ELSEVIER Journal of Economic Dynamics and Control

21 (1997) 273-295

A comparison of numerical and analytic approximate solutions to an intertemporal

consumption choice problem

John Y. Campbell*,“, Hyeng Keun Koob

a Department of Economics. Harvard University, Cambridge, MA 02138, USA

b Pohang University of Science and Technology, Kyungbuk, 790-784, Korea

(Received July 1992; final version received June 1996)

Abstract

Nystrom’s method is a popular way to obtain approximate numerical solutions to problems involving integral equations. This paper applies the method to solve numer- ically for the optimal consumption choice of an agent with recursive utility facing an exogenous loglinear market return process. The paper studies the convergence rate of numerical solutions and compares them with approximate analytical solutions proposed by Campbell (1993).

Key words: Integral equations; Intertemporal consumption choice; Nystrom’s method; Numerical quadrature JEL classification: C61

1. Introduction

This paper considers the intertemporal consumption choice problem of an economic agent facing time-varying expected returns. We assume that the agent

*Corresponding author.

The first draft of this paper circulated under the title ‘A Numerical Solution for the Consump-

tion-wealth Ratio in a Model with Nonexpected Utility Using Nystrom’s Method’. We are grateful for helpful discussions with Kerry Back, and for useful comments by the editor and two anonymous

referees. Campbell acknowledges the financial support of the National Science Foundation.

0165-1889/97/$15.00 0 1997 Elsevier Science B.V. All rights reserved

PII SO165-1889(96)00932-3

274 J. Y Campbell, H.K. Koo /Journal of Economic Dynamics and Control 21 (1997) 273-295

has the recursive objective function proposed by Epstein and Zin (1989, 1991) and Weil (1990) because, by contrast with traditional time-separable von Neumann-Morgenstern utility, this objective function allows us to distinguish the coefficient of relative risk aversion from the elasticity of intertemporal substitution. We can thus compare the roles of these two parameters in the intertemporal choice problem.

The nonlinearity of the intertemporal consumption choice problem makes it impossible to obtain an exact closed form solution when expected returns are time-varying. In response to this, Campbell (1993) has proposed using a log- linear approximation to the intertemporal budget constraint. This enables Campbell to derive approximate analytical solutions when the market return follows a homoskedastic lognormal process.

The intertemporal consumption choice problem considered here, like many problems arising in economics and finance, involves integral equations. Tauchen and Hussey (1991) have suggested using Nystrom’s method to solve these problems numerically. We apply Nystrom’s method to the problem at hand and solve for the optimal consumption level of the agent, assuming that the expected log market return in an AR(l) process. We test the accuracy of the numerical solutions using the test proposed by Den Haan and Marcet (1994), study their rate of convergence, and compare them to Campbell’s approximate loglinear solutions.

The next section of the paper reviews Nystrom’s method, while Section 3 explains the intertemporal consumption choice problem. Section 4 presents numerical solutions and compares them to Campbell’s approximate loglinear solutions.

2. Nystrom’s method

In this section we present Nystrom’s method, following the exposition in Cryer (1982). Tauchen and Hussey (1991) give a similar exposition. Nystrom’s method is designed to solve an equation of the form

(S b

V(x) = K ~(y)aQx,.dw(y)dy . >

(1) n

Here, the known function w(x) and unknown function V(x) are continuous functions on a subset 9 of the real line (9 = (a, b), (a, b], [a, b), or [a, b]) and the function k&y) is a known continuous function on 9 x 9. The numbers a and b can be negative and positive infinity, respectively. The known functional K is a continuous functional on an appropriate Banach space of continuous functions.

J.Y. Campbell, H.K. Koo /Journal of Economic Dynamics and Control 21 (1997) 273-295 275

Nystrom’s method is based on a numerical quadrature rule which approxi- mates an integral on (a,b) by a finite sum. An n-point quadrature rule is an operator Q of the form

Q(w) E i: W(xi)wi ~ i=l s

b W(x)w(x)dx, (2) (1

where W is an integrable function with respect to the measure p that satisfies p(A) = fA w(x)dx for every real Bore1 set A. The numbers xi, x2, . . , x, are known as the abscissa for the quadrature rule and wi, w2, , w, are called the weights for it. The function w(x) is called the weight function for the integral.

A very popular quadrature rule is the Gaussian quadrature rule where the abscissa are the roots of the nth orthogonal polynomial with respect to the weight function w. In the following discussion we suppose that all the moments of the weight function w(x) exist, i.e., Jf: x”w(x)dx < co for all n. This allows w(x) to be the pdf of a normal distribution or a gamma distribution, but w(x) cannot be the pdf of a r-distribution or any other distribution which has infinite moments.

The set of orthogonal polynomials consists of polynomials (P,,(x)},“=~ such that

s

b

J’,,,(x)P,(x)w(x)dx = 0 if n # m, (3) (I

and the degree of P,(x) is n. The nth orthogonal polynomial is P,(x). If w(x) > 0, it can be verified that each equation P. = 0 has n different real roots. It can also be verified that there are wi, i = 1, . . . , n, where (2.2) holds exactly for the functions V(x) = xj, j = 1, . . . ,2n - 1, i.e., the rule matches the first 2n - 1 moments of the weight function w (Chihara, 1978). This rule is called the n-point Gaussian quadrature rule for the weight function w(x).

We now return to Nystrom’s method. Using an n-point quadrature rule, it approximates Eq. (1) first by the system of n algebraic equations

V(Xi) = K f: ( V(Xj)ak(Xi,Xj)Wj . j=l >

After finding V(xi), i = 1, . . , n, the value of V at a general x is given by

(4)

(5)

216 J.Y. Campbell, H.K. Koo /Journal ofEconomic Dynamics and Control 21 (1997) 273-295

The basic idea of Nystrom’s method is to approximate the original integral equation by a system of algebraic equations (4) with the help of a numerical quadrature rule.

In the case where k(x, y) is the conditional probability distribution function of y given x, Tauchen and Hussey (1991) suggest the following modification to Nystrom’s method: instead of solving Eq. (4), we solve

V(Xj) = K i V(Xj).% Wj

(

)

j=l n I >

where

n

S,(X) = C k(X,Xj)Wj. j

(6)

(7)

This approach also modifies Eq. (5) by replacing k(x, xj) with k(x, xj)/sn(x). If we define tij by

t,. ~ k(xi9xj) V ~ w.

sn(xi) ” (8)

then we have

j$l tij = 1 for all i = 1, . . . ,n; (9)

therefore, tij can be regarded as transition probabilities of the states

x1,x2, ..*, x,. In our application of Nystrom’s method we follow this approach.

