Inflation adjustment in the open economy: an I(2) analysis of UK prices
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DOI: 10.1007/s00181-005-0030-9
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- Nielsen, H.B. & Bowdler, C. Empirical Economics (2006) 31: 569. doi:10.1007/s00181-005-0030-9
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Abstract
This paper estimates a cointegrated vector autoregressive (VAR) model for UK data on consumer prices, unit labour costs, import prices and real consumption growth. The estimated VAR indicates that the nominal variables are characterised by I(2) trends, and that a linear combination of these processes cointegrate to I(1). This supports an analysis in which I(1) and I(2) restrictions are imposed. A key finding is that an increase in real import prices reduces productivity adjusted real wages, such that the change in domestic inflation is moderated. This may explain why the depreciation of sterling in 1992 left inflation unchanged.
Keywords
CointegrationI(2)InflationImport pricesJEL Classification
C32C51C53E31F0Introduction
Empirical models of inflation in the United Kingdom typically assign a central role to import prices; see, e.g., Bank of England (1999). In recent years, however, the link between import prices and domestic inflation appears to have been quite weak. For instance, the 10% increase in the ratio of import prices to consumer prices that followed the UK’s exit from the European Exchange Rate Mechanism (ERM) in 1992, did not lead to an increase in consumer price inflation. Such an episode stands in stark contrast to the British experience of the 1970s, in which large increases in import prices induced bouts of high inflation.
In this paper we investigate some potential explanations for these episodes through conducting a joint analysis of consumer prices, import prices, unit labour costs and real consumption growth. The econometric approach uses a vector autoregressive (VAR) model parametrised in a way that allows for variables integrated of order one and order two, I(1) and I(2), respectively; see inter alia Johansen (1996) and Haldrup (1998). The VAR model for I(2) variables imposes additional restrictions on the well known I(1) VAR model, and likelihood ratio testing indicates strong support for the presence of I(2) trends in the data that we analyse. In the I(2) framework stationary relations are in general obtained by combining levels and first differences in so-called polynomially cointegrating relations, and in the present context that allows for the identification of permanent effects of real import prices and real unit labour costs on consumer price inflation. Throughout we apply full information maximum likelihood (FIML) estimation; see Johansen (1996) and Johansen (1997); to the best of our knowledge this is the first application of the FIML estimator in an I(2) framework.
The results show that a homogeneity restriction can be imposed on the three nominal variables, namely consumer prices, import prices and unit labour costs, such that the system can be written in terms of real unit labour costs, real import prices, consumer price inflation and the growth rate of real consumption. This transformed system is then estimated in I(1) space, and two stationary relations are identified, one polynomially cointegrating relation originating from the I(2) system that links inflation to real unit labour costs and real import prices, and one simpler relation linking consumption growth to its constant steady-state value. The error correction structure for the system implies that increases in real import prices are associated with downward movements in productivity adjusted real wages, such that inflation adjustment is moderated. This real wage accommodation effect has theoretical foundations in the competing claims models proposed by Layard et al. (1991), and provides one possible explanation for the damped response of UK inflation to the post-ERM depreciation of sterling.
The remainder of the paper expands on these points and has the following structure. Section 2 illustrates how a markup model of prices can be interpreted given different assumptions concerning the time series properties of the price level. Section 3 presents the econometric framework, i.e., the VAR model with I(2) and I(1) restrictions. Section 4 presents quarterly data on the key variables and Sections 5 and 6 present results from the econometric analysis. Finally, Section 7 concludes.
Theoretical framework and time series interpretation
In Eq. 1 lower case letters denote logarithms of variables, μ_{t} measures the equilibrium or steady-state markup factor and the elasticities of P_{t} with respect to ULC_{t} and IP_{t} are γ and (1−γ), respectively.
The economic interpretation of the price-setting rule in Eq. 1 depends upon the integration and cointegration properties of the different series. If the price variables (p_{t} : ulc_{t}: ip_{t})′ are integrated of first order, I(1), as is often assumed in econometric analyses, then they could potentially cointegrate to stationarity and Eq. 1 would represent an I(1)-to-I(0) cointegrating relation; see inter alia de Brouwer and Ericsson (1998). In this case the equilibrium markup factor μ_{t} is fixed over time, μ_{t}=μ, and fluctuations in the observed markup of prices over costs are temporary phenomena associated with disequilibrium in the pricing relation.
