Financial Econometrics_Edmond_Farah

FORECASTING STOCK RETURNS
FINANCIAL ECONOMETRICS
Edmond Farah
Tel: 416-454-8397
edmond.farah1@gmail.com

TABLE OF CONTENTS
Contents
Introduction…………………………………………………………………………………………………………………………………….……….1
Technical Methodology…………………………………………………………………………………………………………………………….1
Overview…………………………………………………………………………………………………………………………………...…………….1
Random Walk One…………………………………………………………………………………………………………………………………….2
RW1 Analysis via Portmanteau Statistics……………………………………………………………………………………..……………3
Random Walk Three………………………………………………………………………………………………………………………...……….5
The Sharpe-Lintner Capital Asset Pricing Model…………………………………………………………………………..……………7
The Wald Test…………………………………………………………………………………………………………………………………….….….7
The Fama-MacBeth Cross-Sectional Regression Approach…………………………………………………………………………8
Conclusion…………………………………………………………………………………………………………………………………………….…..8
Appendix……………………………………………………………………………………………………………………………………….…………10

ECONOMETRICS OF FINANCIAL MARKETS
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Introduction
Financial institutions and investors involved with investments in equity securities would want to know
if asset prices are forecastable. This paper attempts to construct forecasting models for 4 NASDAQ-listed
individual stocks and the SPDR S&P 500 ETF (SPY). We extracted their daily historical adjusted close prices
from the Yahoo Finance database, each from January 1, 2015 to December 31, 2015. The Sharpe-Lintner
Capital Asset Pricing Model is tested using 20 individual NASDAQ-listed stocks to determine arbitrage
opportunities. A thorough cleaning and manipulation of the data sets were done through Microsoft Excel
and imported into statistical programs: GAUSS and Stata in order to conduct the econometric analysis. All
analyses were tested at the 5% significance level.
Technical Methodology
In order to construct forecasting models we must begin our analysis by determining the distributional
characteristics of the daily returns. We initially test for a Random Walk – the theory in which the returns have the
same distribution and are independent of each other. This would imply that historical returns or trends cannot be
used to predict future returns. We thoroughly use three different methods to assess the Random Walk 1 (RW1)
process: Cowles-Jones statistic, autocorrelation coefficients, and Portmanteau statistics. A critical necessary
assumption for the use of the Cowles-Jones statistic is that we require the returns distribution to be statistically
normal. The Jarque-Bera statistic is used to assess for normality. In practicality, we should observe the returns to
have a Random Walk 3 (RW3) process – in which the returns are not independent and identically distributed and
they are not serially correlated. We use the Variance-Ratio statistic to test for RW3. The final part of the paper
will test the Sharpe-Lintner Capital Asset Pricing Model (SL-CAPM). We use a total of 40 stocks to test whether
the SL-CAPM holds. The assessment is done through the Wald-test and the Fama-Macbeth cross-sectional
regression approach. Please see the appendix for the hypothesis and statistical methodology.
Overview
TABLE 1: List of Selected Securities
EXCHANGE NASDAQ COMPANY TICKER MARKET CAP
Region North America Apple Inc. AAPL 537.33B
Facebook, Inc. FB 307.17B
Microsoft Corporation MSFT 405.75B
Alphabet Inc. GOOGL 498.94B
ETF SPDR S&P 500 ETF SPY N/A

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FIGURE 1: SPDR S&P 500 ETF vs. Apple and Facebook Log Returns for 2015
FIGURE 2: SPDR S&P 500 ETF vs. Microsoft and Alphabet Inc. Log Returns for 2015
As per FIGURE 1, the returns of Facebook and Apple have exhibited a greater magnitude of volatility relative to
those of Microsoft and Alphabet Inc. The SPDR S&P 500 ETF returns have remained relatively stable within the
band of -5% to +5% throughout 2015. All security returns fell more than -5% in mid-August.
Random Walk One
This section tests whether the log returns are independent and identically distributed over time and there are
uncorrelated. Furthermore, the variance and mean are the same over time. Hence, the returns are random.
Please see the Appendix section for how the following assessments are tested statistically.
-8.00%
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
SPY Apple Facebook
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
SPY Microsoft Alphabet Inc. (Google)

