# Summary of Hypotheses and Tests for all components of strategy

### Background

It was hypothesized by Han, Yang, and Zhou (2013) that high-volatility stocks yield better opportunities for technical analysis, insofar as these stocks exhibit greater uncertainty, due to the embedded uncertainty and behaviour of market participants. They argued that, since investors know little about the high volatility stocks, technical analysis was more likely to be used instead of fundamental analysis. However, Biceroglu (2016b) found that low volatility stocks also perform well. The rationale behind this conclusion is that Moving Average [MA] based indicators potentially contain information about level and slope of the underlying price or index. Assuming that indices or prices are affected by random noise (where the average noise is 0), then using a MA based indicator may reduce noise (as it averages it away), thereby potentially exposing any signals contained in the MA indicator time series.

### Filter

One core hypothesis that was not refuted, and supported in the paper replication of Biceroglu (2016b), is that low volatility portfolios yield superior risk-adjusted performance vs. higher volatility portfolios (Xi Li and Luis Garcia-Feijóo (2016)). Therefore, this filter was applied outright to the investment universe to focus on a portfolio that most likely yields higher risk-adjusted returns.

### Indicator

It is then hypothesized that an indicator based on an Exponentially-Weighed Moving Average [EMA] of index prices may be able to measure the slope of an index trend, as well as the de-noised index level. To test the level hypothesis, correlation between the EMA index and low volatility portfolio level are verified for strength (i.e. a high correlation) as well as statistical significance. To test that the EMA index is capturing slope of the trend, it was compared to a “true” measure of slope, which used a symmetric filter of length equal to the lag-length of the EMA indicator, where half of the data-window contained future index levels. Then, a linear regression was run to extract the slope of the symmetric filter. This was compared to the differences of the EMA index. What was required to adequately conclude that slope was being captured, was high and statistically significant correlation between the regression slope, and EMA slope, as well as co-integration between the two slope measures. Lastly, it was verified that the difference between the two series is not statistically significantly different from zero, to ensure that no information was being lost.

It was also hypothesized that this EMA indicator could contain predictive power about the future level of prices, potentially providing the opportunity for market timing. This is verified using confusion matrices, since the forecasting ability would only be able to assign a direction, and not a magnitude. Forecasting strength is assessed over a 5 day period, since over the long term, risk premia provide returns for holding assets, thus with long window horizon, these risk premia would contaminate the ability to decipher the source of returns, i.e. market timing vs. risk premia.

### Signal

The statistical significance of the confusion matrix is verified to ensure that the matrix contains information different from randomness. The prediction is based on where the index is relative to the EMA level, and what the slope of the EMA index is. Both cases are verified, one with only the comparison of index level to EMA level, and one with both the level comparison, as well as the sign of the EMA slope. Once this is assessed, the Heidke Skill Score [HSS] is verified for its magnitude, as well as its statistical significance, to isolate the lag lengths that contain predictive power, as well as the number of days ahead that the predictability persists. Ultimately, not surprisingly, lower lag lengths have a higher relative HSS over a lower number of days ahead. Using the slope of the EMA index had virtually no impact on predictability; the only factor that was relevant was - where the index level was relative to the EMA level.

### Rules

Having successfully validated that the EMA exhibits predictive ability, and following a similar approach to Han, Yang, and Zhou (2013), the market entry and exit rules were assigned such that a position would be entered when the index level crosses and exceeds the EMA level, and any positions would be exited if the index level crosses and is less than the EMA level. The rationale behind these rules is, if the index level crosses the EMA level such that the index is higher, given that the EMA level contains stale information, it will get updated with the information contained in the new index level, and therefore increase. If the EMA level increases, and it accurately captures the de-noised price level, then this implies that the index level is likely to increase. The converse is thought to apply for a index level decrease.

