Does Purchasing Power Parity Hold for the German-US Real Exchange Rate? A Bayesian Dynamic Linear Model

In the purchasing power parity (PPP) literature, most studies do not take dynamics into account when they try to explain the deviation of PPP, and none use the Bayesian approach. This paper closes this gap by using the Bayesian dynamic linear model (DLM) to examine the German-US real exchange rate and test whether the Balassa-Samuelson effect explains the deviation from PPP. The results show that PPP does not hold during the period examined and the negative sign of the coefficient of German productivity differential between tradable and nontradable sectors and the positive effect of the productivity differential violate the assumption of the Balassa-Samuelson effect.


Introduction
Purchasing power parity (PPP) is one of the most controversial topics in international economics.Froot and Rogoff (1995) gave a thorough literature review of what we know and do not know about PPP.They classified PPP tests into three periods.In stage one, people examined the relationship between the nominal exchange rate and price levels and tested whether beta equals one.The tests in the second period assumed that beta is the unit and tested the stationarity of the real exchange rate.In stage three, researchers used cointegration techniques to test a weaker version of PPP, a linear combination of prices and the nominal exchange rate.
However, empirically, PPP does not fit the data well.Many studies tried to explain this PPP deviation by including fundamental variables such as productivity and government spending.From the supply side, the Balassa-Samuelson effect provides a plausible explanation for the change in relative internal prices and thus the real exchange rate.Among those attempts, Hsieh (1982) was the first to examine the time series properties and confirmed the effect.Marston (1987) and Edison and Klovland (1987) also endorsed the effect.However, Froot and Rogoff (1991) and Asea and Mendoza (1994) showed that the effect is weak at the best.In addition to the mixed empirical results, as Froot and Rogoff (1995) pointed out, most of the studies did not include dynamics explicitly.
Researchers continued using panel data (e.g., Canzoneri et al., 1999;Koedijk et al., 1999;Chinn & Johnson, 1997) and cointegration techniques (e.g., Bahmani-Oskooee & Rhee, 1996;Dibooglu, 1996), and tried different models (e.g., Weber, 1997;SVAR model) to examine the determination of PPP from the supply side along with the Balassa-Samuelson effect tradition.In addition to the productivity differentials, Strauss and Ferris (1996) showed that the real wage gap between tradable and nontradable sectors may influence real exchange rates.Kakkar and Ogaki (1999) made the point that the relative price of nontradables and tradables does not necessarily move together with the real exchange rate because of different time periods, countries, and measures of relative prices.
In one of two more recent papers, De Carvalho (2002) presented a modified version of PPP and examined the yen-dollar rate over the years 1976-1991.He found that the increase in Japanese labor productivity is one cause of the appreciation of the yen.Chang (2002) introduced real wages in the estimation using the mean-squared error decomposition, and found that the real exchange rate and relative real wages are negatively related in the short run but positively related in the long run.
Since Froot and Rogoff's 1995 paper, to the best of my knowledge, few studies have explicitly taken dynamics into account, and none used the Bayesian approach.The purpose of this paper is to close this gap in the PPP literature by using the Bayesian dynamic linear model (DLM) to examine the dynamics of the real exchange rate under the flexible exchange rate regime, and try to explain the deviation of the real exchange rate using the Balassa-Samuelson effect.
The major advantage of the DLM over other traditional static models is that it takes dynamics explicitly in the model.For static models, the quantitative relationship is globally applicable.In other words, the parameters are constant through time.However, this assumption is especially dangerous in time series analysis.As time passes, the value of the information will decrease, and thus, the parameters will change over time.This time-varying parameter or dynamics of the model is realized through the system equation in the DLM.In addition, the DLM is a linear model, which is simpler and easier to use than nonlinear models, but it captures the nonlinear relationship between the dependent and explanatory variables in a dynamic way.Since the process updates consistently, the relationship between any two adjacent points is linear.Another advantage of the model is that the retrospective analysis fits the data better by using all the information available, not only the information up to time t but also all the information in the entire series.Moreover, the Bayesian approach takes external shocks in the analysis by adjusting the priors directly.By using probability distribution to represent knowledge about the parameters, the Bayesian approach makes the results easy to interpret.In addition, with robust priors, we get good results even from a small sample, which is always an issue in empirical macroeconomic analysis.This paper uses quarterly data for the German real exchange rate from 1974:1 to 2001:1 and steady DLM to test whether (Consumer Price Index, CPI) PPP holds.Based on the result that (CPI) PPP did not hold during this period, this study continues using annual data (1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000) for German and American productivity in tradable and nontradable sectors and a regression DLM to investigate whether the Balassa-Samuelson effect can explain this deviation.The negative sign of the coefficient of German productivity differential between tradable and nontradable sectors and the positive effect of the productivity differential violate the assumption that productivity grows faster in tradable sectors.
The organization of the remainder of this paper is as follows.In the next section, the theoretical background of PPP and the Balassa-Samuelson effect is discussed.In Section 3, the Bayesian methodological framework is outlined.Data is presented in Section 4. In Section 5, the empirical results are presented, and the last section concludes.

