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 Sterling to USS will be explored and investigated. The U.S. dollar is fiat money. It is the currency most used in international transactions and is the world’s primary reserve currency. Several countries use it as their official currency, and in many others it is the de facto currency. Besides the United States, it is also used as the sole currency in two British Overseas Territories in the Caribbean: the British Virgin Islands and Turks and Caicos Islands.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has 2 columns and 116 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. 
##   1.343   1.536   1.601   1.643   1.660   2.077

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 2.0772746 , 25% of data is greater than 1.660854, 50% of data is greater than 1.6007687 (median value) and the smallest value is 1.3428226. The standard deviation is 0.1761663 and the amplitude is 0.734452. It is of interest to analise how the variable behaves along time, as it can be seen in the plot below.

The 2008/09 global crisis can be clearly observed in the series: a big decline happens in a short period of time. After that, the series do not present any visible kind of tendency or sazonality.

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:  base_index ~ base_date
## DW = 0.11287, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

Something also 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 holds 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 0.2852206. So, in a 5% level of significance, the series is non-stationary.

To observe and compare the variation among the years of the Sterling to USS the plot below shows a Box Plot for the index in each year separately.

Normally, in exchange rates, big variations among short periods of time are not expected and that can be easily seen in the plot above: the boxes from 2008 and 2009 presented the biggest lengths and, consequently, the biggest variations. In the other hand, the other boxes tend to present the same behavior of variability.

As explained in section 1, the aim of the report is to verify the relationship between the Sterling to USS 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 = 6.6263, df = 112, p-value = 1.253e-09
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3842749 0.6510555
## sample estimates:
##       cor 
## 0.5306842

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 (0.5306842) and the p-value (1.252954710^{-9}). 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 if 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 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:  base.5[, names(base.5) == variavel[2]] and house.sold.5$numSoldHouses
## t = -3.9775, df = 58, p-value = 0.0001957
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.6414577 -0.2368550
## sample estimates:
##        cor 
## -0.4629371

The correlation coefficient is significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 1.957343610^{-4}), so there is significant linear relationship between the two variables.

When the most influential points are excluded and only the last 5 years are analysed the correlation becomes negative and it’s moderate. It means that, when the exchange rate grows the Number of Sold Houses tends to decline, as it can be seen in the plot above.

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.3[, names(base.3) == variavel[2]] and house.sold.3$numSoldHouses
## t = -3.9526, df = 34, p-value = 0.0003709
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.7511662 -0.2851271
## sample estimates:
##        cor 
## -0.5611019

Using only the last 3 years, the correlation coefficient is significant and the linear relationship between the variables is negative and moderate. It seems that the exchange rate between the American Dollar and the Pound Sterling is a reasonable indicator of the Number of Sold Houses.