ﬁrst aspect of cointegration modelling is interesting from the perspective of “hedging” assets against each other. The estimated equilibrium relationship will be one in whichthe effect of common risk factors is neutralised or at least minimised, allowing low-riskApplied Quantitative Methods for Trading and Investment. 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
Edited by C.L. Dunis, J. Laws and P. Naı̈m
42
Applied Quantitative Methods for Trading and Investment
combinations of assets to be created. The second aspect is interesting as a potential sourceof statistical arbitrage strategies. Deviations from the long-term “fair price” relationshipcan be considered as statistical “mispricings” and error-correction models can be used tocapture any predictable component in the tendency of these mispricings to revert towardsthe longer term equilibrium.Whilst the econometric methods used in cointegration modelling form the basis of ourapproach, they involve a number of restrictive assumptions which limit the extent to whichthey can be applied in practice. From our somewhat contrasting perspective, the use oftools from cointegration modelling is seen as a “means to an end”, with the “end” beingthe creation of successful trading strategies. In this chapter we explore the application ofcointegration-inspired tools to the task of trading and hedging international equities.For both trading and hedging, the cointegration perspective can be viewed as an extension and generalisation of more established methods. In the case of statistical arbitrage,cointegration can be thought of as a principled extension of the relative value strategies,such as “pairs trading”, which are in common use by market practitioners. In the caseof hedging, the use of a cointegration approach can be viewed as extending factor-modelhedging to include situations where the underlying risk factors are not measurable directly,but are instead manifested implicitly through their effect on asset prices.The structure of the rest of the chapter is as follows. In Section 2.2 we provide amore detailed description of the econometric basis of our approach and illustrate theway in which cointegration models are constructed and how variance ratio tests can beused as a means of identifying potentially predictable components in asset price dynamics. In Section 2.3 we explain how cointegration can be used to perform implicit factorhedging. In Section 2.4 we explain how cointegration can be used to construct sophisticated relative-value models as a potential basis for statistical arbitrage trading strategies.In Section 2.5 we present a controlled simulation in which we show how cointegrationmethods can be used to “reverse engineer” certain aspects of the underlying dynamicsof a set of time series. In Section 2.6 we describe the application of cointegration techniques to a particular set of asset prices, namely the daily closing prices of the 50 equitieswhich constituted the STOXX 50 index as of 4 July 2002; a detailed description of themethodology is provided along with a discussion of the accompanying spreadsheet whichcontains the analysis itself. Finally, Section 2.7 contains a brief discussion of furtherpractical issues together with a concluding summary of the chapter.
2.2 TIME SERIES MODELLING AND COINTEGRATIONIn this section we review alternative methods for representing and modelling time series.Whilst often overlooked, the choice of problem representation can play a decisive role indetermining the success or failure of any subsequent modelling or forecasting procedure.In particular, the representation will determine the extent to which the statistical propertiesof the data are stable over time, or “stationary”.Stable statistical properties are important because most types of model are more suitedto tasks of interpolation (queries within the range of past data) rather than extrapolation(queries outside the range of known data). Where the statistical properties of a systemare “nonstationary”, i.e. changing over time, future queries may lie in regions outside theknown data range, resulting in a degradation in the performance of any associated model.The most common solution to the problems posed by nonstationarity is to attempt toidentify a representation of the data which minimises these effects. Figure 2.1 illustrates
43
Value (yt )
Value (yt )
Cointegration to Hedge and Trade International Equities
Time (t )Value (yt )
Value (yt )
Time (t )
Time (t )
Time (t )
Figure 2.1 Time series with different characteristics, particularly with regard to stationarity: (topleft) stationary time series; (top right) trend-stationary time series; (bottom left) integrated timeseries; (bottom right) cointegrated time series
different classes of time series from the viewpoint of the transformations that are requiredto achieve stationarity.A naturally stationary series, such as that shown in the top-left chart, is one which has astable range of values over time. Such a series can be directly included in a model, eitheras a dependent or independent variable, without creating any undue risk of extrapolation.The top-right chart shows an example of a “trend-stationary” variable; it is stationaryaround a known trend which is a deterministic function of time. A stationary representationof such a variable can be obtained by “de-trending” the variable relative to the underlyingtrend. Some economic time series fall into this category.Series such as that in the bottom-left chart are known as “difference stationary” becausethe period-to-period differences in the series are stationary although the series itself is not.Turning this around, such series can also be viewed as “integrated series”, which representthe integration (sum) of a stationary time series. Artiﬁcial random-walk series and mostasset prices fall into this category, i.e. prices are nonstationary but price differences,returns, are stationary.The two series in the bottom-right chart represent a so-called cointegrated set of variables. Whilst the individual series are nonstationary we can construct a combined series(in this case the difference between the two) which is stationary. As we shall demonstratebelow, some sets of asset prices exhibit cointegration to a greater or lesser degree, leadingto interesting and valuable opportunities for both trading and hedging the assets withinthe set. Another way of looking at cointegration is that we are “de-trending” the seriesagainst each other, rather than against time.The class into which a time series or set of time series fall, whether stationary, integrated, or cointegrated, has important implications both for the modelling approach whichshould be adopted and the nature of any potentially predictable components that the timeseries may contain. Details of a wide range of statistical tests, for identifying both the typeof time series (stationary, nonstationary, cointegrated) and the presence of any potentiallypredictable component in the time series dynamics, are provided in Burgess (1999). Inthis chapter we will concentrate on two main tests: regression-based tests for the presenceof cointegration, and variance ratio tests for the presence of potential predictability.
