Instrument choice

Economics Letters 6 (1980) 37-42 North-Holland Publishing Company

INSTRUMENT CHOICE Empirical Implications and Analysis Michael Depurtment Received

E. BURNS

and Edgar J. WILSON

of Economics 5 August

Mmush

Untoersity,

Cluyton,

Vict.. 3168, Austruhu

1980

A neglected consequence of instrument choice is that standard regression approaches are invalid in macroeconomic research. Analysis accommodating policy objective and reaction functions yields non-technical solutions to this problem. Exploratory empirical analysis undertaken has implications for monetary targetting strategies.

1. Introduction Consider a conventional macroeconomic modelling framework in which there are (n - 1) basic independent relationships linking the values of (n + m) variables, (y,, . . . ,y,) and (x,, _. . ,xm). Excluding accounting identities these relationships will not be exact so that the structural form of the system may be given in vector notation by YT+Xp+e.

(1)

The instrument choice dilemma is characterised by assuming that the central authority may control y, leaving y, to be determined endogenously within the system, or vice versa. All other y, are always current and endogenous while X are exogenous (or lagger dependent) variables. The system (1) may be ‘solved’ for any (n - 1) elements of Y in terms of X and the remaining element of Y. One such set of ‘solution’ equations would be y, = “, y, +

XY, +

0,)

(24

if& 37

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h4.E. Burns, E.J. Wilson / Instrument

choice

which contains

Here y, is a vector and the stochastic element oi are linear combinations of the structural errors e, weighted by structural parameters in I. Where y, is fixed by the central authorities (2a) are the true reduced form equations relevant to that case. Substitutions of (2b) into (2a) yields

which are the true reduced form equations where y, is controlled. Unless (2b) is exact the choice of instrument has a differential effect upon the unexplained variability of the endogenous variables, as indicated by the variance expressions VdYilY,)

Var(YilY2)=V4Vi),

-

varb, - qf-+/q).

@a), (3b)

Since any pair v, and v, will in general be correlated the expansion of (3b) will contain a covariance term that may be either positive or negative. The control of y, as against y, may thus reduce the unexplained variability of some y, and increase that of others. Instrument choice should therefore not be based upon stability analysis of a single final target variable. The formal choice of instrument and selection of an instruments value would be derived through maximisation of a policy-makers objective function U= U(E(Y),

var Y),

(4)

subject to the structural constraints imposed by either (2a) or (2~). Such maximisation would yield instrument settings (reaction functions) Yl =rm

or

Y2 =YzV)

(54t(5b)

depending upon whether y, or y, was chosen to be controlled. Instrument choice and setting would thus depend upon slope coefficients and variances in (2a) and (2~). Since these are linear combinations of structural parameters instrument choice could also be determined (much less straight forwardly) from these structural properties.

M.E. Burns. E.J. Wilson / Insm.ment

2. An alternative conceptualisation

choice

39

of the problem

Leaving aside the information contained in the ‘solution’ equation relating y, and y,, (2b), the system contains (n - 2) linearly independent elements of information. This residual information could be used to derive Y, =

S,,Y, +&Y,

+

xx + CL,

if

12,

(6)

where yi are vectors different from y, and the stochastic elements 6, differ from 0,. For given X, (4) and (5) together enable the policy-maker to ranks all possible outcomes in ( y,, y2) space on an expected utility basis. In fig. 1 are shown possible configurations of indifference curves that conform to such rankings, denoted I, to I,. Also shown by the locus AB is the ‘solution’ eq. (2b), the broken lines bounding AB characterising the (range of) uncertainty in this relation as incorporated in the stochastic element u, . An informal but general demonstration of the effects of instrument choice upon expected utility is obtained by noting that setting y, = y; would result in variations of y, between y; and y;‘, and of expected utility between I, and Z4_ However, setting y, =yz leads to variations of y, between y; and y;’ and of expected utility between Z, and I,. Fig. 1 also yields insights into the following complex notions: the overall expected utility associated with setting y2 = y; is the sum of the expected utilities

Fig. I.

