Testing Residual Normality in Regression

This assignment explores the concept of residual normality in linear regression. It explains why checking for normality is important and provides a step-by-step guide on how to perform tests for normality using various Excel functions, including the Jarque-Bera, Shapiro-Wilk, and Anderson-Darling tests. The document also includes examples and interpretations of the results obtained from these tests.

Running head: STOCK RETURNS ANALYSIS
Stock Returns: Boeing & IBM Analysis
Name
Course Number
Date
Faculty Name

STOCK RETURNS ANALYSIS 2
Stock Returns: Boeing & IBM Analysis
1. Line Charts for the Stocks
Figure 1: S&P 500 price against time
The S&P 500 time series chart shown in figure 1 shows that the stock prices have been
significantly increasing from 2010 to the end of 2015. However, the prices seem to have experienced
stationary processes on the last quarter of the year 2011. The prices increased from around 1,000 in
mid-year 2010 to around 2,000 at the end of 20151.
1 R Hyndman, "6.1 Time series components", in Otexts.org, , 2017,
<https://www.otexts.org/fpp/6/1> [accessed 10 October 2017].

STOCK RETURNS ANALYSIS 3
Figure 2: Boeing stock price line chart
Between July 2010 and February 2012, Boeing stock prices trend was stationary, indicating
that they were not affected by seasonal or cyclic factors. After January 2013, an upward trend of the
prices was experienced, hence moving from around 75 to 140in February 2014. This was a double
improvement of Boeing stock prices, which might have been improved by policy changes within the
institution. Thereafter, the prices show a stationary trend with high levels of seasonality. The
maximum price achieved for the Boeing stock was around 160.
Figure 3: IBM stock prices line chart

STOCK RETURNS ANALYSIS 4
IBM stock prices show small variation between July 2010 and December 2015. The lowest
stock price achieved within this period is around 125 and the highest is slightly above 210.
Throughout this specified period, IBM stock has been experiencing seasonality effects, leading to
unstable prices. However, the general trend of the stock prices is stationary, with few times when
upward and downward trends were experienced.
2. Stock Returns for the Three Series
Summary Statistics
Table 1: Table for the Stock returns
Re-BA Re-S&P Re-IBM
Mean 0.06117 0.05011 0.00872
Standard Error 0.03905 0.02618 0.03231
Median 0.07640 0.05921 0.01573
Mode 0 4.63174 0
Standard Deviation 1.45322 0.97435 1.20233
Sample Variance 2.11184 0.94935 1.44560
Kurtosis 2.52241 4.66029 5.58741
Skewness -0.22490 -0.45674 -0.78094
Range 14.23780 11.52758 14.15247
Minimum -8.23848 -6.89584 -8.64191
Maximum 5.99932 4.63174 5.51057
Sum 84.72524 69.40879 12.08377
Count 1385 1385 1385
On average, Boeing has higher returns compared to S&P 500 and IBM stocks. However,
Boeing stock has higher variation, which might negatively affect investments compared to the other
two series. The variation can be affirmed based on the range statistic, which also indicates that
Boeing stock returns are sparsely distributed.
Jarque-Berra test of Normality
H0: x follows a normal distribution
Ha: x does not follow a normal distribution

STOCK RETURNS ANALYSIS 5
JB – Jarque-Berra statistic2
JB - Boeing 378.8468
JB - IBM 1942.384
Jarque-Berra follows a Chi-square statistic with 2 degrees of freedom
ꭓ2 2, 0.05 = 3.841
For both Boeing and IBM stock, we conclude that the returns are not normally distributed
because the calculated values are greater than the tabulated.
3. Test of Hypothesis
Testing the hypothesis that the average stock return of Boeing is at least 3%. One sample t-
test is used to test this hypothesis because the statistic is based on the average value.
2 Spider Financial, "Normality Test in Excel | Jarque-Bera | Doornick and Hansen | Shapiro-Wilk |
Anderson Darling| NumXL", in NumXL | Spider Financial, , 2017,
<http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-
tests/normalitytest> [accessed 10 October 2017].