Theoretical Background
The basic equation for PPP is where Q is the real exchange rate; P is the domestic price level; S is the nominal exchange rate; and P* is the foreign price level.There are two versions of PPP: absolute and relative.Built on the law of one price (LOP), absolute PPP has the real exchange rate 1.Under relative PPP, Q is a constant.Balassa (1964) posited that to fully understand the PPP, nontraded goods should also be considered.Balassa (1964) and Samuelson (1964) argued that historically productivity in traded goods sectors grows faster than that of nontraded goods sectors.Because nontraded goods sectors are more labor intensive, the rise in productivity in the traded goods sectors will bid up the wage rate in the whole economy.As the result, the relative price of nontraded to traded goods and thus the CPI-based real exchange rate will increase.We can decompose the price level into two parts: the prices of tradables and nontradables.Specifically, suppose the overall price levels take the form where P T is the domestic price level of the tradables and PN T is the price level of the nontradables.Foreign country is denoted with "*".
To test the PPP and examine the Balassa-Samuelson effect, it is useful to substitute equation (2) into equation (1) and take the logarithm form, which gives If we assume the price of the tradables is 1, then With the Cobb-Douglas production functions for traded and nontraded goods, we have where μ L is the labor share of income in the sector and α is the log of the productivity of the sector.Assuming the labor shares of the income are the same for domestic and foreign countries, we then have If LT LN μ μ >1, that is, if the nontraded goods sectors are more labor intensive, then an increase in productivity in domestic traded goods will lead to an increase in the domestic relative price of nontraded goods, and thus the real exchange rate.

Methodological Framework
In this study, the most general model with regressors is as follows: ). , 0 ( , With the notation from the DLM framework, the response, regression vector, and system matrix are defined as: Using the framework and notation developed by Pole et al. (1994), we can write the DLM as Observation Equation ), , 0 ( , System Equation ), , 0 ( , Where Y t is the observation series at time t; Ft is the regression vector; θt denotes the vector of unknown parameters; v t is white noise with zero mean and variance v t and is normally distributed; G t is a matrix of the known coefficients that determine how the state vectors evolve systematically over time; and w t is white noise with zero mean and covariance matrix w t is normally distributed.
Theoretically, it is assumed that variances V and W are known.However, in empirical analysis, they are not.The Bayesian approach deals with this problem by using the information-discounting factor, which is between zero and one.The idea is very simple: The farther away the information is from today, the less useful the information is for forming a future forecast.For example, a discount factor at 0.95 means that 5% of the information today will be lost tomorrow.In the DLM, this process is modeled through the system equation.When time passes, the uncertainty will increase, and thus, the variance will become larger.Then the key problem is choosing an appropriate discount factor.In practice, we choose the discount factor with the best forecast performance among a set of discount values.
After a model is selected, the first step for researchers is to form their priors, that is, to quantify their knowledge and assumptions in terms of the probability distribution.The DLM assumes normality; therefore, the mean and variance are enough to characterize the distribution.These priors, combined with information from the data, will yield the posterior distribution at time t using the Bayes' theorem.We then have This Bayesian learning process is the mechanism though which we can modify our uncertainty about the future when new information is available.One of the major advantages of the Bayesian methodology is that it incorporates new external information into the prior and thus into the formulation of the posterior.
The goal of Bayesian analysis is to obtain the parameter distribution conditional on all the information and beliefs.During the process from prior to posterior, observations with low probability may occur from unusual circumstances.We should be aware of those circumstances and decide how we should deal with them considering whether the "shock" is profound or not.We may want to adjust the model to produce more accurate forecasts.This monitoring process is part of the assessment of model specification and will be repeated whenever there is a sign of departure from the current model.
In addition to forward-looking processes, we can also start analysis from the end of the period and use all the information available to better understand the development of the time series.For example, our data set of the German real exchange rate spans 1974:1 to 2001:4.When we estimate the level of the real exchange rate of 1984:3 backwards, we take into account all the information not only up to and including 1984:3 but also include the data from 1984:4 until 2001:4.This backward-looking process is also called smoothing or retrospective analysis.

