Summary Statistics, Normality Test, Hypothesis Test, And Risk Comparison For Boeing And IBM Stocks

It used to display information as a data point series that are connected by straight line segments (Kumar & Phrommathed, 2005). Line charts can be used to depict the features of time series, thereby, there are referred to as time series graphs when the component of time is included. As a result, time series carts are vital in determining the patterns of a model over a period of time (Enders, 2007). The time series charts to be developed for S&P 500, BA and IBM are needed to describe vital features of the time series pattern, to forecast future series’ values, explain how the past impacts the future, and to maybe provide a controlled standard for a variable. The vital characteristics considered in a time series include the trend, long-run cycle, seasonality, outliers, constant variance and abrupt changes (Shumway & Stoffer, 2006).  

It can be seen that the S&P 500 has a trend which is increasing over time. The trend can be observed by the straight line connecting the start point and the end point. However, the trend does not show any seasonality as there are no significant outliers. Other observations that can be made from the time series are that the S&P 500 has no abrupt changes and no period unrelated to seasonality factors.

Like the S&P time series, it can be seen that the trend for Boeing Company is increasing over time. The figure shows that the trend line has outliers. The outliers are values which are far away from the trend line. However, it can be seen that the time series has no abrupt changes, no seasonality or any long-run-cycle.

It is seen that IBM has no trend. Thus, the price of stocks can be seen to rise from 7/1/2010 to1/1/2013. From 1/1/2013, the prices of the stock have been on the decline. Moreover, due to a lack of a trend, the stocks can be seen to have no seasonality (Koopman & Lee, 2009). Thus, there are no regular repeating patterns of high and lows. Conversely, the time series has no abrupt changes, outliers, and long-run cycle. 

Table 1: Returns summary statistics

Table 1 shows that Boeing Company (1.157) has the highest return compared to IBM (0.107). Moreover, Boeing Company has the highest standard deviation (5.77) compared to IBM (4.7). It should be noted that the standard deviation of a stock measures its risk (Pastor & Robert, 2003). In addition, when the standard deviation is high, it shows that the stock is riskier. Thus, Boeing Company is riskier than IBM even though the returns are higher. As a result, it can be concluded that the average return of a stock has a linear relationship with the risk (Whitelaw, 2000). The S&P has an average return of 0.95 with a standard deviation of 3.5. Thus, the low standard deviation of the index shows that the market returns are less volatile. On the other hand, the average returns are relatively high as they are recorded at 0.95. 

Summary statistics for returns series

Table 2: Jarque-Berra of return normality test derivation

From table 2, it can be seen that the p-values for the Jarque-Berra test are 0.67 for Boeing Company and 0.37 for IBM. Since the p-value is greater than 0.5, we choose to fail to not accept the null hypothesis that the returns are normally distributed. Therefore, Boeing Company and IBM returns are normally distributed. Normality tests are carried out in order to determine the appropriate tests that should be applied to the data (Jarque, 2011). 

To test the hypothesis that the average return of Boeing Company is greater than 3%, we choose to perform a one-tailed z-test. Moreover, the test opts to check one direction, which is greater than 3% (Franz et al., 2009).

The null hypothesis developed states that the average return on Boeing company stock is at least 3% while the alternate hypothesis states that the average return on Boeing company stock is less than 3%.

Based on the formula; Z = (x? – µ)/(σ/(√n)), the z tests statistics derived will be equal to 1.574. The critical value for α = 0.05 for a one-tailed test is 1.645. Since 1.574 is less than 1.645, it is in the acceptance region. Therefore, the average return of Boeing Company is at least 3%. 

To compare the risk associated to each of the two stocks, a chi-square test was adopted. The chi-square test was adapted since it tests the relationship between categorical variables (Zibran, 2007).

The null hypothesis developed states that the standard devotions of the two stocks are equal. On the other hand, the alternate hypothesis states that the standard deviations of the two stocks are not equal.

The α to be used for this test is equal to 0.05. On the other hand, the numerator degree of freedom and the denominator degree of freedom are both 64 that is 65-1.

Thus, using the formula; F = , the derived F test statistics is 1.506.

