Introduction

RPubs link:http://rpubs.com/bgav/501300
To analyse the performance of the Istanbul Stock Exchange (ISE) index relative to the S&P500 index.
To observe the association between the positive/negative returns of the ISE index to the positive/negative returns of the S&P500 index.
To observe correlations between the Istanbul Stock Exchange’s index and global stock indexes.

Why is the Istanbul Stock Exchange the subject of this analysis?

Turkey’s economy is one of the fastest emerging economies[1] in the world. It has the 17th largest nominal GDP[2] and is a newly industrialised country. Given this, Turkey’s financial markets is subject to more growth than the financial markets of an already developed country.

Why did we use global stock indexes?

The best indicator of the overall growth, performance and strength of a countries financial markets, is its stock index[3]. The evaluation and comparison of the returns of each stock index over the same period in time could give us an insight into the performance of the ISE relative to global stock exchanges. This in turn would also give us an insight into Turkey’s financial performance relative to the finacial performance of global economies.

Problem Statement

Hypothesis Test 1

A two sample (paired) t-test was used to test the difference between the returns of the Istanbul Stock Exchange and the S&P500 stock index, and, to compare the financial performance between the Istanbul stock market and the American stock market. We expected to see a positive difference between the returns of the two indexes (‘ISE_USD’ - ‘SP’ > 0) because Turkey is an emerging economy (high growth prospects) and the U.S.A. is a stable economy[3].

Hypothesis Test 2

A Chi-squared test of association was used to check if the relationship between two categorical variables was significant. The two categorical variables referred to were the positive/negative returns of the ISE index and the S&P500 index. We expected to see a significant association between ‘ISE_CAT’ and ‘SP_CAT’.

Correlation between the stock indexes

We then observed Pearson’s correlation matrix for the indices of our dataset. We were able to examine the ‘Spillover effect’, where the performance of one countries economy affects another countries economy without any direct relationship (thus, making most stock markets correlated)[4].

Data

The data was accessed on: https://archive.ics.uci.edu/ml/datasets/ISTANBUL+STOCK+EXCHANGE. When clicked, the ‘Data Folder’ led us to a downloadable link for an excel file. Once downloaded, the first row of the dataset was gotten rid of manually, using excel. In addition, the headers of the second and third columns were renamed to ‘ISE_TL’ and ‘ISE_USD’, respectively (also done manually on excel). The file was then imported into R-Studio after setting the appropriate working directory.
The dataset had 536 observations and 0 missing values.
The attributes included: ‘date’, ‘ISE_TL’, ‘ISE_USD’, ‘SP’, ‘DAX’, ‘FTSE’, ‘NIKKEI’, ‘BOVESPA’, ‘EU’, ‘EM’. ISE_TL and ISE_USD represented returns of the Istanbul Stock Exchange in Turkish Lira and U.S. dollars, respectively. SP, DAX, FTSE, NIKKEI, BOVESPA, EU, EM represented the returns (in U.S. dollars) of the stock market indexes of U.S.A., Germany, U.K., Japan, Brazil, European Union and emerging markets, respectively. The returns were reported each weekday from the 5th of January, 2009 to the 22nd of February, 2011.
The sampling was carried out by Akbilgic, et. al[5].

Data Cont.

For Hypothesis Test’s 1 and 2, the variables of concern were ISE_USD and SP. Being one of the best performing indexes worldwide, the S&P500 index was used as a benchmark for the performance of the ISE_USD [6].
The class of each attribute of our dataset was examined. Most of the attributes consisted of numeric instances, as expected.
Each index’s returns were recorded in U.S. dollar value (except for ISE_TL which was in terms of Turkish Lira).
On the next slide, it was observed that the ISE index’s mean returns may have been similar if not greater than the S&P500 index’s mean returns. ISE index’s returns had a greater Interquartile Range in comparison to S&P500 index’s mean returns. This was suggestive that the returns for the ISE index were more spread out and could have had greater volatility.

