Empirical Economics

, Volume 31, Issue 3, pp 569–586

Inflation adjustment in the open economy: an I(2) analysis of UK prices


    • Economics DepartmentUniversity of Copenhagen
  • Christopher Bowdler
    • Nuffield CollegeUniversity of Oxford
Original Paper

DOI: 10.1007/s00181-005-0030-9

Cite this article as:
Nielsen, H.B. & Bowdler, C. Empirical Economics (2006) 31: 569. doi:10.1007/s00181-005-0030-9


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.


CointegrationI(2)InflationImport prices

JEL Classification



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

Empirical analyses of inflation fluctuations are often based upon markup models of the price level; see, for example, de Brouwer and Ericsson (1998). In such models the price level, Pt, is a markup over total unit costs, which we take to be a combination of unit labour costs, ULCt, and import prices, IPt. If it is assumed that the price level is linearly homogeneous in input costs then an expression for the price level can be written as follows:
$$ p_{t} = p_{t} + \gamma \cdot ulc_{t} + {\left( {1 - \gamma } \right)} \cdot ip_{t} . $$

In Eq. 1 lower case letters denote logarithms of variables, μt measures the equilibrium or steady-state markup factor and the elasticities of Pt with respect to ULCt and IPt 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 (pt : ulct: ipt)′ 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 μt is fixed over time, μt=μ, and fluctuations in the observed markup of prices over costs are temporary phenomena associated with disequilibrium in the pricing relation.

An alternative scenario is that the price variables (pt : ulct : ipt)′ are I(2) processes in which case the first differences, Δpt, Δulct, and Δipt, are non-stationary I(1) variables. The price measures may still cointegrate, in this case from I(2) to I(1) in general, such that the linearly homogeneous relation μt = ptγ·ulct−(1 − γipt is an I(1) process; see Banerjee et al. (2001) and Banerjee and Russell (2001). If this markup formula is then rewritten as a function of real unit labour costs,1 (ulc−p)t, and real import prices, (ip−p)t, a potential second layer of cointegration is the following polynomially cointegrating relation
$$Δp_{t} - \theta \cdot {\left[ {\gamma \cdot {\left( {ulc_{t} - p_{t} } \right)} + {\left( {1 - \gamma } \right)} \cdot {\left( {ip_{t} - p_{t} } \right)}} \right]} \sim I{\left( 0 \right)}, $$
which can be interpreted as a dynamic steady state relation. In this setting the equilibrium markup factor is not fixed. Rather, a 1% increase in inflation is associated with a reduction in the equilibrium markup factor of \( \frac{1} {\theta }\)%. Banerjee et al. (2001) argue that this occurs because firms have imperfect information on market prices and face a comparatively large loss if prices are set too high, e.g. due to the presence of a kinked demand curve. If firms are risk averse, the target markup will depend negatively on the level of uncertainty. If inflation is a measure of uncertainty, high levels of inflation will be accompanied by a relatively low markup. In order to identify this link between inflation and the markup the price variables (pt : ulct : ipt)′ must be treated as I(2) processes.2

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, Δ2pt. 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 Δct (ct is the log of real consumer expenditure) in order to capture cyclical effects. If Δct 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 starting point for analysing the p--dimensional vector Xt, t = 1,...,T, is a VAR model of order k, which can be parametrised as:
$$ \Delta ^{2} X_{t} = {\text{IL}}X_{{t - 1}} - \Gamma \Delta X_{{t - 1}} + {\sum\limits_{i = 1}^{k - 2} {\Psi _{i} \Delta ^{2} } }X_{{t - i}} + \mu _{0} + \mu _{1} t + \phi D_{t} + \varepsilon _{t} . $$

The innovations are assumed to be identically and independently Gaussian,  ɛtN (0, Ω), and the initial values, Xk+1,..., X0, are taken to be fixed. The k matrices of autoregressive parameters,Π, Γ, Ψ1,...,Ψk−2, are each of dimension p × p and the deterministic specification is given by a constant, μ0, a linear drift term, μ1t, and a set of dummy variables, Dt, with unrestricted coefficients, φ.

