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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 959, November 2008 --- Screen Reader
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Abstract:

Robustness and fragility in Leamer's sense are defined with respect to a particular coefficient over a class of models. This paper shows that inclusion of the data generation process in that class of models is neither necessary nor sufficient for robustness. This result holds even if the properly specified model has well-determined, statistically significant coefficients. The encompassing principle explains how this result can occur. Encompassing also provides a link to a more common-sense notion of robustness, which is still a desirable property empirically; and encompassing clarifies recent discussion on model averaging and the pooling of forecasts.

Keywords: Encompassing, exogeneity, extreme bounds analysis, model averaging, parameter nonconstancy, pooling of forecasts, robustness, regime shifts, sensitivity analysis

JEL classification: C52, E41

Economists want their empirical results to be robust, and with
good reason. Especially after Goldfeld's (1976) missing money and
Lucas's (1976) critique, economists have been all too aware of the
fragility of many empirical models to the choice of (e.g.)
explanatory variables, sample period, and dynamics. In an attempt
to measure a coefficient's sensitivity to model selection, Leamer's
(1983) extreme bounds analysis (also known as sensitivity analysis)
calculates the range of potential coefficient estimates over a
class of models. Extreme bounds analysis thus offers an appealing
and intuitive methodology for determining whether an empirical
result is robust or fragile. Extreme bounds analysis also
formalizes a common practice in empirical studies, and its explicit
implementation is widespread; cf. Allen and Connolly (1989),
Cooley and LeRoy (1981), Levine and Renelt (1992), Leamer (1997),
Sala-i-Martin (1997), Serra (2006), Freille et al. (2007), and
Bjø rnskov et al. (2008) *inter alia*.

Robustness and fragility in Leamer's sense are defined with
respect to a particular coefficient or set of coefficients over a
class of models. This paper shows that the inclusion of the data
generation process in that class of models is neither necessary nor
sufficient for robustness. This result holds even if the properly
specified model has well-determined, statistically significant
coefficients. The encompassing principle--pioneered by Mizon (1984)
and Mizon and Richard (1986)--explains how this result can occur.
Encompassing also provides a link to a more common-sense notion of
robustness that *is* a desirable property empirically; and
encompassing clarifies recent discussion on model averaging and the
pooling of forecasts.

Put somewhat differently, extreme bounds analysis focuses on the
variation of estimated coefficients across model specifications.
While that coefficient variation is of interest, it is so only in
light of its causes. Extreme bounds analysis considers *all*
coefficient variation, regardless of its causes. The encompassing
principle discerns between coefficient variation that is not
explainable by the model at hand, and coefficient variation that
is. Encompassing thus clarifies when to worry about coefficient
variation and when not to, whereas extreme bounds analysis worries
about all coefficient variation. In essence, extreme bounds
analysis is not sensitive enough to the data's nuances, whereas the
encompassing principle is.

Section 2 briefly describes extreme bounds analysis, including modifications proposed by Leamer and Leonard (1983) and Levine and Renelt (1992). Section 3 establishes the lack of necessity and the lack of sufficiency through illustrations with classical regression models. Section 4 demonstrates how robustness and fragility in Leamer's sense are themselves fragile in practice, employing Hendry and Ericsson's (1991) model of U.K. money demand. Section 5 re-interprets the fragility of sensitivity analysis in light of the encompassing principle. In so doing, Section 5 clarifies the meaning of robustness and illuminates recent discussion on model averaging and the pooling of forecasts. Section 6 concludes.

In order to make the results readily accessible, examples--rather than proofs--are employed. Also, proofs are readily apparent, given the examples.

This section summarizes extreme bounds analysis (EBA), including modifications proposed by Leamer and Leonard (1983) and Levine and Renelt (1992). The classical regression model serves to illustrate. See Leamer (1978), Chapter 5 for an initial discussion of Bayesian sensitivity analysis; and see Leamer and Leonard (1983), Leamer (1985), and Levine and Renelt (1992) for more detailed descriptions of EBA .

