Uncover Latent PPP by Dynamic Factor Error Correction Model (Df-Ecm) Approach: Evidence from Five OECD Countries

This study explores a new modelling approach that bridges the gap between multilateral country-level data and the bilateral-model based, goods-market specific purchasing power parity (PPP) hypothesis. Under this approach, PPP is embedded in latent common factors, extractable from a large set of bilateral price disparities, and tested via an error-correction model where the factors act as error-correction leading indicators for exchange rate and inflation. Significant modelling results for five OECD countries using monthly data suggest that the extant finding of insignificant PPP using similar data should be due to errors-in-variables attenuation and that its correction lies in effective construction of latent variables. --


Introduction
It is widely observed that real exchange rates exhibit slow mean reversion and weak equilibrating power over the dynamic adjustments of nominal rates. The phenomenon forms the basis of the PPP (purchasing power parity) puzzle, i.e. empirical verification of the 'Law of One Price' (LOP) underlying PPP is much weaker than expected, cf. Obstfeld and Rogoff (2000). The puzzle has been attributed to the considerable gap between what the PPP theory assumes and the conditions of available data, especially macro data (e.g. see Taylor and Taylor, 2004;Sarno, 2005). Two issues have come to the fore -aggregation and dynamics. Concerns over aggregation are focused on the fact that heterogeneity among types of traded goods, rates of trading costs as well as heterogeneity between traded and non-traded goods across different countries is simply too pronounced to assume away empirically. A direct solution is to test the theory at a micro level, e.g. the studies carried out by Barrett (2001), Barrett and Li (2002), and Parsley and Wei (2004); a more elaborate method is to try to filter out the heterogeneous features considered to be highly significant from disaggregate panel data before inferences on PPP at a certain aggregate level are made (see e.g. Crucini et al. 2005;Imbs et al. 2005). As for dynamics, time-series studies show that different dynamic features exist not only between exchange rate and price but also among prices of different countries. Nonlinear models are used by Taylor et al. (2001) and Sarno et al. (2004) to characterise the complicated price dynamics; various VAR models and dynamic panel methods are exploited to study the exchange rate pass-through to different prices and in different countries. The literature is still growing (see e.g. Bussière 2007 for a recent survey). The present study attempts to tackle the two issues together via a novel route. The key contention here is that it is inadequate to attack dynamics alone without considering the attenuation issue due to aggregation when country-level data are used. In fact, the source of the problem is wider than aggregation. The theoretical base of PPP is a bilateral, goods-market model, in which a domestic economy trades with a 'foreign' entity. In reality, all single countries face multilateral purchasing power disparities and interest rate disparities with numerous foreign economies each with different resource endowments, goods and capital market traditions, as well as different policies that interfere frequently with its market conditions. The gap between theory and countrylevel data is simply too substantive to ignore. We propose to treat the gap as an 'errorsin-variables' issue and to deal with it by taking the bilateral-model based 'foreign' variables in PPP as latent. 1 Specifically, we assume that PPP is embodied in the common factors of a dynamic factor model (DFM) comprising bilateral purchasing _________________________ power disparities of a home country with a large number of foreign economies. This amounts to filtering the heterogeneous part of the data into the idiosyncratic errors of the DFM and discarding them as measurement errors. Once the common factors are extracted, they are postulated as proxies of the disparities driving the price and exchange rate adjustment of the home country. The PPP postulate is then tested via the error-correction model (ECM): a convenient form of dynamic models as it not only facilitates the commonly adopted presentation of PPP as a long-run equilibrium condition but also verifies the condition in a much more stringent manner than what mean-reversion tests or simple cointegration analysis can achieve (see e.g. Johansen 2006).
The above procedure is referred to as the dynamic-factor error-correction model (DF-ECM) approach. The DF-ECM approach is initially explored by Qin et al. (2007) for the purpose of measuring regional market integration, and its trial application to the developing Asian region has yielded encouraging results. The present study develops the approach by applying it to the verification of PPP for five OECD countries. Thirty foreign economies are chosen to represent the world market and their price disparities vis-à-vis each of the five countries form the basis of dynamic factor analysis (DFA). Monthly data for the period of 1975-2005 are used.
The rest of the paper is organised as follows: The next section presents the DF-ECM approach; Section 3 describes practical issues pertinent to the implementation of the approach; Section 4 discusses the main findings from the five cases; The last section concludes with a short summary.

