Abstract
In this chapter the identification problem in the most meaningful single-equation linear stationary models with rational expectation (RE) discussed in the previous pages is considered. As pointed out in Pesaran (1989, p. 119) “identification is fundamental to the empirical analysis of structural models. Unless a model is identified it will not be possible to give a meaningful interpretation to its parameter estimates.”1 In all the single equation models “the substitution of unobserved expectational variables by functions of observables poses the problem of whether the unknown parameters can be identified from the knowledge of the observables” (Pesaran, 1989, p. 119). In those containing future expectations, like the Cagan and Taylor type models, “the new problem which emerges ... is the multiplicity of solutions ... (which depends) ... on the arbitrary choice of martingale differences. The number of arbitrary processes depends on the structure of the model (the horizon of the expectations and the size of the model) and on the values of the structural matrices. ... Consequently in the case of linear stationary models identification concerns both kinds of parameters: the structural ones and the additional, or auxiliary, ones” (Broze and Szafarz, 1991, pp. 184–185).
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Notes
See, e.g., Pesaran (1989, pp. 121–122) or Judge et al. (1985, p. 574) for a formal introduction to the identification problem. An example of semi-parametric approach to the identification problem can be found in Broze and Szafarz (1991, pp. 140–142).
This model is similar to that discussed in Pesaran (1989, pp. 122–126) and Broze and Szafarz (1991, pp. 179–184).
The case with ut following a stationary ARMA process is treated in Pesaran (1989, pp. 126–127).
See, e.g., Harvey (1981a, p. 201). As in the cases studied in Amemiya (1973), Jobson and Fuller (1980) and Judge et al. (1985, pp. 431–439), here the variance is a function of some of the parameters appearing in the regression. These parameters however are associated with a vector of (unknown) constants rather than to a vector of exogenous variables As a consequence the error term is homoscedastic instead of heteroscedastic as in those works.
Indeed they use the concept of ‘first-order strong identifiability’ to stress the fact that the actual value of xt is replaced by xt−1, the expectation of xt computed by the economic agents when forming their expectation on the endogenous variable, in the reduced form equation [9–5].
See, e.g., Judge et al. (1985, pp. 171–172).
The procedure outlined in the text is slightly different from that followed in Pesaran (1989, p. 125) which concentrates on the first-order moments of yt and does not need the conditions on the distribution of the residuals.
When xt is perfectly predictable, i.e. xt xt, only [9–5] holds and system [9–6] reduces to the first 4 equations with 5 unknowns. In this case the structural parameters are not identifiable. “This under-identification of RE models under perfect foresight assumptions on the process generating the predetermined variables, carries over to models with future expectations and simultaneous equation models” notices Pesaran (1989, p. 124). Another important result reported in Pesaran (1989, p. 126) is that “a RE equation with only current expectations which contains both an expected and a realized value of the same exogenous variable cannot be identified… (even though) this result does not necessarily hold in models with future expectations”.
Therefore a test for 0 covariance between vt and ηt boils down to testing the null hypothesis b 0=0 in the auxiliary regression in footnote 7.
See, e.g., Broze and Szafarz (1991, pp. 179–180) for a similar discussion on a slightly different model.
The “identification of the RE model can also be achieved if some of the variables that are included in the process generating xt can be excluded a priori from the RE model itself’ (Pesaran, 1989, p. 127).
See Pesaran (1989, pp. 139–140) for an example in which the necessary condition is satisfied but the structural parameters are not identified.
This is consistent with Pesaran (1989, pp. 122–126) where a model with a vector of non-perfectly-predictable exogenous variables is considered.
When p=1, Eqt. [9–9d] disappears and the structural parameters are no longer identified because there are more unknowns than relations in [9–9]. However β2 and V(ut) can still be identified together with pl and.
See, e.g., Pesaran (1989, Sect. 7.5).
See Section 7.6 in Chapter 7 for details. A generalized vector AR formulation of the exogenous process is suggested in Pesaran (1989, pp. 124–125).
The matrices Rj and Ση can be identified from [9–12].
As pointed out in footnote 12, when the homoscedasticity of the v’s is rejected the researcher is free to proceed in several different ways. To check if the parameters of all the exogenous variables are time-varying in an unrestricted way, the squares of the estimated residuals are regressed on a constant and the squares and cross products of the exogenous variables, both contemporaneous and lagged, included in [9–10]-[9–11] (see, e.g., Harvey, 1981, pp. 201–207). Again it may be useful to use the reparameterization suggested in Schwallie (1982) to avoid the problem of negative estimates of the variances. See also Jobson and Fuller (1980) for the estimation of a model with a similar error term.
This conclusion is consistent with the statement: “in these circumstances only the structural parameters (associated with the non-perfectly-predictable exogenous variables) can be identified. None of the other parameters are identifiable” (Pesaran, 1989, p. 125).