# Description of Constraints, Benchmarks and Objectives of Strategy

## Objectives

Currently, there are funds invested in registered accounts for retirement. These investments are primarily composed of global low-volatility ETFs in the USA, Canada, EAFE, as well as fixed income ETFs. The ETFs are allocated from a risk-based perspective, since loss-aversion is a main driver of the asset allocation strategy. Therefore, for the purposes of this study, the main business objective is to see if a trading strategy could be added to a portfolio, such that it can replace one of the existing ETF holdings in one or both of the registered accounts (after accounting for transaction costs). In order for the trading strategy to be a viable candidate for the portfolio, it would have yield high risk-adjusted returns, in order to align with the low-volatility portfolio. Thus, maximizing a measure of risk-adjusted performance is the objective of this strategy, since leverage can always be applied to increase absolute returns.

$MAX(return)$

$\; \; \; \; s.t. MIN(risk)$

Given that the objective is to maximize risk-adjusted returns, the following statistics can be used to quantify risk-adjusted returns:

• Sharpe Ratio [SR]
• Profit-to-Max Drawdown [PMD]

Defining the size of the allocation of the moving average trading strategy is outside of the scope of this study, as the risk characteristics would need to be obtained first for the “asset allocation exercise”. This can only be determined if a successful strategy is identified. It is also very likely that a successful trading strategy cannot be identified, therefore this study is the first step in the investigation of the feasibility of adding a trading strategy into the registered portfolio of ETFs.

During the trading simulations, PMD will be used to assess risk-adjusted returns as it is more sensitive to the tails than the SR. However, in order to evaluate the viability of the trading strategy for the retirement portfolio, the SR will be used to assess its viability.

## Constraints

Given that the trading strategy will potentially be executed in a registered account, there are a number of constraints:

• Positions are long only
• There is limited capital ($$\leq$$ $100k CAD) • Transaction costs are$5 per share, for buying and selling (through Questrade)
• It is outside of the scope of this study, however, ETFs are free to purchase, and 5 to sell, thus prior to implementation, a search for an ETF proxy will be conducted to see if similar results are obtained, which will significantly reduce transaction costs • No high frequency trading, therefore holding periods are measured in days • Positions can be held indefinitely • Data must be free: use daily end-of-day prices for listed securities - likely from Yahoo! Finance • All traded instruments are US market instruments listed on both the NYSE and NASDAQ • To be consistent with the study outlined by Han, Yang, and Zhou (2013) ## Benchmarks To be consistent with the results from the Han, Yang, and Zhou (2013) paper replication study by Biceroglu (2016b), where the moving average timing strategy exhibited some success on the lowest volatility decile, and to also be consistent with the proposed objectives (i.e. low volatility), as well as make use of the already well-documented low-volatility anomaly (Xi Li and Luis Garcia-Feijóo (2016)), the lowest volatility decile from Han, Yang, and Zhou (2013) will be used as the benchmark. This portfolio is the ‘buy-and-hold’ [BH] version of the index used in the moving-average trading strategy, which will be outlined in the Strategy Description section. This is the “no effort” portfolio with which to measure if the trading signals are adding value. Success is not defined on absolute returns; it may be possible to observe absolute lower returns, however exhibit higher risk-adjusted returns. Leverage can always be applied to a high risk-adjusted portfolio to increase absolute returns. Lastly, given that the objective is to search for a viable candidate in the low-volatility registered accounts, applying the moving average strategy on the low volatility decile is aligned with the strategy for the portfolios in the registered accounts. # Data Description ### Stock Price Data and Symbol Ticker List Volatility decile portfolios are used, which are consistent with those described from the Center for Research in Security Prices [CRSP] between July 1, 1963 and December 31, 2009 for all stocks on the NYSE and NASDAQ exchanges. In order to determine the universe of stocks to include, a list of security tickers that trade on the NYSE and NASDAQ is required. This was located by reading the relevant post by Marascio (2011) at http://quant.stackexchange.com. As per the discussion by Marascio (2011), the list of stock tickers that trade on NYSA and AMEX was then obtained from http://www.nasdaqtrader.com/trader.aspx?id=symboldirdefs on July 8th, 2016. The file format is a pipe-delimited text file. It contains preferred shares and ETFs, which should be excluded from the analysis. Therefore, the file was imported into Microsoft Excel 2010, and then filtered for tickers that were not ETFs, and where the “ACT Symbol” excluded the “” character, as these were predominantly preferred shares. Lastly, the exchange needed to be part of the NYSE, or NASDAQ, and the only exchange code listed on NASDAQ (2016) which is not a part of these exchanges is “Z”, which is the label for BATS Global Markets; therefore any stock with an exchange code “Z” was excluded. After these exclusions, 6,157 ticker symbols remained in the file. The symbols from this cleaned list were exported to a Comma-Separated-Value [CSV] file.

