"Trying to model the complex interdependencies
between financial assets with so restrictive concept of correlation
is like trying to surf the internet with an
IBM AT." Carol Alexander
Just like a a drunk man leaving a bar follows a random walk. His
dog also follows a random walk on its own. The paths will diverge ...
Then they go into a
park where dogs are not allowed to be untied. Therefore the drunk
man puts a strap on his dog and both enter into the park.
Now, they share some common direction, their paths are co-integrated ...
see Murray .
A good intro is also given by
Carol Alexander in 
"Cointegration and asset allocation: A new active hedge fund strategy".
Definition: Two time series xt and yt are co-integrated if, and only if, each is
I(1) and a linear combination Xt-a-b Yt,
where b1¹0 is I(0)
In general, linear combinations of I(1) time series are also I(1).
Co-integration is a
particular feature not displayed between arbitrary pairs of time series.
If two time series
are co-integrated, then the co-integrating vector (b1) is unique
Granger (1981) introduced the case
where the individual time series are I(1) but the error term,
ut, is I(0). That is, the error
term might be autocorrelated but, because it is stationary, the relationship will keep
returning to the equilibrium or long-run equation
Granger (1981) and Engle and Granger (1987) demonstrated that, if a vector of
time series is cointegrated, the long-run parameters can be estimated directly without
specifying the dynamics because, in statistical terms, the estimated long-run parameter
estimates converge to their true values more quickly than those operating on stationary
variables. That is they are super-consistent and a two-step procedure of first estimating
the long-run relationship and estimating the dynamics, conditional on the long run
As a result, simple static models came back in vogue in the late 1980's
but it rapidly became apparent that small sample biases can indeed be large (Banerjee et
Two major problems typically arise in a regression such as
(1). First, it is not always clear
whether one should regress yt on xt or vice versa. Endogeneity is
not an issue asymptotically because the simultaneous equations
bias is of a lower order of importance and, indeed, is dominated
by the non-stationarity of the regressor. However, least squares
is affected by the chosen normalisation and the estimate of one
regression is not the inverse of that in the alternative ordering
unless R2 =1.
Secondly, the coefficient b^ is not asymptotically normal when xt is I(1) without
drift, even if ut is iid. Of course, autocorrelation in the residuals produces a bias in the
least squares standard errors, even when the regressor is non-stationary, and this effect is in
addition to that caused by non-stationarity.
The preceding discussion is based on the assumption that the disturbances are
stationary. In practice, it is necessary to pre-test this assumption. Engle and Granger
suggested a number of alternative tests but that which emerged as the popular method is
the use of an ADF test on the residuals without including a constant or a time trend.
||Stationary and non-stationary variables
yt = r yt-1 + et
If |r| < 1 then the series is stationary (around 0),
if |r| = 1 then it is non-stationary (a random walk in this case).
A stationary series is one for which :
Strictly, this is "weak" or "second order" stationarity but is
good enough for practical purposes.
(i) the mean is constant
- (ii) the variance is constant, and
- (iii) Covariance(yt , yt-s) depends only upon the lag length s.
We can more generally write
yt = a + r yt-1 + et
which is stationary around a/(1-r).
(To see this, set E(yt) = E(yt-1)
hence E(y) = a + r E(y) + 0, hence E(y) = a/(1-r).)
If r=1, we have a random walk with drift.
We can also have a second order AR process, e.g.
yt = r1 yt-1 - r2 yt-2 + et
The conditions for stationarity of this series are given later.
We can also have a time trend incorporated:
yt = b1+ b2 t + r yt-1 + et
This will be stationary (around b1 + b2 t) if r < 1.
This is called a trend stationary series (TSS).
As trend stationary series can look similar to non-stationary series.
This is unfortunate, since we should de-trend the former (take
residuals from a regression on time) and difference the latter
(the latter are also known as difference stationary series for
this reason. It is also known as a stochastic trend). Doing the
wrong transformation leads to biased estimates in regression, so
it's important (but unfortunately difficult) to tell the
Note that, for a non-stationary process
|| a + yt-1 + et we can write:
|| 0 (say)
|| a + 0 + e1
|| a + y1 + e2 = 2a + e1 + e2
|| a t + åe
This implies that
var(y) = t var(varepsilon) = ts2
which tends to infinity as the sample size increases.
For a Trend Stationary Series with r=0:
Note the similarity of
16, apart from the nature of the
error term, i.e. a Difference Stationary Series
can be written as a function of t, like a
Trend Stationary Series , but with a MA error term.
|| b1+ b2 t + et
|| 0 (say)
|| b1 + b2 + e1
|| b1 + b2 2 + e2
|yt = b1 + b2 t + et
In finite samples a Trend Stationary Series can be
approximated arbitrarily well by a Difference Stationary Series , and vice versa.
finite sample distributions are very close together and it can be
hard to tell them apart.
A difference between them is that a shock to the system (D e) has
a temporary effect upon a Trend Series but a permanent effect upon a
If we interpret 'shock' as a change in government policy, then we
can see the importance of finding whether variables are Difference Series or
A non-stationary series is said to be integrated, with the order
of integration being the number of times the series needs to be
differenced before it becomes stationary. A stationary series is
integrated of order zero, I(0).
For the random walk model, y I(1).
Most economic variables are I(0) or I(1). Interest rates are
likely to be I(0), they are not trended. Output, the price
level, investment, etc. are likely to be I(1).
may be I(2). Transforming to logs may affect the order of
The investment strategy we aim at implementing is a market
neutral long/short strategy. This implies that we will try to
find shares with similar betas, where we believe one stock will
outperform the other one in the short term. By simultaneously
taking both a long and short position the beta of the pair equals
zero and the performance generated equals alpha.
Needless to mention, is that the hard part of this strategy is to
find market neutral positions that will deviate in returns. To do
this we can use a statistical tool developed by Schroder Salomon
Smith Barney (SSSB).