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TABLE 2: Descriptive Statistics and Normality Assessment
SECURITY
APPL FB MSFT GOOGL SPY
Mean (0.00008) 0.00115 0.00079 0.00153 0.00005
Variance 0.00028 0.00026 0.00031 0.00042 0.00010
Standard Deviation 0.01687 0.01621 0.01769 0.01774 0.00975
Skewness (0.08196) (0.16955) 0.49339 2.58925 (0.28857
Kurtosis 4.29001 3.76078 12.93871 22.87556 5.10832
Jarque-Bera 17.6851 7.2558 1043.2375 4411.8983 49.9709
p-value 0.0001 0.0266 0.0000 0.0000 0.0000
To test for normality, we use the Jarque-Bera (JB) statistic. It is asymptotically chi-squared distributed with 2
degrees of freedom. This implies that the JB critical value is 5.99. If the JB statistics are greater than 5.99 then we
do not have a normal distribution. The test is conducted at the 5% significance level and we observe that none of
the securities are statistically significant. However, due to the small sample size we see that Facebook’s JB
statistic was closest to the critical value so we would like to analyze its returns further.
TABLE 3: Cowles-Jones Test for Facebook, Inc. (FB)
SECRUITY FACEBOOK, INC. (FB)
Mean 0.00115
Standard Deviation 0.01621
Probability of Sequence (𝝅̂) 0.52825
Probability of Reversal (𝝅̂ 𝒔) 0.50160
Standard Deviation of 𝑪𝑱̂ 0.50037
Cowles-Jones (𝑪𝑱̂)) 1.01
Z-Statistic 0.01
The Cowles-Jones method tests for the RW1 process but requires the asset to have a normal distribution. We see
that Facebook’s Cowles-Jones statistic is 1.01 which is not equal to 1, therefore, there is no RW1 process.
RW1 Analysis via Portmanteau Statistics
TABLE 4: Apple Inc. (AAPL) RW1 Results
k/m 1 2 3 4 5 6 7 8 9 10
𝝆(𝒌) (0.031) (0.072) (0.014) (0.078) (0.008) 0.000 0.055 (0.001) 0.052 0.007
Z-Stat (0.502) (1.142) (0.228) (1.236) (0.119) 0.006 0.869 (0.018) 0.826 0.111
𝑸(𝒎) 0.252 1.556 1.608 3.136 3.150 3.150 3.907 3.907 4.588 4.600
p-value 0.616 0.459 0.658 0.535 0.677 0.789 0.791 0.865 0.869 0.916
Every Portmanteau statistic is statistically insignificant at the 5% significance level. We fail to reject the null
hypothesis and we have a RW1 process. The prediction model: 𝑟𝑡+1 = 𝜇 + 𝜀𝑡+1.

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TABLE 5: Facebook, Inc. (FB) RW1 Analysis
k/m 1 2 3 4 5 6 7 8 9 10
𝝆(𝒌) 0.129 (0.118) (0.194) (0.118) 0.117 (0.031) (0.042) 0.008 0.016 0.046
Z-Stat 2.055 (1.864) (3.076) (1.872) 1.857 (0.489) (0.662) 0.119 0.256 0.730
𝑸(𝒎) 4.222 7.696 17.160 20.666 24.114 24.353 24.791 24.805 24.871 25.404
p-value 0.040 0.021 0.001 0.000 0.000 0.000 0.001 0.002 0.003 0.005
The last statistically significant Portmanteau statistic is the 9th
daily lag. This implies that we reject the null
hypothesis and we do not have a RW1 process. The prediction model: 𝑟𝑡+1 = 𝜇 + 𝜌1 𝑟𝑡 + ⋯ + 𝜌10 𝑟𝑡−9 + 𝜀𝑡+1.
TABLE 6: Microsoft Corp. (MSFT) RW1 Analysis
k/m 1 2 3 4 5 6 7 8 9 10
𝝆(𝒌) 0.073 (0.005) (0.026) (0.082) (0.029) (0.063) 0.004 (0.002) (0.004) 0.065
Z-Stat 1.156 (0.073) (0.411) (1.291) (0.454) (1.001) 0.059 (0.035) (0.068) 1.033
𝑸(𝒎) 1.336 1.341 1.509 3.175 3.382 4.383 4.387 4.388 4.393 5.460
p-value 0.248 0.511 0.680 0.529 0.641 0.625 0.734 0.822 0.884 0.858
TABLE 7: Alphabet Inc. (GOOGL) RW1 Analysis
k/m 1 2 3 4 5 6 7 8 9 10
𝝆(k) 0.113 (0.053) (0.094) (0.056) (0.023) 0.003 (0.066) (0.038) (0.029) (0.008)
Z-Stat 1.785 (0.831) (1.481) (0.884) (0.356) 0.043 (1.042) (0.600) (0.474) (0.131)
𝑸(𝒎) 3.184 3.876 6.068 6.849 6.976 6.978 8.062 8.423 8.647 8.664
p-value 0.074 0.144 0.108 0.144 0.222 0.323 0.327 0.393 0.471 0.564
TABLE 8: SPDR S&P 500 ETF (SPY) RW1 Analysis
k/m 1 2 3 4 5 6 7 8 9 10
𝝆(k) 0.057 (0.099) (0.059) (0.168) 0.008 0.027 (0.041) 0.037 (0.029) (0.072)
Z-Stat 0.904 (1.581) (0.929) (2.656) 0.132 0.428 (0.656) 0.582 (0.474) (1.137)
𝑸(𝒎) 0.817 3.316 4.179 11.231 11.249 11.432 11.863 12.202 12.426 13.719
p-value 0.366 0.191 0.243 0.024 0.047 0.076 0.105 0.142 0.190 0.186
The last statistically significant Portmanteau statistic is the 5th
daily lag. This implies that we reject the null
hypothesis and we do not have a RW1 process. The prediction model: 𝑟𝑡+1 = 𝜇 + 𝜌1 𝑟𝑡 + ⋯ + 𝜌5 𝑟𝑡−4 + 𝜀𝑡+1.