In the I(2) scenario, which turns out to be relevant to the data that we analyse, deviations from the polynomially cointegrating relation (Eq. 2) will be stationary and may explain movements in the stationary second order difference, Δ^{2}p_{t}. If firms do respond to higher inflation by cutting their markups, or if workers moderate real wage demands in order to reduce the unemployment effects of hikes in real import prices (see the models in Layard et al. (1991)), then real unit labour costs may be endogenous to the long-run relation in Eq. 2. To take account of this, the empirical analysis allows for causation in all directions through estimating a system of equations.
A final point to bear in mind when modelling price data is that a likely source of deviations between the observed price-cost markup and its equilibrium value is the cyclical position of the economy, e.g. rapid demand growth may raise markups temporarily if total supply is slow to respond. We include in the analysis the growth rate of real consumer expenditure Δc_{t} (c_{t} is the log of real consumer expenditure) in order to capture cyclical effects. If Δc_{t} is I(1), it may be needed in the cointegrating relations, (1) or (2), in order to make them stationary. The coefficient on consumption growth would then identify the cyclical effect on the equilibrium markup. If, on the other hand, consumption growth is stationary (as proves to be the case in this paper) its effect on prices or inflation cannot be separately identified via the cointegrating relations, since linear combinations of stationary variables are themselves stationary. However, we can still model the effects of the stationary variable within the system through adopting a unit vector as a second long-run relation; see Section 5 for further discussion.
The cointegrated VAR model
The innovations are assumed to be identically and independently Gaussian, ɛ_{t} ∼ N (0, Ω), and the initial values, X_{−k+1},..., X_{0}, are taken to be fixed. The k matrices of autoregressive parameters,Π, Γ, Ψ_{1},...,Ψ_{k−2}, are each of dimension p × p and the deterministic specification is given by a constant, μ_{0}, a linear drift term, μ_{1}t, and a set of dummy variables, D_{t}, with unrestricted coefficients, φ.
If r > p − r − s then δ_{ }′S_{t} will be combinations of the levels alone, allowing for direct I(2)-to-I(0) cointegration.
The data
In the empirical analysis we study the data set X_{t} = (p_{t} : ulc_{t} : ip_{t} : Δc_{t})′ for the effective sample t=1969:1–2000:4. We use the natural log of the implicit deflator for household consumption in measuring p_{t}, the log of average unit labour costs for ulc_{t}, the log of the implicit deflator for imports of goods and services for ip_{t}, and the log of real household consumption for c_{t}, see the Appendix for further details. All of the data are seasonally adjusted. Ericsson et al. (1994) discuss how seasonal adjustment can affect estimated dynamics and the calculation of degrees of freedom, and this caveat should be kept in mind when interpreting the results that follow.
The final variable in our empirical analysis is the growth rate of real consumption, Δc_{t}. This measure of cyclical conditions may be more appropriate than alternatives based on GDP, for the inflation rate that we study is based on the deflator for total consumer expenditure, and is therefore more likely to be a function of fluctuations in the consumer sector than those in the aggregate economy.^{3}
Based on a visual inspection of graph (B)–(D) the first differences do not appear to be stationary, as there are several persistent movements in each series during the sample period. This implies that the levels in graph (A) may be driven by stochastic I(2) trends. A priori and given the similar movements in the price levels in graph (A), the most likely scenario in terms of cointegration is that in which there is a single I(2) trend affecting (p_{t} : ulc_{t} : ip_{t})′ proportionately. In this case the polynomially cointegrating relation (Eq. 2) is a likely candidate for a stationary relation. From graph (F) consumption growth appears stationary suggesting a unit vector as a second cointegrating relation. These conjectures are confirmed by formal statistical testing in the next section of the paper.
Analysis of the long-run structure
The location of the dummies was determined from their impact on the likelihood function along the lines of Nielsen (2004a); we note that the results of the analysis are not sensitive to particular dummies, and, by and large, similar results are obtained for a model with no dummies. By construction, the dummies produce level shifts in the I(2) linear combinations of the data, β_{2}′X_{t}, but they do not imply changes in the slopes of the deterministic linear trends. D73q1 accounts for a surge in consumption growth, possibly driven by the fiscal expansions undertaken by the Heath administration. D74q1 controls for the fluctuations following the first oil price shock, while D75q1 and D75q3 control for the uneven nature of earnings growth due to the Wilson–Callaghan ‘social contract’ applied to labour market bargaining. Finally, D79q2 and D80q1 control for, respectively, the second oil price shock and the effects of increases in Value Added Tax under the first Thatcher administration.
