in a random walk.Armed with this information, we can conclude that the Kalman smoothing approach may be applied to the random walk sequence sampled at multiple frequencies to achieve varying degrees of coarseness. In our example,we calculate the state values by sampling the random walk at half the frequency of the available data. Suppose we need to calculate the value of thestate at time t = 22. We use the observations y22 ,y20 ,y18 , . . . . Similarly, to estimate the state value at time t = 23, we use the observation valuesy23 ,y21 ,y19 , . . . . The resulting smoothing on the logarithm of the S&P pricesis shown in Figure 4.4.
SUMMARYThe Kalman filter is an optimal state estimation process applied to a dynamic system that involves random perturbations.Inherent in any discussion on the Kalman filter are the notions of stateand observation.Kalman filtering may be summarized as a three-step process comprisingprediction, observation, and correction, or reconciliation of the prediction with the observation.The simplest case of the Kalman filter reduces to finding the average ofn numbers.The recursive least squares method is also a special case of the Kalmanfilter that may be applied to filtering random walks.When the state and observation variances are the same, that is, the signal-to-noise ratio is unity, then the estimation of the Kalman states fora random walk boils down to a weighted average of the observations,with the weights formed by ratios of Fibonacci numbers.The degree of smoothness to be achieved in a random walk can be controlled by varying the sampling rate of the random walk sequence.
FURTHER READING MATERIALKalman FilterMaybeck, Peter S. Stochastic Model, Estimation and Control. (San Diego, California:Academic Press, 1979).
68
BACKGROUND MATERIAL
Harvey, A. C. Forecasting, Structural Time Series Model and the Kalman Filter.(Cambridge, UK. Cambridge University Press, 1991).Variance CalculationIsmail, Malik Magdon and Amir Atiya. “A Maximum Likelihood Approach to Variance Estimation for a Brownian Motion Using the High Low and Close.” Quantitative Finance (forthcoming).Rogers, L. and S. Satchell. “Estimating Variance from High Low and ClosingPrices.” Annals of Applied Probability 1, no. 4 (1991): 504–512.Rogers, L., S. Satchell and Y. Yoon. “Estimating the Volatility of Stock Prices: AComparison of Methods That Use High and Low Prices.” Applied FinancialEconomics 4 (1994): 241–247.Fibonacci SeriesHuntley, H. E. The Divine Proportion: A Study in Mathematical Beauty. (New York:Dover Publications, 1970).Fischer, Robert. Fibonacci Applications and Strategies for Traders. (New York: JohnWiley & Sons, Inc., 1993).
Kalman Filtering
69
APPENDIXWe describe the formulas for the Kalman filtering steps here. The notationwe use is the same as that discussed in the section on the Kalman filter.1. Evaluate Xˆ t | t −1 and Pˆt | t −1 using the state equation.Xˆ t | t −1 = AXˆ t −1| t −1Pˆ= APˆATt | t −1
t −1| t −1
2. Find the observation Yt and R by observing the system. Note that wehave the matrix H defined as follows:Yt = HXt + vt3. Compute the Kalman gain Kt.
(
Kt = Pˆt H T HPˆt H T + R
(
)
−1
)
4. Evaluate Xˆ t | t given by Xˆ t | t −1 + Kt Yt − HXˆ t |t −1 .5. Evaluate Pˆt | t .
(
)
(
)
TPˆt | t = 1 − KH Pˆt | t −1 1 − KH + KRK T
PART
TwoStatistical ArbitragePairs
CHAPTER
5
Overview
HISTORYThe first practice of statistical pairs trading is attributed to Wall Streetquant Nunzio Tartaglia, who was at Morgan Stanley in the mid 1980s. Atthe time, he assembled a group of mathematicians, physicists, and computerscientists. Their mission was to develop quantitative arbitrage strategiesusing state-of-the-art statistical techniques. The strategies developed by thegroup were automated to the point where they could generate trades in a73
74
STATISTICAL ARBITRAGE PAIRS
mechanical fashion and, if needed, execute them seamlessly through automated trading systems. At that time, trading systems of this kind were considered the cutting edge of technology.One of the techniques they used for trading involved trading securitiesin pairs. The process involved identifying pairs of securities whose pricestended to move together. Whenever an anomaly in the relationship was noticed, the pair would be traded with the idea that the anomaly would correctitself. This came to be known on the street as “pairs trading.” Tartaglia andhis group employed pairs trading with great success in 1987. The group,however, disbanded in 1989. Members of the group found themselves invarious other trading firms, and knowledge of the idea of pairs trading gradually spread. Pairs trading has since increased in popularity and has becomea common trading strategy used by hedge funds and institutional investors.