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Random Walk Three
RW3 is a more general test than that of RW1 and RW2 because it does not assume identically or independent normally distributed asset returns.
A market is said to be efficient when all past and current information has been absorbed by the current price of the asset. Therefore, rejecting
RW3 implies the market is inefficient for that particular asset and past information can be used to forecast future prices of the asset. To test for
RW3 use the Variance-Ratio (VR) statistic. Initially, we calculated the 10 lags for each security and by looking at the PSI values I am able to
determine the amount of lags, if any, are used in the model. The VR statistic is standardized through the PSI-statistic to be compared with the
critical value of a normal distribution. If any of the PSI values are greater than 1.96, then the specified lag is included in the model. The RW3 test
was conducted at the 5% significance level for all securities. See the Appendix for methodology.
TABLE 9: Apple Inc. (AAPL) RW3 Analysis
k 1 2 3 4 5 6 7 8 9 10
𝝆(𝒌) (0.031) (0.072) (0.014) (0.078) (0.008) 0.000 0.055 (0.001) 0.052 0.007
q 2 3 4 5 6 7 8 9 10 11
VR(q): 1.0571 1.010 0.957 0.858 0.795 0.757 0.719 0.697 0.665 0.626
Theta(q) 2.290 5.139 8.101 11.047 13.858 17.091 19.008 21.419 23.743 25.988
PSI(q) 0.597 0.067 (0.242) (0.679) (0.875) (0.931) (1.022) (1.037) (1.090) (1.162)
Past information cannot be used to forecast log returns. The market for this stock is efficient. We have a RW3 model: 𝑟𝑡+1 = 𝜇 + 𝜀𝑡+1.
Table 10: Facebook, Inc. (FB) RW3 Analysis
k 1 2 3 4 5 6 7 8 9 10
𝝆(𝒌) 0.129 (0.118) (0.194) (0.118) 0.117 (0.031) (0.042) 0.008 0.016 0.046
q 2 3 4 5 6 7 8 9 10 11
VR(q): 1.130 1.095 0.979 0.864 0.825 0.789 0.751 0.724 0.711 0.709
Theta(q) 1.799 3.813 5.750 7.632 9.474 11.745 12.962 14.598 16.173 17.693
PSI(q) 1.532 0.767 (0.133) (0.782) (0.899) (0.974) (1.093) (1.144) (1.140) (1.096)

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TABLE 11: Microsoft Corp. (MSFT) RW3 Analysis
k 1 2 3 4 5 6 7 8 9 10
𝝆(k) 0.073 (0.005) (0.026) (0.082) (0.029) (0.063) 0.004 (0.002) (0.004) 0.065
q 2 3 4 5 6 7 8 9 10 11
VR(q): 1.073 1.094 1.092 1.058 1.026 0.985 0.955 0.931 0.925 0.932
Theta(q) 1.674 3.466 5.294 7.179 8.992 10.969 12.236 13.737 15.174 16.550
PSI(q) 0.893 0.802 0.633 0.342 0.136 (0.074) (0.205) (0.295) (0.305) (0.264)
TABLE 12: Alphabet Inc. (GOOGL) RW3 Analysis
k 1 2 3 4 5 6 7 8 9 10
𝝆(k) 0.113 (0.053) (0.094) (0.056) (0.023) 0.003 (0.066) (0.038) (0.029) (0.008)
q 2 3 4 5 6 7 8 9 10 11
VR(q): 1.113 1.115 1.070 1.020 0.980 0.951 0.914 0.876 0.844 0.820
Theta(q) 1.539 3.081 4.644 6.437 8.412 10.923 12.269 14.035 15.688 17.231
PSI(q) 1.439 1.040 0.513 0.130 (0.111) (0.233) (0.390) (0.524) (0.622) (0.699)
TABLE 13: SPDR S&P 500 ETF (SPY) RW3 Analysis
k 1 2 3 4 5 6 7 8 9 10
𝝆(k) 0.057 (0.099) (0.059) (0.168) 0.008 0.027 (0.041) 0.037 (0.029) (0.072)
q 2 3 4 5 6 7 8 9 10 11
VR(q): 1.057 1.009 0.957 0.858 0.795 0.757 0.719 0.697 0.665 0.626
Theta(q) 2.290 5.140 8.101 11.047 13.858 17.091 19.001 21.419 23.743 25.988
PSI(q) 0.597 0.067 (0.242) (0.679) (0.875) (0.931) (1.022) (1.037) (1.088) (1.162)

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The Sharpe-Lintner Capital Asset Pricing Model
There are three methods that can be used to arrive at the estimates for this model. The first one is through
Ordinary Least Squares (OLS), second is the Maximum Likelihood (ML), and lastly, Generalized Method of
Moments (GMM). OLS and ML assume that the data is identically and independently normally distributed but
their standard errors are smaller than that of GMM. However, all three methods provide the same point
estimates. Any method can be used as long as standard errors are not necessary. To test the CAPM, an estimate
of the alpha (intercept) values in the regression must all equal to zero simultaneously for each stock. When alpha
is not equal to zero then the CAPM does not hold for a given portfolio and cannot be used for forecasting. Testing
for 𝛼𝑖 is where OLS presents a problem because OLS estimates the values of alpha separately for each security,
whereas ML estimates them simultaneously. Therefore, if the OLS method is used, all the estimated alpha’s must
be placed into a vector and then I can test the CAPM model.
We assess the CAPM with a total of 40 (20 large cap and 20 small cap) stocks and using the S&P 500 ETF as the
proxy for the market portfolio. None of the stocks have normally distributed returns so we use GMM to estimate
their intercepts simultaneously. By looking at p-values, FIGURE 3 exhibits all the intercepts do not equal zero
simultaneously.
FIGURE 3: CAPM Individual Stock Intercept Significance
The Wald Test
To Test the CAPM, we use the Wald test for finite samples to see if the CAPM holds jointly for our set of 40 stocks.
We estimate the CAPM model 40 times using OLS by regressing historical net returns for each stock against those