STOCK RETURNS ANALYSIS 6
Figure 4: Histogram of Boeing stock returns
According to figure 4 above, the stock returns for Boeing are not highly skewed, hence we
can conclude that they are approximately normal. Therefore, we can proceed with one sample t-test
because the normality assumption is approved.
Null hypothesis: Average returns of Boeing stock equal to 3%
Alternative hypothesis: Average returns of Boeing stock is greater or equal to 3%
Table 2: One sample t-test
One-Sample t-test
Null hypothesis: mean = 3%
Alternative hypothesis: mean ≥ 3%
Mean (Xbar) 0.061173
Standard Deviations 1.453216
Sample size (n) 1385
t statistic 0.798325
t statistic (tabulated) 1.645
Decision We fail to reject the null
hypothesis

STOCK RETURNS ANALYSIS 7
We fail to reject the null hypothesis because the calculated t statistics is less than tabulated.
Therefore, we concluded that the average return of Boeing stock is not equal to 3%3.
4. Comparing risk associated with Boeing and IBM
To compare the risk associated with Boeing and IBM stock, we will use an independent
sample t-test to determine whether there is a difference investing in either of the two.
Null Hypothesis: There is no difference between average returns of Boeing and IBM stocks
Alternative hypothesis: There is a difference between average returns of Boeing and BM stocks
Table 3: Two-sample test
t-Test: Two-Sample Assuming Unequal Variances
Re-BA Re-IBM
Mean 0.061173 0.008725
Variance 2.111837 1.445596
Observations 1385 1385
Hypothesized Mean Difference 0
Degrees of freedom 2674
t Stat 1.034883
P(T<=t) one-tail 0.150409
t Critical one-tail 1.645424
P(T<=t) two-tail 0.300817
t Critical two-tail 1.960852
Decision We fail to reject the null
hypothesis
Since the p-value is greater than 0.05 (significance level), we conclude that there is no
difference between investing in Boeing or IBM stocks4. However, an investor can concentrate on
other factors such as the variance and distribution.
3 J List, A Shaikh & Y Xu, Multiple Hypothesis Testing in Experimental Economics, in , Cambridge,
Mass., National Bureau of Economic Research, 2016.
4 J List, A Shaikh & Y Xu, Multiple Hypothesis Testing in Experimental Economics, in , Cambridge,
Mass., National Bureau of Economic Research, 2016.

STOCK RETURNS ANALYSIS 8
5. Testing whether Boeing and IBM have same population average
Table 4: Summary statistics for sampled 65 observations
Re-BA Re-IBM
Mean 0.1828 0.2669
Standard Error 0.1596 0.1174
Median 0.0425 0.2276
Standard Deviation 1.2868 0.9463
Sample Variance 1.6558 0.8955
Kurtosis 1.0045 0.2794
Skewness 0.6672 0.2883
Range 6.8967 4.2716
Minimum -2.5361 -1.5077
Maximum 4.3605 2.7639
Sum 11.8809 17.3484
Count 65 65
Null hypothesis: There is no difference between Boeing and IBM of average stock returns
Alternative hypothesis: There is a difference in means of stock returns between Boeing and IBM
Table 5: Two-sample test for Boeing and IBM
t-Test: Two-Sample Assuming Unequal Variances
Re-BA Re-IBM
Mean 0.182783 0.266898
Variance 1.65576 0.895516
Observations 65 65
Hypothesized Mean Difference 0
Degree of freedom 118
t Stat -0.42458
P(T<=t) one-tail 0.33596
t Critical one-tail 1.65787
P(T<=t) two-tail 0.671919
t Critical two-tail 1.980272

