Applied Statistics for Finance and Economics Project Report - Coursework Example

APPLIED STATISTICS FOR FINANCE & ECONOMICS PROJECT REPORT Abstract The aim of this paper is to apply the econometrics knowledge learnt in the practical application. Two main variables of interest were used. STATA software was employed for the analysis of the data. Results showed that there exists a positive relationship between log returns of the variables y (‘dly’) and that of sap (‘dlsap’). Introduction We began by declaring the data as a time series data. This was followed by generating the log of the variables y (‘ly’) and sap (‘lsap’). We also generated the log returns of the variables y (‘dly’) and sap (‘dlsap’). The other sections present the results of the analysis starting with the descriptive statistics. Summary (Descriptive) statistics In this part, we present the summary statistics (which includes the mean, standard deviation, skewness, kurtosis and the plots) for the log returns. The for the log returns for sap (dlsap) is given as 1.9539 while that of end-of-week share price for a particular stock (dly) is given as 0.4551. The skewness for dlsap is 0.2553 while that of dly is 0.1481; the two values are greater than zero which implies that we have a Right skewed distribution in both cases - most values are concentrated on left of the mean, with extreme values to the right. In regard to the kurtosis, the dlsap has a kurtosis of 2.2926 while that of dly is 2.225; the two values are less than 3 implying that we have a Platykurtic distribution, flatter than a normal distribution with a wider peak. The probability for extreme values is less than for a normal distribution, and the values are wider spread around the mean. Time series plots To understand the flow of data over time, we constructed a time series plot and it can clearly be observed that the values for dly have been on decrease from 2002 to end of 2004 before it began to rise again. On the other hand, the values for dlsap began to rise from 2004 and it maintained the rise all through. Box plots In this part we present the box plots for both dly and dlsap. The graphs show no presence of any outliers for any of the graphs. This also indicates there is no much variation in the data set. Figure 1: Box Plots Histogram The two histograms for dly and dlsap are presented below. The histograms are in an attempt to check for the normality of the data. It is clear that dly does not show element of being normally distributed while dlsap is close to be normal as from the shape. Statistical test for normality Apart from the graphical visualization, we conducted a statistical test to check for normality of the data. Shapiro-Wilk test was used to check for the normality. The table below gives the results for the Shapiro-Wilk test; From the table, we observe the p-value for both the dly and dlsap to be less than 5% significance level; we thus reject the null hypothesis of normality for both dly and dlsap and conclude that there is no evidence of normality distribution in the two variables presented at 5% significance level. Significance test for mean differences in the log returns between the y stock and the S&P The difference in the mean of the two variables (log returns between the y stock and the S&P) was checked using a t-test. The test checks for any significance difference that exists between two variables. We present the t-test results in the table below; From the table, we observe the p-value to be 0.000 (a value less than α=0.05), we thus reject the null hypothesis of equal mean and conclude that indeed the for log returns for y stock and that of S&P are statistically different at 5% significance level. Important to the study was looking at the mean for the log returns for the y stock between the first half and the second-half of the sample. The results are presented in the table above. It is clear that there is a statistically significant mean differences in the log returns for the y stock between the first half and the second-half of the sample. This can be explained by the fact that the p-value is less than 5% significance level leading to the rejection of the null hypothesis. Test for median differences Apart from the mean difference, it is was important to check for the median difference. In this section we present the results and analysis of the difference in the median in the log returns for the y stock between the first half and the second-half of the sample. From the table, we observe the p-value to be 0.000 (a value less than α=0.05), we thus reject the null hypothesis and conclude that the median test is indeed significant at 5% significance level and that there exists a statistically significant difference in the medians values for the log returns for the y stock between the first half and the second-half of the sample. 72.4%% of the observations in the first half had values greater than the median while only 27.7% of the observations in the second-half had values greater than the median; this indeed shows a clear difference in the two categories. Testing for randomness or non-randomness Randomness refers to the lack of pattern or predictability in events [Joh03]. The run test (Bradley, 1968) is used to decide whether a data set is from randomly distributed. In section, we present the results and analysis of the test for randomness for the log returns for the y stock is random or not. We present the results for the run test (test for randomness) in the table below; The p-value is given as 0.000 (a value less than α=0.05), we thus reject the null hypothesis of non-random and conclude that the data (time series sequence of log returns for the y stock) are randomly distributed. Capital Asset Pricing Model (CAPM) Ordinary Least Squares Estimation (OLS) was employed to estimate a simple Capital Asset Pricing Model (CAPM). To do this, a simple linear regression equation was modelled as shown below; The first vitl part is to chek for the goodness of fit; the p-value for the F-statistic is 0.0239; comparing with the α value-it is less than α=0.05, we thus reject the null hypothesis and conclude that the model is significant