The starting point of this strategy is that
stocks that have historically had the same trading patterns (i.e.
constant price ratio) will have so in the future as well. If
there is a deviation from the historical mean, this creates a
trading opportunity, which can be exploited. Gains are earned
when the price relationship is restored. The historical
calculations of betas and the millions of tests executed are done
by SSSB, but it is our job as portfolio managers to interpret the
signals and execute the trades.
find two stocks prices of which have historically
- mean reversion in the ratio of the prices, correlation is not key
- Gains earned when the historical price relationship is restored
- Free resources invested in risk-free interest rate
||Testing for the mean reversion
The challenge in this strategy is identifying stocks that tend to
move together and therefore make potential pairs. Our aim is to
identify pairs of stocks with mean-reverting relative prices. To
find out if two stocks are mean-reverting the test conducted is
the Dickey-Fuller test of the log ratio of the pair. In the
A Dickey-Fuller test for determining stationarity in the log-ratio
yt=logAt-log Bt of share prices A and B
D yt = µ + g yt-1 + et
In other words, we are regressing
D yt on lagged values of yt.
the null hypothesis is that g=0,
which means that the process is not mean reverting.
If the null hypothesis can be rejected on the 99% confidence
level the price ratio is following a weak stationary process and
is thereby mean-reverting. Research has shown that if the
confidence level is relaxed, the pairs do not mean-revert good
enough to generate satisfactory returns. This implies that a very
large number of regressions will be run to identify the pairs. If
you have 200 stocks, you will have to run 19 900 regressions,
which makes this quite computer-power and time consuming.
Schroder Salomon Smith Barney provide such calculation
By conducting this procedure, a large number of pairs will be
generated. The problem is that all of them do not have the same
or similar betas, which makes it difficult for us to stay market
neutral. Therefore a trading rule is introduced regarding the
spread of betas within a pair. The beta spread must be no larger
than 0.2, in order for a trade to be executed. The betas are
measured on a two-year rolling window on daily data.
We now have mean-reverting pairs with a limited beta spread, but
to further eliminate the risk we also want to stay sector
neutral. This implies that we only want to open a position in a
pair that is within the same sector. Due to the different
volatility within different sectors, we expect sectors showing
high volatility to produce very few pairs, while sectors with low
volatility to generate more pairs. Another factor influencing the
number of pairs generated is the homogeneity of the sector. A
sector like Commercial services is expected to generate very few
pairs, but Financials on the other hand should give many trading
opportunities. The reason why, is that companies within the
Financial sector have more homogenous operations and earnings.
The screening process described gives us a large set of pairs
that are both market and sector neutral, which can be used to
take positions. This should not be done randomly, since timing is
an important issue. We will therefore introduce several trading
All the calculations described above will be updated on a daily
basis. However, we will not have to do this ourselves, but we
will be provided with updated numbers every day, showing pairs
that are likely mean revert within the next couple of weeks. In
order to execute the strategy we need a couple of trading rules
to follow, i.e. to clarify when to open and when to close a
trade. Our basic rule will be to open a position when the ratio
of two share prices hits the 2 rolling standard deviation and
close it when the ratio returns to the mean. However, we do not
want to open a position in a pair with a spread that is wide and
getting wider. This can partly be avoided by the following
procedure: We actually want to open a position when the price
ratio deviates with more than two standard deviations from the
130 days rolling mean. The position is not opened when the ratio
breaks the two-standard-deviations limit for the first time, but
rather when it crosses it to revert to the mean again. You can
say that we have an open position when the pair is on its way
back again (see the picture below).
Figure 1: Pairs Trading rules
Open position when the ratio hits the 2 standard deviation
band for two consecutive times
- Close position when the ratio hits the mean
Furthermore, there will be some additional rules to prevent us
from loosing too much money on one single trade. If the ratio
develops in an unfavourable way, we will use a stop-loss and
close the position as we have lost 20% of the initial size of the
position. Finally, we will never keep a position for more that 50
days. On average, the mean reversion will occur in approximately
35 days , and there is no reason to wait for a pair to revert
fully, if there is very little return to be earned. The potential
return to be earned must always be higher than the return earned
on the benchmark or in the fixed income market. The maximum
of a position is therefore set to 50 days. This should be enough
time for the pairs to revert, but also a short enough time not to
loose time value.
The rules described are totally based on statistics and
predetermined numbers. In addition, there is a possibility for us
to make our own decisions. If we for example are aware of
fundamentals that are not taken into account in the calculations
and that indicates that there will be no mean reversion for a
specific pairs, we can of course avoid investing in such pairs.
From the rules it can be concluded that we will open our last
position no later than 50 days before the trading game ends. The
last 50 days we will spend trying to close the trades at the most
optimal points of time.
Stop loss at 20% of position value
- Beta spread <0.2
- Sector neutrality
- Maximum holding period < 50 trading days
- 10 equally weighted positions
As already mentioned, through this strategy we do almost totally
avoid the systematic market risk. The reason there is still some
market risk exposure, is that a minor beta spread is allowed for.
In order to find a sufficient number of pairs, we have to accept
this beta spread, but the spread is so small that in practise the
market risk we are exposed to is ignorable. Also the industry
risk is eliminated, since we are only investing in pairs
belonging to the same industry.
The main risk we are being exposed to is then the risk of stock
specific events, that is the risk of fundamental changes implying
that the prices may never mean revert again, or at least not
within 50 days. In order to control for this risk we use the
rules of stop-loss and maximum holding period. This risk is
further reduced through diversification, which is obtained by
simultaneously investing in several pairs. Initially we plan to
open approximately 10 different positions. Finally, we do face
the risk that the trading game does not last long enough. It
might be the case that our strategy is successful in the long
run, but that a few short run failures will ruin our overall
excess return possibilities.
||General Discussion on pairs trading
generally two types of pairs trading: statistical arbitrage
convergence/divergence trades, and fundamentally-driven
valuation trades. In the former, the driving force for the trade
is a aberration in the long-term spread between the two
securities, and to realize the mean-reversion back to the norm,
you short one and go long the other.
The trick is creating a program to find the pairs,
and for the relationship to hold.
