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MA Advanced Econometrics:Spurious Regressions and Cointegration

Karl Whelan

School of Economics, UCD

February 22, 2011

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 1 / 18

A More Serious Problem: Spurious Regressions

When estimating a univariate time series process, we are often interested incalculating the value of and whether this value equals one or somethingless can be of interest: We may be interested in whether shocks havepermanent or temporary effects and, if temporary, how long they take tofade away. This is one reason to teach about the non-standard distributionsthat occur when a time series is nonstationary.

However, there is a deeper problem when analysing nonstationary time series.

Most of econometrics is concerned with assessing relationships betweenvariables: Usually, we are asking the question Does x have an effect on y?But when two different unrelated nonstationary series are regressed on eachother, the result is usually a so-called spurious regression, in which the OLSestimates and t statistics indicate that a relationship exists when, in reality,there is no such relationship.

The modern literature on this dates from a famous paper by Granger andNewbold from 1974. However, the nature of the problem was known at leastas far back as 1926.

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 2 / 18

Yule (1926) on Nonsense Correlations

In 1926, Georges Udny Yule wrote a paper in the Journal of the RoyalStatistical Society called Why Do We Sometimes get NonsenseCorrelations between Time-Series?

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 3 / 18

George Udny Yules Chart from 1926

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 4 / 18

Yules Discussion of His Chart

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 5 / 18

Spurious Regressions: Unit Roots with Drifts

When discussing spurious regressions, econometric textbooks tend to focuson what happens when we take processes that are unit roots without drift(i.e. yt = yt1 + t with no constant term) and regress them on each other.

In applied econometric work, however, unit root without drift processes arenot very common. Generally, we work with series that tend to be stationaryor else with series that have a clear upward trend and which may be unitroot processes with drift (i.e. take the form yt = + yt1 + t .)

While explanations of how the spurious regression problem works fornon-drifting unit root processes are quite complex, the spurious regressionproblem is far more relevant in the case where the processes have drift. Italso turns out that the problem is easier to explain in this case.

A property of drifting unit root processes that we will use is the following

yt = + yt1 + t (1)

= + + yt2 + t + t1 (2)

= t +t

k=1

k + y0 (3)

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 6 / 18

Useful Results About Infinite Sums

Establishing properties about regressions involving drifting unit root serieswill require figuring out properties of sums of the form

Tt=1 t and

Tt=1 t

2.

Note that 1 + 2 + 3 = 6 = (3)(4)2 and 1 + 2 + 3 + 4 = 10 =(4)(5)

2 . Thegeneral rule is

Tt=1

t =T (T + 1)

2=

1

2

(T 2 + T

)(4)

For sums of squares, we have

Tt=1

t2 =T (T + 1) (2T + 1)

6=

1

6

(2T 3 + 3T 2 + T

)(5)

This means that as T

1

T 2

Tt=1

t 12

(6)

1

T 3

Tt=1

t2 13

(7)

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 7 / 18

Regressions Featuring Unit Roots with Drifts

Consider regressing yt on the completely unrelated series xt where

yt = y + yt1 + yt (8)

xt = x + xt1 + xt (9)

The OLS estimator is

=

Tt=1 xtytTt=1 x

2t

(10)

=

Tt=1

(x t +

tk=1

xk + x0

) (y t +

tk=1

yk + y0

)Tt=1

(x t +

tk=1

xk + x0

)2 (11)As T gets large, the terms in t2 will dominate all other terms. Re-writingthis as

=1

T 3

Tt=1

(x t +

tk=1

xk + x0

) (y t +

tk=1

yk + y0

)1

T 3

Tt=1

(x t +

tk=1

xk + x0

)2 (12)then all of the terms that are not of the form 1T 3

Tt=1 t

2 will go to zero.

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 8 / 18

Spurious Regression Results

This means that as T gets large

p xy

2x=

yx

(13)

In other words, the OLS estimator will tend towards the ratio of the twodrift terms. In addition, the t statistics will generally indicate that there is ahighly statistically significant relationship.