3. An intertemporal consumption choice problem

In this section we present our consumption-investment problem for an economic agent with recursive intertemporal utility. Campbell (1993) poses this problem and derives a loglinear approximate solution. Here we show how to solve the problem using Nystrom’s method.

An agent has a recursive objective function of the following form:

u, = ((1 - p)C: -lja + P(E,&--;)‘I -lla)/(l-y)}l/(l-l/u). UO)

Here C, is the agent’s consumption during time period t, o is the agent’s elasticity of intertemporal substitution, and y is the agent’s coefficient of relative risk

J.Y. Campbell, H.K. Koo /Journal of Economic Dynamics and Cont~oi 2J (1997) 273-295 277

aversion. E, denotes the conditional expectation taken at the end of time period t. This obejctive function has been proposed by Epstein and Zin (1989, 1991) and Weil(l990) as a generalization of time-separable power utility. Power utility is the special case of (10) where cr = l/y.

We do not try to solve the agent’s full portfolio and consumption choice problem. Rather, we take the agent’s portfolio at each time as given and assume a time-series process for the return on this ‘market’ portfolio (the name arises from the case where the agent is a representative agent for the economy as a whole). We solve for the optimal allocation of the agent’s wealth between consumption and investment.

We assume that the log market return I,,,+ 1 = log R,,,, 1 is given by

rm,t+1 = m(x*) + E1.t+1, Xt+1 = 4% + K + -%.t+1> (11)

where m(x) is a real-valued continuous function defined on the real line, s1 and s2 are random variables with a zero mean, and IC is a constant. Eq. (11) implies that the unconditional mean of the observable variable xt is ~/(l - 4). The expected market return E, T,,,~+ 1 is a function of the observable variable xr,

Et rm,t+l =4x,). (12)

The random variable sl is the unexpected return, and a2 denotes the innova- tion to the observable variable x,. We allow these shocks to be correlated with each other but assume that they are serially uncorrelated and stationary with variances 1, 1 and Czz and covariance C12. Eq. (11) says that the variable which determines the expected log market return follows a univariate normal AR(l) process, and, in particular, if E, I,, f + 1 = x,, then the expected log market return itself follows a univariate normal AR(l) process (Campbell, 1991; Fama and French, 1988; Poterba and Summers, 1988). We can express the unexpected return s1 as follows:

El*t+l = &2,r+1) + %+1, (13)

where u is a random variable independent of s2 with a zero mean, and b( .) is the optimal forecast of .sl,t+l given E~,~+~. We assume that b( .) is a continuous function. If Ed and sz are jointly normally distributed, then b(x) = bl * x, where b1 is the regression coefficient in the theoretical linear regression of s1 on s2. Eq. (13) will be used later to compute expectations.

Eq. (10) has been normalized so that the agent’s indirect utility function is homogeneous of degree one. Therefore we can solve the problem by finding optimal consumption-wealth ratios and the indirect value function for unit wealth, i.e., the agent’s maximized utility when his or her wealth is one. The single state variable for the problem is x, which can take a continuum of values

218 J. Y. Campbell, H.K. Koo / Journal of Economic Dynamics and Control 21 (1997) 273-295

and is normally distributed. We simplify notation by defining x 3 x,, y E x,+ 1, and R, E R,,,.,. These notational conventions allow us to drop time subscripts from the problem. We also denote the unit wealth indirect value function by V(x) and the consumption-wealth ratio by a(x). Then our problem can be rewritten as

V(x) = max [(l - /+(x)‘-‘/~ 06a(x)Ql

+ p(1 _a(x))‘-‘/~(E[T/(y)‘-YR~-YIX]}(1-l/~)/(l-Y)]1/(1-l/~).

By the first-order condition the optimal consumption-wealth as follows:

(14)

ratio a(x) is given

A u(x) = -

l+A where A = u{~~~(y)l-~R~-YI~])(l--a)l(l-y). (15)

Therefore, the optimal consumption-wealth ratio is equal to 1 - p if cr = 1. By Eqs. (ll), (12), and (13) we know that

s m

V(Y) 1 y (l-y)[m(x)+b(~-d~-~)lE~(l-~)”

- e f(YlX) = - w(Y)dY.

-OX W(Y) (16)

Here f(y 1 x) denotes the pdf of next period’s state variable conditional on a current value of x and u is the random variable in (13). If .sl and s2 are jointly normally distributed, then u is a normal random variable and Ee(’ -y)” = ei” -Y)‘zUU, where Z:,, is the variance of u. Substituting Eqs. (15) and (16) into Eq. (14), we can delete the max operator in the equation and obtain an expression that is in the same form as (1).

We can now apply Nystrom’s method to our problem. The first task is to choose an appropriate weight function w(x).l Tauchen and Hussey (1991) suggest using either the unconditional pdff(x) or the conditional pdff(x 1 m) evaluated at the unconditional mean m of the state variable x. In numerical work for the case where .sl and s2 are normally distributed and m(x) = x, we have found that the choice between these two densities makes little difference; we report results based on the unconditional pdf.

’ One essential property of a weight function is that the function f(y (x)/w(x) has a tail which does not grow too fast. We require it to be O(eklxl) for some positive k when x tends to infinity. Tauchen and Hussey (1991) discuss the choice of a weight function in greater detail.

