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 Japanese Yen to Sterling will be explored and investigated. The yen is the official currency of Japan. It is the third most traded currency in the foreign exchange market after the United States dollar and the euro. It is also widely used as a reserve currency after the U.S. dollar, the euro, and the pound sterling.

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. 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. 
##   119.6   132.9   155.3   163.6   185.4   247.8

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 247.8105 , 25% of data is greater than 185.36275, 50% of data is greater than 155.34945 (median value) and the smallest value is 119.6372. The standard deviation is 34.8624359 and the amplitude is 128.1733. It is of interest to analise how the variable behaves along time, as it can be seen in the plot below.

The series represents a declining tendency till middle 2012, when it starts to increase. Between 2008 and 2009 it’s possible to observe the global crisis effects in the exchange of the currencies. It seems that the Japanese economy took longer to establish itself after the crisis, if compared to the British one.

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 = 0.043583, 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 0.4752785. So, in a 5% level of significance, the series is non-stationary

To observe and compare the variation among the years of the Japanese Yen to Sterling the plot below shows a BoxPlot for the index in each year separately.

As seen previously, 2008 and 2009 are marked as a big decline in the series and, in this new plot, It can be seen a bigger variation in these years. In texchange rates, big variation among short periods of times are not expected.

As explained in section 1, the aim of the report is to verify the relationship between the Japanese Yen 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 = 6.9087, df = 112, p-value = 3.138e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4032698 0.6638191
## sample estimates:
##       cor 
## 0.5466417

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

In the output there are two main informations: the correlation coefficient (0.5466417) and the p-value (3.138226210^{-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 = 0.98969, df = 58, p-value = 0.3264
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.1292868  0.3706641
## sample estimates:
##       cor 
## 0.1288686

The correlation coefficient is not significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 0.3264385), so there is no significant linear relationship between the two variables.

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.3[, names(base.3) == variavel[2]] and house.sold.3$numSoldHouses
## t = -0.052812, df = 34, p-value = 0.9582
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.3365911  0.3204323
## sample estimates:
##         cor 
## -0.00905688

Using only the last 3 years, the correlation coefficient is not significant and the linear relationship between the variables is negative and weak.