44
Applied Quantitative Methods for Trading and Investment
The most popular method of testing for cointegration is that introduced by Granger(1983) and is based upon the concept of a “cointegrating regression”. In this approach aparticular time series (the “target series”) y0,t is regressed upon the remainder of the setof time series (the “cointegrating series”) y1,t , . . . , yn,t :y0,t = α + β1 y1,t + β2 y2,t + · · · + βn yn,t + dt
(2.1)
If the series are cointegrated then statistical tests will indicate that dt is stationary and theparameter vector α = (1, −α, −β1 , −β2 , . . . , −βn ) is referred to as the cointegrating vector. Two standard tests recommended by Engle and Granger (1987) are the Dickey–Fuller(DF) and the Cointegrating Regression Durbin–Watson (CRDW). The Dickey–Fuller testis described later in this chapter, as part of the controlled simulation in Section 2.5. Anextensive review of approaches to constructing and testing for cointegrating relationshipsis contained in Burgess (1999).Variance ratio tests are a powerful way of testing for potential predictability in timeseries dynamics. They are derived from a property of unpredictable series where thevariance of the differences in the series grows linearly with the length of the period overwhich they are measured. A simple intuition for this property is presented in Figure 2.2.In the limiting case where all steps are in the same direction the variance of the series willgrow as a function of time squared, at the other extreme of pure reversion the variance of theseries will be independent of time (and close to zero). A random diffusion will be a weightedcombination of both behaviours and will exhibit variance which grows linearly with time.This effect has been used as the basis of statistical tests for deviations from random-walkbehaviour by a number of authors starting with Lo and MacKinlay (1988) and Cochrane(1988). The motivation for testing for deviations from random-walk behaviour is thatthey suggest the presence of a potentially predictable component in the dynamics of atime series. The τ -period variance ratio is simply the normalised ratio of the variance ofτ -period differences to the variance of single-period differences:VR(τ ) =
t
τ
2
(τ yt − τ y)

(yt − y)
(2.2)
2
t
1-periodPerfecttrend:
r
Price
Variance2-period
r2
(r + r )2 = 4r 2
r2
(r − r )2 = 0
r2
2r 2
rTimePerfectreversion:
Price
r
−r
TimeRandom:
Figure 2.2
50/50 mix of trend + reversion
The relationship between variance and time for a simple diffusion process
Cointegration to Hedge and Trade International Equities
45
5
Value
Variance ratio
TrendingRandom walkMean reverting
4TrendingRandom walkMean reverting
32101 2 3 4 5 6 7 8 9 10
Time
Period
Figure 2.3 Example time series with different characteristics (left) and their variance ratiofunctions (right)
By viewing the variance ratio statistics for different periods collectively, we form thevariance ratio function (VRF) of the time series (Burgess, 1999). A positive gradient tothe VRF indicates positive autocorrelation in the time series dynamics and hence trendingbehaviour; conversely a negative gradient to the VRF indicates negative autocorrelation and mean-reverting or cyclical behaviour. Figure 2.3 shows examples of time serieswith different characteristics, together with their associated VRFs. Further examples arecontained in Burgess (1999).For the random walk series, the variance grows linearly with the period τ and hencethe VRF remains close to one. For a trending series the variance grows at a greater thanlinear rate and so the VRF rises as the period over which the differences are calculatedincreases. Finally, for the mean-reverting series the converse is true: the variance growssublinearly and hence the VRF falls below one.
