Abstract
We examine a two country model of the EU and the US. Each has a small sector of the labour and product markets in which there is wage/price rigidity, but otherwise enjoys flexible wages and prices with a one quarter information lag. Using a VAR to represent the data, we find the model as a whole is rejected. However it is accepted for real variables, output and the real exchange rate, suggesting mis-specification lies in monetary relationships. The model highlights a lack of spillovers between the US and the EU.
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Notes
In this study to achieve stationary data suitable for this analysis we follow SW’s practice in their EU paper of detrending each series with a linear trend and constant. This achieves stationarity for all our data, EU and US, according to standard ADF tests. Meenagh et al. checked whether the EU study was robust to the widely-used alternative Hodrick-Prescott filter and found little difference to the results, though the filter does take a lot more of the variation out of the data. Thus the linear detrending method used here has the advantage of suppressing as little of the data as possible. However, filtering data at all is a concern and we are working on methods that can use the raw nonstationary data.
We use separate linear time trends to detrend each variable on the grounds that in a short sample such as ours here a number of separate trends are present, affecting each variable differently. Thus there are output productivity, labour supply, and inflation target trends, as well as trend developments not explicitly modelled such as terms of trade and net foreign assets. Of course the same ad hoc approach is taken with other filters that ‘take out trend’ according to the series’ own movements.
Another issue arises in the choice of a VAR(1). Keeping the order of the VAR as low as possible reduces the complexity of the dynamics to be matched by the model, much as reducing the number of variables in the VAR does. Clearly it is possible to raise the order and the number, and so increase the challenge for the DSGE model. However, it appears from our work that the challenge from what we have chosen is quite enough.
For equations in which expectations do not enter these residuals are simply backed out. Where expectations enter the residuals must be estimated. For this we followed a procedure of robust estimation of the structural residuals along the lines suggested by McCallum (1976) and Wickens (1982) under which the expectations on the right hand side of each equation are generated by an instrumental variable regression that is implied by the model. The instruments chosen are the lagged values of the endogenous variables. Thus, in effect, the generated expectations used in deriving the residuals are the predictions of the data-estimated VAR.
It should also be noted that we excluded the first 20 error observations from the sample because of extreme values; we also smoothed two extreme error values in Q. Thus our sample for both bootstraps and data estimation was 98 quarters, i.e. 1975(1)–1999(2).
Formally, we model this as follows. We assume that firms producing intermediate goods have a production function that combines in a fixed proportion labour in imperfect competition (‘unionised’) with labour from competitive markets—thus the labour used by intermediate firms becomes \(n_{t}=n_{1t}+n_{2t}= \left\{ \left[ {\textstyle\int\nolimits_{0}^{1}} (n_{1it})^{\frac{1}{1+\lambda_{w,t}}}di\right] ^{1+\lambda_{w,t}}+\left[ {\textstyle\int\nolimits_{0}^{1}} (n_{2it})di\right] \right\} \) where n 1it is the unionised, n 2it the competitive labour provided by the ith household at t; we can think of n t as representing the activities of an intermediary ‘labour bundler’. Note that n 1t = v w n t , n 2t = (1 − v w )n t so that W t = v w W 1t + (1 − v w )W 2t . Each household’s utility includes the two sorts of labour in the same way, that is \(U_{it}=...-\frac{n_{1it} ^{1+\sigma_{n}}\epsilon_{1nt}}{1+\sigma_{n}}-\frac{n_{2it}^{1+\sigma_{n} }\epsilon_{2nt}}{1+\sigma_{n}}...\) W 1t is now set according to the Calvo wage-setting equation, while W 2t is set equal to current expected marginal monetary disutility of work; in the latter case a 1-quarter information lag is assumed for current inflation but for convenience this is ignored in the usual way as unimportant in the Calvo setting over the whole future horizon.
These wages are then passed to the labour bundler who offers a labour unit as above at this weighted average wage. Firms then buy these labour units off the manager for use in the firm.