The I(2) model, denoted Hr,s, is a sub-model of Eq. 3 defined by the two reduced rank restrictions
$$ \Pi = \alpha \beta \prime $$
$$ \alpha \prime _{ \bot } \Gamma \beta _{ \bot } = \xi \eta \prime , $$
where α and β are matrices of dimension p × r, ξ and η are matrices of dimension (prs, and α and β are the orthogonal complements to α and β, respectively. Under the additional assumption that the characteristic polynomial corresponding to Eq. 3 has 2(pr) − s roots at the point z = 1 and the remaining roots are outside the unit circle, Xt is an I(2) process; see Johansen (1992) for the full representation. To characterise the cointegration properties we define the matrices \( \beta _{1} = \beta _{ \bot } \eta , \)\( \beta _{2} = \beta _{ \bot } \eta _{ \bot } \) and \( \delta = \overline{\alpha } \prime \Gamma \overline{\beta } _{2} , \) where for a matrix β we define \( \overline{\beta } = \beta {\left( {\beta \prime \beta } \right)}^{{ - 1}} \) Using this notation the prs linear combinations β2Xt are I(2) and non-cointegrating. The r+s combinations (β:β1)′Xt cointegrate from I(2) to I(1). They can be further divided into s combinations, β1Xt, that remain I(1), and r combinations that cointegrate to stationarity with the first differences through the polynomially cointegrating relations:
$$ S_{t} = \beta \prime X_{t} - \delta \beta ^{\prime }_{2} \Delta X_{t} . $$

If r > prs then δ St will be combinations of the levels alone, allowing for direct I(2)-to-I(0) cointegration.

Previous applications of the I(2) model have employed the two-step estimator of Johansen (1995); see inter alia Juselius (1998), Diamandis et al. (2000), Banerjee et al. (2001), Banerjee and Russell (2001) and Nielsen (2002) for examples. The two-step estimator is a sequential application of the reduced rank regression known from the analysis of I(1) VAR models, and is not the ML estimator for the I(2) model. In this paper we rely on the ML estimation algorithm of Johansen (1997), based on the parametrisation
$$ \Delta ^{2} X_{t} = \alpha {\left( {\rho \prime \tau \prime X_{{t - 1}} + \psi \prime \Delta X_{{t - 1}} } \right)} + \Omega \alpha _{ \bot } {\left( {\alpha ^{\prime }_{ \bot } \Omega \alpha _{ \bot } } \right)}^{{ - 1}} \kappa \prime \tau \prime \Delta X_{{t - 1}} + {\sum\limits_{i = 1}^{k - 2} {\Psi _{i} \Delta ^{2} } }X_{{t - i}} + \mu _{0} + \mu _{1} t + \phi D_{t} + \varepsilon _{t} ,$$
where the parameters (α, ρ, τ, ψ, κ, Ψ1,...,Ψk−2, Ω) are all freely varying. The parameters of the previous notation can be derived from the new parameters using β=τρ,\(\xi {\text{ = }} - \kappa \prime \overline{\rho } _{ \bot }\), \( \eta = \beta ^{\prime }_{ \bot } \tau \rho _{ \bot } \), and \( \Gamma = - \Omega \alpha _{ \bot } {\left( {\alpha ^{\prime }_{ \bot } \Omega \alpha _{ \bot } } \right)}^{{ - 1}} \kappa \prime \tau \prime - \alpha \psi \prime \). Likelihood ratio (LR) tests for the cointegration ranks (r, s) based on this parametrisation are suggested in Nielsen and Rahbek (2003), and using a Monte Carlo simulation they illustrate that the rejection frequencies for true hypotheses are much closer to the nominal sizes than are those based on the two-step estimation procedure. We therefore prefer to use the ML estimator and the associated LR tests in this paper.
Due to unit roots, unrestricted deterministic components in Eqs. 3 or 6 will accumulate in the levels of the variables, e.g. an unrestricted constant will produce a quadratic trend in the I(2) combinations, β2Xt. In the empirical analysis we want to allow for linear trends in all linear combinations of the variables, including the stationary polynomially cointegrating relations, St, since deterministic trends may provide an alternative to stochastic trends in describing non-stationarities in the data. One potential interpretation of the linear trends is that they control for different methods of quality adjustment applied in the construction of consumer and import price deflators. We exclude a priori quadratic deterministic trends. This approach can be implemented using the specification proposed in Rahbek et al. (1999), which entails the restrictions:
$$ \mu _{1} = \alpha \rho \prime \tau ^{\prime }_{0} \,{\text{and}}\,\mu _{0} = \alpha \psi ^{\prime }_{0} + \Omega \alpha _{ \bot } {\left( {\alpha ^{\prime }_{ \bot } \Omega \alpha _{ \bot } } \right)}^{{ - 1}} \kappa \prime \tau ^{\prime }_{0} , $$
where τ0 and ψ0 are matrices of dimension 1 × (r + s) and 1 × r, respectively. This specification is asymptotically similar, such that the actual coefficients on the linear trends in different linear combinations do not appear as nuisance parameters in the asymptotic distributions of estimators of other coefficients; see Rahbek et al. (1999) and Nielsen and Rahbek (2000).

