THE ECONOMETRICS OF MACROECONOMIC MODELLING
Cointegration and identification
In Chapter 3, we made the following assumptions about the time-series properties of the variables we introduced: nominal and real wages and productivity are I (1), while, possibly after removal of deterministic shifts, the rate of unemployment is without a unit root. A main concern is clearly how the theoretical wage curve model can be reconciled with these properties of the data. In other words: how should the long-run wage equation be specified to attain a true cointegrating relationship for real wages, and to avoid the pitfall of spurious regressions?
As we have seen, according to the bargaining theory, the term mq, t in (5.5) depends on average productivity, At. A Having assumed ut — I(0), and keeping in mind the possibility that ш = 0, it is seen that it follows directly from cointegration that productivity has to be an important variable in the relationship. In other words, a positive elasticity EAWq is required to balance the I(1) trend in the product real wage on the left-hand side of the expression.
Thus, the general long-run wage equation implied by the wage bargaining approach becomes
w t = mb + iat + upqt — wut, 0 < i < 1, 0 < ш < 1, w > 0, (5.10)
where wbqt = wq — qt denotes the ‘bargained real wage’ as before. The intercept mb is redefined without the productivity term, which is now singled out as an I(1) variable on the left-hand side of the expression, and with the other determinants assumed to be constant. Finally, defining
ecmq, t = Wq, t — wq, t - I(0)
allows us to write the hypothesised cointegrating wage equation as
Wq, t = mq + iat + upq, t — w щ + ecmq, t. (5.11)
Some writers prefer to include the reservation wage (the wage equivalent of being unemployed) in (5.10). For example, from Blanchard and Katz (1999) (but using our own notation to express their ideas):
wbq t = mq + i'at + (1 — i')wrt + upq, t — wut, 0 < i < 1, (5.12)
where wrt denotes the reservation wage. However, since real wages are integrated, any meaningful operational measure of w must logically cointegrate with wq, t directly. In fact, Blanchard and Katz hypothesise that wtt is a linear function of the real wage and the level of productivity.[32] [33] Using that (second) cointegrating relationship to substitute out w from (5.12) implies a relationship which is observationally equivalent to (5.11).
The cointegration relationship stemming from price-setting is anchored in equation (5.6). In the same way as for wage-setting, it becomes important in
applied work to represent the productivity term explicitly in the relationship. We therefore rewrite the long-term price-setting schedule as
wft = mf + at + dut, (5.13)
where the composite term mf in (5.6) has been replaced by mf+at. Introducing ecm f, t = wq t — wft ~ I (0), the second implied cointegration relationship becomes
Wq, t = mf + at + dut + ecmf, t. (5.14)
While the two cointegrating relationships are not identified in general, identifying restrictions can be shown to apply in specific situations that occur frequently in applied work. From our own experience with modelling both disaggregate and aggregate data, the following three ‘identification schemes’ have proven themselves useful:
One cointegrating vector. In many applications, especially on sectorial data, formal tests of cointegration support only one cointegration relationship, thus either one of ecmb, t and ecm ft is I(1), instead of both being I(0). In this case it is usually possible to identify the single cointegrating equation economically by restricting the coefficients, and by testing the weak exogeneity of one or more of the variables in the system.
No wedge. Second, and still thinking in terms of a sectorial wage-price system: assume that the price markup is not constant as assumed above, but is a function of the relative price (via the price elasticity є). In this case, the price equation (5.14) is augmented by the real exchange rate pt — pit. If we furthermore assume that ш = 0 (no wedge in wage formation) and d = 0 (normal cost pricing), identification of both long-run schedules is logically possible.
Aggregate price-wage model. The third cointegrating identification scheme is suited for the case of aggregated wages and prices. The long-run model is
wt = mb + (1 — w)qt + iat + wpt — шщ + ecmb, t, (5.15)
qt = —mf + wt — at — dut — ecm ft, (5.16)
pt = Фqt + (1 — Ф)рч,
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solving out for producer prices qt then gives a model in wages wt and consumer prices pt only,
(1 — ш)(1 — ф)
Ф pt = —фmf + ф(wt — at) — фdut + (1 — ф)рч — фecmf, t, that implicitly implies non-linear cross-equation restrictions in terms of ф.
By simply viewing (5.17) and (5.18) as a pair of simultaneous equations, it is clear that the system is unidentified in general. However, if the high level of aggregation means that ш can be set to unity (while retaining cointegration),
and there is normal cost pricing in the aggregated price relationship, identification is again possible. Thus ш =1 and d = 0 represent one set of necessary (order) restrictions for identification in this case:
wt = mb + pt + iat - wut + ecmb, t, (5.19)
pt = - фт} + ф(wt - at) + (1 - ф)рч - фест$^,- (5.20)
We next give examples of how the first and third schemes can be used to identify cointegrating relationships in Norwegian manufacturing and in a model of aggregate United Kingdom wages and prices.
