ated stock pairs can be compiled by the process of identifying similar stocks.The notion of similarity is formalized using a distance measure betweentwo stocks.The distance measure is based on an APT model possibly with fundamental risk factors.The candidate list of pairs is determined by choosing pairs with distancevalues within a certain threshold.The distance measure is the absolute value of the common factor correlation between the two stocks.If the common factor correlation is +1 or –1 and the integration of thespecific return series of the stocks involved are stationary, then conditions for cointegration are satisfied.It may be possible to trade pairs of stocks even though they deviate fromideal conditions of cointegration.The signal-to-noise ratio as defined is a measure of the deviation fromthe ideal condition of cointegration.
FURTHER READING MATERIALCointegration PropertiesPhilips, P. C. B. and S. N. Durlauf. “Multiple Time Series Regression with IntegratedProcesses,” Review of Economic Studies 53 (1986): 473–495.Park, J. Y. S. Oularis and B. Choi. “Spurious Regressions and Tests for Co-integration” (CAE working paper 88–07, Cornell University, Ithaca, New York(1988).)Near CointegrationHaldrup, Niels and Michael Jansson. “Spurious Regression, Co-integration andNear Co-integration: A Unifying Approach” (working papers, DK-8000, Department of Economics, University of Aarhus, Denmark, 1999.)
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APPENDIX: EIGENVALUE DECOMPOSITIONConsider a scalar l and a corresponding vector v. They are an eigenvalue,eigenvector pair of a square matrix A if they satisfy the equationAv = lvThe equation means that the vector v is special with respect to the matrix A.Multiplying the vector with the matrix does not change the direction or orientation of the vector. Its magnitude, however, is multiplied by the scalar l.A square n × n matrix may have n eigenvalues l1,l2,...,ln and n corresponding eigenvectors v1,v2,...,vnThereforeAv1 = l1v1Av2 = lv2MAvn = lvnLet us write this in matrix form. We construct a diagonal matrix D with theeigenvalues forming the diagonal and a matrix U with each column corresponding to an eigenvectors. ThenAU = UDorA = UDU–1Note that we now have the matrix A in terms of its eigenvalues and eigenvectors. This is the eigenvalue decomposition of A. The covariance matrix Fhas special properties, and in that caseU–1 = UTandF = UDUT
CHAPTER
7
Testing For Tradability
INTRODUCTIONIn Chapter 6 we discussed the process of choosing potential stock pairs. Inthis chapter we will focus on whether the identified candidate pairs are actually tradable. Based on the discussions so far, we can state that a pair istradable if the stocks making up the pair are cointegrated. We need to bearin mind, however, that in most cases we are dealing with systems that are notexactly cointegrated. As a matter of fact, in the course of examining the candidate pairs, we can almost always expect to have a residual factor exposurecausing the phenomenon of mean drift, thereby resulting in the signal to benonstationary. However, if the signal-to-noise ratio is good enough, then forall practical purposes we could treat the residual series as stationary for thetime period of the trade. Based on the preceding observations we could refinethe phrasing of our question as follows: How do we decide that a pair istradable even though it deviates from ideal conditions of cointegration? Toseek out an approach to answer that, we could draw on the insights gleanedfrom cointegration testing procedures. With that in mind, let us outline theprocess of verifying that two stocks are indeed cointegrated.Most verification processes are based on the “If it walks like a duck andquacks like a duck, it must be a duck” philosophy. Needless to add, the approach to cointegration testing is also along the same lines. We identify theproperties that must be satisfied if cointegration were indeed true and checkto see if the properties hold. If they do, then the stock pair in question is declared cointegrated. Let us therefore review some properties of cointegratedsystems that could be potentially of use in the testing process.We start off with the Stock-Watson characterization of cointegratedsystems. Each individual stock series is modeled as a sum of a trend component and a stationary component. The property characteristic of cointegrated systems is that the trend components of the two stocks must be thesame. If the trend component is the same and the stationary component justoscillates about some value, then there must be a strong correlation between104
