A Quarterly Econometric Model of Portfolio Choice—Part I: Specification and Estimation Problems

A Quarterly Econometric Model of Portfolio Choice- Part I: Specification and Estimation Problems*

The purpose of this study is to develop a short-run quarterly model of portfolio choice. Primarily we are interested in determining which variables influence portfolio composition of a financial intermediary and in analysing the manner in which it adjusts to changes in such variables. As portfolio changes ultimately affect the flow of funds into alternative investment forms, knowledge of the causal factors is of utmost importance for the efficient operation of monetary policy. It is hoped that the study will enable us to formulate answers to such questions as: Do portfolio decisions take account of the liquidity and profitability characteristics of different assets? Are asset decisions interdependent? And, how do financial intermediaries react to portfolio disequilibrium ? This latter question is extremely important in monetary analysis : Friedman, Tobin, and Brunner and Meltzer all appear to agree ‘that the general process by which a change in the money supply affects the real economy is a process operating through a broad range of market-determined and implicit interest rates, with the public adjusting and re-adjusting its asset holdings (including holdings of real assets) in response to changes in relative interest rates’ [28, p. 1621.

The study has been split into two parts. The first part is a rather technical discussion of the problems of specification and estimation of a portfolio choice model; the second part, which is the topic of a following paper, illustrates the application of the model to the portfolio behaviour of Australian savings banks.

Recent Contributions in Financial Model-building Recent papers by Brainard and Tobin [l] and Christ [2] emphasize

the interdependence of financial markets and suggest certain desirable properties of financial models. First, they point out that the portfolio behaviour equations of each sector are functionally dependent in that they must always satisfy the balance sheet identity, total assets equal total liabilities plus net worth. In most econometric models the balance sheet constraint is satisfied by allowing one of the assets in the portfolio to play

* A second paper, ‘A Quarterly Econometria Model of Portfolio Choice-Pert II: Portfolio Behaviour of Amtrelien Savings Banks’, will be publiahed in the next issue.

518

DEC., 19 i3 PORTFOLIO CROICE 519

a residual role.’ That is, assuming the portfolio consists of n endogenous assets and liabilities, with net worth exogenously given, then n - 1 asset and liability equations would be estimated while the nth item would be determined from the balance sheet identity. However, a serious objection can be levelled a t this approach in that the results are not invariant as to the choice of asset used as residual.’

A second desirable property of a financial model is that, for each sector, the entire list of relevant interest rates should appear in each behavioural equation. If an interest rate only appears in one equation and the residual method of satisfying the balance sheet constraint applied, then an offsetting effect of the opposite sign and magnitude is implicitly assigned to the residual asset. While such an assumption may a t times be completely justified, the practice of incorporating all interest rates in each equation, irrespective of whether they are statistically significant or not, could avoid an inadvertent assumption. It is also desirable that the total effect of an interest rate change when summed over all the assets in a sector’s portfolio should, ceteris paribus, be zero. But this property is automatically achieved in a model in which the balance sheet constraint is always binding,3 so we need only concentrate our attention on the balance sheet constraint.

Brainard and Tobin [l, p. 1051 further emphasize that the balance sheet constraint must apply whether the model is in equilibrium or not. Many financial models use a formulation, which we shall refer to through- out this study as a ‘simple stock adjustment model’, in which it is assumed that the institution has a ‘desired’ level for each asset in the portfolio. It is further assumed that in each period the institution reduces the gap between its ‘desired’ and actual asset levels by some proportion, not necessarily the same for each asset. If the model ia to be consistent, then,

For example, de Leeuw [4], Goldfeld [Q], Silber [23, Appendix C]. end Norton ef 01. [17].

2 To illustrate this fact, aasume that TI = 3 and the b a h c e sheet constraint 3 Z Y, = holds, where P, refers to the ith asset and NW net worth. Further

i-1 m u m e that Y1 = a1 + B,R, where R is the independent variable and a1 and PI am the ordinary leaet squ8ree estimators. B0C8W

/11 = 3, then Y 3 = a3 + [S 3 1.. On the other hand,

Y , = NW- (YI + Yz) = NW - (a1 + az) - (A + Bz)R

= NW - (al + a t ) - [ Z R ( Y 1 I 3 f: Y 2 ) 1 B = NW - (al + az ) + [ rJi( ’&; NW)]R.

