1/4/2009 1

Cointegration in Theory and Practice

A Tribute to Clive Granger

ASSA Meetings January 5, 2010

James H. Stock Department of Economics, Harvard University and the NBER

1/4/2009 2

Cointegration: The Historical Setting Granger and Newbold (Journal of Econometrics, 1974) It is very common to see reported in applied econometric literature time series regression equations with an apparently high degree of fit, as measured by the coefficient of multiple correlation R2 or the corrected coefficient 2R , but with an extremely low value for the Durbin-Watson statistic. We find it very curious that whereas virtually every textbook on econometric methodology contains explicit warnings of the dangers of autocorrelated errors, this phenomenon crops up so frequently in well-respected work… (p. 111)

1/4/2009 3

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Dickey-Fuller (JASA, 1979):

Yt = ρYt–1 + εt

The hypothesis that ρ = 1 is of some interest in applications because it corresponds to the hypothesis that it is appropriate to transform the time series by differencing. Currently, practitioners may decide to difference a time series on the basis of visual inspection of the autocorrelation function… (p. 427)

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Davidson, Hendry, Srba and Yeo (Economics Journal, 1978)

Δ4ct = 0.49Δ4yt – 0.17ΔΔ4yt – .06(ct-4 – yt–4) + 0.01Dt (41) (.04) (.05) (.01) (.004) cf. Hall (1978)

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Cointegration: Econometric Theory Triumphs and Disappointments

Triangular model – cointegrated (1,1) with n=2, r=1: Δxt = vt yt = θxt + ut Main aims of initial econometric theory:

ˆ1.Superconsistency of OLS estimator θ ˆ2.Use of estimated ECM term ˆtz = yt – θ xt as a regressor – without the

“generated regressor” problem 3.Testing for cointegration (e.g. EG-ADF test on OLS residual) 4.Efficient (Gaussian) estimation of θ 5.Inference for θ

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Simple expositional model: Δxt = vt yt = θxt + ut, Assume: (vt, ut) ~ (0,Σ), serially uncorrelated, and

[ .]

1[ .]

1

1

1

T

ttT

tt

vT

uT

=

=

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ ⇒ ⎟ , B is BM(Σ)

(.)(.)

v

u

BB⎛ ⎞⎜⎝ ⎠

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Superconsistency and distribution

ˆ( )T θ θ− = 1

2

1

1

1

T

t tt

T

tt

x uT

xT

=

=

∑

∑ ⇒

2

uv v u

v

B dB

B

σ + ∫∫

ECM as a regressor wt = βzt + ζt = β ˆtz + [β(zt – ˆtz ) + ζt]

= βzt + ζt = β ˆtz + [(θ̂ – θ)βxt + ζt] Efficient estimation and inference MLE: f(Y,X|θ,γ) = f(Y|X,θ,γ1)f(X|γ2) so yt = θxt + γ1(L)Δxt + tu⊥

ˆ( )T θ θ− = 1

2

1

1

1

T

t tt

T

tt

x uT

xT

⊥

=

=

∑

∑ + op(1) ⇒

2

v u

v

B dB

B⊥∫

∫ ~ 2

2

2(0, )

v

uB

v

N dFB

σ ⊥

∫∫ ∫

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Which of these results are robust to changes in assumptions about long run properties? xt = αxt–1 + vt, α = 1 + c/T (local to unity model) yt = θxt + ut, (vt, ut) satisfy same assumptions; c is unknown Superconsistency and distribution

ˆ( )T θ θ− = 1

22

1

1

1

T

t tt

T

tt

x uT

xT

=

=

∑

∑ ⇒

2

uv v u

v

J dB

J

σ + ∫∫

, dJv = cJv + dWv

ECM as a regressor wt = βzt + ζt = β ˆtz + [β(zt – ˆtz ) + ζt]

= βzt + ζt = β ˆtz + [(θ̂ – θ)βxt + ζt]

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Efficient estimation and inference MLE: f(Y,X|θ,γ) = f(Y|X,θ,γ1)f(X|γ2) so yt = θxt + γ1(L)(xt – αxt–1) + tu⊥

= θxt + γ1(L)Δxt + [γ(L)(1–α)xt–1 + tu⊥]

= θxt + γ1(L)Δxt + [T–1γ(L)cxt–1 + tu⊥] so (Elliott (1998))

ˆ( )T θ θ− = ( )1

1

22

1

1 (1)

1

T

t t tt

T

tt

x T cx uT

xT

γ− ⊥

=

=

+∑

∑ + op(1)

= γ(1)c + 1

22

1

1

1

T

t tt

T

tt

x uT

xT

=

=

∑

∑ ⇒ γ(1)c +

2

v u

v

J dB

J⊥∫

∫ ~ 2

2

2( (1) , )

v

uJ

v

N c dFJ

σγ⊥

∫∫ ∫

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Some recent work on this problem Jansson and Moreira (2006) The OLS and MLE distributions are sensitive to other models of long-run behavior e.g. fractional integration – essentially have nuisance parameters that are not estimable. See Müller and Watson (2008) Testing has the same issues

1/4/2009 14

Cointegration: Empirical Legacy Two examples (1) Consumption/income and consumption/income/wealth DHSY (1978)…. Lettau and Ludvigson (2004)

8.6

8.7

8.8

8.9

99.

1ln

(a/y

)

-.05

0.0

5.1

.15

ln(c

/y)

1950q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1time

ln(c/y) ln(a/y)

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-.05

0.0

5.1

cay

1950q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1time

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-.05

0.0

5.1

cay

2000q1 2002q3 2005q1 2007q3 2010q1time

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(2) Housing values and median income

-.05

0.0

5.1

hous

ing

pric

e - i

ncom

e E

C te

rm

1970q1 1980q1 1990q1 2000q1time

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-.10

.1.2

.3ho

usin

g pr

ice

- inc

ome

EC

term

1970q1 1980q1 1990q1 2000q1 2010q1time

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