The next pages show s and t-stats from regressing yt on xt where

yt = 0.2 + yt1 + yt (14)

xt = 0.1 + xt1 + xt (15)

where the error terms are i.i.d. normally distributed errors. They show OLScoefficients averaging 2 and highly significant t-stats.

Note that the key terms driving these results were the time trends. Theseresults also apply to trend stationary series like yt = t + yt1 + t , sothe problem is not specific to the unit root series.

Similar results apply to regressions featuring unit roots without drifts butderiving these results analytically is beyond the scope of this module.

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 9 / 18

Distribution of from Regressions of Unrelated Unit RootsWith Drift (T = 500)

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 Mean 1.99986 Std Error 0.26284 Skewness 0.09493 Exc Kurtosis 0.00404

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 10 / 18

Distribution of t Statistics from Regressions of UnrelatedUnit Roots With Drift (T = 500)

0 100 200 300 400 500 600 7000.000

0.001

0.002

0.003

0.004

0.005

0.006 Mean 232.95066 Std Error 69.49500 Skewness 0.62136 Exc Kurtosis 0.50443

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 11 / 18

The I (k) Terminology and Cointegration

Unit root series such as yt = + yt1 + t are nonstationary: Sums of ytdont settle down at a stable mean and its covariances change over time.

However, once you calculate the first difference of this series yt = + t , itbecomes a covariance stationary series.

We say a series is integrated of order k (denoted I (k)) if it has to bedifferenced k times before it becomes stationary. Sometimes, one can cancome across examples involving I (2) series, but generally the time series inpractical applications are either I (1) or I (0).

The spurious regression problem can be stated as the fact that unrelatedI (1) series regressed upon each other tend to appear to be related accordingto the usual OLS diagnostics.

However, what if there really is a relationship? For example, what if yt andxt are both I (1) series but there existed a coefficient such thatyt xt I (0). In this case, there is a common trend across the series andwe say that the series yt and xt are cointegrated.

In this case, it turns out that OLS estimates of are consistent.

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 12 / 18

Consistency of OLS Under Cointegration

Consider again the case where xt is a unit root with drift

xt = x + xt1 + xt (16)

but in this case the variable yt is cointegrated with xt so that

yt = xt + ut (17)

where ut is mean-zero I (0) series.

We can calculate the properties of the OLS estimator as follows:

= +

Tt=1 xtuttt=1 x

2t

(18)

= +

Tt=1

(x t +

tk=1

xk + x0

)utT

t=1

(x t +

tk=1

xk + x0

)2 (19)In this case, the terms in t2 will dominate as T so that thedenominator of the last term will grow faster than the numerator. This

means that p . (One could show this more formally using the formulae

for infinite sums derived earlier.)

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 13 / 18

Super-Consistency!

Not only is the OLS estimator of a cointegrating regression consistent, inthe sense that it is likely to get ever-closer to the true value of as samplesget larger, it turns out its superconsistent. Whats that mean?

Now multiply both sides of (19) by T but do this to the right hand side bydividing the numerator by T 2 and the denominator by T 3:

T(

)=

1T 2

Tt=1

(x t +

tk=1

xk + x0

)ut

1T 3

Tt=1

(x t +

tk=1

xk + x0

)2 (20)The numerator converges to zero (the sum 1T 2

Tt=1 x t

x2 but is

multiplied by an uncorrelated mean zero series ut) while the denominator

converges to2x3 . So T

(

)p 0.

We have seen cases before where converges in distribution to a meanzero series when multiplied by

T . In this case, the gap between the

estimator and the true value converges in probability to zero even whenmultiplied by T . This property is known as superconsistency.

Karl Whelan (UCD) Spurious Regressions and Cointegration February 22, 2011 14 / 18

The Error-Correction Representation

Consider two I (1) series, y