Slides-08 Modeling Long Run relationship
Modeling Long Run relationship
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School of Economics
Lecture Plan
• Long-run relationship: co-movement in trending time series
• Cointegration and common trend
• Interest rate and inflation
• Long and short term interest rates
• Regression with I(1) series under cointegration and dynamic OLS
• Spurious regression
• Test for cointegration
• Error correction models
• Information & price discovery
Long-run relationships
Long-run relationships
• Co-movement among time series
eg. US zero coupon rates: 3-month vs 9-month
Topic 4. Modelling Long-run Relationships
• Long-run relationships
– Co-movement among time series
eg. US zero coupon rates: 3-month vs 9-month
(1946:12-1987:2, 483 monthly observations)
Both appear non-stationary but move together.
School of Economics, UNSW Slides-06, Financial Econometrics 3
1950 1955 1960 1965 1970 1975 1980 1985
3-month coupon rate 9-month coupon rate
1950 1955 1960 1965 1970 1975 1980 1985
(3 Month Rate – 9 Month Rate)
(1946:12-1987:2, 483 monthly observations)
Both appear non-stationary but move together.
Long-run relationships
Long-run relationships
• Co-movement among time series
eg. US zero coupon rates: 3-month vs 9-month
Topic 4. Modelling Long-run Relationships
• Long-run relationships
– Co-movement among time series
eg. NYSE log Composite & Industrial indices
Both are non-stationary but move together.
– How to characterise such “co-movement”?
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LCOMP LINDU
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(LINDU – LCOMP)
Long-run relationships
Long-run relationships
• Co-movement among time series
• Two (or more) time series move together over time and never
depart for long.
• The time series are individually I(1) and vary a great deal. But their
long-run relationship appears stable over time.
• There must be a common trend that drives both time series.
• Important to exploit long-run relationships in finance eg.
pairs-trading; rational bubbles; bi-listed stocks
• We introduce basic facts on modelling long-run relationships, mainly
with bi-variate cases.
Long-run relationships
pairs=trading
12/09/2018 1:13 pmPairs Trading: Introduction | Investopedia
Page 1 of 3https://www.investopedia.com/university/guide-pairs-trading/
Guide to Pairs Trading
The origin of Pairs Trading
1. Pairs Trading: Introduction
2. Pairs Trading: Market Neutral Investing
3. Pairs Trading: Correlation
4. Arbitrage and Pairs Trading
5. Fundamental and Technical Analysis for Pairs
Pairs trading is a market-neutral trading strategy that matches a long position with a short position
in a pair of highly correlated instruments such as two stocks, exchange-traded funds (ETFs),
currencies, commodities or options. Pairs traders wait for weakness in the correlation and then go
long the under-performer while simultaneously short selling the over-performer, closing the
positions as the relationship returns to statistical norms.
The strategy’s profit is derived from the di!erence in price change between the two instruments,
rather than from the direction each moves. Therefore, a profit can be realized if the long position
goes up more than the short, or the short position goes down more than the long (in a perfect
situation, the long position rises and the short position falls, but that’s not a requirement for making
a profit). It’s possible for pairs traders to profit during a variety of market conditions, including
periods when the market goes up, down or sideways – and during periods of either low or high
volatility. (See also: 4 Factors That shape Market Trends.)
By Jean Folger | Updated February 21, 2018 — 8:30 AM EST SHARE
Spurious Regression
The spurious regression problem
I General result: a linear combination zt of a set of variables xit,
with order xit ∼ I(1), will have an order of integration equal to 1, if
there exists a linear combination, zt =
i=1 αixit ∼ I(0)
I Example: consider two series yt and xt, with
yt ∼ I(1); xt ∼ I(1)
and a linear combination zt thereof, i.e.
zt = α0 + α1yt + α2xt ∼ I(0)
Spurious Regression
The spurious regression problem
• Example: NYSE log Composite index vs Simulated RW
Symptom: the residual looks like RW
Topic 4. Modelling Long-run Relationships
• Spurious regression
– What if 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated?
• If two I(1) series 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated, the
linear regression
𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + 𝜀𝜀𝑡𝑡
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Residual Actual Fitted
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LCOMP SIMUL
Topic 4. Modelling Long-run Relationships
• Spurious regression
– What if 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated?