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of the SPDR S&P 500 ETF. For each regression we store the intercepts and residuals. After constructing the 20x1
column vector of intercepts, we use the 𝐽1 statistic through the Wald test to determine if the CAPM holds jointly
for our set of stocks. We Please see Appendix for statistically background of the analysis. The p-value result for
the 𝐽1 statistic is 0.316937 which is far greater than 0.05 - the 5% significance level. Therefore, we reject the null
hypothesis in which the CAPM does not hold.
The Fama-MacBeth Cross-Sectional Regression Approach
The next two tests are done through the Fama-Macbeth (FM) approach. The FM approach is used for running
cross-sectional regressions and thus testing whether CAPM holds and whether or not the stock betas explain the
cross-sectional variation in the expected excess returns on the stocks. This is quite useful because it also enables
us to determine whether other factors, such as macroeconomic factors, are needed for this model to hold or
whether only the proxy for the market portfolio is sufficient. Please see Appendix for statistical methodology.
We construct the 95% confidence interval using the adjusted R-squared values from the FM approach. After
obtained the adjusted R-squared values, I organized them in ascending order and eliminated the first and last 6
observations to construct the 95% confidence interval. I thus obtained the following confidence interval:
[0.000002, 0.329947]. This allows us to compare various models. This result shows that this CAPM with only 1
factor, would have an adjusted R-squared that falls within this interval. It can be seen that the interval is close to
zero and thus infers that this specific model is not very good. Empirical tests however have shown that by
including other factors, up until 5 factors, the model betters and thus should be used instead of a single-factor
model.
TABLE 14: Fama-Macbeth Cross-Sectional Regression
Statistics Point Estimate t-Statistic p-value 𝑹 𝟐
95% Confidence Interval
𝜸 𝟎 (0.000765) (0.610515) 0.542075 [0.000002, 0.329947]
𝜸 𝟏 0.001173 0.797391 0.425980
Conclusion
Initially, we tested if log returns of all 5 securities follow a RW3 model. We also proved that
RW1 is a special and unrealistic case of RW3. If returns on a security follow a RW3 then the market for
this security is efficient. Furthermore, the returns of this security are independently and identically
distributed. If the returns on a particular security does not follow a RW3 then an AR(p) model is plausible.
In this case the market for this security is not efficient and past information can be used to forecast the
returns.

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We then introduced Sharpe-Lintner Capital Asset Pricing Model (CAPM) and how to empirically test it. The
CAPM attempts to capture contemporaneous information. It is observed that asset prices usually move in the
same direction therefore it’s reasonable to assume that there are systematic common risk factors that influence
the prices of all securities. The excess net returns on the market portfolio are meant to capture the systematic
common risk factors. We also explained why such a portfolio is unobservable and that is why we use a proxy for
the market portfolio. Furthermore, we explained we can only determine CAPM holds for a set of stocks if we run
3 different tests: the estimated intercepts should jointly be equal to zero, the stocks betas’ coefficient in the
Fama-MacBeth regression should equal the mean return of market portfolio proxy, and the intercept of the
Fama-MacBeth regression should equal to zero. Only then can we state that CAPM holds for our particular set of
stocks for the specific time period. We constructed a 95%confidence interval of the adjusted R^2 to analyze how
well the CAPM can explain the excess net asset returns. We explained that the low confidence interval is intuitive
since CAPM is a single factor model.
We discussed that due to the large trading volume of large cap stocks, it is most likely that large cap stock
markets are efficient. This should not be the case with small cap stocks due to the low trading volume. We
analyzed our results when we tested for RW3 and when we tested if CAPM holds individually for each
stock. We did not find observe the size effect in the RW3 and CAPM results. Prior to testing CAPM individually for
each stock, we explained when to use OLS and when to use GMM. In order to use OLS we must check if the
assumptions of OLS are satisfied: returns on the security must be normally, independently, and identically
distributed. If this is the case, we must use OLS because OLS is more efficient than GMM. For all our asset returns
we tested for normality; if the returns were normally distributed we checked if the returns follow a RW3. By
testing for RW3 we are determining if returns are independently and identically distributed.
We also discussed that when we tested for RW3 we are only used a subset of past information. The past
information we considered were historical asset prices of the stock we are trying to forecast. We can use a
bigger subset of past information; for example, we can use historical prices of other assets or historical
information on macroeconomic variables. We also discussed that CAPM is a single factor model; we explained
that we could build on CAPM and generate a multifactor model. However, we also explained that we should
keep the excess net returns on the market portfolio as an explanatory variable because its inclusion was not
arbitrary. When we generate a multifactor model we should limit our explanatory factors to 5 or 6 factors;
otherwise, we are running the risk of data snooping. In the case of data snooping, our model will be able to
explain historical phenomena perfectly at the expense of poor forecasts.