Consider the standard linear regression model:

(1) |

where is the dependent variable, and are and
vectors of "free" and "doubtful" explanatory variables, and
are and
vectors of their coefficients,
is a normal independently distributed
disturbance with mean zero and variance
, and is the time
index for the interval . A variable is
free if *a priori* its coefficient is believed to be different
from zero, and a variable is doubtful if *a priori* its
coefficient is believed to be equal to zero.

Extreme bounds analysis centers on a "focus" coefficient, denoted . Commonly, is an element of and so is associated with a free variable, although could be any linear combination of , as in Breusch (1990). Below, is always an element of , purely for expository convenience.

Extreme bounds analysis determines the range of the least squares estimates of over variations in the set of doubtful variables . To illustrate, consider equation (1) with two doubtful variables ( and ) and one free variable (), where is also the focus variable:

(2) |

Bounds can be calculated from the four estimates of that are obtained by including or excluding each of the doubtful variables in equation (2). However, these bounds generally are not invariant to nonsingular linear transformations of the doubtful variables. For instance, a different set of bounds might arise if and (rather than and ) were used as the doubtful variables.

Leamer and Leonard (1983) propose a simple solution to this conundrum: calculate the bounds over all linear combinations of the doubtful variables. While these bounds might appear complicated to compute, they are not; and Breusch (1990, equation (17)) provides analytical formulae for them in terms of standard regression output from (1), estimated with and without the restriction . The bounds are:

(3) |

where the circumflex ˆ and tilde ˜ denote unrestricted and restricted estimates respectively, is the estimated standard error, and is the -statistic for testing the exclusion of all doubtful variables. Below, these bounds are called and . As Breusch notes, these bounds are narrow when exclusion of the doubtful variables results in either little loss of fit ( is small) or little change in the precision of the estimate of when adjusted for possible changes in error variance. See Stewart (1984) for an alternative approach.

Estimation of is "robust" if the range of
inferences about is small over the interval
.
Estimation is "fragile" if the range of inferences is large.
Specifically, the result is designated as fragile if
includes zero, and as robust if
excludes zero.^{1} Robustness and fragility in this sense
are called L-robustness and L-fragility ("L" for Leamer) so as
to distinguish them from other senses of robustness and fragility
used below.

Some values of in the range may fit the data very poorly relative to the unrestricted estimate . These values correspond to points in the parameter space that have very small likelihoods in terms of the observed data. To address this problem, Leamer and Leonard (1983, p. 311) propose restricting the extreme bounds to those points that lie within (e.g.) a 95% likelihood ellipsoid relative to . Cooley and LeRoy (1981) implement this modified EBA in their analysis of money demand equations. Granger and Uhlig (1990, 1992) propose a similar modification to EBA by limiting consideration to only those models with a sufficiently high , relative to the of the unrestricted model.

McAleer and Veall (1989) criticize EBA for not accounting for the uncertainty in the extreme bounds estimates themselves. McAleer and Veall use bootstrap techniques to estimate the standard errors of the bounds and find that those standard errors can be large empirically. [Magee 1990] derives the asymptotic variance of the extreme bounds:

(4) |

where the two terms resulting from the "" are the variances of the upper () and lower () bounds, denoted and . As with the bounds in (3), the variances in (4) are calculable from standard regression output for the restricted and unrestricted models.

With the estimated bounds' uncertainty in mind, Levine and Renelt (1992) propose a modified EBA, which solves for "... the widest range of coefficient estimates on the [focus variable] that standard hypothesis tests do not reject" (p. 944). In practice, Levine and Renelt use 95% critical values for calculating the extreme bounds. If the variance of the estimate of is insensitive to the linear combination of chosen, Levine and Renelt's bounds are approximately:

(5) |

Thus, while pure EBA ignores the plausibility of the bounds (in terms of the data) and the uncertainty of the bounds themselves, computationally feasible solutions exist for addressing both shortcomings.

This section establishes that L-robustness is neither necessary nor sufficient for the data generation process (DGP) to be included as one of the models in extreme bounds analysis (Section 3.1). To simplify discussion, only the "pure" form of EBA is examined initially; the modifications to EBA are considered at the end of Section 3.1. A simple DGP and several classical regression models illustrate the four propositions associated with lack of necessity and lack of sufficiency (Section 3.2). In the examples, bounds are calculated at population values for the various estimates in equation (3). This permits a clearer exposition and in no way invalidates the results. In fact, the four propositions hold both in finite samples and asymptotically, and they are not restricted to the examples in Section 3.2 Hendry and Mizon (1990) provide the essential framework for this section's approach.