2.3 IMPLICIT HEDGING OF UNKNOWN COMMONRISK FACTORSThe relevance of cointegration to hedging is based upon the recognition that much of the“risk” or stochastic component in asset returns is caused by variations in factors which havea common effect on many assets. This viewpoint forms the basis of traditional asset pricingmodels such as the CAPM (Capital Asset Pricing Model) of Sharpe (1964) and the APT(Arbitrage Pricing Theory) of Ross (1976). Essentially these pricing models take the form:yi,t = αi + βi,Mkt Mktt + βi,1 f1,t + · · · + βi,n fn,t + εi,t
(2.3)
This general formulation relates changes in asset prices yt to sources of systematic risk(changes in the market, Mktt , and in other economic “risk factors”, fj,t ) together withan idiosyncratic asset-speciﬁc component εi,t .The presence of market-wide risk factors creates the possibility of hedging or reducingrisk through the construction of appropriate combinations of assets. Consider a portfolioconsisting of a long (bought) position in an asset y1 and a short (sold) position in an assety2 . If the asset price dynamics in each case follow a data-generating process of the formshown in equation (2.3), then the combined returns y1,t − y2,t are given by:y1,t − y2,t = (α1 − α2 )+ (β1,Mkt − β2,Mkt )Mktt + (β1,1 − β2,1 )f1,t + · · · + (β1,n − β2,n )fn,t+ (ε1,t − ε2,t )(2.4)
46
Applied Quantitative Methods for Trading and Investment
100 %90 %80 %70 %60 %50 %40 %30 %20 %10 %0%
Asset Y1Asset Y2Synthetic Y1 − Y2
basset, Mkt
basset, 1
basset, 2
Idiosyncratic 1 Idiosyncratic 2
Totalvariance
Variance Idiosyncraticfromvariancemarketfactors
Figure 2.4 Attribution of price variance across risk factors: whilst the individual assets Y1 andY2 are primarily inﬂuenced by changes in market-wide risk factors, the price changes of the“synthetic asset” Y1 − Y2 are largely immunised from such effects
If the factor exposures are similar, i.e. β1,j ≈ β2,j , then the proportion of variance whichis caused by market-wide factors will be correspondingly reduced. This effect is illustratedin Figure 2.4.A common approach to hedging is to assume that we can explicitly identify at leastreasonable approximations to the underlying risk factors fj,t and factor sensitivities βi,jand then to create portfolios in which the combined exposure to the different risk factorslies within a desired tolerance. However, in cases where this may not be the optimalapproach, cointegration provides an alternative method of implicitly hedging the commonunderlying sources of risk.More speciﬁcally, given an asset universe UA and a particular “target asset”, T ∈ UA , acointegrating regression can be used to create a “synthetic asset” SA(T ) which is a linearcombination of assets which exhibits the maximum possible long-term correlation with thetarget asset T . The coefﬁcients of the linear combination are estimated by regressing thehistorical price of T on the historical prices of a set of “constituent” assets C ⊂ UA − T :SA(T )t =
Ci ∈C
βi Ci,t
s.t.
{βi } = arg min
t=1,...,n
T t −

2βi Ci,t 
(2.5)
Ci ∈C
As the aim of the regression is to minimise the squared differences, this is a standardordinary least squares (OLS) regression, and the optimal “cointegrating vector” β =(β1 , . . . , βnc )T of constituent weights can be calculated directly by:βOLS = (CT C)−1 Ct
(2.6)
where C is the nc (= |C|) × n matrix of historical prices of the constituents and t =(T1 , . . . , Tn )T is the vector of historical prices of the target asset.The standard properties of the OLS procedure used in regression ensure both that thesynthetic asset will be an unbiased estimator for the target asset, i.e. E[Tt ] = SA(T )t , andalso that the deviation between the two price series will be minimal in a mean-squared-error
Cointegration to Hedge and Trade International Equities
47
sense. The synthetic asset can be considered an optimal statistical hedge for the targetseries, given a particular set of constituent assets C.From an economic perspective the set of constituent assets C act as proxies for theunobserved common risk factors. In maximising the correlation between the target assetand the synthetic asset the construction procedure cannot (by deﬁnition) account forthe “asset-speciﬁc” components of price dynamics, but must instead indirectly optimisethe sensitivities to common sources of economic risk. The synthetic asset represents acombination which as closely as possible matches the underlying factor exposures ofthe target asset without requiring either the risk factors or the exposures to be identiﬁedexplicitly. In Section 2.5, this procedure is illustrated in detail by a controlled experimentin which the cointegration approach is applied to simulated data with known properties.