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the two stock series (not to be confused with the correlation between thefirst differences or returns of the series). A strong correlation is usually evidenced by a good fit in the regression of one time series against the other.Additionally, a good fit is characterized by a strong t statistic and r-squaredmeasure. Therefore, deducing from the preceding information, we couldlook for a good value for the t statistic and the r-squared measure from theregression of the two time series, infer a strong correlation between the twoseries, and declare the existence of a common trend. This sounds logical,does it not? However, it is not true. Let us see why that is.Upon careful examination of the preceding argument, it can be seen thatit is actually incomplete. The argument would be complete if we could assertthat the good fit upon regression or strong correlation property is a trait exclusive to cointegrated systems. Only if that were true could we concludethat evidence of the property implies cointegration. Unfortunately for us, thegood fit on regression property is not exclusive to cointegrated systems. Asa matter of fact and rather surprisingly, completely independent randomwalks when regressed against each other can also result in a high r-squaredmeasure. This rather counterintuitive phenomenon was reported in the findings of a simulation experiment conducted by Granger and Newbold, whoaptly coined the phrase “spurious regression” to describe it. Thus, as evidenced by spurious regression, the strong correlation property is not exclusive to cointegrated systems and therefore cannot be used as a definitive testfor cointegration.We now turn to another property of cointegrated systems. This is theexistence of a linear combination of two time series that is stationary andmean reverting. That property is in fact a defining property of cointegrationsystems. We could therefore design a cointegration test based on the verification of this property. Such a test for cointegration was first prescribed byEngle and Granger. The rationale behind the test proceeds somewhat likethis: If two time series are cointegrated, then a simple regression of one timeseries against the other should produce a good estimate of the linear relationship. Also, the spread series resulting from the linear relationship, thatis, the residual series from the regression, must be stationary. Therefore, totest for cointegration all we need to do is estimate the linear relationship between the two series given using simple regression and test for stationarity ofthe residuals. If the residuals form a stationary time series, then we have acointegrated pair. Thus, cointegration testing is a two-step process:11. Determination of the linear relationship.2. Stationarity testing on the residuals.1
There is a one-step test for cointegration originally proposed by Johannsen, whichI do not discuss here. However, references for it are in the appendix.
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We now go back to our original question. Knowing that we are dealingwith systems that are not exactly cointegrated, how do we adapt the cointegration test to test for tradability? With regard to the first step, there is notmuch of a change because in any case we would want to know the linear relationship between the two stocks. The difference, however, is in the secondstep. Contrary to the strict requirement of stationarity of the residual seriesfor cointegration, the pair is deemed tradable as long as the residual exhibitsa sufficient degree of mean reversion. I will seek to quantify this idea ofmean reversion and show how the results may be used to directly verifywhether a stock pair is tradable or not. Thus, similar to the cointegrationtesting, testing for tradability is also a two-step process: estimation of the linear relationship and measuring the degree of mean reversion. We will discussthese two steps in detail in the following section, on the linear relationship.Let us start with the estimation of the linear relationship.
