Financial Econometrics

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Test for Normality in Stock Return

Abstract
The objective of the first assignment is to determine if TJX’s stock returns are normally distributed. We pulled pricing data from 899 days before the corporate data breach and 100 days after the event and then conducted several tests in order to draw a conclusion about the distribution of TJX’s returns. The KS test for the 1,000 observations suggested TJX’s stock returns are not normally distributed. The F test suggested that the volatility of at least two samples is significantly different. We conclude that returns of TJX are not normally distributed. Introduction

Normal distribution is one of the most important statistical distributions as it is used to draw conclusions from sample data about the populations from which theses samples are drawn from. This distribution also has some important characteristics, such as the normal distribution is symmetrical about its mean. Also, the normal distribution provides a benchmark of how the data is dispersed; the normal distribution states that 99.73% of the probability density function lies within three standard deviation of the mean. The test of normality will help analyze other statistical feature of TJX’s stock return. Method

In our analysis, we conducted a KS test to determine if the total number of observations was drawn from a normal distribution firstly. Then we took the 1,000 observations and broke them into subsets to test if these smaller samples were normally distributed. Lastly, we tested the behavior of the sample means and the sample variances to ensure they were both coming from the same distribution. The Kolmogorov-Smirov test is a goodness of fit test; it measures the discrepancies between the observed values and the values expected under the model in question. In this assignment we will use the KS test to determine if TJX’s stock returns are normally distributed. First we will calculate the mean and standard deviation for the 1,000 observations and also for the individual 10 samples of 100 observations. The Mean: i=1nuin

Standard Deviaiton:∑(Xi-X)2n-1
Then we will need to standardize the 1,000 observations to a standard normal distribution and use the Z score to determine the cumulative probably. We then use the mean and variance calculated above to normalize our observations. The Z score transforms the observations into their theoretical cumulative probability. We will then take the maximum difference between the theoretical and actual probably distribution to conduct a KS test to determine if the returns are normal. Normal Z score: z=x-μσ

After normalizing, we took the absolute difference between the actual and theoretical cumulative distribution function. We then will consult the KS table to determine if at a 95% confidence level the critical D value is greater or less than the maximum difference, D. We will then use the below formula to test the difference between two means. This test will look at two means from our sample sizes to see if they are statistically different. t=x1 - x2 - μ1-μ2(s12n1 +s22n2)1/2

Lastly, we will use the F distribution to compare two sample variances to see if they are statistically different.
F= s12s22
Results
Using the KS test on the 1,000 observations we determined this sample was not normally distributed. The actual D value we calculated in our test was .051966. Using a sample size of 1,000 and a 95% confidence interval we determined the critical D value was .043000. Since the actual D value was larger than the critical D value, we reject the null hypothesis and conclude that returns are not normally distributed. (See Appendix 1) We furthered our analysis by using the KS test to determine if smaller subsets of our sample stock returns were normally distributed. At the 95% confidence interval, the critical D value was .1360. Using the output in Appendix 2, we found that 9 out of the 10 sample sizes appear to be normally distributed. Next we tested...
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