EFFECTS OF CLIMATE CHANGE ON PRODUCTIVITY OF SORGHUM YIELD AT MELKASA DISTRICT, EAST SHEWA ZONE: A MULTIVARIATE TIME SERIES MODELLING APPROACH MSc. THESIS BY ABERASH AYLADO HAWASSA UNVERSITY HAWASSA, ETHIOPIA JUNE, 2017 EFFECTS OF CLIMATE CHANGE ON PRODUCTIVITY OF SORGHUM YIELD AT MELKASA DISTRICT, EAST SHEWA ZONE: A MULTIVARIATE TIME SERIES MODELLING APPROACH MSc. THESIS BY ABERASH AYLADO A THESIS SUBMITTED TO THE SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES AT HAWASSA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN APPLIED STATISTICS HAWASSA UNIVERSITY HAWASSA, ETHIOPIA JUNE, 2017 Approval Sheet 1 This is to certify that the thesis entitled “Effects of Climate Change on Productivity of Sorghum Yield at Melkasa District, East Shewa Zone: A Multivariate Time Series Modeling Approach” submitted in partial fulfillment of the requirement for the degree of Master of Science in Applied Statistics to the school of Mathematical and Statistical Sciences, Hawassa University, and record of original research carried out by Aberash Aylado Gelebo, ID No. PGAS/001/08, under my supervision and no part of the thesis has been submitted for another degree or diploma. The assistance and the help received during the course of this investigation have been duly acknowledged. Therefore, I recommended that it may be accepted as fulfilling the thesis requirement. ________________________________ Name of Advisor ________________________________ Name of Co-advisor ___________________ ___________________ Signature Date ___________________ Signature ___________________ Date Approval Sheet 2 We, the undersigned, members of the Board of Examiners of the final open defense by Aberash Aylado Gelebo have read and evaluated her thesis entitled “Effects of Climate Change on Productivity of Sorghum Yield at Melkassa District, East Shewa Zone: A Multivariate Time Series Modeling Approach” and Examined the candidate. This is therefore to certify that the thesis has been accepted in partial fulfillment of the requirement of the degree of Master of Science in Applied Statistics. Approved by: ________________________________ Name of School Head Signature ________________________________ Name of Advisor ___________________ ___________________ Date ___________________ Date ___________________ Signature Date ________________________________ Name of Internal Examiner ___________________ Signature ________________________________ Name of External Examiner ___________________ ___________________ Signature ___________________ Date Declaration This thesis has been submitted to School of Mathematical and Statistical Sciences at Hawassa University in partial fulfillment of the requirements for Master of Science degree in Applied Statistics. I declare that this thesis has not been submitted to any other institution and anywhere for the award of any academic degree, diploma or certificate. ________________________________ Name of Student Hawassa University Hawassa, Ethiopia ___________________ Signature ___________________ Date Acknowledgements First of all, I would like to express my deepest thank to my almighty God for his untold and all time grace that gave me courage to start and finish this thesis work. I am also most grateful to my advisor, Dr. Zeytu Gashaw for his continual advice, useful comments and suggestions throughout the preparation of this paper. I am indebted also to thank my Coadvisor Mr. Nigatu Degu for his critical comments and encouragement in the development of the idea of this thesis. I would also extend my inner thanks to National Meteorology Service Agency of Melkassa district and Melkassa Agricultural Research Center officials and researchers for providing me with the necessary data for this thesis. Lastly, my gratitude goes to my whole family for their moral support. List of Abbreviations ACF Auto Correlation Function ADF Augmented Dickey – Fuller AIC Akaike Information Criteria EIAR Ethiopian Institute of Agricultural Research HQIC Hannan-Quinn Information Criteria IMF International Monetary Fund IRF Impulse Response Function JB Jarque-Bera LM Lagrange Multiplier LSE London School of Economics MARC Melkassa Agricultural Research Center MoARD Ministry of Agriculture and Rural Development MoFED Ministry of Finance and Economic Development NMSA National Meteorology Service Agency PACF Partial Autocorrelation Function PP Phillips – Perron Q/h Quintals per hectare SBIC Schwarz-Bayesian Information Criteria UNSCEB United Nations System Chief Executive Board for Co-ordination USAID United States Agency for International Development USE United States Embassy VAR Vector Autoregressive VEC Vector Error Correction List of Tables Table 4.1.Descriptive Statistics . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . .. . 34 Table 4.2. Correlation between the Variables . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . 35 Table 4.3. Lag order Selection Criteria Values . . . . . . . . . . . . .. . . … . . . . . . . . .. . . . . . .. . . . 37 Table 4.4. ADF and PP Unit Root Tests of original Series . . . . .. . . . . . . .. . . . .. . . . . . .. . . . . 38 Table 4.5. Turning Point Test of Randomness . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . .. . . . 39 Table 4.6. Response of Yield . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Table 4.7. Response of Rainfall . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . 45 Table 4.8. Response of Minimum Temperature . . . . . . . . . . . .. . . . . . . . . . . . ... . . . . . ... . . . . 47 Table 4.9. Response of Maximum Temperature. . . . . . . . . . . . .. . . . . . . . . . . . .. .. . . ... . . . . . . 49 Table 4.10. Response of Wind Speed . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 51 Table 4.11. Response of Relative Humidity . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . .. . . . . 52 Table 4.12. Response of Sunshine Duration . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Table 4.13.Forecast Error Variance Decomposition Function of Yield . . . . . .. . . . . . . . . . . . . 56 Table 4.14.Forecast Error Variance Decomposition Function of Rainfall .. . . . .. . . . . . . . .. . 57 Table 4.15.Forecast Error Variance Decomposition Function of Minimum Temperature . . . ... 58 Table 4.16 .Forecast Error Variance Decomposition Function of Maximum Temperature. .. . . 59 Table 4.17.Forecast Error Variance Decomposition Function of . . . .. . . . . .. . . . . . . . . . . . . . . 60 Table 4.18.Forecast Error Variance Decomposition Function of Relative Humidity . . . . . . . . 61 Table 4.19 .Forecast Error Variance Decomposition Function of Sunshine Duration . . . . . . . . 61 Table 4.20 .Lagrange-multiplier test for Residual Autocorrelation . . . . . . . . . . . . . . . . . . . . .. . 62 Table 4.21.Jarque-Bera test for Normality of Residuals .. . . . . .. . . . . . . . . . . . .. . . . . . . . . .. . 63 Table 4.22 . Stability Condition Test…………………………………………………………… 64 List of Figures Figure 1. Map of the Study Area . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 2. Impulse Response Function graph of Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Figure 3. Impulse Response Function graph of Rainfall . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 46 Figure 4. Impulse Response Function graph of Minimum Temperature . . . . . . . . . . . . .. . . . . 48 Figure 5. Impulse Response Function graph of Maximum Temperature . . . . . . . . . .. . . . . . . . 50 Figure 6. Impulse Response Function graph of Wind Speed . . . . . . . . . . . . . . . . . .. . . . . . . . . 52 Figure 7. Impulse Response Function graph of Relative Humidity . . . . .. . . . . . . . . . . . . . . . . 53 Figure 8. Impulse Response Function graph of Sunshine Duration . . . . . . . . . . . . . . . . . . . . . 54 Figure 9.VAR(1) Forecast Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 List of Appendices APPENDIX A: Values of Yield, Rainfall, Temperature (Both Minimum and Maximum), Wind Speed, Relative Humidity and Sunshine Duration Data for Years 1967-2016 at Melkasa District . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . 76 Appendix B: Grangers’ Causality Test Results . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . 78 APPENDIX C: Time Series Plot of Original Series . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 81 Appendix D: Plot for Test of Randomness of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Appendix E: ACF and PACF Plots of Residuals . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Appendix F: Normal Probability Plots of Residuals .. . . . . . . . ………….. . . . . . .. . . . . . . .. 102 Contents Acknowledgments i List of Abbreviations ii List of Tables iv List of Figures v List of Appendices vi Abstract xi 1. INTRODUCTION. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . 1 1.2. Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 1.3. Objectives of the Study . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1. General Objective of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2. Specific Objectives of the Study . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4. Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5. Scope of the Study . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6. Limitation of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7. Definition of Terms . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . .6 2. LITERATURE REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . 7 2.1. General Overview of Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7 2.2. Causes and Consequences of Climate Change on Crop . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. Effects of Climate Change on Sorghum Production in Africa . . . . . . . . . . .. . . . . . . . . . . . . . ... 9 2.4. Effects of Climate Change on Sorghum Production in Ethiopia . . . . . . . . . . . . . . ... . . . . . . . 10 2.5. Effects of Climate Change on Sorghum Production in Oromia Region . . ... . . . . . . . . . . . . . 11 2.6. Effects of Climate Change on Sorghum Production in Melkassa District . . … . . . . . . . . . . 12 2.7. Review of Empirical Studies . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . 12 3. DATA AND METHODOLOGY. . . . . . . . . . . . . . . . . . . . . . . . . …. . . . . . . . . . . . . . . . . . . . . 15 3.1. Description of the Study Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . 15 3.2.Statistical Data Description . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ……… . . 16 3.3. Study Variables . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . 16 3.4. Source of Data . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . 17 3.5. Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . 17 3.5.1. Model Description . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . 17 3.5.2. Multivariate Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. … 17 3.5.3. Stationary Time Series Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 19 3.5.4.Computation of Autocovariance and Autocorrelations of Stable VAR Processes …... . . 22 3.5.5. Structural Vector Autoregressive (SVAR) Measures . . . . . . . . . . . . . . . . . . . ... . . 23 3.6. Vector Error Correction and Cointegration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6.1. VEC Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6.2. Testing for Cointegration . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 28 3.7. VAR Order Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.8. Assumptions of VAR Model . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . …. . 31 3.9. Model Adequacy Checking . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ….. 31 4. 3.9.1. Checking the Whiteness of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.9.2. Testing for Normality of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . .33 4.1. Descriptive Analysis . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1. Looking Over Nature of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.1.1. Time Series Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.2. Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.3. Test of Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1.4. Auto Correlation and Partial Autocorrelation Functions of the Series . . . . .. . . . . . 39 4.1.5. Cointegration Rank Test . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.6.VAR Order Selection and Estimating Model Parameters . . . . . . . . . . . . . . . . . . . . …. 40 4.1.7.Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . 42 4.1.8.Model Diagnostic Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . … .. . 62 4.1.9. 5. 5.1. Checking for VAR Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 CONCLUSION AND RECOMMENDATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 66 Conclusion . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2. Recommendations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 68 APPENDICES. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . 76 Abstract Climate and other environmental changes in the developing world and the African continent has become a major threat to their agricultural economy as agriculture is the most dominating sector in the national economy. Increasingly, empirical evidences are substantiating the effects of climate change on agricultural production. This study is aimed in examining the impact of climate change on sorghum yield production at Melkassa district. A total of 50 years observation on autumn total rainfall, average wind speed, relative humidity, sunshine duration, minimum and maximum temperatures and sorghum yield for a period of 1967_2016 were used. A multivariate time series model (Vector Autoregressive model) was used to fit the data and structural analysis was also made using impulse response function and forecast variance decomposition. The results obtained revealed that during the autumn cropping season, rainfall, minimum temperature, maximum temperature, wind speed, relative humidity, sunshine duration and yield showed high variability from year to year with 26.3%, 38.6%, 10.3%, 24.3%, 13.6%, 23.1% and 66.1% coefficient of variation respectively. Also, yield, rainfall, minimum and maximum temperatures, wind speed, relative humidity and sunshine duration shocks have significant impacts on the occurrence of one another. Key Words: Climate Change; Sorghum (Gambella#1107 variety); VAR; Melkassa, East Shewa 1. INTRODUCTION 1.1.Background of the Problem In recent times, the issue of climate change through extreme temperature, frequent flooding, drought and increased salinity of water used for irrigation has become a recurrent subject of global debate. The intensity of the debate is on the increase due to the enormity of the challenge posed by the phenomenon especially in the third world. This is as a result of the widespread poverty, prevailing slash-and-burn agriculture, green house emission, erosion and burning of firewood and farm residues that characterize the developing economies (Ajetombi & Abiodun, 2010). Climate is considered as the major controlling factor of life on earth which is continuously changing due to natural forces as well as anthropogenic activities like land use changes and emission of greenhouse gases and aerosols. It is one of the most serious environmental threats facing mankind worldwide; affecting mankind in several ways including its direct impact on food production (Enete & Amusa, 2010). Atmospheric temperature, rainfall, humidity, solar radiation etc. are dominant climatic factors closely linked with agricultural production that forms the economic base of the whole world (Basak, 2011). Africa is among the most vulnerable countries and disproportionately affected region in the world in terms of climate change. It is one of the continents that will be hard hit by the impact of climate change, though the continent represents only 3.6% of emissions (Alemneh, 2011). In the region, farming is undertaken mainly under rain-fed conditions, increasing land degradation and low levels of irrigation (6% compared to 38% in Asia) (FAO, 2011). Also, the contribution of agriculture to the gross domestic product in Africa is far higher than in developed regions. This is perhaps nowhere more obvious than in sub Saharan Africa, where economies are extremely sensitive to environmental and/or economic shocks in the agricultural sector. Sub-Saharan Africa is arguably the most vulnerable region to many unpleasant effects of climate change due to a very high dependence on rain-fed agriculture. Thus, the impacts of climate change are likely to fall unreasonably on poorer nations and on poorer households. Among the developing countries in Sub-Saharan Africa that are highly being affected by climate change, Ethiopia is the one with direct impacts of this change on different sectors and areas. According to (World Bank, 2012); (Conway & Schipper, 2011), the country is extremely vulnerable to drought and natural disasters such as flood, heavy rain, frost and heat weaves due to its great reliance on climate vulnerable economy. In the country, climate variability already negatively impacts livelihoods and this is likely to continue. Drought is the single most destructive climate-related natural hazard in Ethiopia. Estimates suggest climate change