MOTIVATIONLet us now explain the idea behind pairs trading. The general theme forinvesting in the marketplace from a valuation point of view is to sell overvalued securities and buy the undervalued ones. However, it is possible todetermine that a security is overvalued or undervalued only if we also knowthe true value of the security in absolute terms. But, this is very hard to do.Pairs trading attempts to resolve this using the idea of relative pricing; thatis, if two securities have similar characteristics, then the prices of both securities must be more or less the same. Note that the specific price of the security is not of importance. The price may be wrong. It is only important thatthe prices of the two securities be the same. If the prices happen to be different, it could be that one of the securities is overpriced, the other securityis underpriced, or the mispricing is a combination of both.Pairs trading involves selling the higher-priced security and buying thelower-priced security with the idea that the mispricing will correct itself inthe future. The mutual mispricing between the two securities is captured bythe notion of spread. The greater the spread, the higher the magnitude ofmispricing and greater the profit potential. A long–short position in the twosecurities is constructed such that it has a negligible beta and therefore minimal exposure to the market. Hence, the returns from the trade are uncorrelated to market returns, a feature typical of market neutral strategies.Based on the discussion so far, it is easy to deduce that the key to success in pairs trading lies in the identification of security pairs. In a study byGatev et al., a purely empirical approach to achieving this end was adopted.They methodically chose pairs based entirely on the historical price movement of securities and checked to see how pairs trading would have fared ina double-blind study. Besides the set of pairs chosen using historical prices,
Overview
75
another set of pairs was created by randomly pairing the securities with oneanother. The trades that would have been executed based on the empiricalpairing approach were then pitted against the trades where the securitieswere randomly paired. The difference in the returns between the two groupswas found to be statistically significant, and the return generated by the methodically paired set was better than the randomly paired sample set.Unlike the purely empirical approach, the methodology that we subscribe to comprises theoretical valuation concepts that are then validatedwith empirical models and data. We will later show that the theoretical valuation approach helps us to easily identify pairs based on the fundamentalsof the firm. It also leads naturally to the formula used to measure the spread,the degree of mispricing between the two securities. Our theoretical explanation for the comovement of security prices stems from arbitrage pricingtheory (APT). According to APT, if two securities have exactly the same riskfactor exposures, then the expected return of the two securities for a giventime frame is the same. The actual return may, however, differ slightly because of different specific returns for the two securities. It is important tonote at this point that APT for the two securities has to be valid in all timeframes. Let the price of securities A and B at time t be ptA and ptB, and at timet + i be ptA+ i and ptB+ i, respectively. The return in the time period i for the twosecurities is given as log(ptA ) − log(ptA+ i ) and log(ptB ) − log(ptB+ i ).1Now let us say that we have the prices of both securities at the currenttime. The return on both securities is expected to be the same in all timeframes. In other words, the increment to the logarithm of the prices at thecurrent time must be about the same for both the securities at all time instances in the future. This, of course, means that the time series of the logarithm of the two prices must move together, and the spread calculationformula is therefore based on the difference in the logarithm of the prices.Having explained our approach, we now need to define in precise termswhat we mean when we say that the price series or the log price series ofthe two securities must move together. Fortunately for us, the idea of comovement of two time series has been well developed in the field of econometrics. We discuss it in the following section on cointegration.
COINTEGRATIONIn the introduction to time series we briefly discussed the preprocessing stepfor nonstationary series. The series is typically transformed into a stationarypt + i − pt. Thus, return canptbe thought of as the increment in the logarithm of the prices.
