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 Canadian Dollar to Sterling will be explored and investigated. Canada’s dollar is the 5th most held reserve currency in the world, accounting for approximately 2% of all global reserves, behind only the U.S. dollar, the euro, the yen and the pound sterling. The Canadian dollar is popular with central banks because of Canada’s relative economic soundness, the Canadian government’s strong sovereign position, and the stability of the country’s legal and political systems.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has 2 columns and 176 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. 
##   1.520   1.599   1.780   1.788   1.932   2.310

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 2.3105 , 25% of data is greater than 1.9324, 50% of data is greater than 1.78005 (median value) and the smallest value is 1.5204. The standard deviation is 0.1977193 and the amplitude is 0.7901. It is of interest to analize how the variable behaves along time, as it can be seen in the plot below.

The series has a negative tendency between January 2007 and May 2010. Furthermore, the exchange of the currencies seems that was not so influenced by the 2008/09 crisis, at least nos as much as the Japanese Yen and the Chinese Yuan, That’s because when European and North American banks teetered on the brink of meltdown in 2008, requiring bailouts and extraordinary central bank intervention, Canadian banks escaped relatively unscathed. So that’s why the series has a negative tendency in this period: the pound sterling was ‘falling’ while Canadian Dollar was ‘stable’. After 2013, the pound sterling starts to grow again.

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 = 0.068224, 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 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 0.6156129. So, in a 5% level of significance, the series is non-stationary.

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

In gereral, the variation among the exchange rate is not big. For example, the Y axis has amplitude of 0.8. Both currencies hold a strong value on the market and the Canadian Dollar tends to be an stable currency, which are possible reasons for the small variation.

As explained in section 1, the aim of the report is to verify the relationship between the Canadian Dollar to Sterling 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 = 6.9248, df = 112, p-value = 2.899e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4043341 0.6645298
## sample estimates:
##       cor 
## 0.5475327

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

In the output there are two main informations: the correlation coefficient (0.5475327) and the p-value (2.898507810^{-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 if 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 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 = 3.5856, df = 58, p-value = 0.0006905
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.1929074 0.6135298
## sample estimates:
##       cor 
## 0.4259665

The correlation coefficient is significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 6.905077210^{-4}), so there is significant linear relationship between the two variables.

The linear relationship is positive and moderate.

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.3[, names(base.3) == variavel[2]] and house.sold.3$numSoldHouses
## t = 3.3784, df = 34, p-value = 0.001841
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2068595 0.7125082
## sample estimates:
##       cor 
## 0.5013248

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