RDP 8601: New Classical Models and Unobserved Aggregates 2. The Model

Although there is no one model that is representative of all variants of the New Classical theory, the central ideas can be captured in a very simple formulation of the demand and supply sides of the economy. In particular, I assume that the supply of aggregate output is determined by a log-linear (Lucas) supply function,

where is the log of the (proportionate) deviation of current aggregate supply from its natural rate, P_t is the log of the nominal price of a bundle of goods, _t−1P_t = E_t−1P_t is the subjective expectation held by producers of (the log of) this price (based on information available at the end of period t−1) and is a random disturbance to supply which is independently and identically distributed .

The hypothesis of rational expectations is an important component of all New Classical models. This hypothesis amounts to assuming that the subjective expectations operator _t−1(.) = E_t−1(.) is equal to the true statistical expectation, conditional on the information set I_t−1, that is

Specific assumptions about I_t−1 will be addressed later.

The demand for aggregate output is assumed to be proportional to the real money stock via the Quantity Theory equation,

where is the log of the (proportionate) deviation of current aggregate demand from its natural rate, M_t is the log of the true, unobservable, monetary aggregate and is a random velocity shock to demand which is assumed to be an , independently and identically distributed variate.^[3]

The specification of activity in the economy is closed by assuming that prices adjust so that all markets clear each period,

where y_t is the observed log of the (proportionate) deviation of real output from its natural rate.

On the nominal side of the model I distinguish between two aggregate money supply indexes. In addition to the unobservable true aggregate, M_t, there is , (the log of) an observable aggregate that measures the true one imperfectly.^[4] Both these aggregates are influenced by the “monetary base”, which the authorities are assumed to be able to control directly. B_t is (the log of) this observable variable. I make these distinctions for two reasons. First, in the New Classical spirit, the authorities cannot be assumed to have an informational advantage over private agents by controlling the true (unobservable) aggregate. Second, because of the central role of monetary information in this model, I do not wish to allow the authorities direct control over the measured aggregate without any control over the true one.

For what follows it will be convenient to define the following identities,

where B_t is (the log of) the monetary base, and z_t and are respectively the true and measured “multipliers”.

While this is a convenient terminology, and one that I shall use throughout the discussion, it should not necessarily be given a behavioral interpretation. In particular, there is no presumption of simple sum aggregation, or that the controlled variable, B_t, is necessarily narrower in definition than either of the two aggregates, M_t and .^[5]

The monetary authorities are assumed to control the monetary base (B_t) via some general reaction function,

where η_t is an independently and identically distributed term.^[6]

The true multiplier is assumed to be determined by the reduced form,

where ν_t is independently and identically distributed as a variate.^[7]

Contemporaneous variables have been explicitly excluded from the right-hand side of equations (6) and (7). This prevents the authorities from having any informational advantage over private agents. It also reflects the usual New Classical assumption that the monetary aggregate (that is important for private behavior) is independent of other contemporaneous variables.^[8] Lagged variables are included because New Classical models almost always assume that the money supply process is dynamic – the simplest process would have θ = φ ≠ 0 and the other parameters set to zero.

The measured multiplier is assumed to be related to the true one by a generalised errors-in-variables equation,

where μ_t is an independently and identically distributed error. This specification of the measurement error, − M_t, allows for both the conceptual error in the definition of the measured aggregate and the statistical errors inherent in its measurement. It is best thought of as a convenient parameterisation of a number of general possibilities, that is also analytically tractable.^[9]

In empirical applications one may wish to use less general parameterisations of equations (6), (7) and (8). For current purposes, it is important to demonstrate how such restrictions affect the results.

I assume that the information set, I_t−1, includes knowledge of the structure of the model (equations (1) through (8)), the values of the parameter set, and the sufficient statistics for the joint distribution of the stochastic disturbance terms. In addition, it is assumed to contain:

where <.> indicates sequences of observations in the interval (−∞,t−1]. It will not contain any variables dated at t and later. That is, I assume that agents can directly observe all past values of aggregate output, prices, the monetary base and the measured money supply. From this they can deduce past values of the measured money multiplier and two of the stochastic shocks. They cannot observe any past values of the true money supply, hence I_t−1, will not contain:

Notice that all observable variables (including the measured money supply) are revealed at the end of the period to which they pertain, as in most New Classical models. There is no notion of preliminary monetary information that is available contemporaneously as in Barro and Hercowitz (1980), Boschen and Grossman (1982) or King (1981). The analysis here will focus on permanent measurement errors and the role of anticipated/unanticipated money and not the temporary errors and perceived/unperceived money with which these authors are concerned. This other body of literature analyses models where the New Classical information set has been expanded to include contemporaneous monetary data. The current paper considers the effects of removing information (lagged values of the true money supply) from agents' information sets.

Substituting for supply and demand (from (1) and (2)) into the equilibrium condition (3) yields

from which

can be deduced by taking E_t−1(.) of both sides. Hence, by substituting back for _t−1P_t, we get

and therefore,

In order to solve these equations, an expression for the conditional expectation of the true money supply (_t−1M_t) is required. In standard New Classical models all lagged variables are elements of the information set, so a simple application of the conditional expectations operator to both sides of equations (6) and (7) would yield the solution. This solution would have the property that the agents' expectational error, and hence output, are white noise disturbance terms. However, in the current model, no lagged values of the true monetary aggregate (or the true multiplier) are in the information set. Thus expressions for the conditional expectations of these unobserved lagged variables (_t−1M_t−1 and _t−1z_t−1) are required before the solution of (9) and (10) can be found. It is to this problem (or at least the more general problem of finding _tz_t) that I now turn.

Footnotes

This specification of activity clearly abstracts from a number of features found in some of the New Classical models. However, generalisations of this model to include intertemporal considerations such as real interest rate effects (as in Barro (1976)) or disaggregated activity (via the Phelps island paradigm as in Lucas (1975)) would only serve to complicate the algebra without adding to the particular information restriction being considered. [3]

Hence this aggregate () shall be referred to as the “measured” aggregate. [4]

Some readers may prefer to think of B_t as a “controlled” variable and z_t and as the (logs of the) proportionate deviations of the true and measured aggregates from this controlled one, respectively. [5]

The solution of this model will exhibit “policy neutrality” because of its New Classical structure. To this extent, one choice of (lagged) deterministic policy rule is as good as any other. A general formulation is used here to demonstrate the robust nature of the results. [6]

More generally z_t could be thought of as being generated by a polynomial distributed lag model. All that is required for the results to follow is that the lag function φ(.) have at least one nonzero parameter – that is, that there is at least one lagged value of z_t appearing in its own reduced form. An alternative way of expressing this requirement is that ∂M_t/∂M_t−1 ≠ 0 for some value of i greater than zero. [7]

This assumption is made so that the model stays within the New Classical framework. It does not affect the qualitative nature of the results in this paper. [8]

In the case of β = γ = δ = 0, the specification collapses to a more standard errors-in-variables situation of a white noise measurement error. When β = γ = 0, the measurement error also depends on lagged variables via the monetary base, but not on the contemporaneous values of other variables. Other restrictions on the values of these parameters can make the measurement error depend only on anticipated movements (apart from µ_t), or only on unanticipated movements or on some linear combination thereof. [9]