may reduce Ethiopia’s GDP up to 10% by 2045, primarily through impacts on agricultural productivity. These changes also hamper economic activity and aggravate existing social and economic problems (LSE,2015); (NCEA, 2015); (USE, 2016); (USAID, 2013); (World Bank, 2016). According to (IMF, 2012), agricultural sector remains a key source of growth in Ethiopia but it continues to face major challenges. Rural livelihoods remain extremely vulnerable to climatic shocks as food production is mainly dependent of natural rainfall and irrigation supports only negligible portion of the country’s total cultivated land. As a result, the amount and temporal variation of rainfall and other climatic factors during the growing season are critical to crop yield and can induce food shortage and famine. This shows that climate change and variability can have greater negative impacts on poor farm households due to high vulnerability leading to food insecurity. Hence, this extreme weather because of the impact of climate change causes the loss of peoples’ and livestock’s live, livelihoods of farmers and their properties disrupts. Ethiopian economy is an agrarian economy as agriculture comprises about 41.3 % of GDP generates 90 % of foreign exchange earnings and employs more than 80 % of the population (MoFED, 2012). Currently, however, the performance of this sector is seriously eroded due to climate change induced problems. It is estimated that in Ethiopia, one drought event in 12 years lowers GDP by 7 to 10 % and increases poverty by 12 to 14 % (Makombe et al., 2011). The projected reduction in the Ethiopian agricultural productivity due to climate change can reduce average income by 30% over the next 50 years (Gebreegziabher et al., 2011). Climate change can also have a significant impact on the urban dweller in terms of higher food prices, limited job opportunities in the agro-processing industries and expensive imported food items due to foreign exchange shortages (Aragie, 2013). In addition to this, it can cause a decline in biodiversity, increases in human and livestock health problems, rural-urban migration and dependency on external supports (Daniel, 2008). In turn, Ethiopia faces numerous development challenges that exacerbate its vulnerability to climate change including high levels of food insecurity and ongoing conflicts over natural resources become a very important development challenge in Ethiopia. In the country, the distribution of rainfall varies over the diverse agro-ecological zones. Mean annual rainfall ranges from about 2,000 millimeters over some areas in the south west to less than 250 millimeters over the Afar lowlands in the northeast and Ogaden in the southeast. Mean annual temperature varies from about 10oC over the highlands of the northwest, central, and southeast to about 35oC on the north-eastern edges. In addition to variations across the country, the climate is characterized by a history of climate extremes such as drought and flood and increasing trends in temperature and a decreasing trend in precipitation. More generally, it is stated by National Meteorological Agency that, in Ethiopia, agriculture, water and range resources, biodiversity and human health are vulnerable to climate variability and change with huge social and economic impacts (EPA, 2011). In Oromia region, many areas are prone to climate effects with reduction in agricultural productivity of the region. According to (Leta, 2011), in West Shewa Zone, climate has been changing and the challenge to rain fed crop production has increased from time to time. The long term temporal trend analysis of climate variables; rain fall and temperature shows considerable variability in the area. Moreover, as rain fed crop is highly dependent on rainfall, small shock in weather has significant impact on production of small holder farmers of the Zone since it is compounded by other exacerbating factors such as land degradation and household income. Melkassa is one of the populated areas in Oromia region experiencing the same challenge. The area is experiencing higher temperature and receives less rainfall. It is the district in central rift valley of Ethiopia which is mostly being affected by rainfall variability and change and other meteorological shocks. According to (Tigist, 2011), in Melkassa district, climate change has affecting the rain fed crop (sorghum yield) productivity with high variability of climate parameters (rainfall and temperature). In the area, rainfall and yield are highly variable and also rainfall shock has significant impact on rainfall, temperature and yield; temperature shock has a significant impact on temperature, rainfall and yield and also yield shock has a significant impact on yield. These are some of the facts that may initiate the researcher to conduct this study in the area. 1.2. Statement of the Problem Melkassa, one of the districts found in East Shewa Zone is vulnerable to the changes and variability in climatic conditions since it is one of the areas found in central rift valley which is characterized by extensive areas of low and erratic rainfall and limited areas receiving adequate rainfall (Jansen & Hube, 2011). In Melkassa district, the localized temporal rainfall and temperature variation during different cropping seasons induces an important challenge to crop production and in turn to food security. Despite unpredictable rainfall, the area has a vital importance for the national food security through production of crops like maize, teff, haricot beans, sorghum etc. According to (Hirut & Kindie, 2015) who analyzed the risks in crop production due to climate change in four districts at central rift valley of Ethiopia of which Melkassa is the one, there is a higher variability in temperature and rainfall during the cropping seasons at the district resulting in significant effects on the crop production of the area. Also, according to (Tigist, 2011) who assessed the effect of climate variability on production of sorghum at Melkassa using multivariate time series approach, there existed high inter annual variability in summer rainfall total which is evidence to climate variability in the study area and also sorghum yield is highly variable with this change. Also, past year and following year rainfall, temperature and sorghum yield are autocorelated and rain water is a principal component in determining sorghum yield production at the area. Thus, since previous studies which were conducted in the district have not addressed the question “what effects climate parameters (rainfall, temperature, humidity, wind and sunshine duration) have in combination on the productivity of sorghum?” but focused onlyon the effects that rainfall and temperature have on this yield, there is greatest motivation towards this study in filling this gap and hence, at the end of this study, the following questions were answered. What are the interaction effects among the climatic variables (temperature (minimum and maximum), rain fall, relative humidity, wind speed and sunshine duration) and sorghum yield at Melkassa district? What are the dynamic interrelationships over time among rainfall, temperature, humidity, wind, sunshine duration and sorghum crop? What will the future productivity level of sorghum crop be for the coming 10 years? 1.3. Objectives of the Study 1.3.1. General Objective of the Study The main objective of this study is to assess effects of climate variability on sorghum yield productivity at Melkassa district. 1.3.2. Specific Objectives of the Study The specific objectives of this study include: To see the interaction effects among the climatic variables (temperature (minimum and maximum), rain fall, relative humidity, wind speed and sunshine duration) and sorghum yield at Melkassa district To evaluate the dynamic relationships over time among rainfall, temperature, humidity, wind, sunshine duration and sorghum crop? To predict future productivity level of sorghum crop for the coming 10 years. 1.4. Significance of the Study This study is helpful in providing relevant information on the burden that climate change has and its severity on the sorghum crop productivity at Melkassa district. Furthermore, it can provide governmental, nongovernmental, researchers and policy makers with climate related information about the area and also be a supportive tool used for further study in this area. 1.5. Scope of the Study This study is confined to the impact of key climatic variables such as temperature, rain fall, humidity, wind and sunshine duration to sorghum crop productivity at Melkassa. It used district wise temperature, rain fall, humidity, wind and sunshine duration data which was obtained from Ethiopian National Meteorological Agency of Melkassa weather station and also sorghum yield data from Melkassa Agricultural Research Center. Besides, it utilizes these data for Melkassa district. 1.6. Limitation of the Study One of the limitations of this study is related to the data. As data on some variables which were considered to be determinant factors for the productivity of sorghum yield were not available at the study area as needed, this study is limited to deal only with meteorological variables (rainfall, temperature, relative humidity, wind speed and sunshine duration) that are relevant to the production of sorghum yield. Time deficiency is also one of the constraints to deal the study. 1.7. Definition of Terms Gambella #1107 Variety: is a white seeded variety with semi compact, semi oval and erect panicle. Its height may be within the range of 120-200cm.Usually, part of its head is covered by the flag leaf (not well exerted). Medium Maturing Sorghum: the sorghum variety that grows in intermediate altitude between 1600-1900m above sea level and with maturity date of 120-130. 2. LITERATURE REVIEW 2.1. General Overview of Climate Change Climate change has become a major concern and receiving serious attention at local, national, regional and global levels. While significant debate remains over the extent to which humans have induced climate change, it has generally been accepted that the effect of climate change are manifested in terms of increased weather variability, higher frequency of extreme weather events and decreased predictability. This increased weather variability as a result of climate change results in potentially sudden and irreversible disruptions to life and livelihood sustaining natural systems also resulting economic, social and environmental dislocations (UNSCEB, 2008). Climate change is already putting extra pressure mainly on agriculture and its effects are expected to become more vital in the future (Apata et al., 2009); (Lobell et al., 2011b); (Rosenzweig et al., 2014). It affects agriculture directly and indirectly. Directly, it affects by influencing the weather variables such as rainfall, temperature, solar radiation, wind speed and humidity (Sowunmi, 2010); (Pryor et al., 2014); (Arimi, 2014). Indirectly, it affects through disease and pest outbreak as well as favoring the development of climate related diseases like malaria that affect the workforce (Newton et al., 2011). Despite technological advancements that have already been reached, the agricultural system is still highly dependent on the climatic condition in many areas of the world (Müller et al., 2011). 2.2. Causes and Consequences of Climate Change on Crop Climate change is already affecting rainfall amounts, distribution and intensity in many places. This has direct effects on the timing and duration of crop growing seasons with concomitant impacts on plant growth. Rainfall variability is expected to increase in the future and floods and droughts will become more common. Changes in temperature and rainfall regime may have considerable impacts on agricultural productivity and the ecosystem provisioning services provided by forests and agro forestry systems on which many people depend (Thornton & Lipper, 2014). The negative effects of climate change are threatening to reverse development gains in many parts of the world especially in Sub-Saharan Africa. It is now an accepted scientific phenomenon that the global climate is changing. Precipitation and temperature patterns are changing. In the SubSaharan region rainfall patterns have become less predictable, precipitation has decreased on average and temperatures are rising (Holmgren & Oberg, 2009). Evidence shows that that the upward trend of the already high temperatures and the reduction of precipitation levels will increasingly result in reduced agricultural production in Sub-Saharan Africa (Mano & Nhemachena, 2009). Specially, Africa has been identified as one of the continents most vulnerable to the impacts of climate change. The reasons are the exposure of its population to climate variations and extremes, people’s dependency on natural resources and the underdevelopment of much of the region. Africa is already affected by climatic extremes such as floods and droughts, which will be exacerbated by climate change. Such events are having a negative impact on livelihoods, especially those of the poor. Given the degraded environments, food insecurity, poverty and HIV/AIDS already affecting large parts of Africa, climate change poses a monumental problem for the region (Antle, J., 2010). The change and variability of the issue is likely to impose additional pressures on water availability, water accessibility and water demand in Africa. About 25% of Africa’s population (about 200 million people) currently experience high water stress. The population at risk of increased water stress in Africa is projected to be between 75-250 million and 350-600 million people by 2020 and 2050, respectively (Boko et al, 2009). In Ethiopia, the climate of arid and semi-arid regions is characterized by high rainfall variability and unpredictability, strong winds, high temperature and high evapotranspiration. In 2015, the country faced one of the worst droughts in 30 years caused by the climate conditions, leading to failed harvests and shortages of livestock forage of which about 10.2 million persons have been affected by the drought (Wondifraw, James & Haile, 2016). It is therefore, essential to quantify its effects especially on crop yields because it is likely to be most affected by sudden or gradual adverse change. In the country, studies like (Deressa & Hassan, 2009), (Di Falco et al., 2011a & Di Falco et al., 2011b) have assessed the impacts of climate change on agriculture and determinants of adaptation in case of Nile Basin region. At country level, Gebreegziabher et al. (2011) have modeled the impacts of climate change on overall Ethiopian economy using a countrywide Computable General Equilibrium (CGE) model. These studies find out that both decline in precipitation and increasing in temperature are damaging Ethiopian Agriculture. Also, the study by (Abera et al., 2011) has good indication on food security and climate connection since it tried to address food security from multidimensional perspectives using indicators representing availability, access and stability even if the study focused only on one climatic variable i.e. rainfall and did not consider the impact of temperature. 2.3. Effects of Climate Change on Sorghum Production in Africa Sorghum is one of the crops mostly grown in wide agro-ecological zones throughout the world (Pauw & Thurlow, 2010). It is the second most important cereal after wheat with 22% of total cereal area, followed by millets (pearl and finger) with 19% of the total cereal land coverage (FAO, 2015). However, different environmental conditions and resource constrained low-input farming systems where the crop is grown. Furthermore, in such dry land environments, the issues of climate variability, change and land degradation are acute with a lack of progress the result of neglect, remoteness and weak national institutions. A large number of studies have investigated several aspects of the impact of climate change on sorghum yield in Africa.