The other form of pairs trading would be more
fundamentally-driven variation, which is the purvey of most
market-neutral hedge funds: in essence they short the most
overvalued stock(s) and go long the undervalued stock(s). It's
normal to "pair" up stocks by having the same number per sector
on the long and short side, although the traditional "pairs"
aren't used anymore. Pairs trading had originally been the domain
of BD's in the late 70's, early 80's before it dissipated
somewhat due to the bull market (who would want to be
market-neutral in a rampant bull market), and the impossibility
of assigning "pairs" due to the morphing of traditional sectors
and constituents. Most people don't
perform traditional "pairs trading" anymore (i.e. the selection
of two similar, but mispriced, stocks from the same
industry/sector), but perform a variation. Goetzmann et al wrote
a paper on it a few years back, but at the last firm I worked at,
the research analyst "pooh-paahed" it because he couldn't get the
same results: he thinks Goetzmann
either waived commissions, or
worse, totally ignored slippage (i.e. always took the best price,
not the realistically one). Here's the paper :
source: forum http://www.wilmott.com
some quotations from this paper:
"take a longshort position when they diverge." A test requires
that both of these steps must be parameterized in some way. How
do you identify "stocks that move together?" Need they be in the
same industry? Should they only be liquid stocks? How far do they
have to diverge before a position is put on? When is a position
unwound? We have made some straightforward choices about each of
these questions. We put positions on at a twostandard deviation
spread, which might not always cover transactions costs even when
stock prices converge. Although it is tempting to try potentially
more profitable schemes, the danger in datasnooping refinements
outweigh the potential insights gained about the higher profits
that could result from learning through testing. As it stands
now, datasnooping is a serious concern in our study. Pairs
trading is closely related to a widely studied subject in the
academic literature mean reversion in stock prices. 2 We
consider the possibility that we have simply reformulated a test
of the previously documented tendency of stocks to revert towards
their mean at certain horizons. To address this issue, we develop
a bootstrapping test based upon random pair choice. If
pairstrading profits were simply due to meanreversion, then we
should find that randomly chosen pairs generate profits, i.e.
that buying losers and selling winners in general makes money.
This simple contrarian strategy is unprofitable over the period
that we study, suggesting that mean reversion is not the whole
Although the effect we document is not merely an extension
of previously known anomalies, it is still not immune to the
datasnooping argument. Indeed we have explicitly "snooped" the
data to the extent that we are testing a strategy we know to have
been actively exploited by risk arbitrageurs. As a consequence
we cannot be sure that past trading profits under our simple
strategies will continue in the future. This potential critique
has another side, however. The fact that pairs trading is already
wellknown riskarbitrage strategy means that we can simply test
current practice rather than develop our filterrule ad hoc.
||Optimal Convergence Trading
From , Vladislav Kargin:
"Consider an investment in a mispriced asset. An investor can expect that the
mispricing will be eliminated in the future and play on the convergence of the
asset price to its true value. This play is risky because the convergence is not
immediate and its exact date is uncertain. Often both the expected benefit and
leveraging his positions, that is, by borrowing additional investment funds. An
important question for the investor is what is the optimal leverage policy.
There are two intuitively appealing strategies in this situation. The first
one is to invest only if the mispricing exceeds a threshold and to keep the
position unchanged until the mispricing falls below another threshold (an (s, S)
strategy). For this strategy the relevant questions are what are the optimal
thresholds and what are the properties of the investment portfolio corresponding
to this strategy. The second type of strategy is to continuously change positions
according to the level of mispricing. In this case, we are interested in the optimal
functional form of the dependence of the position on the mispricing."
See also discussion on optimal growth strategies in
||Optimal Trading in presence of mean reversion
In , Thompson define close form of trading
threshold strategies in presence of Ornstein-Uhlenbeck process
and fixed transaction costs c.
if the price of the OU process is
dSt=s d Bt - g St dt
The optimal strategy is
a threshold strategy, ie
to buy if St £ -b/g and to sell
St ³ -b/g whre b satisfies:
with c fixed transaction cost.
|2b-g c = 2 e-b2/(g s2
|| eu2/(g s2 du
||Continuous version of mean reversion : the Ornstein-Uhlenbeck process
Suppose that the dynamic of the mis-pricing can be modelled by
an AR(1) process. RA(1) process is the discrete-time
counterpart to the Ornstein-Uhlenbeck (OU) process in continuous time.
dXt=b(a-Xt) dt + s dWt
where Wt is a standard Wiener process,
s>0 and a, b are constants.
So the Xt process
drifts towards a.
OU process also has a normal transition density function given by :
If the process displays the property of mean reversion (b>0), then
as t0® -¥ or t-t0® +¥, the marginal density
of the process is invariant to time, ie OU process is stationary in the strict sense.
See that there is a time decay term for the variance.
time horizon the variance of this process tends to
unlike the Brownian motion the variance is bounded (not grows to infinite).
The equation describing dXt, the arithmetic Ornstein-Uhlenbeck
equation presented above is a continuous time version of the
first-order autoregressive process, AR(1), in discrete time.
It is the limiting case (D t tends to
zero) of the AR(1) process:
xt - xt-1 = a (1 - e-b) + (e-b - 1) xt - 1 + et
is normally distributed with mean zero and standard deviation
In order to estimate the parameters of mean-reversion, run the regression:
xt - xt - 1 = a + b xt - 1 + et
Calculate the parameters:
|| -a/b ;
|| - ln(1 + b);
The choice of the representation may depend on the data.
For example, with daily data, the AR discrete form is preferable,
with high frequency data, it might preferable to use
the continuous time model.
One important distinction between random walk and stationary
AR(1) processes: for the last one all the shocks are transitory,
whereas for random walk all shocks are permanent
Mean-Reversion Combined with Exponential Drift
It is possible combine Geometric Brownian
Motion (exponential drift) with mean-reverting model.
According to Granger (1981), a times series, Xt, is said to cause
another times series, Yt, if present Y can be predicted better by
using the value of X. The first step in the empirical analysis is
to examine the stationarity of the price series.
We now look at situations where the validity of the linear
regression model is dubious - where variables are trended or,
more formally, non-stationary (not quite the same thing).
Regressions can be spurious when variables are n-s, i.e. you
appear to have 'significant' results when in fact you haven't.
Nelson and Plosser ran an experiment. They generated two random
variables (these are random walks):
||xt-1 + et
||yt-1 + nt
where both errors have the classical properties and are
independent. y and x should therefore be independent.
Regressing y on x, N & P got a 'significant' result (at the 5%
level) 75% of the time !!!