The next step was to download the time series from Yahoo! Finance directly into R using the quantmod package by Ryan, Ulrich, and Thielen (2015). The code to execute this is provided in the Appendix. To ensure that there was no IP address blocking from Yahoo! Finance, single requests in sequence were made for all 6,157 stocks.

Not all ticker requests resulted in a successful download of the time series. There were 242 symbols that failed due to unknown reasons. Given the time constraints, and the fact that the percentage of failed symbols was approximately 3.93%, an attempt to capture any failed symbols was abandoned in pursuit of timely results.

A list of failed symbols can be found in the Appendix. Moreover, a single ticker caused failures during the portfolio return calculations and therefore had to be removed. Once the volatility deciles were created, portfolio returns with rebalancing was calculated using the Return.rebalancing from PerformanceAnalytics, created by Peterson and Carl (2015).

Lastly, for the purpose of this trading strategy, only the lowest volatility decile is used, to align with the registered account portfolio criteria, as well as the well-documented low-volatility anomaly (described by Xi Li and Luis Garcia-Feijóo (2016)).

# Strategy Description

## Filter

For clarity and completeness, the strategy filter will be explicitly stated here, though it has been implicitly described in previous sections. The trading strategy will only be applied to stocks on the NYSE and AMEX that have exhibited volatility less-than-or-equal-to the 10th percentile volatility of the population (over the previous calendar year). This ensures that the trading strategy is applied to the lowest volatility stocks.

The volatility decile portfolio is an equally-weighted portfolio of stocks that meet the criteria outlined in the previous paragraph. The portfolio is rebalanced annually, and the volatility is measured annually. Therefore, 90% of all stocks are filtered out of the investment universe for this proposed trading strategy.

## Strategy Indicators

To extend the original paper replication study from Biceroglu (2016b), in this strategy development project, an Exponentially-weighted Moving Average [EMA] is used instead of the Simple Moving Average [SMA] to assess if this would provide any additional outperformance over Buy-and-Hold [BH] of the volatility decile. The EMA is calculated using functions from the TTR R package (Ulrich (2015)).

In order to test the efficacy of an EMA indicator as the basis for a trading strategy, first, what the EMA indicator is believed to be measuring must be identified, and then secondly, this must be verified statistically.

Here, it is hypothesized that the EMA indicator is attempting to measure two things:

• Actual de-noised level
• Slope of the price level

### Actual de-noised level

To illustrate, given a simple CAPM model, where $$r_{STOCK} \; = \alpha + \beta_{MKT} \; r_{MKT} \; + \epsilon$$, and assuming that $$\epsilon$$ is normally distributed, such that $$N \sim (0,\sigma)$$, then on average, based on that assumption, the noise should cancel out. Therefore, taking a moving average, such as the EMA, conceptually should average out the noise. With a sufficiently small window, any changes to the “actual” price level due to new information should, over time, have minimal impact to the true unknown level, thereby yielding a de-noised estimate of the true level.

To test this hypothesis, the correlation to the index prices will be measured and tested for statistical significance. What is expected is a high correlation, much larger than 0.5. To refute this hypothesis, a low correlation would be expected, such as less than 0.25, or a correlation that exhibits no statistical significance. Some correlation is expected since the EMA time series is derived from the price series.

filterRange <- "1974/1994"

LevelTestResults <-
do.call(cbind,
lapply(nLagSequence
, function(nLag){
decilePortfolioMatrixEMA <- EMA(x = decilePortfolioMatrixLevel[filterRange,1], n = nLag)

testData <- merge(decilePortfolioMatrixLevel[filterRange,1], decilePortfolioMatrixEMA)
testData <- testData[!is.na(testData[,2])]
colnames(testData)[1] <- "Level"

result <-
c(cor = list(cor(coredata(testData))[1,2])
,cor.test(formula = ~ Level +  EMA, data = coredata(testData), na.action = "na.omit")[c("statistic","p.value")]
)

return(result)

})
)
colnames(LevelTestResults) <- nLagSequence

It is clear from the table below, that for lags up to 200 days, the correlation between the index level and the EMA level is very high, as well as statistically significant. Therefore, it can be concluded that EMA for lags up to 200 days is adequately measuring the de-noised level.