The data

In the empirical analysis we study the data set Xt = (pt : ulct : ipt : Δct)′ for the effective sample t=1969:1–2000:4. We use the natural log of the implicit deflator for household consumption in measuring pt, the log of average unit labour costs for ulct, the log of the implicit deflator for imports of goods and services for ipt, and the log of real household consumption for ct, 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 data and some relevant transformations are presented in Fig. 1. Graph (A) depicts the log-levels of nominal prices while graph (B) – (D) depict the first differenced variables. Over the sample period the total increases in consumer prices, pt, and unit labour costs, ulct, have been quite similar, while the increase in import prices, ipt, has been somewhat smaller. These differences are also reflected in Graph (E), in which real import prices, iptpt, are more obviously negatively trended than real unit labour costs, ulctpt. A further interesting feature is that for much of the past 30 years real unit labour costs and real import prices have been negatively correlated. For example, real unit labour costs declined following an increase in real import prices after sterling exited the ERM in 1992. Such co-movements suggest that real wages accommodate the impact of real import price increases; and this effect turns out to be an important part of the empirical findings in Section 5.
Fig. 1

The data used for the empirical analysis 1968:1–2000:4

The final variable in our empirical analysis is the growth rate of real consumption, Δct. 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 (pt : ulct : ipt)′ 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

As a basis for the empirical analysis we first set up a congruent statistical model for the data. The period is characterised by a number of extreme episodes induced by policy interventions or shocks to variables outside the current information set. We do not want our results to be driven by a few extreme shocks and therefore condition on six intervention dummies of the form (0,...,0,1,−1,0,...,0), namely
$$ D_{t} = {\left( {D73q1:D74q1:D75q1:D75q3:D79q2:D80q1} \right)}^{\prime }_{t} . $$

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, β2Xt, 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.

In order to determine the appropriate lag length, k, for the unrestricted VAR model, Table 1 reports LR tests for successive lag deletions and various information criteria; see Lütkepohl (1991). The results clearly point towards two lags, and we set k = 2 in the analysis that follows.
Table 1

Lag length determination


Information criteria

Likelihood ratio test










































SW, HQ and AIC are the values of the Schwarz, Hannan–Quinn, and Akaike information criteria, respectively. The Likelihood Ratio tests km are the F-transforms of the tests for the last mk lags being insignificant. The reported figures are the corresponding p values

The results of a battery of mis-specification tests for the unrestricted VAR(2) are reported in Table 2. These are generally satisfactory, although the multivariate test for no residual autocorrelation up to order eight is formally rejected at the 5% level. There is no evidence of residual autocorrelation in the individual equations and steps that might remedy the problem, for example including additional lags in the model, do not affect the main conclusions of the analysis.4 We therefore take the VAR(2) model as a framework within which to analyse the long-run properties of Xt.
Table 2

