## Saturday, February 24, 2024

### (R) Ethiopian Multiplication

It is unfortunate that Africa is often overlooked as it pertains to the continent’s pre-colonial achievements. In today’s article, we’re going to examine the manner in which Ethiopians once performed multiplication calculations.

Interestingly enough, this method seems to almost purposely avoid the inclusion of decimal figures within its algorithmic components. I’m not sure if this was the result of a greater aversion to irrational figures, or simply just a coincidence of the consequence of its design.

The methodology which I will be referencing throughout this entry can be found below:

“Ethiopian Binary Math | the Engines of Our Ingenuity.” Engines.egr.uh.edu, engines.egr.uh.edu/episode/504. Accessed 24 Feb. 2024.

Example of Method Application

For our example, we will multiply the values of 9 and 115.

If we were to perform this multiplication exercise though the utilization of our modern multiplication algorithm, we would take a far different approach. However, if we were to approach this mathematical inquiry from the Ethiopian perspective, we would take the following steps.

We would begin by recording the first initial value (9), and then record the quotient of that value by dividing by 2, but not before discarding any potential remainder. Next, we would divide that value (4) by 2, recording the quotient after discarding any potential remainder. We would continue this process until our final recorded value was 1.

After having dealt with the first value, we would record the second initial value (115), and then record the product of that value multiplied by 2. Next, we would then record that value’s (115) product after multiplying by 2 (230). We would then continue this process until the number of values within the second column, equals the number of values contained within the first column.

To come to the final sum, we would add all of the values within the Value 2 column which possess a corresponding odd number entry within the Value 1 column.

9 and 1 are both odd numbers, and their corresponding entries are 115 and 920. Therefore, by summing 115 and 920, we reach the value of 1035.

If we were to reverse the order of multiplication, 115 multiplied by 9, as opposed to 9 multiplied by 115, our outcome would not change. Although, the column values would differ.

###########################################################################

# The first value within the multiplication equation #

val1 <- 115

# The second value within the multiplication equation #

val2 <- 9

###########################################################################

vec1 <- c()

vec2 <- c()

while (val1 != 0)

{

vec1 <<- c(vec1, val1)

print(val1)

val1 <- val1 - ceiling(val1/2)

}

# Number of observational entries for val1 #

n <- length(vec1)

while (n != 0)

{

vec2 <<- c(vec2, val2)

print(val2)

val2 <- val2 * 2

n <<- (n - 1)

}

# Create data frame #

Eth_Frame <- data.frame(vec1, vec2)

# Function which identifies odd elements within a vector #

odds <- function(x) subset(x, x %% 2 != 0)

# Identify odd elements #

odd_values <- odds(vec1)

# Match odd elements within odd_values vector to Eth_Frame variable 'vec1' #

Eth_Frame\$odd_vec <- odd_values[match(Eth_Frame\$vec1, odd_values)]

# Remove all observational values in which vec1 = odd_vec within 'Eth_Frame' #

Eth_Frame <- subset(Eth_Frame, Eth_Frame\$vec1 == Eth_Frame\$odd_vec)

# Sum all remaining elements within variable 'vec2', within data frame 'Eth_Frame' #

sum(Eth_Frame\$vec2)

###########################################################################

Corresponding output:

> sum(Eth_Frame\$vec2)
[1] 1035

###########################################################################

If you would like to see the contents of each column prior to the calculation being completed, print the Eth_Frame to the console prior to defining the odds function. Also, try reversing each value’s entry within the code (val1 <- 9 val2 <- 115), to verify that order does not play a role in determining the algorithm’s outcome.

Final Thoughts

I’m not going to pontificate too deeply on the concept detailed within this entry. However, some scholars have found similarities when comparing this method of multiplication, to the way in which digital computing methodologies handle similar tasks. I do find it fascinating that the genesis of many of the concepts which empower modernity, seem to find their un-actualized origins within prior historic periods.

It is also thought provoking in that, the aversion to zero and irrationality in mathematics which began in Greece, continued to haunt the west, and some western adjacent societies for an absurd duration of time.

Matson, John. “The Origin of Zero.” Scientific American, 21 Aug. 2009, www.scientificamerican.com/article/history-of-zero/.

Until next time.

-RD

## Monday, February 12, 2024

### (R) Is Taylor Swift’s Presence Significantly Impacting NFL Game Outcomes?

I hope that everyone enjoyed Super Bowl 58. Regardless of the outcome, I hope that we can all agree that the athleticism displayed by both the 49ers and the Chiefs, made this particular game, one for the ages.

I was planning on writing this article even before the contest was decided. However, I feel that it’s more relevant now, given the game’s results.

Is Taylor Swift’s Presence Significantly Impacting NFL Game Outcomes (for Kansas City)?

To answer this question, we’ll be employing two separate tests and hypotheses.

Both assessments will utilize the Welch Two Sample T-Test methodology.

Our hypotheses are:

H0: (Null) – There was not a significant difference as it pertains to Kansas City’s total offensive yards per game with Taylor Swift in attendance.

HA: (Alternative) – The was a significant difference as it pertains to Kansas City’s total offensive yards per game with Taylor Swift in attendance.

~ AND ~

H0: (Null) – There was not a significant difference as it pertains to Kansas City’s total yards allowed per game with Taylor Swift in attendance.

HA: (Alternative) – The was a significant difference as it pertains to Kansas City’s total yards allowed per game with Taylor Swift in attendance.

Hogan, Kate. “Every Time Taylor Swift Went to a Kansas City Chiefs Game - and Whether or Not the Team Won.” Peoplemag, PEOPLE, 12 Feb. 2024, people.com/every-time-taylor-swift-went-to-kansas-city-chiefs-game-if-they-won-8410511.

“2023 Kansas City Chiefs Rosters, Stats, Schedule, Team Draftees, Injury Reports.” Pro, www.pro-football-reference.com/teams/kan/2023.htm. Accessed 12 Feb. 2024.

###########################################################################

CSV File:

,KC - 2023 NFL Season - Team Statistics,,,
Week,Offense (TotYd),Defense (TotYd),Opponent,Taylor Swift (Y/N)
1,316,368,DET,N
2,399,271,JAX,N
3,456,203,CHI,Y
4,401,336,NYJ,Y
5,333,329,MIN,N
6,389,197,DEN,Y
7,483,358,LAC,Y
8,274,240,DEN,N
9,267,292,MIA,N
10,Bye Week,,,
11,336,238,PHI,N
12,360,358,LVR,N
13,337,382,GNB,Y
14,346,327,BUF,Y
15,326,206,NEW,Y
16,308,205,LVR,Y
17,373,263,CIN,Y
18,268,353,LAC,N

###########################################################################

## Code for Testing and Analysis ##

Offense_TotYd <- c(316, 399, 456, 401, 333, 389, 483, 274, 267, 336, 360, 337, 346, 326, 308, 373, 268)

Defense_TotYd <- c(368, 271, 203, 336, 329, 197, 358, 240, 292, 238, 358, 382, 327, 206, 205, 263, 353)

Offense_TotYd_No <- c(316, 399, 333, 274, 267, 336, 360, 268)

Defense_TotYd_No <- c(368, 271, 329, 240, 292, 238, 358, 353)

Offense_TotYd_Swift <- c(456, 401, 389, 483, 337, 346, 326, 308, 373)

Defense_TotYd_Swift <- c(203, 336, 197, 358, 382, 327, 206, 205, 263)

t.test(Offense_TotYd_No, Offense_TotYd_Swift, paired = FALSE, conf.level = 0.95)

t.test(Defense_TotYd_No, Defense_TotYd_Swift, paired = FALSE, conf.level = 0.95)

sd(Offense_TotYd_No)

sd(Offense_TotYd_Swift)

sd(Defense_TotYd_No)

sd(Defense_TotYd_Swift)

###########################################################################

Findings

There was a significant difference as it pertains to Kansas City’s total offensive yards per game (2023) with Taylor Swift in attendance (n = 9), as compared to total offensive yards per game with Taylor Swift not in attendance (n = 8). Kansas City’s total offensive yards per game with Taylor Swift in attendance (M = 379.89, SD =59.23), Kansas City’s total offensive yards per game with Taylor Swift not in attendance (M = 319.13, SD = 47.67), Conditions; t(14.878) = -2.341, p = 0.03.

There was not a significant difference as it pertains to Kansas City’s total yards allowed per game (2023) with Taylor Swift in attendance (n = 9), as compared to total yards allowed per game with Taylor Swift not in attendance (n = 8). Kansas City’s total yards allowed per game with Taylor Swift in attendance (M = 275.22, SD = 75.69), Kansas City’s total yards allowed per game with Taylor Swift not in attendance (M = 306.13, SD = 53.03), Conditions; t(14.294) = 0.98307, p = 0.34.

Conclusions

While Tay Tay’s presence was generally beneficial to the Kansas City Chiefs as it pertains to the assessed metrics (Offensive Yards per Game, Yards Allowed per Game), only the offensive side of the ball saw a significant differentiation in performance (p = .03). Perhaps T. Swizzle provided the Kansas City Chiefs with some extra oomph via her top 40 mojo. Whatever the case may be, Tayla Swiff continues to be a good luck charm for the Chiefs whenever she is in attendance.

## Monday, January 29, 2024

### (R) Is Motion Prior to the Snap Correlated with Wins per Season?

The year of our Lord 2023, has been an anemic year as it pertains to NFL offensive prowess. There are numerous articles written as to why this may be the case. I personally believe that it may be due to an over reliance on the passing game. While this approach was optimal throughout prior years, recent defensive innovations devised to temper this phenomenon, have since severely limited its present effectiveness.

There are many methodologies which can be utilized to counteract the effectiveness of newly emergent algorithmic football defenses. The tried and true amongst such, is the application of motion prior to the snap.