2
Method of Investigation: The DF-ECM Approach

The DF-ECM procedure
Let us start with the real exchange rate, Q, defined by PPP: where P denotes the aggregate price level of the home economy of interest, P f denotes the price level of the corresponding foreign economy and E is the exchange rate measured in the foreign currency units per unit of the domestic currency. An increasingly common way of testing PPP is to extend (1) into a dynamic model of the following log-linear form and study the mean reversion parameter, β (see, e.g. Koedijk et al. 2004): is a finite-order lag polynomial and u t is the residual term. A shortcoming of (2) is its restriction of the dynamic characteristics being www.economics-ejournal.org identical of, EP and P f , the two price variables denominated in the same currency. 2 We relax this restriction and re-parameterise the model into an ECM: Similar to (2), the variables in small capital letter are logarithms of the corresponding variables in (1). An attractive feature of (3) is that its explanatory variables are presented by two types of structurally interpretable and empirically almost uncorrelated shocks -short-run shocks (the first two terms on the right-hand side) and a long-run disequilibrium shock (the third term), see Qin and Gilbert (2001). Notice that the long-run shock actually plays the role of a leading indicator with error-dampening capacity. Empirical verification of PPP by model (3) entails a significantly negative feedback coefficient, 0 < φ , signifying that the exchange rate adjusts to maintain PPP in the long run. Unfortunately, the coefficient estimates are found to be insignificant in numerous studies where country-level data are used, especially for quarterly or monthly data. When found significant, as with some cases using annual data covering very long periods, the estimates tend to be extremely small. This constitutes the so-called PPP puzzle as described at the beginning of the paper.
Here, we attribute 'errors-in-variables' attenuation as a substantive cause of the problem. As mentioned in the previous section, a considerable gap exists between the extremely abstract PPP hypothesis and the available country-level data. PPP holds only on the basis of a number of conditions -there are only two countries trading; each tradable goods follows 'law of one price'; the factor prices and production functions of the non-tradable parts of the two economies should be identical; their aggregate price indices are perfectly comparable; and, of course, the two goods markets are completely open, without capital market friction or policy interference (see e.g. Isard 1995). Judging by these conditions, errors are inevitably part of the variables f and q, of (3) when these are represented either by data series from one country selected as the 'numéraire' foreign counterpart or by certain weighted aggregates of a group of countries. It is well-known that attenuation becomes non-negligible when the error/noise part of the data is persistent and substantive, as it can bias the OLS estimator in a regression towards zero. p Δ From the standpoint of an applied modeller, an effective way to correct attenuation caused by diverse measurement errors is to construct latent variables via common factor models. Here, we propose to view the foreign variables in (3) as latent and corresponding to certain common shocks of the world. These shocks are extractable by means of DFMs. Let the set of all countries be , the set of foreign countries vis-à-vis country d, the home country of interest, be . Two DFMs are needed for measuring respectively the latent long-run shock, q, and the latent short-run shock, f p Δ , in (3). The first is set to extract common factors from all the observable, bilateral real rates of economy d vis-avis each of the foreign economies, i.e. with . Defining , we assert for country d: In (4), is an m-vector of latent common factors with , which are thereafter referred to as the long-run factors, * Ψ is a parameter matrix and is a vector of lag polynomial, and ( ) L * Λ * ε * ν are error terms with the former being idiosyncratic shocks of the foreign economies vis-a-vis country d. In factor analysis, is commonly referred to as the 'indicator set' or the set of 'manifest variables'. f q Similar to (4), the second DFM for extracting the latent short-run shocks writes as: where the indicator set is a vector of the short-run foreign inflation shocks, and ξ is an l-vector of latent common factors with , thereafter referred to as the short-run factors.  (4) and (5) into (3) leads to a DF-ECM model: Notably, the present DF-ECM approach differs from most of the recent econometric studies involving DFMs, such as the ALI (automated leading indicator) approach linking DFM with VAR (vector auto-regression) by Camba-Mendez et al. (2001), and the extended structural VAR models by common factors explored by Forni et al. (2003), Bernanke et al. (2005), Favero et al. (2005) and Stock and Watson (2005). The common factors in those studies are extracted from indicators of different entities, whereas the indicators are of the same entity in the present case. The DFMs are used here primarily for filtering out measurement errors. In that sense, our approach bears close resemblance to the method of structural equation models with latent variables (SEMWLV) widely used outside econometrics, e.g. see Bedeian et al. (1997), Wansbeek and Meijer (2000), where models like (4) and (5) are referred to as the measurement equations and models such as (6) are labelled as the structural equations. However, (6) is a simpler structural equation in the sense that the modelled endogenous variable is not latent, unlike what is normally assumed in SEMWLV literature. On the other hand, both our measurement equations and our structural equation are dynamic, whereas most models in SEMWLV literature are static. Figure 1 illustrates the static version of our approach via a path diagram.