THE LINEAR RELATIONSHIPThe linear relationship between the two time series is given as
( )
( )
log ptA − γ log ptB = µ + ε t
(7.1)
In Equation 7.1, the left-hand side of the equation represents a linear combination of the two time series; g in the equation is the cointegration coefficient. The right-hand side of the equation therefore represents the residualseries and is expressed as the sum of two components: m is the equilibriumvalue, and et is a time series with zero mean, which may be construed as thedisturbance term in the equilibrium. If the series were mean reverting, thenwe would expect its value to oscillate about the equilibrium value. Owing tothis, the linear relationship between the two series is also termed the equilibrium relationship, characterized by the two values g and m. It is thereforeimportant to remind ourselves of the economic interpretation of these twovalues.The interpretation of g as the common factor beta between the twostocks was already discussed in Chapter 6. We are now left with the interpretation of m. To do that, let us consider a portfolio long one share of stockA and short g shares of stock B. The g shares of B represents a position interms of stock B that is equivalent to one share of A. Now, according to theequilibrium relationship, such a portfolio yields an average cash flow of m,which is given back when the position is reversed. Thus m represents the premium paid for holding stock A over an equivalent position of stock B. But
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do such premiums exist in real life? As a matter of fact, stocks do trade at apremium for a variety of reasons. Greater relative liquidity (liquidity premium), the possibility of the firm being a takeover target (takeover premium), and pure charisma on the part of some stocks are some reasons thatcome to mind. Therefore, in the evaluation of the equilibrium relationshipcare must be taken to estimate both the values g the cointegration coefficientand m the premium.We will discuss two approaches to estimating the equilibrium relationship. The first approach is based on the multifactor framework. The secondapproach is the regression approach. Let us begin with the multifactorapproach.
ESTIMATING THE LINEAR RELATIONSHIP:THE MULTIFACTOR APPROACHIn Chapter 6 we mentioned that the cointegration coefficient could be estimated by performing a regression of the common factor returns of one stockagainst the other. The estimated value from the regression is the cointegration coefficient. Also, the formula for the regression coefficient can be expressed in terms of the covariance and variance of the stocks involved. Now,exploiting the multifactor framework the variance and covariance may beexpressed in terms of the factor exposures and the factor covariance matrix.Thus, the regression formula may be expressed completely in terms of theconstructs of the multifactor framework and may be used to evaluate thecointegration coefficient.It is important to note here that we have two values for the cointegration coefficient depending on the choice of the independent variable. If thelinear relationship is expressed assuming stock B to be the independent variable, we haverest = log ptA − γ log ptB
( )
( )
(7.2)
(
) = e Fe
(7.3)
γ = β AB =
cov rA , rB
( )
var rB
TABeBT FeB
Alternately, if the equilibrium relationship is represented assuming stock Ato be the independent variable, we have
( )
( )
rest = log ptB − γ ′ log ptA ,
(7.4)
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γ ′ = β BA =
(
cov rA , rB
( )
var rA
) = e Fe
TBAe TA FeA
(7.5)
We now have two relationships and two values for the cointegration coefficient, bringing us to the question of which value to use in our tests. We suggest going with the larger of the two. From a purely numerical viewpoint interms of reducing precision errors, we are better off estimating the larger ofthe two numbers. This choice has some additional implications. To see that,let us suppose that our choice was g (stock B is the independent variable) because g > g ′, then it follows from the formulas that var rB < var rA . Thus,by choosing the larger of the two values for the cointegration coefficient, weare by implication designating the stock with lower volatility as the independent variable.Once the value of the cointegration coefficient is determined, we canvery easily evaluate the residual time series. From the earlier discussion onthe linear relationship, the equilibrium value m is the mean value of theresidual time series. If this is significantly different from zero, we have anonzero equilibrium value. Otherwise, we could assume that it is zero. Tosummarize, the steps involved in estimating the equilibrium relationship areas follows:
( )
( )
1. Calculate the two values g and g ′ using multifactor model constructs.2. Determine which value must be used for the cointegration coefficient.Our choice is guided by the larger of the two values g and g ′.3. Construct the time series corresponding to the appropriate linear combination and evaluate its mean. If it is significant, we have a nonzeroequilibrium value; otherwise, it is zero.