This is worrying!
This is a spurious regression and it occurs because
of the common trends in y and x.
In these circumstances, the t and F statistics do not have
the standard distributions. Unfortunately, the problem generally
gets worse with a larger sample size.
Such problems tend to occur with non-stationary variables and,
unfortunately, many economic variables are like this.
||Unit Root Hypohtesis Testing
Formally, a stochastic process is stationary if all the roots of
its characteristic equation are >1 in absolute value. Solving is
the same as solving a difference equation.
yt = r yt-1 + et
we rewrite it as yt - r yt-1 = et
or (1-r L)yt=et.
Hence this will be stationary if the root of the
1-r L = 0 is >1.
The root is L = 1/r which is >1 if r <1.
This is the condition for stationarity.
Example II: yt = 2.8yt-1 -1.6yt-2 + et
The characteristic equation is 1 - 2.8L + 1.6L2 = 0.
From this we get L = 1.25 and L = 0.5 are the roots. Both
roots need to be greater than 1 in absolute value for
stationarity. This does not apply here, so the process is
However, in practice we do not know the r values, we have to
estimate them and then test whether the roots are all >1. We
H0: r = 1 (non stationary)
H1: r < 1 (stationary)
using a t-test.
Unfortunately, if r=1, the
estimate of r is biased downwards (even in large samples) and
also the t-distribution is inappropriate. Hence can't use
Instead we use the Dickey-Fuller test. Re-write
where r* = r-1.
Now we test
H0: r* = 0 (non stationary)
H1: r* < 0 (stationary)
We cannot use critical values from the
t-distribution, but D-F provide alternative tables to use.
The D-F equation only tests for first order autocorrelation of
y. If the order is higher, the test is invalid and the D-F
equation suffers from residual correlation. To counter this, add
lagged values of D y to the equation before testing.
the Augmented Dickey-Fuller test. Sufficient lags should be
added to ensure e is white noise.
95% critical value for the augmented Dickey-Fuller statistic ADF=-3.0199
It is important to know the order of integration of
non-stationary variables, so they may be differenced before being
included in a regression equation. The ADF test does this, but
it should be noted that it tends to have low power (i.e. it fails
to reject H0 of non-stationarity even when false) against the
alternative of a stationary series with r near to 1.
The Dickey-Fuller test requires that the us be uncorrelated. But suppose we
have a model like the following, where the first difference of Y is a stationary
This model yields a model for Yt that is:
||Variants on the Dickey-Fuller Test
If this is really what's going on in our series, and we estimate a standard
|Yt = Yt-1 +
|| di D Yt-i+ut
the term åi=1p di D Yt-i
gets lumped into the errors ut. This induces an AR(p)
structure in the us, and the standard D.F. test statistics will be wrong.
There are two ways of dealing with this problem:
Change the model (known as the augmented Dickey-Fuller test), or
- Change the test statistic (the Phillips-Perron test).
Rather than estimating the simple model, we can instead estimate:
||The Augmented Dickey-Fuller Test
and test whether or not r = 0.
This is the Augmented Dickey-Fuller test.
As with the D-F test, we can include a constant/trend term to differentiate
between a series with a unit root and one with a deterministic trend.
The purpose of the lags of D Yt
is to ensure that the us are white noise.
This means that in choosing p
(the number of lagged D Yt-i terms to include),
we have to consider two things:
|Yt = a+b t +Yt-1 +
|| di D Yt-i+ut
This suggests, as a practical matter, a couple different ways to go about
determining the value of p:
Too few lags will leave autocorrelation in the errors, while
- Too many lags will reduce the power of the test statistic.
A sidenote: AIC and BIC tests:
Start with a large value of p, and reduce it if the values of di are
insignificant at long lags - This is generally a pretty good approach.
- Start with a small value of p, and increase it if values of di are
significant. This is a less-good approach...
- Estimate models with a range of values for p, and use an AIC/BIC/
F-test to determine which is the best option. This is probably the best
option of all...
The Akaike Information Criterion (AIC) and Bayes Information Criterion
(BIC) are general tests for model specification. They can be applied across a
range of different areas, and are like F-tests in that they allow for the testing
of the relative power of nested models. Each, however, does so by penalizing
models which are overspecified (i.e., those with "too many" parameters). The
AIC statistic is:
where N is the number of observations in the regression, p is the number of
parameters in the model (including r and a),
and sp2 is the estimated s2 for
the regression including p total parameters.
Similarly, the BIC statistic is
2.1 The KPSS test
One potential problem with all the unit root tests so far described is that they
take a unit root as the null hypothesis. Kwiatkowski et. al. (1992) provide
an alternative test (which has come to be known as the KPSS test) for testing
the null of stationarity against the alternative of a unit root. This method
considers models with constant terms, and either with or
without a deterministic trend term. Thus, the KPSS test tests the null
of a level- or trend-stationary process against the alternative of a unit root.
Formally, the KPSS test is equal to:
estimated error variance from the regression:
Yt =a+ et
Yt =a+ b t + et
for the model with a trend.
The practical advantages to the KPSS test are twofold. First, they provide
an alternative to the DF/ADF/PP tests in which the null hypothesis is sta-tionarity.
They are thus good "complements" for the tests we have focused on
so far. A common strategy is to present results of both ADF/PP and KPSS
tests, and show that the results are consistent (e.g., that the former reject
the null while the latter fails to do so, or vice-versa). In cases where the two
tests diverge (e.g., both fail to reject the null), the possibility of "fractional
integration" should be considered (e.g. Baillie 1989; Box-Steffensmeier and
Smith 1996, 1998).
General Issues in Unit Root Testing
The Sims (1988) article I assigned is to point out an issue with unit root
econometrics in general: that classicists and Bayesians have very different
ideas about the value of knife-edge unit root tests like the ones here.1
Unlike classical statisticians, Bayesians regard (the "true" value of the
parameter) as a random variable, and the goal to describe the
distribution of this variable, making use of the information contained in the
data. One result of this is that, unlike the classical approach (where the
distribution is skewed), the Bayesian perspective allows testing using
standard t distributions. For more on why this is, see the discussion in
Another issue has to do with lag lengths. As in the case of ARIMA models,
choosing different lag lengths (e.g. in the ADF, PP and KPSS tests) can lead
to different conclusions. This is an element of subjectivity that one needs to
be aware of, and sensitivity testing across numerous different lags is almost
always a good idea.