(Elodie, 2012) assessed the Impacts of Climate Change on Crop Yields in Sub-Saharan Africa using regression analysis. He made his focus on four most commonly grown crops (millet, maize, sorghum and cassava) in SubSaharan Africa and standard weather variables, such as temperature and precipitation and sophisticated weather measures such as evapotranspiration and the standardized precipitation index (SPI). The analysis result revealed that, there is a significant impact of weather variability on these yields. More specifically, regression analyses using temperature and precipitation provided significant and sensible effects on these yields. Another study by (Gbetibouo & Hassan, 2009) used a Ricardian model to measure the impact of climate change on South Africa’s field crops and analyzed potential future impacts of further changes in the climate. A regression of farm net revenue on climate, soil and other socioeconomic variables was conducted to capture farmer-adapted responses to climate variations. The analysis was based on agricultural data for seven field crops (maize, wheat, sorghum, sugarcane, groundnut, sunflower and soybean), climate and edaphic data across 300 districts in South Africa. Results from the study indicated that production of field crops was sensitive to marginal changes in temperature as compared to changes in precipitation. Temperature rise positively affects net revenue whereas the effect of reduction in rainfall is negative. The study also highlighted the importance of season and location in dealing with climate change showing that the spatial distribution of climate change impact and consequently needed adaptations will not be uniform across the different agro-ecological regions of South Africa. Results from the climate change scenarios indicated that there is a need for shifting farming practices and patterns in different regions such as shifts in crop calendars and growing seasons, switching between crops to the possibility of complete disappearance of some field crops from some regions. Also, (D.S. Maccarthy & P.L.G. Vlek, 2012) evaluated the impact of climate change on sorghum production under different nutrient and crop residue management in semi-arid region of Ghana through Agricultural Production Systems Simulator (APSIM). The outcome shows climate change poses potential risk more to low input small holder farmers who provide a significant proportion of sorghum crop, hence, results in a potential threat to food security in the region.(Jane & Millicent, 2015) assessed climate change and food security in Kenya using Atmospheric Oceanic Global Circulation Models based on county-level panel data for yields of four major crops (Sorghum, Maize, Bean and Millet) and daily climate variables (precipitation, temperature, runoff, and total cloud cover data spanning over three decades. The results show that rainfall during short seasonal spells, as well as during long vs. short rains, exhibit an inverted U-shaped relationship with most food crops; and the effects are most pronounced for maize and sorghum. 2.4. Effects of Climate Change on Sorghum Production in Ethiopia Cereals are the major crops produced in Ethiopia and they constitute the largest share of domestic food production. In 2010/11 main cropping season, cereals were cultivated on 9.9 million hectares producing 17.2 million ton of food grains (CSA, 2010). This represented 82.3% and 87.7% of the total area and production of food grains in the country respectively. Among these cereals, sorghum took up 13.82% (nearly 1.5 million hectares) of the grain crop area. This crop is considered as a potential adaptation option for millions of farmers hit hard by climate change. The crop appeared to have been domesticated in Ethiopia about 5000 years ago(Taylor, 2009). Currently, large part of sorghum production areas in Ethiopia fall under the arid and semi-arid regions of the country that are characterized by high rainfall variability and low soil water storage capacity. The crop is widely grown in low moisture areas due to its high capacity to tolerate soil water deficit and wide range of ecological diversity (MoARD, 2010). Despite its significant area coverage however, the national average sorghum productivity is estimated to be less than one tone per hectare (Mesfin e et al., 2009). Ethiopia is the sixth largest producer of sorghum in Africa. Sorghum is a drought resistant cereal crop and also an important crop for overall food security. It is grown primarily in the eastern highlands of Ethiopia. It is the most important cereal in terms of production, with a national average of 1.7 tons per hectare produced in 2012/2013 (CSA, 2013).However, the productivity of this crop is very low despite its large production area (accounting 47% of cultivated grain crop area in combination with maize, wheat and finger millet) and also available evidences suggests that, this yield of major cereal crops under farmers’ management is still far lower than what can be obtained under on-station and on-farm research managed plots. According to (Woldeamlak, 2009), in Amhara region in which historical rainfall records from 12 stations and time series data on area coverage, production and yield of cereals during the meher season of years 1994-2003 were used as inputs, the inter-annual and seasonal variability of rainfall is a major cause of fluctuations in production of cereals (sorghum, teff, barley, wheat, maize and millet) in the region. The analysis also revealed that sorghum shows the largest year-to-year variability as it is cultivated in semi-arid and arid parts of the region where rainfall variability is high and also sorghum production is more strongly correlated with belg rainfall. Also, as assessed by (Amare, 2015),in Ethiopia cereals, oilseeds, pulses, coffee and other crops are highly affected by rainfall variability; that is rainfall variability has significant and negative impact on all crop types in the country. (Oumer, 2016) estimated impact of weather variations on cereal productivity and influence of agro-ecological differences in Ethiopian through the help of regression analysis. The results shows that weather variables, temperature and rainfall both annually and seasonally were found to be significant determinants of cereal crop productivity in the country, implying that climate has a non-linear effect on cereal crop productivity. 2.5. Effects of Climate Change on Sorghum Production in Oromia Region Sorghum is mainly grown in four big regions of Ethiopia among which Oromia is the one. Sorghum productivity can be affected by different factors such as natural and manmade.(Gutu, Bezabih & Mengistu, 2012) analyzed the impact of climate change factors on food production in North Shewa Zone using the co-integrated Vector Auto Regressive and Error Correction Models. Their estimated results show that food production in the zone was significantly affected by improved technology, area under irrigation, manure usage, Meherrain and temperature, while fertilizer application and Belg rain were found to be less significant in the model. Also as pointed out by (Fekede et al., 2016), in Mieso and DaroLebu districts of Hararghe Zone, due to climate change induced factors, the productivity of agriculture was reduced from time to time. The findings also revealed that majority of the communities in the area response to the effect of climate change through practicing planting drought tolerant and early maturing crop variety, shifting from maize production to sorghum and groundnut production, participating on non-farming activities, adjusting cropping time (from April to June), shifting from cattle raring to shoat and camel production, reducing livestock flock, migration to search feed & water and migration to other area and serve as daily laborer. 2.6. Effects of Climate Change on Sorghum Production in Melkassa District Sorghum crop is widely grown in low moisture areas due to its high capacity to tolerate soil water deficit and wide range of ecological diversity. Melkassa is one of the districts found in East Shewa Zone of Oromia region which is known by sorghum production with localized temporal weather variation during the cropping seasons that induces an important challenge to this crop production and hence in turn to food security. (Abebe, 2012) assessed water requirement and crop coefficient for sorghum (sorghum bicolor L.) at Melkassa district. The result revealed that, sorghum which is an important and stable food crop for most of the people who live in the district is affected by early and terminal water stress imposed by climate variability. 2.7. Review of Empirical Studies Many studies have been conducted at regional and country levels to estimate economic impacts of climate change on agriculture and factors affecting adaptation strategies. For instance, (Thurlow et al., 2009) has examined the impacts of climate variability and change on economic growth and poverty in case of Zambia under different scenarios. The results of their study indicated that climate variability has imposed significant cost to Zambian economy. Specifically, the estimated cost to the economy was USD 4.3 billion over a 10-year period and USD 7.1 billion under worstcase rainfall scenario. Another study by (Zhai et al., 2009) modeled the potential long-term impacts of global climate change on agricultural production and trade in the case of China. They employed an economywide, global CGE model, and simulation scenarios of how global agricultural productivity may be affected by climate change up to 2080. The interesting finding of their study is that as the share of agriculture in GDP decline, the impact of climate change on the overall economy become less intense. In Tanzania, (Muamba & Kraybill, 2010) examined climate change impact on yields of maize, banana and coffee in Mt. Kilimanjaro area using Ricardian framework. The study estimates yield reaction to a 1%, 2%, and 3% annual precipitation decrease. For a 1% precipitation decrease, their simulation predicts that maize, coffee and banana yield will decrease by 74.8%, 76%, and 8.4% respectively. For 2% precipitation decease, the simulation predicts that maize, coffee, and banana yield will decrease by 94%, 95%, and 11% respectively. For a 3% decrease in rainfall, the model predicts that maize, coffee, and banana yield will decrease by 98.7%, 99%, and 23.3% respectively. These results indicate strong evidence of a negative impact of climate change on all three crops. In the Ethiopian context, there are few studies that have examined the issue of climate change. (Deressa & Hassan, 2009) assessed the vulnerability of Ethiopian farmers to climate change in broad region of Nile Basin by using Ricardian approach based on household socioeconomic data collected from 1000 households selected from different agro-ecological settings. They conducted regression of net farm revenue on climate, household and soil variables. The results indicate that vulnerability to climate shocks is not uniform across agro-ecological zones. Also, marginal increase in precipitation during spring would increase revenue, while marginal increase in temperature during summer and winter would reduce net revenue. After forecasting future climate using three climate scenario models, they predict that there would be a reduction of net farm revenue in 2050 and 2100. However, this study did not examine the effectiveness of adaptation strategies adopted by farmers to cope with climate change so that they can maximize their net revenue. Also,(Yesuf et al., 2010), using the same household data set, but monthly collected meteorological station data analyzed the impact of climate change on food production in low income countries. Their results indicate that adaptations to climate change have a significant impact on farm productivity. Their results also show that extension services, both formal and informal, access to credit, and information about future changes in climate variables significantly and positively affect adaptation to climate change. (Adugna, 2009) attempted to show patterns of rainfall and provides insight into the preparation of an early warning system in Ethiopia using time series analysis techniques. Auto-Regressive Moving Average (ARMA) and Vector Auto-Regressive (VAR) models are used to see the pattern of rainfall and response of yield to rainfall as well as to previous yield shocks. Results from estimation of VAR show that current levels of yield respond to previous levels of yield even more than responses to rainfall in most provinces. Also, (Tigist, 2011) tried to evaluate impacts of climate parameters (rainfall and temperature) on sorghum yield at Melkassa using Vector AutoRegressive (VAR) model. The results obtained show that rainfall and yield are highly variable. Rainfall shock has significant impact on rainfall temperature and yield, temperature shock has a significant impact on temperature, rainfall and yield and also yield shock has a significant impact on yield. It was also observed that rainfall variation could fully be explained by its own innovations. For temperature, 56% variation has resulted from the shock of its own innovation and 44% variation resulted from the change in rainfall while yield variation of up to 48.1% is explained by changes in rainfall amounts and the percentage contribution of yield shock for its forecast variance is about 48.6%. More significantly, what makes this study different from previous ones is that it relates rainfall, temperature, humidity, wind, sunshine duration and sorghum yield dynamically using a time series technique called VAR model since no studies have been conducted using this model over the effects of climate change on this yield type using all these climatic variables. 3. DATA AND METHODOLOGY 3.1. Description of the Study Area Melkassa is one of the populated places in the state of Oromia with an estimated population of 16,715. It is about 104 km far away from Addis Ababa and 21 km from Adama and lies on the Addis Ababa-Djibouti railway. It is also called Awash and located between 8°24'0" N and 39°19'60" E with an altitude of 1531 meters above sea level and found in the East Shewa Zone along the Rift valley. Flood, drought, soil erosion and rainfall deficiency are some of the natural hazards that are frequent in the area. Melkassa constitutes the heart and corridor of the Ethiopian Rift Valley that extends from the Afar triangle in the North to the Chew Bahir in southern Ethiopia. Physiographical, Central Rift Valley is characterized by almost level to gentle slope and a benched rift valley without sedimentary surface features. It has also volcanic lacustrine terraces formed in quaternary lacustrine siltstone, sand stone, inter-bedded pumice and stuffs with fault topography bordering the major lakes plus parallels and low coastal ridges. It also has quaternary alluvial landforms, mostly bordering the main river valley or located at the foot of the higher plateaus, as alluvial colluvial cones. Despite the variability in rainfall and the prevalence of the long established spiral of land degradation in the district, there is considerable opportunity for raising the level of farmer’s returns through transfer of improved technologies (material and knowledge). The main rainy season at Melkassa is during the summer from June to September (Kiremt or Ganna) which contributes about 69% of its annual rainfall and the second short rainy season (Belg or Arfaasa) is from March to May which covers nearly 24%. The third season, which is from October to January (Bega or Bona), is dry most of the time but contributes around 7% of the annual rainfall especially during October and January for the late cessation of Kiremt and early onset of Belg seasons respectively. Fig. 1.Map of the Study Area 3.2. Statistical Data Description The data that were used to undertake this study were both climatic data (rainfall, temperature, relative humidity, wind speed and sunshine duration) which were obtained from NMSA (at Melkassa weather station) and yield (sorghum variety of Gambella #1107) data that were taken from Melkassa Agricultural Research Center (MARC). The yield data were taken on this yield category since it is one of the major cereal crops mostly being produced in the area and also drought resistant crop. Both data were taken for the years 1967 to 2016 G.C with the total of 50 years belg or autumn season observations of total rainfall, average minimum and maximum temperatures, average relative humidity, average wind speed, average sunshine duration and sorghum yield. The belg (autumn) season is selected due to the reason that sorghum yield is a long cycle crop being produced during this season. 3.3. Study Variables Climate change or variability can be explicated by its vital indicators that are rainfall and temperature and its impacts on agricultural production can be seen from the historical effects of rainfall and temperature on crop yield. Thus, the variables of interest under this study are sorghum yield (Gambella #1107) measured in quintal/hectare, rainfall (in mm), minimum and maximum temperatures (in oC), relative humidity (in %), wind speed (in m/s) and sunshine duration (in hours). 3.4. Source of Data All the information used to conduct this study has obtained from secondary sources. The data on rainfall, temperature, relative humidity, wind speed and sunshine duration values were obtained from the records at NMSA of Melkassa weather station while the crop data were documented at Melkassa Agricultural Research Center of Ethiopian Institute of Agricultural Research. 3.5. Statistical Model 3.5.1. Model Description The statistical model used to fit the data is time series model. Time series refers to a sequence of observations ordered by a time parameter. It may be measured continuously or discretely. One of the special futures of time series is that the data ordered with respect to time and successive objection is assumed to be dependent, which facilitates to give reliable forecast. For observations, Yit; i= 1,…….., n; t =1, …,T being taken sequentially over time, where i is indexes of measurements made at each time point t, n the number of variables being observed and T the number of observations made, if n is equal to one then the time series is referred to as univariate (Chatfield, 1989), and if it is greater than one the time series is referred to as multivariate (Hannan, 1970). Under this work, multivariate time series analysis takes place using vector autoregressive (VAR) model. 