Lag length determination
k | Information criteria | Likelihood ratio test | |||||
---|---|---|---|---|---|---|---|
SW | HQ | AIC | k∣5 | k∣4 | k∣3 | k∣2 | |
5 | −33.8910 | −35.3726 | −36.3866 | ||||
4 | −34.3735 | −35.6435 | −36.5126 | 0.72 | |||
3 | −34.8064 | −35.8646 | −36.5889 | 0.56 | 0.33 | ||
2 | −35.2225 | −36.0691 | −36.6485 | 0.37 | 0.20 | 0.20 | |
1 | −34.7829 | −35.4179 | −35.8524 | 0.00 | 0.00 | 0.00 | 0.00 |
Tests for mis-specification of the unrestricted VAR(2) model
| AR(1) | AR(1–8) | ARCH(8) | Normality |
---|---|---|---|---|
p_{t} | 0.00 [0.99] | 0.87 [0.54] | 0.55 [0.82] | 3.22 [0.20] |
ulc_{t} | 0.67 [0.41] | 1.69 [0.11] | 1.77 [0.09] | 1.38 [0.50] |
ip_{t} | 1.04 [0.31] | 1.62 [0.13] | 0.97 [0.46] | 3.19 [0.20] |
Δc_{t} | 0.00 [0.99] | 0.41 [0.91] | 0.52 [0.84] | 1.62 [0.45] |
Multivariate tests: | 1.02 [0.44] | 1.37 [0.02] | 10.91 [0.21] |
I(2) Analysis
Test for the rank indices of the I(2) model
r | Partial nesting structure | ||||||||
---|---|---|---|---|---|---|---|---|---|
0 | H_{0,0} | ⊂ | H_{0,1} | ⊂ | H_{0,2} | ⊂ | H_{0,3} | ⊂ | H_{0,4} |
∩ | |||||||||
1 | H_{1,0} | ⊂ | H_{1,1} | ⊂ | H_{1,2} | ⊂ | H_{1,3} | ||
∩ | |||||||||
2 | H_{2,0} | ⊂ | H_{2,1} | H_{2,2} | |||||
∩ | |||||||||
3 | H_{3,0} | ⊂ | H_{3,1} | ||||||
∩ | |||||||||
H_{4,0} |
r | LR tests of H_{r,s}∣H_{4} | ||||
---|---|---|---|---|---|
0 | 519.30 [0.00] | 238.46 [0.00] | 165.79 [0.00] | 126.44 [0.00] | 117.42 [0.00] |
1 | 102.30 [0.00] | 157.58 [0.00] | 70.35 [0.00] | 64.56 [0.00] | |
2 | 55.43 [0.01] | 23.92 [0.49] | 20.12 [0.22] | ||
3 | 11.00 [0.56] | 7.24 [0.33] | |||
p−r−s | 4 | 3 | 2 | 1 | 0 |
The rank indices (r,s) can be determined via repeated applications of Likelihood Ratio (LR) tests for H_{r,s}∣H_{4,0}, see Johansen (1995) and Nielsen and Rahbek (2003). The idea is to first test the most restricted model, H_{0,0}, against the unrestricted VAR, then H_{0,1}, and so on, row-wise, rejecting a model only if each of the more restricted models have also been rejected.
The lower part of Table 3 reports test statistics for each of the restricted models, H_{r,s}, against the unrestricted model, H_{4,0}, as well as the corresponding asymptotic p values. All models with no stationary relations, r = 0, are safely rejected, and the same is the case for models with r = 1. The model H_{2,1} generates a test statistic against H_{4,0} of 23.92 and a p value of 0.49. The acceptance of this hypothesis indicates the presence of r = 2 stationary relations, s =1 I(1) trend, and hence p − r − s = 1 I(2) trend. A more direct test for the presence of the I(2) trend implied by model H_{2,1} against the alternative of an I(1) model is suggested in Nielsen (2004b) as the LR test for the hypothesis H_{2,1}∣H_{2,2}. The LR statistic for this hypothesis is 3.80, and this corresponds to an asymptotic p value of 0.45, clearly indicating the presence of one I(2) trend in the data.^{5} We note that adopting H_{2,2} as the preferred framework, which implies that the data is characterised by I(1) trends but not I(2) trends, leads to an unrestricted eigenvalue of 0.94 in the characteristic polynomial for the estimated model, i.e., such a framework appears to leave some important non-stationarity unexplained.
