APPLICATION OF ARIMA MODEL IN PREDICTION OF REAL ESTATE PRICES IN KENYABYLAURETTAH NYAMBURA KARIUKIKMA 003/15A RESEARCH PROPOSAL SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN MATHEMATICS IN THE FACULTY OF SCIENCE IN KIRIRI WOMEN’S UNIVERSITY OF SCIENCE AND TECHNOLOGY

APPLICATION OF ARIMA MODEL IN PREDICTION OF REAL ESTATE PRICES IN KENYABYLAURETTAH NYAMBURA KARIUKIKMA 003/15A RESEARCH PROPOSAL SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN MATHEMATICS IN THE FACULTY OF SCIENCE IN KIRIRI WOMEN’S UNIVERSITY OF SCIENCE AND TECHNOLOGY.SEPTEMBER 2017
Declarations
This Proposal is my original work and has not been presented for a Degree in any other University.
Signature: ___________________ Date: __________________
Name: Laurettah Nyambura Kariuki – KMA 003/15Supervisor: This proposal has been submitted for review with our approval as University supervisors.Signature: ___________________ Date: ________________________
Name: ___________________________
Department _____________________
Kiriri Women’s University of Science and Technology
Abbreviations and acronymsGDP- Gross Domestic Product at market prices
CBK-Central Bank of Kenya
ARIMA- Autoregressive Integrated Moving Average model
ACF-Auto correlated Function
AR-Autoregressive model
MA-Moving Average model
ARMA- Autoregressive Moving Average model
Operational definition of terms
Red loans- a discriminatory practice by which banks, insurance companies, among others refuse or limit loans, mortgages, insurance, within specific geographical areas especially inner-city neighbourhoods.

Abstract
Real estate industry has been one of the most resilient, vibrant and profitable in Kenya today. Nairobi has experienced one of the most rapid growths in urban centres. This is unlikely to slow down as the population in Kenya increases every day. This study uses ARIMA model to provide forecasts for Nairobi housing prices from available samples. Housing prices have positive GDP, diaspora remittances, lending rates, loans to real estate sector and cost of construction. However, a negative relationship exists between the housing prices and inflation. Interest rates affect housing affordability and thus demand for new and resale homes. When this happens, there is an increase in GDP which leads to increased investment in real estates. This increases the supply of houses hence reducing real estate prices. An increase in money supply gives rise to greater inflation growth uncertainty. This creates an adverse impact on real estate market hence increasing the prices. An application of ARIMA model in predicting real estate prices will provide an indication of short term market direction and a sense of whether there will be a small or large movement (of direction). This will serve as an advance warning well ahead of any turning points for investment strategy.
Table of Contents
TOC o “1-3” h z u Chapter 1: Introduction………………………………………………………………….. PAGEREF _Toc516897110 h 11.1 Background to the Study PAGEREF _Toc516897111 h 11.2 Statement of the Problem PAGEREF _Toc516897112 h 21.3 Purpose of the study PAGEREF _Toc516897113 h 21.4 Objectives of the Study PAGEREF _Toc516897114 h 21.4.1 General objective PAGEREF _Toc516897115 h 21.4.2 Specific objectives PAGEREF _Toc516897116 h 21.5 Significance of the study PAGEREF _Toc516897117 h 31.6 Justification PAGEREF _Toc516897118 h 4Chapter 2: Literature review…………………………………………………………….. PAGEREF _Toc516897119 h 52.2.1 Theoretical findings PAGEREF _Toc516897120 h 52.2 Empirical findings PAGEREF _Toc516897121 h 52.2.1 AR Models PAGEREF _Toc516897122 h 72.2.2. MA Models PAGEREF _Toc516897123 h 82.2.3. ARMA Models PAGEREF _Toc516897124 h 92.2.4. ARIMA Models PAGEREF _Toc516897125 h 9Chapter 3: Methodology…………………………………………………………………. PAGEREF _Toc516897126 h 113.1 Introduction PAGEREF _Toc516897127 h 113.2 Research Design PAGEREF _Toc516897128 h 113.3 Study Area PAGEREF _Toc516897129 h 113.4 Target Population PAGEREF _Toc516897130 h 113.4 Sampling Techniques PAGEREF _Toc516897131 h 123.5 Validity and Reliability PAGEREF _Toc516897132 h 123.6 Data collection technique PAGEREF _Toc516897133 h 133.7 Data Analysis PAGEREF _Toc516897134 h 133.7.1 Source PAGEREF _Toc516897135 h 133.7.2 Correlation Analysis and ARIMA Modelling PAGEREF _Toc516897136 h 133.8 Box-Jenkins Method PAGEREF _Toc516897137 h 14Chapter 4: Findings and analysis………………………………………………………. PAGEREF _Toc516897138 h 164.1 Introduction PAGEREF _Toc516897139 h 164.2 Analysis of the trends of real estate prices in Kenya PAGEREF _Toc516897140 h 164.3 Modelling of inflation rates of real estate prices in Kenya PAGEREF _Toc516897141 h 194.3.1 Twelve month forecast of inflation rates of real estate prices in Kenya PAGEREF _Toc516897142 h 194.3.2 Investigation of the relationship between property prices and inflation rates PAGEREF _Toc516897143 h 204.3.3 Discussion of findings PAGEREF _Toc516897144 h 264.4 Summary PAGEREF _Toc516897145 h 27Chapter 5: Conclusions………………………………………………………………….. PAGEREF _Toc516897146 h 285.1 Introduction PAGEREF _Toc516897147 h 285.2 Conclusions PAGEREF _Toc516897148 h 285.3 Recommendations PAGEREF _Toc516897149 h 295.4 Limitations of this research PAGEREF _Toc516897150 h 305.5 Directions for future research PAGEREF _Toc516897151 h 31References………………………………………………………………………………… PAGEREF _Toc516897152 h 32Appendix 1: Gantt chart…………………………………………………………………. PAGEREF _Toc516897153 h 34Appendix 2: Budget…………………………………………………………………………………………………………. PAGEREF _Toc516897154 h 35