3.5.2. Multivariate Time Series Analysis Multivariate time series (MTS) analysis is a powerful tool for the analysis of time series data. It is of considerable interest in a variety of fields such as engineering, the physical sciences- particularly the earth sciences (e.g. meteorology and geophysics), and economics and business (Reinsel, 1997). The method is used when one wants to model and explain the interactions and co-movements among a group of time series variables. In analogy with the univariate case, it is one major objective of multiple time series analyses to determine suitable functions that may be used to obtain forecasts with “good” properties. It is also often of interest to learn about the dynamic interrelationships between a number of variables. 3.5.2.1. Vector Autoregressive (VAR) Model The vector autoregressive (VAR) model is one of the most successful, flexible and easy to use models for the analysis of multivariate time series. It is a natural extension of the univariate autoregressive model to multivariate time series. The model was made famous in Chris Sims’s paper in the year 1980 for macro-economic forecasts. The term auto regressive is used due to the fact that the variables are regressed on their own past values while the term vector is used due to the fact that we are dealing with a vector of two or more variables. The VAR model has established to be especially useful for describing the dynamic behavior of economic and financial time series and for forecasting purpose. It often provides superior forecasts to those from univariate time series models and elaborate theory-based simultaneous equations models. The following section gives the brief description over the analysis of covariance stationary multivariate time series using VAR models. Let Yt = (y1t, y2t, . . …,ynt)' represent an (n×1) random vector of time series variables. The basic p-lag vector autoregressive (VAR (p)) model has the form (Hamilton, 1994) Yt= c + Π1Yt−1+Π2Yt−2+· · · +ΠpYt−p + εt, t = 1, . . . ..,T … … … … … … … … … . . . … … … . … . … (1) where Πi’s are (n×n) fixed coefficient matrices, c= (c1,…….,cn )' is a fixed (nx1) vector of intercept terms allowing for the possibility of a nonzero mean E(Yt) and εt = (ε1t,……., εnt)' is an (n×1) unobservable zero-mean white noise vector process (serially uncorrelated or independent) with time invariant covariance matrix Σ, that is E(εt)=0, E(εtεs') = 0 for s ≠ t. Note that the covariance matrix Σ is assumed to be nonsingular. The VAR (p) can be expressed in lag operator form as follows: Π(L)Y𝑡 = c + εt … … … … … … … … … . . . … … … . … . … … … … … … … … … … … … … … … … … … (2) where Π(L) = In − Π1L1 − ... – ΠpL pand LpYt =Yt-p The VAR (p) is stable if the roots of det(In − Π1 Z − · · · −Π𝑝 𝑍 𝑝 ) … … … … … … … … … …. (3) lie outside the complex unit circle (have modulus greater than one) for complex z, |z|<1, or, equivalently, if the eigen values of the companion matrix have modulus less than one. Assuming that the process has been initialized in the infinite past, then a stable VAR (p) process is stationary with time invariant means, variances and autocovariances. If Y𝑡 in equation (1) is covariance stationary, then the unconditional mean is given by: µ = (In – Π1 − · · · −Π𝑝 )−1 𝑐 … … … … … … … … … … … … … … … … … … … … … . … … … … … . (4) The mean-adjusted form of the VAR (p) is then obtained as: Yt − µ = Π1 (Yt−1 − µ) + Π2 (Yt−2 − µ) + ⋯ + Πp (Yt−p − µ) + εt … … … … . … … … … … … (5) 3.5.3. Stationary Time Series Processes A stochastic process 𝐘t is weak stationary if its first and second moments are time invariant. In other words, a stochastic process is stationary if E(Yt ) = µ. … … … … … … … … . … … … … … …(6) for all t and E[(Yt - µ)(Yt−h - µ)'] = Γy (h) = y(-h)'… … … … . … … … … … … … … . … … … … … …(7) for all t and h = 0,1,2, ………………. Condition (6) means that all Yt have the same finite mean vector µand (7) requires that the autocovariances of the process do not depend on t but just on the time period h the two vectors Yt and Yt−h are apart. Note that all quantities are assumed to be finite. For instance, µ is a vector of finite mean terms and Γy (h) is a matrix of finite covariances. Thus, a stable VAR (p) processYt , t=1,2, … , is stationary. Since stability implies stationarity, the stability condition (3) is often referred to as stationarity condition in the time series literature. But the converse is not true. In other words, an unstable process is not necessarily non-stationary (Lütkepohl, 2005). 3.5.3.1. Unit Root Tests of Stationarity To check whether the given series is stationary or not, different tests called unit root tests are helpful. These include Augmented Dickey–Fuller (ADF) test, Elliott–Rothenberg–Stock test, Kwiatkowski Phillips-Schmidt-Shin (KPSS) unit root test, Phillips–Perron test (PPT), Schmidt– Phillips test and Zivot–Andrews test. Among these tests, the ADF, KPSS unit root, PPT and Schmidt– Phillips tests were the most commonly used ones (Hamilton 1994). In this study, the presence of stationarity of the data for the variables considered will be checked using Augmented Dickey-Fuller (ADF) (Dickey and Fuller, 1979) and Phillips-Perron (PP) (Hamilton, 1994) tests. For the derivation of Dickey-Fuller test of an arbitrary seriesYt , consider the following model: Yt = β0 + β1 𝑡 + ut … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (8) ut = αut−1+εt … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (9) wheret is time and εt is a zero mean covariance stationary process. Using (9) the reduced form of (8) can be written as: Yt = δ0 + δ1 t + αyt−1+εt … … … … … … … … … … … … … … … … … … … … … … … … … (8a) where δ0 = β0 (1 − α) and δ1 = β1 (1 − α) The Dickey-Fuller test tests the hypothesis: H0: The series has unit root (α = 1) Vs H1: The series has no unit root(α < 1) Note that α is the term in equation (8a). Dickey-Fuller test is based on the assumptions that residuals are white noise and the data generating process is autoregressive of order one (AR (1)). It may lead to wrong conclusions if the data generating process is autoregressive of higher order or if the errors are autocorrelated. The Augmented Dickey-Fuller test (Dickey and Fuller, 1981) includes additional higher order lagged differences to the Dickey-Fuller test model. Inclusion of the lagged difference term allows autoregressive moving average (ARMA) errors (Maddala, 2011). ADF test is specified based on the model: … … … … … … … … … . . . … … … . … . … … …(10) where ∆ is a first difference operator and n is the lag length in the model. The lag length is determined based on Akaike Information Criterion. ADF test is biased towards accepting the null hypothesis of unit root in the series (Badawi, 2009). The Phillips-Perron test is specified based on the model: … … … … … … … … … . . . … … … . …. (11) where T is the number of observations and m is lag length. The lag length is determined based on (Newey & West, 1987) suggestions. Although differencing may transform non-stationary series into stationary ones, it leads to the loss of important long run information about the variables. To deal with this problem of differencing, (Engle & Granger, 1987) recommend cointegration. 3.5.3.2. Test of Randomness This test is used to check whether or not the data is random. There are certain types of tests which are helpful in checking the randomness of data. These are turning point test, rank test and phase length test. Here, turning point test will takes place for this purpose. 3.5.3.2.1. Turning Points Test of Randomness It is a type of test based on counting the number of turning points that means the number of times there is a local maximum or minimum in the series. A local maximum is defined to be any observation Yt such that Yt> Yt-1 and also Yt > Yt+1. The converse is true for local minimum. If the series is really random, one can work out the expected number of turning points and compare it with the observed value and also count the number of peaks (a value greater than its two neighbors) or troughs (a value less than its two neighbors) in the time series plot. The peak and trough together are termed as turning points (Pollock, 1993). To conduct the test, it should be necessary to define a counting number, C as follows. Ci = 1, if Yi< Yi+1> Yi+2 or Yi< Yi+1< Yi+2 … … … … … … … … … . . . … … … . … . … … … … … … …(12) 0, otherwise Hence, the number of turning points denoted by p in the series is given by p = ∑ni=1 𝐶𝑖 and the probability of finding turning points in n consecutive values is given as: E(p) = E(𝐶𝑖 ) = 2(n − 2) 3 The test can be done through the following hypothesis testing procedures. 1. Ho: Yt , t = 1,2,3,.....,n are independently and identically distributed or the data is random. Vs H1: not Ho. 2. Selecting the level of significance (α). 3. Determining the expected values and variance of the turning point p for the given set of observations. (16n − 29) 90 2(n − 2) E(p) = E(Ci) = 3 V(p) = 4. Computing test statistic. The test statistic to be used is Z, Zcal = p−E(p) √V(p) ~ N (0, 1) 5. Finding the critical or table value for the selected test statistic i.e. computing value of 𝑍α⁄2 . 6. Making decision. Decision can be made as follows: Reject Ho if │Zcal│>𝑍α⁄2 and do not otherwise. 7. Drawing conclusion (the conclusioncan be drawn based on what is decided). 3.5.4. Computation of Autocovariance and Autocorrelations of Stable VAR Processes 3.5.4.1. Autocovariance of a Stable VAR (p) Processes Autocovariance is the covariance between two observations separated by k units of time in a time series. For a higher order stable VAR (p) process, … … … … … … … … … . . . … … … . … . … … … …(13) post multiplying with (Yt-h- µ) and taking expectations gives: … … … . … . …14) Thus for h=0, … … … … … … … … … . . . … … … . … … … … … … … … … 15) These equations are usually referred to as Yule-Walker equations. 3.5.4.1. Autocorrelations of Stable VAR (p) Processes Autocorrelation is the correlation of the observations in a time series, usually expressed as a function of the time lag between observations. Because the autocovariance depend on the unit measurement used for the variables of the system, they are sometimes difficult to interpret. Therefore, the autocorrelations Ry(h) = D-1Γy(h)D-1… … … … … … … … … . . . … … … . … . . . . … … … . … . … … … … … … … … … …. (16) are usually more convenient to work with as they are scale invariant measures of the linear dependencies among the variables of the system. Here D is a diagonal matrix with the standard deviations of the components of Yt on the main diagonal. That is, the diagonal elements of D are the square roots of the diagonal elements of Γy(0). Denoting the covariance between yi,t and yj,t-h by γij(h) (i.e., γij(h)) is the ijth element of Γy(h)) the diagonal elements γ11(0), … , γnn (0) of Γy(0) are the variances of y1t, … ……,ynt. Thus And the correlation between yi,t and yj,t-h is … … … … … … … … … . . . … … … . … . … … … … … … … … … … … . … … … … … .. (17) which is just the ijth element of Ry(h) given in equation 15 above. 3.5.5. Structural Vector Autoregressive (SVAR) Measures The general VAR (p) model has many parameters, and they may be difficult to interpret due to complex interactions and feedback between the variables in the model. As a result, the dynamic properties of a VAR (p) are often summarized using various types of structural analysis. The three main types of structural analysis summaries are Granger causality tests, impulse response functions and forecast error variance decompositions. The following sections give brief descriptions of these summary measures (Lütkepohl, 2005). 3.5.5.1. Granger Causality The structure of the VAR model provides information about a variable’s or a group of variables’ forecasting ability for other variables. The following intuitive notion of a variable’s forecasting ability is due to Granger (1969). If a variable, or group of variables, Y1t is found to be helpful for predicting another variable, or group of variables, Y2t then Y1t is said to Granger-cause Y2t; otherwise it is said to fail to Granger-cause Y2t. Formally, Y1tfails to Granger-cause Y2t if for all s > 0 the MSE of a forecast of Y2,t+s based on (Y2,t, Y2,t−1, . . .) is the same as the mean squared error (MSE) of a forecast of Y2,t+sbased on (Y2,t, Y2,t−1, . . .) and (Y1,t, Y1,t−1, . . .). Clearly, the notion of Granger causality does not imply true causality. It only implies forecasting ability. If Y1t causes Y2t and Y2t also causes Y1t the process (Y1t', Y2t' )' is called a feedback system. In a bivariate VAR(p) model for Yt= (Y1t, Y2t)', Y2tfails to Granger-cause Y1tif all of the p VAR coefficient matrices Π1, . . . , Πp are lower triangular. That is, the VAR (p) model has the form So that all of the coefficients on lagged values of Y2t are zero in the equation for Y1t. Similarly, Y1t fails to Granger-cause Y2t if all of the coefficients on lagged values of Y1t are zero in the equation for Y2t. Granger non-causality may be tested using the Wald statistic. 3.5.5.2. Impulse Response Functions Any covariance stationary VAR (p) process has a Wold representation of the form Yt= µ+ εt+ Ψ1εt−1+ Ψ2εt−2 + … … … … … … … … … . . . … … … … … … … … … … . … . … … … … .. (18) where the (n × n) moving average matrices Ψs are determined recursively using It is tempting to interpret the (i, j)th element, Ψijs, of the matrix Ψs as the dynamic multiplier or impulse response i.e. Ψijs represent the effects of unit shocks in the variables of the system. However, this interpretation is only possible if var(εt) = Σ is a diagonal matrix so that the elements of εt are uncorrelated. One way to make the errors uncorrelated is to follow Sims (1980) and estimate the triangular structural VAR(p) model defined by: y1t = c1+ γ'11Yt−1 + · · · +γ'1pYt−p + η1t y2t = c2 + β21y1t + γ'21Yt−1+· · · +γ'2pYt−p+ η2t y3t = c3 + β31y1t + β32y2t + γ'31Yt−1+· · · +γ'3pYt−p + η3t… … … … … … … … … . . . … … … …. (19) . . . . ynt = cn + βn1y1t + · · · +βn,n−1yn−1,t + γ'n1Yt−1+· · · +γ'npYt−p + ηnt In matrix form, the triangular structural VAR (p) model is given by: BYt= C + Γ1Yt−1+Γ2Yt−2+· · · +ΓpYt−p + ηt… … … … … … … … . . . … … … … … … . … … . …. (20) where … … … … … … … … … . . . … … … … … … … … . … … . ….(21) C= [c1 c2 ….. cn]' and Γi = [γ'1i γ'2i …. γ'ni]' for i= 1,2, …,p is a lower triangular matrix with 1's along the diagonal. The algebra of least squares will ensure that the estimated covariance matrix of the error vector ηt is diagonal. The uncorrelated or orthogonal errors ηt are referred to as structural errors. The triangular structural model (18) imposes the recursive causal ordering: y1 → y2 → · · ·→ yn… … … … … … … … … . . . … … … … … … . … … . … . … … … … … … … … … . .. (22) The ordering (21) means that the contemporaneous values of the variables to the left of the arrow → affect the contemporaneous values of the variables to the right of the arrow but not vice-versa. These contemporaneous effects are captured by the coefficients βij in (18). For a VAR (p) with n variables there are n! possible recursive causal orderings. Which ordering to use in practice depends on the context and whether prior theory can be used to justify a particular ordering. Results from alternative orderings can always be compared to determine the sensitivity of results to the imposed ordering. Once a recursive ordering has been established, the Wold representation of Yt based on the orthogonal errors ηt is given by: Yt= µ + Θ0ηt + Θ1ηt –1+Θ 2ηt-2 + … … … … … … … … … . . . … … … … … … . … … … … . . … . … …(23) where Θ0 = B -1is a lower triangular matrix. The impulse responses to the orthogonal shocks ηit are … … … … … … … … … . . . … … … … … … . … … … …(24) s s where θij is the (i, j)th element of Θs. A plot of θij against s is called the orthogonal impulse response function (IRF) of yi with respect to ηj. With n variables there are n2 possible impulse response functions. In practice, the orthogonal IRF (23) based on the triangular VAR (p) (18) may be computed directly from the parameters of the non-triangular VAR (p) in (1) as follows. First, decompose the residual covariance matrix Σ as: Σ = ADA' where A is an invertible lower triangular matrix with 1's along the diagonal and D is a diagonal matrix with positive diagonal elements. Next, define the structural errors as: ηt= A−1εt These structural errors are orthogonal by construction since var(ηt) =A−1ΣA−1'= A−1ADA'A−1'= D. Finally, re-expressing the Wold representation (17) as: Yt = µ +AA−1εt+ Ψ1 AA−1εt−1+ Ψ 2 AA−1εt−2 + · · · = µ + Θ0ηt + Θ1ηt−1+Θ2ηt−2 + · · · where Θ j= ΨjA. Notice that the structural B matrix in (19) is equal to A−1. 3.5.5.3. Forecasting Error Variance Decompositions The forecast error variance decomposition (FEVD) answers the question: “what portion of the variance of the forecast error in predicting yi,T+h is due to the structural shock ηj?”