2.6. COINTEGRATION

63

improving the power of unit root tests, Schwert [126] shows that there are regions of the parameter space under which the size of the augmented Dickey—Fuller test is wrong in small samples. Since the panel tests are based on the augmented Dickey—Fuller test in some way or another, it is probably the case that this size distortion will get impounded into the panel test. To the extent that size distortion is an issue, however, it is not a problem that is speciÞc to the panel tests.

2.6Cointegration

The unit root processes {qt} and {ft} will be cointegrated if there exists a linear combination of the two time-series that is stationary. To understand the implications of cointegration, let’s Þrst look at what happens when the observations are not cointegrated.

No cointegration. Let ξqt = ξqt−1 + uqt and ξft		= ξft−1 + uft be	(38)
		iid		(39)
two independent random walk processes where uqt N(0, σq2) and
iid
uft N(0, σf2). Let zt = (zqt, zft)0 follow a stationary bivariate pro-			(40)
cess such as a VAR. The exact process for zt doesn’t need to explicitly
modeled at this point. Now consider the two unit root series built up
from these components
qt	= ξqt + zqt,
ft	= ξft + zft.	(2.87)

Since qt and ft are driven by independent random walks, they will drift arbitrarily far apart from each other over time. If you try to Þnd a value of β to form a stationary linear combination of qt and ft, you will fail because

qt − βft = (ξqt − βξft) + (zqt − βzft).

(2.88)

For any value of β, ξqt −βξft = (˜u1 + u˜2 + · · · u˜t) where u˜t ≡ uqt −βuft so the linear combination is itself a random walk. qt and ft clearly do not share a long run relationship. There may, however, be short-run interactions between their Þrst di erences

à ∆ft	! =	à ∆zft	! +	à ²ft	! .	(2.89)
∆qt		∆zqt		²qt

64 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS

By analogy to the derivation of (2.58), if zt follows a Þrst-order VAR, you can show that (2.89) follows a vector ARMA process. Thus, when both {qt} and {ft} are unit root processes but are driven by independent random walks, they can be Þrst di erenced to induce stationarity and their Þrst di erences modeled as a stationary vector process.

Cointegration. {qt} and {ft} will be cointegrated if they are driven by

	iid
the same random walk, ξt = ξt−1 +²t, where ²t N(0, σ2). For example
if
qt	= ξt + zqt,
ft	= φ(ξt + zft),	(2.90)
and you look for a value of β that renders
qt − βft = (1 − βφ)ξt + zqt − βφzft,		(2.91)

stationary, you will succeed by choosing β = φ1 since qt − fφt = zqt − zft is the di erence between two stationary processes so it will itself be stationary. {qt} and {ft} share a long-run relationship. We say that they are cointegrated with cointegrating vector (1, −φ1 ). Since random walks are sometimes referred to as stochastic trend processes, when two series are cointegrated we sometimes say that they share a common trend.28

The Vector Error-Correction Representation

Recall that for the univariate AR(2) process, you can rewrite qt = ρ1qt−1 + ρ2qt−2 + ut in augmented Dickey—Fuller test equation form as

∆qt = (ρ1 + ρ2 − 1)qt−1 − ρ2∆qt−1 + ut,	(2.92)
iid	+ ρ2 −1) = 0,
where ut N(0, σu2). If qt is a unit root process, then (ρ1

and (ρ1 +ρ2 −1)−1 clearly doesn’t exist. There is in a sense a singularity

28Suppose you are analyzing three variables (q1t, q2t, q3t). If they are cointegrated, there can be at most 2 independent random walks driving the series. If there are 2 random walks, there can be only 1 cointegrating vector. If there is only 1 random walk, there can be as many as 2 cointegrating vectors.

2.6. COINTEGRATION

65

in qt−1 because ∆qt is stationary and this can be true only if qt−1 drops out from the right side of (2.92).

