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 Indian Rupee to Sterling will be explored and investigated. The Rupee, or more specifically the Indian rupee is the official currency of the Republic of India. The issuance of the currency is controlled by the Reserve Bank of India.

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. 
##   66.98   76.42   82.47   84.95   96.04  103.50

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 103.489 , 25% of data is greater than 96.042375, 50% of data is greater than 82.46915 (median value) and the smallest value is 66.9813. The standard deviation is 10.7390582 and the amplitude is 36.5077. It is of interest to analise how the variable behaves along time, as it can be seen in the plot below.

After 2010 the series presents a positive tendency. The Indian Rupee has fallen in value against a basket of currencies since independence in 1947. In recent years, the Indian Rupee has continued to depreciate in value. The long term decline in the value of the Rupee reflects India’s relative decline in competitiveness. In particular, India has a higher inflation rate than its international competitors. In November 2013, Indian inflation reached 11.24%. Therefore, there is relatively less demand for the rising price of Indian goods; this reduction in demand causes a fall in the value of the Rupee.

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 = 0.11225, 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 0.5710336. So, in a 5% level of significance, the series is non-stationary

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

2013 is marked as a big variation in the exchange rate between the Rupee and the Sterling and it is also marked by a huge inclination on the time series presented before. Due to stagnant reforms, and declining foreign investment, rupee started depreciating in the early 2013. Measures were announced by the government before this drop to prevent it from dropping further, but none managed to slow down the depreciation. After continued depreciation, and high inflation, the then Prime Minister of India, Manmohan Singh, made a statement in the Parliament of India on the issue. He was of the view that, the present depreciation is partly led by global factors as well as domestic factors. He also asked the political parties to help his Government, tide over the crisis that the country was facing with rupee losing its value.

As explained in section 1, the aim of the report is to verify the relationship between the Indian Rupee 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 = -2.5985, df = 112, p-value = 0.01062
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.40462312 -0.05703976
## sample estimates:
##        cor 
## -0.2384531

It can be seen that a linear relationship is not well fitted to the dataset. Also, although the correlation coefficient is significant, it has a small absolute value and can be considered very weak

In the output there are two main informations: the correlation coefficient (-0.2384531) and the p-value (0.0106221). 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 weak. 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 = 0.9609, df = 58, p-value = 0.3406
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.1329718  0.3674252
## sample estimates:
##       cor 
## 0.1251796

The correlation coefficient is not significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 0.3405927), 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 = -1.2824, df = 34, p-value = 0.2084
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.5075206  0.1223710
## sample estimates:
##        cor 
## -0.2148004

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