2.4 RELATIVE VALUE AND STATISTICAL ARBITRAGEIn the previous section we saw that appropriately constructed combinations of prices canbe largely immunised against market-wide sources of risk. Such combinations of assetsare potentially amenable to statistical arbitrage because they represent opportunities toexploit predictable components in asset-speciﬁc price dynamics in a manner which is(statistically) independent of changes in the level of the market as a whole, or other marketwide sources of risk. Furthermore, as the asset-speciﬁc component of the dynamics is notdirectly observable by market participants it is plausible that regularities in the dynamicsmay exist from this perspective which have not yet been “arbitraged away” by marketparticipants.To motivate the use of statistical arbitrage strategies, we brieﬂy relate the opportunitiesthey offer to those of more traditional “riskless” arbitrage strategies. The basic conceptof riskless arbitrage is that where the future cash-ﬂows of an asset can be replicated bya combination of other assets, the price of forming the replicating portfolio should beapproximately the same as the price of the original asset. Thus the no-arbitrage conditioncan be represented in a general form as:|payoff(Xt − SA(Xt ))| < Transaction cost
(2.7)
where Xt is an arbitrary asset (or combination of assets), SA(Xt ) is a “synthetic asset”which is constructed to replicate the payoff of Xt and “transaction cost” represents the netcosts involved in constructing (buying) the synthetic asset and selling the “underlying”Xt (or vice versa). This general relationship forms the basis of the “no-arbitrage” pricingapproach used in the pricing of ﬁnancial “derivatives” such as options, forwards andfutures.1 From this perspective, the price difference Xt − SA(Xt ) can be thought of asthe mispricing between the two (sets of) assets.A speciﬁc example of riskless arbitrage is index arbitrage in the UK equities market.Index arbitrage (see for example Hull (1993)) occurs between the equities constituting aparticular market index, and the associated futures contract on the index itself. Typicallythe futures contract Ft will be deﬁned so as to pay a value equal to the level of the index1
See Hull (1993) for a good introduction to derivative securities and no-arbitrage relationships.
48
Applied Quantitative Methods for Trading and Investment
at some future “expiration date” T . Denoting the current (spot) stock prices as Sti , theno-arbitrage relationship, specialising the general case in equation (2.7), is given by:i (r−qi )(T −t) (2.8)wi St e < costFt −i
100
5280
80
5260
522040520020
Index level
5240
60
5180
3:59:47 PM
3:42:35 PM
3:22:55 PM
3:07:25 PM
2:54:43 PM
2:41:10 PM
2:20:11 PM
1:54:09 PM
1:35:49 PM
1:12:45 PM
12:51:25 PM
12:27:14 PM
12:04:25 PM
5140
11:47:47 AM
−20
11:26:10 AM
516011:02:19 AM
0
10:40:56 AM
Basis (future − spot)
where wi is the weight of stock i in determining the market index, r is the risk-freeinterest rate, and qi is the dividend rate for stock i. In the context of equation (2.7) theweighted combination of constituent equities can be considered as the synthetic assetwhich replicates the indexfutures contract.When the “basis” Ft − i wi Sti e(r−qi )(T −t) exceeds the transaction costs of a particulartrader, the arbitrageur can “lock in” a riskless proﬁt by selling the (overpriced) futurescontract Ft and buying the (underpriced) combination of constituent equities. When themagnitude of the mispricing between the spot and future grows, there are frequently largecorrections in the basis which are caused by index arbitrage activity, as illustrated inFigure 2.5 for the UK FTSE 100 index.Many complex arbitrage relationships exist and “riskless” arbitrage is an importantsubject in its own right. However such strategies are inherently self-limiting – as competition amongst arbitrageurs grows, the magnitude and duration of mispricings decreases.Furthermore, in practice, even arbitrage which is technically “riskless” will still involve acertain level of risk due to uncertain future dividend rates qi , trading risks, and so on. Fromthis perspective the true attraction of index arbitrage strategies lies less in the theoreticalprice relationship than in a favourable property of the mispricing dynamics – namely atendency for the basis risk to “mean revert” or ﬂuctuate around a stable level.
