Direct multiperiod forecasting for algorithmic trading

• DOI: http://doi.org/10.1002/for.2488
• Date: 01 January 2018
• Author: Hiroyuki Kawakatsu
• Published date: 01 January 2018

Received: 3 July 2016 Revised: 8 July 2017 Accepted: 9 August 2017

DOI: 10.1002/for.2488

RESEARCH ARTICLE

Direct multiperiod forecasting for algorithmic trading

Hiroyuki Kawakatsu

Business School, Dublin City University,

Dublin 9, Ireland

Correspondence

Hiroyuki Kawakatsu, Business School,

Dublin City University, Dublin 9, Ireland.

Email: hiroyuki.kawakatsu@dcu.ie

Abstract

This paper examines the performance of iterated and direct forecasts for the

number of shares traded in high-frequency intraday data. Constructing direct

forecasts in the context of formulating volume weighted average price trading

strategies requires the generation of a sequence of multistep-ahead forecasts.

I discuss nonlinear transformations to ensure nonnegative forecasts and lag

length selection for generating a sequence of direct forecasts. In contrast to

the literature based on low-frequency macroeconomic data, I find that direct

multiperiod forecasts can outperform iterated forecasts when the conditioning

information set is dynamically updated in real time.

KEYWORDS

direct multistep forecasting, intraday forecasting, lag length selection, volume weight averageprice

1INTRODUCTION

There is a large and active literature on time series anal-

ysis of high-frequency data from financial markets. While

several types of financial time series have been the sub-

ject of analysis, this paper contributes to the literature

on modeling and forecasting of intraday transaction vol-

ume (Białkowski, Darolles, & Le Fol, 2008; Brownlees,

Cipollini, & Gallo, 2011; Humphery-Jenner, 2011). The

interest in analyzing transaction volume data arises from

its application to computing a benchmark price, known

as the volume weighted average price (VWAP), commonly

used in the industry to measure the market price impact of

financial transactions. VWAPappears to be formally intro-

duced in the academic literature by Berkowitz, Logue, and

Noser (1988).

To define VWAP, denote the number of sharestraded at

transaction ion trading day tby xit and by pit the trans-

action price per share where i=1,…,ntand ntis the

number of transactions on trading day t=1,…,T.The

VWAP vtfor tradingday tis defined as

vt=

nt



i=1

witpit ,wit ≡xi,t

nt

j=1xj,t

.

Rather than work with raw transactions data, the exist-

ing literature aggregatesthe raw data into bins of fixed time

intervals, for example 15 minutes, perhaps to smooth out

various microstructure noises. Divide each tradingday into

b=1,…,Bbins, each of equal (calendar) time length and

define the bin aggregate quantities

ab,t≡

i∈b

xit,vb,t≡1

ab,t

i∈b

xitpit ,

wb,t≡ab,t

B

b′=1ab′,t

=ab,t

nt

i=1xit

=wit

xit

ab,t,

where ab,tare the shares traded in the bth bin and vb,tis the

VWAP within the bth bin. The VWAP vtfor trading day t

can then be computed from binned data as

vt=

B



b=1

i∈b

witpit =

B



b=1

i∈b

wb,t

ab,t

xitpit

=

B



b=1wb,t

ab,t

i∈b

xitpit =

B



b=1

wb,tvb,t.

A key component for formulating a VWAP trading strategy

is to obtain a good forecast for the volume weights wb,t.In

particular,the trader does not need to predict the sequence

Journal of Forecasting. 2018;37:83–101. wileyonlinelibrary.com/journal/for Copyright © 2017 John Wiley & Sons, Ltd. 83

84 KAWAKATSU

of prices pit (Białkowski et al., 2008). The volume weights

wb,tin turn depend on the trading volume ab,tin each bin

b=1,…,Bover the trading day t. As the denominator

in wb,tis the sum of trading volume over the trading day,

computation of wb,tfor bin brequires knowledge of trading

volume ab,tfor all bins b=1,…,Bon trading day t.Asa

forecasting problem, a VWAPstrategy therefore requires a

sequence of forecasts for different step sizes over the trad-

ing day.Białkowski et al. (2008) and Brownlees et al. (2011)

generate these forecasts from a one-step-ahead model by

recursively iterating on the previous ahead forecasts. This

method of obtaining multistep forecasts by recursively iter-

ating on one-step forecasts is also known as the plug-in

method.

