David F. Hendry 39that assumption corresponds to agents knowing precisely what the conditioning
operator is. In a stationary world, one could imagine learning mechanisms that
eventually led to its discovery (see, e.g., Evans and Honkapohja, 2001). However,
in a wide-sense non-stationary environment, an explicit statement of the form of
(1.34) is:
ytre+ 1 =Et+ 1[
yt+ 1 |It]
=∫
yt+ 1 ft+ 1(
yt+ 1 |It)
dyt+ 1. (1.35)Thus, whenft(·) =ft+ 1 (·), agents need toknow the futureconditional density
functionft+ 1 (yt+ 1 |It), given present information, to obtain the appropriate con-
ditioning relation, since only then willyret+ 1 be an unbiased predictor ofyt+ 1. That
ft(·) =ft+ 1 (·)is precisely why forecasting is so prone to problems. Unfortunately,
knowingft+ 1 (·)virtually requires agents to have crystal balls that genuinely “see
into the future.” When distributions are changing over time, agents can at best
form “sensible expectations,”ytse+ 1 , based on forecastingft+ 1 (·)bŷft+ 1 (·)from
some rule, such that:
yset+ 1 =∫
yt+ 1 ̂ft+ 1(
yt+ 1 |It)
dyt+ 1. (1.36)There are no guaranteed good rules for estimatingft+ 1 (yt+ 1 |It)when
{
yt}
is
wide-sense non-stationary. In particular, when the conditional moments of
ft+ 1 (yt+ 1 |It)are changing in unanticipated ways, settinĝft+ 1 (·)=ft(·)could be a
poor choice, yet that underlies most of the formal derivations of RE, which rarely
distinguish betweenft(·)andft+ 1 (·). Outside a stationary environment, agents
cannot solve (1.34), or often even (1.35). The drawbacks of (1.34) and (1.35), and
the relative success of robust forecasting rules (see, e.g., Clements and Hendry,
1999; Hendry, 2006), suggest agents should use them, an example of imperfect-
knowledge expectations (IKE) (see Aghionet al., 2002; Frydman and Goldberg,
2007). IKE acknowledges that agents cannot know howItentersft(·)when pro-
cesses are evolving in a non-stationary manner, let aloneft+ 1 (·), which still lies in
the future. Collecting systematic evidence on agents’ expectations to replace the
unobservables by estimates, rather than postulates, deserves greater investment
(see, e.g., Nerlove, 1983).
Finally, take expectations conditional on the available information setIt− 1 in a
regression model with valid weak exogeneity:
yt=β′zt+ (^) t, (1.37)
so that:
E
[
yt|It− 1
]
=β′E
[
zt|It− 1
]
, (1.38)
asE
[
(^) t|It− 1
]
=0. Writing (1.38) asyte = β′zet, the conditional model (1.37)
always has an expectations representation, although the converse is false. Impor-
tantly, therefore, contemporaneous conditioning variables can also be expectations
variables, and some robust forecasting rules likêpt+ 1 =pthave that property.