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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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(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

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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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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 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com