#### EMA Correlation Results Table

Moving Average Length Correlation t-Stat p-value
5 99.9973% 9835.76 0
10 99.9925% 5935.81 0
15 99.9873% 4562.39 0
20 99.9818% 3809.89 0
25 99.976% 3320.50 0
30 99.9701% 2971.96 0
40 99.9579% 2502.35 0
50 99.9454% 2195.16 0
60 99.9327% 1974.50 0
70 99.9197% 1805.78 0
80 99.9065% 1671.23 0
90 99.893% 1561.12 0
100 99.8795% 1469.23 0
110 99.8658% 1390.90 0
120 99.852% 1322.83 0
140 99.8246% 1212.72 0
160 99.7971% 1125.14 0
180 99.7704% 1055.30 0
200 99.7439% 997.15 0

To further visualize this, in the plot below, the index time series for both the lowest volatility decile is plotted against an EMA of 200 days. Optically, the EMA “follows” the actual index level. If a divergence between the series was observed, it would then be inconsistent with the above results. Therefore, numerical results agree with optical observations.

Lastly, a scatter plot of the EMA(200) level vs. the volatility decile index level is plotted below. If these time series were to track each other perfectly, a scatter plot along the diagonal would be observed. In this case, the graph shows a nearly perfect line closely following the diagonal, suggesting that EMA(200) is adequately capturing the index level, again agreeing with numerical results obtained above.

### Slope of the price level

Assuming that an EMA averages noise such that it cancels out, this generates a smooth(er) time series when compared to the actual index level. A consequence of a smoothed time series is that trends are more optically visible/observable. Thus, it is hypothesized that the EMA is also capturing the slope of the trend line. In order to test this hypothesis, first a measure for slope needs to be defined.

$Slope = \frac{Rise}{Run}$

Thus, using information in the EMA time series, a slope can be approximated, when using the smallest difference possible between data points (i.e. 1 day):

$Slope_{EMA} = \frac{Rise_{EMA}}{Run_{EMA}} = \frac{EMA_{t} - EMA_{t-1}}{(t - (t - 1))} = \frac{EMA_{t} - EMA_{t-1}}{1} = EMA_{t} - EMA_{t-1}$

The “true” slope is assumed to be the $$\beta$$ of a linear regression of index levels over a period equal to the MA lag-length. In order to capture the “true”, forward-looking slope, a symmetric window will be used, such that half of the linear regression window will contain future index level data, and the other half will contain previous index level data.

To conclude that the indicator is measuring slope, both correlation and co-integration, at minimum, are required. Correlation by itself is insufficient since the levels of correlated time series can diverge. Co-integration by itself is insufficient, since two time series can be co-integrated and negatively correlated. However, if the time series are both highly correlated, and co-integrated, then it is difficult to conclude that they are not measuring the same market characteristic. High and statistically significant correlation (> 0.8) is expected between the EMA slope and the “true” slope. Furthermore, the series should also be co-integrated (i.e. contain no unit root with statistical significance). Observing little correlation, or concluding that there is no co-integration would lead to a rejection of this hypothesis. A test to see if the difference between the regressions (the residuals) is different from zero will also be conducted. If there is a statistically significant difference between the differences of the series, this would imply that not all of the information between the “true” slope and the EMA slope proxy is being captured, thereby refuting the hypothesis.