J. Y. Campbell, H.K. Koo / Journal of Economic Dynamics and Control 21 (1997) 273-295 279

Taking either the unconditional pdff(x) or the conditional pdff(x 1 m) as the

weight function, the next step is to use the n-point Gaussian quadrature rule for

the integral in (16). For many probability distribution functions including the pdf of a normal distribution, the abscissa x1, . , x, and weights wr, . , w, can be found in handbooks of mathematical functions. (For the normal distribution, these appear under the heading of Gauss-Hermite rules.) Using the n-point

quadrature rule, Eq. (14) can be approximated by the following system of n algebraic equations:

V(Xi) =

[

(1 - p)U(Xi)l - lia + CB( 1 - U(Xi))’ - l”

~~(l-~)[“(“i)+b(xj-~x;-~)l ftxj I xi) w 1/e l/(1 -l/a)

f txj) J I 1 (17)

for i= l,..., n. Here C is a constant

C = (Ee”-Y’“}l/e, (18)

which is equal to ef(l-Y)(l-l/~)~uu when u is a normal random variable, 0 = (1 - y)/( 1 - 1 /o) following the notation in Giovannini and Weil(1989), and U(xi) is given by

A a(xJ = -

l+A’ (19)

where

J=c-” I j$l v(x_i) - 1 ~e(l-y)[“(Xi)+b(xj-dx~-~)l f(xj I-&) w, (1-@/(1-y)

Stxj) J ’

The value function at other values of x can be found as in (5) and is given as follows:

V(x) = [

(1 - p)u(x)‘-“~ + /?C(l - u(x))‘_“6

x ~~(1 -v)b(x)+b(xj-$x-@l . (20)

280 J Y. Campbell, H.K. Koo /Journal of Economic Dynamics and Control 21 (1997) 273-295

In our application we follow Tauchen and Hussey’s (1991) suggested variation of Nystrom’s method and include s,(x) as explained in the last paragraph in Section 2; therefore we modify (20) as

V(x) = [

(1 - #@a(x)‘-“” + CP(l - a(x))‘-“C

x 1 j$l v(xj) - 1 ~~(1 -~)bW+t&-9x-W f txj Ix) 1/e l/(1-1/0)

S,(X)W(Xj) wj II ,

(21)

where a(x) is as in (19) and s,(x) is as in (7).

4. Numerical results

In this section we report numerical solutions, obtained using Nystrom’s method, for the consumption choice problem defined by Eqs. (10) and (11). We study the case where s1 and s2 are jointly normally distributed and m(x) = x. The parameters of the problem are the utility parameters /3, y, and 0, and the market return parameters K, 4, Zll, Zzz, and

4.1. Parameterization

c 12.

The first task is to pick reasonable values for these parameters. Campbell (1991) estimates persistence measures for expected returns using CRSP monthly data on real value-weighted U.S. stock returns over the period 1927-1988. The persistence measures are estimated using a VAR system, rather than the simple AR(l) model (11). However it is straightforward to calculate the AR(l) para- meter that would correspond to any VAR persistence measure. Campbell’s estimates imply 4 = 0.79 over the full 1927-1988 sample period, and a similar value of 4 = 0.83 in the postwar 1952-1988 sample period. Thus 4 = 0.8 is a realistic benchmark value to use in the AR(l) model. This value of 4 implies a half-life for expected return shocks of just over three months.

Having chosen a value for 4 we can pick K to match the unconditional mean real return on the value-weighted stock index, as (11) implies that Er,,,,,+l = K/(I - d). W e c h oose K to give an unconditional mean stock return of 6.5% at an annual rate.

Given 4, we can also find the innovation variance of expected returns by using the fact that the variance of expected returns var (E, I,,,+ 1) = C,,/( 1 - 4’). The variance of expected returns is given by the variance of the fitted value in a regression of returns on the relevant information, or equivalently by the

J.Y. Campbell, H.K. Koo /Journal ofEconomic Dynamics and Control 21 (1997) 273-295 281

overall variance of returns and the R* statistic from the regression. Campbell’s (1991) estimates imply that C,, = (0.0563)* and ,IZ2 = (0.0063)’ in the 1927-1988 sample period. The implied standard deviation for monthly expected

returns when 4 = 0.8 is G/,/m = 0.0063/0.6, just over 1% at a monthly rate. This compares to an overall standard deviation of the monthly ex post return of 5.7%, implying a monthly R* statistic ofjust over 3%. Finally, the correlation between innovations in expected returns and realized returns is estimated to be about -0.8, and we pick this value for the example here. These estimates imply substantial variability in expected returns, and also considerable mean reversion in the market return process.

The remaining parameters of the model are p, the time discount factor, y, the coefficient of relative risk aversion, and 0, the elasticity of intertemporal substi- tution. We set /? equal to the twelfth root of 0.94, implying a time discount rate of 6% at an annual rate, and consider various values for y and (T. We first set y = 0 and c = 2, then set y = 2 and g = 0.5, and finally present some results for a wide range of y and 0 values.

4.2. Convergence of Nystrom’s method

Our first exercise is to study the rate of convergence of Nystrom’s method solutions using n grid points with n = 2,3, . . . , 10,12. We use the notation Xl . ..x.(xr <x2 < ... < x,) for the abscissas of the n-point rule. We calculated value functions and consumption-wealth ratios for all points in the union of the abscissas for the 2,3, . . . , 10,12-point Gaussian quadrature rules. For each n-point rule, we obtained the value function and the consumption-wealth ratio according to (21). For each n = 2,3,. . _ , 10, we computed the maximum absolute error over [x1,x,] between the solution obtained from the n-point rule and the solution obtained from the 1Zpoint rule.2

Our numerical results for the case y = 0 and cr = 2 show that convergence errors shrink rapidly as the number of grid points increases. The percentage convergence error for the value function declines from 24.8% with n = 2 to 11.6% with n = 3, 5.85% with n = 4, 3.11% with n = 5, 1.63% with n = 6, 0.80% with n = 7, 0.36% with n = 8, 0.17% with n = 9, and 0.091% with n = 10. The corresponding percentage convergence errors for the log consump- tion-wealth ratio are 33.0%, 12.3%, 5.49%, 2.69%, 1.33%, 0.62%, 0.27%, 0.14%, and 0.075%.

‘The reason we use this metric for error calculation is because we expect the numerical solution

obtained from the n-point rule to be fairly accurate within the range [x1,x.]. Convergence in this

metric is stronger than uniform convergence on every bounded interval.

282 J. Y. Campbell, H.K. Koo 1 Journal of Economic Dynamics and Control 21 (1997) 273-295

=: 6 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08

Demeaned Expected Market Return

Fig. 1. Optimal consumption policy.

This figure plots alternative solutions for the optimal percentage monthly consumption-wealth ratio as a function of the conditional expected percentage monthly return on the market portfolio. The market return process is calibrated to monthly U.S. data in the manner described in the text, while the utility parameters are u = 0.5 and y = 2. The consumption-wealth ratio is plotted over a range from 5.50 standard deviations below to 5.50 standard deviations above the unconditional mean of the market return, corresponding to the range from the lowest to the highest abscissa of a 12-point Nystrom solution. The solid line is the 1Zpoint Nystrom solution, the dotted lines are 3-, 5-, 7-, and 9-point Nystrom solutions, and the dashed line is the approximate loglinear solution (22) with the parameter p taken from (24).