See, e.g., Pesaran (1989, p. 133–146) and Broze and Szafarz (1991, pp. 185–196). As stressed by the former author, different assumptions concerning the martingale process and the information set can lead to different identifiability requirements. See also sections 7.2 and 7.3.
See Section 7.7 for details.
The parameters Ø0 and k0 are δu and δx, respectively, in Pesaran (1989, p. 142).
In general the error term is heteroscedastic and serially correlated in a time-varying parameter model (see, e.g., Harvey, 1981, pp. 201–203).
This is consistent with Theorem 5.111 in Broze and Szafarz (1991, p. 195) where it is stated that “if the endogenous process yt corresponds to a linear stationary solution for which the structural parameters are identifiable then (all) the auxiliary parameters are also identifiable”. Given that a model with time-varying parameters can always be rewritten as a constant coefficient model with heteroscedastic and serially correlated disturbances, an alternative proof of the same result may be based on Kelejian (1974). He shows that the necessary and sufficient condition for the identifiability of the structural parameters in a system of simultaneous equations with random parameters is identical to that relative to the constant coefficient version of the model. Then in the case discussed in the text if the structual parameters, i.e. the constant coefficients, are identified the auxiliary parameters appearing in the composite error term are also identified provided a sufficient number of observations is available.
When vt includes a term incorporating the “innovations in variables representing boot-strap or bubbles effects, assuming of course that observations of these variables are allowed to enter the information set” (Pesaran, 1989, p. 142), the identification of the structural and auxiliary parameters is more troublesome. In this case Ø0, σ2 u and the variance of the newly introduced ‘bubble’ term are not identifiable. See also Evans and Honkapohja (2000) for a discussion of speculative ‘bubbles’ and ‘sunspots’ equilibria in the presence of ‘learning’ agents.
See Pesaran (1989, pp. 133–137).
See footnote 19 above.
Obviously the two procedures generate the same constraints.
Alternatively it can said that the RE model [9–15] is observationally equivalent to the non RE model [9–24] (Pesaran, 1989, p. 138).
“For other choices (of the initial conditions) it is not legitimate to cancel the common lag polynomial since there are additional ‘transient’ term(s) associated with the initial condition(s).… (These terms disappear asymptotically if the roots are stable but) if this condition is not met then the (transient) term(s) remain non-negligible for all time (periods)” (Evans and Honkapohja, 2000, Sect. 8.2, fn. 3).
This condition is more restrictive than that seen for the Muth type model. As shown in Pesaran (1989, p. 199) this is true even for higher order RE models with lagged dependent variables.
See also footnote 27 in Pesaran (1989, p. 203).
See Section 8.2 for details.
See also Pesaran (1989, p. 145).
See footnote 20 above.
See Appendix 9A for the derivation.
As before Ση and the R’s can be estimated from [9–12].
In this special case it is not necessary to know a priori how many roots are explosive as in Pesaran (1989, p. 202). Given that the identifiability condition(s) is(are) stricter when the number of unstable roots is higher, it is sufficient to check the identifiability requirement for n=m. If the structural parameters are identified in this ‘most unfavorable’ case they are certainly identified for all the other, less demanding, cases.
See footnotes 7, 10 and 11 above. When homoscedasticity is rejected the researcher may proceed in a number of alternative ways as pointed out when discussing the Cagan type model. Again the best reference is Harvey (1981a, pp. 201–207) for a quick look to some of them.
See Section 7.9 for details.
The restrictions implied by [9–36] are identical to those implied by [9–37].
The number of unknowns reflects the fact that Ø1 and kl are constrained.
It is sufficient to concentrate on the identifiability requirements of the Cagan type model because they are generally stricter than those for a Taylor type model.
See Section 8.4 for details.
Similarly to what noticed for the simple Taylor type model, this model contains K-1 overidentifying restrictions for ß2 which can be estimated either from [9–40b] or [9–40j].
See Appendix 9B for the derivation.
See Appendix 9B.
When m= n=1, Eqt. [9–28] collapses to the case discussed in Pesaran (1989, pp. 144–146).
For an application of the Jordan canonical form to RE models see, e.g., Broze and Szafarz (1991, Ch. 4).
As well known given a square matrix A, not necessarily symmetric it is possible to find a square matrix B with distinct roots arbitrarily close to the former. See, e.g., Bellman (1960, p. 199).
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Tucci, M.P. (2004). The Identification Problem in Single-Equation RE Models. In: The Rational Expectation Hypothesis, Time-Varying Parameters and Adaptive Control. Advances in Computational Economics, vol 19. Springer, Boston, MA. https://doi.org/10.1007/978-1-4020-2874-8_9
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