ESTIMATING THE LINEAR RELATIONSHIP:THE REGRESSION APPROACHThe linear relationship may also be estimated using a regression approach.The use of regression to estimate the linear relationship is based on thepremise that if two series are cointegrated, then a simple regression of onetime series against the other should give us the cointegration coefficient andthe value of the premium. The slope of the regression line is the cointegration coefficient, and the intercept is the premium.The attractiveness of the regression approach is that the general methodology of regression is well known and has found ready application in innumerable situations. Implementations of the ordinary least squares approachis part of most software packages and may be readily used on the prepared
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data set. In fact, it has often been argued that the simplicity of the regressionprocess is probably its most powerful feature. Then why do we even attemptto discuss it at length? Well, it is also true that although the general idea ofregression is relatively simple, the simplicity of the process lends itself tohasty application without much thought. Thus, ironically, the pervasivenessand simplicity of the regression approach is the reason to discuss aspects ofthe regression approach in some detail.Proper use of regression is possible only if we thoroughly understand thestandard regression scenario and the deviations of our situation from thestandard. Let us therefore examine the standard regression scenario as applied to physical systems. Central to physical systems is the study of causeand effect; that is, the response of the system to a particular stimulus. If theexpectation is that the response is proportional to the stimulus, the processof linear regression comes in very handy in measuring this constant of proportionality. The typical experiment involves subjecting the system to a series of inputs or a stimulus in a specified range and measuring the responseof the system to those inputs. The input–output pairs then form the data seton which the regression is run. The independent variable in this case is theinput or stimulus, and the slope of the regression is the constant of proportionality connecting the stimulus to the response.Let us now highlight some aspects of the experimental process. Notethat in this case, since the experimenter is the one administering the stimulus to the system, he or she can design it to be accurate with a very smallmagnitude of error. We can therefore assume that there is no error in theinput data. However, the output data come from the response of the systemand may not be known accurately due to imperfect experimental conditions.Also, if the experimental conditions do not change dramatically during thecourse of the trials, we may assume that the error manifest in each observation of the response is drawn from a common probability distribution. Thus,here is a situation where the input is known relatively accurately, and thesource of error is only in the output with an error standard deviation that isconstant across all observations.Now let us see how stock price data relates to the scenario describedabove. First, it may be argued that the price of a stock is known exactly.Then where is the source of error? Note that we use just one representativevalue for the price in a given time period, while in actuality the price changesconstantly within the period itself. Therefore, a reasonable case may bemade for the existence of uncertainty or error associated with our choice forthe price in the time period. Next, in our situation, there is no distinct separation of cause and effect. The prices of the two stocks may very well befeeding off each other, as discussed in the error correction model for cointegration in Chapter 5. Since it is not possible to easily separate cause from effect in this scenario, and both of the price values are read as outputs, we are
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now faced with a situation where there could be uncertainty or error inboth variables. This is vastly different from the standard linear regressioncase, which assumes errors only in one variable.Also mentioned earlier, each stock moves constantly, reaching its ownhighs and lows within a time period, with the high and low price characterizing the range of price movement. Although we allow ourselves to chooseonly one representative price for a given time period for each stock withinthis range, we also admit that there is a certain amount of uncertainty associated with our choice. But data points in each time period are chosen fromthe price ranges of different magnitudes. Under such circumstances, it wouldbe rather facetious to assume a constant probability distribution for the uncertainty or errors that are intricately linked to the magnitude of the pricerange in a given period. We can strongly assert that our situation is onewhere the uncertainty associated with each data point is different. This isalso different from the standard regression scenario of constant variance inthe observations.Therefore, to sum up, we have a situation where there is error associatedwith both observations, and the variance of the observation error is also avarying quantity. Although the differences of our situation with the standardregression scenario are substantial, they are by no means novel. Such situations have been encountered in a variety of other applications and may beapplied to ours without change. Going that route, however, adds to the complexity of the process. We will briefly discuss the solution approach in thiscase to highlight the issues and proceed to suggest a much simpler approach.The situation of nonconstant error distributions coupled with errors inboth variables can be handled by minimizing the chi-squared merit function,given as