Tests for mis-specification of the unrestricted VAR(2) model







0.00 [0.99]

0.87 [0.54]

0.55 [0.82]

3.22 [0.20]


0.67 [0.41]

1.69 [0.11]

1.77 [0.09]

1.38 [0.50]


1.04 [0.31]

1.62 [0.13]

0.97 [0.46]

3.19 [0.20]


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]

Figures in square brackets are p values. AR(1) are the F-tests for first order autocorrelation and are distributed as F(1,111) and F(16,321) for the single equation and vector tests, respectively. AR(1–8) are tests for up to eighth order autocorrelation and are distributed as F(8,104) and F(128,308), respectively. ARCH (8) tests for ARCH effects up to the eighth order and is distributed as F(8,96). The last column reports the Jarque–Bera asymptotic tests for normality, which are distributed as χ2(2) and χ2(8), respectively

I(2) Analysis

The rank indices (r, s) determine the cointegrating properties of the VAR system and define the sequence of partially nested models illustrated in the upper part of Table 3. The columns contain models with the same number of I(2) trends but different numbers of I(1) trends; the last column sets prs = 0 and thus contains the I(1) models. It should be noted that the presence of I(2) trends in the data imposes the additional reduced rank restrictions given in Eq. 5.
Table 3

Test for the rank indices of the I(2) model


Partial nesting structure






























LR tests of Hr,sH4


519.30 [0.00]

238.46 [0.00]

165.79 [0.00]

126.44 [0.00]

117.42 [0.00]



102.30 [0.00]

157.58 [0.00]

70.35 [0.00]

64.56 [0.00]



55.43 [0.01]

23.92 [0.49]

20.12 [0.22]



11.00 [0.56]

7.24 [0.33]







Likelihood Ratio tests for the rank indices (r, s), see Nielsen and Rahbek (2003). The appropriate asymptotic distributions are given in e.g. Rahbek et al. (1999). The p values in square brackets are derived from approximate Γ−distributions, see Doornik (1998).

The rank indices (r,s) can be determined via repeated applications of Likelihood Ratio (LR) tests for Hr,sH4,0, see Johansen (1995) and Nielsen and Rahbek (2003). The idea is to first test the most restricted model, H0,0, against the unrestricted VAR, then H0,1, 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, Hr,s, against the unrestricted model, H4,0, 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 H2,1 generates a test statistic against H4,0 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 prs = 1 I(2) trend. A more direct test for the presence of the I(2) trend implied by model H2,1 against the alternative of an I(1) model is suggested in Nielsen (2004b) as the LR test for the hypothesis H2,1H2,2. 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 H2,2 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, H2,1, 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 Xt = (pt : ulct :ipt : Δct)′ implies that the loadings applied to the I(2) trend in the nominal variables are proportional, i.e., span(β2) = 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 β2 is orthogonal to τ=(β:β1), 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 χ2(3) distribution.

Given homogeneity we can reparametrise the I(2) model to the well-known vector error correction form
$$ \Delta Y_{t} = \widetilde{\alpha }\widetilde{\beta } * \prime {\left( {\begin{array}{*{20}c} {{Y_{{t - 1}} }} \\ {t} \\ \end{array} } \right)} + \widetilde{\Gamma }_{1} \Delta Y_{{t - 1}} + \widetilde{\mu }_{0} + \widetilde{\phi }D_{t} + \varepsilon _{t} , $$
for the transformed I(1) data Yt = (XtBXtv)′, where B = b and ∣bv∣≠0. This simplifies hypothesis testing relating to the long-run structure considerably; see Kongsted and Nielsen (2004). In the present paper we choose Yt=(ulctpt : iptpt : Δpt : Δct)′, to obtain a measure of real wages, ulctpt, real import prices, iptpt, and consumer price inflation, Δpt, as well as consumption growth, Δct. The cointegration rank, r, determined in the I(2) analysis carries over to Eq. 7, and the polynomially cointegrating relations from the I(2) model, St, are embedded in the new system as I(1) cointegrating relations.

