) | ARIMA (2,1,0) | ARIMA (1,1,1) | ARIMA (2,1,2) | |

AIC | 601.55 | 601.68 | 602.01 | 602.48 | 602.67 | 602.68 | 603.09 |

therefore be written as

X ⌢ t = 0.830 X t − 1 − 1.324 ε t − 1 + 0.404 ε t − 2 + 0.529 ε t − 3 − 0.827 ε t − 4 + 0.634 ε t − 5

Step 4: Diagnostic Checking

Goodness of fit for time series models involves testing if the model residuals form a white noise process. It is through diagnostic checks that a model can be declared statistically adequate and thereafter can be used to forecast. According to [

The Plots of ACF, Normal Q-Q and Histogram of Residuals show that the residual are a white noise process. Thus, diagnostic check for an ARIMA (1,1,5) model in

The results in

Forecasting results plays a vital role to policy makers in creating good policies and coming up with suitable strategic plans on FDI. R output of ARIMA (1,1,5) forecasts for the next 10 years of annual net Zambia’s FDI’s inflow is shown in

forecasts in the period from 2014 to 2024 is shown in

Forecasting is key to every field of science. ARIMA (1, 1, 5) can be used to forecast the annual net inflows of FDI to Zambia. This model can be used for both

Variable | Coefficient | p-value |
---|---|---|

Constant | ||

AR (1) | 0.830 | 0.000000036 |

MA (1) | −1.324 | 0.000000000060 |

MA (2) | 0.404 | 0.10 |

MA (3) | 0.529 | 0.019 |

MA (4) | −0.827 | 0.0013 |

MA (5) | 0.634 | 0.00015 |

Note: *p value < 0.05.

Measure of accuracy | SES | HWES | ARIMA (1, 1, 5) |
---|---|---|---|

MPE | −4.573 | 266.981 | 165.380 |

RMSE | 213.696 | 133.025 | 76.74 |

MAE | 111.912 | 2.659 | −9.478 |

MAPE | 45.413 | 47.617 | 39.960 |

MASE | 0.929 | 1.103 | 0.637 |

Year | Point Forecast | Lo 80 | Hi 80 | Lo 95 | Hi 95 |
---|---|---|---|---|---|

2015 | 2353.979 | 2135.315 | 2572.644 | 2019.561 | 2688.398 |

2016 | 2442.925 | 2194.337 | 2691.512 | 2062.743 | 2823.106 |

2017 | 2757.575 | 2482.147 | 3033.004 | 2336.343 | 3178.807 |

2018 | 2665.618 | 2304.685 | 3026.552 | 2113.619 | 3217.618 |

2019 | 2806.015 | 2415.494 | 3196.537 | 2208.764 | 3403.267 |

2020 | 2922.545 | 2477.628 | 3367.463 | 2242.103 | 3602.988 |

2021 | 3019.266 | 2500.801 | 3537.732 | 2226.341 | 3812.191 |

2022 | 3099.545 | 2495.403 | 3703.686 | 2175.590 | 4023.499 |

2023 | 3166.176 | 2469.682 | 3862.670 | 2100.980 | 4231.372 |

2024 | 3221.481 | 2429.585 | 4013.377 | 2010.380 | 4432.581 |

short and long term forecasting. Best strategies can only be created with accurate forecasting results. Studies have shown that FDI affects the growth of GDP. Therefore, the importance of FDI is acknowledged world over. FDI also helps diversify the country’s economy (through job creation and increase in productivity), increase export in host country, improve efficiency and have technological spillovers on the already existing firms.

Three models of univariate time-series analysis were considered in this study: SES, HWES and ARIMA models. The best fit of the three models used in this study was picked based on the model indicating minimum errors. The ARIMA (1,1,5) showed smallest error than that of the SES or HWES models. Forecasting results give a gradual increase in annual net FDI inflows of about 44.36% by 2024. Policy makers use accurate forecasts to come up good policies. Therefore, the Zambian government should use such forecasts in formulating policies and making strategies that will promote FDI industry. Future research should go further and consider non-linear models such as Autoregressive Conditional Heteroscedasticity (ARCH), Generalized Autoregressive Conditional Heteroscedasticity (GARCH).

The authors are thankful to Zambia Development Agency (ZDA) for providing time series data on FDI. Many thanks also go to the Dean, School of Science, Engineering and Technology Dr Douglas Kunda for the encouragements. Not forgetting Mulungushi University for making it possible through provision of resources to come up with this research work. Also many other colleagues who made good comments on this paper.

Jere, S., Kasense, B. and Chilyabanyama, O. (2017) Forecasting Foreign Direct Investment to Zambia: A Time Series Analysis. Open Journal of Statistics, 7, 122-131. https://doi.org/10.4236/ojs.2017.71010