Visualisation of the returns of the ISE index and the S&P500 index

boxplot(index$ISE_USD,index$SP, title = "Boxplot for Stock Index Returns",
        ylab= "Returns", xlab="")
axis(1, at=1:2, labels=c('ISE','SP'))

Descriptive Statistics

We created a new column ‘d’, that represented the difference between the return of the ISE index and the returns of the S&P500 index i.e., (ISE_USD - SP). Followed by an examination of the summary statistics of the difference in returns between the two indexes i.e., d.

index <- index %>% mutate(d= (ISE_USD-SP))
index %>% summarise(Min = min(d,na.rm = TRUE),
                  Q1 = quantile(d,probs = .25,na.rm = TRUE),
                  Median = median(d, na.rm = TRUE),
                  Q3 = quantile(d,probs = .75,na.rm = TRUE),
                  Max = max(d,na.rm = TRUE),
                  Mean = mean(d, na.rm = TRUE),
                  SD = sd(d, na.rm = TRUE),
                  n = n(),
                  Missing = sum(is.na(d))) -> diff_descriptive_stats
knitr::kable(diff_descriptive_stats)

Min	Q1	Median	Q3	Max	Mean	SD	n	Missing
-0.0881073	-0.010171	0.001646	0.0122443	0.0681678	0.0009089	0.0194192	536	0

Hence, it was observed that the mean return of the ISE index was on average higher than the mean return of the S&P500 index, by $0.0009089 (i.e., d>0).

Hypothesis Test 1

Are the returns of the ISE index on average, statistically significantly greater than the returns of the S&P500 index?
We used a paired sample t-test as each return instance was for two different indexes on the same day. Given that our sample had 536 obervations, i.e., (n=536)>30, the normality assumption was approximately held, according to the Central Limit Theorem. We further examined this by constructing a qqPlot on the difference between ISE_USD and SP i.e., d.

index <- index %>% mutate(d= (ISE_USD-SP))
qqPlot(index$d, dist="norm")

[1]  4 95

It was observed that the difference in returns between the ISE index and the S&P500 index was approximately normally distributed.

Hypthesis Test 1 Cont.

Given that the difference in returns was normally distributed, we proceeded to perform the t-test. $H_0: µ_d = 0 $ and $H_A: µ_d > 0$. The t-test was right tailed as we were concerned with whether or not the difference between ISE_USD and SP was positive (thus, implying d>0). The test had $= 0.05 $ significance level. The t-statistic for the paired sample t-test was given by $t =d/(s_∆/√n_∆)$. The results were as follows:

t.test(index$d,mu=0, alternative="two.sided")


    One Sample t-test

data:  index$d
t = 1.0836, df = 535, p-value = 0.279
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.0007387824  0.0025566354
sample estimates:
   mean of x 
0.0009089265

The mean difference between ISE and SP was 0.0009089 with a standard deviation of 0.0194192. As the sample consisted of n>30 observations, the distribution was assumed to follow a normal distribution. The paired sample t-test found the mean difference between the return of the ISE index and the S&P500 index to not be statistically significant.The results of the test (t(df=535)=1.0836, p-value=0.279, confidence interval = 95% [-0.000739, 0.002557]) meant that we failed to reject $H_0: µ_d = 0$.The p-value of 0.279 was greater than the alpha of 5%. The confidence interval succeeded in capturing the mean of our sample i.e., 0.0009089.

Hypothesis Test 2

Was there a statistically significant association between the positive/negative returns of the ISE index and the S&P500 index?
We first created two categorical variables; ‘ISE_CAT’ and ‘SP_CAT’ that consisted of the factors ‘positive’ and ‘negative’.
We then proceeded to perform the test where $H_0$: ‘There is no association in the population between the categorcial variables’ and, $H_A$: ‘There is association between the populations categorical variables’. The test was a Chi-squared test of association. The test had $= 0.05 $ significance level. The test statistic followed a Chi-squared distribution with (n-2 = 1) degrees of freedom.
The test of association (next slide) found, that the association between positive/negative returns of the ISE index and the S&P500 index was statistically significant.The results of the test ($X^2$(df=1) = 22.571, p-value=<0.001 and $X^2$ > $X^2$ critical) implied that we should have rejectd the $H_0$. The p-value of <0.001 was lesser than the alpha of 5% and, $X^2$ > $X^2$ critical of 3.841459.

Hypothesis Test 2 Contd.

chi1 <- chisq.test(table(index$ISE_CAT , index$SP_CAT))
chi1


    Pearson's Chi-squared test with Yates' continuity correction

data:  table(index$ISE_CAT, index$SP_CAT)
X-squared = 22.571, df = 1, p-value = 2.025e-06

qchisq(p=0.05, df=1, lower.tail = FALSE)

[1] 3.841459

Reggression Model 1

We first examine the scatterplot of the dependent variable (ISE_USD) and the predictor variable (SP).

plot(index$ISE_USD~index$SP, xlab='S&P500 Returns', ylab = 'ISE Index Returnes', title='Scatterplot of Returns')

- It was observed that the returns for the two indeces depicted an approximate positive linear relationship. Hence, we could proceed with the creation of a linear regression model.