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Appendix
Jarque-Bera statistic and its components:
Jarque-Bera statistic: 𝐽𝐵 =
𝑇
6
(𝑆̂2
+
(𝐾̂−3)2
4
) ~𝜒(2)
2
(1)
where
Sample Mean: 𝜇̂ ≡
1
𝑇
∑ 𝑧𝑡
𝑇
𝑡=1 (2)
Sample Variance: 𝜎̂2
≡
1
𝑇−1
∑ (𝑧𝑡 − 𝜇̂)2𝑇
𝑡=1
(3)
Sample Skewness: 𝑆̂(𝑧) ≡
1
(𝑇−1)𝜎̂3
∑ (𝑧𝑡 − 𝜇̂)3
~𝑇
𝑡=1 𝑁(0,
6
𝑇
) (4)
Sample Kurtosis: 𝐾̂(𝑧) ≡
1
(𝑇−1)𝜎̂4
∑ (𝑧𝑡 − 𝜇̂)4
~𝑇
𝑡=1 𝑁 (3,
24
𝑇
) (5)
To have a normal distribution the skewness which is the measure of asymmetry of the probability distribution
and kurtosis – the degree of peakedness of a distribution should be 0 and 3, respectively.
The Jarque-Bera statistic has a asymptotically chi-squared distribution with 2 degrees of freedom and the null and
alternative hypothesis for normality testing are:
𝐻0: JB = 0 ⇒ (𝑟𝑡~𝑁) and 𝐻1: 𝐽𝐵 ≠ 0 ⇒ (𝑟𝑡 ≇ 𝑁)
Under the alternative hypothesis, either skewness is not equal to 0 or kurtosis is not equal to 3.
The null hypothesis for normality is then rejected if the calculated test statistic exceeds a critical value from the
chi-squared distribution with two degrees of freedom which is 5.99. Once the daily gross and log returns for each
asset underwent this analysis at the 5% significance level and selection process, only the ones with a suspected
normal distribution were chosen to undergo an analysis of its respective Cowles Jones statistic for Random Walk
1. The Random Walk 1 model is:
Random Walk 1: 𝑟𝑡 = 𝜇 + 𝜖 𝑡 (6)
where 𝑟𝑡 = 𝑃𝑡 − 𝑃𝑡−1
(7)
𝑟𝑡 ~ IID(𝜇, 𝜎2
) and 𝜖 𝑡~IID(0, 𝜎2
) (8)
Statement (8) implies that the distribution of log return of the asset is independent and identically distributed
over time which means if you take the time series of daily returns they then will be independent of each other
and their correlation will be equal to zero, and their variance is the same over time with the same mean.

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Ultimately meaning that 𝑟𝑡 is completely random. And if that’s the case for returns, then it would be assumed to
be true for 𝜖 𝑡. The Cowles-Jones statistic is then used at the 5% significance level to determine if there is a
Random Walk 1 process and is presented here as:
Cowles-Jones statistic: CJ =
𝜋 𝑆
1−𝜋 𝑆
=
𝜋2+(1−𝜋)2
2𝜋(1−𝜋)
(9)
𝐶𝐽̂ ~ 𝑁 (
𝜋 𝑆
1−𝜋 𝑆
,
𝜋 𝑠(1−𝜋 𝑠)+2(𝜋3+(1−𝜋)3−𝜋 𝑠
2)
𝑛(1−𝜋 𝑠)4 ) (Asymptotically) (10)
𝜋̂ = 𝜙 (
𝜇̂
𝜎̂
) and n = T - 1 (11)
The Cowles-Jones statistic is the probability of a sequence (𝜋 𝑆) divided by the probability of a reversal (1-𝜋 𝑆). The
Cowles-Jones statistic is now incorporated into the following test:
𝐻0: =𝐶𝐽̂=1 ⇒ (Random Walk 1) and 𝐻1: 𝐶𝐽̂ ≠ 1 ⇒ (Not a Random Walk 1)
The analysis for Random Walk 1 continued further with two more tests. Based off the analysis of the individual
autocorrelation coefficients of order 1-10 on the 5 securities and also by using the Portmanteau statistic. Through
the method of looking at the autocorrelation coefficients, the ones that are significant (Z-stat > +/- 1.96) by
looking at their corresponding Z-statistic at the 5% significance level then we would reject the null hypothesis
that there exists a RW1 process. If there is at least one autocorrelation coefficient that is not equal to zero and
their Z-stat is less than +/- 1.96 (not significant) then we would reject the null hypothesis and we do not have a
RW1 process and the model would be constructed incorporating lags that are up to the last significant lag.
Rho (Autocorrelation coefficient): 𝜌̂ =
𝐶𝑜𝑣(𝑟𝑡,𝑟 𝑡+𝑘)
𝑆𝐷(𝑟𝑡)x 𝑆𝐷(𝑟 𝑡+𝑘)
k = 1…10 (12)
z-stat = 𝜌̂(𝑘)/ √1
𝑇⁄
(13)
𝐻0: 𝜌̂(𝑘)= 0 and 𝐻1: 𝜌̂(𝑘) ≠ 0 for k = 1...10
Furthermore, the Portmanteau statistic (𝑄(𝑚) is then used to assess if there exists a RW1 process. If one lag (in
set m) is not significant (p-value > 0.05) then we would fail to reject the null hypothesis and we would have a
RW1 process. However, for one lag (in set m) is significant (p-value < 0.05) then we would reject the null
hypothesis in favor of the alternative such that the does not exist a RW1 process.
The statistic and hypothesis is as follows:
𝑄(𝑚) = 𝑇 ∑ 𝜌(𝑘)210
𝑘=1 (Large sample formula) (14)
where 𝐻0: 𝑄(𝑚)= 0 ⇒ RW1 and 𝐻1: 𝑄(𝑚) ≠ 0 ⇒ No RW1 for m = 1...10 (15)

