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 Standard Variable Rate will be explored and investigated. A variable interest rate is an interest rate on a loan or security that fluctuates over time, because it is based on an underlying benchmark interest rate or index that changes periodically. The obvious advantage of a variable interest rate is that if the underlying interest rate or index declines, the borrower’s interest payments also fall. Conversely, if the underlying index rises, interest payments increase.

It might be interesting to compare the results of this report with the Fixed 2Y Interest Rate and the Fixed 5Y Interest Rate.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has 2 columns and 115 observations. 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.

3. Analysis

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

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   3.820   4.100   4.380   4.871   4.560   7.740

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 7.74 , 25% of data is greater than 4.56, 50% of data is greater than 4.38 (median value) and the smallest value is 3.82. The standard deviation is 1.2331135 and the amplitude is 3.92.

It is of interest to analise how the variable behaves along time, as it can be seen in the plot below. There is some techniques in the analysis of time series which support the identification of dynamic changes in the behavior of the series. In the plot below, the orange line(s) identifies the point(s) classified as “changing point(s)”.

The observation from December 2008 is defined as a changing point. It is a reflection of the 2008 crisis which started in the U.S house market and affected the entire world. After the ‘big decline’ the series presents a positive tendency with small variability

In the analysis of time series it is comum 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 coeficiente among time (lags). The p-value is smaller than 0,05, so we assume that autocorrelation is significant.

## 
##  Durbin-Watson test
## 
## data:  base_index ~ base_date
## DW = 0.021861, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

Something algo very important in the analysis of time series is to check if the series has a stationary behavior or not. The test used in this report is the Dickey-Fuller test. Stationarity is one way of modeling the dependence structure. It turns out that a lot of nice results which holds for independent random variables hold for stationary random variables. And of course it turns out that a lot of data can be considered stationary, so the concept of stationarity is very important in modeling non-independent data.

The Dickey-Fuller test for this case presents p-value of 0.6099988. So, in a 5% level of significance, the series is non-stationary

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

It is interesting to observe distribution of the box in 2008 and how massive is the decrease in the rate. The following years present very small variations, as expected.

As explained in section 1, the aim of the report is to verify the relationship between the Standard Variable Rate 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:  base_index and house.sold$numSoldHouses
## t = 7.025, df = 112, p-value = 1.763e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4109300 0.6689238
## sample estimates:
##       cor 
## 0.5530476

It can be seen that the biggest values are very influential and they are from 2007 and 2008.

In the output there are two main informations: the correlation coefficient (0.5530476) and the p-value (1.763259410^{-10}). 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 positive and moderate. 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 coeficiente 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:  base.5[, names(base.5) == variavel[2]] and house.sold.5$numSoldHouses
## t = 2.0294, df = 58, p-value = 0.04702
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.003809773 0.480025133
## sample estimates:
##       cor 
## 0.2574854

The correlation coefficient is significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 0.0470172), so there is significant linear relationship between the two variables.

However, the p-value was very close to the non-rejection area (p-value >0.05) and the strength of the CC is very weak. So, the linear relationship can be neglected in this case

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.3[, names(base.3) == variavel[2]] and house.sold.3$numSoldHouses
## t = 3.9146, df = 34, p-value = 0.0004134
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2801522 0.7488006
## sample estimates:
##       cor 
## 0.5573861

Using only the last 3 years, the correlation coefficient is significant and the linear relationship between the variables is positive and moderate.