Our model: $y=\beta_0 + \beta_1x_1 =>$ ISE_USD = $\beta_0 + \beta_1$ SP

model <- lm(index$ISE_USD~index$SP)
model %>% summary()


Call:
lm(formula = index$ISE_USD ~ index$SP)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.088120 -0.010764  0.000579  0.011136  0.070506 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.0011188  0.0008166    1.37    0.171    
index$SP    0.6737831  0.0579340   11.63   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.01888 on 534 degrees of freedom
Multiple R-squared:  0.2021,    Adjusted R-squared:  0.2006 
F-statistic: 135.3 on 1 and 534 DF,  p-value: < 2.2e-16

2*pt(q = 1.37,df = 534, lower.tail=FALSE)

[1] 0.1712626

Interpretation

$\beta_0$=0.0011188. ISE_USD was on average 0.0011188, when SP was 0.
$\beta_1$=0.6737831. ISE_USD increased by 0.6737831 dollars when SP increased by 1 dollar. This was an indication of the correlation between the returns of the two indexes, most likely due to spillover effects (touched upon earlier).
$R^2$ of 0.2021 implied that only 20.21% of the variation in ISE_USD could be explained by a linear relationship with SP.

Significance

The intercept of the model was tested using a t-statistic (df=534) where the $H_0$ was $\beta_0$=0 and $H_A$ was $\beta_0$ ≠ 0. The p-value of this test was 0.171 and the t-statistic was 1.37. Hence, as the p-value of our test was greater than $\alpha$=0.05, we failed to reject the $H_0$ and concluded that $\beta_0$ was not statistically significantly different from 0.
The slope of the model was tested using a t-statistic (df=534) where the $H_0$ was $\beta_1$=0 and $H_A$ was $\beta_1$ ≠ 0. The p-value of this test was <0.001 and the t-statistic was 11.63. Hence, as the p-value of our test was lesser than $\alpha$=0.05 and the t-value of our test was greater than the t-critical value of 0.1712626, we rejected the $H_0$ and concluded that $\beta_1$ was statistically significantly different from 0.
The F-test results were also observed, for which the $H_0$ was, ‘the data does not fit the linear regression model’ and, the $H_A$ was, ‘the data does fit the linear regression model’. The p-value for the F-test was <0.001, suggesting that we should have rejected $H_0$ at the 5% significance level thus suggesting that statistically, the data fit the linear regression model.

Assumptions of our Regression Model

plot(model)

Residuals vs. Fitted: There seemed to be a linear trend. The regression line was relatively falt and the variance did not seem to indicate any hetergoneous qualities.
Normal Q-Q: Residuals seemed to approximately follow a normal distribution.
Scale-Location: There seemed to be a linear trend. The regression line is relatively falt and the variance did not seem to indicate any hetergoneous qualities
Residuals vs. Leverage: All points were within Cook’s distance. Hence, there were no obvious outliers that effected our regression model. Therefore, the assumptions on which our model was based, were held.

The Correlation Matrix

The correlation between the returns of all eight stock market indexes was used to examine the spillover effect. It was observed that the returns of any given stock market index is positively correlated with the returns of other stock market indexes suggesting that spillover effect is in play. It was also observed that the p-value of these correlations was <0.001 (except for NIKKEI). Hence, as the p-value of each correlations was below the significance level of 5%, we rejected $H_0: r=0 $, for each correlation and concluded that the Pearsons correlations were statistically significantly different from 0.

res2 <- as.matrix(dplyr::select(index, ISE_USD,SP,DAX,FTSE,NIKKEI,BOVESPA,EU,EM))
rcorr(res2, type=c("pearson"))