12 | P a g e
If any of the Portmanteau statistics become significant then we will have to reject the Random Walk 1.
Since we are working with daily data, the daily risk free net return is negligible and almost equal to zero. In
this case we only calculated the real net returns for all securities.
Using the adjusted close prices of the 20 large and 20 small cap stocks alongside the prices for the S&P 500 ETF
(SPDR) we tested for RW3 initially. The hypothesis of a random walk entails that continuously compounded returns
𝑟𝑡 are uncorrelated at all leads and lags. We test the random walk hypothesis by testing whether the null hypothesis
that the autocorrelation coefficients at various lags are all zero. RW3 on the other hand does not have the returns
independent and identically distributed. To test for RW3 we use the Variance-Ratio (VR) statistic. The statistic’s
formations are below.
𝑉𝑅̂ (𝑞) = 1 + ∑ 2 (1 −
𝑘
𝑞
) 𝜌̂(𝑘)
𝑞−1
𝑘=1 (asymptotically) (16)
Where the VR statistic is asymptotically normally distributed with mean 1 and variance 𝜃̂(𝑞)/𝑛𝑞 under the null.
To assess whether this statistic is significant or not we must calculate the standardized test statistic 𝜑∗
(q). To
arrive at its derivation we must first determine:
𝛿̂(𝑘) =
𝑛𝑞 ∑ (𝑟 𝑗−𝑟̅)2(𝑟 𝑗−𝑘−𝑟̅)2𝑛𝑞
𝑗=𝑘+1
[∑ (𝑟 𝑗−𝑟̅)2𝑛𝑞
𝑗=1
]
2 and 𝜃̂(𝑞) = ∑ [2(1 − 𝑘 𝑞)⁄ ]2𝑛𝑞
𝑘=1 𝛿̂(𝑘) (17)
where the standardized test statistic is:
𝜑∗(𝑞) = √ 𝑛𝑞(𝑉𝑅̂ (𝑞)−1)
√𝜃
~ 𝑁(0,1) (asymptotically) (18)
Under the hypothesis that:
𝐻0: 𝑉𝑅 = 1 (𝑅𝑊3 𝑒𝑥𝑖𝑠𝑡𝑠) vs. 𝐻1: 𝑉𝑅 ≠ 1 (𝑅𝑊3 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡) (for any q)
The importance of testing for RW3 is when the analysis and significance testing for RW3 is complete we assess
whether the returns are normally distributed and follow RW3. If they do, then OLS assumptions are satisfied and
we must use OLS to create point estimates for the SL-CAPM. Then we test whether we reject or fail to reject that
the SL-CAPM holds for large and small cap stocks and compare the frequency to determine if there is a size effect.
The size effect is when we would reject the SL-CAPM for small cap stocks more times than for large cap stocks.
The Sharpe-Lintner Capital Asset Pricing Model
This model incorporates the existence of a risk-free asset. The function form of the model is:
𝐸[𝑅𝑖] = 𝑅𝑓 + 𝛽𝑖 𝐸[𝑅 𝑚 − 𝑅𝑓]
(19)

13 | P a g e
Where 𝐸[𝑅𝑖] is the net return on asset i, 𝑅 𝑚 being the excess return of the market portfolio and 𝑅𝑓 being the
risk-free rate of return. This is a 2-period model in which this should hold for any “t”. To test the model on
particular stocks we let 𝐸[𝑅𝑖𝑡 − 𝑅𝑓𝑡] = 𝑍𝑖𝑡 and [𝑅 𝑚𝑡 − 𝑅𝑓𝑡] = 𝑍 𝑚𝑡, where 𝑍𝑖𝑡 is the excess return of asset i.
Putting those two together we get relation (20):
𝐸[𝑍𝑖𝑡] = 𝛽𝑖 𝐸[𝑍 𝑚𝑡]
(20)
The idea of this is to see the expected return of any asset should be on the straight line of the CAPM. To test the
CAPM we estimate equation (21) under the null hypothesis that the CAPM holds (which means the intercept ∝1
must be equal to zero) for a particular stock. Once its equal to zero the CAPM holds.
𝑍𝑖𝑡 = ∝𝑖+ 𝛽𝑖 𝑍 𝑚𝑡 + 𝜀𝑡
(21)
FIGURE 4: SL-CAPM and The Security Market Line
FIGURE 4 shows that at point (a) the equilibrium price is equal to what the price is for the current time period, so
there is no opportunity for arbitrage in this case. However, at point (b) we see that the price in the current period
is lower than what it should be in equilibrium (on the line), but in a few seconds it will revert back to equilibrium.
The idea would be to purchase the asset and capitalize on the arbitrage opportunity. The null hypothesis for
testing for the CAPM is stated below:
𝐻0: ∝𝑖 = 0 (𝐶𝐴𝑃𝑀 ℎ𝑜𝑙𝑑𝑠) vs. 𝐻1: ∝𝑖 ≠ 0 (𝐶𝐴𝑃𝑀 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 ℎ𝑜𝑙𝑑)

