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.
 National Metrological Agency Services of the Melkassa district need to establish robust
climate forecasting tactic with investment on the stations.
 Melkassa Agricultural Research Center should work in collaboration with the weather station
of the area on how to increase the productivity level of sorghum yield and overcome the
increased weather variability.
 More interventions should be facilitated by researchers in regard with impacts of climatic
change on sorghum yield production of the study area using additional factors rather than those
considered under this study since the study considers only natural factors and do not include
the man-made ones.
REFERENCES
[1] Abera, B., Alwin, K., Manfred, Z. (2011). Using Panel Data to Estimate the Effect of Rainfall
Shocks on Smallholders‟ Food Security and Vulnerability in Rural Ethiopia. Climatic
Change, 108:185–206.
[2] Ajetombi, J. & A. Abiodun (2010). Climate Change Impacts on Cowpea Productivity in
Nigeria. African Journal of Food, Agriculture, Nutrition and Development, 10(1): 2258-2271.
[3] Alemneh, D. (2011). Strengthening capacity for climate change adaptation in the agriculture
sector in Ethiopia. Proceedings from National Workshop Held in Nazareth, Ethiopia 5–6 July
2010, Proceedings from National Workshop Held in Nazareth, Ethiopia 5–6 July 2010.
[4] Amare, A. (2015). Review on the Impact of Climate Change on Crop Production in Ethiopia.
Journal of Biology, Agriculture and Healthcare, Vol.5 (13).
[5] Antle, J. (2010). Adaptation of Agriculture and the Food System to Climate Change: Policy
Issues. Issue Brief 10-03. Washington, DC: Resources for the Future.
[6] Apata, T. G., Samuel, K. D. & Adeola, A. O. (2009). Analysis of climate change perception
and adaptation among arable food crop farmers in South Western Nigeria. Contributed paper
prepared for presentation at the International Association of Agricultural Economists’ 2009
conference, Beijing, China, August 16 (Vol. 22).
[7] Aragie, E. A. (2013). Climate change, growth and poverty in Ethiopia. The Robert S. Strauss
Center for International Security and Law, University of Texas at Austin.
[8]
Aye, G.C. & Ater, P.I. (2012). Impact of Climate Change on Grain Yield and Variability in
Nigeria: A Stochastic Production Model Approach. Mediterranean Journal of Social
Sciences, Vol 3 (16):142-150.
[9] Arimi, K. (2014). Determinants of climate change adaptation strategies used by rice farmers
in Southwestern, Nigeria. Journal of Agriculture and Rural Development in the Tropics and
Subtropics, 115 (2), 91–99.
[10] Badawi, A. (2009). “Testing Stationarity in Selected Macroeconomic Series from Sudan.”
African Journal of Economic Policy.
[11] Basak, J.K. (2011). Implications of Climate Change on Crop Production in Bangladesh and
Possible Adaptation Techniques. In: International Conference on United Nations
Framework Convention on Climate Change (UNFCCC), Durban Exhibition Centre,
Durban, COP 17. Unnayan Onneshan - The Innovators, Dhaka-1215, Bangladesh.
[12] Boko et al. (2009). Africa Climate change 2007: impacts, adaptation and vulnerability. In:
Parry ML, Canziani OF, Palutik of JP, vd Linden PJ, Hanson CE (eds) Contribution of
working Group II to the fourth assessment report of the intergovernmental panel on climate
change. Cambridge University Press, Cambridge, pp 433–467.
[13] Chatfield C. (1988). What is the ’best’ method of forecasting? Journal of Applied Statistics,
15(1), 19-38.
[14] Chikezie et al. (2015). Effect Of Climate Change On Food Crop Production In Southeast,
Nigeria: A Co-Integration Model Approach. International Journal of Weather Climate
Change and Conservation Research. Vol.2, No.1, pp.47-56.
[15] Conway, D. & Schipper, E.L.F. (2011). Adaptation to climate change in Africa: Challenges
and opportunities identified from Ethiopia. Global Environmental Change, 21 (1), 227–237.
[16] Central Statistical Agency (CSA) (2010). Reports on area and crop production forecasts for
major grain crops (For private peasant holding, Meher Season). The FDRE Statistical
Bulletins (1990–2010), CSA, Addis Ababa, Ethiopia.
[17] Central Statistics Agency (CSA) (2013). Agricultural Sample Survey 2012/2013 (2005 E.C.).
Volume I. Report on Area and Production of Major Crops (Private Peasant Holdings,
Meher Season). Statistical Bulletin 532, Addis Ababa CSA (Central Statistics Agency),
2012.
[18] D.S. Maccarthy & P.L.G. Vlek (2012). Impact of Climate Change on Sorghum Production
Under Different Nutrient and Crop Residue Management in Semi-Arid Region Of Ghana:
A Modeling Perspective. African Crop Science journal. vol. 20, issue supplement s2, 243 –
259.