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Chapter 1: Introduction
1.1 Background to the StudyReal estate is broadly defined as the property that consists of land, buildings and the natural resources that are present on the land (Crowe et al., 2013). These natural resources may include and are not limited to minerals, crops, water and immovable property. According to Mulunda (2017) real estate remains an attractive investment to many Kenyans. With a growth in demand for housing due to a rise in urbanisation and the onset of devolution, the Kenyan government has failed in meeting the annual housing demand across major cities and towns in Kenya. Resultantly, Anyanzwa (2018) observes that this has given rise to a number of business opportunities to private sector players that are buying land and developing houses that cater to different tastes and budgets of Kenyans.
Increased participation of the private sector in the Kenyan real estate industry has resulted in the market forces of demand and supply to determine the prices for residential real estate properties across Kenya. Central to this growth, investment analysts at Britam explicate that Nairobi has been the primary contributor to the growth of the Kenyan real estate sector (Mureithi, 2018). This contribution is against the backdrop of Nairobi county being the smallest yet the most populous with 3,375,000 residents as of the 2009 census. Its rapid growth is unlikely to slow down any time soon as the population in Kenya increases by an average of about 3% that is 1 million each year. Real estate industry has been one of the most resilient, vibrant and profitable in Nairobi today. The growth has been mainly driven by urbanization, a strong economy and stable legal environment, significant credit expansion and increased spending on infrastructure by the government. According to Kenya National Bureau of Statistics in Kenya, the real estate has been a driver of growth in the past five years. Real estate markets are continually adjusting to equilibrium where price range is adjusted according to variation in supply influenced by changes in national and regional economies. Inflation has pushed up the cost of doing business contributing to the increased number of properties. In Nairobi real estate industry has played a key role in the growth of the economy due to its high multiplier effect through increased investments in production and marketing of building materials, employment generation and wealth creation.1.2 Statement of the Problem Kenya’s real estate market like any other in the world operates in cycles. Cycles in Kenya have a tendency of starting after a general election and ending with the next general election.
2017 being an election year, there has been a tell-tale sign that the market is adopting a wait and see attitude. This can be clearly seen as there has been a slowdown in uptake, drop in prices and the increase in number of vacancies. This in one way or another makes it possible to forecast real estate prices in Kenya. However, there are other factors that affect real estate prices: demographics, income, urbanization, rising interests, changing lifestyles, government policy, infrastructure, market stakeholders, and availability and cost of credit.
In this study, the problem is to determine inflation of real estate prices in Nairobi, Kenya using time series Autoregressive Integrated Moving Average (ARIMA) models. When it comes to forecasting, there are extensive number of methods and approaches available and their relative success or failure to outperform each other is in general conditional to the problem at hand. The rational for choosing this type of model is contingent on the behaviour of the time series data. Also in the history of inflation forecasting, this model has proved to perform better than other models.1.3 Purpose of the studyForecasting of real estate prices in Kenya will play a crucial role in business, industry, government, and institutional planning because many important decisions depend on the anticipated future values of inflation rates. This study will help ensure that the real estate industry grows and its prices be affordable to every Kenyan.1.4 Objectives of the StudyThe objectives of this study are1.4.1 General objectiveTo apply ARIMA model in prediction of inflation of real estate prices. 1.4.2 Specific objectivesTo analyse the trends of real estate prices in Kenya.To model inflation rates of real estate prices in KenyaTo perform 12 months forecast of inflation rates of real estate prices in Kenya.

To investigate the relationship between property prices and the inflation rates. 1.5 Significance of the study
It is hoped that the findings of this study will add to the body of knowledge that is already in existence in the field of real estate valuation and also form a basis for further research by researchers. As such, the study will further make a contribution to the literature on determinants of real estate properly prices which will be part of articles that will be useful to researchers who want to further in this study and to other wider stakeholders in the field. Real estate agents and brokers will also benefit from this study by getting information concerning real estate patterns and thus be able to advise their clients on these patterns. It is also expected that the results of this study will be of importance to the lenders, especially those who advance mortgages as they would find it useful to assist them in fine tuning loan advancement decisions to real estate investors. Financial analysts should also find this study useful in providing information necessary in advising their clients in financial decisions. Additionally, since the study will draw attention to the determinants of residential real estate prices, the study will help real estate investors to make informed choices in the real estate property investment. The findings of this study may also be used by the government and other policy making bodies as a guideline in formulation and development of policies that are concerned with real estate sector of the economy. The government as the regulator of real estate sector would benefit with the findings of this study as it would be enlightened on the various approaches that real estate firms can adopt to determine the prices of properties. Information gathered through this study would help the government to formulate policies beneficial in the best approaches in the real estate sector in Kenya. In order to make a fair and comprehensive analysis and forecast, we set out to analyse five (5) years of monthly real estate prices in Kenya. The forecasting time horizon for this research is short-range because most ARIMA models have short term memories and tend to be more accurate than 5 longer-range forecast. Also some of the traditional model identification techniques are subjective and the reliability of the chosen model can depend on the skill and experience of the forecaster (although this criticism often applies to other modelling approaches as well). Furthermore, the economic significance of the chosen model is not clear therefore it is not possible to run policy simulations with the model.
1.6 Justification Inflation is a challenge facing most countries in Africa with Kenya not being exception. It is a major focus of economic policy worldwide. Real estate markets are continually adjusting to equilibrium where price range is adjusted to variation in supply which is influenced by changes in national and regional economies. Inflation has pushed up the cost of doing business contributing to the number of properties. It is important to forecast real estate prices as there will be low-cost but good quality housing for the common Kenyan. An application of ARIMA model in predicting real estate prices will provide an indication of short term market direction and a sense of whether there will be a small or large movement (of direction). This will serve as an advance warning well ahead of any turning points for investment strategy.

