THE ECONOMETRICS OF MACROECONOMIC MODELLING

# Cointegration, causality, and the Phillips curve natural rate

As indicated earlier, there are many ways that a Phillips curve for an open economy can be derived from economic theory. Our appraisal of the Phillips curve in this section builds on Calmfors (1977), who reconciled the Phillips curve with the Scandinavian model of inflation. We want to go one step further, however, and incorporate the Phillips curve in a framework that allows for integrated wage and price series. Reconstructing the model in terms of cointegration and causality reveals that the Phillips curve version of the main-course model forces a particular equilibrium correction mechanism on the system. Thus, while it is consistent with Aukrust’s main-course theory, the Phillips curve is also a special model thereof, since it includes only one of the many wage stabilising mechanisms discussed by Aukrust.

Without loss of generality we concentrate on the wage Phillips curve and recall that, according to Aukrust’s theory, it is assumed that (using the same symbols as in Chapter 3):

1. (we, t — qe, t —ae, t) ~ I(0) and ut ~ I(0), possibly after removal of deterministic shifts in their means; and

2. the causal structure is ‘one way’ as represented by H4mc and H5mc in Chapter 3.

Consistency with the assumed cointegration and causality requires that there exists an equilibrium-correction model (EqCM hereafter) for the nominal wage rate in the exposed sector. Assuming first-order dynamics for simplicity, a Phillips curve EqCM system is defined by the following two equations:

Awt — pwQ — PwlUt + Pw2 Дat + pw^Aqt + ew, t,

0 < 0wl, 0 <pw2 < 1, 0 < @w3 < 1, (4.1)

ДЩ — Pu0 Pu1Ut-1 + ви2 (w q a)t-1 Pu3zu, t + єu, t,

0 < (3ui < 1, Put > 0, Pu3 > 0, (4.2)

where we have simplified the notation somewhat by dropping the ‘e’ subscript.[19] [20] Д is the difference operator. ew, t and eu, t are innovations with respect to an information set available in period t — 1, denoted It-1.A Equation (4.1) is the short-run Phillips curve, while (4.2) represents the basic idea that profitability (in the e-sector) is a factor that explains changes in the economy-wide rate of unemployment. zu, t represents (a vector of) other I(0) variables (and deterministic terms) which ceteris paribus lower the rate of unemployment. zu, t will typically include a measure of the growth rate of the domestic economy, and possibly factors connected with the supply of labour. Insertion of (4.2) into (4.1) is seen to give an explicit EqCM for wages.

To establish the main-course rate of equilibrium unemployment, we rewrite (4.1) as

Дwt — —pwi (ut — U) + fЗw‘2Дat + Pw3 Дqt + ew, t, (4.3)

where

w Pw0

U — ~7,--------

Hw 1

is the rate of unemployment which does not put upward or downward pressure on wage growth. Taking unconditional means, denoted by E, on both sides

of (4.3) gives

E[Aw(] - gf - ga = - PwiEiut - u] + (3w2 - l)ga + (Pw3 - l)gf ■

Using the assumption of a stationary wage share, the left-hand side is zero. Thus, using ga and gf to denote the constant steady-state growth rates of productivity and foreign prices, we obtain

as the solution for the main-course equilibrium rate of unemployment which we denote uphil. The long-run mean of the wage share is consequently

Moreover, uphil and wshphi represent the unique and stable steady state of the corresponding pair of deterministic difference equations.

The well-known dynamics of the Phillips curve is illustrated in Figure 4.1. Assume that the economy is initially running at a low level of unemployment, that is, u0 in the figure. The short-run Phillips curve (4.1) determines the rate of wage inflation Aw0. The wage share consistent with equation (4.2) is above its long-run equilibrium, implying that unemployment starts to rise and wage growth is reduced along the Phillips curve. The steep Phillips curve is defined for the case of Awt = Aqt + Aat. The slope of this curve is given by - pw1/(1 - Pw3), and it has been dubbed the long-run Phillips curve in the literature. The stable equilibrium is attained when wage growth is equal to the steady-state growth of the main course, that is, gf + ga and the corresponding

level of unemployment is given by nphl1. The issue about the slope of the long - run Phillips curve is seen to hinge on the coefficient (3w3, the elasticity of wage growth with respect to the product price. In the figure, the long-run curve is downward sloping, corresponding to (3w3 < 1 which is conventionally referred to as dynamic inhomogeneity in wage-setting. The converse, dynamic homogeneity, implies (3w3 = 1 and a vertical Phillips curve. Subject to dynamic homogeneity, the equilibrium rate nphl1 is independent of world inflation gf.