In doing research for this article, I stumbled upon the following data:

So, in the spirit of upcoming Super Bowl LVIII (49ers vs Chiefs), let’s analyze this information in order to discern whether team wins are correlated with pre-snap motion.

###########################################################################

# Getting the Libraries in Order (for Graphical Output) #

library(ggpubr)

library(tseries)

# Motion Prior to Snap (through Week 6) #

# Data Collected by ESPNStatsInfo #

# Populate Data Frame #

Team <- c('Dolphins', 'Rams', '49ers', 'Lions', 'Packers', 'Chargers', 'Seahawks', 'Falcons', 'Ravens', 'Titans', 'Giants', 'Bears', 'Chiefs', 'Colts', 'Steelers', 'Texans', 'Jaguars', 'Jets', 'Broncos', 'Vikings', 'Washington', 'Bengals', 'Patriots', 'Buccaneers', 'Cardinals', 'Bills', 'Browns', 'Panthers', 'Saints', 'Raiders', 'Eagles', 'Cowboys')

Motion_Percentage <- c(80.2, 65.4, 77.5, 62.3, 58.3, 55.6, 48.9, 61.1, 49.5, 48, 44.3, 56.1, 68.3, 45.4, 45.4, 52.3, 44.9, 35.5, 38.1, 46.3, 53.2, 44.6, 51.1, 43.9, 39.9, 48.7, 47.3, 35.4, 28.5, 52.9, 21.7, 42.1)

Wins_2023 <- c(11, 10, 12, 12, 9, 5, 9, 7, 13, 6, 6, 7, 11, 9, 10, 10, 9, 7, 8, 7, 4, 9, 4, 9, 4, 11, 11, 2, 9, 8, 11, 12)

Motion_Report <- data.frame(Team, Wins_2023, Motion_Percentage)

# Derive Mean and Standard Deviation #

mean(Motion_Report\$Motion_Percentage)

sd(Motion_Report\$Motion_Percentage)

mean(Motion_Report\$Wins_2023)

sd(Motion_Report\$Wins_2023)

# Apply Correlation Methodology \$

cor.test(Motion_Report\$Motion_Percentage, Motion_Report\$Wins_2023)

# Create Graphic Visualizations #

data <- data.frame(Motion_Percentage, Wins_2023)

ggscatter(data, x = "Motion_Percentage", y = "Wins_2023",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "Rate of Motion (Through Week 6)", ylab = "Season Wins (2023)")

###########################################################################

Findings

There was a positive correlation between the two variables: Season Wins (2023) (n = 32) and Rate of Motion (Through Week 6) (n = 32). Season Wins (2023) (M = 8.5, SD = 2.747), Rate of Motion (Through Week 6) (M = 49.772, SD = 12.526), Conditions; t(30) = 1.4809, p = .15. Pearson Product-Moment Correlation Coefficient: r = .26.

###########################################################################

Conclusions

While the p-value findings (p = .15) can be viewed as non-significant at the alpha level of .05, we must take into account that there are certain experimental limitations which will innately confound our results. For one, wins per season is zero sum. Meaning, that a win for one team, is always loss for another. Also, as both motion and wins per season are discrete variables, there is a limited predefined range of differentiation which exists between each variable. This, combined with wins per team being non-independent, reduces the test result to a generalization.

Success of motion implementation is being assessed solely on wins, and the mechanism for generating such is being assessed by offensive motion alone. Our assessment does not account for strength of schedule, team defensive prowess, player fundamentals, etc.

However, that being said, with a p = .15, and a correlation coefficient value of r = .26, it likely is more fortuitous, all things being equal, to implement an offense which possesses pre-snap motion. There certainly are many other factors which can determine outcome which are not assessed within this model, but in all likelihood, they will not have a large impact upon the overall findings.

## Tuesday, November 7, 2023

### (R) Why Economic Forecasts are Nonsense

I’m tired of witnessing the infestation of financial news sources by prognostication.

So in today’s entry, we will test for correlation between common financial variables.

Why do these ghosts continue to haunt the thoughts of otherwise rational readers? Perhaps it's the illusion of control. Or perhaps, it’s a desire to be in possession of exclusive knowledge. Knowledge which grants a higher status to those whom possess it. Or maybe it’s a way of justifying one’s superiority. A survivorship bias of sorts.

The truth is that all economic prediction models are fundamentally flawed. As their predictive capacity is always assessed post-hoc.

That also holds true for prognosticators, who can only contribute one successful prediction throughout the entirety of their careers. After achieving such, they remain a trusted source, with all of their previous crack-pot predictions washed away.

Yet despite what I have just described, the madness of this charade continues.

Is it just for the fun of it all? The belief that through ritualization, the impossible could potentially become possible? Or maybe it's just the human need to create experts.

Whatever the case may be, I refuse to consider these male astrology exercises to be anything more than the board room equivalent of camp fire stories.

The Data

# S&P 500 Total Returns by Year (Percent change) #

# https://www.slickcharts.com/sp500/returns #

sp_close_72_22 <- c(-18.11, 28.71, 18.4, 31.49, -4.38, 21.83, 11.96, 1.38, 13.69, 32.39, 16, 2.11, 15.06, 26.46, -37, 5.49, 15.79, 4.91, 10.88, 28.68, -22.1, -11.89, -9.1, 21.04, 28.58, 33.36, 22.96, 37.58, 1.32, 10.08, 7.62, 30.47, -3.1, 31.69, 16.61, 5.25, 18.67, 31.73, 6.27, 22.56, 21.55, -4.91, 32.42, 18.44, 6.56, -7.18, 23.84, 37.2, -26.47, -14.66, 18.98)

# CPI Percentage Change Dec-Dec #

# https://www.usinflationcalculator.com/inflation/consumer-price-index-and-annual-percent-changes-from-1913-to-2008/ #

cpi_72_22 <- c(6.5, 7, 1.4, 2.3, 1.9, 2.1, 2.1, 0.7, 0.8, 1.5, 1.7, 3, 1.5, 2.7, 0.1, 4.1, 2.5, 3.4, 3.3, 1.9, 2.4, 1.6, 3.4, 2.7, 1.6, 1.7, 3.3, 2.5, 2.7, 2.7, 2.9, 3.1, 6.1, 4.6, 4.4, 4.4, 1.1, 3.8, 3.9, 3.8, 3.8, 8.9, 12.5, 13.3, 9, 6.7, 4.9, 6.9, 12.3, 8.7, 3.4)

# 30-year fixed-rate average #

# https://www.bankrate.com/mortgages/historical-mortgage-rates/#current-rates #

thirty_year_mor_72_22 <- c(0.0553, 0.0315, 0.0338, 0.0413, 0.047, 0.0414, 0.0379, 0.0399, 0.0431, 0.0416, 0.0388, 0.0465, 0.0486, 0.0538, 0.0623, 0.064, 0.0647, 0.0593, 0.0588, 0.0589, 0.0657, 0.0701, 0.0808, 0.0746, 0.0691, 0.0757, 0.0776, 0.0786, 0.0828, 0.0717, 0.0827, 0.0909, 0.0997, 0.1025, 0.1038, 0.104, 0.1039, 0.1243, 0.1388, 0.1324, 0.1604, 0.1664, 0.1374, 0.112, 0.0964, 0.0885, 0.0887, 0.0905, 0.0919, 0.0804, 0.0738)

# M2 Percent Change from Year Ago #

# https://fred.stlouisfed.org/graph/?graph_id=248494 #

m2_72_22 <- c(5.11903, 16.28949, 19.11447, 5.06449, 3.77566, 5.68219, 6.76589, 5.76268, 6.167, 6.73845, 8.57838, 7.30235, 2.49613, 8.02493, 6.75141, 6.11739, 5.23708, 4.28224, 4.7706, 6.97704, 7.44801, 8.63346, 6.04183, 7.38864, 7.15112, 5.27695, 4.8759, 2.07367, 1.26729, 1.03366, 1.86908, 3.71999, 5.50776, 4.19323, 5.35027, 6.48672, 8.12228, 8.90245, 8.00654, 12.20922, 9.0296, 9.03193, 8.0325, 7.82262, 8.27402, 12.39339, 12.76942, 9.41921, 5.86709, 9.70773, 12.41968)

The Sources

“S&P 500 Total Returns.” S&P 500 Total Returns by Year Since 1926,                    www.slickcharts.com/sp500/returns. Accessed 5 Nov. 2023.

“Consumer Price Index Data from 1913 to 2023.” US Inflation Calculator | Easily Calculate How the Buying Power of the U.S. Dollar Has Changed from 1913 to 2023. Get Inflation Rates and U.S. Inflation News., 12 Oct. 2023, www.usinflationcalculator.com/inflation/consumer-price-index-and-annual-percent-changes-from-1913-to-2008.

“M2.” FRED, fred.stlouisfed.org/graph/?graph_id=248494. Accessed 5 Nov. 2023.