. Using the orthogonal shocks ηt the h-step ahead forecast error vector with known VAR coefficients, may be expressed as: WhereYT+h|T is h-step forecasts based on information available at time T. For a particular variable yi,T+h,this forecast error has the form: Since the structural errors are orthogonal, the variance of the h-step forecast error is: Whereσ2ηjt= var(ηjt). The portion of var(yi,T+h − YT+h|T) due to shock ηj is then: … … … … … … … … … . . . … … … … …(25) In a VAR with n variables there will be n2FEVDi,j(h) values. It must be kept in mind that the FEVD in (24) depends on the recursive causal ordering used to identify the structural shocks ηt and is not unique. That is, different causal orderings will produce different FEVD values. 3.6. Vector Error Correction and Cointegration Theory 3.6.1. VEC Models The fact that many time series contain a unit root has spurred the development of the theory of non-stationary time series analysis. (Engle & Granger, 1987) pointed out that a linear combination of two or more non-stationary series may be stationary. If such a stationary, or I(0), linear combination exists, the non-stationary (with a unit root), time series are said to be cointegrated. The linear combination which is stationary is called the cointegrating equation and may be interpreted as a long-run equilibrium relationship between the variables. For example, in income consumption analysis, consumption and income are likely to be cointegrated. If they were not, then in the long-run consumption might drift above or below income, so that consumers were irrationally spending or piling up savings. A vector error correction (VEC) model is a restricted VAR that has cointegration restrictions built in to the specification, so that it is designed for use with non-stationary series that are known to be cointegrated. The VEC specification restricts the long run behavior of the endogenous variables to converge to their cointegrating relationships while allowing a wide range of short-run dynamics. The cointegration term is known as the error correction term since the deviation from long run equilibrium is corrected gradually through a series of partial short run adjustments. 3.6.2. Testing for Cointegration Given a group of non-stationary series, we may be interested in determining whether the series are cointegrated, and if they are, identify the cointegration (long-run equilibrium) relationships. We can interpret the long run paths of cointegrating variables as interdependent. Application of cointegration tests in estimation are analyzed by (Johanson & Jusselius, 1990). VAR-based cointegration tests using the methodology developed by (Johansen, 1988) is the most common method. Johanson’s method is to test the restrictions imposed by cointegration on the unrestricted VAR involving the series. It applies the maximum likelihood method to determine the presence of cointegrating vectors in non-stationary time series. The trace tests and eigen value tests are used to determine the number of cointegrating vectors. This implies a stationary long- run equilibrium relationship between the variables. The maximum lag length of the VAR model which is used in Johanson’s procedure is determined by the Likelihood Ratio (LR) statistics. Consider a VAR of order p Yt= A1Yt-1 + … + ApYt-p +εt… … … … … … … … … … … . . . … … … … … … . … … … … . . … . … …(26) Where Yt is an n-vector of non-stationary I(1) variables, and εt is a vector of innovations. We can rewrite the VAR as: p−1 ∆Yt = ∏ Yt−1 + ∑ ΓY∆ t−1 + εt … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (27) i=1 and ∆ is the difference operator. Granger’s representation theorem asserts that if the coefficient (parameters) matrix ∏ has reduced rank r < n, then there exist nxr matrices Θ and Φ each with rank r such that ∏= ΘΦ ' and Φ 'Yt is stationary. The letter r denotes the number of cointegrating relations (the cointegrating rank) and each column of Φ is the cointegrating vector. The elements of Θ are known as the adjustment parameters in the vector error correction model. Johanson’s method is to estimate the ∏ matrix in an restricted form, then test whether we can reject the restrictions implied by the reduced rank of ∏. If we have n endogenous variables, each of which has one unit root, there can be from zero to n-1 linearly independent, cointegrating relations. If there are no cointegrating relations, standard time series analysis such as the (unrestricted) VAR may be applied to the first differences of the data. Since there are n separate integrated elements driving the series, levels of the series do not appear in the VAR in this case. Conversely, if there is one cointegrating equation in the system, then a single linear combination of the levels of the endogenous series ΦYt-1, should be added to each equation in the VAR. Each column of the Φ matrix gives an estimate of a cointegrating vector. The cointegrating vector is not identified unless we impose some arbitrary normalization. We can adopt the normalization so that the r cointegrating relations are solved for the first r variables in the Yt vector as a function of the remaining n-r variables. When multiplied by a coefficient for an equation, the resulting term ΘΦ 'Yt-1, is referred to as an error correction term. If there are additional cointegrating equations, each will contribute an additional error correction term involving a different linear combination of the levels of the series. The null hypothesis of at most r cointegrating vectors against a general alternative hypothesis of more than r cointegrating vectors is tested by trace statistics. The trace statistic is given by: … … … … … … … … … . . . … … … … … … . … … … … . . … . … … … …(28) where, T is the number of observations and is the eigen values. The null hypothesis of r cointegrating vector against the alternative of r+1 is tested by maximum eigen value statistic. The maximum eigen value is given by: … … … … … … … … … . . . … … … … … … . … … … … . . … . … … …(29) 3.7. VAR Order Selection The lag length for the VAR (p) model may be determined using model selection criteria. The general approach is to fit VAR (p) models with orders p = 0, ... ,pmax and choose the value of p which minimizes some model selection criteria. Model selection criteria for VAR (p) models have the form: … … … … … … … … … . . . … … … … … … . … … … … . . … . … … (30) Where is the residual covariance matrix without a degrees of freedom correction from a VAR (p) model, cT is a sequence indexed by the sample size T and φ(n, p) is a penalty function which penalizes large VAR(p) models. The three most common information criteria are the Akaike (AIC), Schwarz-Bayesian (BIC) and Hannan-Quinn (HQIC) and defined respectively as: … … … … … … … … … . . . … … … … … … . … … … … . . … . … … . … (31) The AIC criterion asymptotically overestimates the order with positive probability, whereas the BIC and HQ criteria estimate the order consistently under fairly general conditions if the true order p is less than or equal to pmax. 3.8. Assumptions of VAR Model The VAR model has the following basic assumptions: Each variable yi,t is I(0) or I(1). i. ii. εt has a multivariate normal distribution. iii. For all “inverse roots” or ”characteristic roots” λi, |λi| ≤1. In particular, VAR only has unit roots, λi = 1 and/or ”stable roots”. 3.9. Model Adequacy Checking 3.9.1. Checking the Whiteness of Residuals It is assumed that εt is an n-dimensional white noise process with nonsingular covariance matrix Σ. For instance, εt may represent the residuals of a VAR (p) process. The Lagrange multiplier test is a popular statistic for checking the overall significance of the residual autocorrelations. In Lagrange multiplier tests, we wish to test: H0: D1 = · · · · · · = Dh = 0 Against H1 :Dj = 0 for at least one j ∈ {1, . . . , h} Where the error vector, εt= D1εt−1+ … + Dhεt−h + vt, where vt is white noise. It is equal to εt if there is no residual autocorrelation. 3.9.2. Testing for Normality of Residuals A stationary, stable VAR (p) process is Gaussian (normally distributed) if and only if the white noise process εt is Gaussian. Therefore, the normality of the yt’s may be checked via the εt’s. In practice, the εt’s are replaced by estimated residuals. Normality of the underlying data generating process is needed for instance, in setting up forecast intervals. Non normal residuals can also indicate more generally that the model is not a good representation of the data generation process. Therefore, testing this distributional assumption is desirable. (Lütkepohl, 1993) suggests using the multivariate generalization of the Jarque-Bera test. (Jarque & Bera, 1987) established a test statistic to test for the normality of observations. This statistic is based on the skewness and kurtosis properties of the residuals, (3rd& 4th moments). In this study it is used to test the null hypothesis that the disturbances are normally distributed. The Jarque-Bera test statistic is given by the formula: where s is a measure of skewness, k is a measure of kurtosis and n is the sample size. Under H0, JB has a χ2distribution with 2 degrees of freedom asymptotically, and the null hypothesis is rejected if the computed value exceeds a χ2critical value (small p-value). 4. RESULTS AND DISCUSSIONS In this chapter, the analysis results which are done using STATA software and are divided into the descriptive results and inferential results (results of VAR model) were presented. 4.1. Descriptive Analysis In this section, descriptive analysis results for total autumn rainfall in mm, average temperature (both minimum and maximum) in °C, average relative humidity in %, average wind speed in m/s, average sunshine duration in hrs and sorghum yield of medium maturing Gambella#1107 variety which is measured in quintal per hectare(Q/h) were discussed. From the summary statistics given below in table 4.1, the total amount of autumn rainfall is ranged from 218.7 mm to 690 mm indicating that there is higher inter-annual changeability of the autumn total rainfall in the area. The standard deviation (s.d) is 123.58 mm showing the higher variability of the autumn total rainfall. Also, the average minimum and maximum temperatures are fluctuating from 0.5oc to 9.2oc and 20oc to 30.5oc respectively, revealing that the inter-annual average minimum and maximum temperatures of autumn season are changing at high rate with respective s.d of 2.19oc and 2.67oc which indicates that the year to year variability in average autumn temperature (both minimum and maximum) is immense. The average wind speed varying from 0.4 m/s to 1 m/s displays that there is extreme wind speed occurrence in the district having s.d of 0.19 m/s that indicates its year to year inconsistency is high while the average relative humidity is varying between 39 % to 75 % whose s.d is 8.44 % proving that there is higher yearly variation in average relative humidity. Also, the incidence of average sunshine duration is ranging from 4 hrs to 9.5hrs and the s.d is 1.61 hrs which shows its advanced change over years. Sorghum yield also shows the highest sparseness from year to year, with a range of 3 Q/h to 70 Q/h and s.d of 17.98 Q/h. The results also revealed that rainfall, minimum temperature, maximum temperature, wind speed, relative humidity, sunshine duration and yield showed high variability from year to year with 26.3%, 38.6%, 10.3%, 24.3%, 13.6%, 23.1% and 66.1%coefficients of variation respectively. Table 4.1.Descriptive Statistics Descriptive Rainfall statistics (mm) Minimum Maximum Temperature Temperature (oC) (oC) Wind Relative Sunshine Yield speed humidity duration (m/s) (%) (hrs) (Q/h) Mean 468.95 5.66 25.71 0.79 61.84 6.96 27.20 Median 499 5.9 25.75 0.8 63 7.05 21.25 Minimum 218.7 0.5 20 0.4 39 4 3 Maximum 690 9.2 30.5 1 75 9.5 70 Standard deviation 123.58 2.19 2.67 0.19 8.44 1.61 17.98 Coefficient of variation 0.66103 0.10382 0.24339 0.13660 0.23176 0.2635348 0.3865681 The correlation analysis of seasonal variation in all the climatic variables and sorghum yield are shown below in Table 4.2, indicating that sorghum production has significant positive correlation with the autumn total rainfall and average relative humidity while it is significantly negatively correlated with average maximum temperature and average sunshine duration of the season. Also, sorghum production shows positive correlation with average wind speed while its correlation is negative with that of average minimum temperature. Table 4.2. Correlation between the Variables Variable Rainfall Min temperature Max temperature Wind speed Relative humidity Sunshine Yield duration Rainfall 1 -0.273 -0.603* 0.030 0.608* -0.260 0.486* Min temperature -0.273 1 -0.021 0.179 -0.218 0.078 -0.236 Max temperature -0.603* -0.021 1 -0.029 -0.530* 0.256 -0.397* Wind speed 0.030 0.179 -0.029 1 0.028 -0.140 0.222 Relative humidity 0.608* -0.218 -0.530* 0.028 1 -0.165 0.330* Sunshine duration -0.260 0.078 0.256 -0.140 -0.165 1 -0.359* Yield 0.486* -0.236 -0.397* 0.222 0.330* -0.359* 1 *Correlation is significant at 0.05 level (two-tailed) In general, the finding of this descriptive analysis reveals that sorghum yield production has higher year to year variation and also there is higher correlation between yield production and all the above climate variables at the Melkassa district of East Shewa Zone. These descriptive results were similar with the findings by (Tigist, 2011) in the way that yield and rainfall are highly variable with 63.38% and 18.43% coefficients of variation and yearly change of 3 Q/h to 70 Q/h and 320 to 690 mm respectively but opposite in the case that the variation in annual average temperature is not large (accounts for about 0.75◦C). Moreover, (Elodie, 2012) who assessed the impacts of climate change (change in standard weather variables, such as temperature and precipitation and sophisticated weather measures such as evapotranspiration and the standardized precipitation index (SPI)) on four most commonly grown crops (millet, maize, sorghum and cassava) in Sub-Saharan Africa using regression analysis, found the same results in the way that there is significant correlation between weather parameters and these yields, specifically sorghum yield. As found by (Woldeamlak, 2009),in Amhara region, sorghum yield shows the largest year-to-year variability as it is cultivated in semi-arid and arid parts of the region where rainfall variability is high and also sorghum production is more strongly correlated with belg rainfall; which is also the supportive finding of this study. 4.1.1. Looking Over Nature of the Data 4.1.1.1. Time Series Plot Before applying different multivariate time series analysis techniques on the data, it is necessary to check for the stationarity of the series since many of the time series methods assume that the data is stationary with respect to the mean and variance. One of the methods used to check this condition is time series plot of the series. The time series plots of the original data for all the variables under study, as displayed in figure 1 of appendix C indicates clearly that all the series revealed that stationarity was inherent, that is the movement of all the variables for the years 1967 through 2016 was constant and do not varied from one year to the other with systematically visible pattern and identifiable trend components in the time series data. 4.1.2. Unit Root Tests Unit root tests provide a more formal approach to determining whether the series is stationary or not such as the ADF and PP tests which were applied under this study. The main difference among these tests is the way they treat serial correlation in testing regressions. ADF tests use a parametric autoregressive structure to capture serial correlation while PP tests use non-parametric corrections based on estimates of the long-run variance of the differences ∆yt. The hypothesis of interest is that: H0: The series is not stationary (there is a unit root) versus H1: The series is stationary. Before conducting the unit root tests, it is important to make lag order selection as in Stata, lag order selection is preliminary condition in order to identify lag number to be used under these tests. 