Other authors have also pointed out difficulties with EBA . McAleer et al. (1985) show that the bounds may lie in implausible parts of the parameter space, that other information may be relevant to the model's usefulness (e.g., white-noise errors), and that L-robustness is sensitive to the choice of the parameter's prior mean and to its classification as free or doubtful. McAleer and Veall (1989) also show that accounting for the uncertainty in the estimated bounds can affect the results of EBA .

L-robustness is neither necessary nor sufficient for the DGP to be included as one of the models in extreme bounds analysis. It is helpful to examine this statement as four separate propositions. No formal proofs need be given; the examples in Section 3.2 are sufficient.

Propositions 1 and 2 pertain to L-robustness.

Propositions 1 and 2 are unsurprising. Still, together they imply that L-robustness says nothing about whether any of the models in the EBA are the DGP, and so says nothing about how close or far away any of the models in the EBA are to the DGP .

Propositions 3 and 4 pertain to L-fragility, or the lack of L-robustness.

From the perspective of an empirical modeler, Proposition 3 is problematic: correct specification of the unrestricted model does not ensure L-robustness.

Propositions 3 and 4 together imply that L-fragility says nothing about whether any of the models in the EBA are the DGP, paralleling the implication about L-robustness from Propositions 1 and 2.

Propositions 1-4 extend immediately to EBA that is modified to account for uncertainty in the estimated bounds. For Propositions 1 and 2, a large enough sample size always exists such that the uncertainty in the estimated bounds is negligible. For Propositions 3 and 4, the results remain L-fragile when accounting for uncertainty in the estimated bounds because that uncertainty must increase the range spanned by the bounds.

Propositions 1 and 2 also extend to EBA that is restricted to lie within a likelihood ellipsoid, as when the doubtful variable has no explanatory power; see Examples 1 and 2 below. Section 5 discusses implications for Propositions 3 and 4.

Examples 1,2,3, and 4 below illustrate Propositions 1,2,3, and 4 respectively. A simple framework is used to characterize the implications as clearly and directly as possible.

Suppose that the data ( ) are generated by the conditional process:

(6) |

and that the marginal variables ( ) are normal, independent, and identically distributed:

(7) |

where is the variance-covariance matrix of . The dependent variable in (6) is the same as that in (2). For simplicity, is an identity matrix and is unity, unless otherwise stated.

To evaluate the properties of EBA, the right-hand side variables in the conditional process (6) must be mapped into focus, free, and doubtful variables. Also, it is possible that some may be omitted from all models in the EBA . In Examples 1-4, is the focus variable and is also the only free variable, is the doubtful variable, and is excluded from all models in the EBA. Thus, in these examples, only two models need to be estimated in order to calculate the bounds in equation (3) and the asymptotic variance of the bounds in equation (4). One model is the unrestricted model, which has both and as regressors. The other model is the restricted model, which has as its only regressor.

The EBA will satisfy any of Propositions 1-4, depending upon the values of the parameters in the DGP (6)-(7). Thus, the examples are stated in terms of the parameters of the DGP .

Example 1 illustrates Proposition 1.

*
(the focus variable has a
nonzero coefficient),*

*
(the doubtful variable is
unnecessary for explaining ), and*

*
(the excluded variable is
unnecessary for explaining ).*

*Then Proposition 1 holds for large
enough .*

In Example 1, expectations of relevant estimators are , , and , where the approximate equalities indicate evaluation at population values (i.e., the asymptotic approximation). These equalities imply that the term in square brackets in equation (3) is zero when evaluated at population values. Hence, the bounds are likely to be narrow in finite samples. For large enough (or for small enough ), the bounds can be arbitrarily narrow.

Example 2 is the same as Example 1 except that the omitted variable is important in the DGP .