• If two I(1) series 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated, the
linear regression
𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + 𝜀𝜀𝑡𝑡
is called “spurious”.
eg. NYSE log Composite index
vs Simulated RW
School of Economics, UNSW Slides-06, Financial Econometrics 10
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Residual Actual Fitted
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LCOMP SIMUL
Spurious Regression
Examples of Spurious Regression
Multivariate Time Series Analysis: Cointegration analysis
Basic concepts
Spurious regression
Examples of spurious regression
I Egyptian infant mortality rate (Yt), 1971-1990, annual data,
on gross aggregate income of American farmers (It) and total
Honduran money supply (Mt)
Ŷt = 179.9
R2 = 0.918; F = 95.17; DW = 0.475
I US export index (Yt), 1960-1990, annual data, on Australian
males life expectancy (Xt)
Ŷt = −2943
R2 = 0.916; F = 315.2; DW = 0.360
Spurious Regression
The spurious regression problem
yt = β1 + β2xt + �t
• The spurious regression problem is characterized by
• Highly significant value for β2
• Fairly high R2
• Reason: distribution of the conventional test statistics are very
different from conventional case (stationary data)
• OLS estimator does not converge in probability as T →∞
• t−stats do not have well-defined asymptotic distributions
• Estimated stdv strongly underestimates true stdv (b/c
autocorrelation)
• Sign something is wrong:
• Highly autocorrelated residuals
Spurious Regression
Implication
The spurious regression problem implies that when regressing
non-stationary variables, the estimation results should not be taken
too seriously!!!
I Take first-differences of I(1) variables (GLS correction for
autocorrelation)
An important exception arises when the non-stationary series have a
common stochastic trend: cointegration.
I Don’t take first-differences
– specification error!
– advantage of I(1) variables (superconsistency)
Cointegration
Definition cointegration
The k variables of the k × 1 vector xt = (x1t, x2t, · · · , xkt)′ are said to
be cointegrated of order one, denoted as x1 ∼ CI(1) if
1 All variables in xt are integrated of the same order one, i.e.
xit ∼ I(1), for all i
2 There exists at least one vector β = (β1, β2, · · · , βk)′ of coefficients,
called the cointegrating vector, such that the linear combination
x′tβ = (β1x1t + β2x2t + · · · + βkxkt)
is integrated of a order zero, i.e. xt ∼ I(0)
Cointegration
In practice, xt ∼ CI(1) is most common.
Consider for instance two variables, yt and xt , which are both I(1).
If the residuals �t of the regression
yt = β1 + β2xt + �t
are I(0), i.e. �t ∼ I(0), then yt and xt are said to be cointegrated of
order CI(1) with cointegrating vector β = (1,−β1,−β2) as
yt − β1 − β2xt = �t ∼ I(0)
• eg. When (9monthRate − 3monthRate) is stationary, they are
cointegrated with cointegrating vector β = [1,−1].
• eg. When (logIndustrial − 0.98 logComposite) is stationary, they are
cointegrated with cointegrating vector β = [1,−0.98].
& common trend
• Common trend
eg. A model of interest rates (Fisher equation)
• Short & long term interest rates (rst , rlt) are directly influenced by
the inflation πt), subject to stationary shocks (�
• Both will be I(1) when the πt is I(1).
Here πt acts as the common trend that represents the trend
(non-stationary part) in both rst and r
• (rst , rlt) are cointegrated with β = [1,−1]′ because
rst − rlt = as − al + �st − �lt is I(0).
Cointegration
Economic Interpretation
If two (or more) series are linked to form an equilibrium relation
yt = β1 + β2xt
then even though the series themselves are non-stationary they will
nevertheless move closely together over time, i.e. they have a common
trend, such that deviations from the equilibrium
�t = yt − (β1 + β2xt)
are stationary.
I The concept of cointegration indicates the existence of a long-run
equilibrium to which an economic system converges over time and �t
can be interpreted as the equilibrium error, i.e. the distance the
system is away from the equilibrium at time t. As equilibrium errors
should be temporary, �t should be stationary.
Cointegration
Economic Interpretation
I The concept of spurious regression indicates that there is no
long-run equilibrium relation between yt and xt as the error term �t
is non-stationary, implying that deviations from the presumed
relation between yt and xt are permanent such that this relation is
not a long-run equilibrium relation.
Cointegration
Econometric implication
I If non-stationary variables are cointegrated, regression analysis
imparts meaningful information about the long-run relationship
between the variables.
In fact, it can be shown that in this case, the OLS estimator β̂ is
even a super consistent estimator for β, i.e. β̂ converges to β at a
much faster rate than with conventional asymptotics (i.e. for
stationary variables).
I If non-stationary variables are not cointegrated, regression results are
not meaningful, i.e. spurious regression problem.