Chapter 2: Literature review 2.2.1 Theoretical findings
In time series analysis, an autoregressive integrated moving average (ARIMA) model is a class of statistical models for analyzing and forecasting time series data. It is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time series data either to better understand the data or to predict future points in the series (forecasting). ARIMA models are applied in some cases where data show evidence of non-stationary, where an initial differencing step (corresponding to the “integrated” part of the model) can be applied one or more times to eliminate the non-stationary. The AR part of ARIMA indicates that the evolving variable of interest is regressed on its own lagged (i.e., prior) values. The MA part indicates that the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past. The I (for “integrated”) indicates that the data values have been replaced with the difference between their values and the previous values (and this differencing process may have been performed more than once). The purpose of each of these features is to make the model fit the data as well as possible. Non-seasonal ARIMA models are generally denoted ARIMA (p, d, q) where parameters p, d, and q are non-negative integers, p is the order (number of time lags) of the autoregressive model, d is the degree of differencing (the number of times the data have had past values subtracted), and q is the order of the moving-average model. Seasonal ARIMA models are usually denoted ARIMA( p ,d ,q)( P, D, Q) m, where m refers to the number of periods in each season, and the uppercase P ,D ,Q refer to the autoregressive, differencing, and moving average terms for the seasonal part of the ARIMA model. When two out of the three terms are zeros, the model may be referred to base on the non-zero parameter, dropping “AR”, “I” or “MA” from the acronym describing the model. For example, ARIMA (1,0,0) is AR(1), ARIMA(0,1,0) is I(1), and ARIMA(0,0,1) is MA(1). ARIMA models can be estimated following the Box–Jenkins.

2.2 Empirical findings
Investigating the empirical relationship between real estate prices and inflation using data from Nairobi. Several researchers have been interested in the real estate sector in terms of development and management. In Kenya, very little literature has been churned in financial aspects of the real estate industry with regard to pricing.
The real estate sector in Kenya has seen a boom that began somewhere in the mid to late 2000s because the property market is responding to increased demand. In Nairobi, there is one of the largest expatriate communities in the continent due to the significant number of multinationals who have chosen Nairobi as either their African hub or East and Central African hub. CITATION wk l 2057 (why Kenya’s real estate is robust, 2013) The rebirth of property development in Nairobi has attracted global attention. In its 2012 Wealth Report, real estate management company, Knight Frank, ranked Nairobi as the fastest growing real estate market in the world. CITATION ab l 2057 (Abacus: Nairobi’s changing skyline) Real estate prices in Nairobi rose 25 percent between January and December 2011. Nairobi was also voted as one of the top 10 cities to watch by global real estate firm, Jones Lang LaSalle, out of 150 cities globally.

MARKETS
Office
The office market in Kenya has moved from a position of oversupply and the market is finally stabilising, as Nairobi reinforces its position as the regional commercial hub of Sub-Saharan Africa. CITATION grr l 2057 (Knight Frank)
Retail
Kenya continues to experience some of the most decentralised growth of the retail market as marketers move outside Nairobi and go to other urban centres in Kenya such as Mombasa, Kisumu, Eldoret and Nakuru.
Industrial
Most industrial companies in Kenya tend to avoid renting or leasing space. However, over the recent years there has been significant development especially along Mombasa Road. Rent and take up is low so the sector might need a couple more years to mature.