In Example 2, , , and . While these equalities differ from those in Example 1, they still imply that the term in square brackets in equation (3) is zero at population values. So, the bounds are again likely to be narrow, and they can be arbitrarily narrow for a large enough . Worryingly, in general if and are correlated. That is, the restricted and unrestricted estimates have the same expectations, and both estimates are biased (and in fact are typically inconsistent) for the true parameter.

For the next two examples--Examples 3 and 4--the marginal process (7) is modified slightly to include a nonzero correlation () between and .

Example 3 is the same as Example 1 except that the doubtful variable is necessary for explaining , and the doubtful variable is correlated with the focus variable. In Example 3, , , , , , and . Further, is a noncentral -statistic, with , which is essentially the noncentrality parameter divided by . Thus, the extreme bounds are approximately:

(8) |

For the parameter values in Example 3, equation (8) implies extreme bounds of approximately . In practice, the bounds may be even larger, noting that and . Even with these wide bounds, the unrestricted estimator is unbiased for ; and, for (e.g.) , is precisely estimated by , with an approximate estimated standard error of , implying a typical -ratio of over eight.

The DGP in Example 4 is the same as in Example 3, except that is nonzero and so the unrestricted model for EBA does not include the DGP .

Because is independent of and , all results for Example carry through, but with redefined as .

To highlight the empirical consequences of the four propositions
in Section 3,
the current section re-examines an empirical model of narrow money
demand in the United Kingdom from Hendry and Ericsson (1991). This
model is described, and its history and properties summarized.
Several alternatives to this model are then estimated, and extreme
bounds are calculated for various model pairs. Treating the
original model as the DGP, examples of all four propositions can be
found empirically. Of course, the original model is *not* the
DGP . However, that model does appear well-specified when examined
with a wide range of diagnostic statistics, so it behaves like a
local DGP, thus making these extreme bounds analyses of substantive
interest.

The data are quarterly seasonally adjusted nominal M1 (), real total final expenditure (TFE) at 1985 prices (), the TFE deflator (), and the net interest rate (), which measures the opportunity cost of holding money. The net interest rate is the differential between the three-month local authority interest rate and the learning-adjusted retail sight-deposit interest rate. Equation (6) in Hendry and Ericsson (1991) is the following equilibrium correction model (EqCM):

(9) |

where variables in lowercase are in logarithms, is the one-period difference operator, and estimated standard errors are in parentheses. Equation (9) is an empirically constant parsimonious simplification of an autoregressive distributed lag in the money demand variables. Equation (9) has long-run unit price and income elasticities but near-zero short-run ones; and the long-run interest rate elasticity is large and negative. Hendry and Ericsson (1991) discuss the economic and statistical merits of (9) in greater detail.

The history of (9) provides a perspective on its empirical validity, which motivates treating (9) as the DGP in the examples below. Hacche (1974) and Coghlan (1978) developed some of the first models of U.K. narrow money demand. Hendry (1979) noted problems in the dynamic specification of those models and obtained a better-specified model much like (9) as a simplification from an autoregressive distributed lag model on data through 1977. Equation (9) differs from Hendry's (1979) model by having the interest rate in levels rather than logs (a formulation proposed by Trundle (1982)), by having slightly simpler dynamics, and by having the net interest rate rather than the local authority rate. The net interest rate helps properly measure the economic concept of the opportunity cost when narrow money started earning interest in the 1980s.

The empirical specification in (9) has been extensively analyzed. Aspects examined include parameter constancy [Hendry (1985), Hendry and Ericsson (1991)], cointegration [Johansen (1992), Hendry and Mizon (1993), Paruolo (1996)], weak exogeneity [Johansen (1992), Boswijk (1992)], super exogeneity [Cuthbertson (1988), Hendry (1988), Hendry and Ericsson (1991), Engle and Hendry (1993)], dynamic specification [Ericsson et al. (1990)], finite-sample biases in estimation [Kiviet and Phillips (1994)], and seasonality [Ericsson et al. (1994)]. Only the last (seasonality) provides any evidence of mis-specification, and the magnitude of that mis-specification appears relatively small.