The Analysis

###########################################################################

# Getting the Libraries in Order #

library("ggpubr")

library(tseries)

###########################################################################

cor.test(sp_close_72_22, cpi_72_22)

cor.test(sp_close_72_22, thirty_year_mor_72_22)

cor.test(sp_close_72_22, m2_72_22)

cor.test(cpi_72_22, m2_72_22)

###########################################################################

# Visual Creation Output #

###########################################################################

my_data <- data.frame(sp_close_72_22, cpi_72_22)

ggscatter(my_data, x = "sp_close_72_22", y = "cpi_72_22",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "S&P 500 Total Returns by Year (Percent change)", ylab = "CPI Percent Change Dec-Dec")

my_data <- data.frame(sp_close_72_22, thirty_year_mor_72_22)

ggscatter(my_data, x = "sp_close_72_22", y = "thirty_year_mor_72_22",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "S&P 500 Total Returns by Year (Percent change)", ylab = "30-year fixed-rate average")

my_data <- data.frame(sp_close_72_22, m2_72_22)

ggscatter(my_data, x = "sp_close_72_22", y = "m2_72_22",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "S&P 500 Total Returns by Year (Percent change)", ylab = "M2 Percent Change from Year Ago")

my_data <- data.frame(m2_72_22, cpi_72_22)

ggscatter(my_data, x = "m2_72_22", y = "cpi_72_22",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "M2 Percent Change from Year Ago", ylab = "CPI Percent Change Dec-Dec")

###########################################################################

Findings

There was a negative correlation between the two variables: S&P 500 Total Returns by Year (Percent change) (n = 51) and CPI Percent Change Dec-Dec (n = 51). S&P 500 Total Returns by Year (M = 11.983, SD = 17.536), CPI Percent Change Dec-Dec (M = 3.992, SD = 3.032), Conditions; t(49) = -0.82177, p = .42. Pearson Product-Moment Correlation Coefficient: r = -.12.

There was a positive correlation between the two variables: S&P 500 Total Returns by Year (Percent change) (n = 51) and 30-year fixed-rate average (n = 51). S&P 500 Total Returns by Year (Percent change) (M = 11.983, SD = 17.536), 30-year fixed-rate average (M = .078, SD = .032), Conditions; t(49) = 0.29162, p = .77. Pearson Product-Moment Correlation Coefficient: r = .04.

There was a positive correlation between the two variables: S&P 500 Total Returns by Year (Percent change) (n = 51) and M2 Percent Change from Year Ago (n = 51). S&P 500 Total Returns by Year (Percent change) (M = 11.983, SD = 17.536), M2 Percent Change from Year Ago (M = 7.085, SD = 3.464), Conditions; t(49) = 0.58362, p = .56. Pearson Product-Moment Correlation Coefficient: r = .08.

There was a positive correlation between the two variables: CPI Percent Change Dec-Dec (n = 51) and M2 Percent Change from Year Ago (n = 51). CPI Percent Change Dec-Dec (M = 3.992, SD = 3.032), M2 Percent Change from Year Ago (M = 7.085, SD = 3.464), Conditions; t(49) = 1.4331, p = .16. Pearson Product-Moment Correlation Coefficient: r = .20.

Conclusions

As is evident from our findings, there is not a single economic indicator which demonstrates a 1:1 correlation of any significance (alpha = .05). Therefore, we can infer that news headlines which contains hyperbolic language such as:

The Jobs Market Is A Recession Warning

Major U.S. indicators point to a recession starting

XYZ strategist sees year-end equities rally

Economic Indicator Says Recession Imminent

Are ultimately the work of some vaudeville spirit medium summoning imaginary ghosts to a drawing room.

The closest that we came to discerning anything of interest, was the non-significant (p = .16), but still somewhat insightful, correlative (r = .20) assessment of CPI and M2. The impact of money supply on CPI figures should be somewhat self-evident. However, with the coefficient of determination being less than .50, we can only make the assessment that the increase of currency within a financial system might increase the rate of inflation. Sometimes, but not always.

All these revelations should shine a light in the darkness as it pertains to the definitive pronouncements of various opaque economic models. Almost as if by design, the failure of such doomsday warnings are rarely revisited the day after our purported judgement. However, this does little to inhibit the same modeling systems from being trotted out time and again.

Once a system has been established, all the evidence which is contra to its continued existence is explained away in terms of circumstance, with no fault being assigned to the system itself. The same holds true for reputations, and un-verified phenomenon.

I read somewhere that the ancient Greeks assessed their heroes not upon their fates, but as to how those heroes behaved within heroic situations. The qualities of temperance and mastery of the self, were what ultimately determined the differentiating attributes of heroism. Perhaps the modern world can learn something from this perspective. That honest action within itself, regardless of the outcome, is that which ultimately provides the truth.

-RD

## Thursday, October 26, 2023

### (R) Does Trading Volume Impact Stock Index Pricing?

In a previous article, we concluded that US equity markets are random. At least that is what was determined based upon our research. I was willing to mark this matter as being solved, however, prior to doing such, a reader of this site sent a message inquiring about the Winner’s Curse, and how this phenomenon might be applied to stock pricing.

Disproving the Winner’s Curse

The winner's curse is a phenomenon that may occur in common value auctions, where all bidders have the same (ex post) value for an item but receive different private (ex ante) signals about this value and wherein the winner is the bidder with the most optimistic evaluation of the asset and therefore will tend to overestimate and overpay.

“Winner’s Curse.” Wikipedia, Wikimedia Foundation, 17 Oct. 2023,         en.wikipedia.org/wiki/Winner%27s_curse.

In plain terms, this basically states that the number of bids within an auction, ought to drive up the price of the item being sold. Let’s take a look at this phenomenon as it is applied to the stock market. After all, one probably should expect to observe such a phenomenon within this medium.

To perform the appropriate analysis, I gathered the monthly closing percentage figures for three different stock indices (S&P 500, Nikkei 225, STOXX 600). The months included within this study are November 2003 – September 2023. Data was queried from Yahoo Finance.

I will be utilizing the Pearson's product-moment correlation to test for relationships as it pertains to the index volume for each month, as assessed in comparison to the percentage differentiation of closing price.

I will also test for randomness as it pertains to monthly volume figures through the utilization of the Phillips-Perron Unit Root Test.

Finally, I will test for stationarity amongst monthly volume figures through the application of the Augmented Dicky-Fuller methodology.

###########################################################################

# Getting the Libraries in Order #

library("ggpubr")

library(tseries)

###########################################################################

# S&P 500 (USA) #

###########################################################################

sp_close <- c(0.007, 0.051, 0.017, 0.012, -0.016, -0.017, 0.012, 0.018, -0.034, 0.002, 0.009, 0.014, 0.039, 0.032, -0.025, 0.019, -0.019, -0.02, 0.03, 0, 0.036, -0.011, 0.007, -0.018, 0.035, -0.001, 0.025, 0, 0.011, 0.006, -0.031, 0, 0.005, 0.02, 0.025, 0.032, 0.017, 0.013, 0.014, -0.022, 0.01, 0.043, 0.033, -0.018, -0.033, 0.013, 0.036, 0.014, -0.042, -0.008, -0.061, -0.035, -0.006, 0.045, 0.01, -0.085, -0.007, 0.011, -0.094, -0.168, -0.075, 0.016, -0.085, -0.107, 0.094, 0.1, 0.053, -0.004, 0.072, 0.031, 0.037, -0.018, 0.057, 0.015, -0.038, 0.028, 0.058, 0.013, -0.083, -0.052, 0.068, -0.053, 0.087, 0.035, -0.004, 0.06, 0.023, 0.03, -0.002, 0.026, -0.015, -0.018, -0.021, -0.057, -0.072, 0.108, -0.003, 0.009, 0.043, 0.041, 0.031, -0.007, -0.063, 0.04, 0.012, 0.02, 0.024, -0.02, 0.003, 0.007, 0.05, 0.011, 0.036, 0.018, 0.021, -0.016, 0.047, -0.033, 0.028, 0.044, 0.027, 0.023, -0.034, 0.043, 0.008, 0.005, 0.021, 0.019, -0.016, 0.038, -0.016, 0.024, 0.024, -0.003, -0.031, 0.054, -0.018, 0.009, 0.01, -0.022, 0.018, -0.063, -0.025, 0.083, 0, -0.019, -0.048, -0.002, 0.063, 0.004, 0.014, 0.002, 0.035, -0.001, -0.001, -0.018, 0.033, 0.018, 0.012, 0.034, -0.007, 0.009, 0.01, 0.003, 0.016, -0.002, 0.018, 0.021, 0.025, 0.011, 0.052, -0.036, -0.027, 0.006, 0.024, 0, 0.041, 0.028, 0.006, -0.073, 0.016, -0.102, 0.092, 0.03, 0.013, 0.034, -0.068, 0.069, 0.003, -0.018, 0.023, 0.018, 0.03, 0.028, -0.006, -0.087, -0.131, 0.166, 0.061, 0.02, 0.053, 0.064, -0.041, -0.034, 0.099, 0.03, -0.013, 0.021, 0.034, 0.047, 0.003, 0.019, 0.022, 0.026, -0.049, 0.067, -0.009, 0.035, -0.055, -0.032, 0.038, -0.09, 0, -0.088, 0.092, -0.038, -0.089, 0.073, 0.046, -0.061, 0.058, -0.025, 0.037, 0.016, 0.003, 0.064, 0.031, -0.016, -0.054)