Consequently, for N W # 0, the two expression8 for Y 3 differ, indicating that the model varies according to whether Y 3 is obtained 88 8 ‘ residd’ or from 8n estim8ted beheviourel equation.

Assume that we have a model in which the balance sheet constraint is dways binding end thmt the net worth of the institution is constant. If one interest rate should increase, thereby causing the institution to adjust the proportion of its given net worth among assets, then the fact that the balmce sheet is binding meam that the net effect of such chengea when summed across all Bssets must be zero.

520 THE ECONOMIC RECORD DEC.

ceteris p a r h , a desire to increase one asset must be matched by 8 desire to reduce one or more other assets. That is, deviatiom of desired from actual asset levels must always sum to zero.

The simple stock adjustment model as formulated by Feige [7] and incorporated in studies by de Leeuw [4], Goldfeld [9], Silber [23], and Norton et al. [17] may be criticized on the grounds that it imposes a rather simple and uniform asset adjustment mechanism whereby the long-run effect is some multiple of the initial effect. One method of obtain. ing a more flexible adjustment using the simple stock adjustment formulation ia to include the lagged value or change in one of the important liabilities in the portfolio as an explanatory variable. It is then argued that the change in the liability acts as a short-run constraint on the institution’s adjustment process. However, this procedure appears very ad hoc and is theoretically less pleasing than an alternative suggested by Brainard and Tobin. That is, the adjustment of any one asset should depend not only on its own gap between actual and desired levels but also on the deviations of other assets in the portfolio from their desired levels. This formulation will be referred to as the ‘general stock adjustment model’.

The following example will clarify the theoretical difference between the simple and general stock adjustment formulations. Assume an institution holds current deposits and two illiquid assets in its portfolio and that there is a change in relative yields which leads the institution, in the long run, to increase one illiquid asset a t the expense of the other while leaving current deposits unaltered. As the change has not altered the current deposit deviation of actual from desired levels, the simple stock adjustment model would not allow any change in current deposits while the two illiquid assets were altering. On the other hand, the general model would allow the possibility of temporarily altering the stock of demand deposits in order to facilitate the adjustment of the two illiquid assets. Thus the general model permits greater flexibility in the adjustment process.

It should be emphasized that neither Brainard and Tobin nor Christ appear to have a particular economic model in mind which yields these desirable properties. Consequently, in the following sections a model based on utility maximization and cost minimization is derived. The model overcomes the weaknesses of the ‘residual’ method of satisfying the balance sheet constraint, the entire list of relevant interest rates for each sector appears in each behavioural equation, and the adjustment of any one asset does indeed depend on the deviations from desired levels of all endogenous assets in the portfolio.

Theoretical Framework The model to be developed in the following pages may be viewed as

an approximation to the solution of an optimal control problem in which the financial intermediary is assumed to maximize expected utility when there are explicit and implicit adjustment costs associated with changes in the portfolio asset levels. Optimal adjustment paths of the portfolio assets are determined subject to the constraint that the balance sheet identity is always binding.

1973 PORTFOLIO CHOICE 521

It is convenient to approach the problem in two stages.' First, we shall derive a stationary long-run solution with corresponding long-run asset demand equations. These longrun demand equationa correspond to equations for the desired stocks of assets in the simple stock adjustment model. Second, we analyse the approach or adjustment of assets to this long-run solution. We may illustrate' this two-stage procedure in Figure 1 which corresponds to a simple two-asset example (PI and Y,) . The balance sheet constraint is wealth, W = Y , + Y,. Each combination of P, and Y , yields the institution certain expected utility which is assumed to depend on the mean and variance of the distribution of profits. The dotted line, AB, is the locus of efficient expansion points along which expected utility is maximized in the long run. However, in the short run there is reason to believe that institutions will diverge from AB and expand along a path such as CGD. Reasons for such divergence could include poorly organized secondary markets or adjustment costs involved in purchasing additional assets.