To illustrate the four propositions above, consider the following four variants on equation (9).

equation (9) itself; | |

equation (9), excluding ; | |

equation (9), excluding and ; and | |

equation (9), excluding and . |

Table 1 summarizes the estimation results for these four models.

For EBA, choices of excluded, free, doubtful, and focus variables must be made; and the following choices aim to illustrate Propositions 1-4. Treating as the DGP, the model pairs , , , and correspond to Propositions 1,2,3, and 4 respectively in terms of whether the DGP is included in the EBA . Each model pair determines which (if any) variable the EBA excludes, relative to the DGP . For each model pair, the free variables are all of the variables included in the restricted model, and the doubtful variables are all variables excluded from the restricted model but included in the unrestricted model. The interest rate is the focus variable for the first two model pairs, and the dynamic response to disequilibrium is the focus variable for the second two model pairs.

Table 2 presents the extreme bounds analysis for each of the four model pairs. The interest rate is L-robust, whether or not is included in the models analyzed: see the first two columns of results. Conversely, the variable is L-fragile, whether or not is included: see the last two columns of results. Both variables are highly statistically significant in the original model , with -statistics of for , for , and for and jointly.

To summarize, inclusion or exclusion of the DGP in the models examined has no bearing on the determination of L-robustness or L-fragility in extreme bounds analysis. Additionally, a coefficient can be highly statistically significant, yet be either L-robust or L-fragile. As this section and the previous section show, these negative results are easily demonstrated in principle and in empirical practice.

This section interprets Propositions 1-4 in light of the encompassing literature (Section 5.1), relates Leamer and Leonard's modified EBA to encompassing (Section 5.2), and re-interprets several encompassing tests and diagnostic tests as tests of robustness (Section 5.3). See Mizon (1984), Mizon and Richard (1986), Hendry and Richard (1989), and Bontemps and Mizon (2003) for key references on encompassing.

Whether or not a variable is L-robust depends *inter alia*
upon the correlations between the free and doubtful variables. That
dependence leads to an encompassing interpretation of EBA. A simple
regression model illustrates.

Returning to equations (1) and (2), consider the case with one focus variable and one doubtful variable :

(10) |

With only one doubtful variable in (10), the extreme bounds are given by the unrestricted and restricted estimates and . The restricted estimate can be expressed in terms of the unrestricted estimate as:

(11) |

where is the sample correlation between the focus variable and the doubtful variable. From (11), the distance between the bounds is the product of that correlation, the square root of the ratio of the regressors' sample second moments, and the unrestricted coefficient estimate for the doubtful variable. For similar interpretations, see McAleer et al. (1985) and Breusch (1990).

Thus, for restricted models with statistically *invalid*
restrictions, loose bounds (i.e., implying L-fragility) are
unworrying. For example, for the model pair
in Table 2, the deletion of
switches the sign of the
coefficient on
, resulting in
L-fragility. However, the exclusion of
is statistically invalid; and
the exclusion of
generates omitted variable bias
in the estimation of , where the omitted
variable bias is
, as
from (11). The restricted
model fails to parsimoniously encompass
the more general model because
is statistically significant in
.

For restricted models with statistically *valid*
restrictions, the restricted model parsimoniously encompasses the
unrestricted model because those restrictions *are*
statistically valid. If EBA obtains loose bounds in such a
situation, those loose bounds must arise from the uncertainty in
the estimated coefficients: the corresponding statistical reduction
appears valid, implying no omitted variable bias.

The examples in the previous two paragraphs compare a given model with a more general, nesting model. A given model can also be compared with a non-nested model or with a more restricted model. Non-nested comparisons generate variance-encompassing and parameter-encompassing test statistics. General-to-specific comparisons generate the standard -statistic, interpretable as a statistic for encompassing. On the latter, Gouriéroux and Monfort (1995, Proposition 8) demonstrate that (counterintuitively) a general model need not always encompass a model nested within it. However, Bontemps and Mizon (2003) find "... that the congruence of a model is a sufficient condition for it to nest and encompass a simplification (parametric or nonparametric) of itself, and that consequently [congruence] plays a crucial role in the application of the encompassing principle." (p. 355)

Leamer and Leonard's (1983) modified EBA restricts the bounds to lie within some specified likelihood ellipsoid relative to the unrestricted model. This statistical modification is very classical in nature: modified EBA considers only those models that are statistically valid simplifications of the unrestricted model. This modification is very much in the spirit of the encompassing literature, given the discussion in Section 5.1; and it motivates orthogonalization of variables in model design, as discussed below.