14 | P a g e
if ∝𝑖 > 0, the best choice would be to buy and if ∝𝑖 < 0 then the decision should be to sell. With the data collected
on the 20 large cap and 20 small cap stocks we test if all intercepts are equal to zero.
The Wald Test
Essentially we will run the estimates of the ∝𝑖 ′𝑠 through the vector ∝ simultaneously for all stocks to see if all of
them are equal to zero. If they are all equal to zero, then the CAPM holds for our particular set of stocks (20 large
cap and 20 small cap). Moreover, we allow the market portfolio to be the S&P 500 ETF (SPDR). We use the Wald
test through the 𝐽0 and 𝐽1 statistics what are defined as:
𝐽0 = 𝑇 [1 +
𝜇̂ 𝑚
2
𝜎̂ 𝑚
2 ]
−1
∝̂ ′∑−1
∝̂ and 𝐽1 =
(𝑇−𝑁−1)
𝑁
𝑇 [1 +
𝜇̂ 𝑚
2
𝜎̂ 𝑚
2 ]
−1
∝̂ ′∑−1
∝̂ (22)
Under the null hypothesis for the 𝐽0 statistic; 𝐽0 is asymptotically chi-squared distributed with N degrees of
freedom. The 𝐽1 statistic (for finite samples) under the null hypothesis is distributed F(N, T-N-1). In this model we
have N=40 and 210 (251 – 40 – 1) degrees of freedom and at the 5% significance level.
We then converted the F-statistic to its corresponding p-value and if its greater than 0.05 we then fail to reject
the null hypothesis that the ∝𝑖 ′𝑠 equal to zero, so the CAPM hold for this particular set of stocks.
The hypothesis for testing if the intercepts are equal to zero through the Wald test is:
𝐻0: ∝ = 0 (𝐶𝐴𝑃𝑀 ℎ𝑜𝑙𝑑𝑠) vs. 𝐻1: ∝ ≠ 0 (𝐶𝐴𝑃𝑀 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 ℎ𝑜𝑙𝑑)
The Size Effect
The size effect is when we reject the CAPM for small cap stocks far more times than for large cap stocks.
The intercepts (∝ ′𝑠) for each stock is tested to see if they are statistically significant at the 5% significance level.
Based on the interpretation of the results of the intercepts we can then determine if we reject or fail to reject the
null hypothesis for the intercept to be equal to zero (CAPM holds). To estimate the CAPM for each stock we must
first accurately look at how the returns are distributed. This would give us the direction in which estimator we
should use get the point estimate and its corresponding standard error. If the net returns are not serially
correlated (RW3 holds) such that the returns are IID and moreover, if the net returns are normally distributed
then the ordinary least squared (OLS) assumptions are satisfied and we must use OLS to estimate. However, the
OLS is only used for one particular stock at a time. To run them simultaneously in a vector then we will use the
maximum likelihood (ML) method. If they are not normally distributed and not IID (RW3 fails to hold) then we
must use general method of moments (GMM). The difference will be in the standard errors as they will be larger
under GMM than in OLS and ML. However, the point estimates will be the same. So we would reject the null less
often under the GMM because of the standard errors. One thing to note, it is prohibited to use OLS and ML with
not normally distributed and IID returns. The orthogonality conditions for a portfolio of N assets are:

15 | P a g e
𝐸[𝜀𝑡] = 0 and 𝐸[𝑍 𝑚𝑡, 𝜀𝑡] = 0
(23)
The Fama-MacBeth cross-sectional regression approach
The Fama-MacBeth cross-sectional approach to test whether the betas of the previously mentioned stocks
explain the cross-sectional variation in the expected excess returns. The CAPM is transformed from equation (24)
to (25).
𝐸[𝑍𝑡] = ∝ + 𝛽 𝑚 𝐸[𝑍 𝑚𝑡] + 𝜀𝑡
(24)
𝒁𝑡 = 𝛾0𝑡 + 𝛾1𝑡 𝜷 𝑚 + 𝜂 𝑡 (25)
Where 𝒁𝑡 is a vector of excess returns and 𝜷 𝑚 is the vector of betas that were previously estimated. In the model
(25), 𝛾1is the estimator of the expected value of market portfolio. It is required that it should be positive as the
beta’s will by positive on average from the co-movement of the market portfolio given the stock. Hence, a null
hypothesis is that 𝛾0 = 0 versus the alternative that 𝛾0 ≠ 0. The second test that must occur is that the null
hypothesis: 𝛾1 = 0 against the alternative that 𝛾1 > 0. If both null hypotheses hold (jointly not rejecting the null)
then the CAPM holds. Otherwise if they don’t jointly hold, then we reject the hypothesis that the CAPM holds. To
see whether the 𝛾0 and 𝛾1 are significant we use the t-statistics. The value of ∝ is given by 𝛾0 and 𝛾1is in terms
of CAPM: expected return of mean value 𝑍 𝑚𝑡.
𝛾0 =
1
𝑇
∑ 𝛾0,𝑡
𝑇
𝑡=1 and 𝛾1 =
1
𝑇
∑ 𝛾1,𝑡
𝑇
𝑡=1
(26)
𝑡𝛾̂𝑗 =
𝛾̂0−𝜇
𝜎 𝛾 𝑗̂
=
𝑇−1 ∑ 𝛾̂ 𝑗𝑡
𝑇
𝑡=1
[
1
(𝑇−1)𝑇
∑ (𝛾̂ 𝑗𝑡−𝛾̂ 𝑗)
2𝑇
𝑡=1 ]
1/2
(27)
GAUSS Coding Sample
GMM Code
library optmum
optset;
#include optmum.ext;
T=N;
iteration=1e+5;
precision=1e-5;
_opstep = 3;
_opgtol = 1e-5;
_opmiter= 1e+5;