We repeated these calculations for the case where CJ = 0.5 and y = 2. The percentage convergence error for the value function declines from 24.9% with n = 2 to 11.7% with n = 3, 5.89% with n = 4, 3.13% with n = 5, 1.65% with n = 6,0.82% with n = 7,0.37% with n = 8,0.17% with n = 9, and 0.089% with n = 10. The corresponding percentage convergence errors for the log consump- tion-wealth ratio are 13.0%, 5.74%, 2.81%, 1.47%, 0.76%, 0.37%, 0.17%, 0.079%, and 0.040%.

A visual impression of these results is given in Fig. 1. For c = 0.5 and y = 2, the figure plots the optimal consumption-wealth ratio as a function of the deviation of the expected market return from its mean. Both the consump- tion-wealth ratio and the expected return are expressed in percentage points at a monthly rate, since the model has been calibrated to monthly data. The solid line is the Nystrom’s solution from a 12-point rule, while the dotted lines are the Nystrom’s solutions from 3-point, 5-point, 7-point, and g-point rules. These lines are plotted over the range from the smallest abscissa to the largest abscissa of the 12-point rule, which are 5.50 standard deviations below the unconditional

J.Y. Campbell, H.K. Koo 1 Journal of Economic D.ynamics and Control 21 (1997) 273-295 283

mean and 5.50 standard deviations above the unconditional mean, respectively. For these parameter values it is essential to go beyond a 5-point rule, but the 7-point rule, the 9-point rule, and 12-point rule give very similar optimal consumption policies.

4.3. Comparison with approximate analytical solutions

We now compare numerical solutions obtained from the 9-point version of Nystrom’s method with the approximate loglinear solutions derived in Campbell (1993). We have already seen that the 9-point rule gives almost exactly the same solution as the 12-point rule; below we also show that the 9-point rule passes the Den Haan-Marcet test for solution accuracy (Den Haan and Marcet, 1994). Therefore, a comparison of approximate loglinear solutions with the solutions from the 9-point rule enables us to judge the accuracy and usefulness of the approximate loglinear solutions.

Campbell’s (1993) approximate solution for the log consumption-wealth ratio is

c _w _(l-)pE r

fN 1 _p4 f m.t+l+

(1 - a)p% f

1-P

+ P(k - P(m) l-p .

(22)

Here p = 1 - exp(c - w), where c - w is the mean log consumption-wealth ratio, k = log(p) + (1 - p) log(1 - p)/p, and

p,,, = ologfi + $(l - a)2

=alog/?++(l-0)2

A detailed discussion of (22) is given in Campbell (1993), but the simple intuition is that an increase in the expected market return raises consumption when c < 1 (since then the income effect dominates the substitution effect) and lowers consumption when cr > 1 (since then the substitution effect dominates). When (r = 1, we have the well-known result that the consumption-wealth ratio is a constant. The magnitude of the change in consumption resulting from a change in the expected return depends not only on c but also on the persistence of the expected return process 4. A more persistent change has a bigger effect on consumption.

The consumption growth rate corresponding to (22) is

(1 - dP

284 J.Y. Campbell, H.K. Koo 1 Journal of Economic Dynamics and Control 21 (1997) 273-295

Note that negative correlation Zr2 between E~,~+ 1 and sZ,*+ 1 makes consump- tion smoother if D < 1. This is because negative correlation between the shocks creates mean reversion in market returns, so that the income effect of a positive return (which dominates consumption behavior when c < 1) is less than one- for-one.

To implement (22) and (23) in practice one must pick a value for p. One approach is to solve the model numerically, find the unconditional mean of the numerical log consumption-wealth ratio, and calculate the implied value of p. This gives an accurate approximation but of course cannot be implemented unless the model has already been solved numerically.

A second approach is to take unconditional expectations of (22), substitute in the definitions of k and ,u,,,, and solve the resulting equation for p.3 If we use the approximation log p = p - 1, we obtain the following cubic equation:

a3p3+a2p2+alp+a~=o, (24)

where

(1 - a)ic

ao=m- 1 -alog/Lj(l -a)2 ; err,

0

ur = 1 + 24 - 24(1 - b)K

I_4 +2&logB+W-a)2 t Zll

0

u2 = (1 - 4d’~ l-4

- 4’ - 24 - C#&JlogP - &#?(l - a)2 ; cir 0

- f(1 - a)2 0

; z22 + 40 - aI2 ; z12,

0

a3 = I$‘.

A third approach uses the fact that when 0 = 1, then p = j?, the utility discount factor. One can set p to this value even for B greater or less than one, as a rough approximation.

3 We are grateful to a referee for suggesting this approach.

J. Y. Campbell, H.K. Koo /Journal ofEconomic Dynamics and Control 21 (1997) 273-295 285

In Tables 1 through 4, which are similar to those in Campbell (1993) we compare these approximate solutions with Nystrom solutions calculated using a 9-point rule. Table 1 first shows the mean of the Nystrom value function for a wealth level of 100 (that is, the mean value function as a percentage of wealth), calculated numerically for a grid of six values of 0 and six values of y. g is set equal to 0.05,0.50, 1.00, 2.00, 3.00, and 4.00, while y is 0.00,0.50, 1.00, 2.00, 3.00, and 4.00. The smallest value of D is 0.05 rather than 0.00 because the program encounters some numerical difficulties when Q is extremely small.

Table 1 confirms the usual economic intuition. The mean value function falls when relative risk aversion increases, because a risk-averse agent is reluctant to make risky investments and hence investment opportunities are less valuable on average. On the other hand the value function increases with the elasticity of intertemporal substitution, because an agent who is willing to substitute inter- temporally can take advantage of time-varying investment opportunities.

Table 1 also shows the percentage losses in the mean value function that result from using the loglinear consumption rule (22). Since the objective function has been normalized to be linear in wealth, these losses have the same units as reductions in wealth. The first number in parentheses is the percentage utility loss when p is calculated using the mean of the Nystrom log consump- tion-wealth ratio. The second number in parentheses is the percentage utility loss when p is calculated using Eq. (24). The third number in parentheses is the percentage utility loss when p is set equal to j3, its value for the 0 = 1 case.