Equation (11) highlights an advantage to having nearly orthogonal regressors: they help minimize the potential for omitted variable bias. Because linear models are invariant to nonsingular linear transformations of the regressors, orthogonalization of the variables in the unrestricted model could be obtained by construction. For typical (i.e., highly autocorrelated) economic time series, near orthogonalization can often be obtained by using two economically interpretable transformations: differencing, and differentials. For example, in equation (9), inflation is a differenced variable, lagged inverse velocity and the net interest rate are differentials, and is a differenced differential. Transformation of the original level variables in the unrestricted autoregressive distributed lag into near-orthogonal variables in an equilibrium correction representation provides some insurance against omitted variable bias for the estimates in a restricted model, where the omitted variables are those variables that are deleted in the reduction from the unrestricted model to the restricted model. See Ericsson et al. (1990) for an example of such transformations and reductions with the U.K. money data.

While this sense of robustness is often achievable by design, no
procedure appears capable of ensuring orthogonalization with
respect to variables that are not included in the
*unrestricted* model. This implication emphasizes the
importance of starting with a general enough model. [Leamer and
Leonard 1983, p. 306] are sympathetic to this view, given
their concern for obtaining robust inferences over a broad family
of models. For detailed discussions of general-to-specific modeling
and model design, see Hendry (1983), Hendry et al. (1984), Gilbert (1986), Spanos (1986), Ericsson et al. (1990), Mizon
(1995), Hoover and Perez (1999), Hendry and Krolzig (1999, 2005),
Campos et al. (2005), and Doornik (2008).

L-robustness focuses on how coefficient estimates alter as the
information set for the model changes. From the discussion above,
L-robustness is only statistically or economically interesting if
the information that is excluded--relative to the unrestricted
model--is *validly* excluded. Tests of encompassing are tests
of that exclusion; hence tests of encompassing are interpretable as
tests of robustness. Put slightly differently, an encompassing test
of a given model evaluates whether or not the information in the
*other* model is redundant, conditional on the information in
the given model. If encompassing holds, then the given model is
robust to that additional information.

At a more general level, robustness (and so encompassing) can be defined in terms of generic changes to the model's information set, and not just in terms of changes associated with the additional variables in another model; see Mizon (1995, pp. 121-122; 2008) and Lu and Mizon (1996). In a partition of information sources similar to the one in [Hendry 1983] for test statistics, consider the following four sources of information: the model itself, other models, other sample periods, and other regimes.

*Information from the model itself.* Robustness to data in
the model itself corresponds to satisfying a range of standard
diagnostic tests, such as those for white-noise residuals and
homoscedasticity. In this spirit, Edwards et al. (2006)
propose a further refinement to Leamer and Leonard's modified EBA
by requiring the bounds to satisfy not just the standard likelihood
ratio test but also a battery of diagnostic tests. By focusing on
congruence, this refinement parallels the generalized concept of
encompassing.

*Information from other models.* Robustness to data in
another model corresponds to standard encompassing: in particular,
variance encompassing and parameter encompassing for non-nested
models, and parsimonious encompassing for nested models.
Differences in coefficient estimates across models are unimportant
*per se*. Rather, the interest is in the ability of a given
model to explain why other models obtain the results that they do.
The formula for omitted variable bias provides one way for such an
accounting. The relationship of the given model to the alternative
model formally defines the type of encompassing statistic.
Non-nested models generate variance-encompassing and
parameter-encompassing statistics; nested models generate
parsimonious-encompassing statistics.

If the two models differ in their dynamic specification, special attention must be given to the construction of the encompassing statistic, even although the comparison of models may appear conceptually equivalent to the one generating the usual encompassing statistics. See Hendry and Richard (1989) and Govaerts et al. (1994) for details.