4.1.2.1. Lag Order Selection In pre lag order selection of VAR model, four maximum number of lags (pmax = 4) were used and the following table shows the results of lag order selection criteria. The results suggest that the model is significant at lag one (1) since all the lag selection criteria were smaller at lag one than at the rest lags and also tells us that we can conduct the unit root tests using lag one as follows. Table 4.3. Lag order Selection Criteria Values Lag P 0 AIC HQIC SBIC 39.3927 40.2266 41.6189 1 0.002 39.0672* 39.1714* 39.3454* 2 0.074 39.9276 41.4913 44.1017 3 0.091 39.9682 42.2616 46.0902 4 0.130 39.971 42.994 48.0409 The values of ADF and PP tests were displayed in the following table. Based on ADF and PP test results as displayed below, one can decide that all the variables satisfy the stationarity assumption at their level since their p-values are not larger than the given level of significance (5%) which results in rejection of H0.Hence it is clear from the time series plots and the unit root test of the series that all of the variables are stationary and enables us to fit the model. Table 4.4. ADF and PP Unit Root Tests of original Series Critical value (5% Variable Test Statistics ADF PP Yield -2.948 -3.801 Rainfall -3.089 Min significance level) ADF P-Value Decision PP ADF PP -2.936 -2.933 0.0417 0.0029 Stationary -4.474 -2.936 -2.933 0.0274 0.0002 Stationary -4.677 -7.633 -2.936 -2.933 0.0001 0.0000 Stationary -4.420 -6.224 -2.936 -2.933 0.0003 0.0000 Stationary Wind speed -5.657 -10.410 -2.936 -2.933 0.0000 0.0000 Stationary Relative -6.241 -6.722 -2.936 -2.933 0.0000 0.0000 Stationary -3.821 -7.123 -2.936 -2.933 0.0027 0.0000 Stationary Temperature Max Temperature humidity Sunshine duration 4.1.3. Test of Randomness The test for randomness of data for this study has checked by a very simple diagnostic test called turning point test, which examines the series whether it is purely random or not. The idea is that if the series is purely random, then at most three successive values are equally likely to occur in any of the six possible orders. The following table shows the results obtained from the test of randomness of the series using a turning point test. The hypothesis of interest is: Ho: Yt , t = 1,2,3,.....,n are independently and identically distributed or the data is random. Vs H1: not Ho. Based on the results of table 4.5, since all the test statistic values for all the variables are smaller than the critical value, we fail to reject the null hypothesis and conclude that the data are random. Also, the p-values for the variables reveal the same information since they all are larger than the default 5% (0.05) significance level. The plots showing the turning points test are also displayed in appendix D. Table 4.5. Turning Point Test of Randomness Variable Test statistic Critical value (5%) P-value Number of turning points or runs Yield -1.71 1.96 0.09 20 Total Rainfall -0.86 1.96 0.39 23 Average Minimum Temperature 1.14 1.96 0.25 30 Average Maximum Temperature -1.71 1.96 0.09 20 Average Wind Speed 0.91 1.96 0.36 29 Average Relative Humidity 1.16 1.96 0.25 30 Average Sunshine Duration -0.86 1.96 0.39 23 4.1.4. Auto Correlation and Partial Autocorrelation Functions of the Series Autocorrelation plots are widely used tools for checking the randomness or non-stationarity in any time series. The autocorrelation plots of stationary data drops to zero relatively quickly while for non-stationary data, they become significantly different from zero for several lags and PACF will have a large spike close to 1 at lag 1.The ACF and PACF plots of the raw data for this study are given on appendix E suggesting that the plots of all the variables show no evidence of significant spikes (the spikes are within the 95% confidence limits) indicating that the series seem to be uncorrelated. Hence we can apply time series techniques to model the data. 4.1.5. Cointegration Rank Test This section describes the cointegrating rank (rank of matrix П) which is estimated using Johansen’s methodology. In performing the cointegration rank test, if the rank is zero, then there is no cointegrating relationship among the variables and if the rank is one there is one, if it is two there are two and so on. According to (Engle & Granger, 1987), cointegration tests are only to be done when one have two or more I(1) variables or non-stationary series in order to examine the existence of co-movements (long-run equilibrium relationship) among these originally nonstationary time series, but happen to attain stationarity after first or second differencing. Thus, as seen from the unit root test results, since all the variables are stationary at their level, conducting cointegration test is not needed and hence no need of fitting VEC model but rather fitting VAR model for the series is appropriate. 4.1.6. VAR Order Selection and Estimating Model Parameters 4.1.6.1. Model Selection The VAR model considered under this study can be indicated as a seven variable system for a period 1967 to 20016. Generally, the VAR model for this study is given as: Yt−1 C1t ε1t Y1t Y R ε2t C2t Y2t t−i R p T ε3t C Y3t min t−i Tmin 3t Yt = Y4t = Tmax = C4t + ∑ πi Tmax t−i + ε4t , t = 1,2, … … … … … … . ,50 ε5t Y5t C5t W i=1 Wt−i ε6t Y6t C6t H Ht−i [ ε7t ] [ ] [Y7t ] [C7t ] S [ St−i ] Where Y– yield R– rainfall Tmin– minimum temperature Tmax– maximum temperature W– wind speed H–relative humidity S– sunshine duration In the pre-lag order selection using four maximum numbers of lags as given in table 4.3, the suggested model is VAR (1) in all model selection criteria since it has the minimum AIC, SBIC and HQIC values. The estimated VAR model is thus given as follows. , where coefficient matrix 1 and vector of constants C were estimated by the method of least squares. Thus, the model for all the seven variables can explicitly be written as follows. Yt= 15.1518+2.1958Yt-1+ 0 .4189Rt-1+ 0.0447Tmint-1− 0.8359Tmaxt-1 + 0.7694 Wt-1+ 8.0101Ht-1−0.5794 St-1 Rt=23.2717+0.9897Rt-1+ 0.9464Yt-1+ 1− 0.6131Tmint-1−14.3175Tmaxt-1+4.4194Wt-1+19.8657Ht- 4.5617St-1 Tmint= 4.355554−0.0935Tmint-1+0.0013Yt-1 −0.0021Rt-1−0.0793Tmaxt-1 +0.1607Wt-1 −1.6456Ht1− 0.0085St-1 Tmaxt=22.95657+0.2343Tmaxt-1 1+ −0.0062Yt-1+0.0086Rt-1−0.1061Tmint-1−0.0029Wt-1−2.3627Ht- 0.1287St-1 Wt=1.3396−0.0171Wt-1+0.0008Yt-1+0.0006Rt-1−0.0057Tmint-1+0.0111Tmaxt-1 −0.4022Ht1−0.0107 St-1 Ht = 85.643− 0.3049Ht-1 +0.0263Yt-1 +0.0109Rt-1 +0.4397Tmint-1− 0.8237Tmaxt-1+4.1921Wt-1 − 0.176S t-1 St=1.7331−0.1228St-1−0.0293Yt-1−0.0030Rt-1−0.2388Tmint-1−0.1280Tmaxt-1+ 2.669Wt- 1 +0.0131H t-1 Note that it is rare to report and interpret the estimated VAR coefficients since the number of parameters is large and presenting and interpreting all of them is cumbersome. Furthermore, they are poorly estimated except for the first own lag. In general, they are all insignificant. It is therefore typical to report functions of the VAR coefficients instead of interpreting them which summarize information better, have some economic meaning and hopefully, are more precisely estimated. Among the many possible functions, impulse response functions and forecast variance decompositions are the most ones. Therefore, the coefficients of our fitted VAR(1) model shown above were interpreted using these functions in the following section. 4.1.7. Structural Analysis 4.1.7.1. Granger Causality Test Structural analysis is used to deal with the dynamic properties of a VAR(p) model. Variable y1 is said to granger cause variable y2, if the lags of y1 can improve a forecast for variable y2, and so on. The following output is the pair wise granger causality test among all the seven variables. A Wald test is most commonly used to perform the Granger causality test. From the output given in the table at appendix B, each row reports a Wald test that the coefficients on the lags of the variable in the "excluded" column are zero in the equation for the variable in the "equation" column. Therefore, as seen from the result, maximum temperature granger causes relative humidity, wind speed granger causes both rainfall and relative humidity and also sunshine granger causes yield, minimum temperature and wind speed as they all are statistically significant at 5% significance level due to their smaller p-values. 4.1.7.2. Impulse Response Function Impulse response functions show the effects of shocks on the adjustment path of the variables. It indicates the response of an endogenous variable to a change in one of the innovations in the VAR system and is also standard tool for investigating the relations between the variables in a VAR model. Usually, the response is rendered graphically with horizon on the horizontal and response on the vertical. It proposes the effect of a one standard deviation shock to one of the innovations on current and future values of the dependent variables through the dynamic structure of the VAR model. The response of all the endogenous variables to a change in one of the innovations in the given VAR model were given and discussed below. Results from IRF of yield as shown below indicates that, the response of a yield for a one unit of its own innovations has a positive significant impact on its own for all the variables included in the analysis. Also, for a one unit change in wind speed, relative humidity and sunshine duration, yield has opposite response while it has direct response to rainfall, minimum temperature and maximum temperature. Table 4.6. Response of Yield Min Max Rainfall Temperature Temperature Period Yield 1 0.41892 2 0.30034 0.04298 1.06742 3 0.17549 0.658801 4 0.08648 0.00883 5 44799 0.769314 Relative Sunshine Humidity Duration - 8.010 -0.5794 -2.1958 -10.459 -0.3537 -0.5526 0.833904 -0.7916 -0.0005 -0.0957 0.262194 0.313069 -1.2698 -0.0544 -0.2419 0.05046 0.00539 0.185882 0.208421 -0.6658 -0.0278 -0.0818 6 0.02777 0.0029 0.093592 0.1179 -0.3541 -0.0144 -0.0548 7 0.01546 0.0016 0.053568 0.06384 -0.2167 -0.0077 -0.0278 8 0.0085 0.00087 0.029211 0.034871 -0.1028 -0.0042 -0.0161 9 0.0047 0.00049 0.016128 0.019385 -0.0645 -0.0025 -0.0091 10 0.00261 0.00027 0.009038 0.010839 -0.0337 -0.0013 -0.0048 0.016 0.835994 Wind Speed 1.49991 The following graphs also support the idea which is stated under the above table. From the graphs, we can clearly understand that a positive shock to yield causes an increase in yield and the effect dies out after roughly 4 periods. Also, a positive shock in yield causes an increase in rainfall, minimum and maximum temperatures followed by decrease until the effect dies out after roughly 3, 4 and 4 periods respectively while its positive shock causes a decrease which is followed by increase and the effect died after the 3rd period for wind speed, decrease and died after periods 3 and 4 for relative humidity and sunshine duration respectively. varbasic: Yield, Yield varbasic: Rainfall, Yield 5 15 10 0 5 0 -5 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf varbasic: Min Temp, Yield 8 oirf varbasic: Max Temp, Yield 6 6 4 4 2 2 0 0 -2 -2 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf 8 oirf Graphs by irfname, impulse variable, and response variable varbasic: Wind Speed, Yield varbasic: Relative Humidity, Yield 4 2 2 0 0 -2 -4 -2 -6 -4 0 2 4 step 6 95% CI for oirf 8 oirf 0 2 4 step 95% CI for oirf 6 8 oirf varbasic: Sunshine Duration, Yield 2 0 -2 -4 -6 0 2 4 step 95% CI for oirf 6 8 oirf Graphs by irfname, impulse variable, and response variable Figure 2. Impulse Response Function graph of Yield From table 4.7, we can understand that the response of a rainfall for a one unit change of its own innovations has positively significant impact on its own for all the variables included in the system while it showed a negative response for a one unit change in wind speed, relative humidity and sunshine duration but positively respond to a one unit change in yield, minimum and maximum temperatures. Table 4.7.Response of Rainfall Period Yield Rainfall 1 0.946469 0.61311 2 1.13372 3 0.79029 4 Min Max Wind Temperature Temperature Speed 14.3175 Relative Sunshine Humidity Duration 4.4195 19.8658 -4.56175 -0.98972 0.313738 6.08112 9.83209 -43.998 -2.07587 -1.81562 0.101055 3.68613 4.6605 -13.775 0.044027 0.544189 0.374013 0.039502 1.28851 1.37441 -1.5619 -0.14003 -0.93262 5 0.21691 0.026662 0.800212 0.882808 -4.0351 -0.17977 -0.47477 6 0.126993 0.013733 0.463308 0.574741 -1.5333 -0.05806 -0.18845 7 0.06901 0.007143 0.233263 0.286087 -0.9600 -0.03405 -0.13399 8 0.038227 0.003946 0.133604 0.155849 -0.4794 -0.0192 -0.07010 9 0.021118 0.002208 0.072355 0.087327 -0.2782 -0.01133 -0.04114 10 0.011763 0.001221 0.040688 0.04889 -0.1578 -0.00600 -0.02158 From the graphs shown below, we can see that a positive shock to rainfall causes an increase in rainfall and the effect dies out after roughly 3 periods. Also, a positive shock in rainfall causes an increase in yield, minimum and maximum temperatures that is followed by decrease until the effect dies out after period 4, increase which is died out after 4th period and also increase which is followed by decrease and died after the 4th period for yield, minimum and maximum temperatures respectively. The rainfall shocks have also significant impact on wind speed, relative humidity and sunshine duration. That means, its shock causes a decrease in wind speed at the 1st period and increase at period two followed by dying effect at 5th step and similarly in the first and 3rd periods, an increase in rainfall shocks cause sunshine duration to decrease whose effect dies quickly after the 5th period while relative humidity is caused to be decreased at 3rd period and died out at period five. varbasic: Yield, Rainfall varbasic:Rainfall, Rainfall 80 150 60 100 40 50 20 0 0 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf varbasic: Min Temp, Rainfall 8 oirf varbasic: Max Temp, Rainfall 60 40 40 20 20 0 0 -20 -20 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf 8 oirf Graphs by irfname, impulse variable, and response variable varbasic: Wind Speed, Rainfall varbasic: Relative Humidity, Rainfall 20 20 0 0 -20 -40 -20 -60 0 2 4 step 6 95% CI for oirf 8 oirf 0 2 4 step 95% CI for oirf 6 8 oirf varbasic: Sunshine Duration, Rainfall 20 0 -20 0 2 4 step 95% CI for oirf 6 8 oirf Graphs by irfname, impulse variable, and response variable Figure 3. Impulse Response Function graph of Rainfall The effect of minimum temperature for a change in one unit is negatively significant towards its own innovations. Minimum temperature also responds negatively to a one unit change in yield, total rainfall and maximum temperature although it has positive reply towards wind speed, relative humidity and sunshine duration. Table 4.8.Response of Minimum Temperature Min Period Yield Rainfall Temperature Max Temperature Wind Speed Relative Humidity Sunshine Duration 1 0.0013 -0.0021 -0.079321 0.160772 -1.6456 0.008555 -0.09354 2 -0.0014 -0.0031 -0.005343 -0.035204 0.1662 0.044106 0.081326 3 -0.0067 -0.0013 -0.042484 -0.06048 0.3538 0.001936 -0.02780 4 -0.0026 -0.0001 -0.009239 -0.011759 -0.0757 -0.00274 0.00267 5 -0.0012 -0.0001 -0.004166 -0.003218 0.0440 0.002091 0.006 6 -0.0009 -0.0001 -0.00389 -0.004496 0.0073 0.000404 0.00054 7 -0.0004 -0.0005 -0.001485 -0.002098 0.0078 0.000253 0.00108 8 -0.0002 -0.0000 -0.000981 -0.001085 0.0033 0.000111 0.0004 9 -0.0001 -0.0000 -0.000485 -0.000587 0.0016 0.000081 0.000314 10 -0.0000 -8.7e-06 -0.000285 -0.000342 0.0012 0.000045 0.000149 The IRF graph of minimum temperature as displayed below reveals that, a negative shock in minimum temperature causes positive impact on yield at 1st period however the effect deceased out the 3rdand retro. Similarly, the negative shock in minimum temperature impacts rainfall to decrease and then increase at periods one and 2-3 respectively and finally departed out at its 4thperiod whereas it has negatively significant effect on maximum temperature at first period with an increase periods 2-4 which quickly died out at 5th period and it causes decreasing effect towards itself. Likewise, the impact that negative shocks in minimum temperature have on wind speed decreases at first time and increases at 2nd and 3rd periods and again decreases at 4th period. However, relative humidity is caused to be increased and decreased at periods 1-3 and 4 respectively whereas sunshine duration rises and then falls down at times one and three respectively. varbasic: Yield, Min Temp varbasic: Rainfall, Min Temp .5 .5 0 0 -.5 -.5 -1 -1 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf varbasic: Min Temp, Min Temp 8 oirf varbasic: Max Temp, Min Temp 1 2 .5 1 0 0 -1 -.5 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf 8 oirf Graphs by irfname, impulse variable, and response variable varbasic: Wind Speed, Min Temp varbasic:Relative Humidity, Min Temp .5 .5 0 0 -.5 -1 -.5 0 2 4 step 6 95% CI for oirf 8 oirf 0 2 4 step 6 95% CI for oirf 8 oirf varbasic: Sunshine Duration, MinTemp .5 0 -.5 0 2 4 step 95% CI for oirf 6 8 oirf Graphs by irfname, impulse variable, and response variable Figure 4. Impulse Response Function graph of Minimum Temperature From the response of maximum temperature given in table 4.9, we can clearly observe that maximum temperature responded adversely towards a one unit change in yield, rainfall and minimum temperature but supportively to wind speed, relative humidity and sunshine duration and also has inverse impact or effect on its own innovations for a one unit change of its shocks. Table 4.9.Response of Maximum Temperature Min Temperature Max Temperature Wind Speed Relative Humidity Sunshine Duration -0.002919 -2.3627 0.128754 0.234354 -0.00606 -0.106231 -0.222494 2.07496 0.047586 0.003991 -0.0138 -0.00083 -0.065983 -0.077637 -0.2296 -0.01910 -0.03589 4 -0.0048 -0.00055 -0.008554 -0.00695 0.08098 0.00663 0.03339 5 -0.0036 -0.00045 -0.016078 -0.016351 0.06517 0.002759 0.002804 6 -0.0019 -0.00020 -0.00639 -0.009239 0.01997 0.000785 0.003732 7 -0.0010 -0.00011 -0.003745 -0.004318 0.01777 0.000524 0.001906 8 -0.0005 -0.00006 -0.002055 -0.002386 0.00565 0.000284 0.001122 9 -0.0003 -0.00003 -0.001111 -0.001357 0.00501 0.000195 0.000667 10 -0.0001 -0.00001 -0.000645 -0.000771 0.00229 0.000086 0.000313 Period Yield Rainfall 1 -0.0062 -0.00865 -0.106196 2 -0.0231 3 From the graph shown below, we can note the following facts. Over the ten years considered, shocks in maximum temperature have significantly negative impact on yield and rainfall up to three years into the future and then the impact dies out quickly while the effect decreases at the first period, increases at third to fourth periods and slightly died out at the future periods for wind speed, relative humidity