In the chosen model, H_{2,1}, one of the two stationary relations is directly cointegrating to I(0) and one is a polynomially cointegrating relation involving first differences. This is potentially consistent with the relationships set out in Section 2, in which the polynomially cointegrating relation is given by Eq. 2, and the second cointegrating relation is the unit vector (0 : 0 : 0 : 1)′, which implies stationarity of consumption growth.
A nominal-to-real transformation of the system to I(1)
If the nominal variables of the system are found to be first-order homogeneous, it follows that relative magnitudes are invariant to the values taken by nominal aggregates, ruling out such phenomena as ‘long-run permanent money illusion’. Thus, a test for first-order homogeneity also constitutes a check of consistency with Neo-Classical economic theory. Furthermore, homogeneity permits a transformation of the I(2) model to one expressed in I(1) space; see Kongsted (2003) and Kongsted and Nielsen (2004).
Homogeneity of the nominal levels of the first three variables in X_{t} = (p_{t} : ulc_{t} :ip_{t} : Δc_{t})′ implies that the loadings applied to the I(2) trend in the nominal variables are proportional, i.e., span(β_{2}) = span(b), where b = (1 : 1 : 1 : 0)′. The estimate of the loadings matrix is given by \(\widehat{\beta }_{2} = {\left( {1.000\,:\,0.976\,:\,1.304\,:\,0.003} \right)}\prime \), which is not too far from the theoretical vector. Since β_{2} is orthogonal to τ=(β:β_{1}), homogeneity can be tested as the restriction b′τ=0; see Johansen (2004). For the present data set we obtain a LR statistic of 7.54, which is not significant at a 5% level according to a χ^{2}(3) distribution.
Identifying the long-run structure within the I(1) Model
Identification of the long-run structure
| \({\cal H}_{0} \) | \({\cal H}_{1} \) | \({\cal H}_{2} \) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\widetilde{\beta } {^* } \) | \({\tilde {\text Î±} }\) | \(\widetilde{\beta } ^* \) | \({\tilde {\text Î±} }\) | \(\widetilde{\beta } ^* \) | \({\tilde {\text Î±} }\) | |||||||
ulc_{t}−p_{t} | 0.123 (0.031) | −0.231 (0.029) | 0.060 (0.135) | 0.674 (0.138) | 0 (...) | −0.213 (0.028) | 0.198 (0.116) | 0.706 (0.134) | 0 (...) | −0.208 (0.028) | 0.205 (0.114) | 0.710 (0.129) |
ip_{t}−p_{t} | 0.007 (0.009) | −0.056 (0.008) | 0.086 (0.300) | 0.128 (0.304) | 0 (...) | −0.055 (0.008) | 0.008 (0.255) | 0.105 (0.294) | 0 (...) | −0.053 (0.008) | 0 (...) | 0 (...) |
Δp_{t} | 0 (...) | 1 (...) | 0.182 (0.92) | −0.476 (0.094) | 0 (...) | 1 (...) | 0.166 (0.078) | −0.509 (0.090) | 0 (...) | 1 (...) | 0.161 (0.078) | −0.503 (0.083) |
Δc_{t} | 1 (...) | 0 (...) | −0.757 (0.112) | −0.098 (0.115) | 1 (...) | 0 (...) | −0.529 (0.102) | 0.035 (0.117) | 1 (...) | 0 (...) | −0.539 (0.096) | 0 (...) |
trend | 0.100 (0.047) | −0.018 (0.004) | 0 (...) | −0.017 (0.004) | 0 (...) | −0.016 (0.004) | ||||||
Long-likelihood value | 2395.97955 |
| 2391.34757 |
| 2391.25303 |
| ||||||
Test statistics | – | 9.26 | 9.45 | |||||||||
Asymp. p value | – | 0.026 | 0.150 | |||||||||
Asymp. distribution | – | χ^{2} (3) | χ^{2} (6) | |||||||||
Bootstrap p value^{a} | – | 0.085 | 0.303 |
Next, under \( {\user1{\mathcal{H}}}_{1} \), we test the hypothesis that the first cointegrating relation is a unit vector. The hypothesis implies three over-identifying restrictions on \( {\text{span}}{\left( {\overline{\beta } * } \right)} \) and produces a LR test statistic of 9.26, which corresponds to a p-- value of 0.03 when using the asymptotic χ^{2}(3) distribution. However, LR tests pertaining to cointegrating coefficients are often found to reject a true null hypothesis too often; see inter alia Li and Maddala (1997), Jacobson et al. (1998) and Gredenhoff and Jacobson (2001). We therefore also estimate the finite sample distribution using the Bootstrap principle as proposed in Gredenhoff and Jacobson (2001). The Bootstrap p value of the test for the reduction of \({\cal H}_{0}\) to \({\cal H}_{1}\) is 0.09 indicating borderline acceptance. This is the result that one would expect from a visual inspection of Fig. 1 (F).