As one would expect, the utility losses from using loglinear consumption rules increase as c moves further from unity. The losses are zero when 0 = 1, for then the approximations are exact. When cr = 0.5 or 2.00, the approximate consump- tion functions cost no more than 0.01% of wealth unless y is at its highest value of 4.00. When 0 = 0.05, 3.00, or 4.00, the approximate consumption functions with the Nystrom value of p or with p calculated from Eq. (24) still cost no more than 0.01% of wealth, but the approximate consumption function setting p = /l is noticeably less accurate. As an extreme case, when both 0 and y are at their maximum values of 4.00, this approximation costs 1.2% of wealth.

Table 2 shows 9-point Nystrom solutions for mean optimal consumption as a percentage of wealth. Since fl and the expected return process are fixed, the mean consumption-wealth ratio varies considerably as the parameters y and 0 vary. The smallest value in the table is 0.237% (2.84% at an annual rate) when 0 = 4.00 and y = 0.00. The largest value is 1.05% (12.6% at an annual rate) when 0 = 4.00 and y = 4.00. There are no monotonic relationships between the parameters y and (T and the mean consumption-wealth ratio; an increase in rr, for example, increases the mean consumption-wealth ratio when y is large but decreases it when y is small.

Table 2 also reports the differences between the means of the Nystrom and loglinear consumption-wealth ratios, expressed as percentages of the mean Nystrom consumption-wealth ratio. These differences are again zero when

286 J. Y. Campbell, H.K. Koo /Journal of Economic Dynamics and Control .?I (1997) 273-295

Table 1 Mean value function with percentage losses from approximate consumption rules

rJ

Y 0.05 0.50 1.00 2.00 3.00 4.00

0.00 0.608 0.612 0.617 0.629 0.646 0.670

(0.002) (0.000) (0.000) (0.000) (0.001) (0.003) (0.002) (0.000) (0.000) (0.000) (0.001) (0.003) (0.238) (0.002) (0.000) (0.010) (0.137) (0.724)

0.50 0.574 0.576 0.578 0.583 0.589 0.596

(0.002) (0.000) (0.000) (0.000) (0.001) (0.003) (0.002) (0.000) (0.000) (0.000) (0.001) (0.003) (0.048) (0.000) (0.000) (0.002) (0.023) (0.102)

1.00 0.540 0.541 0.542 0.543 0.544 0.546

(0.002) (0.000) (0.000) (O.OW (0.001) (0.002) (0.002) (O.ooo) (0.000) (0.000) (0.001) (0.002) (0.006) (0.000) (0.000) (0.000) (0.002) (0.008)

2.00 0.473

(0.004)

(0.004) (0.021)

3.00 0.407

(0.007) (0.007) (0.867)

0.474

(0.000)

(0.000) (0.000)

0.412

(0.000) (0.000) (0.004)

0.475

(0.000)

(O.OW (0.000)

0.417

(0.000) (0.000) (0.000)

0.477

(0.000)

(0.000)

(0.000)

0.426

(0.000)

(0.000) (0.010)

0.479

(0.001) (0.001) (0.004)

0.433

(0.000)

(0.000) (0.085)

0.48 1

(0.002)

(0.002)

(0.016)

0.438

(0.002)

(0.002)

(0.268)

4.00 0.341 0.354 0.366 0.384 0.397 0.407

(0.010) (0.000) (0.000) (0.000) (0.000) (0.001) (0.010) (0.000) (0.000) (0.000) (0.000) (0.001)

(-) (0.029) (0.000) (0.055) (0.426) (1.21)

The first number in each block is the mean of the 9-point Nystrom numerical solution for the value function, expressed as a percentage of wealth, of an agent facing the return process (11) with m(x) = x. The remaining numbers, in parentheses, are losses from using approximate consumption rules, expressed as percentages of the mean Nystrom value function. The second number in each block is the loss from using the loglinear approximate consumption rule (22) with p calculated from the Nystrom solution, the third number is the loss from using (22) with p calculated from (24). and the fourth number is the loss from using (22) with p = /I.

’ Calculation of loss from approximate consumption rule failed to converge.

J. Y. Campbell. H.K. Km / Journal of Economic Dynamics and Control 21 (1997) 273-?95 287

Table 2 Mean optimal consumption-wealth ratio with percentage errors of approximate consumption rules

> 0.05 0.50 1.00 2.00 3.00 4.00

0.00 0.603 0.561 0.514 0.421 0.329 0.237

(0.090) (0.017) (0.000) (0.170) (0.635) (1.43) (0.091) (0.017) (0.000) (0.171) (0.640) (1.45) (1.40) (0.396) (0.000) (2.01) (9.55) (27.9)

0.50 0.571 0.544 0.514 0.455 0.396 0.336

(0.103) (0.024) (0.000) (0.153) (0.583) (1.30) (0.103) (0.024) (0.000) (0.154) (0.585) (1.31) (0.659) (0.183) (0.000) (0.868) (3.76) (9.45)

1 .oo 0.539 0.527 0.514 0.488 0.463 0.437

(0.117) (0.032) (0.000) (0.134) (0.535) (1.21) (0.117) (0.032) (O.fW (0.134) (0.536) (1.21) (0.228) (0.063) (0.000) (0.265) (1.08) (2.49)

2.00 0.475 0.494 0.514 0.556 0.597 0.639

(0.151) (0.049) (0.000) (0.103) (0.470) (1.10) (0.150) (0.049) (0.000) (0.103) (0.470) (1.10) (0.439) (0.127) (0.000) (0.393) (1.59) (3.54)

3.00 0.412 0.460 0.514 0.623 0.733 0.843

(0.195) (0.068) (0.000) (0.078) (0.427) (1.04) (0.193) (0.068) (O.OOQ (0.078) (0.429) (1.04) (2.44) (0.642) (0.000) (1.97) (7.42) (15.9)

4.00 0.348 0.427 0.514 0.691 0.869 1.05

(0.254) (0.091) (O.ooo) (0.059) (0.397) (0.991) (0.252) (0.090) (O.ooo) (0.059) (0.401) (1.01) (6.95) (1.68) (0.000) (4.77) (17.5) (37.6)

The first number in each block is the mean of the 9-point Nystrom numerical solution for the optimal consumption-wealth ratio, expressed as a percentage of wealth, of an agent facing the return process (11) with m(x) = x. The remaining numbers, in parentheses, are the differences between mean consumption-wealth ratios when approximate consumption rules are used and the mean Nystrom consumption-wealth ratio, expressed as percentages of the mean Nystrom consump- tion-wealth ratio. The second number corresponds to the loglinear consumption rule (22) with the Nystrom value of p, the third number corresponds to (22) with p taken from (24), and the fourth number corresponds to (22) with p = fi.