sp_vol <- c(24463220000, 27839130000, 32820000000, 27985600000, 33597900000, 31611900000, 29326400000, 27529500000, 29285600000, 26586800000, 26829870000, 31511000000, 30460280000, 31102500000, 31498800000, 29297410000, 39014150000, 43424270000, 39321990000, 40334040000, 37464670000, 42030090000, 44777510000, 49793790000, 45102870000, 41756130000, 49211650000, 42859940000, 50905040000, 43308430000, 54312830000, 54873260000, 46348220000, 50485620000, 49001440000, 56793620000, 55343930000, 47578780000, 56686200000, 51844990000, 67622250000, 57032470000, 64958050000, 65322800000, 70337430000, 91381760000, 57809700000, 76022580000, 86246950000, 64821670000, 98475340000, 78536130000, 93189170000, 85978630000, 80990480000, 96614040000, 124980570000, 86266010000, 140007320000, 159823030000, 115660210000, 112884470000, 112090640000, 124492210000, 161843640000, 138855320000, 131614940000, 112653150000, 106635790000, 116059270000, 112295490000, 113410990000, 84981530000, 89515330000, 90947580000, 84561340000, 103683550000, 116741910000, 127662780000, 110106750000, 94778110000, 85738250000, 79589450000, 89536270000, 87151070000, 80984530000, 92164940000, 59223660000, 89507640000, 77364810000, 81708980000, 86122730000, 81102170000, 108419170000, 102786820000, 98063670000, 84275050000, 74742430000, 79567560000, 78385710000, 83899660000, 74761710000, 86920490000, 81582440000, 73103810000, 70283810000, 69784280000, 71752320000, 71489310000, 66388180000, 75848510000, 69273480000, 68527110000, 77098000000, 76447250000, 74946790000, 68106820000, 64802810000, 66174410000, 76647400000, 63628190000, 64958820000, 75871910000, 69725590000, 71885030000, 71595810000, 63623630000, 63283380000, 66524690000, 58131140000, 66706000000, 93714040000, 63600190000, 80743820000, 77330040000, 68775560000, 76675850000, 72060940000, 65187730000, 73213980000, 77920590000, 84626790000, 79989370000, 85844900000, 75943590000, 83649260000, 92409770000, 93049560000, 92639420000, 81124990000, 78883600000, 86852700000, 69530250000, 75610310000, 77023620000, 72915530000, 88445380000, 75344550000, 70576420000, 69260940000, 81664010000, 65369860000, 79719460000, 81078810000, 63348090000, 70784900000, 66624120000, 71088550000, 73416960000, 65531700000, 77318690000, 79933970000, 76803890000, 70194700000, 76011820000, 77891360000, 64898300000, 69523070000, 63031510000, 91930980000, 80620020000, 84162180000, 80859870000, 70638770000, 79159660000, 70090370000, 77250740000, 71250630000, 70599470000, 80269220000, 74178980000, 77720640000, 72410620000, 72325540000, 77287980000, 84436590000, 162185380000, 123608160000, 107135190000, 131458880000, 96928130000, 82466520000, 92310780000, 89938980000, 101247180000, 96375680000, 106117800000, 99082320000, 122371150000, 83124090000, 88321860000, 102544180000, 84255620000, 80500760000, 85528860000, 80253600000, 88268840000, 92750180000, 95562890000, 92667710000, 123546260000, 90367840000, 108860390000, 106116710000, 81688320000, 92252350000, 94241020000, 95823760000, 92671910000, 85249330000, 80763810000, 80392280000, 113094800000, 70861260000, 88929200000, 87983140000, 75063200000, 86840820000, 73482980000)

cor.test(sp_close, sp_vol)

PP.test(sp_vol)

###########################################################################

# Nikkei 225 (JAPAN) #

###########################################################################

nik_vol <- c(2103300000, 1973300000, 1923900000, 2093700000, 1779800000, 1305200000, 1782200000, 1369700000, 1364600000, 1484200000, 1555800000, 1435500000, 1377200000, 1397800000, 1339300000, 1637700000, 1582900000, 1345900000, 1882500000, 1390900000, 1387900000, 1284100000, 1433700000, 1601900000, 1561500000, 1235400000, 1108100000, 1258700000, 1313600000, 1231300000, 1956500000, 1406000000, 1388600000, 1340300000, 1614800000, 1148600000, 1362400000, 1357300000, 1454700000, 1754000000, 1539400000, 1837200000, 3101700000, 1396200000, 1133600000, 1173000000, 1356700000, 1347900000, 1392600000, 1373500000, 1168700000, 1195400000, 1458100000, 1187800000, 1396900000, 1276100000, 1275600000, 1529200000, 1534700000, 1822800000, 1447600000, 1515200000, 1523800000, 1513500000, 1607600000, 1386000000, 1732700000, 1840200000, 1606300000, 1539000000, 1981200000, 1872800000, 1807500000, 1876000000, 1790600000, 2326500000, 2230200000, 2178500000, 2646000000, 2606300000, 2440400000, 3038700000, 3226000000, 2222800000, 2493200000, 2675600000, 2816900000, 3093000000, 2563800000, 3230600000, 3307500000, 4174100000, 3235900000, 2749100000, 2559500000, 2979600000, 3132300000, 3723100000, 3318200000, 3459500000, 3291500000, 3047400000, 3191800000, 2975400000, 2691500000, 2962900000, 3155400000, 3436700000, 2530100000, 2300800000, 2467400000, 2715300000, 2560200000, 2575500000, 3011800000, 3390300000, 3734900000, 3151000000, 3175900000, 3095600000, 2908600000, 3158800000, 4145700000, 4755100000, 7190600000, 6211600000, 4251700000, 5299900000, 4335800000, 3432700000, 2687500000, 2738800000, 2403400000, 2571500000, 2457400000, 2548800000, 2713200000, 2547300000, 3340600000, 3455900000, 2153800000, 2086100000, 2276100000, 2371900000, 2664200000, 3428500000, 2450800000, 2889600000, 2475600000, 2991400000, 4792200000, 3125000000, 2736000000, 2638500000, 2736400000, 2922000000, 2480200000, 2551700000, 2938300000, 2699700000, 3169300000, 3039100000, 2840200000, 2486300000, 3461300000, 3335300000, 2760300000, 3133500000, 2585200000, 2927600000, 3393000000, 3795800000, 3006700000, 3798500000, 3376900000, 2809500000, 2655200000, 2663800000, 2850200000, 4267300000, 2905700000, 2443300000, 2658600000, 2825900000, 2699700000, 2572700000, 2821400000, 2916400000, 3114100000, 2330800000, 3089900000, 2633300000, 2068700000, 3199300000, 2635500000, 2726700000, 2759200000, 2597500000, 3402500000, 3256500000, 2668700000, 2415700000, 2143600000, 2207200000, 2062500000, 2278400000, 2085700000, 2526300000, 2337600000, 2145500000, 2389600000, 2767300000, 2680600000, 3239400000, 3468100000, 3283900000, 3272000000, 2344500000, 1432300000, 1617600000, 1446100000, 1675000000, 2032200000, 1665700000, 1616200000, 1437800000, 1400600000, 1588400000, 1666700000, 1481800000, 1373300000, 1704500000, 1696200000, 2430200000, 2429300000, 1282100000, 1436200000, 1274000000, 1273100000)

nik_close <- c(-0.021, -0.021, -0.01, 0.069, 0.059, 0.023, 0.024, -0.001, 0.055, -0.083, 0.013, 0.066, -0.072, 0.01, 0.048, -0.034, 0.016, -0.029, 0.035, -0.024, -0.078, 0.032, -0.054, -0.012, 0.043, 0.021, -0.057, -0.007, -0.006, -0.022, -0.008, 0.045, 0.003, 0.03, 0.126, -0.009, 0.004, 0.052, -0.029, 0.017, 0.086, 0.075, -0.102, -0.082, -0.005, 0.011, 0.024, 0.048, 0.052, -0.032, -0.002, 0.045, -0.077, 0.034, -0.013, 0.028, 0.054, -0.131, 0.02, -0.103, 0.054, 0.01, 0.014, 0.008, -0.011, 0.046, -0.021, -0.055, 0.001, -0.007, 0.026, 0.073, 0.031, -0.013, -0.007, 0.017, 0.025, 0.011, -0.017, 0.01, -0.014, 0.03, 0.051, 0.049, -0.026, 0.028, 0.053, -0.098, 0.051, -0.003, 0.044, -0.104, -0.074, -0.04, 0.047, 0.084, -0.079, -0.087, 0.014, -0.01, 0.051, 0.02, 0.018, 0.067, 0.02, -0.001, 0.042, 0.015, 0.044, -0.006, 0.028, 0.025, 0.02, -0.04, 0.011, 0.004, -0.083, 0.039, 0.08, -0.013, 0.07, -0.021, -0.006, 0.009, -0.005, 0.107, 0.075, 0.032, 0.048, 0.088, 0.054, 0.013, 0.004, 0.025, -0.047, 0.06, -0.109, -0.067, 0.031, 0.096, 0.029, -0.015, -0.053, 0.047, -0.036, -0.106, -0.005, 0.011, -0.028, 0.009, -0.094, 0.032, -0.011, 0.028, 0.078, -0.026, 0.057, -0.085, 0.025, -0.039, -0.11, -0.011, 0.087, -0.009, -0.04, 0.12, -0.06, -0.004, -0.032, 0.013, 0.045, 0.044, 0.071, 0.074, 0.081, -0.045, -0.125, 0.045, -0.022, -0.329, -0.149, -0.016, -0.01, -0.064, 0.037, 0.095, -0.071, 0.006, -0.115, -0.029, -0.072, -0.002, 0.016, -0.036, -0.052, 0.01, 0.027, 0.003, -0.015, 0.013, 0.004, 0.053, -0.004, 0.014, 0.003, 0.047, -0.008, -0.006, -0.095, -0.013, 0.061, -0.024, 0.021, 0.074, 0.078, 0.003, 0.079, 0.041, 0.027, 0.031, 0.029, -0.053, -0.006, 0.027, -0.006, 0.061, 0.015, -0.011, -0.026, -0.017, -0.054, 0.055, -0.048, 0, 0.051, 0.023, 0, 0.063, -0.059)

cor.test(nik_close, nik_vol)

PP.test(nik_vol)