Error-Correction Mechanism
Cointegration and Error-Correction Mechanisms
The existence of a long-run equilibrium relationship also has its
implications for the short-run behaviour of the I(1) variables
• The Granger representation theorem states that if a set of
variables is cointegrated, there has to be a mechanism that drives
the variables back to their long-run equilibrium relationship after the
equilibrium has been disturbed by a shock
• This mechanism is called an error-correction model
Error-Correction Mechanism
Example of an error-correction model
Consider two variables yt and xt which are cointegrated with
cointegrating vector β = (1,−β1,−β2).
A simple error-correction model (ECM) is given by
∆yt = γ1∆xt − α(yt−1 − β1 − β2xt−1) + µt (1)
= γ1∆xt − α�t−1 + µt (2)
The ECM incorporates both short-run and long-run effects
I The long-run equilibrium is obtained by imposing the ’no change’
condition ∆yt = ∆xt = µt = 0 and solve for yt
yt = β1 + β2xt
Thus, the long-run impact of xt on yt is given by β2.
I The contemporaneous impact of xt on yt is given by γ1.
Error-Correction Mechanism
Error correction mechanism
I The term −α�t−1 captures the error-correction mechanism. If yt
and xt are cointegrated, the Granger representation theorem implies
that α > 0.
I When yt is below its equilibrium value implied by xt, �t < 0 such
that yt increases back the equilibrium
I When yt is above its equilibrium value implied by xt , �t > 0 such
that yt decreases back to the equilibrium
Note that α measures the speed of adjustment towards the
equilibrium. The smaller α (i.e. the closer to zero), the lower this
speed of adjustment.
• When yt and xt are cointegrated, �t is the deviation from their
long-run equilibrium.
• yt+1 and xt+1 must move toward eliminating the deviation, or
correcting the cointegation error �t.
• Hence, �t is useful for predicting ∆yt+1 and ∆xt+1 and the models
for ∆yt+1 and ∆xt+1 should include �t as an explanatory variable.
Vector Error Correction Model
Vector Error correction VEC
• Vector error correction (VEC) model:
�t−1 = yt−1 − β0 − β1xt−1 (3)
∆xt = c1 + α1�t−1 + φ11∆xt−1 + φ12∆yt−1 + u1t (4)
∆yt = c2 + α2�t−1 + φ21∆xt−1 + φ22∆yt−1 + u2t (5)
• Eg, when α1 = 0, the adjustment toward equilibrium is all done by
yt and the common trend is xt.
Call α1 and α2 adjustment coefficients.
What happens when both α1 and α2 are zero?
Vector Error Correction Model
Price discovery in parallel markets
How information is incorporated into prices?
• Examples (usually require intraday price series)
• Bi-listed stock: which market sets the price?
• Spot & futures prices: does spot follows futures?
• For two log prices, yt and xt, on the same asset, the rule-of-one-price
dictates that �t = yt − xt can only fluctuate around zero.
• Hence, yt and xt are cointegrated with [1,−1] being the
cointegrating vector. The error correction model is applicable.
• The relative magnitudes of α1 and α2 can tell us to what extent xt
acts as price setter, sx =
Vector Error Correction Model
Example: Price discovery in parallel marketsTopic 4. Modelling Long-run Relationships
• Price discovery in parallel markets
eg. SP500 spot & futures indices: VEC
(20100104-20120810, 656 obs.)
– The adjustment coefficients:
𝛼𝛼futures is insignificant (t-stat = 0.46).
𝛼𝛼spot is significant (t-stat = -2.34).
– Futures appears to be the price-setter.
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LSPT LFUT DLSF
Vector Error Correction Model
Example: US and Canadian 10-years bond yeilds
Topic 4. Modelling Long-run Relationships
• Error correction & cointegration
eg. US and Canadian 10-year bond yields
Error correction model:
dca = ca – ca(-1), dus = us – us(-1),
e = ca – b0 – b1∙us .
School of Economics, UNSW Slides-06, Financial Econometrics 18
Correction is
done by CA, not US.
US acts as
the common trend.
Super consistency
Properties of OLS : Super consistency
Consider two time series yt and xt which are both I(1). Estimating the
static equation
yt = β1 + β2xt + �t
using OLS yields super consistent estimates of the long-run parameters
β1 and β2 when �t is I(0).