Identifying the long-run structure within the I(1) Model

In the model (7) the space spanned by the columns in \(\widetilde{\beta }\) is identified but individual coefficients are not, and in order to exactly identify the coefficients we have to impose one normalisation and one restriction on each relation. Table 4 reports, under the heading \({\cal H}_{0}\), the coefficients of two exactly identified long-run relationships. The corresponding asymptotic standard errors are reported in parentheses beneath the coefficients. The first relation is normalised on consumption growth and a zero restriction is placed on the inflation term, consistent with one of the long-run relations being directly cointegrating. The second relation resembles Eq. 2. The loading coefficients of this exactly identified structure, \(\widetilde{\alpha }\), clearly suggest that the real import price is weakly exogenous for the long-run parameters of the model.
Table 4

Identification of the long-run structure


\({\cal H}_{0} \)

\({\cal H}_{1} \)

\({\cal H}_{2} \)

\(\widetilde{\beta } {^* } \)

\({\tilde {\text α} }\)

\(\widetilde{\beta } ^* \)

\({\tilde {\text α} }\)

\(\widetilde{\beta } ^* \)

\({\tilde {\text α} }\)


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)


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 (...)


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)


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 (...)


0.100 (0.047)

−0.018 (0.004)


0 (...)

−0.017 (0.004)


0 (...)

−0.016 (0.004)


Long-likelihood value







Test statistics



Asymp. p value



Asymp. distribution

χ2 (3)

χ2 (6)

Bootstrap p valuea



Asymptotic standard errors in parentheses. The trend is divided by 100, i.e. with increments of 0.01

aBootstrap p values are constructed by parametric resampling as proposed in Gredenhoff and Jacobson (2001). The estimated model under the null hypothesis is used as a data generating process to construct 10,000 pseudo samples with innovations drawn from \(N{\left( {\widehat{\Omega }} \right)}\), where \(\widehat{\Omega }\) is the estimated covariance matrix. On each sample the hypothesis of interest is then tested and the distribution of test statistics is used as an estimate of the small sample distribution

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 χ2(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).

We next impose the restrictions that real import prices are weakly exogenous, and that consumption growth does not react to disequilibrium in the pricing relation, leading to the structure \( {\user1{\mathcal{H}}}_{2} \). The test statistic for \( {\user1{\mathcal{H}}}_{2} \) against \({\cal H}_{0}\) is 9.45 and follows a χ2(6) distribution under the null. The marginal restrictions embodied in \({\cal H}_{2}\) compared to \({\cal H}_{1}\) are thus easily accepted, and the total structure is accepted with an asymptotic p value of 0.15 and a Bootstrap p value of 0.30. Under \({\cal H}_{2}\) the long-run inflation relation can be written as
$$ Δ p_{t} = - .261 \cdot {\left( {p_{t} - .797 \cdot ulc_{t} - .203 \cdot ip_{t} - .016 \cdot trend} \right)}, $$
indicating an import share of just above 20%, which is a plausible estimate for the sample period that we have used.

