Forecasting intraday S&P 500 index returns: A functional time series approach

• Author: Han Lin Shang
• Date: 01 November 2017
• DOI: http://doi.org/10.1002/for.2467
• Published date: 01 November 2017

Received: 17 March 2016 Revised: 18 October 2016 Accepted: 25 February 2017

DOI: 10.1002/for.2467

RESEARCH ARTICLE

Forecasting intraday S&P 500 index returns: A functional time

series approach

Han Lin Shang

Research School of Finance, Actuarial

Studies and Statistics, Australian National

University,Canber ra, Australia

Correspondence

Han Lin Shang, Research School of Finance,

Actuarial Studies and Statistics, Australian

National University,Canber ra, ACT2601,

Australia.

Email: hanlin.shang@anu.edu.au

Abstract

Financial data often take the form of a collection of curves that can be observed

sequentially over time; forexample, intraday stock price curves and intraday volatil-

ity curves. These curves can be viewed as a time series of functions that can be

observed on equally spaced and dense grids. Owing to the so-called curse of dimen-

sionality, the nature of high-dimensional data poses challenges from a statistical

perspective; however, it also provides opportunities to analyze a rich source of infor-

mation, so that the dynamic changes of short time intervals can be better understood.

In this paper, we consider forecastinga time series of functions and propose a number

of statistical methods that can be used to forecast 1-day-ahead intradaystock returns.

As we sequentially observe new data, we also consider the use of dynamic updat-

ing in updating point and interval forecasts for achieving improved accuracy. The

forecasting methods were validated through an empirical study of 5-minute intraday

S&P 500 index returns.

KEYWORDS

dynamic updating, functional principal component regression, functional linear regression, ordinary least

squares, penalize least squares, ridge regression

1INTRODUCTION

Traditional financial models (e.g., capital asset pricing model)

consider the time points at which prices are taken, but ignores

the underlying dynamic changes of the price curves, that is,

how a stock price shifts from a time point t−1tot,where

tdenotes a time variable (e.g., an hour, a day, or a month). If

the price at tis equal to the price at t−1, the return will be

zero; however, the path from t−1totcomprises much infor-

mation about price changes. From a statistical perspective,

if prices are considered and analyzed as discrete time series,

the underlying stochastic process that generates these obser-

vations cannot be ascertained. Conversely, functional data

analysis techniques can often be used to extract additional

information possessed in a time series of functions (e.g., their

derivatives) to measure the velocity and acceleration of price

curves (Ramsay & Silverman, 2005, chapter 18).

We consider the daily curvesof cumulative intraday returns

(CIDRs) that Gabrys, Horváth, and Kokoszka (2010) intro-

duced to transform daily nonstationary price curves to daily

stationary price curves. Horváth, Kokoszka, and Rice (2014)

developed a stationarity test for a functional time series. Fur-

thermore, Kokoszka, Miao, and Zhang (2015) showed that

CIDRs are stationary. Let Pi(tj),i=1,…,n,j=1,…,mbe

the price of a financial asset at time tjon day i. CIDRs can be

defined as

Ri(tj)=100 ×[lnPi(tj)−ln Pi(t1)],(1)

where ln(·) represents the natural logarithm and Pi(tj)repre-

sents 5-minute closing prices on a given day i. The curved

shape of CIDRs are similar to the shape of the original price

curves. Thus CIDRs were considered to be continuous curves

that could be constructed from

Xi(t)=Ri(tj),t∈(5(j−1),5j],for j=1,…,,

Journal of Forecasting.2017;36:741–755. wileyonlinelibrary.com/journal/for Copyright © 2017 John Wiley & Sons, Ltd. 741