The document discusses tracking error in exchange-traded funds (ETFs) and the use of linear regression to analyze the relationship between an ETF's returns and the returns of its underlying assets. Specifically: 1) Tracking error measures how closely an ETF's performance tracks that of its underlying index or assets, with a lower error indicating better tracking. Linear regression is used to analyze the relationship. 2) The document examines the SPDR Morgan Stanley Technology ETF (MTK) and finds a statistically significant correlation between the expected returns of MTK's top 10 holdings and its actual returns, though regression only indicates correlation, not causation. 3) While ETFs aim to track underlying assets,

1. ETF Tracking Error and Linear Regression Executive Summary 1
1. Tacking Error Description
ETF Fund managers, financial advisors, and introductory investors considering entry into the
stock market are bombarded with different securities to add to their portfolio. The issue
arises when there are many different caveats to a particular product or the price exceeds
allocated costs. ETFs provide a lower cost alternative to traditional investments. An example
is shown using the SPDR Morgan Stanley Technology ETF (MTK) in Appendices A - E.
1.1 Exchange-traded funds (ETFs) are securities that take the nature of a mutual fund and
trades like an equity on the stock market. ETFs are typically used to maximize wealth
creation by maximizing investor diversification in many different markets at a reduced
cost. This is done by linking its performance to an underlying index of securities.
Considering share prices of blue-chip companies and transaction fees and risks associated
with foreign investment, ETF Funds offer product variety to reduce risk in a portfolio and
allow for investors to take advantage of exchange rate and interest rate fluctuations. State
Street Global Advisors SPDR funds are among some of the most notable ETFs.
1.2 Because ETFs do not follow one asset performance, its quality is dependent on its
tracking error, accuracy of performance mirroring that of the underlying index. A higher
tracking error indicates a larger variance on how the underlying securities perform while
a lower error indicates marginal correctness per percentage of return (Hougan, 2014). In
order to reduce the variance between fund performance and underlying securities
performance, an OLS estimator is used and a simple regression is run.
1.3 All ETFs are not equal. ETFs come with different leverage and direction for underlying
securities. Common ETF adjectives describe its path as being long (upside growth of the
underlying) or short/inverse (downside of the underlying), and the leverage (the multiple

2. ETF Tracking Error and Linear Regression Executive Summary 2
of what was borrowed to increase the underlying security shares and track performance).
The path seems to be an important factor when determining ETF performance. While
only dealing specifically with leveraged ETFs, Avellaneda and Zhang (2010) found that
the path had an effect on the fund’s ability to track performance.
1.4 Linear regressions are used to find the link between assets and returns of those assets.
Due to changes in the underlying assets over time, the return of the ETF might not
accurately reflect that of the true return of the underlying portfolio. For this reason, ETFs
are generally not recommended for long-term passive investing (DiLellio, Hesse, &
Stanley, 2014). This revelation implies that even though a forecast predicts a certain
return from an ETF does not mean that it will happen. It also leads to the solution of
periodic portfolio rebalancing, in order to reduce loss on the aggregate portfolio.
2. Linear Regression appropriateness
2.1 Linear regressions have the following assumptions in order for the process to work:
validity, linearity, error independence, error equivalence, and error normality. It is used to
get create a simple relationship based on observed data and the correlation between
points and sets. The problem is the expectation of the linear regression. Due to its heavy
use, it has become the hypothesis testing standard in science and economics.
2.2 Linear regressions are based on the correlations of each data point within the sheet. It
merely shows how variables act in association with another variable. Inherently, linear
regression do not indicate causality, due to the existence of the error term in the
regression equation.
2.3 Similarly to mean distribution analysis, linear regressions feature hypothesis testing. The
testable hypothesis is the correlated coefficient that the regression returns. When

3. ETF Tracking Error and Linear Regression Executive Summary 3
specified to a certain confidence level, the accuracy of the prediction is accepted or
rejected. In addition to the hypothesis test for correlation, regressions also feature a
measure of data fitness, which can often times lead to confusion on whether or not the
model predicts accurately. In the example of MTK’s tracking error (Appendix A – E), all
information is shown.
3. Conclusion. The distribution and linear regression testing agreed with one another. In the
case of the Morgan Stanley Technology ETF, hypothesis to test was whether or not the
proposed hypothetical return from MTK aligned with what was expected. Using a two-tailed
Student’s t distribution and confidence level of 95%, the mean return for MTK was outside
the rejection region, allowing for conclusion of the fund mean has a similar nature to the top
10 underlying securities. This is supported by the regression. Using the quadratic regression,
a change of 1% in expected value correlates with an almost 3% return in the fund. Under the
two tailed 99% confidence level, the probability value of the correlated coefficient was less
than the allocated Type I error, leading to a statistical significance between Expected Return
and MTK’s actual return. However, as aforementioned, the regression only indicates a
correlation and not a causality.

4. ETF Tracking Error and Linear Regression Executive Summary 4
References
Hougan, M. (2014, September 8). The Key Statistic When Evaluating ETFs. Retrieved April 5,
2015, from http://www.etf.com/sections/blog/23214-the-key-statistic-when-evaluating-
etfs.html?nopaging=1
Avellaneda, M., & Zhang, S. (2010). Path-dependence of leveraged ETF returns*. SIAM Journal
on Financial Mathematics, 1(1), 586-603. Retrieved from
http://ezproxy.snhu.edu/login?url=http://search.proquest.com/docview/880106700?accou
ntid=3783
DiLellio, J. A., Hesse, R., & Stanley, D. J. (2014). Portfolio performance with inverse and
leveraged ETFs. Financial Services Review, 23(2), 123-149. Retrieved from
http://ezproxy.snhu.edu/login?url=http://search.proquest.com/docview/1551368971?acco
untid=3783

5. ETF Tracking Error and Linear Regression Executive Summary 5
Appendix
Appendix A
Appendix B
15.00%12.50%10.00%7.50%5.00%2.50%0.00%-2.50%
1800
1600
1400
1200
1000
800
600
400
200
0
Mean 0.0003305
StDev 0.005951
N 3646
Expexted Return
Frequency
Return of Top 10 MTK Holdings distribution (weighted)
Normal
16.00%12.00%8.00%4.00%0.00%-4.00%-8.00%
700
600
500
400
300
200
100
0
Mean 0.0001 775
StDev 0.01 806
N 3646
MTK Return
Frequency
Mean Return of MTK
Normal

6. ETF Tracking Error and Linear Regression Executive Summary 6
Appendix C
Appendix D
70
60
50
40
30
20
10
0
X
Density
-0.01133
0.025
0.01199
0.025
0.0003305
0.00018
Distribution Plot
Normal, Mean=0.0003305, StDev=0.005951
Red Line indicates mean of MTK
R-squared (adjusted) 62.32% 53.08%
P-value, model <0.005* <0.005*
P-value, linear term <0.005* <0.005*
P-value, quadratic term <0.005* —
Residual standard deviation 0.011 0.012
Statistics Quadratic
Selected Model
Linear
Alternative Model
1 5.00%1 0.00%5.00%0.00%-5.00%
1 5.00%
1 0.00%
5.00%
0.00%
-5.00%
-1 0.00%
Expexted Return
MTKReturn
Large residual
Unusual X
Large residual and unusual X
Y: MTK Return
X: Expexted Return
Fitted Line Plot for Quadratic Model
Y = - 0.000175 + 2.659 X - 14.83 X^2
* Statistically significant (p < 0.05)
Regression for MTK Return vs Expexted Return
Model Selection Report
Data from Yahoo! Finance

7. ETF Tracking Error and Linear Regression Executive Summary 7
Appendix E
Appendix F