Finally, the whole reason we do unit root tests will become clearer when we
talk about cointegration in a few weeks.
||Error Correction Model
Error Correction Model (ECM) is a step forward to determine how variables are
1. Test the variables for order of integration. They must both (all) be I(d).
2. Estimate the parameters of the long run relationship. For example,
yt=a+b xt + et
when yt and zt are cointegrated OLS is super consistent.
That is, the rate of convergence is T2 rather than just T in Chebyshev's inequality.
3. Denote the residuals from step 2 as and fit the model
The null and alternate hypotheses are
| H0 : a = 0 => unit root = no cointegration
|H1 : a ¹ 0 =>no unit root = cointegration
Interpretation: Rejection of the Null implies the residual is
If the residual series is stationary then yt and xt
must be cointegrated.
4. If you reject the null in step 3 then estimate the parameters of the
Error Correction Model
| D yt
| a1 + ay (yt-1-b xt-1
|| a11(i)D yt-i
|| a12(i)D xt-i
| a2 + ax (yt-1-b xt-1
|| a21(i)D yt-i
|| a22(i)D xt-i
ECM is generalized to vectors:
The components of the vector xt = (x1t, x2t, .. , xnt) are
cointegrated of order (d,b), denoted by xt CI(d,b), if
All components of xt are I(d) and
There exists a vector b = (b1, b2,...,bn)
such that b xt is I(d-b), where b > 0.
Note b is called the cointegrating vector.
Points to remember:
To make b unique we must normalize on one of the coefficients.
All variables must be cointegrated of the same order. But, all
variables of the same I(d) are not necessarily cointegrated.
If xt is n x 1
then there may be as many as n-1 cointegrating vectors. The
number of cointegrating vectors is called the cointegrating rank.
An interpretation of cointegrated variables is that they share a
common stochastic trend.
Given our notions of equilibrium in economics, we must conclude
that the time paths of cointegrated variables are determined in
part by how far we are from equilibrium. That is, if the
variables wander from each other, there must be some way for them
to get back together. This is
the notion of error correction.
Granger Representation theorem :
"Cointegration implies Error Correction Model (ECM)."
||Aaron at willmot.com
see original discussion link
Cointegration is covered in any good econometrics textbook. If
you need more depth, Johansen's Likelihood-Based Inference in
Cointegrated Vector Autoregressive Models, Oxford University
Press, 1995, is good.
I do not recommend the original Engle and Granger paper (1987).
Two series are said to be (linearly) "cointegrated" if a (linear)
combination of them is stationary. The practical effect in
Finance is we deal with asset prices directly instead of asset
For example, suppose I want to hold a market neutral investment
in stock A (I think stock A will outperform the general market,
but I have no view about the direction of the overall market).
Traditionally, I buy 1,000,000 of stock A and short 1,000,000
times Beta of the Index. Beta is derived from the covariance of
returns between stock A and the Index. Looking at things another
way, I choose the portfolio of A and the Index that would have
had minimum variance of return in the past (over the estimation
interval I used for Beta, and subject to the constraint that the
holding of stock A is $1,000,000).
A linear cointegration approach is to select the portfolio in the
past that would have been most stationary. There are a variety of
ways of defining this (just as there are a variety of ways of
estimating Beta) but the simplest one is to select the portfolio
with the minimum extreme profit or loss over the interval. Note
that the criterion is based on P&L of the portfolio (price) not
return (derivative of price).
The key difference is correlation is extremely sensitive to small
time deviations, cointegration is not. Leads or lags in either
price reaction or data measurement make correlations useless. For
example, suppose you shifted the data in the problem above, so
your stock quotes were one day earlier than you Index quotes.
That would completely change the Beta, probably sending it near
zero, but would have little effect on the cointegration analysis.
Economists need cointegration because they deal with bad data,
and their theories incorporate lots of unpredictable leads and
lags. Finance, in theory, deals with perfect data with no leads
or lags. If you have really good data of execution prices,
cointregation throws out your most valuable (in a money-making
sense) information. If you can really execute, you don't care if
there's only a few seconds in which to do so. But if you have bad
data, either in the sense that the time is not well-determined or
that you may not be able to execute, cointegration is much safer.
In a sense, people have been using cointegration for asset
management as long as they have been computing historical pro
forma strategy returns and looking at the entire chart, not just
the mean and standard deviation (or other assumed-stationary
My feeling is cointegration is essential for risk management and
hedging, but useless for trading and pricing. Correlation is easy
and well-understood. You can use it for risk-management and
hedging, but only if you backtest (which essentially checks the
results against a co-integration approach) to find the
appropriate adjustments and estimation techniques. Correlation is
useful for trading and pricing (sorry Paul) but only if you allow
stochastic covariance matrices.
More formally, if a vector
of time series is I(d) but a linear combination is integrated to a lower order, the time
series are said to be co-integrated.
from : Econometric Forecasting
These are the arguments in favor of testing whether a series has a unit root:
(1) It gives information about the nature of the series that
should be helpful in model specification, particularly whether to
express the variable in levels or in differences.
(2) For two or more variables to be cointegrated each must
possess a unit root (or more than one).
These are the arguments against testing:
(1) Unit root tests are fairly blunt tools. They have low power
and often conclude that a unit root is present when in fact it is
not. Therefore, the finding that a variable does not possess a
unit root is a strong result. What is perhaps less well known is
that many unit-root tests suffer from size distortions. The
actual chance of rejecting the null hypothesis of a unit root,
when it is true, is much higher than implied by the nominal
significance level. These findings are based on 15 or more Monte
Carlo studies, of which Schwert (1989) is the most influential
(Stock 1994, p. 2777).
(2) The testing strategy needed is quite complex.
In practice, a nonseasonal economic variable rarely has more than
a single unit root and is made stationary by taking first
differences. Dickey and Fuller (1979) recognized that they could
test for the presence of a unit root by regressing the
first-differenced series on lagged values of the original series.