Figure 1. Static Path Diagram for Equations (4), (5) and (6)
Notice that (6) can be extended into two variants through relaxing the term , which effectively allows for different dynamic pass-through of t ( ) p e + Δ e Δ and t . This is useful when two types of exchange rate regimes are considered. When exchange rate is fixed or under tight control, PPP works primarily via domestic price changes. Hence we have: Whereas under the regime of a free-floating currency, the nominal exchange rate is expected to shoulder most of the adjustment with respect to PPP: As the number of parameters in (6a) or (6b) rapidly increases when m and l are larger than two or three, the computer-automated model reduction software, PcGets, is employed for primary model simplification search, or, using the software's terminology, 'testimation'. The key advantage of PcGets is that it carries out testimation by the general → specific approach in a consistent and efficient manner. This means that the specific model thus produced is guaranteed to be data-coherent and parsimoniously encompass the general model at the starting point, see Hendry (1995), Hendry and Krolzig (2001), Owen (2003), Phillips (2005). In other words, the specific model has survived all the commonly used diagnostic tests.

Useful Statistic Indicators
A number of statistics and parameter estimates are particularly useful for informing us about the power of PPP. Some are from the ECM procedure, and others from the DFMs. The first and foremost is the vector of the feedback coefficients, Ф, in (6). Note that the signs of these coefficients depend upon the signs of the relevant coefficients in of (4), e.g. . Since there is more than one long-run factor in most cases, a simple linear combination of the surviving factors from PcGets testimation is carried to yield one EC (error correction) term (see the next section). The next sets of statistics are summaries of the model fit from the PcGets testimation. These include, respectively, the adjusted R 2 , Schwarz information criterion, the numbers of parameters of the general model at the start of testimation and of the specific model at the end. Since PcGets conducts testimation based on an array of parsimonious encompassing tests, there is no need for us to check and report these diagnostic tests here.
A popular means of verifying PPP empirically is unit-root analyses of real exchange rates. However, it has been shown that different testing methods can generate conflicting results, e.g. by Pipatchaipoom and Norrbin (2006), and that the unit-root approach may be too restrictive with respect to economic reasoning, e.g. by Coakley et al. (2005). We believe that the present ECM approach is more stringent than unit-root tests. Nevertheless, several unit-root tests are performed on the EC terms of the DF-ECMs at the final stage.
Two useful statistics are derived from the DFMs. The first is the correlation coefficient of each indicator variable, , with its fitted value by the DFMs. This statistic is referred to as 'communality' in factor analysis when all the indicator variables are standardised.  if based on (4). This statistics exploits the fact that all indicators are of the same entity. We refer to this statistics as the covariation coefficient. A time series of these coefficients is expected to show how the panel of bilateral PPPs for one economy co-moves with the set of the common factors over time. It also serves as an indication of the size of the measurement errors in the form of idiosyncratic shocks.

