Dow Jones Stock Index Analysis - part 6

Introduction

I go on with the definition of the ARMA-GARCH model for Dow Jones Industrial Average (DJIA) daily log-returns.

Packages

The packages being used in this post series are herein listed.

suppressPackageStartupMessages(library(lubridate))
suppressPackageStartupMessages(library(fBasics))
suppressPackageStartupMessages(library(lmtest))
suppressPackageStartupMessages(library(urca))
suppressPackageStartupMessages(library(ggplot2))
suppressPackageStartupMessages(library(quantmod))
suppressPackageStartupMessages(library(PerformanceAnalytics))
suppressPackageStartupMessages(library(rugarch))
suppressPackageStartupMessages(library(FinTS))
suppressPackageStartupMessages(library(forecast))
suppressPackageStartupMessages(library(strucchange))
suppressPackageStartupMessages(library(TSA))

Getting Data

We upload the environment status as saved at the end of part 2.

load(file='DowEnvironment.RData')

We show the original DJIA log-returns time series with the mean model fit (red line) and the conditional volatility (blue line).

par(mfrow=c(1,1))
cond_volatility <- sigma(garchfit)
mean_model_fit <- fitted(garchfit)
p <- plot(dj_ret, col = "grey")
p <- addSeries(mean_model_fit, col = 'red', on = 1)
p <- addSeries(cond_volatility, col = 'blue', on = 1)
p

Model Equation

Combining both ARMA(2,2) and eGARCH models we have:

\[ \begin{equation} \begin{cases} y_{t}\ -\ \phi_{1}y_{t-1}\ -\ \phi_{2}y_{t-2} =\ \phi_{0}\ +\ u_{t}\ +\ \theta_{1}u_{t-1}\ +\ \theta_{2}u_{t-2} \\ \\ u_{t}\ =\ \sigma_{t}\epsilon_{t},\ \ \ \ \ \epsilon_{t}=N(0,1) \\ \\ \ln(\sigma_{t}^2)\ =\ \omega\ + \sum_{j=1}^{q} (\alpha_{j} \epsilon_{t-j}^2\ + \gamma (\epsilon_{t-j} - E|\epsilon_{t-j}|)) +\ \sum_{i=1}^{p} \beta_{i} ln(\sigma_{t-1}^2) \end{cases} \end{equation} \]

Using the model resulting coefficients, it results as follows.

\[ \begin{equation} \begin{cases} y_{t}\ + 0.476\ y_{t-1}\ + 0.575\ y_{t-2} = \ u_{t}\ + 0.429\ u_{t-1}\ + 0.563\ u_{t-2} \\ \\ u_{t}\ =\ \sigma_{t}\epsilon_{t},\ \ \ \ \ \epsilon_{t}=N(0,1) \\ \\ \ln(\sigma_{t}^2)\ =\ -0.313\ -0.174 \epsilon_{t-1}^2\ + 0.189\ (\epsilon_{t-1} - E|\epsilon_{t-1}|)) +\ 0.966\ ln(\sigma_{t-1}^2) \end{cases} \end{equation} \]

Saving the current enviroment for further analysis.

save.image(file='DowEnvironment.RData')

References

[1] Dow Jones Industrial Average [https://en.wikipedia.org/wiki/Dow_Jones_Industrial_Average]

[2] Skewness [https://en.wikipedia.org/wiki/Skewness]

[3] Kurtosis [https://en.wikipedia.org/wiki/Kurtosis]

[4] An introduction to analysis of financial data with R, Wiley, Ruey S. Tsay [https://www.wiley.com/en-us/An+Introduction+to+Analysis+of+Financial+Data+with+R-p-9780470890813]

[5] Time series analysis and its applications, Springer ed., R.H. Shumway, D.S. Stoffer [https://www.springer.com/gp/book/9783319524511]

[6] Applied Econometric Time Series, Wiley, W. Enders, 4th ed. [https://www.wiley.com/en-us/Applied+Econometric+Time+Series%2C+4th+Edition-p-9781118808566]

[7] Forecasting - Principle and Practice, Texts, R.J. Hyndman [https://otexts.org/fpp2/]

[8] Options, Futures and other Derivatives, Pearson ed., J.C. Hull[https://www.pearson.com/us/higher-education/product/Hull-Options-Futures-and-Other-Derivatives-9th-Edition/9780133456318.html]

[9] An introduction to rugarch package [https://cran.r-project.org/web/packages/rugarch/vignettes/Introduction_to_the_rugarch_package.pdf]

[10] Applied Econometrics with R, Achim Zeileis, Christian Kleiber - Springer Ed. [http://www.springer.com/la/book/9780387773162]

[11] GARCH modeling: diagnostic tests [https://logicalerrors.wordpress.com/2017/08/14/garch-modeling-conditional-variance-useful-diagnostic-tests/]

Disclaimer

Any securities or databases referred in this post are solely for illustration purposes, and under no regard should the findings presented here be interpreted as investment advice or a promotion of any particular security or source.