The Gaussian Estimation Method of Continuous-Time Models with Discrete Data Discrete Data

Most of the continuous-time models developed during the 1960s and 1970s were either of first order or their estimation involved the use of an approximate discrete model.

More realistic and flexible continuous-time models were largely acknowledged as most appropriate in order to capture the dynamics observed in many economic phenomena.

However, using more sophisticated models has led to increasingly complex estimation procedures; hence, the importance of developing a rigorous theoretical framework of statistical inference that could be applied for open and closed linear higher order continuous-time systems with both stock and flow data.

Continuous-time formulations that intensively use economic theory in the attempt to model the relationship between economic variables were at the centre of Rex Bergstrom’s research agenda. In a seminal paper, Bergstrom (1983) presented the first theoretical elements of the Gaussian econometric methodology applied to linear stochastic differential equations systems with discrete data. This article represented the foundational study that led, through a systematic approach, to major subsequent developments in all areas of research related to the Gaussian estimation methodology.

Following Bergstrom (1983), the general formulation of a closed linear continuous model of orderk is: ,

1 1

1 1

d D ^k x t( )   A( ) D^kx t( ) ... A_k ( ) Dx t( )A_k( ) ( ) x t b( ) dt( )dt (3.4) where x(t)is a real continuous n - dimensional random process,  is a ^p- dimensional vector of structural parameters, A₁,A₂,...A_kare n n coefficient matrices whose elements are known functions of  , bis an n1vector whose elements are also known functions of

 , and D represents the mean square differential operator. It is assumed that the vector of disturbances (dt) satisfies the following Assumption 1 in Bergstrom (1983):

ASSUMPTION 1:  



₁,...,_n



^ is a vector of random measures with a finite Lebesgue measure on all subsets of the real line, such that E[(dt)]0, E[(dt)(dt)](dt) where   ( ) is a positive definite matrix whose elements are known functions of  another vector of parameters and E[(_i(₁)_j(₂)]0for all i, j 1,...,n and any disjoint sets ₁and ₂. (See Bergstrom (1984a) for a discussion of random measures).

66 In order to avoid the restriction of stationarity, Bergstrom (1983) considered some boundary conditions that x(t) should satisfy:

(0)x y Dx₁, (0)y₂, ... , D^k1x(0) y_k , (3.5) where y₁,...,y_{k are} n - dimensional random vectors verifying the following assumption:

ASSUMPTION 2: E[yⁱ()]0, (i 1,...,k)for any set  in the half line ^{0 t} with finite Lebesgue measure.

Therefore, the continuous-time model comprises the differential equation system (3.4), the boundary conditions (3.5) and the Assumptions 1-2 and it will be hereafter referred to as the basic continuous model. The complete vector parameter to be estimated will comprise   and, y, whereyincludes the unobservable elements of the initial state vector corresponding to the flow data and y₂,...,y defined above. The generality of the _k model has expectedly increased the level of complexity in the estimation procedure. The higher order feature brought an element of complication as the quantities

( ),..., ^k 1 ( )

Dx t D ^x t are unobservable quantities, while new sources of autocorrelation were created in the residual vector by considering mixed data. As a consequence, the estimates of the parameters would be affected by a temporal aggregation bias.

The central theoretical development in Bergstrom (1983) was a fundamental theoretical theorem that proved the existence and uniqueness of a discrete solution for the basic continuous model. Despite the additional complexities, the exact discrete model that is satisfied by the discrete data generated from the continuous model with mixed data was shown to be a vector autoregressive moving average model. In the case when only stock variables are considered the solution is given by a VARMA k k 



^{, 1}



. For continuous models involving only flow data, where the variables are measured as integrals over the observation period ( 



^t

t x r dr t

x( ) 1 ( ) ), the solution becomes a VARMA k k model. When

 

^,

the data is mixed the exact discrete model implied by the basic continuous-time model still maintains the VARMA k k form. The exact solution of the basic continuous model

 

^,

on the domain [ T0, ] could be expressed mathematically as:

1 1 1

( ) ( ) ( 1) .... _k( ) ( ) ( ) _t ( , ) _t ... _k( , ) _{t k}

x t F  x t  F  x t k g   G   _  G   _ (3.6) where the coefficient matrices F ,...,₁ F_k are highly nonlinear transcendental (matrix exponential) functions involving the parameter vectors  and  . The innovations ^{{ }} t t

have the following properties: E(_t)0, E(_t_t)K(,), E(_s_t)0, for s  . t

67 The form of the exact discrete model given by (3.6) is used for analysing the asymptotic sampling properties of the maximum likelihood estimator, however for the purpose of computing the Gaussian estimate and likelihood function an intermediary version (with only the autoregressive coefficients) was considered. The moving average side was embedded in the wide sense stationary process of the disturbances_t. According to Bergstrom (1983, Theorem 2), the form of the exact discrete solution implied by a closed continuous-time model of order k is:

( )x t F₁( ) ( x t  1) .... F_k( ) ( x t k ) g( )  _t, t k 1,...,T (3.7) where { }_{t t}is a wide sense stationary first-order vector moving average disturbance with the following properties:

E(_t)0,

E(_t_t_r)_r(,) for r0,...,k1 E(_t_t_r)0 for r  k1

The complexity of the explicit form of the matrix functions coefficients of the autoregressive part F₁(),…, F_k() and g(), ₀(,),…,_k1(,) increases as the order of the system gets higher ( Bergstrom, 1983 and 1984a,b).

For the estimation of the vector parameter (,),Bergstrom suggested two alternatives that implied additional assumptions. For computational reasons, the procedure followed assumed the wide sense stationarity property for the n- dimensional process



x ),(t  t 



. The exact Gaussian estimates were obtained by minimizing Lˆ with respect to (,), where Lˆ equals minus twice the maximum likelihood function and is computed as:

^{( 1)}



²



1

ˆ 2 log

n T

i ii

i

L z m









 (3.8) This simplified expression for ˆL was derived using a common procedure in Bergstrom’s algorithms for computing the maximum likelihood function: the Cholesky factorization of the correlation matrix valued function V(,). Mathematically, there is a lower triangular real matrix M with positive elements m_ii on the first diagonal, such that

. M M

V   The vector z[ ,...z ]z_{1 nT} can be recursively determined from the Mz. The algorithm for computing Lˆ was rigorously outlaid in Bergstrom (1983, 1985, 1986) and

68 it will be reapplied in the current study and presented at a later stage in the estimation section. Once ˆL has been computed, the Gaussian estimates could be obtained by various optimization procedures which involve repetitive evaluation of ˆL for a set of parameter values. Bergstrom (1983) suggested two such numerical procedures: the maximum gradient method and approximation method for larger continuous models using the spectral density function. While the former method is simpler it does not necessarily lead to the minimum of ˆL in case there are multiple local minima. Therefore, the procedure should be repeated using different inputs for the initial values of the vector parameter ( , )  . The latter method concerned finding an approximation to ˆL in order to avoid the computation of high dimension (nTnT ) matrices. Bergstrom (1983) computed these approximations using the spectral density function of the stationary process { ( ),x t    t }, that after a specific decomposition can be represented as an autoregressive process.

3.2.5 The Development of Computational Algorithms, Hypothesis Testing, Forecasting and Control

Bergstrom (1985) derived a new efficient algorithm for computing the exact Gaussian likelihood for the parameters of a higher order closed continuous-time dynamic model with flow data. A new set of parameters was added for estimation, as the initial state vector y is unobservable⁶. An approximate estimate (extrapolation) for y is accepted instead of an exact one. With y fixed in this way then the estimation procedure will provide asymptotically efficient estimates for ( , )  . Bergstrom (1985) conjectured that the new efficient estimation based on the VARMA type discrete models, should considerably increase the precision of estimates and reduce the prediction errors of future observations.