If a unit root is present, the coefficient on the lagged values
should not differ significantly from zero. They also developed
the special tables of critical values needed for the test.
Since the publication of the original unit root test there has
been an avalanche of modifications, alternatives, and
comparisons. Banerjee, Dolado, Galbraith, and Hendry (1993,
chapter 4) give details of the more popular methods. The standard
test today is the augmented Dickey-Fuller test (ADF), in which
lagged dependent variables are added to the regression. This is
intended to improve the properties of the disturbances, which the
test requires to be independent with constant variance, but
adding too many lagged variables weakens an already low-powered
Two problems must be solved to perform an ADF unit-root test: How
many lagged variables should be used? And should the series be
modeled with a constant and deterministic trend which, if
present, distort the test statistics? Taking the second problem
first, the ADF-GLS test proposed by Elliott, Rothenberg, and
Stock (1996) has a straightforward strategy that is easy to
implement and uses the same tables of critical values as the
regular ADF test.
First, estimate the coefficients of an ordinary trend regression
but use generalized least squares rather than ordinary least
squares. Form the detrended series, yd, given by
ytd = yt - b0 - b1 t , where
b0 and b1 are the coefficients just estimated.
In the second stage, conduct a unit root test with the standard
ADF approach with no constant and no deterministic trend but use
yd instead of the original series.
To solve the problem of how many lagged variables to use, start
with a fairly high lag order, for example, eight lags for annual,
16 for quarterly, and 24 for monthly data. Test successively
shorter lags to find the length that gives the best compromise
between keeping the power of the test up and keeping the
desirable properties of the disturbances. Monte Carlo experiments
reported by Stock (1994) and Elliott, Rothenberg and Stock (1996)
favor the Schwartz BIC over a likelihood-ratio criterion but both
increased the power of the unit-root test compared with using an
arbitrarily fixed lag length. We suspect that this difference has
little consequence in practice. Cheung and Chinn (1997) give an
example of using the ADF-GLS test on US GNP.
Although the ADF-GLS test has so far been little used it does
seem to have several advantages over competing unit-root tests:
(1) It has a simple strategy that avoids the need for sequential
testing starting with the most general form of ADF equation (as
described by Dolado, Jenkinson, and Sosvilla-Rivero (1990, p.
(2) It performs as well as or better than other unit-root tests.
Monte Carlo studies show that its size distortion (the difference
between actual and nominal significance levels) is almost as good
as the ADF t-test (Elliott, Rothenberg & Stock 1996; Stock 1994)
and much less than the Phillips-Perron Z test (Schwert 1989).
Also, the power of the ADF-GLS statistic is often much greater
than that of the ADF t-test, particularly in borderline
Elliott et. al (1996) showed that there is no uniformly most
powerful test for this problem and derived tests that were
approximately most powerful in the sense that they have
asymptotic power close to the envelope of most powerful tests for
The variance ratio methodology tests the hypothesis that the variance of multi-period
returns increases linearly with time.
Based on the idea that, if a series is stationary, the variance of the
series is not increasing over time; while a series with a unit root has
- Intuition: Compare the variance of a subset of the data "early" in
the series with a similarly-sized subset "later" in the process. In the
limit, for a stationary series, these two values should be the same, while
they will be different for an I(1) series. Thus, the null hypothesis is
stationarity, as for the KPSS test.
- There's a good, brief discussion of these tests in Hamilton (p. 531-32).
Other cites are Cochrane (1988), Lo and McKinlay (1988), and
Cecchetti and Lam (1991).
Hence if we calculate the variance s2 of a series of returns
every D t periods, the null hypothesis suggests that sampling
every k*D t periods will lead to a variance k s2:
Variance(rk D t) = k Variance(rD t)
The variance ratio is significantly below
one under mean reversion, and above one under random walk:
||=1 under random walk
|Variance(rk D t)/k
||<1 under mean reversion
||>1 under mean aversion
More precisely, in the usual fixed k asymptotic treatment,
under the null hypothesis that the xt follow a random
walk with possible drift, given by
where µ is areal number and et is a sequence of
et is a sequence of zero mean independent random variables, it is
possible to show
where sk2 is some simple function
of k (this is not the variance, due to overlapping
observations, to make sample size sufficiently large for large k, and to
correct the bias in variance estimators)
(VR(k)-1) ® N(0,sk2)
note that this result is quite general, and stands under the
simple hypothesis that
et is a sequence of zero mean independent random variables.
Any significant deviation from 53
means that et are not independent random variables.
This result extends to the case where the et
are a martingale difference series
with conditional heteroscedasticity
though the variance sk2 has to be adjusted a little.
The use of the VR statistic
can be advantageous when testing against several interesting
alternatives to the random walk model, most notably those
hypotheses associated with mean reversion.
In fact, a number of authors (e.g., Lo and Mackinlay (1989),
Faust (1992) and Richardson and Smith (1991)) have found that the
VR statistic has optimal power against such alternatives.
Note that VR(k) can be writen as :
This expression can be expanded in :
where ri is the i'th term in the autocorrelation
of returns. This expression holds asymptotically.
Note that this expression can be used to calculate
the ACF at various lags.
For example, for k=2
Note that if VR(2) is significantly under one,is the same as
as a negative autocorrelation at lag one : r1<0
Let xi i=0,N be the series of increments (log of returns for example)
xk is the log price at time k, N, the sample size,
M=k(N-k+1)(1-k/N) and µ^=(xN-x0)/N estimate of the mean.
VR(k) is defined as sa^2/sc^2
Testing for the null hypothesis is to test if
is normally distributed.
Nomality classical tests can be applied, like z scores,
Kolmogorov Smirnov test (or rather a Lillifors test).
||Absolute Return Ratio Test
source: Groenendijky & Al., A Hybrid Joint Moment Ratio Test for Financial Time Series
If one augments the martingale assumption for financial
asset prices with the condition that the martingale differences have constant
(conditional) variance, it follows that the variance of asset returns is directly
proportional to the holding period. This property has been used to construct
formal testing procedures for the martingale hypothesis, known as variance
Variance ratio tests are especially good at detecting linear dependence
in the returns.
While the variance ratio statistic describes
one aspect of asset returns, the idea behind this statistic can be generalized
to provide a more complete characterization of asset return data.