        ISE_USD   SP  DAX FTSE NIKKEI BOVESPA   EU   EM
ISE_USD    1.00 0.45 0.63 0.65   0.39    0.45 0.69 0.70
SP         0.45 1.00 0.69 0.66   0.13    0.72 0.69 0.53
DAX        0.63 0.69 1.00 0.87   0.26    0.59 0.94 0.67
FTSE       0.65 0.66 0.87 1.00   0.26    0.60 0.95 0.69
NIKKEI     0.39 0.13 0.26 0.26   1.00    0.17 0.28 0.55
BOVESPA    0.45 0.72 0.59 0.60   0.17    1.00 0.62 0.69
EU         0.69 0.69 0.94 0.95   0.28    0.62 1.00 0.72
EM         0.70 0.53 0.67 0.69   0.55    0.69 0.72 1.00

n= 536 


P
        ISE_USD SP     DAX    FTSE   NIKKEI BOVESPA EU     EM    
ISE_USD         0.0000 0.0000 0.0000 0.0000 0.0000  0.0000 0.0000
SP      0.0000         0.0000 0.0000 0.0023 0.0000  0.0000 0.0000
DAX     0.0000  0.0000        0.0000 0.0000 0.0000  0.0000 0.0000
FTSE    0.0000  0.0000 0.0000        0.0000 0.0000  0.0000 0.0000
NIKKEI  0.0000  0.0023 0.0000 0.0000        0.0000  0.0000 0.0000
BOVESPA 0.0000  0.0000 0.0000 0.0000 0.0000         0.0000 0.0000
EU      0.0000  0.0000 0.0000 0.0000 0.0000 0.0000         0.0000
EM      0.0000  0.0000 0.0000 0.0000 0.0000 0.0000  0.0000

Discussion & Conclusions:

Hypothesis Test 1: Although our sample mean was suggestive that ISE index returns were on avaerage greater than S&P500 index returns, the difference between the two average returns was not statistically siginificant.
Hypothesis Test 2: Positive/Negative returns of the ISE index were associated with positive/negative returns of the S&P500.
Regression analysis: The S&P500 returns were capable of explaining only 20% of the variability in the ISE index returns through a linear relationship.
The correlation between the selected indexes is positive and statistically significantly different from 0. Thus, gave an indication of spillover effects.

Limitations:

A multi-linear regressions should have been explored in order to determine a model with a higher $R^2$. The variability in the ISE index returns could have perhaps been better explained by linear relationships with the returns of all 7 indexes, as opposed to just 1. In addition, a low $R^2$ could have been due to a greater reliance of the Turkish economy on the European Union, as opposed to the U.S.A[7]. This was also examined in the correlation matrix. The ISE index returns had a higher positive correlation with the index returns of countries belonging to European Union.
The spillover effect could have been better examined if we had data that included returns during periods of economic turmoil (eg: the Global Financial Crisis). Such crisises can originate in one country, but have global reprecussions. The extent to which the financial market in one country is affected, depends on its correlation to the financial markets of another country.

References

[1] IMF, “IMF Advanced Economies List. World Economic Outlook”, p. 173, 2011.
[2] IMF, “World Economic Outlook Database”, 2019.
[3] The importance of Stock Market Indexes, 2015. [Online]. Accessed on: Jun 1, 2019. Available: http://www.mistakesintrading.com/wiki-stocks/importance-of-stock-market-indexes.php
[4] W, Kenton, Spillover Effects, 2018/ [Online]. Accessed on: Jun 1, 2019. Available: https://www.investopedia.com/terms/s/spillover-effect.asp
[5] Akbilgic, O., Bozdogan, H., Balaban, M.E., “A novel Hybrid RBF Neural Networks model as a forecaster, Statistics and Computing”, 2013.
[6] J. Kuepper, What are the Major World Stock Market Indexes? [Online]. Accessed on: Jun 1, 2019. Available: https://www.thebalance.com/major-world-stock-market-indexes-4148491
[7] OEC, Turkey. [Online]. Accessed on: Jun 1, 2019. Available: https://atlas.media.mit.edu/en/profile/country/tur/

Istanbul Stock Exchange

An analysis of returns of the ISE index.

Introduction

Why is the Istanbul Stock Exchange the subject of this analysis?

Why did we use global stock indexes?

Problem Statement

Hypothesis Test 1

Hypothesis Test 2

Correlation between the stock indexes

Data

Data Cont.

Visualisation of the returns of the ISE index and the S&P500 index

Descriptive Statistics

Hypothesis Test 1

Hypthesis Test 1 Cont.

Hypothesis Test 2

Hypothesis Test 2 Contd.

Reggression Model 1

Interpretation

Significance

The Correlation Matrix

Discussion & Conclusions:

Limitations:

References