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SPURIOUS REGRESSION
Herman J. Bierens
Pennsylvania State University
Consider two independent unit root processes, and , where the u
t 's and yt ' u t xt ' vt
the vt 's are independent, say:
u t
vt - i.i.d . N 2
0
0,
1 0
0 1. (1)
Then yt , with all leads and lags, is independent of xt , with
all leads and lags. If we regress yt on xt
for t = 1,.., n, one would therefore expect the slope
coefficient to be insignificant, but as we show
now, that is not true. First, consider the OLS regression of yt
on xt without an intercept. The OLS
estimate of the slope is:
0 '' n
t ' 1 x t yt ' n
t ' 1 x2t
'(1/ n)' nt ' 1 x0 / n% (1/ n)' t i ' 1vi y0 / n% (1/ n)' t j '
1u j
(1/ n)' nt ' 1 x0 / n% (1/ n)' t i ' 1vi 2
'(1/ n)' nt ' 1 x0 / n% W x,n(t / n) y0 / n% W y,n(t / n)
(1/ n)' t ' 1 x0 / n% W x,n(t / n) 2,
(2)
where
W x,n(r ) ' (1/ n)j[nr ]
j ' 1v j if r 0 [n
& 1,1], W x,n(r ) ' 0 if r 0 [0,n& 1) , (3)
W y,n(r ) ' (1/ n)j[nr ]
j ' 1u j if r 0 [n
& 1,1], W y,n(r ) ' 0 if r 0 [0,n& 1) , (4)
with [ rn ] the largest natural number rn . Since these
functions are step functions, and both#
W x,n(1) and W y,n (1) are standard normally distributed, hence
W x,n(1) = O p(1) and W y,n (1) = O p(1),
we have
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1 Billingsley, Patric: Convergence of Probability Measures ,
John Wiley, New York, 1968
2
(1/ n)jn
t ' 1W x,n(t / n)
2 ' (1/ n)jn& 1
t ' 0m
t % 1
t
W x,n( z / n)2dz % W x,n(1)
2 / n
'
(1/ n)m
n
0W x,n( z / n)
2
dz%
W x,n(1)2
/ n'
mW x,n(r )2
dr %
O p(1/ n) ,
(5)
and similarly
(1/ n)jn
t ' 1W y,n(t / n)
2 ' mW y,n(r )2dr % O p(1/ n) , (6)
(1/ n)jn
t ' 1W x,n(t / n)W y,n(t / n) ' mW x,n(r )W y,n(r )dr % O p(1/
n) . (7)
The integrals in (5) through (7), and in the sequel, are taken
over the unit interval [0,1], unless
otherwise indicated.
It can be shown that the integrals in (5) through (7) converge
jointly in distribution, due to
the functional central limit theorem and the continuous mapping
theorem:
LEMMA 1 . converges in distribution tomW x,n(r )2dr ,mW y,n(r
)2dr ,mW x,n(r )W y,n(r )dr T
, where W x(r ) and W y(r ) are independent standard Wiener mW
x(r )2dr , mW y(r )2dr , mW x(r )W ydr T
processes .
For the proof of Lemma 1, see Billingsley (1968) 1.
Using Lemma 1, we now have
0 'mW y,n(r )W x,n(r )dr
mW x,n(r )2dr % o p(1) 6 0 '
mW y(r )W x(r )dr
mW x(r )2dr (8)
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in distribution. Note that the limiting random variable is
continuously distributed, and in0
particular, P ( = 0) = 0.0
Along the same lines, and using (8), it follows that the
residual sum of squares, RSS0, of
the regression involved, divided by n2, satisfies
RSS0
n 2'
1
n 2j
n
t ' 1 y 2t &
20
1
n 2j
n
t ' 1 x 2t ' mW y,n(r )2dr & 20mW x,n(r )2dr % o p(1)
6 mW y(r )2dr & 20mW x(r )2dr in distr .(9)
hence the t-value , say, of the slope, divided by , satisfies:t
0 n
t 0
n'
0 (1/ n)' nt ' 1 x 2t RRS0 / (n& 1)
' n /(n& 1) 0 (1/ n2)' nt ' 1 x 2t
RRS0 / n2
'
0 mW x,n(r )2dr
mW y,n(r )2dr & 20mW x,n(r )2dr % o p(1) 6
0 mW x(r )2dr
mW y(r )2dr & 20mW x(r )2dr
(10)
in distribution. This result, together with P ( = 0) = 0,
implies that0 plim n64 *t 0* ' 4 .
Similar results as in (8) and (10) hold for slope parameter and
corresponding t-value1
of the regression of yt on xt with intercept:t 1
1 6 1 'mW y(r )W x(r )dr & mW y(r )dr mW x(r )dr
mW x(r )2dr & mW x(r )dr 2(11)
and
t 1
n6 1 mW x(r )
2dr & mW x(r )dr 2
mW y(r )2dr & mW y(r )dr 2 & 21mW x(r )2dr % 21 mW x(r
)dr 2(12)
in distribution. Moreover, the R2 of the regression involved
satisfies:
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R 2 6 21mW x(r )2dr & mW x(r )dr 2
mW y(r )2dr & mW y(r )dr 2(13)
in distribution
The conclusion from these results is that one should be very
cautious when conducting
standard econometric analysis using time series. If the time
series involved are unit root
processes, naive application of regression analysis may yield
nonsense results.