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

In order to examine the pattern of dynamic adjustment in more detail we now develop a parsimonious representation of the short-run structure. We first define the error correction terms as the deviations from the long-run relations obtained from the preferred long-run structure in \( {\user1{\mathcal{H}}}_{2} \), i.e.,
$$ \begin{array}{*{20}c} {{\text{ecm}}1_{t} = \Delta c_{t} } \\ {{\text{ecm}}2_{t} = \Delta p_{t} - 0.208 \cdot {\left( {ulc - p} \right)}_{t} - 0.053 \cdot {\left( {ip - p} \right)}t - 0.016 \cdot trend.} \\ \end{array} $$
The VAR model for (Δ(ulcp)t : Δ(ipp)t : Δ2pt : Δ2ct)′ conditional upon the error correction terms is then estimated by Full Information Maximum Likelihood (FIML). The unrestricted system is highly over-parametrised and we use a general-to-specific strategy to simplify the dynamic structure. More specifically, we impose zero restrictions on individual coefficients if they are not significant at the 5% level (although leaving the intercepts unrestricted). This is done one coefficient at a time, starting with the least significant coefficient, and it turns out that 26 restrictions can be imposed in all. The FIML estimate of the restricted system is given in Table 5. The restrictions imposed are accepted with a LR statistic of 23.0 and a p value of 0.63 in a χ2(26) distribution. It is interesting to note that an automated general-to-simple reduction of a single equation model for Δ2pt implemented using the PcGets programme (Hendry and Krolzig, 2001) gives the same outcome as the system based reduction that we report here. The actual and fitted values for the parsimonious representation are reported in Fig.2 and indicate that the model is capable of tracking the evolution of the four variables quite well.
Table 5

Short-run structure

\( \Delta p_{t} - \theta \cdot {\left[ {\gamma \cdot {\left( {ulc_{t} - p_{t} } \right)} + {\left( {1 - \gamma } \right)} \cdot {\left( {ip_{t} - p_{t} } \right)}} \right]} \sim I{\left( 0 \right)}, \)

Note: Standard errors in parentheses.


Actual and fitted values from the parsimonious reduced form

The structural stability of the parsimonious reduced form system can be assessed by means of recursive estimation, as in Doornik and Hendry (2001). The recursive output is generally satisfactory. In particular, there is no evidence of structural breaks. Figure 3 reports the results of one-step Chow tests and break-point Chow tests for each of the four variables and for the system. In each case the null hypothesis is that the parameters are constant. For test statistics computed at time t, the one-step Chow test evaluates constancy between t−1 and t, and the break-point Chow test evaluates constancy across the two sub-samples running from the start of the full sample to t − 1, and from t to the end of the full sample; see Doornik and Hendry (2001) for further details. As the test statistics are always less than one, the null hypothesis of model stability cannot be rejected from 1975 onwards.
Fig. 3

One-step Chow tests and break-point Chow tests based on recursive estimation of the simplified reduced form, see Doornik and Hendry (2001)

The tested down dynamic model provides insights concerning inflation evolution in the United Kingdom. In the equation for Δ2pt 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(ulcp)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 Δ2pt 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(ulcp)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 correlation matrix for the reduced form residuals casts further light on the interpretation of the short-run structure. This is given below (recall that the ordering of the variables is Δ(ulcp)t, Δ(ipp)t, Δ2pt, Δ2ct).
$$ {\text{Correlation}}\,{\text{matrix}}:{\left( {\begin{array}{*{20}c} {1} & {{}} & {{}} & {{}} \\ {{ - 0.193}} & {1} & {{}} & {{}} \\ {{ - 0.302}} & {{ - 0.083}} & {1} & {{}} \\ {{0.248}} & {{ - 0.021}} & {{ - 0.43}} & {1} \\ \end{array} } \right)} $$

The large negative correlation of −0.43 between the residuals from the Δ2pt and Δ2ct equations is most likely the result of the construction of the transformed variables. Real consumption expenditure is constructed as Ct = NCt / Pt, where NCt 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 Δ2ct = Δ2nct−Δ2pt and Δ2pt. The residual correlation can be transformed into a simultaneous effect between Δ2ct and Δ2pt 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 ecm1t−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 Δ2pt and the equations for Δ(ulcp)t and Δ(ipp)t are most likely the result of the deflator pt appearing in the transformed variables. Once again these correlations can be reduced by including contemporaneous values of Δ2pt in the Δ(ulcp)t and Δ(ipp)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(ulcp)t and Δ(ipp)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(ulcp)t and Δ2ct indicates a small capacity effect operating on impact besides the effect operating trough the error correction term ecm1t−1.


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).



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).

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