In this section we analyze briefly the use of SNPII estimators in the context of dy- namic models. In particular, we suppose that observed data consists of observations from a vector time-series process.
4.3 Basic Formulation for Dynamic Models type xt=θ0(xt−1,t) similar to that considered in the very beginning of this thesis
(in Section 1) are often interesting. In such cases, SNPII estimation should pro- ceed essentially as in the previous section. Indeed, except for a number of details concerning the appropriate choice of auxiliary estimators and sieves, everything else applies essentially in the same way. Most importantly, as suggested in Section 1.4, choices of sieves appropriate for dynamic models (e.g. artificial neural networks) will allowθ0 to be consistently estimated even when it lies in very general spaces (see also Granger and Terasvirta (1993)).
Remark 4.3.1. In nonlinear dynamic models, auxiliary estimators must be chosen
so as to describe appropriately the dynamic features of the data. Furthermore, as pointed out in Section 3.7, sieves should be selected so as to ensure that certain dynamic properties of interest such as stability and fading memory hold.5
Below, we shall focus on making use of SNPII estimators in the context of theory- driven models. Our aim is to analyze the behavior of SNPII estimators when models are formulated “in conjunction with appropriate theories” as suggested by Granger and Terasvirta (1993). For concreteness, let us turn back to the basic RBC model considered in Section 1.6 of Chapter 1. This simple model of an isolated farm abstracts from the complications introduced by the larger and more complex models. Nonetheless, it might provide some important insight into the behavior of SNPII estimators in the context of dynamic theory-driven models in general.
Recall that in the context of Section 1.6, economic theory derives the dynamic behavior of economic variables from the following optimization problem,
max {ct}∞t=1 Et ∞ s=t βs−tu(cs) , s.t. kt+1= f (kt, zt)− ct, zt= g(zt−1) + t. (4.6)
Certain features of this optimization problem might be better described by eco- nomic theory than others, in which case economists will be more confident of some theoretic restrictions than others. For example, the general description of capital accumulation of the ‘isolated farmer’ as the process through which a quantity of cereals is consumed while the remaining stock is used for obtaining new crops in the next season, might be consensual and accurate. Likewise, the general specification of the TFP has having some form of time dependence as described by a nonlinear autoregressive process might be deemed reasonable. However, there is in general, great uncertainty as to which exact form the functions, u, f and g might take. This difficulty is generally accepted in economics.
5Recall that catalogues of conditions for the formulation of nonlinear dynamic models with
appropriate fading memory including near epoch dependence and L0-approximability conditions
can be found in Gallant and White (1988b) and Pötscher and Prucha (1997). Also, Trapletti et al. (1998) provide geometric ergodicity and stationarity conditions directly for artificial neural network sieves.
“One of the main differences between econometrics and the applica- tion of statistical methods in the physical sciences is that the functional forms in the structural equations of an econometric model are seldom given by the theory” in Bergstrom (1985).
Economic theory is often capable of providing very general conditions under which utility functions or production functions are continuous, monotone, concave, etc. (see e.g. Debreu (1959)). These very general results are however far from the restrictions that are typically imposed in empirical work. Unfortunately, in the face of such difficulties, it is common for researchers to proceed by parametrizing the unknown functions u, f and g according to very simplistic (and thus restrictive) forms. Common examples consist of CRRA utility functions u(ct) = c1−θt u/(1−
θu), AK production functions f (kt, zt) = exp(zt)Ak θf
t and linear TFP equations
g(zt−1) = θgzt−1. The choice of such functions is typically justified in an informal
way by the desire to retain algebraic convenience and analytical simplicity. For example, Kydland and Prescott (1982) give the following justification for the form of a production function featured in their RBC model.
“The production function is assumed to have the form f (λ, k, n, y) = λnθ[(1− σ)k−v+ σ−v]−(1−θ)/v where 0 < θ < 1, 0 < σ < 1, and 0 < v <
∞. This form was selected because, among other things, it results in a share θ for labor in the steady state.” in Kydland and Prescott (1982).