Applying Phillips’ (1974, 1976) exogenous variable methods, Bergstrom (1986) extended the efficient algorithm for the closed model to an open model of a higher order continuous-time system with both stock and flow types of data. The exogenous variables introduced into the model are assumed to be generated by polynomials in time of degree not exceeding two. The model can be extended even more: for higher order systems,

6Only the first vectorial component of y, made of the initial states y₁{ (0)}x_{i i1,...,n}is considered known or observable; the rest of the components y_j(0){D^j1x (0)}_{i i1,...,n} for j2,...k1 are unobservable, hence they will be endogeneously estimated.

69 instead of using quadratic interpolation, polynomial interpolation can be used, of a degree dependent on the order of the system. The exact discrete model specification was shown to be a generalized VARMAX model, a convenient form in terms of estimation, testing and forecasting.

According to Phillips (1974, 1976) the biases introduced by these assumptions are smaller (of third order) than those (of second order) obtained by employing Fourier methods developed by Robinson (1976a, b, c). For a model with Gaussian innovations and exogenous variables with such continuous paths it was shown that the method yields exact quasi-maximum likelihood estimates of the structural parameters. Bergstrom provided the exact formulas for the implementation of the Gaussian methodology for the most general second order continuous-time model in which both the endogenous and exogenous variables are a mixture of stock and flow variables. As in Bergstrom (1986), the most general continuous-time linear model allowing for greater dynamics (higher order, considering both types of data, including exogenous variables) would have the following equation:

1 1 2

1 2 1

( ) ( ) ( ) ( ) ( ) .... ( ) ( ) ( ) ( ) ( )

k k k

k k

d D ^ x t    A  D ^ x t A  D ^ x t  A_  Dx t A  x t Bz t  dt ( ),t t 0 (3.9) After the development of a complete theoretical framework of the Gaussian estimation of continuous dynamic systems and robust computational algorithms for its implementation, Bergstrom further looked into various other problems using this type of model, concerning optimal control methods for policy makers in Bergstrom (1987), a forecasting algorithm in Bergstrom (1989), and statistical hypothesis testing in Bergstrom (1990, Chapter 7).

Bergstrom (1987) considered the approximate discrete continuous-time model as the true model and provided a rigorous mathematical solution to the problem of controlling a continuous-time linear stochastic model with the control variables as exogeneous. His approach was extended for control and non-control exogeneous variables. The feedbacks are shown to be optimal in the class of linear feedbacks. The optimal level of control was estimated by minimizing the infinite horizon quadratic cost function for a second order dynamic model. If the estimated optimal feedback is applied from time zero onward, then the costs are minimised. The main advantage of Bergstrom’s method was that it could be applied to higher order systems with more realistic specifications of cost functions and the

70 estimates were still consistent. Bergstrom (1989) presented an optimal forecasting algorithm of discrete mixed data together with a theorem that demonstrates the optimality of the forecasts. They are exact Gaussian estimates of the post-sample observations conditioned by the information contained in the sample.

A final econometric aspect that Bergstrom was determined to address was the statistical testing and model evaluation of the higher order continuous-time models.

Bergstrom (1990) developed some practical procedures for testing hypotheses concerning a specific continuous-time model with a mixed data sample. A more detailed analysis of the VARMA type models satisfied by the exact discrete time models was conducted. The findings, with important practical implications, were proved in a theorem about the behaviour of the moving average coefficient matrices. Bergstrom observed that they are time variant, and he demonstrated that they converge rapidly to a limit set of three matrices that is asymptotically stable stationary. Nowman (1991) suggested for practical applications that after twelve steps the limit matrix is found to seven decimal places.

Following this result, a three-stage testing strategy was presented and its extension to an open and higher order system case was discussed. The strategy was tested for both general and restrictive hypotheses in a VARMA framework. The exact discrete models represented by the VARMA specification provided the basis for exact asymptotic tests of the specification of a continuous-time model and tests of hypothesis of a set of restrictions on the parameters.

3.2.6 Related Other Work on Gaussian Estimation and

In document Multi-Factor Dynamic Modelling and Forecasting of Interest Rates and Equity Markets (Page 84-89)