Implementation of the DF-ECM Approach
The DF-ECM approach is applied to five OECD countries: Canada, France, Germany, Japan and UK. Monthly data are collected for the period of 1975-2005. 4 These include consumer price indices (CPI) and US dollar denominated exchange rates. The Appendix Table A1 gives the details of all the series and their sources.
_________________________ 3 Tucker and MacCallum (1997) give detailed discussion about the statistics. As the number of long-run common factors may vary across different countries, adjusted R 2 is used here instead of the simple R 2 . 4 In the earlier drafts of the paper, quarterly models were also presented as it was uncertain before any experiments whether monthly models would generate any significant results. The quarterly model results are now omitted to make the paper shorter. www.economics-ejournal.org

Implementation of DFMs
Choice of the indicator set: In addition to the above five countries, twenty six economies are selected by the criterion that the selected country set covers over 70%~80% of the total trade for each of the five countries. 5 This makes d contain 30 economies and N=31 for each of the five. All the indicator series are adjusted to zeromean series. The long-run indicator sets are also standardised, but the short-run indicator sets are not as the short-run indicators are already US$ comparable foreign inflation rates.

N −
Determination of the number of factors: Two recently developed procedures of consistent estimators are utilized. One is developed by Bai and Ng (2007) and the other by Onatski (2005). The larger of the two estimates is adopted when they differ. Table 1 reports the estimated results of the two procedures. Factor extraction: DFMs (4) and (5) are estimated using the technique developed by Camba-Mendez et al. (2001). Kalman filter algorithm is used with the initial parameter estimates obtained via principal component analysis. One advantage of this is that the algorithm can handle an unbalanced data panel like ours, where the CPI data series start later than 1975M01 for countries like China and Czech Republic, and quarterly CPI like that of Australia (see the Appendix Table A1). As for the short-run indicator set, there are only 29 indicators when Australia drops out. Determination of the number of lags: The experiment starts from L=1 and moves on to L=2 and L=3. A lag number is then chosen with reference to information criteria, such as Akaike and Schwarz criteria. It is found through numerous DFM experiments that one lag is adequate for the extraction of short-run factors by (5) whereas two or three lags are necessary for long-run factors by (4). The results are given in Table 1.

Implementation of DF-ECMs
Models (6a) and (6b) are the focal point of experiments, though (6) is tried first for each country (to keep this paper brief, the results are not reported here). OLS is used for model estimation. Notice, however, that the estimation method is comparable to a 2SLS _________________________ (two-stage least squares) procedure, where common factors extracted from (4) and (5) are used effectively as IVs (instrumental variables) to circumvent the errors-in-variables problem. As mentioned before, model simplification is a primary task here. We start by trying various lag lengths and found that six lags are generally adequate. The default setting of liberal model selection by Hendry and Krolzig (2001) is used for model testimation. Since coefficient constancy is a major concern, model testimation is performed for different sample periods, starting from the full sample, then for subsamples of 1980-2005 and 1985-2005 respectively. The resulting specific models are further simplified, mainly through reparameterisation and linear combination of the long-run factors, using PcGive (for details on reparameterisation, see e.g. Hendry 1995).
Recursive estimation is used here to monitor coefficient constancy. Hansen parameter instability test (1992) is also calculated.

Application Results
Data series of the both modelled variables, t e Δ and t p Δ , for each of the five countries are plotted in Appendix Figures A1-A5. In order to compare the DF-ECM results with conventional results, standard ECMs for the five countries are run using real effective exchange rates (REERs) as the EC terms (see Appendix Table A1 for data information of these REERs). REERs are chosen here for the main reason that it is more comparable to the multilateral setting of the DFM-based real rate measures than bilateral real rate measures. Besides, REERs have been used in empirical tests of PPP by numerous researchers (see e.g. Corbae and Ouliaris 1991, Bahmani-Oskooee 1995, Ellis 2001, Chinn 2006).