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M.E. Bums, E.J. Wilson / Instrument

chmce

along CD weighted according to the probability density function of v,; the optimal setting of that instrument will be the value of y, that maximises such a weighted sum; this need not be y,*, but even if it was it does not follow that y: would be the optimal control value for that alternative instrument; it is unrealistic to assume that policy will be conducted in accordance with the implied complex decision rules. More realistically, policy-makers might operate a more straightforward twostage approach, equivalent to identifying G( y,, yz) in fig. 1 and choosing the policy instrument by a comparison of the variability associated with setting y, =yT as against y, =y2*. Such a strategy has a role in later considerations. 3. The empirical problem and a solution The slope coefficients and variances of the policy-relevant relations in (2) and (3) will not be known with certainty so attention will be focussed on econometric estimation of the parameters in question. Assuming instrument choice has historically been varied across decision periods, y, will be correlated with the error terms v (and e) in some periods while y, will be so correlated in the remaining periods. The empirical implications are compounded by two factors. First, the econometric data intervals may well differ from decision period intervals so that y, and y, may both have been endogenous during a given data interval. Second, data on historical patterns of instrument choice are unlikely to be available. Thus, in any applied study neither y, nor y, may be treated as exogenous variables over an entire sample period. Where this has been done (for example, by treating an interest rate term or the domestic component of the monetary base as exogenous in regression analysis) biased and inconsistent estimates of slope coefficients and variances will have been generated. In particular these variance terms (fundamental to policy analysis) are likely to have been significantly underestimated. While the development of an optimal statistical procedure would involve technicalities beyond the scope of non-specialist econometricians, applied economists may adopt a simpler strategy. This involves identifying two variables Z, and Z, which will be correlated with y, and y2 respectively but uncorrelated with the error terms. If variables can be found with these properties (holding independently of the choice of policy instrument), the use of both Z, and Z, within a standard instrumental variables approach will generate consistent estimates of the relevant slope and variance parameters.

M.E. Burns, E.J. Wilson / Instrumeni

choice

41

An obvious approach would be to regress y, and y2 on all exogenous (and suitable lagged endogenous) variables specified in the model outlined above, making Z, and Z, the estimated values of the variables so obtained. In fact, the completion of the above model by the inclusion of a policy-makers objective function makes such an approach far less ad hoc than it might at first seem. Here Z, and Z, would relate to the reaction functions (5a,b), any linear approximation involved being less important if policy had been conducted in a simple manner such as outlined above-as if related to some ‘optimal’ point G on the (estimated) locus of AB shown in fig. 1. Providing the policy objectives and the structural form have remained (reasonably) stable over a sample period and also, that the policy-makers estimates of model behaviour are sufficiently accurate that their predicted outcomes of Y are correlated with actual outcomes, than Z, and Z, may be expected to be correlated with yi and y, respectively and independently of whether y, or y, is chosen to be controlled.

4. Exploratory empirical analysis using the suggested technique Investigations were carried using a small-scale macroeconomic model appropriate to a small open economy, and using Australian quarterly data from 1965(l) to 1976(4). The model incorporated two identities (national income, change in monetary base), ten behavioural relations (consumption, investment, exports, imports, money demand, money supply, monetary equilibrium, change in foreign component of base, expectation augmented Phillips relation and price expectations formulation), a relation between disposable income and income, and finally a definition of the price inflation variable. This structure thus contained fourteen relations linking the fifteen conventionally denoted variables

and the exogenous variables in the model. Short-term interest rates and the domestic component of the monetary base were treated as alternative instruments, equivalent to y, and y2 above. All relations were given conventional forms although complex lag structures were not incorporated. Thus, while traditional partial adjustment mechanisms were adopted, only the expectations formation mechanism incorporated variables lagged more than once. To complete the system a policy-makers

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objective function was included and assumed to depend upon current and lagged values of inflation, unemployment, real output, external balance and outstanding government debt. Amongst a range of results obtained were estimates for those relations corresponding to (2a) and (2~) above. The results predicted that, compared to domestic base control the use of an interest rate as a policy instrument would increase the unexplained variability of Ml by 12% and M3, by 3% but reduce that of real income by 4.7%, inflation by 8.7% and unemployment by 13.6%. Estimated relations satisfied standard test criteria. These results, although exploratory and indicating only small variability effects, have provocative implications regarding the current popularity of monetary targetting strategies.To our knowledge they are the only investigations of these kinds of relationship that have explicitly aimed at consistency of slope and variance estimates. Other investigations undertaken confirm the expectation that results obtained in the manner suggested above yield parameter estimates quite different from those derived by conventional but inappropriate econometric methods. More detailed discussion of the material contained in this note, including the complete model and a wider selection of empirical results are given in the full-length manuscript available from the authors at Monash IJniversity.