###########################################################################

# STOXX 600 (EUROPE) #

###########################################################################

stoxx_vol <- c(2082064000, 3593140000, 3623085900, 3227490000, 3963845100, 4537933600, 3364175000, 5644136700, 4315542500, 4190288900, 3429359000, 4428634700, 4324893100, 5044459900, 3673166200, 4209030900, 4403288200, 4668497000, 4080039400, 7015693000, 4665939400, 4219964500, 3725329800, 5141114500, 3873973500, 3990773300, 3214970700, 3353233300, 3727672100, 3913126100, 3529714100, 4851670900, 4375370400, 4120573200, 4414589500, 5990354800, 4457050500, 4378642700, 3472143000, 4493206100, 6506693200, 5685330700, 5778662500, 9781539200, 4717665900, 3747876800, 3687903800, 3914857000, 4634906100, 4167228400, 4371274100, 3994185300, 3822333900, 4370401300, 3655958900, 4272524500, 3844484400, 3935024300, 4010016100, 4369953900, 4636643000, 3756703600, 3675091700, 3686458400, 4898417700, 5329248500, 4001420700, 4761109700, 4563697300, 4414670700, 3765308000, 4558362700, 4396511200, 3962271900, 3806938800, 4136701600, 5288574400, 5776592600, 4559447300, 5731696800, 4727016300, 5108929600, 5481488800, 6189735400, 5284539900, 5715861400, 5162109900, 6441906800, 8798906100, 5814206300, 6047662900, 6569684600, 7243086800, 6170797300, 4829569500, 5156214200, 5810022700, 5920225300, 5598217400, 5667279400, 6016609900, 5530892100, 5447295400, 6784088400, 6455256800, 7130308000, 5601864000, 5783571900, 8068259800, 5715514200, 5061411200, 6041303100, 5109740500, 5664157800, 6260120300, 7557556000, 5391112100, 6532452900, 4626036900, 5155068400, 6300416000, 5596521600, 5120640200, 459936200)

stoxx_close <- c(-0.002, -0.018, -0.028, 0.02, 0.021, -0.034, 0.019, -0.006, 0.016, 0.06, -0.039, 0.061, 0.063, -0.069, -0.054, 0.074, -0.089, -0.011, -0.013, 0.006, -0.038, -0.041, 0.049, -0.028, 0.05, -0.037, 0.018, 0.018, 0.012, 0.02, 0.018, 0.05, 0.02, -0.012, 0.023, 0.119, -0.057, -0.017, 0.028, -0.013, 0.024, 0.033, 0.07, -0.183, -0.094, -0.014, 0.02, 0.025, 0.009, 0.034, -0.014, 0.001, 0.042, -0.061, 0.03, 0.015, 0.037, 0.064, -0.065, -0.012, -0.06, 0.003, -0.025, 0.033, -0.009, -0.006, 0.041, -0.021, -0.044, 0.016, 0.006, -0.024, 0.017, 0.035, -0.012, -0.006, -0.027, 0.008, 0.013, 0.025, 0.024, -0.003, 0.053, 0.006, -0.009, -0.004, 0.003, 0.033, -0.052, 0.017, 0.016, 0.011, -0.029, -0.068, -0.056, 0.029, 0.069, -0.035, -0.095, 0.034, -0.052, 0.012, -0.002, 0.014, 0.064, 0.063, -0.009, 0.03, -0.018, 0.002, 0.018, -0.019, -0.008, 0.019, 0.009, -0.005, 0.047, -0.02, 0.008, 0.008, 0.036, 0.037, -0.01, 0.043)

cor.test(stoxx_close, stoxx_vol)

PP.test(stoxx_vol)

###########################################################################

# Visual Creation Output #

###########################################################################

# S&P 500 (USA) #

my_data <- data.frame(sp_close, sp_vol)

ggscatter(my_data, x = "sp_vol", y = "sp_close",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "S&P 500 Monthly Volume", ylab = "S&P 500 Monthly Close")

# Nikkei 225 (JAPAN) #

my_data <- data.frame(nik_close, nik_vol)

ggscatter(my_data, x = "nik_vol", y = "nik_close",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "Nikkei 225 Monthly Volume", ylab = "Nikkei 225 Monthly Close")

# STOXX 600 (EUROPE) #

my_data <- data.frame(stoxx_close, stoxx_vol)

ggscatter(my_data, x = "stoxx_vol", y = "stoxx_close",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "STOXX 600 Monthly Volume", ylab = "STOXX 600 Monthly Close")

###########################################################################

Initial Findings

There was a negative correlation between the two variables: S&P 500 Monthly Close (n = 239) and S&P 500 Monthly Volume (n = 239). S&P 500 Monthly Close (M = .006, SD = .043), S&P 500 Monthly Volume (M = 77899674812, SD = 24642426931), Conditions; t(237) = -2.1245, p = .03. Pearson Product-Moment Correlation Coefficient: r = -.14.

There was a negative correlation between the two variables: Nikkei 225 Monthly Close (n = 239) and Nikkei 225 Monthly Volume (n = 239). Nikkei 225 Monthly Close (M = .003, SD = .055), Nikkei 225 Monthly Volume (M = 2400008787, SD = 912823479), Conditions; t(237) = -0.48111, p = .63. Pearson Product-Moment Correlation Coefficient: r = -.03.

There was a negative correlation between the two variables: STOXX 600 Monthly Close (n = 124)* and STOXX 600 Monthly Volume (n = 124). STOXX 600 Monthly Close (M = .002, SD = .041), STOXX 600 Monthly Volume (M = 4885890780, SD = 1260418703), Conditions; t(122) = -2.0375, p = .04. Pearson Product-Moment Correlation Coefficient: r = -.18.

* - Only limited historical data was available for the STOXX 600 index.

Initial Conclusions

The results of this analysis indicates that there is a correlative relationship as it pertains to volume and closing price; at least, as it pertains to the S&P 500 and the STOXX 600. These indexes are of American, and European origin, respectively. However, this phenomenon was not evident in the results of the analysis as it pertains to the Nikkei 225, a Japanese stock market index. There is an old adage in the field of economics, which is typically some variation of: treat the Japanese economy as a singular non-universalized phenomenon, in which, observations related to such cannot be applied to other instances within the field as whole. This truism seems to hold merit as it relates to these findings.

Interestingly, each analytical outcome demonstrates a weak modeling potential. The Pearson Product-Moment Correlation Coefficient as it pertains to each index is negative, with the Japanese Nikkei 225 possessing the lowest coefficient figure: - .03.

This would seemingly indicate the inverse conclusion inferred by, “The Winner’s Curse”.

As is observed within the above graphics, the volume which exists within each market, appears to drive down market pricing. Even if the effect is marginal and inconsistent, it exists nevertheless.

Though this really ought to fly in the face of fundamental economic theory, we must ask ourselves, does it really? According to Say’s Law, supply creates its own demand. Therefore, would it stand to reason that the inverse ought to be true? Does demand create its own supply?

With this in mind, I decided to perform some additional analysis in order to either support or disprove the initial findings.

Utilizing my favorite outlier function, I searched for potential outlier values within the volume variables for each individual index.

OutlierFunction <- function(t){

q1 <- fivenum(t)

q1 <- q1[2] #Q1

q3 <- fivenum(t)

q3 <- q3[4] #Q3

iqrange <- q3 - q1

out1 <<- (q1 - (iqrange * 1.5))

out2 <<- (q3 + (iqrange * 1.5))

lowout <<- subset(t, t < out1, na.rm = TRUE)

highout <<- subset(t, t > out2, na.rm = TRUE)

}

OutlierFunction()

This turned up the following values as it pertains to each index.

S&P 500 – 124980570000, 140007320000, 159823030000, 124492210000, 161843640000, 138855320000, 131614940000, 127662780000, 162185380000, 131458880000, 24463220000, 27839130000, 27985600000, 29326400000, 27529500000, 29285600000, 26586800000, 26829870000, 30460280000, 31102500000, 29297410000

Nikkei 225 – 7190600000, 6211600000, 5299900000

STOXX 600 – 9781539200, 8798906100, 459936200

I then matched these volume values to their corresponding closing figures.

S&P 500

Nikkei 225

STOXX 600

Next, I ran additional analysis on the data vectors with outlier values removed.

###########################################################################

# S&P 500 (USA) OUTLIERS! #

###########################################################################

sp_close <- c(0.166, 0.038, 0.034, 0.013, 0.031, -0.075, -0.018, 0.037, 0.016, -0.004, 0.037, -0.085, -0.052, 0, -0.057, 0.061, 0.072, -0.013, -0.088, 0.058, -0.072, 0.019, 0.099, 0.021, -0.061, 0.108, 0.053, -0.085, 0.03, 0.073, -0.055, 0.068, -0.089, 0.024, -0.006, -0.002, 0.035, 0.046, -0.032, 0.063, -0.048, -0.041, -0.038, 0.023, -0.073, 0.013, -0.038, -0.09, -0.034, 0.035, 0.015, -0.002, 0.003, 0.033, 0.003, -0.009, 0.064, -0.004, -0.063, 0.002, -0.016, 0.011, -0.042, -0.018, 0.045, 0.083, -0.053, -0.049, -0.061, 0.057, -0.063, 0.028, -0.087, -0.003, 0.022, -0.102, 0.031, -0.019, 0.047, 0.064, -0.015, 0.092, -0.007, 0.04, 0.004, -0.021, 0.003, 0.01, 0.06, 0.092, 0.058, -0.003, 0.016, 0.026, -0.025, -0.018, 0.067, -0.025, -0.036, 0.01, 0.087, 0.043, 0.013, 0.014, -0.035, 0.041, 0.018, 0, 0.018, 0.026, -0.031, 0.052, -0.006, -0.068, 0.018, -0.001, -0.027, -0.018, 0.044, 0.021, 0.014, 0.024, 0, -0.034, 0.05, -0.001, 0.018, 0.031, -0.016, -0.007, 0.009, 0.023, -0.054, 0.025, -0.022, 0.012, -0.018, 0.03, 0.028, 0.009, 0.008, -0.02, 0.005, 0.003, 0.069, 0.021, 0.016, -0.002, 0.03, 0.003, 0.012, -0.033, 0.02, 0.006, 0.034, 0.024, 0.043, 0.035, 0.028, 0.011, 0.034, 0.054, 0.036, 0.047, 0.01, -0.016, 0.018, -0.016, 0.007, 0.028, 0.011, 0.009, -0.018, 0.01, 0.023, 0.033, 0.041, -0.008, -0.033, 0.027, 0.021, 0.024, 0.016, 0.019, 0.006, 0.03, 0.038, 0.036, 0.043, 0.032, 0.014, 0.017, 0, -0.031, -0.022, 0.011, 0.02, -0.018, 0.025, 0.025, 0.013, 0.005, 0.035, 0.007, -0.02, 0.006, 0, -0.011, -0.001, 0, 0.03, -0.019, 0.036, -0.016, 0.017, -0.017, 0.014, -0.025)