Data
The quarterly German mark and U.S. dollar exchange rates and CPIs spanning 1974:1 to 2001:4 come from International Financial Statistics published by the International Monetary Fund (IMF).
To test the Balassa-Samuelson effect, the productivity data is not available, and it is constructed from the ratio of production to employment.Because construction employment data is not available for (Western) Germany before 1983, the data used to test the effect is from 1984 to 2000.The primary production data comes from the International Sectoral Database (ISDB), which was published by the OECD.The data set is no longer published.The existing database includes annual data about value-added production, employment, and factor returns for 14 countries (Note 1) and 20 sectors (Note 2) from 1960 to 1997.The German production data from 1984 to 1990 is from the ISDB, and the data from 1991 to 2000 is from the OECD STAN Database for Industrial Analysis.The employment data from 1991 to 2000 for Germany is from STAN, and the employment data from 1984 to 1990 is calculated from the Structural Statistics for Industry & Services-Industry Survey, Economic Accounts for Agriculture, and the Main Industrial Indicators (Note 3).
The American production data from 1984 to 1993 is from the ISDB.The production data from 1994 to 2000 and the employment data are from the STAN.
Following the classification by De Gregorio et al. (1994), agriculture, mining, all manufacturing, and transportation aredefined as tradables and other sectors are non-tradables.Then the productivity in the traded and nontraded goods sectors is calculated by the ratio of production to employment.

Results
Using the DLM framework, this study starts with the steady model of the German real exchange rate and then adds productivity differentials to the model to test the Balassa-Samuelson effect.

Steady Model of the Real Exchange Rate
The steady model of the real exchange rate can be written as  Figure 2 shows the one-quarter ahead forecasts with a forecast horizon from 1974:1 to 2001:4.The dotted line is the observation series, the real exchange rate.The line in the middle is the forecast mean.The other two lines are the 90% uncertainty limits about the forecast mean.The values of the real exchange rate are very close to the predicted mean, but the uncertainty is huge.The forecast means are constant at one -the prior mean for the level.Notice the uncertain limits diverge over time.This is exactly the dynamic nature of the model: The uncertainty increases as time passes.Remember we start our forecasts from the beginning of the period and lose information as we move forward.
In the dynamic analysis, the information loss can be realized by the discount factors.We choose the best discount factor by comparing the forecasts results for a set of discount factors.The forecast performance summary is reported in Table 2.In the steady model, there is only one trend discount factor.The best values for measure are those with a trend discount around 0.8.Table 3 shows another set of forecast summary statistics with intervention, and the best discount is 0.8.This time the analyses were performed with BATS (Note 4) level decrease monitor on, which means we make use of the downward shift in the trends.Comparing the measures in Table 2 and Table 3, the measures are generally improved with intervention.From 1981:2, analyses were subject to monitor warning signs and corrected by the BATS default action.Because the run length is greater than one, the observations were not considered outliers and then ignored but treated as a tendency.With use of the downward shift in the trends, Figure 3 to Figure 5 show the results of dynamic analysis with a trend discount factor 0.8 and observation variance discount factor 0.99.  Figure 3 is the one-step forecast of the German real exchange rate, which includes information up to time t-1.In this one-step analysis, assume we have no priors about the data, and let BATS determine the start value after the software examines the series.The forecasts are consistently high from the late 1970s to 1986 and in the late 1990s but consistently low in the late 1980s.The point forecasts miss some of the series structure.The missing structure is confirmed in Figure 6, which indicates a strongly nonrandom autocorrelation pattern in the forecasts residuals.Positive autocorrelation in residuals is typically an indication of insufficient dynamic movement in the          Note.All the coefficients are with 90% uncertainty limits that vary over time.
Figure 14 through Figure 17 show selected results from the intervention analysis with discount factors 1 for the trend and regression components, and 0.99 for the observation variance (Note 6).Table 6 reports the smoothed coefficients of the analysis.The sign of the coefficient of the German productivity differential is negative, not positive as we predicted.The sign of the β2 is negative as the theory suggests.We can see in Figure 15, however, that the effect of German productivity is positive before 1991 and negative thereafter.Since the coefficient of German productivity is negative during the period, the positive effect suggests that the productivity differential between tradables and nontradables is negative from 1984 to 1991.There are two possible explanations for this result.One is that the tradable sector is more labor intensive, that is, μ LN <μ LN , which is less likely; the other Figure 1 exchange Figure 4 Figure 5 Figure 7. S = American pr nontraded goods.
Fig elation coefficient Figure is the prior related to a view of the parameter distribution at time t given the information up to time t-1;

Table 2 .
Forecast performance summary -Steady model Note.MSE is Mean Squared Error.MAD isMean Absolute Deviation.Loglik is Log Predictive Likelihood.* denotes best value for measure.Observation variance discount is 0.99.

Table 3 .
Forecast performance summary with intervention -Steady model

Table 4 .
Forecast performance summary -Regression model The numbers in each cell are Mean Error, Mean Absolute Deviation, and Log Predictive Likihood.*, Best value for measure.Observation variance discount is 0.99.

Table 5 .
Forecast performance summary with intervention -Regression modelNote.The numbers in each cell are Mean Squared Error, Mean Absolute Deviation, and Log Predictive Likihood.*, Best value for measure.Observation variance discount is 0.99.

Table 6 .
Smoothed coefficients of the regression model with intervention