288 J.Y. Campbell, H.K. Koo /Journal of Economic Dynamics and Control 21 (1997) 273-295

(T = 1, but they increase quite rapidly as u moves away from one. When the approximation uses the Nystrom value of p or calculates p from Eq. (24) the error is 0.05% to 0.25% of the exact mean consumption-wealth ratio when 0 = 0.05 or 2.00, increasing to almost 1.5% of the exact mean consump- tion-wealth ratio when cr = 4.00. When the approximation uses p = /I, the errors can be many times larger.

If one is interested in the dynamic behavior of consumption and its implica- tions for asset pricing, errors in the mean consumption-wealth ratio are rela- tively unimportant. What is important is to approximate accurately the vari- ation of the consumption-wealth ratio and the consumption growth rate around their means. Table 3 gives the standard deviation of the 9-point Nys- trom solution for the optimal log consumption-wealth ratio, along with the percentage errors of the three approximations. The log ratio is used because this is approximately normally distributed. The table shows that as rr approaches one, the approximation error for the log standard deviation goes to zero at almost the same rate as the log standard deviation itself. Thus in percentage terms the approximation error is almost constant and small at about 0.75% of the true standard deviation. Table 4 reports the standard deviation of the 9-point Nystrom solution for the optimal log consumption growth rate, again with the percentage errors of the three approximations. The consumption- smoothing effect of mean reversion when 0 is low is clearly visible in this table. The approximations tend to understate the variability of log consumption growth when 0 is low and to overstate the variability when d is high. However, the errors are quite small, never much above 0.5% of the true standard deviation when p is obtained from the Nystrom solution or from Eq. (24). The approxima- tion setting p = p should be used more cautiously, as its maximum error is four times greater than the approximation error using the true p.

Fig. 1 gives a visual impression of these results for the case u = 0.5 and y = 2. The dashed line in Fig. 1 shows the approximate loglinear consumption-wealth ratio where p is taken from Eq. (24). This line is visually indistinguishable from the lines representing the 9-point and 1Zpoint numerical solutions.

These results are encouraging for the acuracy of the loglinear approximation. It is particularly noteworthy that one can use Eq. (24) to solve for p and sacrifice almost no accuraccy relative to an approximation that uses the Nystrom value for p. Furthermore, these results are obtained using a model in which the expected return on the market is highly variable. Less extreme variability in the expected return process would increase approximation accuracy by reducing the variability of the optimal consumption-wealth ratio. Also, a number of authors (Campbell and Mankiw, 1989; Giovannini and Weil, 1989; Hall, 1988; Kandel and Stambaugh, 1991) have argued from direct evidence on consumption and asset price behavior that 0 is more likely to be very small than very large. Tables 3 and 4 show that the loglinear approximation to the optimal consumption function is highly accurate when cr is close to zero.

J.Y. Campbell, H.K. Koo / Journal of Economic Dynamics and Control 21 (1997) 273-295 289

Table 3 Standard deviation of optimal log consumption-wealth ratio with percentage errors of approximate consumption rules

Y 0.05 0.50 1.00 2.00 3.00 4.00

0.00 0.048 0.025

(0.760) (0.756) (0.749) (0.751) (1.20) (0.986)

0.50 0.048 0.025

(0.759) (0.755)

(0.748) (0.750)

(1.04) (0.902)

0.000 0.051 0.103 0.155

(-) (0.750) (0.749) (0.748)

(-) (0.755) (0.756) (0.756)

(-) (0.287) (-0.178) ( - 0.640)

0.000 0.05 1 0.102 0.154

(-) (0.752) (0.755) (0.760)

(-) (0.758) (0.763) (0.767)

(-) (0.454) (0.156) (-0.141)

1.00 0.048 0.025 0.000 0.051 0.102 0.153

(0.758) (0.754) (-) (0.754) (0.760) (0.771) (0.748) (0.750) (-) (0.758) (0.763) (0.767) (0.876) (0.818) (-) (0.622) (0.492) (0.361)

2.00 0.048 0.025 0.000 0.05 1 0.101 0.151

(0.754) (0.753) (-) (0.757) (0.768) (0.788) (0.746) (0.749) (-) (0.759) (0.765) (0.770) (0.558) (0.650) (-) (0.958) (1.16) (1.37)

3.00 0.049

(0.750) (0.743) (0.239)

4.00 0.049

(0.746)

(0.739)

( - 0.078)

0.026

(0.751) (0.747) (0.482)

0.026

(0.749) (0.746) (0.314)

0.000 0.051

(-) (0.759)

(-) (0.760)

(-) (1.30)

0.000 0.050

(-) (0.760)

(-) (0.759)

(-) (1.63)

0.101

(0.773) (0.764) (1.84)

0.100

(0.775) (0.759) (2.52)

0.150

(0.798) (0.765) (2.39)

0.148

(0.802) (0.750) (3.42)

The first number in each block is the standard deviation of the 9-point Nystrom numerical solution for the optimal log consumption-wealth ratio of an agent facing the return process (11) with m(x) = x. The remaining numbers, in parentheses, are the differences between the standard devi- ations of log consumption-wealth ratios when approximate consumption rules are used and the standard deviation of the Nystrom log consumption-wealth ratio, expressed as percentages of the standard deviation of the Nystrom log consumption-wealth ratio. The second number corresponds to the loglinear consumption rule (22) with the Nystrom p, the third number corresponds to (22) with p taken from (24), and the fourth number corresponds to (22) with p = 8.