( )( )(7.6)var(ε ) + γ var(ε )In Equation 7.6, var(ε ) and var(ε ) are the variances of the error in theobservations log( p ) and log( p ) . The errors are assumed to have a zero(
)
N
χ2 γ , µ = ∑
At
At
[log ptA − γ log ptB − µ ]2At
t =1
Bt
2
Bt
Bt
mean and may be calibrated based on the range of movement of the stockwithin each time period. Note that for our purposes it is not important tohave an absolute measure of the variance of the errors, just that the valuesbe proportional to the actual variances.To further understand how the chi-squared function handles the situation of nonconstant error distributions, let us examine it in a little bit of detail. The value in the numerator of the merit function is the squared error inthe regression. If the variance as shown in the denominator was a constant,then the minimization boils down to minimizing the sum of squared errors,
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which is the ordinary least squares procedure. The denominator term, infact, weights each data point in the cost function to be inversely proportional to the variance of the individual error terms. Thus, it may also bethought of as the sum of squared errors each normalized by its variance.This approach to regression using the chi-squared merit function is sometimes also called the weighted least squares approach.However, typical applications adopting a weighted least squares approach assume the error only in the response variable and not in both.Specifically, those applications do not have the term with g 2 in the denominator. The presence of this term in the denominator complicates the minimization of the chi-squared function, in that the derivative of thechi-squared function with respect to g is now nonlinear, and so we may needto resort to numerical methods to solve this. In summary, the regressionprocess can get fairly involved if we are to account for the varying errorprobability distributions in the measurement with errors in measurement forboth the variables.Nevertheless, if there is a way to construct a price series such that the errors associated with the observation in each time period may be assumed tobe the same, then we can do away with these complications, work with justordinary least squares, and arrive at a reasonable answer. Let us see if wemight be able to do that.Note that in the previous paragraph we mentioned that we choose onlyone representative price for a given time period from the range of stockprice movement for that period. In a typical scenario where the length of aperiod is one trading day, the standard convention in the construction ofdaily time series is to use the closing price at the end of the trading day; thatis, the latest price in the process of serial price adjustment. Let us call thismethod for recording the price time series the close-close method. Time series of this type are usually constructed to perform a mark to market of stockinventory. Given the nature of the auction process and the price discoverymechanism, this is definitely a reasonable choice. But is this approach todata construction appropriate for our purposes? It is seductive by habit touse the same construction regardless of purpose even though such an exercise may be ill-advised. Care must be taken to ensure that the process of dataconstruction is a reflection of the specific purpose at hand.Our purpose is to examine the price relationship between two stocks. Inthis quest, to examine price relationships, more important than the closingprice in a given time period is the answer to the question, “At what price inthe time period was the liquidity a maximum?” That would be the consensus price in the time chunk at which the most buyers and sellers agreed thatthe price was right and a lot of shares changed hands. Therefore, conclusionsdrawn on the maximum liquidity price series of two stocks would be morereliable than using the close-close approach. A reasonable proxy for the
112
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maximum liquidity price is the volume weighted average price, commonlytermed as the VWAP price. We could therefore construct our time serieswith VWAP prices.By choosing the VWAP price we are arguably within reason to assumethat the error distributions of the VWAP price from the maximum liquidityprice is about the same regardless of the magnitude of the price range. Wecan therefore resort to the simple ordinary least squares version in our regression analysis. Of course, the manner in which the calculation of standarderror forming the basis for the t statistics tests would still need alteration. Inour scenario of cointegration or tradability testing, the emphasis on the t statistic is rather low, and therefore we contend that this is something we canlive with. Additionally, note that using the VWAP price has the tendency totemper extreme values and therefore has the added benefit of minimizingthe effect of outliers on the regressions. In conclusion, the time series constructed with the VWAP price is better suited to understand equilibrium relationships and should be the preferred approach.However, this does not do away with the need to decide which of thetwo price series we should use for the independent variable. We will avoidbeing repetitive and just say that the same idea as was adopted in the multifactor model case may be applied here, also.
