THE ECONOMETRICS OF MACROECONOMIC MODELLING

Lucas critique

Lucas’s 1976 thesis states that conditional econometric models will be prone to instability and break down whenever non-modelled expectations change. This section establishes the critique for a simple algebraic case. In the fol­lowing section we discuss how the Lucas critique can be confirmed or refuted empirically.

Without loss of generality, consider a single time-series variable yt, which can be split into an explained part yf, and an independent unexplained

parE Cy, t-

yt = УЇ + £y, t - (4.17)

Following Hendry (1995a: ch. 5.2) we think of yf as a plan attributable to agents, and ey, t as the difference between the planned and actual outcome of yt. Thus,

E[yt I yf] = yf, (4.18)

and ey, t is an innovation relative to the plan, hence

E[ey, t 1 yf ]=0. (4.19)

Assume next that agents use an information set It-1 to form rational expectations for a variable xt, that is,

xet = E[xt yit-i] (4.20)

and that expectations are connected to the plan

yf = вхе, (4.21)

which is usually motivated by, or derived from, economic theory.

By construction, E[yf | It-i] = yf, while we assume that ey, t in (4.17) is an innovation

Initially, xf is assumed to follow a first-order AR process (non-stationarity is considered below):

xl = E[xt It-i] = ax—, |ai| < 1. (4.24)

Thus xt = xf + ext, or:

xt = apxt-i + €x, t, E[tx, t xt-i] = 0. (4.25)

For simplicity, we assume that ey, t and ex, t are independent.

Assume next that the single parameter of interest is в in equation (4.21). The reduced form of yt follows from (4.17), (4.21), and (4.24):

yt = aifixt-i + €x, t, (4.26)

where xt is weakly exogenous for £ = aie, but the parameter of interest в is not identifiable from (4.26) alone. Moreover the reduced form equation (4.26), while allowing us to estimate £ consistently in a state of nature characterised by stationarity, is susceptible to the Lucas critique, since £ is not invariant to changes in the autoregressive parameter of the marginal model (4.24).

In practice, the Lucas critique is usually aimed at ‘behavioural equations’ in simultaneous equations systems, for example,

yt = ext + nt (4.27)

with disturbance term:

ht = €y, t Cxtfi. (4.28)

It is straightforward (see Appendix A.1) to show that estimation of (4.27) by OLS on a sample t = 1, 2,...,T, gives

plim eoLs = аїв, (4.29)

T

establishing that, ‘regressing yt on xt’ does not represent the counterpart to yf = x^e in (4.21). Specifically, instead of в, we estimate а2в, and changes in the expectation parameter ai damage the stability of the estimates, thus confirming the Lucas critique.

However, the applicability of the critique rests on the assumptions made. For example, if we change the assumption of |ai| < 1 to ai = 1, so that xt has a unit root but is cointegrated with yt, the Lucas critique does not apply: under cointegration, plim^^,^ /3OLS = в, since the cointegration parameter is unique and can be estimated consistently by OLS.

As another example of the importance of the exact set of assumptions made, consider replacing (4.21) with another economic theory, namely the contingent plan

yp = fat - (4.30)

Equations (4.30) and (4.17) give

yt = Pxt + £y, t, (4.31)

where E[ey t I xt] = 0 ^ cov(ey, t,xt) = 0 and в can be estimated by OLS also in the stationary case of |ai| < 1.