We focus on using a combination of the
variance ratio statistic and the first absolute moment ratio statistic.
The first absolute moment ratio statistic by itself is useful as a measure of linear
dependence if no higher order moments than the variance exist.
In combination with the variance ratio statistic it can be used to disentangle linear
dependence from other deviations of the standard assumption in
unconditionally normally distributed returns. In particular, the absolute
moment ratio statistic provides information concerning the tail of the
distribution and conditional heteroskedasticity.
By using lower order moments of
asset returns in the construction of volatility ratios, e.g., absolute returns,
one relaxes the conditions on the number of moments that need to exist for
standard asymptotic distribution theory to apply.
We formally prove that
our general testing methodology can in principle even be applied for return
distributions that lie in the domain of attraction of a stable law (which
includes the normal distribution as a special case). Stable laws, apart from
the normal distribution, have infinite variance, such that our approach is
applicable outside the finite-variance paradigm.
Since in empirical work there often
exists considerable controversy about the precise nature of the asset return.
The first absolute moment
has been used before as a measure of volatility,
Muller et al. observe a regularity
in the absolute moment estimates which is not in line with the presumption
of i.i.d. normal innovations; this regularity was labelled the scaling law. In
this paper we consider the ratios of these absolute moment estimates, we
obtain their statistical properties under various distributional assumptions,
and we explain the observed regularity behind the `scaling law'.
In particular, we show why the deviations observed by Muller et al. should not
be carelessly interpreted as evidence against the efficient market hypothesis.
Furthermore, we show that the absolute moment ratio statistics contain much
more information than the scaling law. Especially, when the statistic is used
in combination with the variance ratio statistic, most of the characteristic
features of asset returns come to the fore.
Specifically, we advocate the simultaneous use of volatility statistics based
on first (absolute returns) and second order moments (variances).
a way we construct a test which is not only suited to detect linear
dependence in asset returns, but also fat-tailedness and non-linear dependence,
e.g., volatility clustering.
We analytically show why moment ratios based
on absolute returns can be used to detect fat-tailedness and volatility
clustering, while standard variance ratios convey no information in this respect.
Discriminating between the alternative phenomena is important, since they
have different implications for portfolio selection and risk management.
Throughout the paper, we rely on a convenient graphical representation of
the statistics: the moment ratio curves.
The formal testing procedure we propose in this paper heavily builds
on the bootstrap. By performing a non-parametric bootstrap based on the
empirical returns, we construct uniform confidence intervals for the range
of moment ratios considered.
Absolute returns exhibits the highest correlation
(Rama Cont ).
||Multi variate co-integration - Vector Error Correction Modelling
Among the general class of the multivariate ARIMA (AutoRegressive
Integrated Moving Average) model, the Vector Autoregressive (VAR) model turns
out to be particularly convenient for empirical work. Although there are important
reasons to allow also for moving average errors (e.g. L. utkepohl 1991, 1999), the
VAR model has become the dominant work horse in the analysis of multivariate
time series. Furthermore, Engle and Granger (1987) show that VAR model is
an attractive starting point to study the long run relationship between time series
that are stationary in first differences. Since Johansen's (1988) seminal paper, the
co-integrated VAR model has become very popular in empirical macroeconomics.
bibliography list from Petits Déjeuners de la Finance
, Engle and Granger seminal paper:
"Cointegration and Error-Correction: Representation, Estimation and Testing",
- ***** Explaining Cointegration Analysis: David F. Hendry and
part I, cached
part II, cached
- Carol Alexander is specialized in cointegration trading
and index tracking,
"Cointegration and asset allocation: A new active hedge fund strategy"
includes a good intro to cointegration,
see also, http://www.bankingmm.com
and related paper of Alexander 
"Cointegration-based trading strategies:
A new approach to enhanced index tracking and statistical arbitrage"
"Intraday Price Formation in US Equity Index Markets" ,
Joel Hasbrouck is studying relationships between stocks and futures and ETF.
implementation source codes, cached
- Chambers 
This paper analyses the effects of
sampling frequency on the properties of spectral regression
estimators of cointegrating parameters.
- "Numerically Stable Cointegration Analysis" 
is a practical impementation to estimate commen trends:
Cointegration analysis involves the solution of a generalized eigenproblem involving moment matrices
and inverted moment matrices. These formulae are unsuitable for actual computations because the condition
numbers of the resulting matrices are unnecessarily increased. Our note discusses how to use the structure of
the problem to achieve numerically stable computations, based on QR and singular value decompositions.
- , a simple illustration of cointegration
with a drunk man and his dog ...
-  Adrian Trapletti paper on intraday cointegration
- Common stochastic trends, cycles and sectoral fluctuations
- johansen test critical values
This paper uses Reuters high-frequency exchange rate data to investigate the
contributions to the price discovery process by individual banks in the foreign ex-
- "Intraday Lead-Lag Relationships Between the
Futures, Options and Stock Market", F. de Jong and M.W.M. Donders.
Includes interesting method to estimate cross corelation whith
- Cointegration in Single Equations in lectures from
- Vector Error Correction Modeling from SAS online support.
- Variance ration testing: an other test for stationary
aplication to stokc market indices, cached
random walk or mean reversion ..., cached
On the Asymptotic Power of the Variance Ratio Test, cached
- a general discussion on
Econometric Forecasting and methods, by P. Geoffrey Allen and
- a simple presentation of Dickey Fuler test
in French, cached
- The R
tseries package include Augmented Dickey–Fuller
- How to do a 'Regular' Dickey-Fuller Test Using Excel
Alexander, C. (1994): History Debunked
RISK 7 no.12 (1994) pp59-63
- Alexander, C. (1995): Cofeatures in international bond and equity markets
- Alexander, C., Johnson, A. (1992): Are foreign exchange markets really efficient ?