I Super consistency means that the OLS estimator converges to the true
population parameters at a much faster rate than with stationary variables
I This result arises as OLS picks the coefficients β̂ such that the variance of
the estimated residuals �̂t is as small as possible. As setting β̂ 6= β implies
that �t ∼ I(1) such that its variance becomes infinitely large when
T →∞, OLS is very efficient in picking the correct β
I The super consistency property of the OLS estimator implies that in
estimating the long-run relation between cointegrated variables, dynamics
and endogeneity issues can be ignored asymptotically
Super consistency
Properties of OLS : Super consistency
Topic 4. Modelling Long-run Relationships
• Cointegration & common trend
– Cointegration regression
• If two I(1) series 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are cointegrated, they may
be fitted in the linear regression
𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + 𝜀𝜀𝑡𝑡 , 𝜀𝜀𝑡𝑡 being stationary
where [1,−𝛽𝛽1] is the cointegrating vector.
• As long as 𝜀𝜀𝑡𝑡 is stationary, the OLS estimator of 𝛽𝛽1 is
consistent, but generally has a non-standard asymptotic
distribution.
• To make valid inference about 𝛽𝛽1, the “dynamic” OLS
estimator of 𝛽𝛽1 from
𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + ∑ 𝜓𝜓𝑗𝑗Δ𝑥𝑥𝑡𝑡−𝑗𝑗
𝑗𝑗=−𝑞𝑞 + 𝜀𝜀𝑡𝑡 .
See Saikkonen (1992, ET) or Stock & Watson (1993, Etrca).
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Super consistency
Properties of OLS : Super consistency
• The addition of leads and lags removes the deleterious effects that
short-run dynamics of the equilibrium process �t have on the
estimate of the cointegrating vector
• The DOLS estimator is consistent, asymptotically normally
distributed, and efficient.
• Asymptotically valid standard errors for the individual elements of
the estimated cointegration vector are given by their corresponding
HAC (e.g., Newey-West) standard errors.
Testing for cointegration
Consider two time series yt and xt.
Suppose we want to estimate the following equation:
yt = β1 + β2xt + �t
Prior to estimation, test the variables for their order of integration
1 If both are I(0): standard regression analysis is valid
2 If they are integrated of a different order, e.g. yt is I(1) and xt is
I(0): there can be no (long-run) relation between these two variables
3 If both are I(1): use cointegration analysis
Note however that there is almost never certainty about the true order of
integration
The Engle-Granger two-step approach
A popular methodology to test for cointegration and to analyse
cointegrating relationships is the so-called Engle-Granger two-step
1 Estimate the static model and test for cointegration
2 Estimate an ECM to analyse the short-run dynamics
The Engle-Granger two-step approach
Multivariate Time Series Analysis: Cointegration analysis
Testing for cointegration
The Engle-Granger two-step approach
Step 1: estimate static model and test for cointegration
Estimate the model in levels using OLS. Two cases can be
distinguished
1. The regression results are spurious if εt ∼ I (1)
2. OLS is super consistent if εt ∼ I (0)
After estimating a model including non-stationary variables, it is
therefore very important to test the order of integration of the
estimated residuals ε̂t . We consider two alternative tests:
1. The cointegrating regression Durbin-Watson (CRDW) test
2. ADF cointegration test
The Engle-Granger two-step approach
Multivariate Time Series Analysis: Cointegration analysis
Testing for cointegration
The Engle-Granger two-step approach
1. Cointegrating Regression Durbin-Watson (CRDW) test
Tests whether the residuals ε̂t are generated by a unit root
ε̂t = ε̂t−1 + υt
against the alternative that ε̂t is generated by a stationary
AR(1) process:
ε̂t = ρε̂t + υt with |ρ| < 1 using the Durbin-Watson (DW) statistic. As DW ≈ 2(1− ρ̂) this test boils down to testing whether DW is significantly larger than zero. The Engle-Granger two-step approach Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration The Engle-Granger two-step approach I Formally: H0 : ε̂t ∼ I (1) corresponds to ρ = 1 or d = 0 H1 : ε̂t ∼ I (0) corresponds to ρ < 1 or d > 0
I The 5% critical values for the CRDW test are given by
Number of variables Number of observations
(incl. yt) 50 100 250
2 0.72 0.38 0.20
3 0.89 0.48 0.25
4 1.05 0.58 0.30
5 1.19 0.68 0.35
I Drawback: the CRDW test is only valid when εt follows an
AR(1) process as the DW statistic only checks for an AR(1)
pattern in the data.
The Engle-Granger two-step approach
Multivariate Time Series Analysis: Cointegration analysis
Testing for cointegration
The Engle-Granger two-step approach
2. ADF cointegration test
Tests for a unit root in the estimated residuals using the
standard DF specification
∆ε̂t = γε̂t−1 +
αi∆ε̂t−i + ωt
with H0 : γ = 0 → no cointegration
H1 : γ < 0 → cointegration
Important notes
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