The slope of the long-run Phillips curve represented one of the most debated issues in macroeconomics in the 1970s and 1980s. One argument in favour of a vertical long-run Phillips curve is that it is commonly observed that workers are able to obtain full compensation for CPI-inflation. Hence (3w3 = 1 is a reasonable restriction on the Phillips curve, at least if Aqt is interpreted as an expectations variable. The downward sloping long-run Phillips curve has also been denounced on the grounds that it gives a too optimistic picture of the powers of economic policy: namely that the government can permanently reduce the level of unemployment below the natural rate by ‘fixing’ a suitably high level of inflation (see, for example, Romer 1996, ch. 5.5). In the context of an open economy this discussion appears to be somewhat exaggerated, since a long-run tradeoff between inflation and unemployment in any case does not follow from the premise of a downward-sloping long-run curve. Instead, as shown in Figure 4.1, the steady-state level of unemployment is determined by the rate of imported inflation gf and exogenous productivity growth, ga. Neither of these are normally considered as instruments (or intermediate targets) of economic policy.[21]

In the real economy, cost-of-living considerations plays a significant role in wage-setting; see, for example, Carruth and Oswald (1989, ch. 3) for a review. Thus, in applied econometric work, one usually includes current and lagged CPI-inflation, reflecting the weight put on cost-of-living considerations in actual wage bargaining situations. To represent that possibility, consider the following system (4.7)-(4.9):

Awt = $w0 - $w1nt + Pw2Aat + Pw3 Aqt + /3w4Apt + ^wt, (4.7)

Ant = ви0 - виіЩ-i + ви2(w - q - a)t-i - визzt + £ut, (4.8)

Apt = f3pi(Awt - Aat) + /3p2Aqt + £p, t - (4.9)

The first equation augments (4.1) with the change in consumer prices Apt, with coefficient 0 < (3w4 < 1. To distinguish formally between this equation and (4.1), we use an accent above the other coefficients as well (and above the disturbance term). The second equation is identical to the unemployment equation (4.2). The last stochastic price equation combines the stylised definition of consumer prices in (3.8) with the twin assumption of stationarity

of the sheltered sector wages share and wage leadership of the exposed sector.[22]

Z^w2 1 ftw 1 |

$w3 + fiw 4(вр1 + (3p2) — 1 ftw 1 |

Using (4.9) to eliminate Apt in (4.7) brings us back to (4.1), with coefficients and ewt suitably redefined. Thus, the expression for the equilibrium rate uphi1 in (4.5) applies as before. However, it is useful to express uphi1 in terms of the coefficients of the extended system (4.7)-(4.9):

since there are now two homogeneity restrictions needed to ensure a vertical long-run Phillips curve: namely f3w3 + /3w4 = 1 and f3p1 + f3p2 = 1.

Compared to the implicit dynamics of Chapter 3, the open economy wage Phillips curve system represents a full specification of the dynamics of the Norwegian model of inflation. Clearly, the dynamic properties of the model apply to other versions of the Phillips curve as well. In particular, all Phillips curve systems imply that the natural rate (or NAIRU) of unemployment is a stable stationary solution. As a single equation, the Phillips curve equation itself is dynamically unstable for a given rate of unemployment. Dynamic stability of the wage share and the rate of unemployment hinges on the equilibrating mechanism embedded in the equation for the rate of unemployment. In that sense, a Phillips curve specification of wage formation cannot logically accommodate an economic policy that targets the level of (the rate of) unemployment, since only the natural rate of unemployment is consistent with a stable wage share. Any other (targeted) level leads to an ever increasing or ever declining wage share.

The question about the dynamic stability of the natural rate (or NAIRU) is of course of great interest, and cannot be addressed in the incomplete Phillips curve system, that is, by estimating a single-equation Phillips curve model. Nevertheless, as pointed out by Desai (1995), there is a long-standing practice of basing the estimation of the NAIRU on the incomplete system. For United States, the question of correspondence with a steady state may not be an issue: Staiger et al. (1997) is an example of an important study that follows the tradition of estimating only the Phillips curve (leaving the equilibrating mechanism, for example, (4.2) implicit). For other countries, European in particular, where the stationarity of the rate of unemployment is less obvious, the issue about the correspondence between the estimated NAIRUs and the steady state is a more pressing issue.

In the following sections, we turn to two separate aspects of the Phillips curve NAIRU. First, Section 4.3 discusses how much flexibility and time

dependency one can allow to enter into NAIRU estimates, while still claiming consistency with the Phillips curve framework. Second, in Section 4.4 we discuss the statistical problems of measuring the uncertainty of an estimated time independent NAIRU.