The existence of a long-run relationship between inflation and the markup changes the interpretation of the relationship between the price level and its determinants. In the current framework the linear homogeneity is an I(2)-to-I(1) cointegration phenomenon and deviations from the homogeneous linear combination in Eq. 1 is still I(1). As also noted by Banerjee et al. (2001), the relation is only linearly homogeneous in the usual sense after controlling for shifts in inflation. This implies that increases in costs that cause an upward pressure on inflation will reduce the markup, and this will in turn limit the size of the final increase in inflation. According to Eq. 8, a 1% increase in inflation reduces the equilibrium markup by 3.83%. This is larger than the 2.87% estimated by Banerjee and Russell (2001) using the sample period 1961 : 2–1997 : 1, and also a slightly different treatment of the deterministic and cyclical terms in the model.
Lagged consumption growth exerts a positive effect on both the rate of change of inflation and the rate of change of the real wage, suggesting that erosion of spare capacity in the consumer goods sector tends to accelerate price and wage adjustment. However, the asymptotic standard deviations of the estimated parameters are relatively large, reflecting difficulties in identifying the exact channels through which excess demand raises prices. An additional restriction that removes the capacity effect operating via real wages can be imposed on \( {\user1{\mathcal{H}}}_{2} \). This increases the direct capacity effect on inflation from 0.17 to a statistically significant 0.23%. Still, we take \( {\user1{\mathcal{H}}}_{2} \) as the preferred model in the analysis that follows and therefore continue to allow for indirect cyclical effects that operate through real wages.
The short-run structure
Short-run structure
\( \Delta p_{t} - \theta \cdot {\left[ {\gamma \cdot {\left( {ulc_{t} - p_{t} } \right)} + {\left( {1 - \gamma } \right)} \cdot {\left( {ip_{t} - p_{t} } \right)}} \right]} \sim I{\left( 0 \right)}, \) |
The tested down dynamic model provides insights concerning inflation evolution in the United Kingdom. In the equation for Δ^{2}p_{t} the error correction term based on inflation and the markup is highly significant, confirming that increases in productivity adjusted real wages and real import prices Granger cause inflation. However, the highly significant coefficient on ecm2 in the real unit labour cost equation indicates that causation also operates in the opposite direction. The positive coefficient on ecm2 in the Δ(ulc−p)_{t} equation implies that large upward movements in real import prices lead to reductions in real unit labour costs, i.e., there is real wage accommodation that will moderate the increase in inflation arising from higher real import prices. Adjustment along these lines partly explains the lack of any upturn in UK inflation following the substantial increase in real import prices that took place after the devaluation of sterling in 1992.
The model suggests some additional mechanisms through which inflation may have been stabilised following the 1992 experience. First, note from Fig. 1 that consumption growth, i.e., ecm1, was weak during the 2 years leading up to 1992, which implies a reduction in Δ^{2}p_{t} both directly because of the inclusion of ecm1 in the third equation, and indirectly through the role of ecm1 in the real unit labour cost equation. It is important to note that the significance of ecm1 in the unit labour cost equation implies that the real wage accommodation effect is estimated holding constant cyclical influences. This makes it less likely that the real wage accommodation effect is a spurious finding that arises because the Δ(ulc−p)_{t} equation does not control for demand pressures.^{6} Second, note that should real import prices elicit some permanent increase in inflation there will be a reduction in the equilibrium markup factor via the long-run relationship in Eq. 8, which then implies that inflation does not increase by the amount that it would have done had the markup factor remained constant, i.e., the domestic real profit share may also accommodate part of the upturn in real import prices. This is the mechanism discussed in Banerjee et al. (2001) and described above.
The real wage accommodation effect is relevant to the literature on inflation forecasting. Conditional models for UK inflation often predict that upwards trends in real import prices will lead to higher inflation, other things equal; see for example Batini et al. (2005). The real wage accommodation effect suggests that the ‘other things equal’ assumption is unlikely to hold, because upward trends in real import prices will be partially offset by movements in the labour share.