TESTING RESIDUAL FOR TRADABILITYSubsequent to estimating the equilibrium relationship we need to constructthe residual time series. Although we advocated using VWAP prices to estimate the equilibrium relationship, we recommend constructing the residualtime series by applying the equilibrium relationship to the time series ofstocks constructed using the close-close method. If the two series are indeedcointegrated, constructing such a time series provides us with a fairly goodpicture of the oscillations about the equilibrium value.We begin by reviewing the ideal situation. In an ideal situation for tradability, the two stocks would be cointegrated, and the residual series wouldbe stationary. It is therefore desirable that the properties of the residual series exhibit the characteristics of stationary series. One of the properties ofstationary series, the property of mean reversion, is of particular importanceto us. This is relevant because pairs trading is a bet that the residual serieswill revert to its mean or equilibrium value. In other words, deviations fromthe mean are quickly corrected by the series moving back toward the mean.It would therefore be nice if we could quantify the degree of mean reversionof a given time series.It turns out that highly mean-reverting series are also characterized by ahigh frequency of zero-crossings. A zero-crossing is defined as the transitionof the time series across its long-run mean. The frequency of zero-crossing is
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then the number of times we can expect the time series to cross its equilibrium value in unit time. Thus, the zero-crossing frequency provides us witha quantitative characterization for the mean reversion property.Notice that if the zero-crossing rate is very high, then the time to revertto mean is short, implying that the time we need to hold the paired positionis small. The signal-to-noise ratio is bound to be good, and we could bemore comfortable with the idea that the pair is tradable. Thus, a high zerocrossing rate for the residual series is a preferred trait and directly appeals toour requirements.A high zero-crossing rate is also indicative of a stationary series. Tostrengthen this conviction, we make an observation in contrast by considering the example of Brownian motion a nonstationary series. Even though thedistribution of Brownian motion is symmetric about the mean, the zerocrossing event is very infrequent. The theoretical explanation for the phenomenon is well captured by the famous arcsine law for Brownian motiondiscovered by P. Levy. The law provides us with information on the last passage time, or the last time that the Brownian motion visited zero. More precisely, let us consider a Brownian motion starting at zero, or time t = 0 andstopped at time T. If g is the last time when zero is visited, then the probability distribution satisfies
(
)
Pg <u =
2uarcsinTπ
(7.7)
The density function corresponding to this is given as
()
Pu =
11π uT −u
(
)
(7.8)
A quick plot of the density graph in Figure 7.1 shows that we expect the zerocrossing to have occurred with high probability only at the extremes of thetime interval. Thus, we would expect very few zero crossings for Brownianmotion a nonstationary time series.For a stationary ARMA time series with known parameters, the theoretical zero-crossing rate may be calculated using the formula developed byRice. The Rice formula is the sum of a series of zero-crossing probabilitiesat various time steps. The probability at each lag is calculated using the autocorrelation function. Thus, if the residual series is stationary and the ARMAparameters are known, we can apply Rice’s formula to obtain the zerocrossing rate. Note that this requires us to estimate the ARMA parameterswith the assumption that the series is actually stationary. This is somethingwe wish to avoid. We favor a model free approach and will focus on that toobtain an estimate of the zero-crossing rate. Therefore, even though it would
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0.030
Density
0.025
0.020
0.015
0.010
0
20
40
60
80
100
Time Steps
FIGURE 7.1
Density Plot of ArcSin Distribution.
be a worthwhile endeavor, we do not discuss Rice’s formula in detail. Interested readers can find more specifics on this in the references at the end ofthe chapter.As mentioned earlier, our focus remains on the estimation of the zerocrossing rate directly from the given data sample. Obviously, the most directapproach would be to count the number of zero crossings of the residual series and calculate the zero-crossing rate by dividing the number of crossingsby the total time. However, given that we are doing this with a sample sizeof 1, the result is likely to be strongly biased to the residual series at hand.The dilemma facing us is that we have only one realization of the residual series. The general idea of averaging across many sample residual series to ridourselves of the bias is not possible.To resolve this, we resort to a resampling technique popularly known asthe bootstrap. In the bootstrap, however, we look to estimate the distribution of the time between two crossings; that is, the reciprocal of the crossing rate. Note that the time between zero crossings is directly related to thetime that we expect to hold the paired position. Thus, the test for tradability also leads us to an estimate of the holding period, which can be used as abenchmark for time-based stops. The time between two crossings relates directly to the trading horizon and is therefore of direct relevance for trading.