16 | P a g e
_print=1;
_mmg_fonction=zeros(T,param);
_mmg_covariance=zeros(param,param);
mmg_lags=2;
proc moments(beta);
local i,M,Hb,T;
T=N;
M=zeros(T,2);
M[.,1]=dr[.,v]-beta[1]-beta[2]*dr[.,41]; /* e(t)=[Ri(t)-Rf(t)]-alpha
beta*[Rm(t)-Rf(t)] */ M[.,2]=M[.,1].*dr[.,41]; /*
e(t)*[Rm(t)-Rf(t)] */
retp(M);
endp;
/*---------------------------------------------------------------------*/
proc _moyenne(beta);
local fonction,H,G;
fonction=_mmg_fonction;
local fonction:proc;
H=fonction(beta);
G=meanc(H);
retp(G);
endp;
/*--------------------------------------------------------------------*/
proc _mmgf(beta);
local G,S;
G=_moyenne(beta);
S=_mmg_covariance;
retp(G'*invpd(S)*G);
endp;
/*--------------------------------------------------------------------*/
proc (6) = moments_generalises(f,mmg_lags,debut);
local f:proc;
local T,r,k,beta0,S,beta,Qmin,deriv,retcode;
local i,j,_Gamma,H,H_lags,D,ddl;
local covbeta,stderr,tstudent,pvalue,test,ptest;
local corbeta,name,mask,fmt,omat;
debut = 1.0|1.0;
T=rows(f(debut)); r=cols(f(debut)); k=rows(debut); ddl=T-k;
_mmg_fonction=&f; beta0=debut;
S=eye(r); _mmg_covariance=S;
i=1;
do until i > iteration;
{beta,Qmin,deriv,retcode}=optmum(&_mmgf,beta0);
if maxc(abs(beta-beta0))<precision;
break;
endif;
H=f(beta);

17 | P a g e
S=H'H/T;
j=1;
do until j>mmg_lags;
H_lags=shiftr(H',j*ones(r,1),0)';
_Gamma=H'H_lags/T;
S=S+(1-j/(mmg_lags+1))*(_Gamma+_Gamma');
j=j+1;
endo;
_mmg_covariance=S;
beta0=beta;
i=i+1;
endo;
D=gradp(&_moyenne,beta);
if det(D'*invpd(S)*D)==0;/*
beta[1]=0;
beta[2]=0;*/
stderr=beta;
tstudent=beta;
pvalue=beta;
covbeta=eye(param);
corbeta=covbeta;
else;
covbeta=inv(D'*invpd(S)*D)/T;
stderr=sqrt(diag(covbeta));
corbeta=covbeta./stderr./stderr';
tstudent=beta./stderr;
pvalue=2*cdftc(abs(tstudent),ddl);
endif;
/*---------------------------------------------------------------------*/
if r>k;
test=T*Qmin;
ptest=cdfchic(test,r-k);
else;
test=miss(0,0);
ptest=miss(0,0);
endif;
/*---------------------------------------------------------------------*/
if _print==1;
name=0 $+ "B" $+ ftocv(seqa(1,1,k),2,0);
mask=0~1~1~1~1;
let fmt[5,3]=
"-*.*s" 7 8
"*.*lf" 14 6
"*.*lf" 14 6
"*.*lf" 14 6
"*.*lf" 14 6;
omat=name~beta~stderr~tstudent~pvalue;
print;

18 | P a g e
print ftos(T,"Observations: %*.*lf",15,0);
print ftos(r,"Orthogonality conditions: %*.*lf", 3,0);
print ftos(Qmin,"Objective function: %*.*lf",15,5);
print ftos(test,"Test statistic (J): %*.*lf",15,5);
print ftos(r-k,"DF: %*.*lf",25,0);
print ftos(ptest,"Significance level: %*.*lf",15,5);
print"---------------------------------------------------------------";
print" Estimates SE t-student Signif.";
print"---------------------------------------------------------------";
call printfm(omat,mask,fmt);
print;
print "Covariance matrix for estimators:";
mask=ones(1,k);
let fmt[1,3]= "*.*lf" 12 6;
call printfm(covbeta,mask,fmt);
endif;
retp(beta,stderr,covbeta,Qmin,test,ptest);
endp;
/*---------------------------------------------------------------------
*/{beta,stderr,covbeta,Qmin,test,ptest}=moments_generalises(&moments,mmg

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