Before proceeding with the mathematical derivation of the asset equations, it may be useful briefly to relate modem utility theory to portfolio theory. This is done in a recent book by W. F. Sharpe [22, pp. 187-2011 in which he shows that, under a given set of assumptions, we may define a preference curve or utility curve of an investor by asking the person to express his preferences for a hypothetical set of securities

Yl

Wl

WO

A WO Wl Y2 FIGURE 1

4 A similar approach is adopted by Nadiri and Rosen [15] in developing a dynamic model for all input demand functions, allowing for interactions among the factors of production over time.

5Adapted from Nadiri and Rosen [15, p. 4581.

522 THE ECONOMIC RECORD DEC .

v is -chs certain wealth. The assumptions needed are: (1) The investor can identify some gamble that is neither better nor worse than each possible amount of certain wealth. (2) His preferences among portfolios are not affected if money outcomes are replaced by equally desirable gambles. (3) His feelings about each portfolio depend only on the probabilities of various outcomes. (4) His preferences are transitive. Under these conditions it can be shown that the investor will always choose the portfolio with the greatest expected utility. In the following analysis we assume that utility is a quadratic function in wealth.6

S w e 1: A Stationary Long-run Solution A very interesting approach has been suggested by Parkin in his

studies of British discount houses and commercial banks [18], [19, pp. 229-511. Assuming that individual institutions are expected utility maximizers, profits are normally distributed with mean u, and variance n: and several other simplifications, Parkin develops demand equations for each of the assets and liabilities in the portfolio over which the institution has a degree of control. This approach is pursued in deriving the stationary long-run solution.

If we assume that an investor’s utility of money function is positively sloped and concave downward and that his investment strategy is the maximization of expected utility, then, expanding the utility function by the Taylor series and dropping all terms beyond the quadratic, we find [6, p. 211 that the investor will maximize the expression (1) where H is the coeficient of risk aversion and u, and a; the mean and variance respectively of profits. A similar maximand can be derived by assuming a log-normal utility function and that profits are normally distributed with mean u, and variance n2. As Tobin emphasizes [26], either the quadratic utility function assumption or the normality of profits assumption is necessary if we desire to analyse investment decisions in terms of only two parameters, mean and variance, of the investor’s subjective probability distribution.

If we treat a liability as a negative asset, we may write an expression for actual profits, n, as (2) x = RA, where 2 is a k x 1 vector of assets and liabilities in the institution’s portfolio and 2 the corresponding 1 x k vector of yields or borrowing costs. Profits are uncertain because of variability in asset levels, asset yields, and borrowing costs. This suggests that we should decompose the asset levels and yields into an expected component and an error in forecasting term, so that

R = r + e R (4) A = a + eA,

E[U(R)] = u, - an.”,

-

- (3) -

The quadratic function may be criticized because of its limited flexibility in expressing investors’ preferences. It ah0 impliea that aa wealth increeses, the investor will react by taking lees risk and this ia contrary to our impression of risk being a normal good which should increase aa wealth incree~ea [22, pp. 199-2001.


where r is a vector of expected yields, a is a vector of expected asset levels, and eR and eA error vectors. Because of the uncertainty in deposit inflows and mortgage takedowns, most of the assets involved in this study possess a stochastic element.

Following Parkin, assume for the moment that (5 ) E(eR) = 0 (6) E(e,) = 0

the latter assumption implying the independence of asset and yield casting errors. We shall question this independence assumption in pages. Rewriting equation ( 2 ) ) we obtain

Utilizing (5)) (6) and (7)) (9)

(7) E ( e A e R ) = 0)

(8) = ( r + e R ) ( a f e A ) *

u, = E(n) = E(ra + re, + ega + eReA) = ru

fore- later

(10) = rE(e,e;)r' + u'E(e;e,)a + E(e,eke;e,)

But rE(e,e,)a = 0, and we can also show, given the independence assumption of eR and eA, that' (11) a'E(eie,e,) = rE(eAeRe,) = 0. Hence (12) uz = rE(e,e&)r' + a'E(eXe,)u + E(e,eke;e,)

where R,, = E(e,e;), nRR = E(ekeR) and 2 = E(eReXeieA). If we partition the A matrix between those assets which are in choice

set, A , , of the institution and the remaining assets, A,, which are predetermined to the institition, then we may rewrite (9) and (12) as

(13)

u', = E[(n - u,)7 + 2rE(e;e,)a + 2rE(eAeRe,) + 2u'E(eXeRe,).