Encompassing accounts for information in other models. Model
averaging is an alternative approach to accounting for such
information. Early versions of model averaging include pre-test,
Stein-rule, and shrinkage estimators; see Judge and Bock (1978),
Raftery et al. (1997) and Hansen (2007) exemplify recent
directions in model averaging. While model averaging is an
appealing way of combining information, it has several statistical
disadvantages relative to encompassing through general-to-specific
model selection; see Hoover and Perez (2004), Hendry and Krolzig
(2005), and Hendry and Reade (2005) *inter alia*.

For example, consider model averaging across a set of models that includes a well-specified model (e.g., the DGP) and some mis-specified models. As Hendry and Reade (2005) demonstrate, typical rules for model averaging place too much weight on the mis-specified models, in effect mixing too much bad wine with too little good wine. Encompassing through general-to-specific modeling aims to find the well-specified model among the set of models being considered, thus (to continue the analogy) singling out that one bottle of a rare vintage. If the union model is the DGP but none of the individual models are, the distinction between model averaging and encompassing is even sharper. Model averaging places zero weight on the DGP, whereas encompassing through general-to-specific modeling has power to detect the union model as the DGP .

*Information from other sample periods.* Robustness to data
from another sample period corresponds to parameter constancy.
Fisher's (1922) covariance statistic and Chow's (1960) prediction
interval statistic are two early important statistics for testing
this form of robustness. More recent developments have focused on
testing robustness to a range of sample periods: see Brown
et al. (1975), Harvey (1981), and Doornik and Hendry (2007)
*inter alia* on recursive statistics, and Andrews (1993) and
Hansen (1992) on statistics for testing parameter instability when
the breakpoint is unknown.

*Information from other regimes.* Robustness to regime
changes corresponds to valid super exogeneity. Two common tests for
super exogeneity are constructed as follows.

(i) Establish the constancy of the parameters in the conditional model and the nonconstancy of those in the marginal model; cf. Hendry (1988).

(ii) Having established (i), further develop the marginal model by including additional explanatory variables until the marginal model is empirically constant. Then, test for the significance of those additional variables when added to the conditional model. Insignificance in the conditional model demonstrates invariance of the conditional model's parameters to the changes in the marginal process; cf. Engle and Hendry (1993) for this test's initial implementation, Hendry and Santos (2006) for a version based on impulse saturation, and Hendry et al. (2008) and Johansen and Nielsen (2008) for statistical underpinnings of the latter.

These tests use statistics for testing parameter constancy and statistics for omitted variables. Thus, these tests are interpretable as tests of robustness to information from other sample periods and from other models. However, tests of super exogeneity merit separate mention because super exogeneity is central to policy analysis.

Hendry and Ericsson (1991) calculate both types of super exogeneity tests. Hendry and Ericsson show that the EqCM (9) is empirically constant, but that autoregressive models for inflation and the net interest rate are not. The EqCM is constant across regime changes, which were responsible for the nonconstancy of the inflation and interest rate processes. From (i), inflation and the net interest rate are super exogenous in (9). Additionally, functions of the residuals from the marginal processes are insignificant when added to the EqCM, so inflation and the net interest rate are super exogenous from (ii). See Engle et al. (1983) for a general discussion of exogeneity.

*Overlapping sources of information.* Robustness to the
intersection of multiple sources of information is also of
interest. For instance, robustness to another model's data over an
out-of-sample period corresponds to forecast encompassing and
forecast-model encompassing; see Chong and Hendry (1986), Lu and
Mizon (1991), Ericsson (1992), and Ericsson and Marquez (1993). See
also Bates and Granger (1969), Granger (1989), Wright (2003a,
2003b), Hendry and Clements (2004), Hendry and Reade (2006), and
Castle et al. (2008) *inter alia* for discussion on the
related concept of forecast combination.

*Other implications of encompassing.* Encompassing does not
imply that the DGP is included in the set of models being examined.
However, an encompassing model is congruent with respect to the
available information set and thus parsimoniously encompasses the
local DGP . In that specific sense, the encompassing model
establishes a closeness to the DGP . General-to-specific modeling
with diagnostic testing enforces encompassing and generates a
progressive research strategy that converges to the DGP in large
samples; cf. White (1990) and Mizon (1995). Extreme bounds
analysis--at least in its unmodified form--does neither.

