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 Australian Dollar to Sterling will be explored and investigated. The Australian dollar is the currency of the Commonwealth of Australia, including Christmas Island, Cocos (Keeling) Islands, and Norfolk Island, as well as the independent Pacific Island states of Kiribati, Nauru and Tuvalu. As of 2011, the Australian dollar is the fifth most traded currency in the world, accounting for 7.6% of the world’s daily share. It trades in the world foreign exchange markets behind the US dollar, the euro, the yen and the pound sterling. The Australian dollar is popular with currency traders, because of the comparatively high interest rates in Australia, the relative freedom of the foreign exchange market from government intervention, the general stability of Australia’s economy and political system, and the prevailing view that the Australian dollar offers diversification benefits in a portfolio containing the major world currencies, especially because of its greater exposure to Asian economies and the commodities cycle.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has 2 columns and 500 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. 
##   1.456   1.604   1.821   1.871   2.074   2.527

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 2.527 , 25% of data is greater than 2.0743, 50% of data is greater than 1.8212 (median value) and the smallest value is 1.4561. The standard deviation is 0.2898008 and the amplitude is 1.0709. 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 till middle 2011, after that, it presents a positive tendency till the end of 2015. 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. The behavior is more similar to the Canadian Dollar.

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 = 0.070797, 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 0.7511832. So, in a 5% level of significance, the series is non-stationary.

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

It can be seen that in periods of decline or increase, the variation is bigger than is periods of stability. The linear relationship seems to be positive and moderate.

As explained in section 1, the aim of the report is to verify the relationship between the Australian 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 = 7.2417, df = 112, p-value = 5.972e-11
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4249429 0.6781992
## sample estimates:
##       cor 
## 0.5647227

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 (0.5647227) and the p-value (5.972038210^{-11}). 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 = 3.2905, df = 58, p-value = 0.001705
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.1586777 0.5910232
## sample estimates:
##       cor 
## 0.3966217

The correlation coefficient is significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 0.0017048), so there is 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 = 3.4157, df = 34, p-value = 0.001664
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2121408 0.7152171
## sample estimates:
##       cor 
## 0.5054489

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 Australian Dollar and the Pound Sterling is a reasonable indicator of the Number of Sold Houses.