Is modelling subsystems and combining them to a global model a viable procedure?
The traditional approach to building large-scale macroeconometric models has been to estimate one equation (or submodel) at a time and collect the results in the simultaneous setting. Most often this has been done without testing for the adequacy of that procedure. The approach could, however, be defended from the estimation point of view. By adopting limited information maximum likelihood (LIML) methods, one could estimate the parameters of one equation, while leaving the parameters of other equations unrestricted: see Anderson and Rubin (1949) and Koopmans and Hood (1953). It has, however, also been argued that the limited information methods were more robust against mis-specified equations elsewhere in the system in cases where one had better theories or more reliable information about a subset of variables than about the rest (cf. Christ 1966, p. 539). Historically, there is little doubt that limited information methods—like LIML—were adopted out of practical considerations, to avoid the computational burden of full information methods—like full information maximum likelihood (FIML). The problem of sorting out the properties of the system that obtained when the bits and pieces were put together, remained unsolved.
That said, it is no doubt true that we run into uncharted territory when we —after constructing relevant submodels by marginalisation and conditioning—combine the small models of subsectors to a large macroeconometric model. As we alluded to in the Section 2.1, Johansen (2002) points out that a general theory for the validity of the three steps will invariably contain criteria and conditions which are formulated for the full system. The question thus is: given that the full model is too large to be modelled simultaneously, is there a way out?
One solution might be to stay with very aggregated models that are small enough to be analysed as a complete system. Such an approach will necessarily leave out a number of economic mechanisms which we have found to be important and relevant in order to describe the economy adequately.
Our general approach can be seen as one of gradualism—seeking to establish structure (or partial structure) in the submodels. In Section 2.3.2 we gave a formal definition of partial structure as partial models that are (1) invariant to extensions of the sample period; (2) invariant to changes elsewhere in the economy (e. g. due to regime shifts); and (3) remains the same for extensions of the information set.
The first two of these necessary conditions do not require that we know the full model. The most common cause for them to be broken is that there are important omitted explanatory variables. This is detectable within the frame of the submodel once the correlation structure between included and excluded variables changes.
For the last of these conditions we can, at least in principle, think of the full model as the ultimate extension of the information set, and so establishing structure or partial structure represents a way to break free of Spren Johansen’s Catch 22. In practice, however, we know that the full model is not attainable. Nevertheless, we note that the conditional consumption function of Section 2.4.2 is constant when the sample is extended with nine years of additional quarterly observations; it remains unaltered through the period of financial deregulation and it also sustains the experiment of simultaneous modelling of private consumption, household disposable income, household wealth and real housing prices. We have thus found corroborating inductive evidence for the conditional consumption function to represent partial structure. The simultaneous model is in this case hardly an ideal substitute for a better model of the supply-side effects that operate through the labour market, nonetheless it offers a safeguard against really big mistakes of the type that causation ‘goes the other way’, for example, income is in fact equilibrium correcting, not consumption.
There may be an interesting difference in focus between statisticians and macroeconomic modellers. A statistician may be concerned about the estimation perspective, that is, the lack of efficiency by analysing a sequence of submodels instead of a full model, whereas a macroeconomic modeller primarily wants to avoid mis-specified relationships. The latter is due to pragmatic real-world considerations as macroeconomic models are used as a basis for policy-making. From that point of view it is important to model the net coefficients of all relevant explanatory variables by also conditioning on all relevant and applicable knowledge about institutional conditions in the society under study. Relying on more aggregated specifications where gross coefficients pick up the combined effects of the included explanatory variables and correlated omitted variables may lead to misleading policy recommendations. Our conjecture is that such biases are more harmful for policy makers than the simultaneity bias one may incur by combining submodels. Whether this holds true or not is an interesting issue which is tempting to explore by means of Monte Carlo simulations on particular model specifications.
That said, it is of particular importance to get the long-run properties of the submodel right. We know that once a cointegrating equation is found, it is invariant to extensions of the information set. On the other hand, this is a property that needs to be established in each case. We do not know what we do not know. One line of investigation that may shed light on this is associated with the notion of separation in cointegrated systems as described in Granger and Haldrup (1997). Their idea is to decompose each variable into a persistent (long-memory) component and a transitory (short-memory) component. Within the framework of a vector equilibrium correcting model like (2.5), the authors consider two subsystems, where the variables of one subsystem do not enter the cointegrating equations of the other subsystem (cointegration separation). Still, there may be short-term effects of the variables in one subsystem on the variables in the other and the cointegrating equations of one system may also affect the short-term development of the variables in the other. Absence of both types of interaction is called complete separation while if only one of these is present it is referred to as partial separation. These concepts are of course closely related to strong and weak exogeneity of the variables in one subsystem with respect to the parameters of the other. Both partially and completely separated submodels are testable hypotheses, which ought to be tested as part of the cointegration analysis. Hecq et al. (2002) extend the results of Granger and Haldrup (1997). The conclusion of Hecq et al. (2002) is, however, that testing of separation requires that the full system is known, which is in line with Spren Johansen’s observation earlier.
In Chapter 9 we introduce a stepwise procedure for assessing the validity of a submodel for wages and prices for the economy at large. A detailed and carefully modelled core model for wage and price determination (a Model A of Section 2.1) is supplemented with marginal models for the conditioning variables in the core model. The extended model is cruder and more aggregated than the full Model B of Section 2.1. Notwithstanding this, it enables us to test valid conditioning (weak exogeneity) as well as invariance (which together with weak exogeneity defines super exogeneity) of the core model on criteria and conditions formulated within the extended model. The approach features a number of ingredients that are important for establishing an econometrically relevant submodel, and—as in the case of the consumption function—this points to a way to avoid the Catch 22 by establishing partial structure.