STOCK RETURNS ANALYSIS 9
The p-value of greater than the significance level, hence concluding that the average stock
returns are not different between Boeing and IBM. Based on the above analysis, IBM is my
preferred investment stock because it has lower variation and the higher mean of returns.
6. Computing excess returns of IBM stock and Excess Market returns
Table 6: Computed excess returns on market and IBM stock
Re-BA Re-S&P Re-IBM 10-Year Rate Yt Xt
-0.1487 -0.4167 -0.3197 2.6520 -2.9717 -3.0687
2.9565 1.6451 1.5757 2.4560 -0.8803 -0.8109
0.9812 0.4839 0.1781 2.7120 -2.5339 -2.2281
-0.6190 -0.2694 0.5388 2.8640 -2.3252 -3.1334
-0.9370 -0.7497 -1.3186 3.4810 -4.7996 -4.2307
0.1697 0.0771 0.2543 3.3850 -3.1307 -3.3079
0.7683 0.2242 0.0186 3.4140 -3.3954 -3.1898
-0.4019 0.5546 -0.2468 3.4910 -3.7378 -2.9364
0.4356 0.7035 0.9314 3.3120 -2.3806 -2.6085
3.1424 0.3549 0.2404 3.0640 -2.8236 -2.7091
0.4295 0.4073 0.1912 3.0450 -2.8538 -2.6377
0.6541 1.2861 1.4158 2.9510 -1.5352 -1.6649
0.0425 -0.3239 0.2478 2.1880 -1.9402 -2.5119
2.7443 1.5008 2.1272 2.0160 0.1112 -0.5152
1.2341 1.0632 1.8171 2.2030 -0.3859 -1.1398
4.3605 1.0481 0.8887 1.9670 -1.0783 -0.9189
0.6712 -0.2689 -0.5014 2.0090 -2.5104 -2.2779
0.4047 0.0079 0.1082 1.8980 -1.7898 -1.8901
-1.0143 -0.1594 -0.2726 1.9290 -2.2016 -2.0884
-0.0665 0.3351 0.2276 2.1960 -1.9684 -1.8609
-0.6437 -0.4954 0.0531 1.9330 -1.8799 -2.4284
0.3630 0.2411 0.5965 1.7310 -1.1345 -1.4899
0.5698 1.1018 1.1056 1.6210 -0.5154 -0.5192
1.3165 0.8945 0.5455 1.5550 -1.0095 -0.6605
0.7978 1.8901 1.2502 1.6490 -0.3988 0.2411
0.4071 -0.0489 -1.0573 1.6180 -2.6753 -1.6669
1.2462 -0.5753 -0.4792 1.8280 -2.3072 -2.4033
-1.6222 0.2991 0.4604 1.6170 -1.1566 -1.3179
-0.5384 0.7825 0.3914 1.7110 -1.3196 -0.9285
-1.5148 -1.1111 -1.5058 2.0060 -3.5118 -3.1171

STOCK RETURNS ANALYSIS 10
1.2682 -0.3907 -0.1865 1.8530 -2.0395 -2.2437
0.4929 0.2321 1.0304 1.8610 -0.8306 -1.6289
-1.3701 0.5159 0.9280 1.6390 -0.7110 -1.1231
-0.2519 -0.9351 -1.4472 2.1640 -3.6112 -3.0991
-1.5234 -1.4411 -0.6421 2.4900 -3.1321 -3.9311
0.7779 0.5389 0.0889 2.5930 -2.5041 -2.0541
-0.5976 -0.0136 -0.4961 2.7510 -3.2471 -2.7646
1.5947 0.1961 0.2632 2.6150 -2.3518 -2.4189
-1.0498 -0.6047 -0.9352 2.5070 -3.4422 -3.1117
-0.1695 0.5569 2.6541 2.7360 -0.0819 -2.1791
-0.0445 0.2482 0.9319 2.9760 -2.0441 -2.7278
-0.7184 -0.0179 0.7160 2.6930 -1.9770 -2.7109
-2.5361 1.1204 0.5427 2.6070 -2.0643 -1.4866
-0.5444 -0.7406 -0.4927 2.7590 -3.2517 -3.4996
2.1444 0.7015 1.0388 2.6060 -1.5672 -1.9045
-0.4350 -0.0143 -1.5077 2.5340 -4.0417 -2.5483
0.4794 0.0728 0.7188 2.5630 -1.8442 -2.4902
0.7205 0.6656 2.7639 2.5560 0.2079 -1.8904
-1.4911 -2.0202 -1.2083 2.3430 -3.5513 -4.3632
-0.2442 0.3315 0.1561 2.5080 -2.3519 -2.1765
-1.0853 -0.2790 0.1001 2.3230 -2.2229 -2.6020
-0.4297 -0.1386 -0.0856 2.1940 -2.2796 -2.3326
-0.3121 -0.2546 0.1358 2.1900 -2.0542 -2.4446
-0.3483 -0.4901 -0.2870 1.6750 -1.9620 -2.1651
-1.6442 -1.3077 -1.4055 2.1220 -3.5275 -3.4297
1.1443 -0.4549 0.3421 1.8680 -1.5259 -2.3229
-0.9641 -0.3973 -0.8258 2.1170 -2.9428 -2.5143
0.9236 1.0864 1.3799 2.2660 -0.8861 -1.1796
1.4060 -0.1009 -0.3119 2.4180 -2.7299 -2.5189
1.4386 0.6912 1.1188 2.2050 -1.0862 -1.5138
0.8079 -0.2274 0.6379 2.2000 -1.5621 -2.4274
-1.9401 -0.8427 -0.0608 2.0600 -2.1208 -2.9027
1.6943 1.8896 1.7395 2.1730 -0.4335 -0.2834
-0.4745 -0.0450 -0.1990 2.2180 -2.4170 -2.2630
-1.0260 -0.4652 0.6909 2.3030 -1.6121 -2.7682