Economics Letters 40 (1992) 449-453
- Alexander, C., Johnson, A. (1994): Dynamic Links
RISK 7:2 pp56-61
- Alexander, C., Thillainathan, R (1996): the Asian Connections
Emerging Markets Investor 2:6 pp42-47
- Beck, S.E.(1994): Cointegration and market inefficiency in commodities futures markets
Applied Economics 26:3 pp 249-57
- Bradley, M., Lumpkin, S. (1992): The Treasury yield curve as a cointegrated system
Journal of Financial and Quantitative Analysis 27 pp 449-63
- Brenner, R.J., Kroner, K.F. (1995): Arbitrage, cointegration and testing
the unbiasedness hypothesis in financial markets
Journal of Financial and Quantitative Analysis 30:1 pp23-42
- Brenner, R.J., Harjes, Kroner, K.F. (1996): Another look at alternative
models of the short term interest rate
Journal of Financial and Quantitative Analysis 31:1 pp85-108
- Booth, G., Tse, Y. (1995): Long Memory in Interest Rate Futures Markets:
A Fractional Cointegration Analysis
Journal of Futures Markets 15:5
- Campbell, J.Y., Lo, A.W., MacKinley, A.C. (1997): The Econometrics of
Financial Markets Princeton University Press
- Cerchi, M., Havenner, A. (1998): Cointegration and stock prices
Journal of Economic Dynamic and Control 12 pp333-4
- Chowdhury, A.R. (1991): Futures market efficiency: evidence from
The Journal of Futures Markets 11:5 pp577-89
- Chol, l. (1992): Durbin-Haussmann tests for a unit root
Oxford Bulletin of Economics and Statistics 54:3 pp289-304
- Clare, A.D., Maras, M., Thomas, S.H. (1995): The integration and
efficiency of international bond markets
Journal of Business Finance and Accounting 22:2 pp313-22
- Cochrane, J.H. (1991): A critique of the application of unit root tests
Jour. Econ. Dynamics and Control 15 pp275-84
- Dickey, D.A., Fuller, W.A. (1979): Distribution of the estimates for
autoregressive time series with a unit root
Journal of the American Statistical Association 74 pp427-9
- Duan, J.C., Pliska, S. (1998): Option valuation with cointegrated asset prices
- Dwyer, G.P., Wallace, M.S. (1992): Cointegration and market efficiency
Journal of international Money and Finance
- Engle, R.F., Granger, C.W.J. (1987): Cointegration and error correction:
representation, estimation and testing
Econometrica 55:2 pp251-76
- Engle, R.F., Yoo, B.S. (1987): Forecasting and testing in cointegrated systems
Jour. Econometrics 35 pp143-59
- Granger, C.W.J. (1988): Some recent developments on a concept of causality
Jour. Econometrics 39 pp199-211
- Hail, A.D., Anderson, H.M., Granger C.W.J. (1992): A cointegration
analysis of Treasury bill yields
The Review of Economics and Statistics pp116-26
- Hamilton, J.D. (1994): Time Series Analysis
Princeton University Press
- Harris, F.deB., McInish, T.H., Shoesmith, G.L., Wood, R.A. (1995):
Cointegration, Error Correction, And Price Discovery On Informationally Linked
Journal of Financial and Quantitative Analysis 30:4
- Hendry, D.F. (1986): Econometrics modelling with cointegrated variables:
Oxford Bulletin of Economics and Statistics 48:3 pp201-12
- Hendry, D.F. (1995): Dynamic Econometrics
Oxford University Press
- Johansen, S. (1988): Statistical analysis of cointegration vectors
Journal of Economic Dynamics and Control 12 pp231-54
- Johansen, S., Juselius, K. (1990): Maximum likelihood estimation and
inference on cointegration - with applications to the demand for money
Oxford Bulletin of Economics and Statistics 52:2 pp169-210
- Masih, R. (1997): Cointegration of markets since the '87 crash
Quaterly Review of Economics and Finance 37:4
- Proietti, T. (1997): Short-run dynamics in cointegrated systems
Oxford Bulletin of Economics and Statistics 59:3
- Schwartz, T.V., Szakmary, A.C. (1994): Price discovery in petroleum
markets: arbitrage, cointegration and the time interval of analysis
Journal of Futures Markets 14:2 pp147-167
- Schmidt, P., Phillips, P.C.B. (1992): LM tests for a unit root in the
presence of deterministic trends
Oxford Bulletin of Economics and Statistics 54:3 pp257-288
- Wang, G.H.K., Yau, J. (1994): A Time Series Approach To Testing For
Market Linkage: Unit Root And Cointegration Tests
Journal of Futures Markets 14:4
Alexander, C and Dimitriu, A.
''Cointegration-based trading strategies: A new approach to enhanced index
tracking and statistical arbitrage'' , 2002.
Discussion Paper 2002-08, ISMA Centre Discussion Papers in Finance
Alexander, C, Giblin, I, and Weddington, W.
''Cointegration and asset allocation: A new active hedge fund strategy'' ,
Discussion Paper 2003-08, ISMA Centre Discussion Papers in Finance
Chambers, M. J.
''Cointegration and Sampling Frequency'' , 2002.
''Empirical properties of asset returns - stylized facts and statistical
QUANTITATIVE FINANCE, 2000.
Doornik, J. A and O'Brien, R.
''Numerically Stable Cointegration Analysis'', 2001.
Engle, R and Granger, C.
''Cointegration and Error-Correction:
Representation, Estimation and Testing'' .
Econometrica, 55:251--276, 1987.
Goetzmann, W, g. Gatev, E, and Rouwenhorst, K. G.
Trading: Performance of a Relative Value Arbitrage Rule'' , Nov 1998.
''Intraday Price Formation in US Equity Index Markets'' , 2002.
''Croissance optimale'' , 2003.
''Optimal Convergence Trading'' , 2004.
Murray, M. P.
drunk and her dog : An illustration of cointegration and error correction''
The American Statistician, Vol. 48, No. 1, February 1994, p. 37-39.
Stock, J. H and Watson, M.
''Variable Trends in Economic Time Series''.
Journal of Economic Perspectives, 3(3):147--174, Summer 1988.
Trapletti, A, Geyer, A, and Leisch, F.
''Forecasting exchange rates using cointegration models and intra-day data''
Journal of Forecasting, 21:151--166, 2002.
Thompson, G. W. P.
''Optimal trading of an asset driven by a hidden Markov process in the
presence of fixed transaction costs'' , 2002.
source: from Andrei Simonov, no longer available online
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