= rSZAAr' + dRRRa + 2,

u, = [rl : r,] [::I ' It is not intuitively obvious from Parkin's work why such 8 relationehip holds.

A simple proof is as follows:

E [ + ( X . Y)] = Jm Im 4(X, Y) . f ( X , Y)dXdY, - m - m

where + is any function of X and Y, and f the frequency function or probability density of the two variables X and Y. If X and Y are independent random variables then

where f, and fi are the frequency functions of the two marginal distributions. Assume that 4 ( X , Y) = X 2 Y , where X = el and Y = eR. Then

E(X2Y) = I:, Im X a Y . f ( X , Y)dXdl

f(X, Y) =flW .fz(Y)1

- m

- m

= 1" X2f,(X)dX. (" Yf,(Y)dY J - m J - m

= E(X') . E( Y) = 0, beCau80 E( Y) = E(e,) = 0.

See Cramer [3, pp. 66-74].

DEC. 524

(14)

Maximizing u, - Hc2 subject to the balance sheet constraint, il(al + eAl) + i,(u, + eAa) = 0, we obtain the following Lagrangean conditions0

(16) (15) Y l - 2H(n~,,,ai + R R , R , ~ Z ) + Ail = 0, and

il(ul + eAl) + i2& + eAa) = 0. The solution to this set of equations is

I rl - 2HaR1R2u2 k] = [ Z H n f i R i -1

(17) -tl 0 1 [ i l e A ~ + '~('2 + which after block inversion yieldsg

(18) Ul = (2H)-'ch.; - a a ~ ~ ~ ~ a 2 - K(ileAl i2a2 + where the k x k matrix

and the k x 1 vector a-1 i'

R I R I 1

Equation (18) represents the longrun demand of an individual institution so that if we aggregate over all similar institutions and assume that asset level forecmting errors are zero in aggregate,1° we obtain

The sacond-order condition for a maximum is that the principal minors of the Hesaian determinant [ -27RnR i] should alternate in sign beginning poaitive. See Henderson and Quandt [ I l , pp. 271-21.

Let A = [B ' be a nonshgular partitioned matrix, and let B and D = G : H

8-1 = B-'(I + FD-'GB-') d - GB-lF both be nominguler. Then

- B25D-'1 -D-'GB-'

-1 2 H

[ The proof is given in Goldberger [S, pp. 27-81. The conditions for block inversion are mtiaiied aa B = 2HCIRIR1 and D = - i,CI;,'Rlil are nonsingular. Provided that the elements of r, are not linearly dependent random variables, then the covariance matrix is positive definite and thus nonsingular. See Cfoldberger [S, pp. 36, 871 for proof of thia argument.

lo It muat also be eeeumed that all institutions poseesa identical expectationa with respect to interest rates rl and covariance matrix of forecasting errors, C I R I f i I . See Parkin [IS, p. 4791.

525

0 3 3 =

1973 PORTFOLIO CHOICE

(21) as the stationary long-run demand functions.

a, = (2H)-'Gr; - (QnR,R, + Ki,)a,

The solution equation (21) possesses several interesting properties : The yields on all endogenous assets appear in each asset equation.