STOCK RETURNS ANALYSIS 11
7. Estimating Capital Assets Pricing Model
a) Model
Table 7: Regression statistics
Regression Statistics
Multiple R 0.8188
R Square 0.6704
Adjusted R Square 0.6651
Standard Error 0.6222
Observations 65
Table 8: Model's ANOVA
ANOVA
Degree of
Freedom SS MS F
Significance
F
Regression 1 49.6041 49.6041 128.1200 0.0000
Residual 63 24.3917 0.3872
Total 64 73.9958
The model is very significant because the p-value is very small (<0.0001).
Table 9: The model
Coefficients
Standard
Error t Stat P-value Lower 95% Upper 95%
Intercept 0.0327 0.2015 0.1620 0.8718 -0.3701 0.4354
Xt 0.9509 0.0840 11.3190 0.0000 0.7830 1.1187
b) Interpretation of the estimated regression coefficients
An increase in the excess market return by one unit improves the IBM returns by 0.9509.
Therefore, there is a positive relationship between the excess market returns and IBM excess returns.

STOCK RETURNS ANALYSIS 12
Also, the predictor variable is significant in the model at 95% confidence level. Holding the value of
excess market return in the model at point zero, the excess returns for IBM stock will be 0.0327.
Therefore, IBM stock is profitable compared to the market returns.
c) Interpretation of R2
67.04% of the variation of excess returns of IBM stock can be explained by using market returns
calculated using S&P 500 as the only predictor. This makes the model significant and it can be used
for the prediction of IBM excel returns5.
d) Interpreting 95% confidence interval for the slope coefficient
Using sample from the same population for the prediction of excess returns on IBM stock, we
are 95% confident that the slope coefficient will lie within [0.7830, 1.1187] confidence interval.
1. Testing whether IBM stock is neutral
Table 10: Testing neutrality of IBM stock
Null Hypothesis Proportion of IBM returns above 0 = 0.5
Alternative hypothesis Proportion of IBM returns above 0 ≠ 0.5
Sample (n) 65
count of IBM returns above 0 42
Proportion of IBM returns
above 0
0.646154
95% Z-score 1.96
standard error 0.062017
Margin of error 0.121554
Confidence Interval
Lower Bound 0.5246
Upper Bound 0.7677
The hypothesized proportion (0.5) is not included in the 95% confidence interval [0.5246,
0.7677], hence rejecting the null hypothesis and concluding that the IBM stock is not neutral. Based
on the confidence interval, the proportion is greater than 0.5.
5 N Draper, Applied Regression Analysis, in, Wiley-Interscience, 2014.

STOCK RETURNS ANALYSIS 13
6. Normality of the distributed error terms
Figure 5: Error terms probability plot6
Figure 5 shown above shows that the dotted points follow the predictive line. This indicates
that the error terms are approximately normally distributed.
6 M Harmon, "Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling
Test, and Shapiro-Wilk Test For Simple Linear Regression", in Blog.excelmasterseries.com, ,
2014, <http://blog.excelmasterseries.com/2014/05/residual-normality-tests-in-excel.html>
[accessed 10 October 2017].

STOCK RETURNS ANALYSIS 14
References
Draper, N, Applied Regression Analysis. in, Wiley-Interscience, 2014.
Harmon, M, "Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling
Test, and Shapiro-Wilk Test For Simple Linear Regression.". in
Blog.excelmasterseries.com, , 2014, <http://blog.excelmasterseries.com/2014/05/residual-
normality-tests-in-excel.html> [accessed 10 October 2017].
Hyndman, R, "6.1 Time series components.” in Otexts.org, , 2017, <https://www.otexts.org/fpp/6/1>
[accessed 10 October 2017].
List, J, A Shaikh, & Y Xu, Multiple Hypothesis Testing in Experimental Economics. in, Cambridge,
Mass., National Bureau of Economic Research, 2016.
Spider Financial, "Normality Test in Excel | Jarque-Bera | Doornick and Hansen | Shapiro-Wilk |
Anderson Darling| NumXL.". in NumXL | Spider Financial, , 2017,
<http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-
tests/normalitytest> [accessed 10 October 2017].