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A characterization of Granger-Sims exogeneity
Economics Letters 8 (1981) 129-133 North-Holland Publishing Company
A CHARACTERIZATION
129
OF GRANGER-SIMS
EXOGENEITY
R. KOHN University
Received
of Chrcugo, Chicago, IL 60637, USA
21 July 1981
We give a simple characterization of Granger-Sims exogeneity as a zero partial correlation between the present and past values of one series and the future values of another. The derivation also gives a simple proof of the Granger and Sims definitions.
Suppose (x,, n = 0, * 1, ...) and (y,, n = 0, * 1, ...) are two sequences of zero mean, jointly Gaussian r.v.‘s. For an extension to random variables having finite second moments see Remark 1 (i) below. For each n, let
z, = (x,,
X,-l,
. ..>?
w, = (x,+,9 x,+*9 . ..).
u,= (Y,,Y,-,, Following that
4.
Granger
( i) x is exogenous
(1969)
Sims (1972) and Hosoya (1977)
to y in the Granger
sense if for all n,
E(W, I& y,) = E(W, ITA (1) means that for all j 2 1, E(~,+jlX,_k, k>O; yn__k, kaO)=E(x,+jIx,-&, (ii) x is exogenous
we will say
(1) k>O).
to y in the Sims sense if for all n,
WY, 1-L w,) = WI, IX), (2) means that for all j 2 0, all k)=E(yn_,Ixn-k, WY,-, Ix/c,
01651765/81/0000-0000/$02.75
(2) k20).
0 1981 North-Holland
R. Kohtt / Churrrcterizcrtion of Granger-Sims
130
Hosoya (1977) showed equivalent. His conditions below]. Now put u,,
=E(Y,
v,, =E(W,
Iz,J, I&),
exogeneit_v
that definitions (1) and (2) of exogeneity are are similar to ours [see Remark 1 (i) and (ii),
cr,, = Y, - u,,, v,, = w, - Y,,.
We will show that x is exogenous toy if and only if U,,, and V,, are uncorrelated. Now U,, is the residual from regressing Y, on 2,. and V,, is the residual from regressing W, on Z,,. Thus Granger-Sims exogeneity is characterized by zero partial correlation between future x’s and past and present y’s, given present and past x’s. The derivation below also gives a proof of the equivalence of the Granger-Sims definitions of exogeneity, using only simple properties of conditional expectations. The proof seems to be more elementary than Hosoya’s (1977). Under the assumptions made above: (i) x is exogenous toy in Theorem I. the Granger-Sims sense if and only if U,, and Vz, are uncorrelated. (ii) The Granger-Sims definitions of exogeneity are equivalent. Proof E(W,
(i) lz,,
Y,) =E(W,lZ,,
u,,)
(3)
=E(W,lZ,)+E(KIU,,), because
Z,, and U., are uncorrelated.
(4 because E(W,lZ,>
V,, and U,, are uncorrelated.
Substituting
Y,)=E(W,IZn)+E(V,,lU,,).
(4) and (3) we obtain
(5)
Hence eq. (1) holds if and only if E( V,, 1U,,) = 0; i.e., if and only if Vzn and U,, are uncorrelated. This proves (i).
R. Kahn / Choructerizution
(ii)
By switching
of Grunger-Sims
the roles of Y, and W, we can similarly
E(Y, I-%, w,) = E(Y, I-%> +E(%
131
exogeneit_v
show that (6)
I v,n).
Hence, from (6), (2) holds if and only if E(U,, 1V2,,)= 0; i.e., if and only if U,, and V,, are uncorrelated. It now follows from the proof of Part (i) Q.E.D. that the Granger and Sims definitions are equivalent. We now show that (1) is equivalent
and (2) is equivalent
Theorem 2.
to
to
(i) (I) is equivalent
to (7). (ii) (2) is equivalent
to (8).