Residential
The residential market has been one of the key drivers of the property market. There has been a significant number of development projects coming up both within the Greater Nairobi Area and other urban areas such as Mombasa, Naivasha and Kisumu.
Focus has been given to modelling Kenya’s inflation using various determinants in previous reaches as the case of Durevall & Ndungu (2001). The author explains that there are several external shocks that affect inflation and hence finding a stable and parsimonious model that describes Kenya’s inflation as a major challenge.
2.2.1 AR Models
Let {?t} be an entirely random process with mean 0 or variance ?2. Then the process {Xt} is said to be an AR (p) (Autoregressive of order p) process if:
Xt=?1Xt-1+ ?2Xt-2+…+?pXt-pThat is, AR (1) processXt=?1 Xt-1+ et , ?;1 … equation (1)In what follows we express equation (1) as a sum of infinite MA process that is, successively replacing Xt’s in AR (1) process.
Xt = et + ?Xt -1 =et+ ?et-1+ ?Xt-2=et+ ?et-1+?2Xt-1= et+ ?et-1+ ?2 Xt-2+ Xt-3= et+ ?et-1+?2Xt-2+?2Xt-3…=et+ ?et-1+?2Xt-s+?2Xt-s+1 Xt= j=0s?jet-j+ ?s+1Xt-s+1Xt- j=0s?jet-j= ?s+1Xt-s+1We square both sides and get expectations: EXt- j=0s?jet-j2= E ?s+1Xt-s+12… equation (2)From equation (1) we have: EXt= ?EXtAssuming that the process is stationary, then: EXt=? EXt-1= EXt-2Since et are constants EXt- ? j=0s?j=0…
If this holds, then we have 1-?EXt=0 since from equation (1) ??0 and ??1 but |?|<| hence, EXt=0 therefore, from equation (2): EXt- j=0s?jet-j2= ?2(s+1) E Xt-s+12… Equation 3If Xt is stationary, then EXt=EXt-s+1= 0, VarXt=VarXt-s+1=constant EXt- j=0s?jet-j2=?2(s+1)VarXt
Since Xt process is assumed to be stationary, then the variance of Xt is a constant and because|?|<|, then equation 3 approaches 0 as s approaches?. This implies that Xt= j=0s?jet-j , therefore an AR (1) converges to an infinite MA process of random elements with weights ?j that is AR(1) process has been expressed as the sum of infinite MA process.EXt=j=0??jEet-j=0VarXt=j=0??2jVaret-j=r2j=0??2j= r21+?2+?4+?6+?=r21-r2, true as long as |?|<|
rh=EXtXt+h=Ej=0s?jet-jj=0s?iet+h-i=j=0s?ij=0s?iEet-jet+h-i when j=i-h, i=j+h
Eet-jet+h-i=?2 when j?i-h, then Eet-jet+h-i=0. Therefore
rh= r2j=0s?j?j+h, h=0, 1, 2 …
rh=?hr2j=0s?2j= r2?h1-?2, |?|<| r0=?21-r2The acf is ?h=?h?0= ?2?h1-?2?1-?2?2=?h2.2.2. MA Models
Let et be a purely random process with mean 0 and variance ?2. Then the process Xt is said to be a MA process of order q denoted by MA (q).Xt=?0et+ ?1et-1+?+?qet-q=j=0q?jEet-j where ?j’s are constants.
E(Xt)= j=0q?jEet-jVar(Xt)= j=0q?j2jVaret-j= ?2j=0q?j2jThe auto correlation function
rh=EXtXt+h=Ej=0q?jEet-jj=0q?ket+h-k=j=0q?jj=0q?kEet-jet+h-kWhere j=k-h, k=j+h whenk?j+h, Eet-jet+h-k=0 since et is a white noise process hence the autocovariance function:
rh=?2j=0q-h?j0;h;q0h;0r-hh;0The autocovariance does not depend on time t and the mean is a constant.

2.2.3. ARMA Models
An ARMA (p; q) (Autoregressive Moving Average with orders p and q) model is a discrete time linear equations with noise, of the formXt= ?1Xt-1+?+ ?pXt-p+?t+?1 ?t-1+?+?q ?t-qWe may incorporate a non-zero average in this model. If we want that Xt has average?, the natural procedure is to have a zero-average solution Zt of
Zt=?1Zt-1+?+?pZt-p+?t+?1?t-1+?+?q ?t-q And take Xt=Zt+? hence solution of Xt= ?1Xt-1+?+ ?pXt-p+?t+?1 ?t-1+?+?q ?t-q+? with?=?-?1?-?-?p?2.2.4. ARIMA Models
An ARIMA (p; d; q) (Autoregressive Integrated Moving Average with orders (p, d, q) model is a discrete time linear equations with noise, of the form1-k=1p?kLk1-LdXt=1+k=1q?KLk?t
It is a particular case of ARMA models, but with a special structure. SetYt=1-LdXt. Then Yt is an ARMA (p; q) model 1-k=1p?kLkYt=1+k=1q?KLk?t and Xt is obtained from Yt by d successive integrations. The number d is thus the order of integration. The random walk is ARIMA (0; 1; 0). We may incorporate a non-zero average in the auxiliary process Yt and consider the equation
1-k=1p?kLk1-LdXt=1+k=1q?KLk?t+? With ?=?-?1?-?-?p?ARIMA model captures a suite of different standard temporal structures in the time series data. It provides a simple but powerful method for making skilful time series forecasts. Real estate prices in the country are continually changing due to inflation among other factors.

Chapter 3: Methodology
3.1 IntroductionThis chapter covers the methodology and procedures used for collecting and analysing the data in this study. It deals with the type of research design, the study area, target population, sampling technique, sample size, data collection methods and instruments used as well as data analysis and methods suitable to the achievement of the stated objectives.
3.2 Research DesignA research design is the arrangement of conditions for collection and analysis of data in a manner that aims to combine relevance to the research purpose with economy of the procedure. This study used descriptive research design. This research design is suitable where the study seeks to describe the characteristics of a particular individual or a group. It enables the researcher to profile the sample or population by gathering accurate information (Mugenda ; Mugenda, 1999).
3.3 Study AreaThe study area of this research is Nairobi County. This is because Nairobi County has experienced one of the most rapid growths in urban centres which is unlikely to slow down as the population in Kenya increases by an average of about 3% each year. This makes it the best study area as the high population drives growth in demand for both residential and commercial property. This makes property supply to struggle due to the demand arising.

3.4 Target PopulationThe population development of Nairobi as well as related information and services (Wikipedia, Google, images)
Area: 695.1 km² – Density: 4,514.99 inh ./ km²
Nairobi: capital city of Kenya – Inception: 1899–Elevation: +1661 m – Official Website– Local dialing code: 020
Source: Kenya National Bureau of Statistics (web).

