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 Russian Ruble to Sterling will be explored and investigated. The ruble or rouble is the currency of Russia and the two partially recognized republics of Abkhazia and South Ossetia. The ruble was the currency of the Russian Empire and of the Soviet Union before its dissolution.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has 2 columns and 137 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. 
##   42.86   47.64   50.26   57.79   58.12  109.40

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 109.4153 , 25% of data is greater than 58.12425, 50% of data is greater than 50.2643 (median value) and the smallest value is 42.8557. The standard deviation is 17.9561765 and the amplitude is 66.5596. It is of interest to analise how the variable behaves along time, as it can be seen in the plot below.

The series presents quite a stable behavior till the beginning of 2013. From 2013 to middle 2014 the exchange rate starts to increase and in June 2014 the growth becomes monumental. A decline in confidence in the Russian economy caused investors to sell off their Russian assets, which led to a decline in the value of the Russian ruble and sparked fears of a Russian financial crisis. The crisis has affected the Russian economy, both consumers and companies, and regional financial markets, as well as Putin’s ambitions regarding the Eurasian Economic Union. The Russian stock market in particular has experienced large declines, with a 30% drop in the RTS Index from the beginning of December through 16 December 2014.

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 = 0.078202, 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 0.7160228. So, in a 5% level of significance, the series is non-stationary

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

Till 2013 the exchange rate recorded very small variation, as expected in this kind of measure. 2014 and mainly 2015 were very bad years for the Russian Ruble, which was very unappreciated

As explained in section 1, the aim of the report is to verify the relationship between the Russian Ruble 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 = -0.29433, df = 112, p-value = 0.7691
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.2106384  0.1569165
## sample estimates:
##         cor 
## -0.02780059

As the exchange rate has a big and unusual variation it was expected that all the data wouldn’t be correlated to the response variable

In the output there are two main informations: the correlation coefficient (-0.0278006) and the p-value (0.7690524). 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 not significant. 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 = 4.6885, df = 58, p-value = 1.714e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3118342 0.6867523
## sample estimates:
##       cor 
## 0.5242477

The correlation coefficient is significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 1.714247210^{-5}), so there is significant linear relationship between the two variables.

As suspected, when only part of the data is included the CC becomes positive and significant.

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.3[, names(base.3) == variavel[2]] and house.sold.3$numSoldHouses
## t = 4.3387, df = 34, p-value = 0.0001214
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3339094 0.7737453
## sample estimates:
##       cor 
## 0.5969581

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