General results
The most noticeable result from Appendix Tables A2-A6 is that the DFM-based real rates, i.e. the long-run EC terms, are all significant and that their feedback coefficients display a high degree of constancy, as shown by the Hansen test statistics given under the coefficient estimates. The constancy can also be seen from the recursive estimation graphs plotted in the bottom panels of Appendix Figures A1-A5. In contrast, the longrun EC terms in the form of ( ) REER ln of those standard ECMs are all insignificant except for model (6a) in Japan and (6b) in UK, where, however, the Hansen statistics reveal significant coefficient instability. 6 The insignificance of ( ) REER ln is consistent with the extant finding in the literature. The cause is often attributed to the nonstationary feature of REER. This is reconfirmed by the unit-root tests on the series shown in Table 2. In the table, unit-root tests of the DF-based EC terms are also presented. It is easy to see that the nonstationary feature is more pronounced in than in the DF-based EC terms, though the test results on these latter terms are quite mixed, reinforcing the findings by Pipatchaipoom and Norrbin (2006).
Here, we attribute the insignificance to another cause -measurement error attenuation. As seen from the graphs of { } 2 t τ based on DFM (4) in Appendix Figures A1-A5, these covariation coefficient series remain small (mostly well below 30%) and erratic, suggesting that idiosyncratic errors form a major part of the data at each observation point. In other words, substantive measurement errors are present in f q if the set is used directly to construct the theoretical entity of real exchange rates, such as REER. Notably, the measurement error problem may not be unrelated to the nonstationary problem, since one source of nonstationarity is accumulation of independent errors. Inspecting the rescaled plots of the ( ) REER ln series together with those series in Appendix Figures A1-A5, we can see that the * ξ ( ) REER ln series tend to exhibit longer periods of random drifts than the series in general. The unscaled plots show that the series are far less volatile than the series. That explains why the coefficients of the DFM-based EC terms are substantially smaller in magnitude than those of . On the other hand, we see from these graphs that the series are different between models (6a) and (6b) Note: The sample periods used correspond to those used in the model estimation and reduction (see Appendix Tables A2-A6). ADF denotes augmented Dickey-Fuller test; DF-GLS is Elliott-Rothenberg-Stock test (1996); Only MZ t out of the four tests in (Ng-Perron, 2001) is reported to save space. *, ** and *** indicate rejection of the unit-root null hypothesis at 10%, 5% and 1% respectively. The numbers in parentheses are the number of lags used in the tests and these numbers are chosen on the basis of information criteria. The number in the square brackets of Phillip-Perron test (1988) is bandwidth determined by means of Bartlett kernel.
finding further supports the view that the PPP hypothesis is deeply latent in aggregate data which are full of noises interwoven in complicated dynamics. As for the expected signs of the coefficients of the significant long-run factors, these can be checked against Table 3, where (j=m) and the associate standard errors from DFM (4) are reported. Since all the standard errors are fairly large, the implied 95% confidence intervals are generally too wide to restrict any of the feedback coefficients in (6a) or (6b) within the strictly negative range. In terms of the adjustment speed, it is interesting to note that the feedback coefficient estimates of the exchange rate models (6b) are larger in absolute value than those of the inflation models (6a). This evidence is in support of the common view that goods prices are far less responsive than nominal exchange rates to external shocks under the freely floating regime.
Notice that the short-run common factors play an important role in the DF-ECMs as well. This is particularly striking when the R 2 statistics between the DF-ECMs and the corresponding REER-based ECMs are compared (see Appendix Tables A2-A6). On the whole, exchange rates are more responsive than inflation to the short-run factors and react to them in a more instantaneous manner. This feature renders support to the relative version of PPP.
As five short-run factors and five to six long-run factors are found necessary for each country, automated model testimation by PcGets becomes essential, as shown in Table 4. In fact, a great deal more of testimation experiments have been carried out than what is reported here. One particular feature easily revealed during PcGets testimation is that the DF-ECMs do not fit well with subsamples including the prior-1980 data for some countries, e.g. Japan. On the whole, the DF-ECMs fit better with post 1980 subsamples than the full sample. If the adjusted R 2 statistics in Table 4 are compared with those of the DF-ECMs in Appendix Tables A2-A6, one can easily see that further model reduction through reparameterisation helps to improve model fit moderately.
Finally, let us look at the correlation coefficients between the indicator sets and their explained parts by DFMs (4) and (5) respectively. The coefficients are given in Table 5 and Table 6 and ranked by size. Two features are worth commenting on. First, the www.economics-ejournal.org correlation coefficients in Table 5 are substantially larger than those in Table 6, indicating that slow mean reversion must prevail among the bilateral real rate series of the indicator set of DFM (4). Secondly, the correlation rankings across countries are far more similar in Table 6 than in Table 5. This is because the short-run indicator sets differ from each other only by one indicator, namely that of the home country under study. Notice also that France, Germany and Japan rank fairly high in the coefficient sequences of Table 6. This helps to explain why short-run common factors play such a significant role in the DF-ECMs of these three countries. Note: Three-month rates are used here to extract the short-run common factors by DFM (5) because the Australia CPI series is in quarterly only. However, the short-run common factors used in the DF-ECM models are obtained from monthly rates. The Adjusted R 2 is used, instead of the simple R 2 in order to make comparable the cases with different numbers of factors.