This paper considers an alternative direct multiperiod

forecasting approach where forecasts for different step

sizes are generated from different models each specific

to that step size. The iterated and direct multiperiod

approaches to forecasting have been compared both the-

oretically and empirically. The main theoretical result

for autoregressive processes, summarized in Marcellino,

Stock, and Watson (2006), depends on the “true” lag

order p0and the lag order pmassumed in the forecast-

ing model. If p0>pm, the asymptotic mean squared

error criterion (ignoring estimation uncertainty) favors

the direct method over the iterated method. For p0≤

pm, the asymptotic mean squared error is the same but

the iterated method has less estimation uncertainty than

that from the direct method. These theoretical results,

while informative, require knowledge of p0and are asymp-

totic in nature. The empirical finite-sample comparison

of iterated and direct forecasts has been conducted pri-

marily using low-frequency macroeconomic time series.

Marcellino et al. (2006) conduct a large-scale empiri-

cal comparison of iterated and direct forecasts of 170

low-frequency (monthly) US macroeconomic time series

for the period 1959–2002. Their main finding based on

pseudo out-of-sample forecasts from recursive estimation

is that iterated forecasts tend to havesmaller mean squared

forecast errors than direct forecasts but the improvement

is modest. They also find that the performance of direct

forecasts deteriorates with the forecast step size (up to

24-month-ahead forecasts are considered).

The main contribution of this paper is to examine

the performance of iterated and direct forecasts for

high-frequency intraday time series data. In contrastto the

results in Marcellino et al. (2006) based on low-frequency

data, I find that direct multiperiod forecasts can outper-

form iterated forecasts when the conditioning information

set is dynamically updated in real time. The comparison

of iterated and direct forecasts in the context of predict-

ing intraday trading volume ab,tfor VWAP trading strategy

raises some new issues not considered in the existing liter-

ature. To formulate a trading strategy for the next trading

day, one requires a prediction of trading volume ab,tfor

all bins on that day. Therefore, in contrast to the exist-

ing literature, which compares performance for a fixed

step size or forecast horizon, we need to examine per-

formance for a sequence of step sizes. Jordà (2005) also

considers a sequence of direct forecasts in the context of

tracing impulse responses over time. Following Brownlees

et al. (2011) I consider B=26 bins of 15-minute inter-

vals for each trading day. For direct multiperiod forecasts,

this requires estimation of up to B=26 equations for

step sizes h=1,…,B. The cost of estimating a non-

linear model such as the self-extracting threshold autore-

gression of Białkowski et al. (2008) and the component

multiplicative error model of Brownlees et al. (2011) B

times becomes high, especially when we need to repeat this

over a recursive or rolling estimation sample. For this rea-

son the forecasts in this paper are generated from a linear

autoregressive model using a rolling estimation sample.

The model specification and generation of forecasts are

described in detail in Section 2.

Although the focus is on the linear model for which

most of the theoretical results are available,there are some

important issues that need to be addressed when applied to

intraday tradingvolume data. The number of shares traded

are discrete and integer valued.Rather than model number

of shares traded, the empirical analysis follows Brown-

lees et al. (2011) and models volume scaled (divided) by

the number of daily shares outstanding. This scaling can

reduce the effect of abrupt changes in volume due to occa-

sional stock splits in the sample. The scaled volume series

can be treated as continuous and avoid the issue of mod-

eling discreteness in the raw volume series. Another issue

that does not appear to have received much attention in

the literature is that traded volume cannot be negative.

A potentially misspecified linear model may produce a

negative-valued forecast in the pseudo out-of-sample fore-

cast sample. To ensure this nonnegativity condition, I fit

the linear autoregressive model to the (natural) log of trad-

ing volume yb,t≡log(ab,t). To obtain a forecast for the

variable of interest wb,t, we need to transform the fore-

cast for yb,tinto a forecast for the untransformed variable

ab,t. Because the transformation is nonlinear, the naive

(un)transformation exp(̂

yb,t)does not generally yield an

unbiased prediction of ab,teven if ̂

yb,twere an unbiased

prediction of yb,t. This issue of nonlinear transformation is

discussedinSection2.2.AsshowninSection3,another

important benefit of the log transformation is that the dis-

tribution of yb,tis closer to the Gaussian than that of the

untransformed series ab,t. In particular, the kurtosis of yb,t

is closer to the Gaussian than that of ab,tand the least

squares estimates are less likely to be affected by extreme

values in the data.