sp_vol <- c(123608160000, 123546260000, 122371150000, 116741910000, 116059270000, 115660210000, 113410990000, 113094800000, 112884470000, 112653150000, 112295490000, 112090640000, 110106750000, 108860390000, 108419170000, 107135190000, 106635790000, 106117800000, 106116710000, 103683550000, 102786820000, 102544180000, 101247180000, 99082320000, 98475340000, 98063670000, 96928130000, 96614040000, 96375680000, 95823760000, 95562890000, 94778110000, 94241020000, 93714040000, 93189170000, 93049560000, 92750180000, 92671910000, 92667710000, 92639420000, 92409770000, 92310780000, 92252350000, 92164940000, 91930980000, 91381760000, 90947580000, 90367840000, 89938980000, 89536270000, 89515330000, 89507640000, 88929200000, 88445380000, 88321860000, 88268840000, 87983140000, 87151070000, 86920490000, 86852700000, 86840820000, 86266010000, 86246950000, 86122730000, 85978630000, 85844900000, 85738250000, 85528860000, 85249330000, 84981530000, 84626790000, 84561340000, 84436590000, 84275050000, 84255620000, 84162180000, 83899660000, 83649260000, 83124090000, 82466520000, 81708980000, 81688320000, 81664010000, 81582440000, 81124990000, 81102170000, 81078810000, 80990480000, 80984530000, 80859870000, 80763810000, 80743820000, 80620020000, 80500760000, 80392280000, 80269220000, 80253600000, 79989370000, 79933970000, 79719460000, 79589450000, 79567560000, 79159660000, 78883600000, 78536130000, 78385710000, 77920590000, 77891360000, 77720640000, 77364810000, 77330040000, 77318690000, 77287980000, 77250740000, 77098000000, 77023620000, 76803890000, 76675850000, 76647400000, 76447250000, 76022580000, 76011820000, 75943590000, 75871910000, 75848510000, 75610310000, 75344550000, 75063200000, 74946790000, 74761710000, 74742430000, 74178980000, 73482980000, 73416960000, 73213980000, 73103810000, 72915530000, 72410620000, 72325540000, 72060940000, 71885030000, 71752320000, 71595810000, 71489310000, 71250630000, 71088550000, 70861260000, 70784900000, 70638770000, 70599470000, 70576420000, 70337430000, 70283810000, 70194700000, 70090370000, 69784280000, 69725590000, 69530250000, 69523070000, 69273480000, 69260940000, 68775560000, 68527110000, 68106820000, 67622250000, 66706000000, 66624120000, 66524690000, 66388180000, 66174410000, 65531700000, 65369860000, 65322800000, 65187730000, 64958820000, 64958050000, 64898300000, 64821670000, 64802810000, 63628190000, 63623630000, 63600190000, 63348090000, 63283380000, 63031510000, 59223660000, 58131140000, 57809700000, 57032470000, 56793620000, 56686200000, 55343930000, 54873260000, 54312830000, 51844990000, 50905040000, 50485620000, 49793790000, 49211650000, 49001440000, 47578780000, 46348220000, 45102870000, 44777510000, 43424270000, 43308430000, 42859940000, 42030090000, 41756130000, 40334040000, 39321990000, 39014150000, 37464670000, 33597900000, 32820000000, 31611900000, 31511000000, 31498800000)

cor.test(sp_close, sp_vol)

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# Nikkei 225 (JAPAN) OUTLIERS! #

###########################################################################

nik_vol <- c(-0.094, 0.009, 0.048, -0.329, 0.075, -0.104, -0.006, 0.074, 0.044, -0.083, -0.087, 0.078, -0.04, -0.01, 0.096, 0.015, 0.088, -0.106, -0.015, 0.045, 0.004, 0.081, 0.031, 0.12, 0.014, 0.044, 0.051, 0.003, 0.079, 0.013, 0.074, -0.074, -0.003, 0.051, -0.036, 0.018, 0.08, -0.11, -0.021, 0.042, 0.039, -0.004, -0.079, 0.032, -0.115, -0.102, -0.013, -0.098, -0.072, 0.02, -0.011, 0.03, 0.011, 0.071, 0.009, 0.084, 0.067, -0.001, 0.025, 0.013, -0.026, 0.006, 0.07, -0.149, 0.011, -0.022, 0.087, -0.064, -0.071, 0.053, -0.045, -0.024, -0.06, 0.027, -0.04, 0.013, 0.078, -0.011, 0.01, 0.025, -0.109, -0.039, 0.037, 0.02, 0.054, 0.021, 0.028, 0.004, -0.036, 0.045, -0.01, -0.125, -0.017, 0.028, -0.052, -0.002, 0.01, 0.003, -0.032, -0.04, 0.095, 0.025, 0.051, 0.02, 0.047, -0.085, 0.06, -0.067, 0.044, -0.006, -0.026, -0.009, 0.057, -0.028, 0.028, -0.047, -0.005, -0.016, -0.014, 0, 0.051, 0.053, 0.004, 0.061, 0.047, 0.041, -0.095, -0.029, 0.017, -0.006, 0.047, -0.053, 0.025, 0.049, 0.014, 0.011, 0.029, -0.013, -0.004, -0.021, 0.069, -0.015, -0.008, 0.016, 0.003, -0.006, 0.026, -0.021, -0.008, -0.01, 0.035, -0.013, 0.073, -0.055, 0.075, -0.103, 0.031, -0.007, 0.024, 0.059, 0.017, -0.021, 0.055, -0.048, -0.053, -0.026, 0.027, -0.034, 0.031, -0.006, 0.126, -0.011, 0.001, -0.012, -0.011, 0.016, 0.043, 0.013, 0.086, -0.007, 0.02, -0.131, 0.014, 0.01, 0.008, -0.083, -0.017, -0.077, -0.029, 0.054, 0.029, 0.061, 0, 0.066, -0.054, 0.027, 0.045, 0.015, 0.01, -0.013, -0.082, 0.052, -0.024, 0.003, -0.078, 0.046, -0.072, -0.032, -0.054, -0.001, 0.055, 0.004, 0.052, 0.024, 0.048, -0.029, 0.03, 0.048, -0.006, 0.023, 0.032, 0.023, 0.028, 0.054, 0.063, -0.059, -0.007, 0.021, -0.022, 0.045, 0.034, 0.011, -0.002, -0.009, -0.005, -0.057)

nik_close <- c(4792200000, 4755100000, 4335800000, 4267300000, 4251700000, 4174100000, 4145700000, 3798500000, 3795800000, 3734900000, 3723100000, 3468100000, 3461300000, 3459500000, 3455900000, 3436700000, 3432700000, 3428500000, 3402500000, 3393000000, 3390300000, 3376900000, 3340600000, 3335300000, 3318200000, 3307500000, 3291500000, 3283900000, 3272000000, 3256500000, 3239400000, 3235900000, 3230600000, 3226000000, 3199300000, 3191800000, 3175900000, 3169300000, 3158800000, 3155400000, 3151000000, 3133500000, 3132300000, 3125000000, 3114100000, 3101700000, 3095600000, 3093000000, 3089900000, 3047400000, 3039100000, 3038700000, 3011800000, 3006700000, 2991400000, 2979600000, 2975400000, 2962900000, 2938300000, 2927600000, 2922000000, 2916400000, 2908600000, 2905700000, 2889600000, 2850200000, 2840200000, 2825900000, 2821400000, 2816900000, 2809500000, 2767300000, 2760300000, 2759200000, 2749100000, 2738800000, 2736400000, 2736000000, 2726700000, 2715300000, 2713200000, 2699700000, 2699700000, 2691500000, 2687500000, 2680600000, 2675600000, 2668700000, 2664200000, 2663800000, 2658600000, 2655200000, 2646000000, 2638500000, 2635500000, 2633300000, 2606300000, 2597500000, 2585200000, 2575500000, 2572700000, 2571500000, 2563800000, 2560200000, 2559500000, 2551700000, 2548800000, 2547300000, 2530100000, 2526300000, 2493200000, 2486300000, 2480200000, 2475600000, 2467400000, 2457400000, 2450800000, 2443300000, 2440400000, 2430200000, 2429300000, 2415700000, 2403400000, 2389600000, 2371900000, 2344500000, 2337600000, 2330800000, 2326500000, 2300800000, 2278400000, 2276100000, 2230200000, 2222800000, 2207200000, 2178500000, 2153800000, 2145500000, 2143600000, 2103300000, 2093700000, 2086100000, 2085700000, 2068700000, 2062500000, 2032200000, 1981200000, 1973300000, 1956500000, 1923900000, 1882500000, 1876000000, 1872800000, 1840200000, 1837200000, 1822800000, 1807500000, 1790600000, 1782200000, 1779800000, 1754000000, 1732700000, 1704500000, 1696200000, 1675000000, 1666700000, 1665700000, 1637700000, 1617600000, 1616200000, 1614800000, 1607600000, 1606300000, 1601900000, 1588400000, 1582900000, 1561500000, 1555800000, 1539400000, 1539000000, 1534700000, 1529200000, 1523800000, 1515200000, 1513500000, 1484200000, 1481800000, 1458100000, 1454700000, 1447600000, 1446100000, 1437800000, 1436200000, 1435500000, 1433700000, 1432300000, 1406000000, 1400600000, 1397800000, 1396900000, 1396200000, 1392600000, 1390900000, 1388600000, 1387900000, 1386000000, 1377200000, 1373500000, 1373300000, 1369700000, 1364600000, 1362400000, 1357300000, 1356700000, 1347900000, 1345900000, 1340300000, 1339300000, 1313600000, 1305200000, 1284100000, 1282100000, 1276100000, 1275600000, 1274000000, 1273100000, 1258700000, 1235400000, 1231300000, 1195400000, 1187800000, 1173000000, 1168700000, 1148600000, 1133600000, 1108100000)

cor.test(nik_close, nik_vol)