It is thus not surprising to find proponents of such theory-driven models as Lucas (1985) concluding that “Of course, the model is not ‘true’ ”. Under these typical restrictive assumptions, the system of dynamic first-order conditions derived from the optimization problem above, takes the form,
c−θu t = βEt c−θu t+1θfexp(zt+1)Aktθ+1f−1 kt+1= exp(zt)Ak θf t − ct zt= θgzt−1+ t, t∼ N(0, σ2)
Finally, under additional conditions involving rational expectations of agents, the system of first-order conditions above is then ‘solved’ (and typically linearized in the process) and turned into a system of dynamic autoregressive equations determining the behavior of the variables of interest. This dynamic system is then the focus of econometric analysis. However, in the likely event of model misspecification, econometric analysis can be misleading.
Remark 4.3.2. It is precisely the informal justification of functional form found in
4.3 Basic Formulation for Dynamic Models
In essence, SNPII theory exploits the vast body of mathematical results on approxi- mation theory to assist economist and econometricians on the design of functional forms that have greater chances of being coherent with general theory, empirical observation and correct specification axioms.
An example of how SNPII theory can be used in conjunction with economic the- ory consists precisely of letting Approximation Theory guide the process of choosing functional forms for u, f and g. In particular, sieves can be chosen in conjunction
with theory so as to obtain an SNPII estimator that is indeed capable of consistently
estimating any element within a class of functions suggested by theory, e.g. functions that are continuous, increasing, monotone, concave, and others.
A particular example consisting of polynomial approximations to these functions leads to a an optimization problem given by (4.6) where,6
u(ct)≈ kTu i=0 θu,i(ct− css)i , g(zt−1)≈ kTg i=0 θg,i(zt−1− zss)i, f (kt, zt)≈ kfT i=0 kfT j=0 θf,i,j(kt− kss)i(zt− zss)j.
In the spirit of SNPII estimation, important generality might be gained by letting the truncation orders ku
T, k g Tand k
f
T diverge to infinity at appropriate rates. As usual,
a system of first-order conditions can once again be derived,
kuT
i=1
iθu,ici−1p,t = βEt
kTf i=1 kTf j=0
iθf,i,j(kt+1− kss)i−1(zt+1− zss)j kuT
i=1
iθu,ici−1p,t
kt+1= kfT i=0 kfT j=0 θf,i,j(kt− kss)i(zt− zss)j− ct zt= kgT i=0 θg,izt−1+ t, t∼ N(0, σ2).
Finally, in the context of rational expectation models, SNPII estimation can proceed by applying appropriate nonlinear solution methods that approximate the consumption policy function with any desired level of accuracy. In particular, per-
turbation methods (see e.g. Judd Judd (1998)) might be especially well suited in
this context, since they also approximate the policy function by a truncated power series. Various other methods, including spectral projection, finite element or spline methods, might be preferable depending on the choice of sieves. An issue that seems
6Here, c
to have been ignored in this literature concerns however the fact that (just like poly- nomial sieves) perturbation solution methods do not confer the dynamic model with appropriate stability properties. The same applies to several other solution methods if appropriate conditions are not imposed.
Some limited simulation based experiences suggest nonetheless that the SNPII estimator works very well with Chebyshev spectral projection and perturbation solu- tion methods. The computational requirements prevent us however from developing a fully fledged Monte Carlo exercise. In Section 4.4 we shall thus avoid the ‘solution’ part of the modeling process.
In what follows we keep working with a weighted quadratic criterion function
QT as in Gourieroux et al. (1993), QT,S(θ) = μT ˆ βT, ˜βT,S(θ) = i∈N wT,i ˆ βiT− ˜β i T,S(θ) 2 . (4.7)
This time however, auxiliary estimators are chosen to be least-squares estimators obtained by regressing ytoverTk(yt−1) for k = 1, ..., kT, whereTk(yt−1) denotes the
k-th order Chebyshev polynomial transformation of the lag yt−1. Multiple lags are
also considered. This choice seems to offer enough “information” about the nonlin- ear autoregressive structure of the data. As before, alternative auxiliary statistics exploring nonlinearities and asymmetries in the dependence between y and its lags seem to provide virtually identical results.
Simulations were once again performed using the software package MATLAB with a number of Monte Carlo replications of N = 500. For every replication, one set of artificial “observed data” was used to obtain ˆβT and S = 20 sets of
simulated data were used to obtain ˜βT,S(θ). Finally, actual SNPII estimates ˆθT,
were again obtained by minimizing the criterion function using a standard Newton- type algorithm. Alternative initial conditions seem to provide essentially identical results.