177

Economics Letters 20 (1986) 177-181 North-Holland

FINITE STATE MARKOV-CHAIN APPROXIMATIONS UNIVARIATE AND VECTOR AUTOREGRESSIONS

TO

George TAUCHEN Duke Uniuersrtv, Durham, NC 2 7706, USA Received

9 August

1985

The paper develops a procedure for finding a discrete-valued Markov chain whose sample paths approximate well those of a finance, and econometrics where vector autoregression. The procedure has applications in those areas of economics, approximate solutions to integral equations are required.

1. Introduction There is interest in economics, finance, and econometrics in the solutions to functional equations where the arguments of the solution functions are the values of an autoregressive process. A typical problem is to characterize the price of an asset, where the law of motion for the dividend is a logarithmic AR(l) process. For example, with an additively separable intertemporal utility function the functional equation for the asset price p is

(1) where u’(h) is the marginal utility of consumption, h is the dividend, p is the subjective rate of discount, and F( h’ 1h) is the conditional distribution of the dividend. The law of motion for the state variable y = log(h) is y’ = Xy + C, where c is white noise. Under regularity conditions (1) admits a unique solution for the asset price as a function p(v) of the log of the dividend. Such pricing functions are studied by Lucas (1978), Brock (1982) and others. Generally the solutions to (1) are not available in an analytically simple closed form. Instead, the solutions are given as the limit of a sequence of computationally difficult iterations motivated by contraction mapping theorems. However, there are instances where calculation of the exact solution or a good approximation to it - is important. For example, Prescott and Mehra (1985) examine the quantitative aspects of asset pricing in their study of the puzzle of ‘high’ equity returns, while Donaldson and Mehra (1984) study the qualitative features of multivariate asset pricing functions. The strategy adopted in these papers and in other work is to use a finite-state discrete Markov chain for the state variables and to restrict the number of possible values of the state variables to be very small, usually only two or three. In the discrete case the problem of solving the functional equation (1) becomes the simpler problem of just inverting a matrix. If the range space of state variables is small, then one can find ad hoc, though presumably ‘realistic’ numerical values for the transition probabilities and the state variables. However, the 0165-1765/86/$3.50

0 1986, Elsevier Science Publishers

B.V. (North-Holland)

178

G. Tauchen / Mm-km-chain

approximations

difficulty with using a small state space is that one is never sure that unusual and interesting characteristics of the solution are not simply artifacts of the coarseness of the state space. This suggests using finer state spaces. Of course, the size of the matrix to be inverted will become much larger, but with the large-scale computational resources that are, or will soon be available [National Science Foundation (1984)], the inversion of very large matrices is practicable. This paper develops a method for choosing values for the state variables and the transition probabilities so that the resulting finite-state Markov chain mimics closely an underlying continuousvalued autoregression. The motivation for the method is the well-known fact that the statistical properties of many economic time series are captured adequately by vector autoregressions, after an adjustment for trend. Thus it is possible to use the method to calculate explicitly the solutions to functional equations like (1) using dynamics for the state variables that are close to those actually observed in the economy. In fact, as the state space becomes finer and finer the solution to the functional equation will, under regularity conditions, become arbitrarily close to the solution for the continuous case. This is the Kantorovich approach [Wouk (1979, pp. 120&142)] to solving functional equations. Below, evidence is presented indicating that the approximation error from the method should be small for moderate sized state spaces. Also, the method works well in an ongoing project [Tauchen (1985)] that investigates the small sample properties of generalized method of moments estimator [Hansen (1982)].

2. The scalar case Let Y, be generated Y, =

by the autoregressive

scheme,

(2)

AY,-I + et,

where E, is a white noise process with variance 0,‘. Let the distribution function of cr be Pr[c, < U] = F(u/u,), where F is a cumulative distribution with unit variance. Let jjl denote the discrete-valued process that approximates the continuous-valued process (2) and let y’
y”-hyJ+ww/2 i

0,

_F i

yk-hy’-w/2 0,

i

i?

otherwise, y’ - xy’ + w/2 P,I =F

0,

and

P,~ =

1- F

-N - xyi - w/2 Y 0, !

