Palgrave Handbook of Econometrics: Applied Econometrics

David F. Hendry 23

thenxt=N−t^1
∑Nt
i= 1 xi,t∼IN

[

μt,Nt−^1 σt^2

]

. Log transforms of totals and means,

x>0, only differ by that population size as ln
∑Nt
i= 1 xi,t=lnxt+lnNt, so stan-
dard deviations of log aggregates are proportional to scaled standard deviations of

means:SD[ln

∑Nt

i= 1 xi,t]N

− 1
t σt/μt(see, e.g., Hendry, 1995a, Ch. 2). Thus logs
of aggregates can be well behaved, independently of the underlying individual
economic behavior.

Data transformations. Most econometric models also analyze data after transfor-

mations (such as logs, growth rates, etc.), written here asW^1 T=g(V^1 T). Again,

the key impact is onφT^1 →φT^1 and the consequences on the constancy of, and
cross-links between, the resulting parameters. At this stage we have created:

DW

(

W^1 T|U 0 ,QT^1 ,φ^1 T

)

. (1.2)

The functional form of the resulting representation is determined here by the
choice ofg(·). Many economic variables are intrinsically positive in levels, a prop-
erty imposed in models by taking logs, which also ensures that the error standard
deviation is proportional to the level.

Data partition. No reduction is involved in specifying thatWT^1 =(W^1 T :R^1 T),

whereR^1 Tdenotes then×Tdata to be analyzed andW
1
Tthe rest. However, this
decision is a fundamental one for the success of the modeling exercise, in that the
parameters of whatever process determinesR^1 Tmust deliver the objectives of the
analysis.

Marginalizing. To implement the choice ofRT^1 as the data under analysis neces-
sitates discarding all the other potential variables, which corresponds to the

statistical operation of marginalizing (1.2) with respect toW^1 T:

DW

(

W

1

T,R

1

T|U 0 ,Q

1

T,φ

1

T

)

=DW

(

W

1

T|R

1

T,U 0 ,Q

1

T,φ

1

T

)

DR

(

R^1 T|U 0 ,QT^1 ,ω^1 T

)

.

(1.3)

While such a conditional-marginal factorization is always possible, a viable

analysis requires no loss of information from just retainingω^1 T. That will occur

only if

(

φ^1 T,ωT^1

)
satisfy a cut, so their joint parameter space is the cross-product
of their individual spaces, precluding links across those parameters. At first sight,
such a condition may seem innocuous, but it is very far from being so: implic-

itly, it entails Granger non-causality of (all lagged values of)W^1 TinDR(·), which
is obviously a demanding requirement (see Granger, 1969; Hendry and Mizon,
1999). Spanos (1989) calls the marginal distributionDR(·)in (1.3) the Haavelmo
distribution.