1. Introduction

In this project, possible associated variables with the number of sold houses between the \(1^{st}\) of January 2007 and the \(30^{th}\) of June 2016 on the Stirling Ackroyd real estate company were analysed.

In this report the index FTSE 250 will be explored and investigated. The FTSE 250 Index is a capitalisation-weighted index consisting of the 101st to the 350th largest companies listed on the London Stock Exchange. Promotions and demotions to and from this index take place quarterly in March, June, September and December. This Index is calculated in real-time and published every minute.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has six columns and 119 observations (from October 2006 to August 2016). As described above, the original data were manipulated and reduced to 114 observations (to match the time of interest) and a column containing only the year of the observation was added. In this report it will only be considered the closing value of the index.

3. Analysis

The dataset were summarized and some results can be seen below:

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    6049   10050   11550   12270   15500   18150

The present dataset has standard deviation of 3254.596 and amplitude of 12105.28 (values considerably higher than the FSTE 100). It is of interest to analise how the variable behaves along time, as it can be seen in the plot below.

Some remarkable economic moments can be identified in the time series. For example, the global crisis which started in the US housing market between 2008 and 2009 reflected the decline of the index in the plot. Also, between 2012 and 2013 the series presented a reasonable increase on the index.

In the analysis of time series it is common that the observations are correlated among time,this characteristic is called autocorrelation. The Durbin-Watson test is a popular option to check the hypothesis of autocorrelation. In a confidence level of 95% the output of the test is a value (p-value) between 0 and 1: if (p-value \(>\) 0,05) the null hypothesis of non-autocorrelation is not rejected, otherwise (p-valor \(\leq\) 0.05) we assume that the observations of the series are correlated among time. Also, the autocorrelation function (ACF) tests the significance of the coefficient among time (lags). The p-value is smaller than 0,05, so we assume that autocorrelation is significant.

## 
##  Durbin-Watson test
## 
## data:  ftse.250$Close ~ ftse.250$Date
## DW = 0.085614, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

To observe and compare the variation among the years of the FTSE 250 the plot below shows a BoxPlot for the index in each year separately.

As seen previously, 2008 and 2009 are marked as a decline in the series and, in this new plot, It can be seen a bigger variation in these years. In this kind of index, big variation among short periods of times are not expected. The index tends to establish among time, as the boxes became shorter. It seems that the values and variations in this index are smaller than the FTSE 100, as it can be seen here.

As explained in section 1, the aim of the report is to verify the relationship between the FTSE 250 index and the Number of Sold Houses on the Stirling Ackroyd real estate company. The first exploratory analysis is to check the Pearson Correlation between the two variables, as shown below.

## 
##  Pearson's product-moment correlation
## 
## data:  ftse.250$Close and house.sold$numSoldHouses
## t = -1.9965, df = 112, p-value = 0.0483
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.357124234 -0.001522756
## sample estimates:
##        cor 
## -0.1853856

In the output there are two main informations: the correlation coefficient (-0.1853856) and the p-value (0.0483). The correlation coefficient (CC) can be interpreted as a measure of the degree of linear relationship between two variables and the p-value is a test to check the CC is significant: if (p-value \(>\) 0,05) the null hypothesis of correlation equals to 0 is not rejected, otherwise (p-valor \(\leq\) 0.05) the correlation is significant. In this case, the linear relationship between the two variables is negative and weak. As a complementary analysis, the plot shown above is a scatter plot between the variables and a regression line.

A point that has to be considered is that the data from the two variables are from quite a long time (9 years) and economic changes can be notice in short periods of time. Taking this into account the correlation coefficient will be analized considering different periods of time: i) the last 5 years and ii) the last 3 years. The output from the correlation test and coefficient can be seen below and right beneath that a scatter plot is displayed, aiming to observe the existence of linear relationship.

i) Last 5 years

## 
##  Pearson's product-moment correlation
## 
## data:  ftse.250.5$Close and house.sold.5$numSoldHouses
## t = 1.8768, df = 58, p-value = 0.06558
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.01559774  0.46495004
## sample estimates:
##       cor 
## 0.2392747

The correlation coefficient is not significant, in a 95% confidence level, when only the 5 last years are considered (p-value = 0.06558), so there is no significant linear relationship between the two variables.

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  ftse.250.3$Close and house.sold.3$numSoldHouses
## t = 2.789, df = 34, p-value = 0.0086
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.1199569 0.6656600
## sample estimates:
##       cor 
## 0.4314876

The results from the last 3 years are more interesting, a signifitcant and weak to moderated correlation coefficient can be observed.