• Thesis: Analyzed Amazon’s monthly opening price

• Scope: Jan 2009 to Oct 2021

• Personal Motivation: Improved financial literacy, valuable market knowledge

• Target Audience: Investors, economists, media analysts

• R and RStudio (knitr, forecast, lubridate)

• Exponential Smoothing

• Holt / Holt-Winters

• “Weighted Averaging” ~ think a step above Moving Average

• Sourced from Kaggle

• High data workability

  • No missing values, consistent data entry and variable encoding

• Came with a daily observation for every market-open day from May 15, 1997 to October 27, 2021

  • First few observations seen below

• Research Question: What is the history and how can I predict the future?

• Constraints: Only looking at market opens on the first day of every month on timeline from January 2009 to October of 2021

  • Opening price = strong understanding colloquially

  • Day-to-day = too many observations

  • Year-to-year = too few observations

  • Chosen timeline: n = 154

  • First few observations of dataset after programming manipulation

    Month	Year	Open	Season
    Jan	2009	52.01	Winter
    Feb	2009	62.87	Winter
    Mar	2009	68.35	Spring
    Apr	2009	78.28	Spring
    May	2009	77.84	Spring
    Jun	2009	83.19	Summer

• Priority 1: Observe Trend

• What we see from plot below:

  • Positive
  • Not linear
  • (Increase ~ 2017 to 2022) >>> (Increase ~ 2009 to 2016)

• Priority 2: Observe Seasonality
  • Seasonal differences exacerbate as time goes on

• Training Data ≠ Testing Data
  • Poses risk of overfitting
  • Avoids true “real world” application
• Training → Jan 2009 to Oct 2020 (142 month period)
• Testing → Nov 2020 to Oct 2021 (12 month period)

• Accredit these ideas to Charles Holt, Professor of Business and Finance at University of Austin

• Developed throughout the mid to late 1950’s

• Holt expanded on the foundations of SES (discovered in previous decade) by considering trend

• Note: Holt Multiplicative without Damp is rarely ever used
  • Approaches \(\infty\)
  • Computationally unstable
  • Not realistic

• Holt and a student of his, Peter-Winters, then went even further and developed a method that also factored in seasonality

• Officially published these findings in a 1960 paper “Forecasting Sales by Exponentially Weighted Moving Averages”

• The non-damped Holt and HW models drastically overestimated

• The damped HW models overestimated but not as much

• Most accurate were SES and damped Holt models

• Before plotting, wanted to use a table to review each model’s accuracy measures

• Using R’s accuracy function (forecast package), calculated error metrics below

  • Note that calculations are based on training data

• Model’s with the best ranking in said metric are marked

• Damped HW Multiplicative model had best ranking in three of the seven error measurements

• Next step was to look at plots (real and recorded values versus what models predicted)

• SES and damped Holt models are only ones that remotely resemble the true values

• The SES model:
  • Four overestimations
  • Eight underestimations
  • Range of Error [19.96 to 371.36]
• The Holt Additive Damped model:
  • Eleven overestimations
  • One underestimation
  • Range of Error [28.72 to 328.45]
• Slight edge to SES model, still need to dig further

• Will calculate forecasts from each of these two models versus the known values (test data) to make ultimate decision

• Two measures of absolute error (ME and MSE)

• Two measures of relative error (MPE and MAPE)

  	SES	Holt Add W/ Damp
  ME	-54.23	111.60
  MSE	21039.21	24131.76
  MPE	-1.49	3.51
  MAPE	3.28	4.26

• Simple Exponential Smoothing (SES) was the most effective and accurate for the purposes of this analysis

Why did the simplest model perform the best?

1.) The massive spike in the rate of Amazon’s stock increase from about 2018 to October of 2021

• Because of the “out of norm” rate of increase, all the non-damped models (Holt and HW) drastically overforecasted

2.) Seasonal oscillations mirrored the overall data trend, disparities grew at exponential rate over the last ~ 4 years of data set

• Damped HW models assume depreciation over time in trend, not in season-by-season disparities
  • *This is because usually seasonal oscillations are constant (Additive) or proportional (Multiplicative) independently of whether or not trend is increasing or decreasing

3.) Damping parameter phi (pronounced “fee”) (\(\phi\)) was too high

• \(\phi\) always exists between 0 and 1
  • If \(\phi\) = 1, then the damp is non-existent
• Automatically calculated via R’s forecast function
• Per $ selector, \(\phi\)) ~ 0.80 for damped Holt Additive and damped Holt Multiplicative models

• I did attempt to manually rebuild these two Holt models with a lower phi value, but was consistently met with errors
  • The value of phi, alpha and beta are all interdependent on one another
  • I could not even lower phi’s value to 0.79

• Subsetting into training vs testing sets is very important

• “More complex” does not necessarily = better fit for situation at hand

  • Sometimes more accuracy is retained by not considering all possible factors

• The value of investing long term in companies you believe in

  • Say you bought 100 shares of Amazon in January of 2009 for $5,201
    • Those shares would be worth $30,288 by January of 2015
    • By January of 2018. they would be worth $130,138
    • If you continued to hold, they would be worth $332,598 by October of 2022

• Perform an associative and external analysis, looking at what specific factors across timeline (Jan 2009 - October 2021) contributed to stock’s monthly increase/decrease

• Perform a time series analysis on more recent data and on a tighter timeline for Amazon

  • Ex: Jan 2022 to Dec 2025, observations at weekly intervals (sample size ~ 150)
    • As we saw, the year of 2020 had a massive impact on both trend and seasonality

• Perform a time series analysis on the same timeline but with a different company

Original Dataset Source:

• https://www.kaggle.com/datasets/kannan1314/amazon-stock-price-all-time/data

My Complete Report:

• https://rpubs.com/Chris_Bahm/1372994