1
The value as calculated here is approximately equal to
STATISTICAL ARBITRAGE PAIRS
76
time series by differencing. By extension, when analyzing multivariate timeseries where each of the component series is nonstationary, it would thenmake sense to difference each component and then subject them to examination. However, that need not be the case.In the course of examining multivariate series to determine statisticallyif there is a cause–effect relationship between the variables represented bythe time series, the econometricians Engle and Granger observed a rather interesting phenomenon. Even though two time series are nonstationary, it ispossible that in some instances a specific linear combination of the two is actually stationary; that is, the two series move together in somewhat of alockstep. Engle and Granger coined the term cointegration and proposedthe idea in an article, the reference for which is at the end of this chapter.Notably, this was one of the ideas for which they won the Nobel Prize ineconomics in 2003.Let us now state the idea of cointegration more formally. Let yt, and xtbe two nonstationary time series. If for a certain value g, the series yt – gxt isstationary, then the two series are said to be cointegrated. Real-life examplesof cointegration abound in economics. In fact, the first demonstrations andtests of cointegration involved economic variable pairs like consumptionand income, short-term and long-term rates, the M2 money supply andGDP, and so forth.The explanation for cointegration dynamics is captured by the notion oferror correction. The idea behind error correction is that cointegrated systems have a long-run equilibrium; that is, the long-run mean of the linearcombination of the two time series. If there is a deviation from the long-runmean, then one or both time series adjust themselves to restore the long-runequilibrium. The formal theorem stating that error correction and cointegration are essentially equivalent representations is called the Granger representation theorem. We shall not attempt to discuss the proof of thetheorem, but simply present here the error correction representation.Let ext be the white noise process corresponding to time series xt .Let ε yt be the white noise process corresponding to the time series yt . Theerror correction representation is
{ }
(= α (y
)−γx )+ε
yt − yt −1 = α y yt −1 − γ xt −1 + ε ytxt − xt −1
x
t −1
t −1
{ }
(5.1)
xt
Let us interpret the Equations 5.1. The left-hand side is the increment to thetime series at each time step. The right-hand side is the sum of two expressions, the error correction part and the white noise part. Let us look at theerror correction part ay(yt–1 – gxt–1) from the first equation. The term yt–1 –gxt–1 is representative of the deviation from the long-run equilibrium (equi-
Overview
77
librium value is zero in this case), and g is the coefficient of cointegration. ayis the error correction rate, indicative of the speed with which the time seriescorrects itself to maintain equilibrium. Thus, as the two series evolve withtime, deviations from the long-run equilibrium are caused by white noise,and these deviations are subsequently corrected in future time steps.We will now illustrate that the idea of error correction does indeed leadto a stationary time series for the spread. Two independent white noise series with zero mean and unit standard deviation were generated to representε yt and ε xt, respectively. The other values were set as ay = –0.2, ax = 0.2, andg = 1.0. Note that it is important to have the two coefficients ay and ax setto opposite signs for error-correcting behavior. The values for the two timeseries xt and yt were then generated using the simulated data and theequations from the error correction representation. A plot of the two seriesis shown in Figure 5.1.Subsequently, the spread at each time instance was calculated using theknown value for g. A plot of the spread series and its autocorrelation isshown in Figures 5.2a and b. It is easy to appreciate from the autocorrelation function that the spread series is indeed stationary.A more direct approach to model cointegration is attributed to Stock andWatson, called the common trends model. The primary idea of the common
{ }
{ }
1086420Time Series 1Time Series 2
–2
10
30
FIGURE 5.1
50
70
Cointegrated Time Series.
90
STATISTICAL ARBITRAGE PAIRS
78
86420–2–4–610
30
50
FIGURE 5.2A
70
90
Spread.
Auto Correlation
1.00.80.60.40.2–0.0–0.20
10Lag
5
FIGURE 5.2B
15
20
Spread ACF.
trends model is that of a time series being expressed as a simple sum of twocomponent time series: a stationary component and a nonstationary component. If two series are cointegrated, then the cointegrating linear composition acts to nullify the nonstationary components, leaving only thestationary components. To see what we mean, consider two time seriesyt = nyt + ε yt
(5.2)
z t = n zt + ε ztwhere nyt and n zt are the random walk (nonstationary ) components of thetwo time series, and ε yt and ε zt are the stationary components of the timeseries. Also, let the linear combination yt – gzt be the cointegrating combi-
Overview
79
nation that results in a stationary time series. Expanding the linear combination and rearranging some terms, we haveyt − γ z t = (nyt − γ n zt ) + (ε yt − γ ε zt )
(5.3)
If the combination in Equation 5.3 must be stationary, the nonstationarycomponent must be zero, implying that nyt = γ n zt , or the trend componentof one series must be a scalar multiple of the trend component in the otherseries. Therefore, for two series to be cointegrated, the trends must be identical up to a scalar. In later chapters, we will rely on the Stock-Watsonmodel to establish links between arbitrage pricing theory and cointegration.