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The process works as follows:1. First, we get the sample small size population of time between crossingsby counting the time between subsequent crossings in the residual series.2. A probability distribution is then constructed by resampling repeatedlyfrom the existing sample. The large sample obtained as a result of the resampling exercise is then used to construct the probability distribution.3. Percentile levels may then be constructed for the population. We canthen check to see if the rates at the desired percentile levels on either sideof the median satisfy our trading requirements. If they do, then we declare the pair tradable and vice versa.
ExampleWe now illustrate with an example the application of the process just described. For illustrative purposes, we carefully picked two stocks from thesemiconductor sector and sampled their prices as of the day’s close for 90days. Although we advocate the use of VWAP prices to do the regression, forsake of expediency we ran an ordinary least squares method on it two times,changing the independent variable. Results from the regression for the largerof the two g values are as follows:Equilibrium value
m = –0.6971
Cointegration coefficient
g = 1.0617
R-squared from the regression is 0.7965Figure 7.2a is a scatter plot of the log-price series of the two stocks againsteach other. Figure 7.2b is the residual time series. A cursory look at the
Log ( Stock B)
2.4
2.1
1.4
1.5
1.6Log (Stock A)
FIGURE 7.2A
Scatter Plot.
1.7
1.8
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Residual
0.2
0.1
0.0
–0.10
20
40
60
80
Time
FIGURE 7.2B
Residual Time Series.
residual time series seems to suggest that it is indeed tradable. We now runthe bootstrap procedure on it and obtain the population. A quick look suggests that in the extreme the time between zero crossings can be as long as14 to 26 days with a median value of about 5 days. The pair definitelyseems to be in tradable category. However, it might be better to liquidate theposition on reversion to zero instead of waiting for an excursion of magnitude in the other direction.
SUMMARYTradability testing is a two-step process consisting of evaluating the linear relationship and measuring the degree of mean reversion of theresidual.The linear relationship between the log-price series of the two stocks ischaracterized by the cointegration coefficient and the stock premium.They may be estimated in a multifactor framework or by ordinary leastsquares regression.The spread series can be calculated by applying the linear relationship.The degree of mean reversion of a series is quantifiable in terms of thezero-crossing frequency.The zero-crossing frequency can be directly estimated using the bootstrap procedure.The reciprocal of the zero-crossing frequency is indicative of the tradingperiod, and a pair may be deemed tradable if we are satisfied with therange of trade periods or zero crossing frequencies generated by thebootstrap.
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FURTHER READING MATERIALArcsine Law and Zero CrossingLevy, P. Processus Stochastiques et Mouvement Brownien. Paris: Editions JaquesGaby, 1948.Rice, S. O. “Mathematical Analysis of Random Noise.” Bell System Technical Journal 24 (1945): 46–156.Linear RegressionPress, W. H., and others. Numerical Recipes in C, (Cambridge, UK: 2nd Edition.Cambridge University Press, 1992): 656–670.Spurious RegressionGranger, C. W. J. and P. Newbold “Spurious Regression in Econometrics,” Journalof Econometrics 2 (1974): 111–120.Philips, P. C. B. “Understanding Spurious Regression in Econometrics,” Journal ofEconometrics 33 (1986): 311–340.Resampling Methods and BootStrappingGood, Philip I. Resampling Methods, A Practical Guide To Data Analysis. (Cambridge, Mass.: Birkhauser Boston, 1999