at 5% significance level hence the model is good and appropriate at 5% significance level. However, the percent of variation in dly explained by dlsap is very minimal (1.99%). In the table, we observe (the coefficient for dlsap) to be 2.013; this is a clear indication that there exists a positive relationship between dly and dlsap. It further shows that if dlsap increases by a unit then we would expect the response variable (dly) to increase by 2.013. The vice versa is true for the decrease in the explanatory variable (dlsap). In the table below, we present the regression analysis but with an exclusion of the constant term (intercept); Comparing the beta estimates in the case with constant term included and the one with no constant term included; we see that there is a difference in the beta values; the beta value without constant term is much larger than the one with the constant term. Works Cited Joh03: , (John , 2003), APPENDIX Figure 2: Kernel Plots Figure 3: P-P Plots Do Files gen ly=log(y) gen lsap=log(sap) gen dly=log(ly) gen dlsap=log(lsap) graph box dly kdensity dly histogram dlsap ttest dly=dlsap ttest dly, by( time_dum) median dly, by(time_dum) runtest dly reg dly dlsap reg dly dlsap, noconstant summarize dly dlsap, detail swilk dly dlsap graph box dlsap kdensity dlsap histogram dly Log Files ___ ____ ____ ____ ____ (R) /__ / ____/ / ____/ ___/ / /___/ / /___/ 11.2 Copyright 1985-2009 StataCorp LP Statistics/Data Analysis StataCorp 4905 Lakeway Drive Special Edition College Station, Texas 77845 USA 800-STATA-PC http://www.stata.com 979-696-4600 stata@stata.com 979-696-4601 (fax) Single-user Stata perpetual license: Serial number: 40110571722 Licensed to: Centre for the Study of African Economies University of Oxford Notes: 1. (/m# option or -set memory-) 50.00 MB allocated to data 2. (/v# option or -set maxvar-) 5000 maximum variables . use "C:\Users\albert\Downloads\EK.dta", clear . do "C:\Users\albert~1\AppData\Local\Temp\STD0m000000.tmp" . gen ly=log(y) . gen lsap=log(sap) . gen dly=log(ly) . gen dlsap=log(lsap) . . graph box dly . . kdensity dly . . histogram dlsap (bin=16, start=1.935872, width=.00236788) . . ttest dly=dlsap Paired t test ------------------------------------------------------------------------------ Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- dly | 257 .4551141 .0079944 .1281599 .439371 .4708573 dlsap | 257 1.953924 .0005596 .0089706 1.952822 1.955026 ---------+-------------------------------------------------------------------- diff | 257 -1.49881 .0079349 .1272063 -1.514436 -1.483184 ------------------------------------------------------------------------------ mean(diff) = mean(dly - dlsap) t = -1.9e+02 Ho: mean(diff) = 0 degrees of freedom = 256 Ha: mean(diff) < 0 Ha: mean(diff) != 0 Ha: mean(diff) > 0 Pr(T < t) = 0.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 1.0000 . ttest dly, by( time_dum) Two-sample t test with equal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- 0 | 127 .5175042 .0110185 .1241724 .4956989 .5393095 1 | 130 .3941638 .0087526 .0997955 .3768465 .4114812 ---------+-------------------------------------------------------------------- combined | 257 .4551141 .0079944 .1281599 .439371 .4708573 ---------+-------------------------------------------------------------------- diff | .1233404 .0140364 .0956983 .1509824 ------------------------------------------------------------------------------ diff = mean(0) - mean(1) t = 8.7872 Ho: diff = 0 degrees of freedom = 255 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 1.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 0.0000 . median dly, by(time_dum) Median test Greater | than the | time_dum median | 0 1 | Total -----------+----------------------+---------- no | 35 94 | 129 yes | 92 36 | 128 -----------+----------------------+---------- Total | 127 130 | 257 Pearson chi2(1) = 51.4565 Pr = 0.000 Continuity corrected: Pearson chi2(1) = 49.6821 Pr = 0.000 . runtest dly N(dly .4515195190906525) = 128 obs = 257 N(runs) = 24 z = -13.19 Prob>|z| = 0 . reg dly dlsap Source | SS df MS Number of obs = 257 -------------+------------------------------ F( 1, 255) = 5.17 Model | .083484326 1 .083484326 Prob > F = 0.0239 Residual | 4.12130523 255 .016161981 R-squared = 0.0199 -------------+------------------------------ Adj R-squared = 0.0160 Total | 4.20478955 256 .016424959 Root MSE = .12713 ------------------------------------------------------------------------------ dly | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- dlsap | 2.013083 .8857403 2.27 0.024 .2687848 3.75738 _cons | -3.478297 1.730688 -2.01 0.046 -6.886558 -.0700352 ------------------------------------------------------------------------------ . reg dly dlsap, noconstant Source | SS df MS Number of obs = 257 -------------+------------------------------ F( 1, 256) = 3256.13 Model | 53.2503237 1 53.2503237 Prob > F = 0.0000 Residual | 4.18658678 256 .016353855 R-squared = 0.9271 -------------+------------------------------ Adj R-squared = 0.9268 Total | 57.4369105 257 .223489924 Root MSE = .12788 ------------------------------------------------------------------------------ dly | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- dlsap | .2329605 .0040825 57.06 0.000 .2249209 .2410002 ------------------------------------------------------------------------------ . summarize dly dlsap, detail dly ------------------------------------------------------------- Percentiles Smallest 1% .2454155 .2253514 5% .260411 .241035 10% .2789611 .2454155 Obs 257 25% .3355553 .2454155 Sum of Wgt. 257 50% .4515195 Mean .4551141 Largest Std. Dev. .1281599 75% .5511164 .7320994 90% .5868996 .7519175 Variance .016425 95% .6938869 .7524823 Skewness .1481466 99% .7519175 .7820968 Kurtosis 2.225 dlsap ------------------------------------------------------------- Percentiles Smallest 1% 1.936546 1.935872 5% 1.94046 1.935872 10% 1.942618 1.936546 Obs 257 25% 1.946745 1.936643 Sum of Wgt. 257 50% 1.9539 Mean 1.953924 Largest Std. Dev. .0089706 75% 1.959536 1.972891 90% 1.967357 1.97301 Variance .0000805 95% 1.969873 1.973341 Skewness .2552852 99% 1.97301 1.973758 Kurtosis 2.29262 --more-- Read More