HIGHLIGHTING THE POINTThis is an anecdote about an ingenious little kid. He was asked on atest to say a few sentences about a cow. The poor lad knew only howto say a few sentences about a tree. Thinking for a moment, the kid inhis first sentence tied the cow to a tree and then went on to talk aboutthe tree. The example is probably a little tongue in cheek. It is, however, true that there is a strong urge to relate the unknown to something familiar and enhance understanding through association.To further underscore the point, it is worth mentioning that thespirit of that approach actually forms the basis for formal proof techniques in mathematics. Proof by induction, a technique attributed toCantor, relies on forming a series of logical relationships from themost general to the most trivial. Proofs of NP completeness, used toclassify algorithms in the area of computational complexity theory (attributed to Richard Karp), also rely on transformations to a knownproblem. It is probably safe to say that in almost all fields of humanendeavor there is a common tendency to relate the intractable to something manageable and leverage existing knowledge to arrive at meaningful conclusions.We are no different from everyone else in this respect. When facedwith the prospect of having to work with nonstationary time series, weimmediately look for ways to construct portfolios that can be relatedto stationary time series. The transformation to stationarity is typicallyachieved using cointegration ideas and strict parity relationships.Needless to say, this approach appears as a recurrent theme in the design of trading strategies across all asset classes.
STATISTICAL ARBITRAGE PAIRS
80
APPLYING THE MODELIn this section, we fit the cointegration model to the logarithm of stockprices. For the cointegration model to apply, we would require the logarithmof stock prices to be a nonstationary series. The assumption that the logarithm of stock prices is a random walk (read as nonstationary) is a ratherstandard one. It has been used fairly extensively in option-pricing modelswith satisfactory results. We are therefore good on that assumption andready to proceed further.Let us say that two stocks A and B are cointegrated with the nonstationary time series corresponding to them being log ptA and log ptB ,respectively. Applying the error correction representation described here,we have
{ ( )}
( ) ( )( )( )log( p ) − log( p ) = α log( p ) − γ log( p ) + ε
{ ( )}
log ptA − log ptA−1 = α A log ptA−1 − γ log ptB−1 + ε ABt
Bt −1
B
At −1
Bt −1
(5.4)
B
The parameters that uniquely determine the model are the cointegration coefficient g and the two error correction constants aA and aB. Therefore, estimating the model involves determining the appropriate values for aA, aB,and g. The left-hand side of the Equations 5.4 is the return of the stocks inthe current time period. On the right-hand side, note the expression for thelong-run equilibrium, log(ptA−1 ) − γ . log(ptB−1 ), in both the equations. Inwords, it is the scaled difference of the logarithm of price. Incidentally, thiscoincides with what we termed the spread in our earlier discussion. Also notice that the subscripts for stock prices in the expression for the long-runequilibrium is t – 1. The past deviation from equilibrium plays a role in deciding the next point in the time series. Therefore, knowledge of the past realizations may be used to give us an edge in predicting the increments to thelogarithm of prices; that is, returns. This is important and exciting. Eventhough both stocks follow a log-normal process, one can eke out some predictability in their returns based on past realizations. Thus, one can attemptto trade either of the stocks in the pair based on predictions using the estimated values from the error correction representation.Let us now focus on the cointegration part of the representation theorem.This is the assertion that the time series of the long-run equilibrium (alsotermed spread in our case) is stationary and mean reverting. Now, we definitely know a lot about predicting stationary time series. If only we could associate the time series of the spread to a portfolio, we could consider tradingthe portfolio based on our prediction of the time series values. We proceed
Overview
81
to do exactly that. Consider a portfolio with long one share of A and shortg shares of B. The return of the portfolio for a given time period is given as
[log(p ) − log(p )] − γ [log(p ) − log(p )] .At +i
At
Bt +i
Bt
(5.5)
COINTEGRATION AND TRACKING ERRORTracking error is an idea that arises naturally in the context of an investment technique called indexing. The basic premise for indexing isthe notion that it is extremely hard to time the market. Therefore, thestrategy adopted is to passively invest in the market index. Alternately,to reduce trading costs, the investment is made in portfolios designedto mimic index returns. These portfolios are sometimes referred to astracking baskets. The ability of the tracking basket to mimic the market returns is characterized by its tracking error. The tracking errormay therefore be thought of as a measure of discrepancy or margin oferror that one can expect in the tracking process.Implicit in the definition of tracking error is the time interval forwhich the returns are measured. It is typically one year (assumed tohave 252 trading days). If the holding period is different from oneyear, then the tracking error needs to be scaled accordingly. The convention is to assume that the tracking error is a random walk seriesand scale the tracking error using the formula below. If T is the holding period for the portfolio, thenerr(T ) =
err(1 year)
T
(5.6)
252
The tricky part here is in the random walk assumption for trackingerror in the process of scaling. In the case of long–short portfolios consisting of cointegrated stocks, the tracking error is not a random walkseries. As a matter of fact, it is a stationary time series. The variance orstandard deviation in this case is independent of the holding period. Inother words, the tracking error remains the same regardless of theholding period, and scaling formulas are not required.We therefore caution against making the random walk assumption blindly, particularly in indexing situations, because they tend todistort the tracking error.