Critical values: F _{(0.975, 64, 64)} = 0.54

                        F _{(0.025, 64, 64)} =1.0

Rejection region: Reject H0 if F < 0.54 or F > 1.0

Thus, there is enough evidence to not accept the null hypothesis since the F statistic is within the acceptance region. Therefore, the risks associated with the stock are similar to each other. 

To determine whether the population average returns are equal, an ANOVA analysis was chosen. The results of the ANOVA analysis are shown below:

Jarque-Berra test of normally distributed returns

The null hypothesis developed states that the population average returns are equal while the alternate hypothesis states that the population averages are not equal.

Table 3: Summary

Table 4: ANOVA 

The P-value of the ANOVA test is greater than the significance level of 5%. Thus, the decision is to reject the null hypothesis. Therefore, the population average returns of the two stocks are not the same. 

Computation of excess returns as seen in the attached excel document. 

The preferred stock is Boeing Company 

The beta coefficient for Boeing Company is 0.007. Thus, the stock is less volatile than the market since it is less than 1 (Young-tao, 2004). Thus, the Boeing company stock is 0.7% less volatile than the market.

The R square of the stock is 0.908. Since the R squared is between 85% and 100%, it implies that the performance of the stock is in line with the market (Van Rooji et al., 2011).

The 95% confidence interval of the slope coefficient is between -2.48 and 0.32.

The value of a neutral stock is equivalent to 1 (Fama & French, 2012). Thus, the developed null hypothesis states that the beta coefficient is equal to one while the alternate hypothesis states that the beta coefficient is not equal to 1.

Z-score at 95% confidence level = 1.96

Z = (1 – β) / σ

1.96 = (β – 1) / ± 5.67

1.96 * ± 5.67 = (β -1)

β = 1 + (1.96 ± 5.67)

β = 12.11 or -10.11

Thus, -10.11 < β < 12.11

Since the beta of Boeing Company is within the accepted region of -10.11 and 12.11, we choose not to reject the null hypothesis. Therefore, Boeing Company stock is a neutral stock. 

To find whether the error term in the model is normally distributed, a Jarque-Berra test was adopted the table below shows the process of the Jarque Berra-test:

Table 5: Jarque-Berra test for error term normality

It can be seen that the p-values for the Jarque-Berra test are 0.71 for the residuals. Since the p-value is greater than 0.5, we choose to fail to reject the null hypothesis and conclude that the standard error follows a normal distribution.    

References:

Enders, W. (2004). Applied econometric time series, by walter. Technometrics, 46(2), 264.

Fama, E. F., & French, K. R. (2012). Size, value, and momentum in international stock returns. Journal of financial economics, 105(3), 457-472.

Faul, F., Erdfelder, E., Buchner, A., & Lang, A. G. (2009). Statistical power analyses using G* Power 3.1: Tests for correlation and regression analyses. Behavior research methods, 41(4), 1149-1160.

Koopman, S. J., & Lee, K. M. (2009). Seasonality with trend and cycle interactions in unobserved components models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 58(4), 427-448.

Jarque, C. M. (2011). Jarque-Bera test. In International Encyclopedia of Statistical Science (pp. 701-702). Springer Berlin Heidelberg.

Kumar, S., & Phrommathed, P. (2005). Research methodology (pp. 43-50). Springer US.

Pástor, ?., & Stambaugh, R. F. (2003). Liquidity risk and expected stock returns. Journal of Political economy, 111(3), 642-685.

Shumway, R. H., & Stoffer, D. S. (2006). Time series analysis and its applications: with R examples. Springer Science & Business Media.

Van Rooij, M., Lusardi, A., & Alessie, R. (2011). Financial literacy and stock market participation. Journal of Financial Economics, 101(2), 449-472.

Whitelaw, R. F. (2000). Stock market risk and return: An equilibrium approach. The Review of Financial Studies, 13(3), 521-547.

Yong-tao, L. I. U. (2004). An Empirical Study on Beta Coefficient and Its Related Characteristic in Shanghai Stock Market [J]. Policy-making Reference, 1.

Zibran, M. F. (2007). Chi-squared test of independence. Department of Computer Science, University of Calgary, Alberta, Canada.

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