Individual countries
Canada: The DF-ECMs show reasonable fit (see Appendix Table A2) with fairly constant long-run coefficients (see Appendix Figure A1). The long-run coefficients in (6a) are clearly consistent with the positive coefficient estimates of , and from DFM (4) in Table 3. As for , the large standard error of 1.132 (Table 3) makes its 95% confidence interval cover as low as -1.65, well allowing for the positive feedback coefficient of +0.0002 in (6a) of Appendix Table A2. The feedback coefficient estimate of (6b) is about three times of that of (6a), indicating a much stronger PPP response in the exchange rate dynamics than the inflation dynamics.  Table A3). The two series for (6a) and (6b) are almost identical. The signs of the feedback coefficients are consistent with those of the factor loading coefficients from (4) implied in Table 3. * ξ Germany: Only survives the testimation in (6b), though the model fits remarkably well, even better than (6a), mainly due to the explanatory power of the short-run common factors (see Appendix Table A4). The relatively weak EC term here is also reflected in the unit-root test results in Table 2. * 3 ξ Japan: PcGets testimation reveals that sensible DF-ECMs only become possible for the post-1980 periods. In fact, only in the current-period does the first short-run factor survive in the full-sample testimation of model (6b) (see Table 4). This is also discernible from the recursive estimation graphs in Appendix Figure A4, where convergence to constancy of the feedback coefficients occurs around the end of the 1980s. The term is significant in model (6a) but its coefficient fails the constancy test (see Appendix Table A5).
(REER ln ) UK: Noticeably from Appendix Figure A5, the dynamic pattern of ( ) REER ln resembles that of the series of model (6b), except for the post-2000 period. This may help to explain why the * ξ ( ) REER ln -based EC term is significant in the comparable model. But the coefficients suffer from non-constancy (see Appendix Table A6).

Concluding Comments
This study explores a new modelling approach to empirically verify PPP. Under the new approach, PPP is embodied in latent common factors, extractable from a large set of bilateral price disparities, and tested via an error-correction model where the factors act as error-correction leading indicators for exchange rate and inflation. The indicators are found significant in monthly inflation and exchange rate models for five OECD countries. The finding reverses the commonly held belief, based on numerous previous results, that PPP is at best a very long-run relationship at the macro level, verifiable only with low-frequency data over very long sample periods.
A key reason for the present PPP evidence is that the new approach provides us with an effective means of correcting attenuation caused by the errors-in-variables problem. The source of the problem is the immense gap between multilateral country-level data and the bilateral-model based, goods-market specific purchasing power parity (PPP) hypothesis. So far, the problem has only been tackled via the use of micro market data. In the present study, country-level data are used and the errors are identified mainly as www.economics-ejournal.org the idiosyncratic shocks in DFMs and filtered out before the dynamic model containing PPP in the form of ECM is estimated. The PPP-based price disparities are treated as latent theoretical constructs.
Another advantage of the new approach is the combination of dynamic factors and the ECM approach. Conceptually, the long-run common factors match with the leading indicator interpretation of the EC term in an ECM, and the ECM lends its structural interpretation conveniently to both the long-run and the short-run factors. Empirically, the ECM and the associate general-to-specific modelling strategy renders more robust results than those by various means of nonstationarity tests. Appendix Table A2. Specific Models of (6a) and ( 2005M12. 2 R denotes the adjusted R 2 . The intercept term is kept in all models irrespective of its statistical significance in order to obtain the R 2 statistics. The statistics in the upper brackets under the coefficient estimates are the standard errors; those in the lower brackets are Hansen parameter instability test statistics. Its 5% critical value is 0.47. Statistical significance at the 5% and 1% levels are marked by * and ** respectively.