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# STOXX 600 (EUROPE) OUTLIERS #

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stoxx_vol <- c(8068259800, 7557556000, 7243086800, 7130308000, 7015693000, 6784088400, 6569684600, 6532452900, 6506693200, 6455256800, 6441906800, 6300416000, 6260120300, 6189735400, 6170797300, 6047662900, 6041303100, 6016609900, 5990354800, 5920225300, 5814206300, 5810022700, 5783571900, 5778662500, 5776592600, 5731696800, 5715861400, 5715514200, 5685330700, 5667279400, 5664157800, 5644136700, 5601864000, 5598217400, 5596521600, 5530892100, 5481488800, 5447295400, 5391112100, 5329248500, 5288574400, 5284539900, 5162109900, 5156214200, 5155068400, 5141114500, 5120640200, 5109740500, 5108929600, 5061411200, 5044459900, 4898417700, 4851670900, 4829569500, 4761109700, 4727016300, 4717665900, 4668497000, 4665939400, 4636643000, 4634906100, 4626036900, 4563697300, 4559447300, 4558362700, 4537933600, 4493206100, 4457050500, 4428634700, 4414670700, 4414589500, 4403288200, 4396511200, 4378642700, 4375370400, 4371274100, 4370401300, 4369953900, 4324893100, 4315542500, 4272524500, 4219964500, 4209030900, 4190288900, 4167228400, 4136701600, 4120573200, 4080039400, 4010016100, 4001420700, 3994185300, 3990773300, 3963845100, 3962271900, 3935024300, 3914857000, 3913126100, 3873973500, 3844484400, 3822333900, 3806938800, 3765308000, 3756703600, 3747876800, 3727672100, 3725329800, 3687903800, 3686458400, 3675091700, 3673166200, 3655958900, 3623085900, 3593140000, 3529714100, 3472143000, 3429359000, 3364175000, 3353233300, 3227490000, 3214970700)

stoxx_close <- c(-0.018, -0.005, -0.029, 0.063, 0.006, 0.014, 0.011, -0.02, 0.024, 0.064, 0.033, 0.036, 0.009, 0.006, -0.068, 0.016, -0.019, -0.052, 0.119, -0.035, 0.017, 0.069, 0.03, 0.07, 0.008, 0.025, -0.004, 0.002, 0.033, 0.034, 0.019, -0.006, -0.009, -0.095, 0.037, 0.012, 0.053, -0.002, 0.047, -0.006, -0.027, -0.009, 0.003, 0.029, 0.008, -0.028, -0.01, -0.008, -0.003, 0.018, -0.069, -0.009, 0.05, -0.056, -0.021, 0.024, -0.094, -0.011, -0.038, -0.06, 0.009, 0.008, -0.044, 0.013, -0.024, -0.034, -0.013, -0.057, 0.061, 0.016, 0.023, -0.089, 0.017, -0.017, 0.02, -0.014, -0.061, -0.012, 0.063, 0.016, 0.015, -0.041, 0.074, 0.06, 0.034, -0.006, -0.012, -0.013, -0.065, 0.041, 0.001, -0.037, 0.021, 0.035, 0.064, 0.025, 0.02, 0.05, 0.037, 0.042, -0.012, 0.006, 0.003, -0.014, 0.012, 0.049, 0.02, 0.033, -0.025, -0.054, 0.03, -0.028, -0.018, 0.018, 0.028, -0.039, 0.019, 0.018, 0.02, 0.018)

cor.test(stoxx_close, stoxx_vol)

###########################################################################

There was a negative correlation between the two variables: S&P 500 Monthly Close (n = 218) and S&P 500 Monthly Volume (n = 218). S&P 500 Monthly Close (M = .007, SD = .039), S&P 500 Monthly Volume (M = 77543082110, SD = 18749895626), Conditions; t(216) = -0.34354, p = .73. Pearson Product-Moment Correlation Coefficient: r = -.02.

There was a negative correlation between the two variables: Nikkei 225 Monthly Close (n = 236) and Nikkei 225 Monthly Volume (n = 236). Nikkei 225 Monthly Close (M = .002, SD = .055), Nikkei 225 Monthly Volume (M = 2351271186, SD = 803886229), Conditions; t(234) = -1.2234, p = .22. Pearson Product-Moment Correlation Coefficient: r = -.08.

There was a positive correlation between the two variables: STOXX 600 Monthly Close (n = 120) and STOXX 600 Monthly Volume (n = 120). STOXX 600 Monthly Close (M = .004, SD = .038), STOXX 600 Monthly Volume (M = 4872733427, SD = 1039773116), Conditions; t(118) = 0.31681, p = .75. Pearson Product-Moment Correlation Coefficient: r = .03.

When outliers are removed and the analysis is performed, there is a slight shift in the correlation coefficient values amongst the various indices. S&P 500 (-.14; -.02), Nikkei 225 (-.03; -.08), STOXX 600 (-.18; .03). There also was a shift in correlation significance. S&P 500 (.03; .73), Nikkei 225 (.63; .22), STOXX 600 (.04; .75).

With the removal of outliers, every index had a decline in correlation significance (p), with the exception being the Nikkei 225. Oddly, as it pertains to correlation coefficient values, the S&P 500 rose and remained negative, the Nikkei 225 declined and remained negative, while the STOXX 600 value became positive.

In almost instance, increases in volume equates for decreases in price. It would seem that the most massive volume increases are the largest contributors to price decline. Seemily, disproportionately so, the Nikkei 500 continues to be an odd duck, in that, it seems to be in a perpetual state of stasis, with smaller upward shifts of volume being more impactful on price than comparatively larger shifts.

Tests for Randomness / Stationarity

Phillips-Perron Unit Root Test

data: sp_vol

Dickey-Fuller = -4.7548, Truncation lag parameter = 4, p-value = 0.01

Phillips-Perron Unit Root Test

data: nik_vol

Dickey-Fuller = -4.2136, Truncation lag parameter = 4, p-value = 0.01

Phillips-Perron Unit Root Test

data: stoxx_vol

Dickey-Fuller = -6.906, Truncation lag parameter = 4, p-value = 0.01

Augmented Dickey-Fuller Test

data: sp_vol

Dickey-Fuller = -2.2459, Lag order = 6, p-value = 0.4725

alternative hypothesis: stationary

Augmented Dickey-Fuller Test

data: nik_vol

Dickey-Fuller = -2.3522, Lag order = 6, p-value = 0.4277

alternative hypothesis: stationary

Augmented Dickey-Fuller Test

data: stoxx_vol

Dickey-Fuller = -2.6845, Lag order = 4, p-value = 0.2921

alternative hypothesis: stationary

Final Thoughts and Conclusions

In every instance, the volume measurements for each index were non-stationary, and non-random. Thus, I suppose that it should be expected, that volume figures should typically increase over time. As was demonstrated by the correlation analysis both as it relates to index/volume assessments (with and without outliers), large unanticipated upward shifts in volume, typically should be expected to cause price decline.

Thus, it might be said, that demand creates its own supply. Or at the very least, impacts supply price in a deflationary manner. Or perhaps some combination of the two.

As it relates to volume’s long term impact on pricing, it would seem that the stationary / random walk analysis that was performed for a prior article, holds true. In that, like a tiny pill bug walking haphazardly across graphing paper, a small push by a human hand here and there may alter the insect’s course, but will otherwise not prove to be impactful, other than demonstrating some significance in the present moment.

In a similar manner, as with a stretched slinky, outside gyrations obviously cause disruption, but only temporarily. Once the interference ceases, the stationarity is maintained.

Markets outside of the United States continue to remain mysterious in comparison to their state side counterparts while comparing growth patterns. Perhaps the limbic bounces of a market’s price have much to do with the culture of an economy. Though, the bounces themselves do remain very much human in origin, regardless.

## Monday, October 16, 2023

Like many of the critical attributes of life, that which is most evident, lies obscured by monotony. This is especially true as it pertains to mathematical paradoxes, as the most enlightening insights within the field, have the habit of appearing both obvious and universally evident after discovery. Like many mystical traditions, these insights are best discovered through contradiction and reduction.

The Friendship Paradox, in simpler terms, identifies the common phenomenon in which an individual, typically possesses less friends than his friends. Additionally, the sum of friends which his friends possess, will be greater than the sum of his total number of friends.

This paradox possesses wide reaching implications, as it describes events which are self-arising and irrefutable. However, before we can detail applicability, we must demonstrate the paradox as it was initially discerned.

First, let’s get some terminology down.

In graph theory, circular graphics are known as nodes, or vertexes. The lines which illustrate relationships between the nodes are known as edges.

Now, let’s utilize this style of graphical representation to demonstrate the relationships between 5 individuals.

The chart below represents the above relationships, but in a different format.

As each relationship is symmetric, if one friend considers himself to be a friend of another individual, that individual also considers the initial individual to be his friend as well. As shown above, A is friends with B and E. B is friends with both A and E, and also friends with C.

If we derive the mean as it relates to the average number of friends that an individual possesses within our experiment, we come to the value of 2.8.

