Modern economies are complicated, dynamic, non-linear, simultaneous, high-dimensional, and evolving entities; social systems alter over time; laws change; and technological innovations occur. The resulting time-series data samples are heterogeneous, non-stationary, time-dependent and inter-dependent. Economic magnitudes are inaccurately measured, subject to revision and important variables are not unobservable: worse still, available samples are relatively short and highly aggregated. Thus, models in economics are bound to be much simpler than data-generating processes. In the face of such problems, econometric modelling of economic time series seeks to discover sustainable and interpretable relationships between observed variables. Several key concepts help in understanding how that is possible, namely, a progressive research strategy; the dichotomy between construction and destruction; an empirical model; and a local data-generating process. The last arises as a reduction of the actual DGP, and the process of reduction delivers a taxonomy of information sets against which models can be evaluated. We now consider these four concepts: for a more complete treatment, see Hendry (1995a).
First, knowledge accumulation is progressive: one does not need to know all the answers at the start (otherwise, no science could have advanced). Although the best empirical model at any point will be supplanted later, it can provide a springboard for further discovery. Thus, tentative specifications can be improved over time. Of course, if there were no constancies in economic behaviour, then sustainable relationships could never be found, so we require their existence despite the non-stationarity of the DGP, and return to that issue in section 9.2.9 below.
The second key is that determining inconsistencies between the implications of any conjectured model and the observed data is reasonably easy. Indeed, the ease of rejection is a powerful advantage of economic data being non-stationary. Conversely, constructive progress is difficult, because we do not know what we don't know, so cannot know how to find it out. The dichotomy between construction and destruction is an old one in the philosophy of science: critically evaluating empirical evidence is a destructive use of econometrics, but can help establish a legitimate basis for models by rejecting inadequate or incomplete representations.
Thirdly, we define the notion of an empirical model, then explain the origins of such models by the theory of reduction.
In an experiment, outputs are `caused' by the inputs, and the process can be treated as if it were a mechanism. Let yt be the observed outcome of an experiment when zt is the experimental input, f(.) is the mapping from input to output, and nt is a small, zero-mean random perturbation, which varies between experiments conducted at the same values of z. Then:
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Given the same inputs {zt}, repeating the experiment generates essentially the same outputs (otherwise, the experiment is not replicable and is deemed invalid). In that sense, the right-hand side generates the left in (eq:9.1).
In an econometric model of observational data, however:
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Any outcome {yt} always can be decomposed into two components, the first of which we denote g(zt) (the part explained) and the second et (the unexplained element). Importantly, such a partition is feasible even when yt does not in fact depend on g(zt). In that sense, the left-hand side is determining the right in (eq:9.2), so it is more appropriate to write:
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The unexplained element results from our choice of zt, so et is a derived entity, not an `autonomous input', implying that models can be designed to achieve pre-assigned criteria. Consequently, design criteria must be analyzed, and doing so leads to the notion of a congruent model: one that matches the data evidence on all the measured attributes. Successive congruent models should be able to explain previous models, which is the concept of encompassing, and thereby progress can be achieved.
The main component now required to deliver a viable approach is to explain the origin of empirical models from the DGP. That occurs in two stages. First, we derive the local DGP (LDGP), which is the generating process in the space of the variables under analysis. The next 10 sub-sections sketch that derivation. Then, we relate the empirical model to that LDGP: §9.3 describes that relation. Next, we see the `excess content' of the theory. First, that many of the main concepts of econometrics (exogeneity, non-causality etc.) correspond to reduction steps. Secondly, that a taxonomy of evaluation information suggests appropriate model-design criteria, by avoiding information losses during reduction, and thereby characterizes the concept of congruence noted above: an empirical model is congruent if it parsimoniously encompasses the LDGP (see Mizon, 1995, and Bontemps and Mizon, 2001).
The data-generating process (DGP) of the complete set of random variables relevant to any economy under investigation over a time period t=1,...,T is far too complicated ever to understand or model. Consequently, reduction to a manageable size is essential for any form of analysis. The sample joint data density is called the Haavelmo distribution after Haavelmo (1944) (see e.g., Spanos, 1989). Operational representations are defined by a sequence of data reductions, organized into 10 stages.
Some of the data series will be measured aggregates to be analyzed, so the rest are disaggregates or other aggregates which are deemed to be sufficiently irrelevant to be eliminated. Equally, logs, ratios, differences, disequilibria, etc., will be created to capture an appropriate interpretation of the outcome. The key effects of transformations (including reductions) are their impacts on the parameters: every operation alters the parameterization. Some transformations will lose constancy in the parameters; others induce constancy in non-constant relations: see section 9.2.9.
The parameters of interest are denoted by yÎY. Parameters which are identifiable, and invariant to an interesting class of interventions, are clearly of interest.