[19] Daniel, K. (2008). Impacts of climate change on Ethiopia: a review of the literature. In: Green
Forum (ed.), Climate Change A Burning Issue for Ethiopia: Proceedings of the 2nd Green
Forum Conference Held in Addis Ababa, 31 October–2 November 2007. Green Forum,
Addis Ababa.
[20] Deressa, T. & R. Hassan (2009). Economic Impact of Climate Change on Crop Production in
Ethiopia: Evidence from Cross-section Measures. Journal of African Economies 18 4:529–
554.
[21] Di Falco, S., Veronesi, M., & Yesuf, M. (2011a). Does Adaptation to Climate Change Provide
Food Security? A Micro-perspective from Ethiopia. American Journal of Agricultural
Economics Advance Access published March 7.
[22] Di Falco, S., Yesuf, M., Kohlin, G. & Ringler, C. (2012b). Estimating the Impact of Climate
Change on Agriculture in Low-Income Countries: Household Level Evidence from the Nile
Basin, Ethiopia. Environ Resource Econ (2012) 52:457–478.
[23] Elodie, B. (2012). The Impact of Climate Change on Crop Yields in Sub-Saharan Africa.
American Journal of Climate Change.
[24]
Enete, A. A. & Amusa, T. A. (2010). Challenges of Agricultural Adaptation to Climate
Change in Nigeria: A Synthesis from the Literature. Field Actions Science Reports (FACTS
Report) Journal of Field Actions, Volume 4.
[25]
Environmental Protection Agency, EPA, (2011). Ethiopia’s program of adaptation to
climate change, first draft, Addis Ababa: EPA.
[26] Engle, R.F., & Granger, C.W.J. (1987). "Co-Integration and Error Correction:
Representation, Estimation, and Testing." Econométrica55:251-76.
[27] Food and Agriculture Organization (FAO) (2011). Two Essays on Climate Change and
Agriculture: A developing country perspective. FAO Economic and social development
paper 145. Rome, Italy.
[28] Fekede, et al. (2016). Effect of Climate Change on Agricultural Production and Community
Response in DaroLebu & Mieso District, West Hararghe Zone, Oromia Region National
State, Ethiopia.Journal of Natural Sciences Research Vol6(24).
[29] Gbetibouo, G.A. (2009). Understanding Farmers’ Perceptions and Adaptation to Climate
Change and Variability: The Case of Limpopo Basin. Discussion Paper No. 849.
International Food Policy Research Institute, South Africa. 52pp.
[30] Gebreegziabher, Z., Stage, J., Mekonnen, A. & Alemu, A. (2011). Climate Change and the
Ethiopian Economy: A Computable General Equilibrium Analysis. Environment for
Development Discussion Paper Series, EfD DP 11-09.
[31] Gutu, T., Bezabih. E & Mengistu, K. (2012). A Time Series Analysis of Climate Variability
And Its Impacts On Food Production In North Shewa Zone In Ethiopia. African Crop Science
Journal, Vol. 20, Issue Supplement S2. 261 – 274.
[32] Hannan, E. J. (1970). Multiple Time Series., Wiley.
[33] IMF (International Monetary Fund) (2012). Ethiopian Economic Outlook.
[34] IPCC (2009). Climate Change Synthesis Report, an Assessment of IPCC.
[35] Hirut, G. & Kindie, T. (2015). Analysis of risks in crop production due to climate change in
four districts at central rift valley of Ethiopia. African Journal of Agricultural Research. Vol
10(16).
[36] Jansen, H. & Huib, H. (2012). Agricultural development in the Central Ethiopian Rift Valley:
A Desk Study on Water-Related Issues, Wageningen. Plant Research International Journal.
[37] Johanson, S. (1988). Likelihood-Based Inference in Cointegrated Vector Autoregressive
Models. Oxford: Oxfored University Press.
[38] Johanson, S. & Juselius, K. (1990). Maximum Likelihood Estimation and Inference on
Cointegration with Applications to the Demand for Money. Oxford Bulletin of Economics
and Statistics, 169-210.
[39] Krishna, P. P. (2011). Economics of Climate Change for Smallholder Farmers in Nepal: A
review. The Journal of Agriculture and Environment Vol. 12.
[40] Leta, A. (2011). Assessment of climate change effects on rain fed crop production and coping
mechanisms: the case of smallholder farmers of west Shoa zone, Oromia, Ethiopia. Master’s
Thesis, Addis Ababa University.
[41] London School of Economics (2015). Ethiopia Country Profile.The Global Climate Change
Legislation Study.
[42] Lütkepohl, H. (1993). Introduction to Multivariate Time Series Analysis., Springer-Verlag,
2nd edition.
[43] Lütkepohl, H. (2005). Introduction to Multivariate Time Series Analysis., Springer-Verlag,
4th edition.
[44] Maddala, G. S. (1992). Introduction to Econometrics, 2nd edition, MACMILLAN Publishing
Company, New York.