STATISTICAL ARBITRAGE PAIRS
82
Rearranging the terms a little bit, we have
[log(p ) − γ log(p )] − [log(p ) − γ log(p )] = spreadAt +i
Bt +i
At
Bt
t +i
− spread t (5.7)
Therefore, the return on the portfolio is the increment to the spread value inthe time period i. We have successfully associated a portfolio with a stationary time series. The one thing that remains is providing an interpretationfor g, the cointegration coefficient. We will discuss this in later chapters.
A TRADING STRATEGYWe now construct a simple trading strategy. The idea is to trade on the oscillations about the equilibrium value for the spread. We could put on atrade upon deviation from the equilibrium value and unwind the trade whenequilibrium is restored. However, note that the equilibrium value is also themean value of the time series. Therefore, given that the spread swingsequally in both directions about the equilibrium value, we could potentiallyunwind the trade when the spread deviates in the other direction. Thiswould reduce the trading frequency on average by a factor of two. Giventhat stocks have a bid-ask spread, we would incur a trading slippage everytime a trade is executed. Reducing the trading frequency reduces the effectof this slippage.Let us therefore consider the strategy where the trades are put on andunwound on a deviation of ∆ on either direction from the long-run equilibrium m. We buy the portfolio (long A and short B) when the time series is ∆below the mean and sell the portfolio (sell A and buy B) when the time series is ∆ above the mean in i time steps.
( )( )log( p ) − γ log( p ) = µ + ∆log ptA − γ log ptB = µ − ∆At +i
(5.8)
Bt +i
The profit on the trade is the incremental change in the spread, 2∆.
ExampleThe DataConsider two stocks A and B that are cointegrated with the following data:Cointegration Ratio = 1.5Delta used for trade signal = 0.045Bid price of A at time t = $19.50Ask price of B at time t = $7.46
Overview
83
Ask price of A at time t + i = $20.10Bid price of B at time t + i = $7.17Average bid-ask spread for A = .0005 percent (5 basis points)Average bid-ask spread for B = .0010 percent ( 10 basis points)The StrategyWe first examine if trading is feasible given the average bid-ask spreads.Average trading slippage = ( 0.0005 + 1.5 × 0.0010)= .002 ( 20 basis points)This is smaller than the delta value of 0.045. Trading is therefore feasible.At time t, buy shares of A and short shares of B in the ratio 1:1.5.Spread at time t
= log (19.50) – 1.5 × log (7.46)= –0.045
At time t + i, sell shares of A and buy back shares the shares of B.Spread at time t + i
= log (20.10) – 1.5 × log (7.17)= 0.045
Total return = return on A + g × return on B= log (20.10) – log(19.50) + 1.5 × (log(7.46) – log(7.17) )= 0.3 + 1.5 × 4.0= .09 (9 percent)
ROAD MAP FOR STRATEGY DESIGNThe discussion so far briefly outlines how we might trade once we know twostocks are cointegrated. We do concede that the course of the discussion sofar has brought up more questions on the details. How do we identify candidate stock pairs? Can we verify that they are indeed cointegrated? How dowe determine the cointegration coefficient? What is the most appropriatevalue for delta? We explore the questions and issues involved in the subsequent chapters. To that end, we provide a road map for the design andanalysis of the pairs trading strategy.The steps involved are as follows:1. Identify stock pairs that could potentially be cointegrated. This processcan be based on the stock fundamentals or alternately on a pure statistical approach based on historical data. Our preferred approach is tomake the stock pair guesses using fundamental information.
STATISTICAL ARBITRAGE PAIRS
84
2. Once the potential pairs are identified, we verify the proposed hypothesis that the stock pairs are indeed cointegrated based on statistical evidence from historical data. This involves determining the cointegrationcoefficient and examining the spread time series to ensure that it is stationary and mean reverting.3. We then examine the cointegrated pairs to determine the delta. A feasible delta that can be traded on will be substantially greater than the slippage encountered due to the bid-ask spreads in the stocks. We alsoindicate methods to compute holding periods.