Test 1: Difference between regressions
$$H_{o}: \; \epsilon_{\beta_{Index}} \: \: = \: \epsilon_{Slope_{EMA}}$$
$$H_{a}: \; \epsilon_{\beta_{Index}} \: \: \neq \: \epsilon_{Slope_{EMA}}$$

Test 2: Correlation
$$H_{o}: \; \rho(\beta_{Index} \; , \; Slope_{EMA} \:) \: \: = \: 0$$
$$H_{a}: \; \rho(\beta_{Index} \; , \; Slope_{EMA} \:) \: \: \neq \: 0$$

Test 3: Augmented Dickey-Fuller Test
$$H_{o}: \; \gamma \: = \: 0$$ (i.e. a unit root is present)
$$H_{a}: \; \gamma \: < \: 0$$ (i.e. no unit root is present and series is stationary/co-integrated)

Results:

#### EMA Hypothesis Test Results

Moving Average Length Residuals between EMA Slope and Index Beta p-value (Test 1) EMA vs. Index Beta Slope Correlation (Test 2) EMA vs. Index Beta Slope Correlation p-value (Test 2) EMA Slope Statistical Significance EMA Slope vs. Index Beta Cointegration p-value (Test 3)
5 0.9679 0.3192 0 35.13214 0.01
10 0.8983 0.6025 0 78.70422 0.01
15 0.8406 0.5927 0 76.71524 0.01
20 0.7789 0.6509 0 89.36752 0.01
25 0.7331 0.6302 0 84.56624 0.01
30 0.7056 0.6624 0 92.12068 0.01
40 0.6110 0.6743 0 95.07468 0.01
50 0.4860 0.6894 0 99.06045 0.01
60 0.3522 0.7004 0 102.08395 0.01
70 0.2450 0.7077 0 104.16465 0.01
80 0.1672 0.7129 0 105.65532 0.01
90 0.1099 0.7154 0 106.37319 0.01
100 0.0681 0.7162 0 106.56368 0.01
110 0.0396 0.7165 0 106.61471 0.01
120 0.0213 0.7165 0 106.56475 0.01
140 0.0055 0.7154 0 106.12563 0.01
160 0.0014 0.7138 0 105.53070 0.01
180 0.0004 0.7117 0 104.79310 0.01
200 0.0001 0.7087 0 103.81942 0.01

Code to generate the results can be found in the Appendix.

Based on the results above, all time series are correlated and co-integrated with statistical significance. The hypothesis starts to break down at a moving average lag length of approximately 100 days, where the difference between the EMA Slope time series and the Index Beta time series start to exhibit a statistically significant difference from 0 (at a 0.05 significance), rejecting the null hypothesis, indicating that some information is not being captured. Therefore, $$\le$$ 100 lag days will be included in the parameter set for the strategy, and > 100 lag days will be excluded, insofar as it indicates that the slope of the trend is not properly being captured.

## Signal Process

The next step in strategy development is to measure the quality of the signal process. The previous section concluded that $$\le$$ 100 lag-days for the EMA measures slope, (whereas level is captured up to 200 EMA lag days).

First, the signal must be defined. Typical MA signals involve the price crossing the MA line, such as the one outlined in Han, Yang, and Zhou (2013). The price contains the latest information, whether either resulting from noise, or an update in aggregate market information. Moving averages by definition contain lagged information, insofar as they contain n-day lagged prices.

If the index level is above the MA, one possible interpretation is that this will increase the smoothed MA level. As the MA is updated with new information, and $$P_{t} > P_{t-n}\;$$ (where $$P$$ is the price, or index level), then the MA level will increase. If the MA level will increase, this is an indication that the de-noised price (i.e. “true” index level) has increased, due to the averaging of the noise to zero. Furthermore, if the MA levels are increasing, and the first difference of the MA index is a proxy for slope, this also implies that the trend is likely to increase, thereby indicating that a decision to interact with the market should be made. The converse is true for a sell signal, where an index level lower than the MA level indicates that the de-noised (“true”) level will decrease, and a lower MA level implies a declining slope, further implying a declining trend, indicating that yet another decision to interact with the market should be made.

Thus, there are two potential signals described above. One, where the index level crosses the MA level, and two, if the differences in MA level are positive or negative. Both will be tested, where in one case, only the index level crossing the MA level will be evaluated, and in the second case, a combination of the index level crossing the MA level and the sign of the slope will be evaluated. It will be shown that adding sign of the slope condition makes virtually no difference to signal predictability results, and thus ultimately it will be ignored when testing the strategy.