The large negative correlation of −0.43 between the residuals from the Δ^{2}p_{t} and Δ^{2}c_{t} equations is most likely the result of the construction of the transformed variables. Real consumption expenditure is constructed as C_{t} = NC_{t} / P_{t}, where NC_{t} is nominal final expenditures, so that any uncertainty in the split between quantity and price in the national accounts data will give a negative correlation between the variables Δ^{2}c_{t} = Δ^{2}nc_{t}−Δ^{2}p_{t} and Δ^{2}p_{t}. The residual correlation can be transformed into a simultaneous effect between Δ^{2}c_{t} and Δ^{2}p_{t} by taking a linear combination of the two equations and identifying the new coefficient for the simultaneous effect by imposing a zero restriction on the coefficient for ecm1_{t−1}. However, given the likely source of the negative correlation the simultaneous equation system (SEM) would not contribute a great deal to the interpretation of the results.
The negative correlations between the residuals from the equations for Δ^{2}p_{t} and the equations for Δ(ulc−p)_{t} and Δ(ip−p)_{t} are most likely the result of the deflator p_{t} appearing in the transformed variables. Once again these correlations can be reduced by including contemporaneous values of Δ^{2}p_{t} in the Δ(ulc−p)_{t} and Δ(ip−p)_{t} equations, or vice versa, but again it does not contribute to the interpretation of the system and the results are not reported. Finally, we note that the negative correlation between the residuals from the Δ(ulc−p)_{t} and Δ(ip−p)_{t} equations may indicate that a real wage accommodation effect operates on impact as well as through the error correction mechanism, and that the positive correlation between the residuals for Δ(ulc−p)_{t} and Δ^{2}c_{t} indicates a small capacity effect operating on impact besides the effect operating trough the error correction term ecm1_{t−1}.
Summary
This paper presented a VAR model for UK data on consumer prices, unit labour costs, import prices and real consumption growth. Likelihood ratio tests for the cointegration rank indices indicated that the nominal variables contained an I(2) trend. It was then shown that these I(2) variables cointegrate to form an I(1) price–cost markup. The cointegrating vector demonstrates that the price level is linearly homogeneous in unit labour costs and import prices, such that the markup can be written as the sum of real unit labour costs and real import prices, which are both I(1). These two variables were then analysed alongside inflation and real consumption growth using the well known I(1) model. A crucial finding was that real unit labour costs error correct with respect to disequilibrium in the long-run relation between inflation and relative prices, suggesting that increases in real import prices may be accommodated through reductions in productivity adjusted real wages. This may help to explain the very small response of inflation to the large upturn in real import prices observed in the UK following the depreciation of sterling in 1992.
Real unit labour costs are equivalent to the productivity adjusted real wage facing consumers. However, as firms are both producers and retailers in this analysis and as we do not model the tax wedge, real unit labour costs are also equivalent to the productivity adjusted real wage facing producers.
An I(1) markup can imply arbitrarily large profits (positive or negative), which is problematic. However, in the results reported in this paper the estimated markup, and hence the profit share, do not drift without limits.
The late 1990s provide a good example of how the choice of cyclical indicator can be important: GDP growth during that period suggested that the economy was expanding at its trend rate, but that masked strong demand pressures in the consumer sector that were offset at the aggregate level by a manufacturing recession, as exporters struggled to cope with the effects of a high sterling exchange rate.
It is possible that the result of the multivariate test is a finite sample phenomenon, particularly given that 128 restrictions are being tested in this case.
Augmented Dickey–Fuller (ADF) tests of order f, where f is the maximum order for the ADF regression in which the final lag is not insignificant at the 5% level indicate that Δp, Δulc and Δip are clearly I(1) such that p, ulc and ip are I(2). All of the ADF regressions included a constant and a time trend. Full details can be obtained from the authors on request.
Equally, we note that the real wage accommodation effect does not depend on the presence of the cyclical term. Estimating a trivariate VAR that excludes real consumption growth also gives a significant real wage accommodation effect (results not reported here).
Acknowledgements
The authors would like to thank the associate editor, an anonymous referee, Dan Knudsen, Hans Christian Kongsted, John Muellbauer, Kamakshya Trivedi and participants at the 58th European Meeting of the Econometric Society, Stockholm, for many helpful comments. The paper was initiated while the first author was visiting Nuffield College, Oxford. Their hospitality and a financing grant from the Euroclear Bank and University of Copenhagen are gratefully acknowledged. The second author acknowledges financial support from a British Academy post-doctoral fellowship. The empirical analysis was carried out using a set of procedures programmed in Ox; see Doornik (2001).