Similarly, retail output is now made up in a fixed proportion of intermediate goods in an imperfectly competitive market and intermediate goods sold competitively. Retail output is therefore \(y_{t}=y_{1t}+y_{2t}=\left\{ \left[ {\textstyle\int\nolimits_{0}^{1}} y_{j1t}^{\frac{1}{1+\lambda_{p,t}}}dj\right] ^{1+\lambda_{p,t}}+\left[ {\textstyle\int\nolimits_{0}^{1}} y_{j2t}dj\right] \right\} \). The intermediary firm prices y 1t according to the Calvo mark-up equation on marginal costs, and y 2t at marginal costs.
Note that y 1t = v p y t , y 2t = (1 − v p )y t so that P t = v p P 1t + (1 − v p )P 2t . The retailer combines these goods as above in a bundle which it sells at this weighted average price.
Notice that apart from these equations the first-order conditions of households and firms will be unaffected by what markets they are operating in.
We form the Lagrangean \(L=\left[ \omega\left( C_{t}^{d}\right) ^{-\varrho}+\left( 1-\omega\right) \left( C_{t}^{f}\right) ^{-\varrho}\right] ^{\left( \frac{-1}{\varrho}\right) }+\mu(\widetilde{C_{t}}-\frac{P_{t}^{d}}{P_{t} }C_{t}^{d}-\frac{P_{t}^{f}}{P_{t}}C_{t}^{f}).\) Thus \(\frac{\partial L}{\partial\widetilde{C_{t}}\phantom{^{l^1}}{\kern-4pt}}=\mu;\) also at its maximum with the constraint binding \(L=\widetilde{C_{t}}\) so that \(\frac{\partial L}{\partial \widetilde{C_{t}}\phantom{^{l^1}}{\kern-4pt}}=1.\) Thus μ = 1—the change in the utility index from a one unit rise in consumption is unity. Substituting this into the first order condition \(0=\frac{\partial L}{\partial C_{t}^{f}.}\) yields Eq. 2. \(0=\frac{\partial L}{\partial C_{t}^{d}.}\) gives the equivalent equation: \(\frac{C_{t}^{d}}{C_{t}}=\omega^{\sigma}(p_{t} ^{d})^{-\sigma}\) where \(p_{t}^{d}=\frac{P_{t}^{d}}{P_{t}}\). Divide Eq. 1 through by C t to obtain \(1=\left[ \omega\left( \frac{C_{t}^{d}}{C_{t}}\right) ^{-\varrho}+\left( 1-\omega\right) \left( \frac{C_{t}^{f}}{C_{t}}\right) ^{-\varrho}\right] ^{\left( \frac {-1}{\varrho}\right)}\); substituting into this for \(\frac{C_{t}^{f}}{C_{t}} \)and \(\frac{C_{t}^{d}}{C_{t}}\)from the previous two equations gives us Eq. 4.
However, as noted by Chari et al. (2002), assuming complete asset markets imposes the condition that the real exchange rate equals the ratio of the two continents’ marginal utilities of consumption at all times. This implies that the expected log change in the real exchange rate equals the expected log change in this ratio, ie the real interest differential- the real UIP condition again. Thus the conditions are in practice similar: under complete markets the real exchange rate exactly moves with relative consumption whereas under incomplete it is only expected to do so, so that random shocks can drive them apart.
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We are grateful for comments to Sumru Altug, James Davidson, Hashem Pesaran, Roman Sustek, Ken Wallis, and other participants at this OER Bank of Greece conference, at seminars in the Hungarian Central Bank and the Bank of England, and at the Birmingham University conference on Macroeconomics and Econometrics in May 2009. We thank the UK Economic and Social Research Council for its support under grants RES-125-25-0020 and PTA-026-27-1623. Tables, figures and other details can be found in the full version at http://www.cf.ac.uk/carbs/econ/workingpapers/index_abstracts.html#2009/03.
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Le, V.P.M., Meenagh, D., Minford, P. et al. Two Orthogonal Continents? Testing a Two-country DSGE Model of the US and the EU Using Indirect Inference. Open Econ Rev 21, 23–44 (2010). https://doi.org/10.1007/s11079-009-9141-9
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DOI: https://doi.org/10.1007/s11079-009-9141-9