The rationale for this assignment of the transition probabilities can be understood by considering a random variable of the form v = AjjJ + c where XYJ is fixed and c is distributed as c,. Then the assignments (3) for p/k make the distribution of jjr conditional on jj,,-, =y’ be a discrete approxima-

G. Tauchen

/ Markou-chain

179

approxmations

Table 1 Case

number

(1) (2) (3) (4)

Number of grid points N

Continuous x

9 9 9 5

0.10 0.80 0.90 0.90

Discrete process

process 0,

x

oi

0.101 0.167 0.229 0.229

0.100 0.798 0.898 0.932

0.103 0.176 0.253 0.291

of the real line formed by the Jk tion to the distribution of the random variable U. If the partitioning closely in is reasonably fine, then the conditional distribution of jr given j,_, = J’ will approximate the sense of weak convergence that of y, given y,-, = XJJ. It is recognized that other integration rules, e.g., Gaussian quadrature, may lead to a placement of the grid points that in principle is more efficient than the equispaced scheme outlined above. The advantages of the above scheme, however, are computational speed and numerical stability, especially in the multivariate case given below. A rule based on Gaussian quadrature would essentially use the method of moments to determine the grid points and the transition probability matrix. This would necessarily entail the inversion of very large Vandermonde matrices, a problem which is notoriously time consuming and numerically unstable. The above scheme, on the other hand, can be coded very easily and the approximating Markov chains have been found to be quickly computable for a large number of sets of parameter values for the underlying autoregression. To assess the adequacy of the approximation (3) note that the discrete process j, admits a representation of the form, j, -xj,-, = Z,, with cov(c,, jr;-,)= 0, where x = COV(~~~;, jr_l)/var(jt), and oc2= (1 -x2) var(j,). The parameters x and uz are functions of the second moments of the j( process, and these moments can be computed from the transition matrix and the { 7’). Table 1 shows the induced population statistics x and yF for a range of values for X and N with of = 0.1. The transition probabilities were computed under a normality assumption for E, and using the value m = 3 for determining the grid width. The approximation of x and oF to h and Us is clearly adequate for most purposes when N = 9. Experimentation showed that the quality of the approximation remains good except when X is very close to unity. Monte Carlo studies showed that fitting standard linear time series models to the discrete-valued approximating process j, gives results very similar to what one would expect if the models had been fitted to realizations of the continuous-valued process y, itself. Generating 51 pseudo observations on .i; for the parameters in case no. (2) in table 1 and then fitting linear autoregressive models to these data gives J = 0.03 + 0.78 j,pl, (0.02) (0.09) _F~= 0.03

+ 0.84 j(;,, -

(0.02) (0.15)

0.08 j,)r2,

s 2 = 0.0095,

NOBS=

50,

s2=0.0096,

NOBS=49,

(44

(0.14)

where standard errors are in parentheses. These regressions are about what one would expect to get if the continuous-valued process yt had been simulated. Interestingly, Kolmorgov-Smirnov tests accept normality of the residuals, indicating that the distance in the K-S metric between the error distribution and the normal distribution is not large.

180

G. Tauchen / Markoo-chain appromnatmts

3. The vector case

Suppose Y, =A.Y-,

the vector process is

+

var(c,)

er3

a diagonal

=Z,,

matrix,

(5)

where y, is an M X 1 vector, A is an M X M matrix, and E, is an M x 1 vector white noise process. It is assumed that the elements E,, of et are mutually independent with distribution Pr[e,, < u] = F,(u/a(c,)), where 6 is a standardized distribution function. After taking appropriate linear combinations any VAR model can be put in the form (5). Assume that after taking such linear combinations the elements of C, are also independent. Let jj, denote the approximating discrete-valued vector Markov chain for y, in (5). Each component J,:, takes on one of N, values: y,r CT,* < . . < J,Ns.As in the univariate case, J,r and y,NJ are set to minus and plus a small integer m times the unconditional standard deviation of y,,. The remaining J,’ satisfy J,‘+ ’ = J, :’ + w,, I= 1, 2 ,..., N, - 1, where w, = 2ma(y,)/(N, - 1). The a(~,) are the square roots of the diagonal elements of the matrix ZY that satisfies Z’,. = AZ,.A’ + Z:,, which can be found by iterating Z.,(Y) = AZ.,(r - 1)A’ + .Z:,, with convergence as r + 00 guaranteed so long as (5) is stationary. There are N* = N, . N, . . . . .NM possible states for the system. Enumerate these states using the index j=l, 2,..., N* as a label for the states. Let i(j) be an M x 1 vector of integers associated with state j such that when the system is in state j at time I- 1 the components J,,,_, assume the i=l,2 ,..., M. values J,:,t_, =J,p, wherep=i,(j),for be the M x 1 vector of values for the J’s when To calculate transition probabilities pJk let J(j) the system is in state j, and put p = AJ( j). For each i let h,( j, I) = Pr[J,, =y/ 1 state j at t - 11 for 1 < I < N,, which, analogously to (3) is taken to be h,(j,