Proofi (i) Clearly, (1) implies (7). Now suppose that (7) holds for all n. We will show that (7) implies (l), the proof proceeding by induction. For k B 1, suppose that for 1 Gj G k,
E(xn+jIYVZJ =E(xn,IZJ
(9)
Then, putting entktl
=
Eb,+k+l
Izn+k,
00)
Y,+k),
it follows that
enfkfl
=
Ebn+k+l
(11)
Izn+k)
of the elements of from (7). It follows that 0,+,+, is a linear combination or a limit of such linear combinations. Hence by the induction hypothesis (9),
Z n+k
E(Bn+k+l
I&,
Y,)
=E(%+k+,
From (lo), and Billingsley E(‘n+k+l
izn,
Y,)
=
E(Xn+k+l
(12)
i&h
(1979, p. 398) lzn,
Y,h
(13)
132
R. Kohn / Chamcterizotion
Similarly,
of Crmger-Sims
exogenert.v
from (1 l), I lz,)
W”+k+
= G%l+k+l
IZJ
(14)
By (12), the right-hand sides of (13) and (14) are equal, so (9) holds for j=/k+ 1. But (9) is true for k = 1, because then (9) coincides with (7). Hence, (9) is true for all k. (ii)
Because (Z,,
( . ..YX.-1,
W,) is equivalent
x Xl x,+1,x,+,,...)-_(...,
by (0 Put h,, = E(y,_, +#L,i
E(%,i
Izn>
=E(Yn-j
x,-,-1,
Wn_,), i.e.,
x,-,3x,-,+I,
... >,
) W,, Z,). Then,
(16)
IZJ = E(Y”FjIZJ
and from the right-hand
to (Zn_,,
side of (15),
lzn-,).
(17)
Thus, from (15), (16) and (17)
E(Yn-j
I W,v zn) rE(Yn-~
Izn>
for j > 0, which is just (2). Remark 1 ‘( i) The above results apply to general random variables having secondorder moments, if we replace a conditional expectation by best linear unbiased prediction. (ii) There is no loss of generality in the above in assuming that all variables have zero means. In the non-zero mean case we would just consider W - E(W) etc.
References Anderson, B.D.O. and J.B. Moore, 1979, Optimal filtering (Prentice Hall, Englewood Cliffs, NJ). Billingsley. P., 1979. Probability and measure (Wiley, New York). Granger, C.W.J., 1969, Investigating causal relations by econometric models and crossspectral methods, Econometrica 37, 424-438. Hosoya, Y., 1977, On the Granger condition for non-causality, Econometrica 45, 1735- 1736. Sims, C.A.. 1972, Money, income, and causality, American Economic Review 62, 540-552.
A CHARACTERIZATION
129
OF GRANGER-SIMS
EXOGENEITY
R. KOHN University
Received
of Chrcugo, Chicago, IL 60637, USA
21 July 1981
We give a simple characterization of Granger-Sims exogeneity as a zero partial correlation between the present and past values of one series and the future values of another. The derivation also gives a simple proof of the Granger and Sims definitions.
Suppose (x,, n = 0, * 1, ...) and (y,, n = 0, * 1, ...) are two sequences of zero mean, jointly Gaussian r.v.‘s. For an extension to random variables having finite second moments see Remark 1 (i) below. For each n, let
z, = (x,,
X,-l,
. ..>?
w, = (x,+,9 x,+*9 . ..).
u,= (Y,,Y,-,, Following that
4.
Granger
( i) x is exogenous
(1969)
Sims (1972) and Hosoya (1977)
to y in the Granger
sense if for all n,
E(W, I& y,) = E(W, ITA (1) means that for all j 2 1, E(~,+jlX,_k, k>O; yn__k, kaO)=E(x,+jIx,-&, (ii) x is exogenous
we will say
(1) k>O).
to y in the Sims sense if for all n,
WY, 1-L w,) = WI, IX), (2) means that for all j 2 0, all k)=E(yn_,Ixn-k, WY,-, Ix/c,
01651765/81/0000-0000/$02.75
(2) k20).
0 1981 North-Holland
R. Kohtt / Churrrcterizcrtion of Granger-Sims
130
Hosoya (1977) showed equivalent. His conditions below]. Now put u,,
=E(Y,
v,, =E(W,
Iz,J, I&),
exogeneit_v
that definitions (1) and (2) of exogeneity are are similar to ours [see Remark 1 (i) and (ii),
cr,, = Y, - u,,, v,, = w, - Y,,.