SUMMARYStatistical pairs trading is a relative value arbitrage on two securities andis based on the premise that there is a long-run equilibrium between theprices of the stocks composing the pair.The degree of deviation from the long-run equilibrium is called thespread and represents the extent of mutual mispricing.Any deviation from the long-run equilibrium is compensated for in subsequent movements of the time series.Pairs trading involves trading on the oscillations about the equilibriumvalue.The econometric paradigm of cointegration and error correction is central to the analysis of the pairs-trading strategy.
FURTHER READING MATERIALPairs TradingGatev, Evan, G., William, N. Goetzmann, and K. Greet Rouwenhorst. Pairs Trading:Performance of a Relative Value Arbitrage Rule. NBER Working Papers 7032,National Bureau of Economic Research Inc., 1999.CointegrationEngle, Robert F. and C. W. Granger. “Co-integration and Error Correction: Representation, Estimation and Testing.” Econometrica 55, no. 2 (March 1987):251–276.Stock, James H. and Mark W. Watson. “Testing for Common Trends.” Journal ofthe American Statistical Association 83, no. 404 (December 1988): 1097–1107.Enders, Walter. Applied Econometric Time Series. (New York: John Wiley & Sons,Inc., 1995).
CHAPTER
6
Pairs Selection inEquity Markets
INTRODUCTIONIn Chapter 5, we explained that strategy design was essentially a three-stepprocess. The three steps are identification of stock pairs, cointegration testing, and trading rule formulation. In this chapter, we focus on the first of thethree steps, the process of identifying potential stock pairs. In this step we essentially short-list the pairs for cointegration testing and further analysis.Why do we need to do that? We could just as well work with a candidate listof all possible pairs, run cointegration tests on all of them, and eliminatepairs that fail the tests. This certainly seems reasonable, except for the factthat in a universe of 5,000 stocks, we have a total of about 12 million pairs.85
86
STATISTICAL ARBITRAGE PAIRS
Running tests on 12 million pairs is definitely not a viable option. Therefore,the natural inclination is to try to reduce the number.The most expedient way (not necessarily the best) to accomplish that isby using heuristics, or rules of thumb. In the heuristics approach, the list ofpairs is explicitly partitioned into two sets: potentially cointegrated and notpotentially cointegrated. This partitioning is accomplished by applying a setof rules. The rules are designed to exclude pairs with a slim chance of beingcointegrated. This limits the number of pairs in the candidate list and reduces the number of cointegration tests that need to be performed. Although this approach seems reasonable, it is best characterized as being adhoc. Different people have different belief systems about the market. Therefore, reasonable people can come up with dramatically different rule setsbased on their personal experiences. This makes the rule sets anecdotal innature. Furthermore, it is possible for individuals to hold opposing views.An aggregation of rules representing the beliefs of multiple individuals mayend up being inconsistent. This is now a case where the whole is lesser thanthe sum of its parts and results in missed opportunities. It might thereforebe useful to put some thought into this to come up with a more definitivemethodology.The methodology we prescribe here is distinctly different from the rulesof thumb or heuristics approach. Instead of attempting to evaluate explicitpartitions, this approach aims to arrive at a relative ordering of the pairsbased on the degree of comovement. Each pair is associated with a score/distance measure. The higher the score, the greater the degree of comovement, and vice versa. Notably, such a structure lends itself to deductive reasoning. If we find that a pair is unsuitable for pairs trading, then we havegood reason to believe that every pair with a score/distance measure worsethan the current pair is also unsuitable. The pair selection process now becomes equivalent to choosing a suitable threshold value for the distancemeasure. Notice that this approach relies solely on the distance measure forordering the pairs. Therefore, a proper choice of the distance measure is keyto the pairs selection process.Let us quickly examine the properties that would be desirable of thescore/distance measure. First, if the evaluation of the score/distance measuretook as much effort as cointegration testing, then it would defeat the purpose. We may as well test for cointegration directly with all the pairs in theexhaustive list. Therefore, at the very least, the evaluation of the distancemeasure must be relatively easy and straightforward. Additionally, it is desirable that the distance measure not be completely empirical. Empirical deductions rely solely on historical data. This comes with an underlyingassumption that the fundamentals of the firm are essentially static. There areno changes that impact the valuation of the firm. Needless to say, this neednot be true. Ideally, we would prefer to tie the evaluation of the distance
Pairs Selection in Equity Markets
87
measure to the fundamentals of the firm. Accomplishing this would helpprovide a theoretical justification for the use of the score/distance measureand therefore a sound economic rationale for our expectation of cointegrated behavior.In