Partition the data set into the two subsets: the m variables {xt} (such that everything about the parameters of interest y is to be learnt from analyzing the {xt} alone), with the remaining variables to be marginalized from the joint density -- retaining only the marginal density of the {xt}.
To do so without loss of information, y must be a function of the parameters of that marginal density alone, and the latter's parameters must be variation free from the parameters of the conditional density. This can only occur if past values of the eliminated variables are irrelevant to the process determining {xt}: i.e., they do not Granger-cause the {xt} (see Granger, 1969, and Hendry and Mizon, 1999).
The next step is to sequentially factorize the complete sample of data on {xt} as:
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where Xt-1=( xt-1,...,x0,...) and {lt} denotes the set of parameters of the data density Dx( .) . The aim of this step is to generate a mean-innovation error process (MIP):
| et=xt-E[ xt½Xt-11] . |
That {et} is at least white noise is easily tested. Further, a necessary condition for no information loss from marginalizing, is that et does not depend on any of the eliminated variables, which is testable for any postulated variable.
Combinations of integrated, but not cointegrated, variables must be eliminated to avoid nonsense relationships. When integrated processes are expressed in terms of cointegrated and differenced variables, a conventional statistical analysis can be implemented. The resulting generic type of linear representation is an equilibrium-correction model, denoted EqCM. The degree of integration of the transformed variables is testable, but not very powerfully. A notation switch to I(0) transformed variables is possible here, but will not be implemented. Many inferences will be valid even if this reduction is not enforced (see Sims, Stock and Watson, 1990).
Partition xt into sets of n and k variables, where m=n+k, such that the yt are endogenous and the zt are non-modelled. Then factorize the sequential density of xt in (eq:9.4) into a conditional and a marginal density:
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Then, zt is weakly exogenous for y if (i) y=f(fa,t) alone; and (ii) (fa,t,fb,t)ÎFa×Fb. Under (i) and (ii), the problem can be reduced to analyzing only the conditional distributionDy|z(yt|zt,Xt-1,fa,t) of yt given zt.
To progress, some restrictions are needed on the heterogeneity over time of the process being observed. One extreme is fa,t=f "t where fÎF, with y=f(f); the other is complete non-constancy. Many mappings are feasible, and most are relatively easy to test. We use the notation fa here, which includes all these cases depending on the dimensionality of fa.
Appropriate formulations of models can induce constant-parameter equations even in non-stationary processes. One important example is that a price index which uses time-varying weights (e.g., a commodity price measure with many individual components) could have a constant coefficient in a model, even though all of the components would enter the same model non-constantly. A second example, called extended constancy in Hendry (1996) and amplified by Ericsson, Hendry and Prestwich (1998), concerns when a constant effect later varies, requiring a model with more parameters to characterize the evidence, yet the change can be `absorbed' by existing parameters to leave the model unaltered with different data definitions. An example is when an institutional change de-restricts a previous constraint and allows it to vary, as with interest-bearing checking accounts becoming legal, necessitating a re-definition of the opportunity cost of holding money from an outside interest rate to a differential over the own rate, after which the model reverts to its previous parameterization.
An important practical reduction is fixing the history of Xt-11 at s earlier periods:
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where et=yt-E[ yt|zt,Xt-1t-s] remains an innovation process. Such a reduction is again testable.
This reduction also affects the form of Dy|z( .) . Map yt into yt*=h(yt), and zt into zt*=g(zt), where:
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such that a linear representation for yt* is obtained in terms of zt*, when Dy*|z*(.) is approximately normal and homoscedastic, denoted Nn[ ht,U] . Such a reduction is testable against some alternatives. Then the right-hand side of (eq:9.7) is the LDGP. Being a formal reduction of the DGP, the LDGP is implicitly nested within that DGP, and its properties are fully explained by the reduction process.
Economies comprise billions of decisions and millions of recorded variables, generated by the DGP of agents' behaviour and the recording procedures. The LDGP is the corresponding representation for the subset of variables under analysis, such that, were the LDGP known, the outcomes could be predicted up to an innovation error. Thus, computer-generated data from the LDGP would only differ randomly from the actual values.
To match these reductions, when s=1, a model usually takes the form in I(0) space (omitting the *):
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which is an EqCM. In (eq:9.8), h ~ denotes `hypothesized to be distributed as', emphasizing that et is a derived, and not an autonomous, process, being defined by:
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Then {et} is autonomous only if there is no loss of information in any reduction, and the model coincides with the LDGP. More generally, we postulate models of the form:
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where et h ~ Nn[ 0,Se] , and A( L) and B( L) are polynomial matrices (i.e., matrices whose elements are polynomials) of order s in the lag operator L. The reduction to the generic econometric equation involves all the stages of aggregation, marginalization, conditioning etc., transforming the parameters from the DGP (which determines the stochastic features of the data), to the coefficients of the empirical model.