Further Population Figures:
Gender (C 2009): males 1,605,219, females 1,533,150
Urbanization (C2009): urban 3,138,369
Name
Population
Census
1999-08-24
Population
Census
2009-08-24
Nairobi 2,143,254 3,138,369
3.4 Sampling TechniquesIn this study, stratified random sampling is used. With stratified sampling, the population is divided into groups based on some characteristic. Then, within each group, a probability sample is selected. The sampling method uses random sampling. The heterogeneous population is divided into non-overlapping relatively homogenous sub-population (strata) which are independent of each other. Since the stratum are homogenous within themselves, the variance is minimal within each stratum. When the strata have been determined, we select a simple random sample from each stratum. If the sample size to be selected is n then we select n1 from N1. That is, i=1Lni=n3.5 Validity and ReliabilityValidity is the extent to which an instrument measures what it is supposed to measure and perform as it is designated to perform. It’s measured in degrees as it is nearly impossible for an instrument to be 100% valid. As a process it involves collecting and analysing data to access the accuracy of an instrument.
3.6 Data collection techniqueThe data will be collected from secondary data from documentation from previous studies, property reports, journals, and data from Housing Finance Corporation, Central Bank of Kenya, Kenya National Bureau of Statistics and Hass Consult Limited.

3.7 Data Analysis3.7.1 SourceThe majority of house price information is derived using HassConsult Sold data as at transaction date, properties sold at true prices. This data is collected monthly at the signing stage and after the price agreed has been completed. Other sources in the public domain, and drawn from more than 20 estate agencies in Nairobi and the propertyleo database, are used to verify the HassConsult position, with a base of offer price data. This additional data source has allowed the development of the Composite Price series which is the Average Offer Price of all properties listed of the relevant property type in the 3months prior to the end of the relevant quarter.
3.7.2 Correlation Analysis and ARIMA ModellingARIMA modelling also makes use of patterns in the data but these patterns might not be easily visible in a plot of the data. Instead, it uses differencing and the autocorrelation and partial autocorrelation functions to help identify an acceptable model. The advantage of the ARIMA model is that it is flexible in fitting data.
3.7.2.1 Differences
Calculates and stores the differences between data values of a time series. This is a common step in assessing likely ARIMA models especially if you want to fit an ARIMA model but the data has a trend or seasonality component. Differencing is used to simplify the correlation structure and to reveal any underlying pattern.

3.7.2.2 Lag
Calculates and stores the lags of a time series. When you lag a time series, Minitab moves the original values down the column, and inserts missing values at the top of the column. The number of missing values inserted depends on the length of the lag.

3.7.2.3 Autocorrelation
Calculates and creates a graph of the autocorrelations of a time series. It is the correlation between observations of a time series separated by k time units. The plot of the autocorrelation is called the ACF (Autocorrelation Function).
3.7.2.4 Partial Autocorrelation
Calculates and creates a graph of the partial autocorrelations of a time series. These are correlations between sets of ordered data pairs of a time series. The PAC at a lag k is the correlation between residuals at time t from an autoregressive model and observations at lag k with terms for all intervening lags in the AR model. The plot for the PAC is called the partial autocorrelation function (PACF).

3.7.2.5 Cross Correlation
Calculates and creates a graph of the correlations between two time series.

3.7.2.6 ARIMA
Fits a Box-Jenkins ARIMA model to a time series. ARIMA refers to filtering steps taken in calculating the ARIMA model until only random noise remains. It models time series behaviour and generates forecasts.

3.8 Box-Jenkins MethodBOX JENKINS PROCEDURE
The Box-Jenkins method, named after the statisticians George Box and Gwilym Jenkins, applies ARMA or ARIMA models to find the best of fit of a time-series model to past values of a time series.

Modelling approach
The original model uses an iterative three-stage modelling approach:
Model identification and model selection: making sure that the variables are stationary’ identifying seasonality in the dependent series and using plots of the autocorrelation and partial correlation functions of the dependent time series to decide which (if any) AR or MA component should be used in the model.

Parameter estimation using computation algorithms to arrive at coefficients that best fit the selected ARIMA model. The most common methods use maximum likelihood estimation or non-linear least-squares estimation.
Model checking by testing whether the estimated model conforms to the specifications of a stationary univariate process.
Estimating the parameters for Box-Jenkins models involves numerically approximating the solutions for non-linear equations. It is common to use statistical software designed to handle to the approach- fortunately, virtually all modern statistical packages feature this capability. The main approaches to fitting Box-Jenkins models are non-linear least squares and maximum likelihood estimation (which is generally the preferred technique).
Chapter 4: Findings and analysis
4.1 Introduction
This chapter critically analyses the findings on the analysis of the trends of real estate prices in Kenya using the ARIMA model. The structure utilised by this chapter is as follows: analysis of the trends of real estate prices in Kenya; modelling of the inflation rates of real estate prices in Kenya.
4.2 Analysis of the trends of real estate prices in Kenya
The figure 1 below shows the trends in Nairobi land prices for Q3 in 2017.

Figure 1: Nairobi Land prices in Q3 2017
Source: Hass Consult (2017)
As shown, land prices in Nairobi rose on aggregate by 3.2% during the Q3 2017 period. Of these, Langata registered the highest year on year rise of 8.5%, followed by Muthaiga (8%), Eastleigh (8%), Donholm (7.2%), Karen (6.8%), Spring Valley (5.9%) and Runda (5.8%). In terms of prices, Upperhill remained Kenya’s most expensive land with an acre averaging US$5.24 million, followed by Kilimani, Westlands and Parklands with average prices per acre at US$4.22 million, US$4.03 million and US$3.95 million respectively.
A global property guide on residential property price change on an annual basis for the period of 2011-2017 suggested that the Kenyan real estate industry was highly speculative. The figure 2 below shows the residential property price changes.