[45] Makombe, G., Namara, R., Hagos, F., Awulachew, S. B., Ayana, M. & Bossio, D. (2011). A
comparative analysis of the technical efficiency of rain-fed and smallholder irrigation in
Ethiopia. IWMI Working Paper 143, International Water Management Institute, Colombo,
Sri Lanka.
[46] Mano, Y., Omori, F., Loaisiga, C.H., & Bird, R. M. (2009). QTL mapping of above ground
adventitious roots during flooding in maize x teosinte “Zeanicaraguensis” backcross
population. Plant Root 3, 3Y9.
[47] Mesfine, T., G. Abebe & A.M. Al-Tuwaha (2009). Effect of Reduced Tillage and Crop
Residue Ground Cover on Yield and Water Use Efficiency of Sorghum (Sorghum Bicolar
L.) Under Semi-Arid Conditions of Ethiopia. World Journal of Agricultural Sciences, vol
1(2):152-160.
[48] MoARD (2010). Ministry of Agriculture and Rural Development Crop Development Crop
Variety Register. Issue No.7. Addis Ababa, Ethiopia. pp104.
[49] MoFED (2012). Growth and Transformation Plan (2010/11–2014/15). Annual Progress
Report for F.Y. 2011/12. Ministry of Finance and Economic Development (MoFED), Addis
Ababa, Ethiopia.
[50] Müller, C., Cramer, W., Hare, W. L. & Lotze-Campen, H. (2011). Climate change risks for
African agriculture. Proceedings of the National Academy of Sciences, 108 (11), 4313–
4315.
[51] Muamba, F. and Kraybill, D. (2010). Weather Vulnerability, Climate Change, and Food
Security in Mt. Kilimanjaro. Poster prepared for presentation at the Agricultural & Applied
Economics Association 2010 AAEA, CAES, & WAEA Joint Annual Meeting. Denver,
Colorado, July 25-27, 2010.
[52] NCEA (2015). Ethiopia Climate Change Profile.
[53] Newey, W. K. & Kenneth, D. W. (1987). "A Simple Positive Semi-Definite, Heteroskedasticity
and Autocorrelation Consistent Covariance Matrix." Econometrica55:703-8.
[54] Newton, A. C., Johnson, S. N. & Gregory, P. J. (2011). Implications of climate change for
diseases, crop yields and food security. Euphytica, 179 (1), 3–18.
[55] Oumer, B. (2016). Impact of Weather Variations on Cereal Productivity and Influence of
Agro-Ecological Differences in Ethiopian Crop Production. PhD Thesis, Addis Ababa
University.
[56] Pryor, S. C., Scavia, D., Downer, C., Gaden, M., Iverson, L., Nordstorm, R., Patz, J. &
Robertson, G. P. (2014). Midwest. In: Melillo, J. M., Terese (T.C.) Richmond & Yohe, G.
W. (eds.), Climate Change Impacts in the United States: The Third National Climate
Assessment. ch. 18, pp. 418–440, U.S. Global Change Research Program, Washington, D.C.
[57] Reinsel G. (1997). Elements of Multivariate Time Series Analysis. Springer-Verlag New
York, Inc.,
[58] Sims, C.A. (1980). “Macroeconomics and Reality,” Econometrica, 48, 1-48.
[59] Sowunmi, F. A. (2010). Effect of climatic variability on maize production in Nigeria.
Research Journal of Environmental and Earth Sciences, 2 (1), 19–30.
[60] Thornton & Lipper (2014). How does climate change Alter Agricultural Strategies to Support
Food Security, IFPRI, Discussion paper 091340.
[61] Tigist, M. (2011). A Multivariate Time Series Analysis of Agro meteorological Data to
Assess the Effect of Climate Variability on Production of Sorghum: The Case of Melkassa,
Ethiopia. Master’s Thesis, Addis Ababa University.
[62] USAID (2013). Ethiopia Climate Vulnerability Profile.
[63] USE (2016). The Humanitarian Response of the USG to the Current Ethiopian Drought.
[64] UNSCEBC (United Nations System Chief Executive Board for Co-ordination) (2008). Acting
on climate Change: The UN system delivering as one, United Nations Headquarter, New
York.
[65] Woldeamlak, B. (2009). Rainfall variability and crop production in Ethiopia Case study in
the Amhara Region. Proceedings of the 16th International Conference of Ethiopian Studies,
2009.
[66] Yesuf, M., Di Falco, S. Ringler, C. & Kohlin, G. (2010). Impact of climate change and
adaptation to climate change on food production in low income countries: Household
survey data evidence from the Nile basin of Ethiopia. Discussion Paper No. 828.
International Food Policy Research Institute (IFPRI), Washington, D.C.
[67] Zhai, F., Lin T., and Byambdorj E. (2009). A General Equilibrium Analysis of the Impact of
Climate Change on Agriculture in the People’s Republic of China. Asian Development
Review 26(1): 206−225.
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