Palgrave Handbook of Econometrics: Applied Econometrics

(Grace) #1

34 Methodology of Empirical Econometric Modeling


For some parameter values in the DGP, the conditional expectation will coincide
with (1.16), whereas for other parameter configurations, (1.16) and (1.21) will dif-
fer, in which case it is unsurprising that (1.16) is not fully informative. However, an
exact match between the equation to be estimated and the conditional expectation
of the dependent variable givenIt− 1 is not sufficient to justify least squares estima-
tion, even when the error is an innovation againstIt− 1. Indeed, whenλ=γ=0,
butρ =0, there is a failure of weak exogeneity ofztforβ, even though the
conditional expectation is:


E

[
yt|zt,It− 1

]
=βzt. (1.22)

Nevertheless,ztis not weakly exogenous forβwhenρ =0 since:


zt=ρ

(
yt− 1 −βzt− 1

)


  • (^2) t, (1.23)
    so a more efficient analysis is feasible by jointly estimating (1.16) (or (1.22)) and
    (1.23). Here the model coincides with both the conditional expectation and the
    DGP equation, but as shown in Phillips and Loretan (1991) and Hendry (1995c),
    the violation of weak exogeneity can lead to important distortions to inference
    when estimating the parameters of (1.16), highlighting the important role of weak
    exogeneity in conditional inference.
    1.4.5.2 Super exogeneity and structural breaks
    Next, processes subject to structural breaks sustain tests for super exogeneity and
    the Lucas (1976) critique (following Frisch, 1938): (see, e.g., Hendry, 1988; Fischer,
    1989; Favero and Hendry, 1992; Engle and Hendry, 1993; Hendry and Santos,
    (2009). Formally, super exogeneity augments weak exogeneity with the require-
    ment that the parameters of the marginal process can change (usually over some
    set) without altering the parameters of the conditional. Reconsider (1.9), written
    with potentially non-constant parameters as:
    Dxt
    (
    xt|Xtt−− 1 s;X^10 −s,qt,ρt
    )
    =Dyt|zt
    (
    yt|zt,Xtt−−s 1 ,X^10 −s,qt,κ1,t
    )
    Dzt
    (
    zt|Xtt−− 1 s;X^10 −s,qt,κ2,t
    )


. (1.24)


Whenθenters bothκ1,tandκ2,tin (1.24), inference can again be distorted if weak
exogeneity is falsely asserted. When conditional models are constant despite data
moments changing considerably, there isprima facieevidence of super exogeneity
for that model’s parameters; whereas, if the model as formulated does not have
constant parameters, resolving that failure ought to take precedence over issues
of exogeneity. However, while super exogeneity tests are powerful in detecting
location shifts, changes to “reaction parameters” of mean-zero stochastic variables
are difficult to detect (see, e.g., Hendry, 2000b). Hendry and Santos (2009) propose
a test for super exogeneity based on impulse saturation (see Hendry, Johansen and
Santos, 2008) to automatically select breaks in the marginal processes, then test
their relevance in the conditional. When none of the breaks enters the conditional
model, that provides evidence in favor ofztcausingyt, since the same response

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