Figure 2: Residential property price change
Source: Hass Consult (2017)
High demand and speculation have driven the Kenyan residential property price inflation up over the past 8 years. This trend is strongly manifested across the rental market. Some of the factors that can be attributed to this tend include the existence of a stable political environment and favourable macroeconomic conditions that have resulted in relatively stable interest rate. These factors are crucial in realising a positive development in the real estate sector. Moreover, a study conducted by Kenyan Property Guide (2017) shows that the Kenyan urban population increases at a rate of 4.2% per year which increases the pressure on housing demand. As of 2010, it was estimated that annually, about 120,000 units were needed whilst the Housing Finance Corporation of Kenya HFC could only deliver 35,000 units. This trend indicated the existence of a demand for affordable housing and office space that could cater to the low and middle income market. Owing to the existence of the shortage in the supply side, property prices in Kenya have been on the rise. The figure 3 below shows the Hass Rental Index, Annual Changes (%).

Figure 3: Hass Rental Index, Annual Change (%)
Source: Hass Consult (2017)
As shown in the figure 3, rents across Kenyan properties are on the decline. After an unprecedented six years of uninterrupted growth, Kenyan residential rents are on the decline. Nationwide, residential rents have dropped 1.8% year on year in Q3 2017 to reach an average of US$1,370 per month. Rents have previously been on the rise to about 70% since 2007. Data shows that amongst Nairobi suburbs, Langata has recorded the highest rent decline of about -10% followed by Westlands at -6.8%, Spring Valley at -5.5%, Muthaiga -4.4%, Kitisuru at -3.1% and Kileleshwa at -3.1%. these trends have completed other Nairobi suburbs that have low rents declines. Some of these suburbs include Donholm at -1.4%, Loresho at -1.3%, Karen at -1.2% and Lavington at 0.1%. However, in other areas of Kenya, rents are on the rise. For instance, in Eastleigh there has been an 8.9% increase during the year to Q3 2017. This was followed by Ridgeways at 2% and Parklands at 1.7%.
The figure 4 below shows the real GDP and inflation in the Kenyan property market.

Figure 4: Real GDP Growth and inflation (%)
Source: International Monetary Fund (2017)
Whilst the Kenyan GSP has grown rapidly, its population has also increased significantly. The Kenyan population grows at an average rate of 2.8% annually over the past 5 years. Resultantly, GDP per capita growth are low for the period of 2012-2017. Kenyan real GDP per capita grew by 2.5% on average and on an annual basis.
4.3 Modelling of inflation rates of real estate prices in Kenya
4.3.1 Twelve month forecast of inflation rates of real estate prices in KenyaThe figure 5 below shows the inflation rates forecast over the past 12 months in Kenya.

Figure 5: 12-month inflation rates of the Kenyan economy
Source: Trading Economics (2018)
As shown in the figure, the 12-month inflation rates of the Kenyan economy show a declining trend from June 2017 towards May 2018. At the height of the inflation rate is May 2017 whereby it was forecast at 9.21% whilst the lowest forecast was in April 2018 whereby it stood at 3.73%. Looking forward, the Central Bank of Kenya (2018) estimates that inflation rate will stand at 4.50% over the next 12 months. Over the long term, the Central Bank forecasts that inflation rate will trend at around 10.50% by the year 2020.
4.3.2 Investigation of the relationship between property prices and inflation ratesUsing the ARIMA model to investigate the relationship between property prices and inflation rates, the table 2 shown below was utilised.
Table 2: Twelve-month average of interest rates and property prices in Nairobi

Source: Summarised by this research
ARIMA Model analysis
Using SPSS statistical analysis, forecasting and sequencing charts, the following findings were collected and are analysed. The figure 6 below shows the details of the sequence plotting.

Figure 6: Sequence plotting for inflation rates and housing prices
Source: Summarised by this research
The figure 7 below shows the graph between inflation rates and housing prices on a monthly basis using zero non-seasonal differencing.

Figure 7: Graph showing the relationship between inflation rates and housing prices in Nairobi using non-seasonal differencing
Source: Summarised by this research
As shown using zero differencing, there is a linear trend between the housing prices in the months of June 2017-May 2018 and prevailing interest rates. There appears to be a trend whereby the housing prices in Nairobi were on the decline as the rate of interest was also on decline over the 12-month period under analysis. The assumption made in the use of the ARIMA model in this case was a time series relationship whereby variables of interest and price were mobile.
The figure 8 below shows the details of the sequence plotting that uses a non-seasonal differencing variable of 1.

Figure 8: Sequence plotting showing relationship between inflation rate and Nairobi housing prices
Source: Summarised by this research
The figure 9 below shows the graph between inflation rates and housing prices on a monthly basis using non-seasonal differencing value of 1.

Figure 9: First order differencing on the relationship between inflation rates and Nairobi housing prices over a 12-month period.
Source: Summarised by this research
As shown using a non-seasonal differencing value of 1, the relationship between Nairobi property prices and inflation rates changes. Using the first order differencing, creates a linearity whereby the trend manifests a constant mean of inflation rates across the 12-month time series period under review. However, it can also be observed that between the months of September 2017-December 2017 (months 5-8), variations in housing prices and inflation rates had a positive relationship. These appear to contrast those of the February 2018-April 2018 (month 9-11) that manifest a stationary interest rate and housing prices.
The figure 10 below shows the details of the autocorrelations that uses a non-seasonal differencing variable of 1.