In this instance, E possesses the most friends, and every individual who is friends with E, possesses less. Therefore, the average number of friends that an individual within a group possesses (2.8), will likely be greater than the actual number of friends that a singular individual possesses.

The Philosophical Implications

If a single individual begins to quantify a particular phenomenon as it relates to their person on an individualized basis, or even as it relates to a novel phenomenon, then the natural consequence of this endeavor is that this individual from the onset will find himself at a disadvantage.

For example, a new creation upon its genesis, possessing autonomy, will immediately be concerned with attaining sustenance. This was not a concern which was possessed within the prior state of non-being. In contemplating one’s beauty, a young woman immediately begins to compare herself to those whom she perceives as being more beautiful. We would never anticipate the inverse to occur.

This is the paradox of living, striving to possess more while the value of that which we possess becomes diminished. This is due to the passage of time, but also due to singular possession of a resource also diminishing in value. Something within our possession loses value from the moment of possession, as both the individual and the possession are diminished by the natural passage of time.

Example (2):

Here is another example, if an individual walks into a crowded elevator filled with random strangers, then there is a greater probability that this individual will have the same number, or a greater number of friends, than each stranger within the elevator. However, if the same individual were invited to a party hosted by a friend, then there is a lesser probability of this individual possessing more friends, or a similar number of friends, as compared to each other party attendees. In the elevator scenario, there is no guarantee that any individual within the elevator possesses a single friend. This also includes the individual entering the already crowded elevator. However, in the party scenario, each party goer has at least one friend, that being - the party’s host. In this case, the count begins at the neutral value of 1, except for the case of the party’s host, who is friends with every individual in attendance.

As described above, the friendship paradox also seeks to demonstrate, “the sum of friends which his friends possess, will be greater than the sum of his total number of friends.”

In the case of our first example, this value would be calculated as follows:

To better illustrate this phenomenon, I’ve constructed a new example relationship diagram below:

In this instance, E has more friends than A, B, C, D.

E has 4 friends. While A, B, C, D each have 1 friend (E).

In total: A, B, C, D possess the same number of friends in sum (1+1+1+1).

If A, B, C, or D possessed one additional friend – F, then in total, they would together possess more friends in sum than E (1 + 1 + 1 + 2).

If this were the case, the paradox would hold, as E would have a total of 4 friends, but the total number of his friends of friends would be greater (5).

Conclusion

That's all for today.

I hope that you enjoyed this entry and will visit again soon.

-RD

### (R) Utilizing Crowd Prediction Methodologies to Draft the Optimal Fantasy Football Team (II)

What would an application be without proof? A notion?

To prove that the ADP drafting strategy is superior to other ranking methodologies, I performed the following analysis.

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Data Source(s):

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Offense

Kickers

DST

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The Analysis (n = 300)

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ADP <- c(1.4, 2.4, 2.7, 4.2, 4.9, 5.8, 7, 7.2, 8.8, 10, 10.6, 12.4, 13.1, 14, 15.3, 16, 16.6, 17.9, 18.5, 19.1, 19.6, 19.8, 21.8, 22.7, 24.7, 25.9, 26.1, 27.3, 27.9, 29.3, 30.5, 31.6, 32.4, 33.2, 33.2, 34.1, 36.1, 38, 39.1, 39.3, 39.4, 39.7, 42.3, 43.2, 43.5, 44.2, 45.1, 46.5, 46.8, 47.3, 48.4, 50.2, 50.9, 51.5, 52.2, 53.9, 54.1, 56, 56.7, 57.9, 59.4, 61.2, 61.5, 62.1, 62.9, 63, 63.8, 63.9, 68, 68.1, 69.2, 69.9, 70.1, 71, 71.4, 71.7, 72, 72.9, 75.3, 75.4, 77.1, 77.9, 82.3, 83.8, 84.4, 84.8, 85, 86.7, 87.9, 89.5, 90.2, 90.2, 90.7, 91.3, 91.7, 91.8, 94.4, 95.9, 96.2, 98.5, 98.7, 99.1, 101.9, 102.6, 102.7, 103.6, 103.7, 105.4, 106.5, 106.7, 107.8, 108.6, 109.7, 109.9, 110.6, 111.6, 113.1, 113.6, 113.9, 114.9, 117.3, 117.7, 118.5, 119, 119.6, 120.5, 121, 121.2, 121.3, 122.7, 124.2, 125, 128.9, 129.8, 130.2, 130.2, 130.3, 130.8, 131.2, 132, 135.1, 135.9, 136.1, 136.3, 137.9, 138.9, 140.1, 140.9, 141.8, 142.7, 143.8, 144.8, 145.3, 145.4, 145.7, 147, 147, 148.1, 148.4, 148.9, 149, 149.4, 151.3, 151.4, 152.1, 152.5, 152.6, 153.8, 154.9, 155.6, 155.7, 156.7, 156.8, 156.8, 158, 158.5, 158.8, 158.9, 159.4, 160.1, 160.5, 160.7, 161, 161.5, 162, 163, 163.5, 164, 164.6, 164.6, 165, 165.2, 165.6, 166, 167, 168, 169, 169.5, 170, 171, 172, 173, 174, 175, 176, 177, 178, 178, 179, 180, 180, 181, 182, 183, 184, 185, 186, 187, 187, 188, 189, 190, 190, 191, 192, 192, 193, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265)

PPR <- c(146.4, 356.36, 372.7, 302.76, 368.66, 201.4, 223.46, 237.8, 242.4, 239.5, 211.7, 335.5, 191.1, 316.6, 248.6, 284, 316.3, 301.6, 281.4, 395.52, 168.4, 42, 226.1, 347.2, 216.5, 190.5, 225.4, 185.8, 200.2, 164, 299.6, 75.6, 220.9, 281.26, 141.3, 205.1, 199.1, 229, 417.4, 177.7, 81.2, 176.5, 43.6, 200.5, 180.7, 115.1, 159.4, 259.2, 350.7, 328.3, 167.6, 84.8, 226.8, 236.08, 51.1, 84.9, 98.3, 378.04, 222.8, 145.6, 90.9, 204.2, 216.7, 200.52, 142.7, 52.2, 171.6, 180, 156, 101.5, 126.8, 248.8, 74.2, 166.4, 79, 177.9, 225.76, 267.6, 141.2, 185.3, 249.1, 239.2, 53.5, 246, 87.1, 198.6, 254.6, 271.66, 88.1, 115.6, 227.8, 151.7, 174.8, 105.7, 108.38, 148.7, 69.4, 202.5, 241.9, 88.6, 148.2, 12.46, 168.3, 219.08, 115.7, 178.6, 147.3, 57.3, 87.9, 159.4, 291.58, 55.8, 198.2, 112.7, 237.3, 126, 98.2, 73.5, 135.7, 165.9, 150, 43.4, 135, 105.4, 215.4, 225.9, 70.4, 230.92, 167.12, 139.1, 166.5, 87.7, 88.4, 102, 122.4, 7, 94.1, 114, 105.04, 155.28, 102.9, 160, 103.9, 101.6, 159, 102, 98, 161, 295.98, 295.62, 115, 87.9, 112, 133, 101, 103, 55.2, 43.92, 116, 97, 131, 180.3, 121, 130.6, 143, 142, 119.8, 215.7, 25.5, 125, 123.6, 99, 39, 110, 97.1, 130, 129, 97, 57, 18, 118, 154, 168, 100, 116.9, 20.1, 104, 98.2, 186, 97, 60.2, 110, 101, 52.2, 155.6, 93.8, 3.2, 110, 169.3, 97.6, 34.4, 161.1, 78.9, 25.8, 51.6, 112.3, 115, 176.3, 198.1, 106, 26.4, 82.7, 149.1, 117.8, 88.3, 81.7, 110.1, 83, 4.5, 50.1, 164.1, 73, 12, 57.68, 139, 61.6, 112, 77.4, 45.4, 46.5, 15.1, 35.5, 72.7, 75.2, 84.5, 110, 0.2, 142.1, 34, 12.2, 83, 82.6, 21.1, 77.2, 196.3, 11.4, 51.7, 8.4, 54.1, 161.24, 46.2, 8.6, 8.4, 13.8, 289, 37.8, 170.08, 128.4, 89.6, 112.8, 104.1, 284.32, 154.16, 59.3, 24.9, 114.5, 121.42, 158.5, 59.8, 98.92, 0, 115.1, 10.2, 181.52, 14.8, 4.2, 12.8, 18.8, 103.9, 196.56, 1.3, 11.8, 16.2, 26, 39.1, 43.1, 53.6, 103.8, 27.3, 303.88, 30.8, 64.5, 3.5, 73.88, 110.2, 64.7, 84.3, 30.5, 10.2, 70.3)

Which produces the output:

Pearson's product-moment correlation

t = -13.394, df = 298, p-value < 2.2e-16

alternative hypothesis: true correlation is not equal to 0

95 percent confidence interval:

-0.6791003 -0.5370390

sample estimates:

cor

-0.6130004

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Creating the Visual Output

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library("ggpubr")

ggscatter(my_data, x = "ADP", y = "PPR",

add = "reg.line", conf.int = TRUE,

cor.coef = TRUE, cor.method = "pearson",

xlab = "ADP (2022)", ylab = "Player Score (PPR)")

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Conclusion

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There was a negative correlation between the two variables: ADP (n = 300), and Player Score (PPR). ADP (M = 134.81, SD = 73.31), Player Score (PPR) (M = 133.33, SD = 86.66), Conditions; t(298) = -13.394 , p < .01. Pearson Product-Moment Correlation Coefficient: r = -0.61.

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As the findings indicate, there is a significant negative correlation as it pertains to ADP and player performance (PPR - 2022 Season). In plain terms, this means that we should typically expect to see better fantasy performance from players with lower ADP rankings. I hope that everyone enjoyed this article and my adherence to APA reporting standards. <3

Until next time.