290 J.Y. Campbell, H.K. Koo /Journal of Economic Dynamics and Control 21 (1997) 273-295

Table 4 Standard deviation of optimal log consumption growth rate with percentage errors of approximate consumption rules

0.00 0.037 0.045 0.057 0.085 0.115 0.147

( - 0.25 1) (-0.175) (0.000) (0.260) (0.401) (0.482) ( - 0.247) (-0.173) (0.000) (0.261) (0.404) (0.488) ( - 0.394) ( - 0.228) (0.000) (0.100) ( - 0.088) ( - 0.392)

0.50 0.037 0.045 0.057 0.085 0.115 0.146

( - 0.251) (-0.175) (0.000) (0.260) (0.403) (0.490) ( - 0.247) (-0.174) (0.000) (0.262) (0.406) (0.491) ( - 0.341) ( - 0.209) (0.000) (0.158) (0.087) ( - 0.081)

1.00 0.037 0.045 0.057 0.085 0.115 0.146

( - 0.250) (-0.175) (0.000) (0.260) (0.405) (0.496) ( - 0.247) (-0.174) (0.000) (0.262) (0.407) (0.494) ( - 0.289) (-0.189) (0.000) (0.215) (0.263) (0.235)

2.00 0.037

( - 0.249) ( - 0.246) (-0.184)

3.00 0.037

( - 0.247) ( - 0.245) ( - 0.080)

0.045

(-0.175) (-0.174) (-0.151)

0.045

(-0.174) (-0.174) (-0.112)

0.057

(0.000) (0.000) (0.000)

0.057

(0.000)

(0.000) (0.000)

(0.261) (0.262) (0.331)

0.114

(0.408) (0.407) (0.620)

(0.261) (0.261) (0.447)

0.114

(0.410) (0.405) (0.988)

0.145

(0.506) (0.494) (0.882)

0.144

(0.511) (0.490) (1.56)

4.00 0.037 0.045 0.057 0.085 0.114 0.143

( - 0.246) (-0.174) (0.000) (0.261) (0.410) (0.512) ( - 0.243) (-0.173) (O.ooo) (0.261) (0.401) (0.479)

(0.025) ( - 0.074) (0.000) (0.565) (1.37) (2.29)

The first number in each block is the standard deviation of the g-point Nystrom numerical solution for the optimal log consumption growth rate of an agent facing the return process (11) with m(x) = x. The remaining numbers, in parentheses, are the differences between the standard deviations of log consumption growth rates when approximate consumption rules are used and the standard deviation of the Nystrom log consumption growth rate, expressed as a percentage of the standard deviation of the Nystrom log consumption growth rate. The second number corresponds to the loglinear consumption rule (22) with the Nystrom p, the third number corresponds to (22) with p taken from (24), and the fourth number corresponds to (22) with p = /I.

J.Y. Campbell, H.K. Koo J Journal of Economic Dynamics and Control 21 (1997) 273-295 291

4.4. The Den Haan-Marcet test for solution accuracy

Our final exercise is to evaluate the accuracy of the Nystrom solutions and approximate loglinear solutions by conducting the statistical test proposed by Den Haan and Marcet (1994).4

The Den Haan-Marcet test exploits the implication of Eq. (15) that

Fat

>

(1 -y)/(l -0)

(1 - BY(l - 4 = E,[I’,:-,YR;,:,].

It follows from (25) that the random variable v,+i,

(1 -d/(1 -0)

vt+1 = ( Pat (1 - PYU - 4) >

_ I/‘-YR’-Y t+1 m.r+1 >

(25)

(26)

is unpredictable with information available at time r. In order to test this orthogonality, we choose a vector of three instrumental

variables h(x,) = (l,x,,x:), where the last instrument is chosen to test whether there is any nonlinear dependence of v f+ 1 on past information. We simulate time series of x, and v, from simulations of a, and V, using one of the rules whose accuracy we want to test. The Den Haan-Marcet test statistic is T&.4; ‘BT, where

(27)

and (i$) and (Xz) are simulated time series of vt and x,. Den Haan and Marcet (1994, Proposition 1) have shown that this statistic converges in probability to a x2(3) distribution under the null hypothesis that the solution is exact.

We first used the Den Haan-Marcet test to evaluate the convergence rate of the Nystrom numerical solutions, using the parameter values y = 0, 0 = 2 and y = 2, 0 = 0.5. Table 5 shows the percentage of 1000 simulations in which the Den Haan-Marcet test statistic lies in the upper and lower 5% tails of the x’(3) distribution. The length T of the time series is 1200 months (= 100 years). The table shows that 2-point, 3-point, and 4-point Nystrom rules are not at all accurate, 5-point rules are moderately accurate, and 6-point, 7-point, 8-point, 9-point, lo-point, and 12-point rules are highly accurate.

We next used the Den Haan-Marcet test to evaluate the accuracy of Campbell’s (1993) approximate analytical solutions, using the same wide range

4 We thank a referee for suggesting this test.

292 J Y Campbell, H.K. Koo /Journal of Economic Dynamics and Control 21 (1997) 273-295

Table 5 Percentage of Den Haan-Marcet statistics in the upper and lower 5% critical tails for various Nystrom rules

n y=o,a=2 y = 2, 0 = 0.5

2 100.0 (0.0)

3 80.8 (0.0)

4 19.4 (1.4)

5 8.1 (4.8)

6 5.6 (5.5)

7 5.3 (5.3)

8 5.2 (4.9)

9 5.0 (5.2)

10 5.2 (5.1)

12 5.1 (5.0)

99.8 (0.0)

82.7 (0.0)

21.6 (1.0)

8.2 (4.6)

6.0 (5.7)

5.1 (5.3)

5.4 (5.3)

5.2 (4.7)

5.2 (4.7)

5.4 (4.7)

The first number in each pair is the percentage of the Den Haan-Marcet (1994) test statistics in 1000 simulations of 1200-observa- tion time series that exceed the 95th percentile of the x2(3) distribution. The second number in each pair, in parentheses, is the percentage of the Den Haan-Marcet statistics that do not exceed the 5th percentile of the x2(3) distribution. These percentages are calculated for n-point Nystrom solutions with n = 2 through 10 and 12.

of parameter values reported in Tables 1-4.5 Tables 6 and 7 report the percent- age of 1000 simulations in which the Den Haan-Marcet statistic lies in the upper and lower 5% tails, respectively, of the x’(3) distribution. Once again the length T of the time series is 1200 months (= 100 years). The percentage for the 9-point Nystrom solution is reported as a benchmark, followed by the percent- age for the approximate solution using the Nystrom value of p, the percentage for the approximate solution with p calculated from Eq. (24), and the percentage for the approximate solution using p = /I?.