Figure 10: ACF analysis using first order differencing for inflation rates and Nairobi housing prices
Source: Summarised by this research.
The table below shows the autocorrelations between the time lags 1-9 that represent property prices and the prevailing inflation rates in the Kenyan economy.
Table 3: Autocorrelations between inflation rates and series lags.

Source: Summarised by this essay.
The figure 11 below shows the autocorrelation graph between interest rate and lag numbers.

Figure 11: ACF graph showing relationship between interest rate and lag numbers
Source: Summarised by this essay.
As shown in the autocorrelation graph, there is a decaying autocorrelation across time t-1 to t-4 lags on the negative side. The most significant autocorrelation is manifested in the time t-6 lag. The findings demonstrate that the most significant correlation between inflation rate and housing prices in Nairobi over the 12-month period is in the month of November.

The table below shows the partial autocorrelations between the time lags 1-9 that represent property prices and the prevailing inflation rates in the Kenyan economy.
Table 4: Partial autocorrelations between inflation rates and series lags.

Source: Summarised by this research
The figure 12 below shows the autocorrelation graph between interest rate and lag numbers.

Figure 12: Partial ACF using first order differencing for inflation rates and lag numbers.
Source: Summarised by this research
As shown, decaying partial-auto correlations is manifested at lags 1, 2 and 3 on the negative PCF. Moreover, as shown, there is a substantial PCF at lags 6, 7 and 8. The findings demonstrate that the most significant correlation between inflation rate and housing prices in Nairobi over the 12-month period is in the month of November.

4.3.3 Discussion of findings
Based on the analysis of the findings above, a number of conclusions can be identified. Firstly, there appears to be a correlation between the housing prices in Nairobi and the economic variables under examination based on the evidence of the ARIMA first order sequencing and the zero sequencing analysis. More specifically, there appears to be a strong and positive correlation between the house prices and the lending rates, loans to the real estate sector and GDP during the month of November (as shown using the autocorrelation and partial autocorrelation analysis). The implication of these findings is that an increase in rate of inflation causes a ripple effect of a rise in the property prices in Kenya. This view contravenes the argument by Andres and Arce (2012) in that a positive relationship between housing prices and lending rates is abnormal. This is because increasing interest rates in the economy has the effect of increasing prices on the demand side thereby lowering demand for housing units. However, as explained using the ARIMA model, in the case of the Kenyan property prices, lending rates have the most significant influence on the supply side as opposed to the demand side in the housing price equation. As such, an increase in the interest charged on lending will have the reciprocal effect of increasing the housing prices in the Kenyan real estate sector.
Secondly, the analysis of the findings shows that a scrutiny of the integration of the variables of property prices and inflation rates there is a differencing parameter that substantially exceeds the value of 1. In the case of the housing prices and the lending rates variables a differential parameter that exceeds 1 suggests that there could be a housing bubble in the making. This view is further reinforced by the observation that the economic variables of inflation, GDP growth and lending rates have a long term relationship towards housing prices. These findings appear to be consistent with the view of Hirata et al. (2012) in that long run relationships between the prices of housing and the GDP growth indicate that the market is stable and there is no sign of a bubble forming. However, on the other hand, the fluctuations of the long run relationships between housing prices and inflation rates indicate the presence of unstable relationships that could result in a housing bubble. In light of this, whilst the co-integration of the findings appears inconclusive in making the argument for the existence of a housing bubble in Kenya, they conclusively indicate that a strong and positive correlation between the property prices and the inflation rates exists.
4.4 Summary
As shown, this chapter has established that there is a positive relationship between Nairobi property prices and inflation rates over the 12-month period of June 2017-May 2018. The findings suggest that as interest rate increases, the property prices in Nairobi also increase. This research has attributed this trend to the unique nature of the Kenyan housing industry whereby a growth in inflation rates tends to have a more profound effect on the supply side in relation to the demand side. As a result, there is a positive correlation between the rise in inflation rates and property prices in Nairobi.
Chapter 5: Conclusions5.1 IntroductionThe goal of this chapter is to reiterate the main findings of the study on the application of the ARIMA model in the prediction of inflation of real estate prices and determine their implication for the research. Other aspects that are addressed in this chapter include the formulation of recommendations based on the conclusions drawn, identification of the limitations of the current research and the formulation n of the direction that future research should take.
5.2 Conclusions
The primary findings based on the ARIMA analysis conducted in the previous chapter indicate that there is a strong correlation between the housing prices in Nairobi and the level of inflation rates experienced in the economy. These findings were consistent with the theory of Andres and Arce (2012) in that inflation rates in Kenya have a significant influence on the supply side in the housing and property market. As a result, regardless of the growth, stagnation or decline of demand, the housing and property prices keep on increasing. This finding indicates that inflation rates across the Kenyan housing and property market had the effect of increasing housing prices because the contractors incurred higher costs in accessing credit and higher costs in the construction of property.
Moreover, the primary findings established from the ARIMA first order sequencing and the zero sequencing analysis, the findings suggested that a strong and positive correlation between the house prices and the lending rates, loans to the real estate sector and GDP during the month of November (as shown using the autocorrelation and partial autocorrelation analysis) existed. These findings resulted in the conclusion that an increase in the rate of inflation had the effect of increasing property prices in Kenya especially in the month of November whereby the highest correlation was manifested. This finding reinforced the view that the unique nature of the Kenyan property maker made it possible that an increase in inflation rates resulted in souring housing and property prices.
Furthermore, the primary findings established that in the relationship between inflation rates and property prices in Kenya, there was a differencing parameter that substantially exceeded the value of 1. In the case of the housing prices and the lending rates variables a differential parameter that exceeds 1 suggests that there could be a housing bubble in the making. This view was further reinforced theoretical viewpoint by Hirata et al. (2012) in that long run relationships between the prices of housing and the GDP growth indicate that the market is stable and there is no sign of a bubble forming. However, on the other hand, the fluctuations of the long run relationships between housing prices and inflation rates indicate the presence of unstable relationships that could result in a housing bubble. Based on these findings, the research concluded that whilst co-integration of the findings appears inconclusive in making the argument for the existence of a housing bubble in Kenya, there always a risk that it could happen unexpectedly and wiping off the fortune of players within the industry.