By analogy, suppose that in the bivariate case the vector (qt, ft) is generated according to

" ft

#

= " a21

a22

# "

ft−1

#+" b21

b22

# "

ft−2

#+"

uft

# , (2.93)

qt

a11

a12

qt 1

b11

b12

qt 2

uqt

−

−

where (uqt, uft)0

iid

N(0, Σu). Rewrite (2.93) as the vector analog of the

augmented Dickey—Fuller test equation

# " ∆ft−1

# +

" uft

# ,

" ∆ft

# = " r21

r22

# " ft−1

# − "

b21

b22

∆qt

r11

r12

qt 1

b11

b12

∆qt 1

uqt

−

−

(2.94)

where

" r21

# "

1 # ≡

r22

a21

+ b21

a22 + b22

r11

r12

=

a11

+ b11 − 1 a12

+ b12

R.

−

If {qt} and {ft} are unit root processes, their Þrst di erences are stationary. This means the terms on the right hand side of (2.94) are stationary. Linear combinations of levels of the variables appear in the system. r11qt−1 + r12ft−1 appears in the equation for ∆qt and r21qt−1 + r22ft−1 appears in the equation for ∆ft.

If {qt} and {ft} do not cointegrate, there are no values of the rij coe cients that can be found to form stationary linear combinations of qt and ft. The level terms must drop out. R is the null matrix, and (∆qt, ∆ft) follows a vector autoregression.

If {qt} and {ft} do cointegrate, then there is a unique combination of the two variables that is stationary. The levels enter on the right side but do so in the same combination in both equations. This means that the columns of R are linearly dependent and the R, which is singular,

can be written as			R =	" r21	−βr21	# .
				r11	βr11
(2.94) can now be written as					−
							# " ∆ft−1	# +	" uft	#
" ∆ft	# =	" r21	# (qt−1 − βft−1) − " b21			b22
∆qt		r11			b11	b12	∆qt 1		uqt

−

66 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS

=	" r21	# zt−1	−	" b21	b22	# " ∆ft−1	# +	" uft	# , (2.95)
	r11			b11	b12	∆qt 1		uqt

−

(41)(eq.2.95)

(42)

(43)(eq.2.96)

where zt−1 ≡ qt−1 −βft−1 is called the error-correction term, and (2.95) is the vector error correction representation (VECM). A VAR in Þrst di erences would be misspeciÞed because it omits the error correction term.

To express the dynamics governing zt, multiply the equation for ∆ft by β and subtract the result from the equation for ∆qt to get

zt = (1 + r11 − βr21)zt−1 − (b11 + βb21)∆qt−1

−(b12 + βb22)∆ft−1 + uqt − βuft.

(44)(eq.2.97) The entire system is then given by

∆ft			=	b21		b22	r12
	∆qt			b11		b12	r11		βr21
	zt			(b11 + βb21)		(b12 + βb22) 1 + r11
			−		−			−

+		uft		.
	uqt	uqt
			βuft
		−

(2.96)

∆qt−1∆ft−1

zt−1

(2.97)

	(∆qt, ∆ft, zt)0 is a stationary vector, and (2.97) looks like a VAR(1) in
	these three variables, except that the columns of the coe cient matrix
	are linearly dependent. In many applications, the cointegration vector
	(1, −β) is given a priori by economic theory and does not need to be
	estimated. In these situations, the linear dependence of the VAR (2.97)
	tells you that all of the information contained in the VECM is preserved
	in a bivariate VAR formed with zt and either ∆qt or ∆ft.
(45)	Suppose you follow this strategy. To get the VAR for (∆qt, zt),
	substitute ft−1 = (qt−1 − zt−1)/β into the equation for ∆qt to get
	∆qt = b11∆qt−1 + b12∆ft−1 + r11zt−1 + uqt
		= a11∆qt−1 + a12zt−1 + a13zt−2 + uqt,
(46)	where a11 = b11 +	b12	, a12 = r11 −	b12	, and a13 =	b12	. Similarly, substitute
		β		β		β
	ft−1 out of the equation for zt to get

zt = a21∆qt−1 + a22zt−1 + a23zt−2 + (uqt − βuft),