The signal predictability will be evaluated on a short-term basis, within a trading week. The rationale behind this is that the buy-and-hold portfolio compensates an investor through market risk premia, which will be realized over the long term. These risk premia are very unlikely to be realized over the short term. Therefore, it can be difficult to distinguish over the medium and long term what the source of returns is for the signal, i.e. if it is in fact due to market timing, or if it is due to capturing market risk premia. Therefore, to maximize the likelihood of measuring EMA signal predictability, the focus will be on 1 - 5 trading days as it is assumed that returns from market risk premia will be negligible, and that short term returns are more likely to be driven by market timing.

### Signal Measurement Description

In order to measure the quality of the signal, the main tool that was used was a confusion matrix, where the main classification was if the market was predicted to move up or down, vs. if it actually moved up or down. Using this matrix, a number of statistics can be calculated, including any potential skill. Both the “Probability of Detection” [POD], since it is more intuitive, and the “Heidke Skill Score” [HSS] will be used to quantify skill, since it measures the “fractional improvement of the forecast over the standard forecast [chance]” (Meteorological Training (2011)). There will be 10 matrices produced, 5 for an up signal, and 5 for a down signal (one for each day forward).

The two tables below (one for up signal, the next for down signal) illustrate that it is irrelevant whether or not the additional criterion for the sign of the EMA slope is added to the signal process, insofar as the statistics presented in the table yield the same results. Confusion matrix calculations will be performed using the verification R package, created by Research Applications Laboratory (2015).

#### Up Signal Results

MA Lag Days Prob of Detection Prob of Detection SE HSS HSS Se Bias Chi Squared Test p-value
Slope Result 5 0.2219 0.0077 0.0658 0.0087 0.3471 0.0000
No Slope Result 5 0.2219 0.0077 0.0658 0.0087 0.3471 0.0000
Slope Result 10 0.1008 0.0056 0.0252 0.0052 0.1614 0.0006
No Slope Result 10 0.1008 0.0056 0.0252 0.0052 0.1614 0.0006
Slope Result 15 0.0648 0.0046 0.0081 0.0028 0.1110 0.1988
No Slope Result 15 0.0648 0.0046 0.0081 0.0028 0.1110 0.1988
Slope Result 20 0.0566 0.0043 0.0105 0.0029 0.0939 0.0683
No Slope Result 20 0.0566 0.0043 0.0105 0.0029 0.0939 0.0683
Slope Result 25 0.0446 0.0038 0.0063 0.0020 0.0756 0.2328
No Slope Result 25 0.0446 0.0038 0.0063 0.0020 0.0756 0.2328

#### Down Signal Results

MA Lag Days Prob of Detection Prob of Detection SE HSS HSS Se Bias Chi Squared Test p-value
Slope Result 5 0.2045 0.0082 0.0740 0.0096 0.3681 0e+00
No Slope Result 5 0.2045 0.0082 0.0740 0.0096 0.3681 0e+00
Slope Result 10 0.1049 0.0063 0.0472 0.0067 0.1793 0e+00
No Slope Result 10 0.1049 0.0063 0.0472 0.0067 0.1793 0e+00
Slope Result 15 0.0782 0.0055 0.0344 0.0055 0.1347 0e+00
No Slope Result 15 0.0782 0.0055 0.0344 0.0055 0.1347 0e+00
Slope Result 20 0.0628 0.0050 0.0277 0.0047 0.1079 0e+00
No Slope Result 20 0.0628 0.0050 0.0277 0.0047 0.1079 0e+00
Slope Result 25 0.0519 0.0045 0.0208 0.0040 0.0917 6e-04
No Slope Result 25 0.0519 0.0045 0.0208 0.0040 0.0917 6e-04

First, the up signal probability of detection is measured across 5-forward days, with up to 100 MA lag days. The contour plot and table below illustrates that the probability of detection is highest for lag days between 5 and 10 days, possibly 15 days, and then drops significantly.