Figure 2.5 Illustration of index arbitrage opportunities in the UK equity market; the data consists of 3200 prices for the FTSE 100 index (in bold) and the derivative futures contract expiringSept. 98; the lower curve shows the so-called “basis”, the deviation from the theoretical fair pricerelationship between the two series; the data sample covers the period from 10.40am to 4pm on15 September 1998; some of the abrupt price shifts will be due to arbitrage activity
Cointegration to Hedge and Trade International Equities
49
Building upon this insight, the premise of “statistical arbitrage” is that regularities incombinations of asset prices can be exploited as the basis of proﬁtable trading strategies,irrespective of the presence or absence of a theoretical fair price relationship between theset of assets involved.Whilst clearly subject to a higher degree of risk than “true” arbitrage strategies, statistical arbitrage opportunities offer the hope of being both more persistent and more prevalentin the markets. More persistent because risk-free arbitrage opportunities are rapidly eliminated by market activity. More prevalent because in principle they may occur betweenany set of assets rather than solely in cases where a suitable “risk-free” hedging strategycan be implemented.A simple form of statistical arbitrage is “pairs trading”, which is in common use by anumber of market participants, such as hedge funds, proprietary trading desks and other“risk arbitrageurs”. Pairs trading is based on a relative value analysis of two asset prices.The two assets might be selected either on the basis of intuition, economic fundamentals,long-term correlations or simply past experience. A promising candidate for a pairs strategy might look like the example in Figure 2.6, between HSBC and Standard Chartered.The pairs in Figure 2.6 show a clear similarity to the riskless arbitrage opportunitiesshown in Figure 2.5. In both cases the two prices “move together” in the long term,with temporary deviations from the long-term correlation which exhibit a strong meanreversion pattern. Note however that in the “statistical arbitrage” case the magnitude ofthe deviations is greater (around ±10% as opposed to <0.5%) and so is the time periodover which the price corrections occur (days or weeks as opposed to seconds or minutes).Opportunities for pairs trading in this simple form, however, are dependent upon theexistence of similar pairs of assets and thus are naturally limited. By constructing synthetic“pairs” in the form of appropriate combinations of two or more assets, cointegrationtechniques provide a sophisticated and powerful method to generalise the relative valueapproach and create a wider range of potential trading opportunities. Once a cointegrating700
200STAN
150
600
HSBC
Deviation
400503000
Prices
500
100
200
29/09/98
24/09/98
22/09/98
17/09/98
15/09/98
09/09/98
07/09/98
02/09/98
0
27/08/98
−100
25/08/98
10020/08/98
−50
Figure 2.6 Illustration of potential statistical arbitrage opportunities in the UK equity market;the chart shows equity prices for Standard Chartered and HSBC, sampled on an hourly basis from20 August to 30 September 1998. Note the mean-reverting nature of the deviation
50
Applied Quantitative Methods for Trading and Investment
regression has been performed to estimate the “fair price” relationship between a set ofassets, tools such as variance ratio analysis can be used to detect deterministic componentsin the mispricing dynamics that could be used as the basis of a “statarb” strategy.In this and the previous section we have provided a motivation for the use of cointegration-based techniques for both hedging and trading. In the following section wesupplement this qualitative motivation with some quantitative results obtained from applying the techniques in a controlled simulation with known time series dynamics.