5.3 Recommendations
The findings resulted in the conclusion that there is no housing bubble in Nairobi presently. However, the nature of this finding indicates that it is highly optimistic since the autocorrelation and partial autocorrelation tests show that it was not conclusive that a housing bubble did not exist in Nairobi. With this set of findings, it is impossible to fully conclude that a housing bubble does or does not exist. As such, based on the possibility that there could be one, investors in the Kenyan real estate should therefore exercise caution when making investments in the sector as the bubble could develop anytime. The reason for this is that the development and subsequent bursting of the housing bubble could be disastrous for investors and the Kenyan economy as well because of the key role that the housing sector plays in the growth of the Kenyan economy. Some of the approaches that can be utilised include the hedging of investments in order to cushion them against adverse changes in macroeconomic conditions or the diversification of investments into other sectors so as to cushion against losses likely to be experienced in the event the Kenyan housing bubble bursts.
Moreover, the policy makers in the Kenyan economy should ensure that the right structures and frameworks exist in order to prevent the occurrence of a housing bubble. Some of the approaches that can be utilised include the re-introduction of a Capital Gains Tax within the budget. The introduction of this budget will be instrumental in taming runaway prices within the housing sector. The rationale for the suggestion that the Kenyan government should take an active role in the taming of fast rising property prices is because of the role that the property market plays within the Kenyan economy. Based on its significant contribution to GDP growth, the Kenyan government should set up policies that not only guarantee but contribute towards the stability and sustainability of the property market in Kenya. As suggested, an introduction of the Capital Gains Tax within the budget will have an effect of taming the inflation rate that has a significant influence on the supply side of the Kenyan property market. The implication of this adjustment is a fairly stable and sustainable housing and property market that contributes to job growth, GDP and makes it easier for the low and middle income earners to access and acquire houses and property.
Furthermore, in the analysis of the trends of real estate prices in Kenya, the findings established that whilst the demand for property amongst the high income earners was on the decline, there remained significant demand amongst the middle and low income earners in the Kenyan economy. On account of this finding, the players in the Kenyan housing and property industry should change tact and consider how to meet housing and property demand amongst the low and middle income earners whilst realising a profit. One of the approaches that can be used include the exploration of alternative housing materials that are not only affordable but durable. This would help cut down the costs incurred in the construction projects whilst also charging the low and middle income earners lower prices for housing products. The implication of this move is that the Kenyan housing and property market players would reduce their overreliance on the wealthy and superrich and also tap into the low and middle income consumer segments whereby the demand for housing and property remains largely untapped.
5.4 Limitations of this research
The findings of this researcher should b taken with caution because of the limited sample size that is utilised. The sample of size of a cross-sectional of 12-month period of interest and average property prices in Nairobi is limited and may not be representative of the behaviour of the inflation rates and its effects on property prices during other time periods.
Another limitations of the current study are the similar results achieved by the use of the autocorrelations and partial-autocorrelations from a qualitative standpoint. Though the findings are different from a quantitative perspective, the results established that a similar conclusion held whereby inflation rates were identified to contribute to higher property prices in Nairobi.
Furthermore, the findings of this research were limited to the extent that they only focused on inflation rate as a determinant of property prices in Kenya. This focus failed to analyse other crucial factors that are likely to have an effect on property prices in Nairobi. For instance, factors such as income and population growth are likely to affect property prices in Nairobi and they were not discussed within the study.
5.5 Directions for future research
In future studies, aspects such as the partially autocorrelation will be discussed in detail in order to create a distinct qualitative disparity of the findings to that of the autocorrelation finding. The importance of this distinction is to create variables that contain nonlinear structures of either a univariate level but can also establish the relationships that exit amongst variables under study.
Moreover, future studies will extensively analyse the impact of inflation rates, income levels and population growth patterns in Kenya on the property prices in Kenya. The aim of this approach will be to engage a more critical study that seeks to establish the extent of the role played by other dimensions such as income and population growth in the determination of housing prices in Kenya. As such, this can help improve the conclusion of this study that inflation rates are the sole determinants of the property prices in Kenya.
Furthermore, future studies should increase the sample size that is utilised in the study. Whilst the current research utilises 12 months, future studies should seek to utilise 36-60 months in order to have a more representative sample size in the study of the impact of inflation rates on property prices. A 36-60month period is more representative when analysing the role that inflation rates can have on property prices because there is a wider range of data to back the main findings of the study.
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Appendix 1: Gantt chart

Appendix 2: BudgetITEM COST IN KSH
Stationery 7000
Airtime for internet bundles 2000
Printing and Binding 3600
Other expenses 5000
TOTAL 17600