#### Up-Signal Probability of Detection Table

5 10 15 20 25 30 40 50 60 70 80 90 100
Forward Day 5 20.38 9.72 6.40 5.45 4.36 3.85 3.38 2.61 2.35 2.45 2.16 2.03 1.68
Forward Day 4 20.29 9.39 6.12 5.28 4.27 3.85 3.11 2.55 2.46 2.52 2.10 1.97 1.54
Forward Day 3 20.93 9.65 6.43 5.48 4.60 3.91 3.25 2.69 2.62 2.76 2.16 2.16 1.90
Forward Day 2 21.71 10.00 6.53 5.59 4.68 4.04 3.16 2.72 2.93 2.90 2.29 2.29 2.12
Forward Day 1 22.19 10.08 6.48 5.66 4.46 3.98 3.25 2.84 2.81 2.74 2.19 2.19 1.85

The HSS is much more unforgiving, where the skill level is relatively low for lag days of 15 or greater, even though the majority of the numbers are statistically significant. Moreover, the skill level is highest for 1 day-forward with a 5 day lag, and drops off relatively rapidly for a 5-day lag when the number of forward days exceed 3 days.

#### Up-Signal HSS Table

5 10 15 20 25 30 40 50 60 70 80 90 100
Forward Day 5 2.93 1.78 0.66 0.63 0.44 0.47 0.69 0.13 -0.52 -0.29 -0.12 0.04 0.09
Forward Day 4 2.71 1.10 0.07 0.28 0.26 0.47 0.14 0.01 -0.31 -0.15 -0.24 -0.09 -0.18
Forward Day 3 4.03 1.64 0.71 0.70 0.91 0.58 0.43 0.29 0.03 0.33 -0.11 0.31 0.54
Forward Day 2 5.61 2.35 0.91 0.91 1.09 0.86 0.26 0.35 0.65 0.61 0.14 0.56 0.98
Forward Day 1 6.58 2.52 0.81 1.05 0.63 0.73 0.44 0.60 0.41 0.30 -0.06 0.36 0.43

#### Up-Signal HSS T-Stat Table

5 10 15 20 25 30 40 50 60 70 80 90 100
Forward Day 5 4.0368 4.0114 2.7017 2.8770 2.7748 3.0490 3.9146 2.6215 -0.8151 -1.4246 -2.0576 2.6597 3.1357
Forward Day 4 3.7635 2.9174 1.6281 2.1655 2.3542 3.0265 2.4548 2.3077 -1.4074 -1.8408 -1.6996 -2.2564 -2.1822
Forward Day 3 5.1147 3.7067 2.7705 2.9688 3.7805 3.2704 3.1824 3.0636 2.3328 3.1497 -2.0997 3.5013 4.7334
Forward Day 2 6.6600 4.6696 3.0821 3.3433 4.0881 3.8473 2.7218 3.2288 3.9680 3.8849 2.8359 4.2499 6.2423
Forward Day 1 7.5487 4.8441 2.8783 3.5626 3.1031 3.5206 3.1580 3.8419 3.3056 3.0352 -2.2866 3.6288 4.2558

These observations are confirmed when examining the statistical significance of the confusion matrix, where the test is to assess if the matrix is statistically significantly different from an equally distributed random matrix. Here, only lag days of 10 or less are statistically significant. This suggests that lag days should not exceed 10 days for the up signal to maintain its predictive power.

#### Up-Signal Confusion Matrix p-values

5 10 15 20 25 30 40 50 60 70 80 90 100
Forward Day 5 0.0024 0.0088 0.2643 0.2469 0.3788 0.3085 0.0975 0.7985 0.1888 0.4796 0.8049 0.9789 0.8357
Forward Day 4 0.0056 0.1159 0.9476 0.6408 0.6376 0.3198 0.7837 1.0000 0.4644 0.7594 0.5431 0.8649 0.6236
Forward Day 3 0.0000 0.0195 0.2380 0.2134 0.0661 0.2198 0.3319 0.5034 1.0000 0.4375 0.8224 0.4173 0.0955
Forward Day 2 0.0000 0.0011 0.1381 0.1080 0.0325 0.0748 0.6000 0.4093 0.1191 0.1401 0.7711 0.1298 0.0024
Forward Day 1 0.0000 0.0006 0.1988 0.0683 0.2328 0.1409 0.3438 0.1572 0.3532 0.5076 0.9520 0.3513 0.2125

Similar results are obtained for the down signal, where the highest probability of detection is concentrated for lag days less than or equal to 15 days.