2.5 ILLUSTRATION OF COINTEGRATIONIN A CONTROLLED SIMULATIONNow that we have described the rationale for applying cointegration-based techniquesin trading, the next sections provide examples of how these techniques can be used inpractice. In Section 2.6 we will explore the application of cointegration techniques to realasset prices. But before we do that, this section highlights the way in which the techniqueswork by means of an artiﬁcial example in which the underlying dynamics of the time seriesare controlled. Consider the example of a set of three assets, each following a two-factorversion of the data-generating process shown in equation (2.3). In this controlled examplewe specify the factor exposures of three assets X, Y and Z as shown in Table 2.1, i.e. pricechanges within the set of three assets X, Y and Z are driven by a total of ﬁve factors, twocommon risk factors f1 and f2 and three asset-speciﬁc components ε1 , ε2 , ε3 . Furthermorelet us specify that f1 and f2 follow random-walk processes whilst the dynamics of theasset-speciﬁc factors contain a mean-reverting component. As discussed in Section 2.4,these dynamics might also be plausibly the case in reality because predictable effects inmarket-wide factors would be easily observed and thus “arbitraged away”, whilst smallpredictable components in asset-speciﬁc dynamics might be less obvious and hence alsomore persistent.Based on the assumptions described above, let us specify the full dynamics of theresulting time series by the following equations:fi,t = ηi,t
i = 1, 2
εj,t = −0.1εj,t + ej,t
ηi,t ∼ N (0,1)
j = 1, 2, 3 ej,t ∼ N (0,0.25)
Xt = f1,t + f2,t + ε1,t
(2.9)
Yt = f1,t + 0.5f2,t + ε2,tZt = 0.5f1,t + f2,t + ε3,tTable 2.1 Price sensitivity of three assets X, Yand Z to changes in common risk factors f1 andf2 and asset-speciﬁc effects ε1 , ε2 , ε3Asset
f1
f2
ε1
ε2
ε3
XYZ
110.5
10.51
100
010
001
Cointegration to Hedge and Trade International Equities
51
2015XYZ
1050−5
1
101
201
301
401
501
Figure 2.7 Realisation of three simulated asset price series which are driven by two underlyingcommon factors in addition to asset-speciﬁc components
i.e. the unobserved “factor” dynamics of f1 and f2 are driven by the pure noise terms ηi,t ;the also-unobserved asset-speciﬁc dynamics εj,t are a combination of noise terms ej,t with“error correction” mean-reversion terms with parameter −0.1; the observed asset dynamicsXt , Yt and Zt are determined by their different exposures to the ﬁve underlying factors.The precise “shapes” of the time series will depend on the sampled innovations ηi,tand ei,t . A particular realisation of the asset prices generated by the system is shown inFigure 2.7 and this is used as the basis of the analysis below. Note that the commonfactor exposures create a broad similarity between the observed price movements of thethree assets.As described in Section 2.3, we estimate the underlying fair price relationship from theobserved data by performing a cointegrating regression. In this case, we arbitrarily selectX as the “target series” and regress on the other two “cointegrating series” Y and Z. Theresulting relationship estimated by the regression is given by:Xt = 0.632Yt + 0.703Zt + mt
(2.10)
Due to sampling error, the estimated relationship differs slightly from the true underlyingrelationship Xt = 2/3Yt + 2/3Zt + m∗t , which would precisely cancel the factor exposuresand leave a pure combination (m∗t ) of the asset-speciﬁc terms. However, it is clear that thecointegrating regression has been able to construct a combination which largely neutralisesthe common risk factors, and that it has done this without any explicit knowledge of (oreven estimation of) the factor exposures shown in Table 2.1. It is because they bypassthe need to estimate explicit factor exposures that we refer to cointegration techniques asperforming “implicit” hedging of market-wide risk factors.In this example, the asset-speciﬁc dynamics have been constructed so as to be meanreverting, so the error term of the regression can be considered as a statistical “mispricing”which represents the temporary deviation from the estimated “fair price” relationshipbetween the three assets. Unlike the nonstationary asset prices X, Y and Z, the estimatedmispricing mt , which is illustrated in Figure 2.8, can clearly be seen to be mean reverting.The mean-reverting nature of the mispricing time series, compared to the close torandom-walk behaviour of the original time series X, Y and Z, is highlighted by thevariance ratio proﬁles shown in Figure 2.9. Whilst the variance ratio for all three originalassets remains close to unity in each case, the variance ratio of the mispricing falls substantially below one as the period over which the differences are calculated increases. Thisindicates that the volatility which is present in the short-term dynamics is not reﬂected in
52
Applied Quantitative Methods for Trading and Investment1.510.50
Mis
−0.5−1−1.5−2
1
101
201
301
401
501
Figure 2.8 The estimated “mispricing” time series, mt = Xt − (0.632Yt + 0.703Zt )1.4Variance ratio
1.21
VR(X)VR(Y)VR(Z)VR(Mis)
0.80.60.40.201
11
21
31
41
51
61
71
81
91
Period
Figure 2.9 Variance ratio proﬁles for the time series X, Y and Z