and sunshine duration and increases up to the second period although it died out on the latter periods for the minimum temperature. varbasic: Yield, Max Temp varbasic: Rainfall, Max Temp .5 1 0 0 -.5 -1 -1 -1.5 -2 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf varbasic: Min Temp, Max Temp 8 oirf varbasic: Max Temp, Max Temp .5 2 0 1 -.5 0 -1 -1 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf 8 oirf Graphs by irfname, impulse variable, and response variable varbasic: Wind Speed, Max Temp varbasic: Relative Humidity, Max Temp 1 1.5 .5 1 0 .5 -.5 0 -1 -.5 0 2 4 step 6 95% CI for oirf 8 oirf 0 2 4 step 6 95% CI for oirf 8 oirf varbasic: Sunshine Duration, Max Temp 1 .5 0 -.5 0 2 4 step 95% CI for oirf 6 8 oirf Graphs by irfname, impulse variable, and response variable Figure 5. Impulse Response Function graph of Maximum Temperature As one can see from the result of IRF result displayed below, wind speed has a positive retort against yield, rainfall, minimum temperature and maximum temperature while it has negative reaction to relative humidity and sunshine duration under their one unit. shock. Moreover, the effect that wind speed has on one unit shock towards its own innovation is significantly negative. Table 4.10. Response of Wind Speed Period Yield Rainfall Min Temperature Max Temperature 1 0.0008 0.00060 0.005728 2 0.0012 0.00001 3 0.0000 4 Wind Speed Relative Humidity Sunshine Duration 0.011127 -0.40222 -0.01073 -0.01710 0.010259 0.008893 -0.07277 -0.00414 -0.01301 0.00001 0.002174 0.001588 -0.00443 -0.00092 -0.00583 0.0000 0.00003 0.001663 0.001248 -0.00933 -0.00021 -0.00071 5 0.0001 9.6e-06 0.000271 0.000602 -0.00180 -6.7e-06 -0.00030 6 0.0006 7.2e-06 0.000211 0.000224 -0.00230 -0.00005 -0.00013 7 0.0003 3.2e-06 0.000136 0.000148 -0.00011 -6.0e-06 -0.00004 8 0.0001 2.1e-06 0.000056 0.00007 -0.00040 -0.00001 -0.00005 9 0.0001 1.1e-06 0.000042 0.000048 -0.00012 -3.9e-06 -0.00001 10 5.8e-06 5.9e-07 0.000019 0.000023 -0.00007 -3.1e-06 -0.00001 The following graph also supports the idea we understand from the above table. varbasic: Yield, Wind Speed varbasic:Rainfall, Wind Speed .1 .1 .05 .05 0 0 -.05 -.05 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf varbasic: Min Temp, Wind Speed 8 oirf varbasic: Max Temp, Wind Speed .1 .05 .05 0 0 -.05 -.05 0 2 4 step 95% CI for oirf 6 8 oirf 0 2 4 step 95% CI for oirf Graphs by irfname, impulse variable, and response variable 6 8 oirf varbasic: Wind Speed, Wind Speed varbasic: Relative Humidity, Wind Speed .2 .05 .1 0 0 -.05 -.1 -.1 0 2 4 step 6 95% CI for oirf 8 0 oirf 2 4 step 95% CI for oirf 6 8 oirf varbasic: Sunshine Duration, Wind Speed .05 0 -.05 0 2 4 step 6 95% CI for oirf 8 oirf Graphs by irfname, impulse variable, and response variable Figure 6. Impulse Response Function graph of Wind Speed Table 4.11. Response of Relative Humidity Period Yield Rainfall Min Temperature 1 0.0263 0.01099 0.439725 2 0.0221 0.01325 3 0.0273 4 Max Temperature Wind Speed Relative Humidity Sunshine Duration 0.82372 -4.19219 -0.17627 -0.30499 0.159228 0.33228 -2.0093 -0.15509 -0.16764 0.00349 0.143716 0.22709 -0.89590 -0.01754 -0.10351 0.0092 0.00042 0.025485 0.021228 -0.22529 -0.00658 -0.02639 5 0.0049 0.00072 0.01744 0.01350 -0.14276 -0.00828 -0.02083 6 0.0034 0.0004 0.013876 0.017699 -0.03928 -0.00158 -0.00257 7 0.0018 0.00018 0.005745 0.00777 -0.02703 -0.00074 -0.00358 8 0.0009 0.00009 0.003514 0.00386 -0.01157 -0.00045 -0.00172 9 0.0005 0.00005 0.001837 0.002216 -0.00685 -0.00031 -0.00114 10 0.0003 0.00003 0.001067 0.00129 -0.00441 -0.00016 -0.00054 The riposte that relative humidity devours is negative for a one units hock in wind speed and sunshine duration but positive for yield, rainfall, minimum and maximum temperatures while its influence on its own innovation is negatively significant. We can also see the graph for more information. varbasic:Rainfall, Relative Humidity varbasic: Yield, Relative Humidity 6 4 4 2 2 0 0 -2 -2 0 2 4 step 0 8 6 8 6 oirf 95% CI for oirf oirf 95% CI for oirf 4 step 2 varbasic: Max Temp, Relative Humidity varbasic: Min Temp, Relative Humidity 4 2 2 0 0 -2 -2 -4 0 2 4 step 0 8 6 8 6 oirf 95% CI for oirf oirf 95% CI for oirf 4 step 2 Graphs by irfname, impulse variable, and response variable varbasic: Wind Speed, Relative Humidity varbasic: Relative Humidity, Relative Humidity 4 10 2 5 0 0 -2 -5 0 2 4 step 6 95% CI for oirf 8 oirf 0 2 4 step 95% CI for oirf 6 8 oirf varbasic: Sunshine Duration, Relative Humidity 2 1 0 -1 -2 0 2 4 step 95% CI for oirf 6 8 oirf Graphs by irfname, impulse variable, and response variable Figure 7. Impulse Response Function graph of Relative Humidity Table 4.12 reveals that, sunshine duration has direct reply for one unit shock in wind speed and relative humidity but negatively replied towards yield, rainfall, minimum and maximum temperatures having positive effect on its own innovation. From the graph, we can observe the same idea that the table tells us. Table 4.12. Response of Sunshine Duration Period Yield Rainfall Min Temperature Max Temperature 1 -0.0293 -0.00307 -0.238852 2 -0.0092 -0.00056 3 -0.0041 4 Wind Speed Relative Humidity Sunshine Duration -0.128007 2.669 0.013193 0.12287 -0.016159 -0.039617 0.94659 0.02009 0.025175 -0.00056 -0.003659 -0.004344 0.42070 0.011611 0.031278 -0.0031 -0.00019 -0.015515 -0.015109 0.06486 0.00179 0.00567 5 -0.0012 -0.00013 -0.001667 -0.004239 0.03578 0.001567 0.007293 6 -0.0008 -0.00008 -0.003699 -0.003657 0.01159 0.000232 0.00007 7 -.00042 -0.00004 -0.001221 -0.001687 0.00268 0.000214 0.00117 8 -0.0002 -0.00002 -0.000864 -0.000993 0.00478 0.000152 0.000425 9 -0.0001 -0.00001 -0.000469 -0.000566 0.00113 0.000054 0.000229 10 -0.0000 -7.7e-06 -0.000244 -0.000297 0.00114 0.000043 0.000156 varbasic: Yield, Sunshine Dration varbasic: Rainfall, Sunshine Dration .5 .5 0 0 -.5 -.5 -1 0 2 4 step 6 95% CI for oirf 8 0 2 oirf 4 step 6 95% CI for oirf varbasic: Min Temp, Sunshine Dration 8 oirf varbasic: Max Temp, Sunshine Dration .5 .5 0 0 -.5 -.5 -1 0 2 4 step 95% CI for oirf 6 8 oirf 0 2 4 step 95% CI for oirf Graphs by irfname, impulse variable, and response variable 6 8 oirf varbasic: Wind Speed, Sunshine Duration varbasic: Relative Humidity, Sunshine Duration 1 .5 .5 0 0 -.5 -.5 0 2 4 step 6 95% CI for oirf 8 0 oirf 2 4 step 95% CI for oirf 6 8 oirf varbasic: Sunshine Duration, Sunshine Duration 1.5 1 .5 0 -.5 0 2 4 step 95% CI for oirf 6 8 oirf Graphs by irfname, impulse variable, and response variable Figure 8. Impulse Response Function graph of Sunshine Duration 4.1.7.3. Forecast Error Variance Decomposition The forecast error variance decomposition (FEVD) measures the fraction of the forecast error variance of an endogenous variable that can be attributed to orthogonalized shocks to itself or to another endogenous variable. The FEVD of a VAR model gives information about the relative importance of each of the random innovations in explaining each endogenous variable in the system. Practically, it is usually observed that own series shocks explain most of the FEVD of the series in a VAR. The results of FEVD were displayed and discussed in the following section. In table 4.13, the FEVD result tells us that in the first step, 100 % change in yield resulted from the shock of its own innovation while in the second step, 90.8% change in yield level is resulted from the shock to the yield innovation and 84% yield variation at the rest steps. The level of yield had explain about 0.15%, 0.38%, 0.042%, 0.43% and 0.44% at rounds two, three, four, five and six, and the remaining four steps respectively; of the forecasting variance of rainfall while it has no effect on the first step on rainfall. Also, it has no effect on the first round on minimum temperature forecasting variance but explains about 1.1% at second round, 1.6% at third round, 1.7% and1.8% at rounds four and all the remaining ones. Furthermore, the effect of change in yield to that of maximum temperature is 0% or it has no effect at the first step, 1.3% at the second step,4.4%and 5.1% at third and fourth steps but remains unchanged for the rest steps accounting for about 5.2%. Yield also has no effect on wind speed, relative humidity and sunshine duration at all the first periods but explains about 0.01%, 1.4%, 1.3%, 1.4% of forecasting variance of wind speed at periods 2, 3, 4 and5-10 respectively, 3.6%, 4.5%, 4.3% at periods 2, 3 and 4-10 respectively for relative humidity and 2.8%, 2.6% for the respective steps 2 and 3-10 for sunshine duration. Table 4.13.Forecast Error Variance Decomposition Function of Yield Period Yield Rainfall Min Temperature Max Temperature 1 0 0 0 2 0.908604 0.001511 3 0.849178 4 Wind Speed Relative Humidity Sunshine Duration 0 0 0 0 0.011363 0.013573 0.000101 0.036243 0.028604 0.003839 0.01605 0.04465 0.014202 0.045245 0.026835 0.842992 0.004225 0.017854 0.051071 0.013845 0.043919 0.026094 5 0.841572 0.004353 0.018147 0.05188 0.014014 0.043844 0.026189 6 0.84099 0.00439 0.018321 0.052289 0.014046 0.043807 0.026156 7 0.840826 0.0044 0.018357 0.052417 0.014055 0.043792 0.026152 8 0.840776 0.004404 0.01837 0.052454 0.014059 0.043788 0.026149 9 0.840762 0.004405 0.018374 0.052465 0.01406 0.043786 0.026149 10 0.840757 0.004405 0.018375 0.052468 0.01406 0.043786 0.026149 As the following table shows, about 84% of the variation in rainfall has resulted from its own shock at first period while71%, 67% and 66% of variation were resulted at periods 2, 3, 4-10 respectively. Also, in round one, rainfall has no effect on minimum and maximum temperature as well as wind speed, relative humidity and sunshine duration while it explains about 15.9% of the variation in yield at both first and second periods. For minimum temperature, 5.8% variation is resulted from the change in rainfall for all rounds except the first and for maximum temperature, 1.4%, 4.1% and 4.6% variation at periods 2, 3, and 4-10 respectively. For wind speed, relative humidity and sunshine duration respectively, the variation due to rainfall is0.03%, 0.59%, 0.61% at times 2, 3, 4 and the rest ones, 4.99%, 5.55%and also 0.01%, 0.04% and 0.05% at rounds two, three and the remaining periods respectively. Table 4.14.Forecast Error Variance Decomposition Function of Rainfall Period Yield Rainfall Min Max Wind Temperature Temperature Speed Relative Sunshine Humidity Duration 1 0.159118 0.840882 0 0 0 0 0 2 0.159468 0.716583 0.058985 0.014573 0.000381 0.049901 0.000109 3 4 0.163605 0.673963 0.058371 0.168407 0.665017 0.058837 0.041763 0.04608 0.005974 0.05588 0.006126 0.055068 0.000443 0.000467 5 0.170023 0.6631 0.058887 0.046378 0.006127 0.054931 0.000554 6 0.170437 0.66243 0.058912 0.046537 0.006171 0.054937 0.000577 7 0.170578 0.66222 0.058917 0.046602 0.006176 0.054925 0.00058 8 0.170623 0.662159 0.058918 0.046618 0.006179 0.054922 0.000582 9 0.170636 0.66214 0.058918 0.046622 0.006179 0.054921 0.000582 10 0.17064 0.662135 0.058919 0.046624 0.006179 0.054921 0.000582 Based on the result in the table below, 96.2%, 90.9% and 88% change in minimum temperature has caused by its own shocks at times t = 1, 2, 3-10 correspondingly. At first, second and third rounds, the effect of the change in minimum temperature towards the change in yield is about 0.89%, 0.13%, and 0.13% respectively but 0.14% at the remaining time periods. Furthermore, minimum temperature’s effect on rainfall is 2.8%, 4.3% and 4.4% respectively for times 1, 2, and 3 to the last ones; however, it doesn’t affect maximum temperature, wind speed, relative humidity and sunshine duration at the first time whereas its shock on these variables is about 1.5% and 1.7%, 0.08% and 1.4%, 0.3% and 0.5% at second and third periods and remains unchanged after the third period separately. Table 4.15.Forecast Error Variance Decomposition Function of Minimum Temperature Period Yield Rainfall Min Max Wind Temperature Temperature Speed Relative Sunshine Humidity Duration 0 1 0.008998 0.02833 0.962672 0 0 0 2 0.013095 0.0437 0.909337 0.015633 0.014292 0.000872 0.003071 3 0.013411 0.044102 0.889374 0.017943 0.015692 0.014205 0.005273 4 0.014264 0.04424 0.884763 0.020903 0.016151 0.014173 0.005507 5 0.014697 0.044232 0.884198 0.020963 0.016182 0.014222 0.005506 6 0.014778 0.044232 0.884053 0.020971 0.016201 0.014248 0.005517 7 0.014809 0.04423 0.884008 0.020987 0.0162 0.014248 0.005517 8 0.014818 0.04423 0.883995 0.02099 0.016201 0.014248 0.005517 9 0.014821 0.04423 0.883991 0.020991 0.016201 0.014248 0.005517 10 0.014822 0.04423 0.88399 0.020991 0.016201 0.014248 0.005517 Table 4.16 reveals the change in maximum temperature which is resulted from its own shock accounting for about 62%, 54% and 53%andalso, 9.1%, 9.2%, 9.6% and 9.7%changes in yield, 22%, 20%, 19% and18% changes in rainfall were resulted from the change in shocks of maximum temperature at each respective time periods; while no change or no influence was caused on wind speed, relative humidity and sunshine duration at first round,0.4%, 2.2% and 2.3% ;7.9%, 8.4% and 8.5%;1.3% and 1.2% were caused by variation in maximum temperature at stages beginning from two and continuing to the last correspondingly. Table 4.16.Forecast Error Variance Decomposition Function of Maximum Temperature Min Temperature Max Temperature Wind Speed Relative Humidity Sunshine Duration 0 0 Period Yield Rainfall 1 0.091 0.22873 0.056287 2 0.09255 0.2036 0.064338 0.542414 0.00431 0.07962 0.01318 3 0.09237 0.19207 0.060723 0.534851 0.02284 0.0849 0.01224 4 0.0966 0.19019 0.060944 0.531193 0.02313 0.08556 0.01239 5 0.09741 0.18994 0.060886 0.530381 0.02318 0.08559 0.01261 6 0.09766 0.18983 0.060926 0.530211 0.02319 0.08557 0.01261 7 0.09775 0.1898 0.060924 0.530165 0.02319 0.08556 0.01261 8 0.09778 0.1898 0.060925 0.530149 0.02319 0.08556 0.01261 9 0.09778 0.18979 0.060925 0.530144 0.02319 0.08556 0.01261 10 0.09779 0.18979 0.060925 0.530142 0.02319 0.08556 0.01261 0.62398 0 From table 4.17, we can see that the shock that wind speed has towards itself is 92%, 79% and 76% whereas its causes towards the levels of yield, rainfall, minimum and maximum temperatures accounts for about 3.6%, 2.5% and 3.0%; 2.7%, 2.4% and 2.3%; 0.8%, 1.4% and 1.8%; and 0.15%, 2.6% and 3.0% at all the periods respectively. Conversely, the effect of the change in wind speed towards the change in relative humidity and sunshine duration is insignificant at step one and 10% and 11%; and 1.3%, 1.9% and 2.0% respectively at the remaining stages. Table 4.17.Forecast Error Variance Decomposition Function of Wind Speed Period Yield Rainfall Min Temperature Max Temperature Wind Speed Relative Humidity Sunshine Duration 1 0.03673 0.02784 2 0.02515 0.02401 3 0.0304 4 0.00847 0.001579 0.92539 0 0 0.014167 0.026308 0.79368 0.10368 0.01302 0.02308 0.018713 0.030252 0.76541 0.11255 0.0196 0.03079 0.02302 0.018869 0.030363 0.76321 0.11277 0.02097 5 0.03095 0.02303 0.019034 0.030537 0.76274 0.11274 0.02098 6 0.03102 0.02303 0.019034 0.030565 0.76265 0.11273 0.02098 7 0.03104 0.02303 0.019035 0.030569 0.76262 0.11272 0.02098 8 0.03105 0.02303 0.019036 0.030571 0.76262 0.11272 0.02098 9 0.03105 0.02303 0.019036 0.030572 0.76261 0.11272 0.02098 10 0.03105 0.02303 0.019036 0.030572 0.76261 0.11272 0.02098 As shown below, for relative humidity, at the first, second, third up to the tenth periods,54%, 51% and 50% variation has occurred due to the shock of its own innovation. Similarly, the change in yield explained by the shock of relative humidityis10.5%, 9.8%, 9.6% and 9.7% at each respective rounds. In the first period, the variation in level of rainfall explained by relative humidity is29%, and 27% at second to tenth periods. The variation in minimum temperature, maximum temperature and wind speed respectively is 0.4%, 2.8% and 2.0% while it has no effect on the variation of sunshine duration at the first step but it explains the deviation of the level of these parameters at the rest times for about 3.7%, 3.6%; 5.1%, 5.6%; 2.0%, 3.0% and 0.2%, 0.2% respectively. Table 4.18.Forecast Error Variance Decomposition Function of Relative Humidity Period Yield Rainfall Min Temperature Max Temperature Wind Speed Relative Humidity Sunshine Duration 1 0.10537 0.29304 0.004538 0.028588 0.02016 0.5483 0 2 0.09831 0.27987 0.037078 0.051828 0.0203 0.5105 0.00212 3 0.09649 0.27339 0.036618 0.059294 0.02335 0.50818 0.00269 4 0.09725 0.27234 0.036701 0.061447 0.02338 0.50597 0.00291 5 0.09759 0.27222 0.036705 0.061428 0.02338 0.50575 0.00293 6 0.09765 0.27218 0.036712 0.061432 0.0234 0.5057 0.00293 7 0.09767 0.27217 0.036715 0.061446 0.0234 0.50567 0.00293 8 0.09768 0.27217 0.036715 0.061448 0.0234 0.50567 0.00294 9 0.09768 0.27217 0.036715 0.061449 0.0234 0.50566 0.00294 10 0.09768 0.27216 0.036715 0.061449 0.0234 0.50566 0.00294 From table 4.19 below, sunshine duration has the effect of about 79%, 65% and 63% on its own innovations at first to the last times respectively while its influence on yield, rainfall, minimum temperature, maximum temperature, wind speed and relative humidity at the same time periods is 16%, 15%, 16%; 0.2%, 0.4%, 0.5%; 0.05%, 5.6%, 5.5%, 5.4%; 1.4%, 4.0%, 3.9%, 4.0%; 1.5%, 7.6%, 8.3%, 8.6%; and 0.8%, 1.0%, 1.6%, 1.8% respectively. Table 4.19.Forecast Error Variance Decomposition Function of Sunshine Duration Period Yield Rainfall Min Temperature Max Temperature 1 0.16713 0.00285 0.000582 0.014509 2 0.1553 0.00449 0.056777 3 4 5 0.16197 0.16207 0.16252 0.00505 0.00507 0.00506 6 0.16262 7 Relative Humidity Sunshine Duration 0.01542 0.00887 0.79063 0.040217 0.07603 0.01086 0.65633 0.055183 0.05486 0.054987 0.04 0.039854 0.0401 0.08399 0.08611 0.08606 0.01638 0.63744 0.01789 0.63415 0.0179 0.63336 0.00507 0.054973 0.040119 0.08607 0.01793 0.63323 0.16266 0.00507 0.05498 0.040135 0.08606 0.01793 0.63318 8 9 0.16267 0.16267 0.00507 0.00507 0.05498 0.05498 0.040138 0.04014 0.08606 0.08606 0.01793 0.63316 0.01793 0.63316 10 0.16268 0.00507 0.05498 0.04014 0.08606 0.01793 0.63316 4.1.8. Model Diagnostic Checking Wind Speed Checking model adequacy is a basis for any statistical analysis since in fitting any statistical model, it should be adequate in the manner that all of its assumptions are met and as a result, the inference being made regarding the model may be interesting. In this section, model diagnostic can be made through the residual analysis so dealing with the nature of residuals before stressing about adequacy of the model is necessary. 