The Den Haan-Marcet test shows that the Nystrom 9-point solution is quite accurate across all parameter values, which justifies our use of this solution as a benchmark in Tables l-4. The test also shows that the loglinear solution with p = p is highly inaccurate; the other loglinear solutions perform better but are not as accurate as the Nystrom solution. The accuracy of the loglinear solutions

’ We do not calculate test statistics for e = 1, because all the model solution procedures are perfectly accurate in this case.

J.Y. Campbell, H.K. Koo 1 Journal ofEconomic Dynamics and Control 21 (1997) 273-295 293

Table 6 Percentage of Den Haan-Marcet statistics in the upper 5% critical tail

Y 0.05 0.50 1.00 2.00 3.00 4.00

0.00 5.2 4.1 - 3.8 4.3 6.1

(10.5) (5.7) (-) (24.8) (68.3) (97.0) (10.5) (5.7) (-) (24.2) (69.0) (97.8)

(100.0) (100.0) (-) (100.0) (100.0) (100.0)

0.50 6.0 4.2 4.4 4.8 4.3

(11.9) (5.9) (-) (18.7) (61.5) (92.8) (11.8) (5.9) (-) (18.8) (62.0) (92.9)

(100.0) (83.2) (-) (100.0) (100.0) (100.0)

1.00 5.3 4.6 4.5 4.7 5.1

(14.1) (8.3) (-) (16.5) (54.3) (90.2) (14.1) (8.3) (-) (16.6) (54.3) (89.8) (48.8) (15.5) (-) (54.4) (98.8) (100.0)

2.00 3.8 5.3 3.7 5.1 4.8

(19.7) (11.2) (-) (10.6) (41.0) (82.3) (19.5) (11.1) (-) (10.6) (41.0) (82.3) (98.2) (52.6) (-) (90.3) (100.0) (100.0)

3.00 5.8 6.2 5.9 4.6 4.9

(32.3) (12.9) (-) (10.9) (36.5) (77.0) (32.7) (12.8) (-) (10.9) (36.8) (77.3)

(100.0) (100.0) (-) (100.0) (100.0) ( 100.0)

4.00 5.3 4.1 4.9 4.2 5.1

(57.9) (25.5) (-) (8.1) (30.9) (74.6) (56.5) (25.3) (-) (8.1) (31.3) (75.8)

(100.0) (100.0) (-) (100.0) (100.0) (100.0)

All numbers reported are the percentage of the Den Haan-Marcet (1994) test statistics in 1000 simulations of 1200-observation time series that exceed the 95th percentile of the x2(3) distribution. The first number in each block is the percentage for a 9-point Nystrom solution, the second number is the percentage for the loglinear approximate consumption rule (22) with p calculated from the Nystrom solution, the third number is the percentage for (22) with p calculated from (24), and the fourth number is the percentage for (22) with p = p. No percentages are calculated for the case where tr = 1, because in this case all the loglinear approximate solutions are identical and exact.

294 J. Y. Campbell, H.K. Koo 1 Journal of Economic Dynamics and Control 21 (1997) 273-295

Table 7 Percentage of Den Haan-Marcet statistics in the lower 5% critical tail

u

Y 0.05 0.50 1.00 2.00 3.00 4.00

0.00 5.3 5.5 - 5.0 4.5 5.8

(2.8) (6.2) (-) (1.5) (0.3) (0.0) (2.9) (6.4) (-) (1.3) (0.2) (0.0) (0.0) (0.0) (-) (0.0) (0.0) (0.0)

0.50 4.8 5.0 - 4.4 3.7 4.4

(2.9) (5.6) (-) (1.8) (0.1) (0.0) (2.8) (5.5) (-) (1.8) (0.1) (0.0) (0.0) (0.1) (-) (0.0) (0.0) (0.0)

1.00 3.2 4.6 - 4.1 4.0 5.5

(3.3) (3.5) (-) (2.5) (0.8) (0.0) (3.3) (3.5) (-) (2.5) (0.8) (0.0) (0.3) (2.5) (-) (0.1) (0.0) (0.0)

2.00 5.3 4.9 - 5.2 6.4 5.9

(1.7) (2.1) (-) (3.2) (0.6) (0.0) (1.7) (2.1) (-) (3.2) (0.6) (0.0) (0.0) (0.0) (-) (0.0) (0.0) (0.0)

3.00 4.5 4.8 5.3 5.3 5.2

(0.9) (2.4) (-) (3.5) (0.3) (0.0) (1.0) (2.4) (-) (3.5) (0.3) (0.0) (0.0) (0.0) (-) (0.0) (0.0) (0.0)

4.00 4.8 5.7 - 4.0 4.4 5.3

(0.0) (0.8) (-) (2.3) (1.0) (0.2) (0.0) (0.9) (-) (2.3) (1.0) (0.2) (0.0) (0.0) (-) (0.0) (0.0) (0.0)

All numbers reported are the percentage of the Den Haan-Marcet (1994) test statistics in 1000 simulations of 1200-observation time series that do not exceed the 5th percentile of the x2(3) distribution. The first number in each block is the percentage for a 9-point Nystrom solution, the second number is the percentage for the loglinear approximate consumption rule (22) with p cal- culated from the Nystrom solution, the third number is the percentage for (22) with p calculated from (24), and the fourth number is the percentage for (22) with p = p. No percentages are calculated for the case where u = 1, because in this case all the loglinear approximate solutions are identical and exact.

J. Y Campbell, H.K. Koo / Journal qf Economic Dynamics and Control 21 (1997) 273-295 295

deteriorates as one moves away from c = 1, particularly when G < 1, 7 > 1 or

C7 > 1, y < 1.

We have already shown in Table 1 that the loglinear solutions give almost the same expected utility as the Nystrom solution, and in Tables 2-4 that some important moments of the loglinear solutions with the Nystrom value of p or

with p calculated from Eq. (24) are very close to the moments of the Nystrom solution. Thus the results in Tables 6 and 7 reveal the ability of the Den Haan-Marcet test to detect small failures of solution accuracy. Den Haan and Marcet (1994) apply their test to a monetary equilibrium model of the macro- economy and conclude that the test is ‘very powerful, in the sense that it detects

small inaccuracies’ (p. 13). Our results support their conclusion.

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