2HR1, 2HQ12 - 1 2HQ12 2HRz2 -1

- 1 - 1 0

2HRI1 2HRIl ZHR,, -1 2HRll 2HR22 2HR13 -1 2HRI3 2HRZ3 2H& -1

- 1 -1 -1 0

Thus, if deposits of th;: institution a& considered to be one of its decision variables, then the yield on deposits must appear in each asset equation. It can be shown that the matrix G is symmetrical." The matrix of interest rate coefficients is therefore symmetrical. Any row or column of G sums to zero.12 This implies that the impact of a change in the yield on any one asset must s u m t o zero across all equations and that if the yield on all assets increase5 by the same amount there will, ceteris paribus, be no change in any of the asset levels. If we pre-multiply R by i,, we obtain the sum of the elements in the K vector. It is obvious that i,R = 1, indicating that the balance sheet identity holds. For a wide range of values for S2RIRr, the vector indicates that our solution implies a diversified portfolio. The exception is when any particular row of the S2,,k1 matrix sums to unity and all other rows total zero. This diversification property is not surprising as Tobin [25] and Markowitz [14], pioneering the mean-variance approach, came to similar conclusions. We can show that the diagonal elements of the fJ matrix are non- negative, indicating that the own-interest coefficients are non- nega t i~e . '~

526 THE ECONOMIC RECORD DEC . However, it seems reasonable to argue that the implicit yield on cash

is related in some way to the variability of deposit liabilities and thus to the variability of cash levels. It also seems likely that an institution would react to an unexpected change in one of its asset yields by adjusting the level of its assets. Hence, we shall now drop the assumption that E(eAeR) = 0. As the notation becomes cumbersome, we shall simplify the analysis by assuming that all the assets of the portfolio except one are in the choice set, where the excluded asset has zero yield which is known with certainty. This avoids the problem of having to partition the asset vector. We now assume that

where a,, is non-symmetrical as there appears to be no reason why E(eA,eRI) should equal E(eAleR,). Under these assumptions we have

a = (r + eR)(a + e,,). (24) E(a) = TU + rn, and

(26) E [ A - E(.)l2 = r a ~ ~ f + + m2 + %f2& + a'RRRa + 2rE(eAeheR) + 2a'E(e;eLe;) - 2m2.

Maximizing u, - Ha2 subject to the 'rational expectations' constraint, ia + f = 0, where f is the exogenous asset, we obtain the solution

Comparing the properties of (26) with those of (21) outlined above, i t is easily shown that properties (a), (d), (e) and ( f ) also apply for (26) . However, while a, is symmetric, the product of a, and the non-symmetric matrix ( I - 2 H a R A ) is non-symmetrical. Thus, if eR and eA are not independent, Parkin's estimation procedure, whereby he constrains the matrix of interest coefficients to be symmetrical, involves a specification error. It appears then that a superior procedure would be to obtain un- constrained long-run interest rate coefficients and then test them to determine whether they are compatible with the hypothesis of symmetrical interest rate effects.


In a f ~ o t n o t e , ' ~ we show that when a matrix with zero column and row sums, such as Go, is post-multiplied by any matrix, then the individual column sums of the resultant matrix are zero. Thus the coeEcients of any interest rate sum to zero across all equations as in property (c) of the earlier results. However, the resultant row sum, except under ex- ceptional circumstances, is not equal to zero thereby invalidating the second portion of property (c). Thus, without the independence assumption, (26) implies that an equal increase in all yields may have an impact on individual asset levels. Parkin has however constrained this row sum to equal zero in his regression results, introducing the possibility of a second specification error.

To summarize, under the assumptions (5 ) , (6), (22), (23) and that institutions maximize expected utility, we derived stationary long-run demand equations of the form (29) = a10 + U l l f + a12r where AT = long-run demand for asset i; f = vector of exogenous or predetermined assets and liabilities; r = a vector of expected yields of all endogenous assets; and cuio = 0, c u l l = 1, c a i 2 = 0.

i i i

Stage 2: Adjustment to th Long-run Solution

Because assets vary in liquidity characteristics and asset transactions are costly, instantaneous adjustment of actual to the stationary long-run value of an asset is not optimal. At any moment of time there are explicit and implicit costs involved in diverging from the optimal long-run portfolio. Such costs encompass the opportunity cost of forgone earnings, and psychological and explicit costs of possessing either a too risky or too conservative portfolio. For example, suppose the stock of liquid assets falls below its desired long-run level and is considered inadequate to cover expected deposit withdrawals. Under these circumstances there would be certain psychological costs, in terms of worry for the portfolio manager, in addition to an opportunity cost as the lack of liquid aseets may prevent the firm from undertaking a lucrative investment. On the other hand, if the portfolio is too liquid there is an opportunity cost in terms of forgone earnings, indicating that there are positive costs involved in having either

l4 Aaaume Y, X, and 2 are n x n matrices and that YX = 2. We can obtain the column and rows s u m of 2 aa follows:

2 1 1 = EYIJXIJ i

2 1 1 = C.YijXji 3

= 0 ifZYIJ = 0 i

# 0 except under strong assumptiom 88 to YIJ and X,J .


excew or insufficient stocks of any asset. Feige [7] auggests that we may approximate such a cost by a quadratic cost function of the form15

(30) a, = A,[A?(t) - A,(t)]2. If these were the only costs involved, it would be advantageous for

the institution to adjust immediately to its long-run equilibrium values. However, in the short run there are certain fixed costs or inadequacies in the secondary security markets that make immediate adjustment very expensive, if not impossible. The more rapid a portfolio adjustment, the greater we would expect the cost of adjustment to be. For example, in Australia there is not a secondary market for mortgages, which means that any institution wishing to increase its stock of mortgages must inveetigate the creditworthiness of an applicant, make an advance commitment, and then wait several months for the advance commitment to be taken down. To bypass or shorten this normal procedure would involve the institution in additional explicit and implicit costs. As there are positive costs involved in buying or selling securities, Feige suggests that we approximate these costs by the quadratic function

(31) c2, = b,[A,( t ) - A,(t - 1)]2.

Equations (30) and (31) merely reflect the costs of adjustment and costa of being out of long-run equilibrium for any particular asset i. However, if net worth remains unchanged, an increase in asset i must be offset by reductions totalling an equal amount in the remaining endogenous assets. Thus, to obtain the total costa of any portfolio mix, we must sum these costs across all assets and then minimize the total cost subject to the balance sheet constraint,16

C A , ( t ) = f.

CAT@) = f,

i (32)

In addition, the long-run asset demands must be consistent so that

i (33)

Is The quadratic cost function is justified primarily on the baais of expediency, for while a quadratic function was found to be a good approximation to such costs aa hiring and lay-off costs, overtime costs, inventory costs, and machine set-up costs in investment theory, there has been little research to date on portfolio adjustment costs. For example, it is doubtful whether portfolio adjustment costs are symmetrical. The quadratic function implies that the cost of having X units too much of en asset ie equal to the cost of having X units too little. Such an assumption is questionable with respect to liquid assets, a shortage of which may c a m bankruptcy of an institution and great financial loss. See Gould [lo].

l6 It would probably be desirable to formulate the problem in terms of a dynamic

cost-minimization problem where discounted future costs, C(0) = J' c-'*C(t)dt,

are minimized subject to the balance sheet constraint. Using calculus of variations and a quadratic cost function, a general stock adjustment model, which ie identical t o that obtained in the following comparative statics analysis, is derived. However, the additional mathematical complexity of calculus of variations fails to yield any additional testable hypotheses, so the following derivation uses the simpler comparative static analysis. The dynamic formulation haa been used in studies of optimal physical investment policy where the firm is constrained by its production function. See Gould [lo]; Lucas [13].

m

529 1973 POBTFQLIO CHOICE

or combining (32) and (33),

(34) zAi ( t ) = Z A t ( t ) . s $

Minimizing total cost, C = C(C1, + C2,), subject to (34), for a t

three-asset case we h d

1 where Ed, = d, + d, + d,, and di = hn. This solution may be

rewritten in the general stock adjustment form i

3

j=1 AAi( t ) = E O i j [ A f ( t ) - Aj(t - l)],

where O,, is defined in the Appendix. Writing (36) in full, i t is easily shown that Coil = 1 and that the cross-adjustment coefficients,