We will show that x is exogenous toy if and only if U,,, and V,, are uncorrelated. Now U,, is the residual from regressing Y, on 2,. and V,, is the residual from regressing W, on Z,,. Thus Granger-Sims exogeneity is characterized by zero partial correlation between future x’s and past and present y’s, given present and past x’s. The derivation below also gives a proof of the equivalence of the Granger-Sims definitions of exogeneity, using only simple properties of conditional expectations. The proof seems to be more elementary than Hosoya’s (1977). Under the assumptions made above: (i) x is exogenous toy in Theorem I. the Granger-Sims sense if and only if U,, and Vz, are uncorrelated. (ii) The Granger-Sims definitions of exogeneity are equivalent. Proof E(W,
(i) lz,,
Y,) =E(W,lZ,,
u,,)
(3)
=E(W,lZ,)+E(KIU,,), because
Z,, and U., are uncorrelated.
(4 because E(W,lZ,>
V,, and U,, are uncorrelated.
Substituting
Y,)=E(W,IZn)+E(V,,lU,,).
(4) and (3) we obtain
(5)
Hence eq. (1) holds if and only if E( V,, 1U,,) = 0; i.e., if and only if Vzn and U,, are uncorrelated. This proves (i).
R. Kahn / Choructerizution
(ii)
By switching
of Grunger-Sims
the roles of Y, and W, we can similarly
E(Y, I-%, w,) = E(Y, I-%> +E(%
131
exogeneit_v
show that (6)
I v,n).
Hence, from (6), (2) holds if and only if E(U,, 1V2,,)= 0; i.e., if and only if U,, and V,, are uncorrelated. It now follows from the proof of Part (i) Q.E.D. that the Granger and Sims definitions are equivalent. We now show that (1) is equivalent
and (2) is equivalent
Theorem 2.
to
to
(i) (I) is equivalent
to (7). (ii) (2) is equivalent
to (8).
Proofi (i) Clearly, (1) implies (7). Now suppose that (7) holds for all n. We will show that (7) implies (l), the proof proceeding by induction. For k B 1, suppose that for 1 Gj G k,
E(xn+jIYVZJ =E(xn,IZJ
(9)
Then, putting entktl
=
Eb,+k+l
Izn+k,
00)
Y,+k),
it follows that
enfkfl
=
Ebn+k+l
(11)
Izn+k)
of the elements of from (7). It follows that 0,+,+, is a linear combination or a limit of such linear combinations. Hence by the induction hypothesis (9),
Z n+k
E(Bn+k+l
I&,
Y,)
=E(%+k+,
From (lo), and Billingsley E(‘n+k+l
izn,
Y,)
=
E(Xn+k+l
(12)
i&h
(1979, p. 398) lzn,
Y,h
(13)
132
R. Kohn / Chamcterizotion
Similarly,
of Crmger-Sims
exogenert.v
from (1 l), I lz,)
W”+k+
= G%l+k+l
IZJ
(14)
By (12), the right-hand sides of (13) and (14) are equal, so (9) holds for j=/k+ 1. But (9) is true for k = 1, because then (9) coincides with (7). Hence, (9) is true for all k. (ii)
Because (Z,,
( . ..YX.-1,
W,) is equivalent
x Xl x,+1,x,+,,...)-_(...,
by (0 Put h,, = E(y,_, +#L,i
E(%,i
Izn>
=E(Yn-j
x,-,-1,
Wn_,), i.e.,
x,-,3x,-,+I,
... >,
) W,, Z,). Then,
(16)
IZJ = E(Y”FjIZJ
and from the right-hand
to (Zn_,,
side of (15),
lzn-,).
(17)
Thus, from (15), (16) and (17)
E(Yn-j
I W,v zn) rE(Yn-~
Izn>
for j > 0, which is just (2). Remark 1 ‘( i) The above results apply to general random variables having secondorder moments, if we replace a conditional expectation by best linear unbiased prediction. (ii) There is no loss of generality in the above in assuming that all variables have zero means. In the non-zero mean case we would just consider W - E(W) etc.
References Anderson, B.D.O. and J.B. Moore, 1979, Optimal filtering (Prentice Hall, Englewood Cliffs, NJ). Billingsley. P., 1979. Probability and measure (Wiley, New York). Granger, C.W.J., 1969, Investigating causal relations by econometric models and crossspectral methods, Econometrica 37, 424-438. Hosoya, Y., 1977, On the Granger condition for non-causality, Econometrica 45, 1735- 1736. Sims, C.A.. 1972, Money, income, and causality, American Economic Review 62, 540-552.