Extreme bounds analysis re-emphasizes the importance of robustness in empirical modeling. The measure of robustness in EBA has several unfortunate properties that render that particular measure useless in practice. Nonetheless, the structure of EBA helps elucidate an important role of encompassing and model design in empirical modeling: encompassing tests and several other diagnostic tests are interpretable as tests of a more appropriately defined notion of robustness.

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Bates, J. M., and C. W. J. Granger (1969) "The
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Table 1. Estimates for Restricted and Unrestricted Models

Variable or Statistic | Model: | Model: | Model: | Model: |
---|---|---|---|---|

-0.687 (0.125) | ||||

-0.175 (0.058) | -0.133 (0.066) | 0.343 (0.090) | ||

-0.630 (0.060) | -0.786 (0.060) | -0.718 (0.051) | ||

-0.093 (0.009) | -0.092 (0.010) | -0.084 (0.009) | -0.006 (0.012) | |

intercept | 0.023 (0.004) | 0.024 (0.005) | 0.022 (0.005) | 0.003 (0.007) |

0.762 | 0.686 | 0.673 | 0.133 | |

1.313% | 1.498% | 1.522% | 2.478% | |

-statistic vs. |
30.01** [0.000] |
17.67** [0.000] |
125.3** [0.000] | |

-statistic vs. |
4.09* [0.046] |
169.3** [0.000] |

Notes:

1. The dependent variable is
, and the sample period is
1964(3)-1989(2) [].

2. Estimated standard errors
of coefficient estimates appear in parentheses.

3. In the last two rows, the
three entries within a given block are the -statistic with degrees of freedom as indicated, the value
of that -statistic, and the tail probability
associated with that value of the -statistic
(in brackets).

4. A single
asterisk * and two asterisks ** indicate rejection at the 5% and 1% levels
respectively.

Table 2. Extreme Bounds Analysis of the Model Pairs , , , and .

Proposition illustrated | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Unrestricted model | ||||

Restricted model | ||||

included | excluded | included | excluded | |

Focus variable | ||||

Bounds | [-0.79, -0.63] (0.05) (0.06) | [-0.79, -0.72] (0.06) (0.05) | [-0.18, +0.34] (0.06) (0.05) | [-0.13, +0.34] (0.07) (0.06) |

Modified bounds | [-0.89, -0.51] | [-0.91,-0.62] | [-0.29, +0.44] | [-0.26, +0.45] |

L-robust or L-fragile | L-robust | L-robust | L-fragile | L-fragile |

Notes:

1. Even although is not the actual DGP, is treated as
such to provide empirical examples of the four
propositions.

2. Estimated asymptotic standard
errors of bounds appear in parentheses underneath the
bounds.

3. The modified bounds are
calculated from equation (5)

* Forthcoming in the *Oxford Bulletin of Economics
and Statistics*, special issue on *Encompassing* (edited by
David F. Hendry, Massimiliano Marcellino, and Grayham E. Mizon).
The author is a staff economist in the Division of International
Finance, Board of Governors of the Federal Reserve System,
Washington, D.C. 20551 U.S.A., and may be reached on the Internet
at ericsson@frb.gov. The views in this paper are solely the
responsibility of the author and should not be interpreted as
reflecting the views of the Board of Governors of the Federal
Reserve System or of any other person associated with the Federal
Reserve System. The author is grateful to Julia Campos, Jon Faust,
John Geweke, Dale Henderson, David Hendry, Ed Leamer, Jaime
Marquez, Grayham Mizon, and two referees for helpful comments and
discussions. All numerical results were obtained using PcGive
Version 12; see Doornik and Hendry (2007). Return to text

1. McAleer et al. (1985, p. 297) denote this definition of fragility as Type B fragility; see also Leamer and Leonard (1983, p. 307). Other similar definitions of fragility in EBA also exist. However, the key results in Section 3 are unaffected by the particular definition chosen, so "Type B" fragility is used throughout as the definition of L-fragility. Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text