Oi,i # j, are gkerally not symmetrical.’’ In the Appendix, i t is also shown that the values of the own- and

cross-adjustment coefficients, O,, and a,,, lie between zero and unity. The requirement that Oi, < 1 somewhat restricts the adjustment process and is inconsistent with empirical results [24]. On the other hand, the restriction that 0 < 0, < 1 implies that a positive gap between longrun equilibrium and actual values of any asset will have a positive impact on all the assets in the portfolio. Assets are thus complementary in the limited sense that a desire to increase any one asset has B positive effect on all others.’*

potential diffi&ltyin Parki;;’8 approach of constre&ing the inter& rate coefficienta to be symmetrical across equations. In the final estimating equation, the inter& coefficients are the result of the product, = alzG, of the long-run interest rate coefficients all and adjustment coefficienta 0. The would be symmetricaJ only if we assumed that both

10 Such a statement ie not inconeistent with the balance eheet constraint. In identity (34) we have imposed the constraint that gaps between actual and long. run equilibrium values s u m to zero across 811 8esets. Hence the positive gap hypo- thesized must be offset by a negative gap in one or more of the remaining -b.

and 8 were symmetrical.


In deriving (36) we implicitly assume that the cost of disequilibrium in any one asset is independent of the distribution of disequilibrium gaps in the remaining assets. The following example illustrates the inadequacy of this assumption. Assume a four-asset portfolio with assets 1 and 2 being liquid assets while assets 3 and 4 are highly illiquid. Let us compare two possible situations:

Situation 1 20 20 - 20 - 20 Situation 2 20 - 20 20 - 20 For illustrative purposes, assume that h, = for all i, so that situations 1 and 2 have identical opportunity costs of being out of long-run equilibrium, equal to 1600$. Situation 2 represents a portfolio disequilibrium in which a shortage of liquid asset 1 is offset by an excess of liquid asset 2. On the other hand, situation 1 represents a very serious liquidity shortage as both liquid aaaets, 1 and 2, are below their optimal long-run levels. It would appear then that situation 1 involves a higher cost of portfolio disequilibrium than situation 2, suggesting that the opportunity cost, h,, varies according to the distribution of gaps between actual and long-run equilibrium levels of the other assets in the portfolio. We therefore assume that the opportunity cost, hi in (301, consists of a constant cost, k,, and a variable cost which depends on the disequilibrium gaps in the remaining assets :

A:@) - A,( t ) A:(t) - A2(4 A:( t ) - A , @ ) - A'#)

(37)

Nnimizing total cost subject to the balance sheet constraint, we once again obtain the general stock adjustment model

where Q,, is defined in the Appendix. It is also shown in this Appendix that, for gU 2 0, the own-adjustment coefficient is positive and may exceed unity while the cross-adjustment coefficients may take either positive or negative values. Also, the cross-adjustment coefficients are generally not symmetrical and ~ O o l , = 1.

i

Problems in Estimatilzg the Portfolio Behuviour Equations

AA,(t) = x O $ [ A f ( t ) - Aj( t - l)] + ui(t), and

Adding disturbance terms to (29) and (38), we have

j (39)

(40) Substituting (40) into (39), we obtain

A f ( t ) = ajo + a l i f ( t ) + aj2r(t) + v j ( t ) .


(41) - C@jAj( t - 1) + Ui(t) + C @ P j V j ( t ) f i

= Pi0 + Pi l f ( t ) + P i 2 W - C@PjAj(t - 1) + er(t) f

where Pa = C@p,ajk, and e i ( t ) = ui(t) + COpjvj(t). From estimates of

equation (41) values of #? and 0' are obtained, while the long-run co- eficients, a, are determined from the formula,

(42 1 a = (O0)-'#?. Generalizing footnote 14, it may be shown that if a square matrix,

whose elements of each column sum to y, is post-multiplied by a vector, whose elements sum to i, then the elements of the resultant vector will s u m to G. As Po = @ao, the product of a square matrix whose columns sum to unity by a column vector which sums to zero, cPi0 = 0. Similarly

we may illustrate that cpil = 1 and each column of the P2 matrix sums

to zero. Hence, in estimating equation (41) i t is important that the regression technique used be such as to satisfy the requirements that C#?,, = 0, CBil = 1, column sums of P2 equal zero, and the column

sums of 0 equal unity. Prais and Houthakker [20, pp. 84-51 have generalized an earlier