I)=~(~~-~,+w,/2)-F;(_$-~,-w,/2)

if

2
-!%+w,/2)

if

/=l,

= 1 - E; ( y,N, - p, - w/2)

if

I= N,.

=F;W

(6)

Given the h’s the transition probabilities p( j, k) = Pr[in state k ]in state j] are, by independence the C’S, the products of the appropriate h’s,

k)=[filh,(j,ii(k))

p(j,

Using this method Y,, = 0.70~,.,-,

forj,

a discrete

= Y

0.332 [ 0.126

0.126 0.185

,...,

approximation

+ 0.30~2,,+1 + 61~~

where e,, and c2( are iid normal this y, process is Z

k=1,2

N*.

was taken to the process

y2, = 0.20~,.,-,

(0, 0.1) random

+ 0.50~~~1

variables.

+

(7)

~2~.

The unconditional

covariance

matrix

of

(8)

1’

The N, were each set to nine,

of

yielding

81 states for j,:- The values

7: were computed

with m = 3

standard deviations. As in the univariate case it is possible to check the accuracy of the approximation by finding the induced representation ,Ff= xj,_, + ?,, where A= (E[_~,~,‘._,])(E[~,i;,,~,‘~,])- ‘, with the expectations computed using the transition matrix. In this case,

0.200

0.299 0.499

1 =_, = ’



0.373 0.139

0.139 0.200

1

(9)

The elements of Aare very close to the corresponding autoregressive parameters in (7). though those of ,Z? are not quite as close to those of 2, in (8). Experimentation showed that increasing moderately the number of grid points N, often improved the accuracy of Zi. Generating 51 observations on this 3, and fitting a VAR. to’ these data gives .F,, = 0.11 + 0.61 j,,,_, (0.06) (0.11)

+ 0.34 A,+,, (0.17)

s2 = 0.15,

NOBS=

50,

.P12,= 0.07 + 0.16 j,,,p, (0.05) (0.08)

+ 0.44 j2,,_,, (0.13)

s2=0.09, NOBS=

50.

Comparing the estimates here with (7)-(9) shows that the discrete the statistical properties underlying a first-order vector process.

Markov

chain

imitates

quite well

4.Conclusion This paper has developed a method for finding a discrete Markov chain that approximates in the sense of weak convergence a continuous-valued univariate or multivariate autoregression. The method should be useful in both economics and finance where discrete state spaces are used for finding numerical solutions to integral equations.

References Brock, W., 1982. Asset prices in a production economy, in: J. McCall, ed., The economics of tnformation and uncertainty (The University of Chicago Press, Chicago, IL). Donaldson, J.B. and R. Mehra, 1984, Comparative dynamics of an intertemporal asset prtcing model. Review of Economic Studies 51. 491-508. Hansen. Lars P., 1982, Generalized method of moments estimators, Econometrica 50. 102991055. Lucas, R.E.. Jr., 1978, Asset prices in an exchange economy, Econometrica 46, 142991445. Michener. R.. 1984. Permanent income in general equilibrium, Journal of Monetary Economics 13, 279-305. National Science Foundation, 1984. Access to supercomputers (NSF Office of Scientific Computtng, Washington. DC). Prescott, E. and R. Mehra. 1985, The equity premium: A puzzle, Journal of Monetary Economics 15, 145-162. Tauchen, George, 1985, Statistical properties of generalized method of moments estimators of structured parameters using financial market data. Working paper (Duke University, Durham, NC). Wouk, A.. 1979, A first course in functional analysis (Wiley, New York).

FINITE STATE MARKOV-CHAIN APPROXIMATIONS ...

discount, and F( h' 1 h) is the conditional distribution of the dividend. The law of motion for the ... unique solution for the asset price as a function p(v) of the log of the dividend. ... If the range space of state variables is small, then one can find ad hoc, though presumably .... Z:,, with convergence as r + 00 guaranteed so long as.

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