Valid reductions lose no relevant information, and many of these have been separately named in econometrics. Indeed, many of the salient concepts of that discipline map onto measures of no information loss as follows.
[ 1] Aggregation entails no loss of information on marginalizing with respect to disaggregates when the retained information comprises a set of sufficient statistics for the parameters of interest y.
[ 2] Transformations per se do not entail any associated reduction, but directly introduce the concept of parameters of interest, and indirectly the notions that parameters should be invariant and identifiable.
[ 3] Data partition is a preliminary step, although the decision about which variables to include and which to omit is perhaps the most fundamental determinant of the success or otherwise of empirical modelling.
[4] Marginalizing with respect to contemporaneous variables is without loss providing the remaining data are sufficient for y, whereas marginalizing without loss with respect to all their lagged values entails both Granger non-causality for xt and variation-free parameters.
[5] Sequential factorization involves no loss if the derived error process is an innovation relative to the history of the random variables.
[6] Integrated data systems can be reduced to I(0) by suitable combinations of cointegration and differencing, allowing conventional inference procedures to be applied to more parsimonious relationships.
[7] Conditional factorization reductions, which eliminate marginal processes, lead to no loss of information relative to the joint analysis when the conditioning variables are weakly exogenous for the parameters of interest.
[ 8] Parameter constancy implicitly relates to invariance, namely constancy across interventions which affect the marginal processes.
[9] Lag truncation involves no loss if the error process remains an innovation despite excluding some of the past of relevant variables.
[10] Functional form approximations need involve no reduction (logs of log-normally distributed variables).
[11] The LDGP, as a reduction of the DGP, is nested within that DGP and its properties are explained by the reduction process: knowledge of the DGP entails knowledge of all reductions thereof. When knowledge of one model entails knowledge of another, the first is said to encompass the second.
Partition the data XT1 used in modelling into the three information sets:
[a] past data;
[b] present data
[c] future data.
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[d] theory information, which often is the source of parameters of interest, and is a creative stimulus in economics;
[ e] measurement information, including price-index theory, constructed identities such as consumption equals income minus savings, data accuracy and so on; and:
[ f] data of rival models, which could be partitioned in turn into past, present and future.The six main criteria which result for selecting an empirical model are:
[a] homoscedastic innovation errors;
[b] weakly-exogenous conditioning variables for the parameters of interest;
[c] constant, invariant parameters of interest;
[d] theory-consistent, identifiable structures;
[e] data-admissible formulations on accurate observations; and
[f] encompassing rival models.
Models which fail to satisfy any of the first five information sets are non-congruent: an encompassing congruent model satisfies all six criteria.
Consider two distinct empirical models, denoted M1 and M2, with mean-innovation processes (MIPs) {nt} and {et} relative to their own information sets, where nt and et have constant, finite variances sn2 and se2 respectively. Then M1 variance dominates M2 if sn2<se2, denoted by M1ÎM2. Variance dominance is transitive since if M1ÎM2 and M2ÎM3 then M1ÎM3; and is anti-symmetric since if M1ÎM2, then it cannot be true that M2ÎM1. A model without a MIP error can be variance dominated by a model with a MIP on a common data set. The DGP cannot be variance dominated in the population by any models thereof (see e.g., Theil, 1971, p543). A model with an innovation error cannot be variance dominated by a model which uses only a subset of the same information.
If et=xt-E[ xt|Xt-1] , then se2 is no larger than the variance of any other empirical model error defined by xt=xt-G[ xt|Xt-1] whatever the choice of G[ .] . The conditional expectation is the minimum mean-square error predictor. These implications favour general rather than simple empirical models, given any choice of information set, and suggest modelling the conditional expectation. A model which nests all contending explanations as special cases must variance dominate in its class. Let model Mj be characterized by parameter vector yj with kj elements, then as in Hendry and Richard (1982):
M1 is parsimoniously undominated in the class {Mi} if "i, k1£ki and no MiÎM1.
Model selection procedures (such as AIC, or the Schwarz criterion: see Judge, Griffiths, Hill, Lütkepohl and Lee (1985)) seek parsimoniously undominated models, but do not check for congruence. When no further feasible reductions can be found, PcGets selects a model by an information criterion. The preferred `final-selection' rule presently is the Schwarz criterion, or BIC, defined as:
| SC=-2 ln L/T+p ln (T)/T, |
where L is the maximized likelihood, p is the number of parameters and T is the sample size. For T=140 and p=40, minimum SC corresponds approximately to the marginal regressor satisfying |t|³1.9.
If knowledge of the model in (eq:9.9) entails knowledge of a second model of the same data, the first is said to encompass the second. Since all models are reductions of the DGP, they are potentially comparable to see which encompasses which. Parsimonious encompassing is defined by a congruent model encompassing a more general congruent model within which it is embedded. §13.11 describes the encompassing test in PcGets.
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