theorem of Nicholson [16] which can be applied to estimates of any set of accounts which have to balance. Applying this theorem to the balance sheet of a financial intermediary, the theorem states that if the same equation form with an identical set of independent variables is fitted to all the endogenous assets of the portfolio, and if the form of the equation consists of the sum of an arbitrarily chosen set of variables with the exception that either one of the variables must be the total exogenous assets and liabilities of the institution in linear form or if there are several exogenous assets these may be included independently but in linear form, then the estimates of the behavioural equations will satisfy the balance sheet constraint, As (41) satisfies the conditions of the theorem, the use of ordinary least squares will satisfy all the u priori constraints on values of #? and 0'.

A second problem is raised by the interdependence of asset decisions which results in an error in forecasting any one asset being offset in the remaining assets. In other words, the error or random terms appearing in any two asset equations are correlated. This is obvious from (41) as the error term in each equation contains a term of the form F@yjv,(t). Dhrymes

[5; pp. 150-611 shows that efficient estimators under these conditions are

i j

a

i

i i

- (43) = (x'+-1x)-1x'+-1Y where + = cov(e), the covariance matrix of ei(t) = u,(t) + COijvj(t) .

However, he goes on to prove that provided the same set of variables is included in each of the asset equations, then the Aitken estimators, j ,

J


identical to the ordinary least squares estimators. Consequently, ordinary least squares are efficient estimators under these conditions. As our estimating equation (41) does indeed satisfy these conditions, we are therefore justified in our use of the ordinary least squares technique.

One final problem remains. Earlier we suggested the desirability of testing the longrun interest coefficients for symmetry. However, in OUT regression estimates of equation (41) we obtain estimates of the variancee and covariances of 0' and and not of a, the long-run coefficients. Klein [12, pp. 258-611 has suggested a method for obtaining the variance and covariance of a. Thus if gL, = fi(zl, x 2 , . . ., x"), where the set [z"] consists of all O,, and BL,, then

is an asymptotic approximation to the variance of gL,. As our regression equations merely estimate covariances within each equation, it is necessary to assume that covariances between estimates in different equations art3 equal to zero.

University of Sydney IAN SHARPE

APPENDIX In this appendix we derive some properties of the adjustment coefficients in the

general stock adjustment model. If we assume independent adjustment costs. we obtain, for a three-asset example, the solution given by (36), where

bl +- hi + 6 , hi

hi + bi =-

(h , + b d h 3 + b3) [(hi + b1)(h2 + b d + (h , + bA(& + 63) + (h2 + b2)(h3 + b 3 ) ] .

Aa h,, b, 2 0, the term in square brackets in the latter expression must be go, but 1. Let this term take its maximum value of unity, then

- 1, ao that 0 5 ell 6 1. hl b Oil = - + 2 - hi + bi hi + b i

Similarly, the cross-adjustment coefficient Q12 may be written

Following the same reasoning aa above, we find that 0 5 012 6 1. When allowance is made for dependent adjustment costs, we obtain the solution

(38) where @pi = O i l i- x~ [$1,dld2 + gl,dld31. As g,, )= 0, the term in square

brackets is positive so that 0, I 2 0 and may now exceed the value of unity. On the other hand,

1

i I

1973 533

where the term in square brackets may be positive or negative depending on the values of g 1 2 , gZ1, d, , d,, and d,. Consequently Opz may be positive or negative.

REFERENCES

[l] Brainard, W. and J. Tobin, ‘Pitfalls in Financial Model Building’, American Economic Review, Vol. LVLII, May 1968.

[2] Christ, C. F.. ‘Econometric Models of the Financial Seotor’, paper presented at the Conference of University Professors, Milwaukee, September 1969.

[3] Cramer, H., The Elementa of Probability Thewy (Wiley. New York, 1968). [4] de Leeuw, F.. ‘A Model of Financial Behavior’, in J. S. Duesenberry et al. (eds.),

T h Brooking8 Quarterly Econometric Model of the United Stolcs (Rand McNally, Chicago, 1965).

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A Quarterly Econometric Model of Portfolio Choice—Part I: Specification and Estimation Problems