4.1.8.1. Whiteness of Residuals For a model to be adequate representation of the data, the residuals should have no significant trend or pattern that means they should be uncorrelated. The test for autocorrelation of the residuals can be performed by Lagrange-multiplier (LM) test of autocorrelation. The test result is given in table 4.20 below indicating that we fail to reject H0and conclude that there is no autocorrelation among the residuals in the model since the p- value (prob>chi2) of the test is larger. The ACF and PACF plots of the residuals were displayed at appendix F and also supports this evidence since all autocoreelation values show no evidence of significant spikes (all the spikes are within the 95% confidence bounds). Table 4.20.Lagrange-multiplier test for Residual Autocorrelation Lag Chi-square (χ2) df Prob>χ2 1 51.7143 49 0.36829 2 44.9587 49 0.63764 H0: no autocorrelation at lag order 4.1.8.2. Test for Normality of Residuals Normality test of the model residuals is necessary for the sake of making valid inference regarding the statistical model under consideration. The test is performed using Jarque-Berra normality test in which the hypotheses were given as H0: The residuals are normally distributed against H1: The residuals are not normally distributed. The test result is as displayed below revealing that the null hypothesis for all the variables is not rejected as their p-values are all large. Therefore, the residuals are normally distributed. Also the normal probability plots of residuals in appendix G indicate that the disturbances are normally distributed. Table 4.21.Jarque-Bera test for Normality of Residuals Chi-square (χ2) df Prob>χ2 Yield 0.186 2 0.91126 Rainfall 0.121 2 0.94113 Min Temperature 3.349 2 0.18744 Max Temperature 1.842 2 0.39807 Wind Speed 1.334 2 0.51315 Relative Humidity 1.479 2 0.47733 Sunshine Duration 1.254 2 0.53419 Equation H0: residuals are normally distributed 4.1.9. Checking for VAR Stability VAR stability can be checked in order to make further structural analysis of the model to deal with its dynamic properties. If the estimated VAR model appears stable, then we can produce IRF and FEVD both in tabular and graphical representations. In order to check for the stability of our selected VAR(1) model, the test for eigen value stability condition should be conducted. The stability condition test results were displayed in table 4.22 below, revealing that all the eigen values lie inside the unit circle, this means that our VAR(1) model has modulus values less than unit and results in stability of the model. Table 4.22. Stability Condition Test Eigen value Modulus 0.7633499 0.76335 0.69532 -0.1048238 + 0.6873727i -0.1048238 -0.6873727i 0.69532 0.69532 0.633759 0.533561 -0.05173975 +0.1303332i -0.5173975 - 0.1303332i -0.4315408 + 0.3126022i -0.4315408 - 0.3126022i 0.533561 0.532867 0.532867 0.530588 0.490336 -0.5305878 0.490336 0.4047825 +0.2767314i 0.303821 0.4047825 -0.2767314i 0.261836 0.3038205 0.261836 -0.02357897 +0.2607721i -0.02357897 -0.2607721i All the eigen values lie inside the unit circle. VAR satisfies stability condition. 4.1.10. Forecasting with VAR Model Forecasting is one of the main objectives of multivariate time series analysis. Forecasting from a VAR model is similar to forecasting from a univariate AR model and the following section gives a brief description of what is forecasted using our VAR(1) model. The graphs shown below indicates that the confidence bands on our forecasts are not large resulting in high forecasting ability or efficiency of our VAR(1) model and hence allows us to forecast the data very confidently. Also, one can observe from the graphs that amount of yield, rainfall, minimum temperature, maximum temperature, wind speed, relative humidity and sunshine duration values will decrease at some years, for instance, at years above 2016 until 2020, they show significant change. After years 2020, yield of the study area is expected to increasing under (increased rainfall, relative humidity and wind speed) and (decreased minimum and maximum temperatures and sunshine duration). Forecast for Min Temp 2 4 6 8 10 200400600800 Forecast for Rainfall 0 -20 0 20 40 60 Forecast for Yield Forecast for Wind Speed Forecast for Relative Humidity 40 50 60 70 80 .4 .6 .8 1 20 25 30 35 1.2 Forecast for Max Temp 2015 2020 2025 2015 4 6 8 10 Forecast for Sunshine Duration 2015 2020 2025 95% CI forecast 2020 2025 Figure 4.9.VAR(1) Forecast Graphs 5. CONCLUSION AND RECOMMENDATION 5.1. Conclusion The aim of this study was to evaluate the effects of climate change (effects of rainfall, minimum temperature, maximum temperature, wind speed, relative humidity and sunshine duration) on production of sorghum yield over the periods 1967 to 2016 at Melkassa district of Eastern Shewa Zone of Oromia region. For assessing and modeling the effects of climate parameters on sorghum yield of Gambella #1107 variety, a total of 50 years observations for the period 1967 to 2016 were included and used in the study. The climate data were taken from weather station of the study area while production data for Gambella#1107 variety were obtained from Melkassa Research Center of EIAR. The data were fitted using a multivariate time series model, Vector Autoregressive (VAR) model was employed in order to fit the data. The findings of this study provided essential numerical evidences on the existence of higher year to year variability in total rainfall, average temperature (both minimum and maximum), average wind speed, average relative humidity and average sunshine duration as well as yield in the study area accounting for 26.3%, 38.6%, 10.3%, 24.3%, 13.6%, 23.1% and 66.1% coefficients of variation respectively. Also, the structural analysis was made in order to assess the dynamic interaction of climate parameters and yield production using VAR(1) model which is selected though model selection criteria (AIC, BIC and HQIC). Based on this analysis, Granger’s causality test shows that maximum temperature granger causes relative humidity, wind speed granger causes both rainfall and relative humidity and also sunshine granger causes yield, minimum temperature and wind speed telling us that all the variables have contribution for the occurrence of each other. Also, the IRF and FEVD analysis results indicate that yield, rainfall, minimum and maximum temperatures, wind speed, relative humidity and sunshine duration shocks have significant impacts on the occurrence of one another. Generally, the findings of this study had arrived on the point that the higher changes in annual rainfall, minimum and maximum temperatures, wind speed, relative humidity and sunshine duration of autumn cropping season significantly changes the productivity level of sorghum yield at the study area. 5.2. Recommendations Based on the results obtained, the study forwarded the following recommendations for all the concerned bodies. 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APPENDICES APPENDIXA: Values of Yield, Rainfall, Temperature (Both Minimum and Maximum), Wind Speed, Relative Humidity and Sunshine Duration Data for Years 1967-2016 At Melkasa District Min Max Average Average Average Average Wind Relative Temp in Temp in Speed Humidity o o C C in m/s in % Average Sunshine Duration in hours Sorghum Yield in quintal/hectar Total RF in mm 1967 8.9 400 8.5 25.7 0.7 64 8.6 1968 10.6 602 7.4 23.7 0.9 70 9.1 1969 20.1 550.8 3.8 25.5 0.6 69 6.9 1970 15.2 254.9 9.2 23.5 1 50 7 1971 20.4 576.3 4.1 20 0.7 66 8.3 1972 10.1 352.5 6.5 27.7 0.9 71 7.7 1973 17.9 607 5 25.9 0.5 65 7.2 1974 6.7 379.2 6.9 28.1 0.8 55 8.1 1975 9.8 332.7 7.3 30.2 1 49 7.7 1976 18 610.1 6 26.1 0.8 73 7.9 1977 20.5 554 4 27.3 1 59 7.1 1978 18.9 445 5.3 25.5 0.8 66 9.3 Year 1979 22.4 320.4 2.2 29.8 1 56 7.5 1980 19.3 516 6.6 22.5 0.7 64 9.4 1981 17.8 511.7 2.5 24.6 0.5 72 5.9 1982 6.1 386 7.9 30 1 58 8.3 1983 7.9 393 5.6 26 0.4 45 9 1984 4.1 385.6 8.3 28 1 50 9.4 1985 19.5 519.1 6.6 25.7 0.7 72 4.7 1986 15.6 405 4.8 24.1 0.9 61 6.2 1987 28.3 440.4 3.4 25.8 0.4 58 5 1988 28.7 560.3 1.5 27.5 1 70 7.9 1989 27 544 7.1 25.9 0.5 63 6.3 1990 21 459 5.8 24.9 0.8 56 4.9 1991 48 535.7 8.2 22 1 68 5.4 1992 7.5 344 5.8 29.5 0.6 49 7.9 1993 47 520 8.8 25.1 1 62 5 1994 47.6 535 6.6 23.9 0.8 72 6 1995 4.6 490.6 8.2 22.6 0.6 59 5.6 1996 53 540.2 7.3 24.8 1 70 4.4 1997 69 575 5.6 26.1 0.7 60 5.1 1998 70 690 8.1 21.3 1 72 7.1 1999 69.5 682 4.2 24.9 0.8 63 4.9 2000 52 599 2.3 26.2 1 59 6.1 2001 45 500 5 22 0.8 67 5.7 2002 3 320 7.5 30 0.4 51 8.3 2003 15.2 498 3 26.6 0.6 62 4 2004 47 572 5.1 24.4 0.9 59 7 2005 48 590 7.4 21 1 65 5.4 2006 49 632.8 0.5 23 0.6 73 9.2 2007 38 615 5.5 24.3 0.8 64 5.5 2008 35 602 4 21.2 1 75 6.4 2009 29 380.4 1.3 30.5 0.7 58 9.5 2010 21.5 258 6.7 28.8 1 39 6.3 2011 28.8 334.5 7.8 26.4 0.9 44 8.9 2012 36.9 435.2 1.7 24.5 1 69 5.1 2013 24.4 305.1 7.2 26.4 0.7 69 9.3 2014 39.4 218.9 7.6 28.1 0.5 57 7.2 2015 9 350.5 6.2 30.4 1 65 4.6 2016 28 218.7 5.4 27.9 0.8 59 8.7 Equation Chi-square (χ2) df Excluded Prob>χ2 Yield Rainfall 2.7537 1 0.097 Yield Min Temperature 0.67692 1 0.411 Yield Max Temperature 0.55526 1 0.456 Yield Wind Speed 0.55324 1 0.457 Yield Relative Humidity 3.3903 1 0.066 Yield Sunshine Duration 2.5751 1 0.109 Rainfall Yield 0.73373 1 0.392 Rainfall Min Temperature 3.4528 1 0.063 Rainfall Max Temperature 0.31867 1 0.572 Rainfall Wind Speed 0.05918 1 0.808 Rainfall Relative Humidity 3.6546 1 0.056 Rainfall Sunshine Duration 0.0091 1 0.924 Min Temperature Yield 0.00397 1 0.950 Min Temperature Rainfall 0.30159 1 0.583 Min Temperature Max Temperature 1.1208 1 0.290 Min Temperature Wind Speed 1.0792 1 0.299 Min Temperature Relative Humidity 0.03416 1 0.853 Min Sunshine Duration 0.21602 1 0.642 0.06334 1 0.801 Temperature Max Temperature Yield Max Temperature Rainfall 3.4838 1 0.062 Max Temperature Min Temperature 0.3705 1 0.543 Max Temperature Wind Speed 1.6328 1 0.201 Max Temperature Relative Humidity 5.6785 1 0.017 Max Temperature Sunshine Duration 0.99493 1 0.319 Wind Speed Yield 0.24781 1 0.619 Wind Speed Rainfall 3.9593 1 0.047 Wind Speed Min Temperature 0.25214 1 0.616 Wind Speed Max Temperature 0.92156 1 0.337 Wind Speed Relative Humidity 9.228 1 0.002 Wind Speed Sunshine Duration 1.2398 1 0.266 Relative Humidity Yield 0.10059 1 0.751 Relative Humidity Relative Humidity Rainfall Min Temperature 0.50931 1 0.475 0.57559 1 0.448 Relative Humidity Max Temperature 1.9565 1 0.162 Relative Humidity Wind Speed 0.46573 1 0.495 Relative Humidity Sunshine Duration 0.15269 1 0.696 Sunshine Duration Yield 3.907 1 0.048 Sunshine Duration Rainfall 1.2543 1 0.263 Sunshine Duration Min Temperature 5.3348 1 0.021 Sunshine Duration Max Temperature 1.4842 1 0.223 Sunshine Duration Wind Speed 5.93 1 0.015 Sunshine Duration Relative Humidity 0.16971 1 0.680 Appendix B: Grangers’ Causality Test Results 0 20 40 60 80 APPENDIX C: Time Series Plot of Original Series 1970 1980 1990 2000 2010 2020 2000 2010 2020 Year 500 400 300 200 Total RF in mm 600 700 Time Series Plot of Total Rainfall 1970 1980 1990 Year 10 8 6 4 2 0 1970 1980 1990 2000 2010 2020 2010 2020 Year 20 22 24 26 28 30 Time Series Plot of Average Minimum Temperature 1970 1980 1990 2000 Year Time Series Plot of Average Maximum Temperature 1 .8 .6 .4 .2 1970 1980 1990 2000 2010 2020 2010 2020 Year 40 50 60 70 80 Time Series Plot of Average Wind Speed 1970 1980 1990 2000 Year Time Series Plot of Average Relative Humidity 10 8 6 4 1970 1980 1990 2000 2010 2020 Year Time Series Plot of Average Sunshine Duration 0 20 40 60 80 Appendix D:Plot for Test of Randomness of Series 1970 1980 1990 2000 Year Plot for Sorghum Yield Test of Randomness 2010 2020 700 600 500 1980 1990 2000 2010 2020 2010 2020 Year Plot for Total Rainfall Test of Randomness 0 2 4 6 8 10 Total RF in mm 400 300 200 1970 1970 1980 1990 2000 Year Plot for Average Minimum Temperature Test of Randomness 30 28 26 24 22 20 1970 1980 1990 2000 2010 2020 Year .2 .4 .6 .8 1 Plot for Average Maximum Temperature Test of Randomness 1970 1980 1990 2000 Year Plot for Average Wind Speed Test of Randomness 2010 2020 80 70 60 50 40 1970 1980 1990 2000 2010 2020 Year 4 6 8 10 Plot for Average Relative Humidity Test of Randomness 1970 1980 1990 2000 2010 Year Plot for Average Sunshine Duration Test of Randomness 2020 -0.40 -0.20 0.00 0.20 0.40 Appendix E: ACF and PACF Plots of Residuals 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Sorghum Yield 0 5 10 15 Lag Bartlett's formula for MA(q) 95% confidence bands ACF Plot of Total Rainfall 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 15 10 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Average Minimum Temperature 0 5 10 15 Lag Bartlett's formula for MA(q) 95% confidence bands ACF Plot of Average Maximum Temperature 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Average Wind Speed 0 5 10 15 Lag Bartlett's formula for MA(q) 95% confidence bands ACF Plot of Average Relative Humidity 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Average Sunshine Duration 0 5 10 15 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Sorghum Yield 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 PACF Plot of Total Rainfall 0 5 10 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Average Minimum Temperature 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag 95% Confidence bands [se = 1/sqrt(n)] -0.40 -0.20 0.00 0.20 0.40 PACF Plot of Average Maximum Temperature 0 5 10 15 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Average Wind Speed 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag 95% Confidence bands [se = 1/sqrt(n)] -0.40 -0.20 0.00 0.20 0.40 PACF Plot of Average Relative Humidity 0 5 10 15 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Average Sunshine Duration 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Sorghum Yield Residual 0 5 10 15 Lag Bartlett's formula for MA(q) 95% confidence bands ACF Plot of Total Rainfall Residual 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Average Minimum Temperature Residual 0 5 10 15 Lag Bartlett's formula for MA(q) 95% confidence bands ACF Plot of Average Maximum Temperature Residual 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Average Wind Speed Residual 0 5 10 15 Lag Bartlett's formula for MA(q) 95% confidence bands ACF Plot of Average Relative Humidity Residual 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag Bartlett's formula for MA(q) 95% confidence bands -0.40 -0.20 0.00 0.20 0.40 ACF Plot of Average Sunshine Duration Residual 0 5 10 15 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Sorghum Yield Residual 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 15 20 25 Lag 95% Confidence bands [se = 1/sqrt(n)] -0.40 -0.20 0.00 0.20 0.40 PACF Plot of Total Rainfall Residual 0 5 10 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Average Minimum Temperature Residual 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag 95% Confidence bands [se = 1/sqrt(n)] -0.40 -0.20 0.00 0.20 0.40 PACF Plot of Average Maximum Temperature Residual 0 5 10 15 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Average Wind Speed Residual 20 25 0.40 0.20 0.00 -0.20 -0.40 0 5 10 15 20 25 Lag 95% Confidence bands [se = 1/sqrt(n)] -0.40 -0.20 0.00 0.20 0.40 PACF Plot of Average Relative Humidity Residual 0 5 10 15 Lag 95% Confidence bands [se = 1/sqrt(n)] PACF Plot of Average Sunshine Duration Residual 20 25 0.00 0.25 0.50 0.75 1.00 Appendix F: Normal Probability Plots of Residuals 0.00 0.25 0.50 Empirical P[i] = i/(N+1) 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Yield Residual Plot 0.00 0.25 0.50 Empirical P[i] = i/(N+1) Rainfall Residual Plot 0.75 1.00 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 Empirical P[i] = i/(N+1) 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Minimum Temperature Residual Plot 0.00 0.25 0.50 Empirical P[i] = i/(N+1) Maximum Temperature Residual Plot 0.75 1.00 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 Empirical P[i] = i/(N+1) 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Wind Speed Residual Plot 0.00 0.25 0.50 Empirical P[i] = i/(N+1) Relative Humidity Residual Plot 0.75 1.00 1.00 0.75 0.50 0.25 0.00 0.00 0.25 Sunshine Duration Residual Plot 0.50 Empirical P[i] = i/(N+1) 0.75 1.00