APPENDIX 1. Estimating Equilibrium Real Exchange Rates in the Baltics in a Cointegration Framework
- Yuan Xiao, Robert Burgess, and Stefania Fabrizio
- Published Date:
- July 2004
This appendix considers the behavior of real effective exchange rates (REERs) in the Baltics over the period 1994–2002. The analysis is based on a theoretical framework that encompasses two principal determinants of movements in equilibrium real exchange rates: (1) the concept of internal equilibrium, based on the Balassa-Samuelson or productivity hypothesis (discussed in Chapter 4 in the main text), and (2) the concept of external balance, based on the asset market view of exchange rate determination. Long-run relationships between real exchange rates and these determinants are estimated empirically using cointegration techniques. An unobserved components analysis is then applied to identify the underlying time-varying equilibrium real exchange rates.
The concept of long-run or equilibrium real effective exchange rates (EREERs) has been widely addressed in the economic literature. One standard and traditionally used approach is the purchasing power parity (PPP) hypothesis. This implies a constant equilibrium exchange rate, as it posits that there is an underlying tendency for movements in the nominal exchange rate to offset inflation differentials with countries’ trading partners, such that deviations from the EREER are transitory. However, persistent exchange rate deviations from PPP equilibrium can be produced by several factors, including technical progress or, more specifically, productivity differentials, which, according to the Balassa-Samuelson hypothesis, can lead to changes in the relative prices of tradable to nontradable goods in an economy.
The Balassa-Samuelson hypothesis assumes that tradable goods produced in different countries are perfect substitutes, and that the nominal exchange rate adjusts to changes in tradable prices such that tradable prices across countries are equal when measured in a common currency. A lack of perfect substitution between traded goods, however, may also lead to deviations from PPP. Theories in this area have focused on the trade balance as the main determinant of the exchange rate, with capital flows being treated as exogenous shocks. With financial liberalization and the increasing volume of international trade in financial assets, modern exchange rate models emphasize financial-asset markets and the role of the exchange rate as one of many prices in the worldwide market of financial assets. Following these theories, trade flows have still a useful role in asset-approach models, since trade flows have implications for financial-asset flows. In fact, the exchange rate must be consistent with a balance of payment position where any current account is compensated by a sustainable flow of international capital. A country running a current account deficit/surplus will accumulate/lose assets, with such imbalances resulting from differing propensities across countries to save and invest, and it is assumed that such factors are not influenced by exchange market developments. In the long run, however, when agents’ assets are at their desired levels, the current account should be balanced.50
As a result, two main lines of research on the determination of EREERs have developed, which emphasize the sectoral (tradable versus nontradable) balance of the economy and the underlying net foreign asset position of the country, respectively. A model that encompasses both approaches is developed in the next section and forms the basis for the subsequent empirical analysis.
A. The Theoretical Framework and the Empirical Model
The theoretical model used follows that developed by Alberola and others (1999) and is based on the decomposition of the exchange rate into two different relative prices: the price of domestic goods relative to foreign tradable goods; and the relative prices of nontradable goods relative to tradable goods within each country. The first component captures the competitiveness of the economy and determines the evolution of the net foreign assets position, and is therefore associated with the external equilibrium of the economy. The second component incorporates the concept of productivity differentials identified in the Balassa-Samuelson hypothesis, and since these prices determine the allocation of resources within the economy, it is associated with the internal equilibrium of the economy. The long-run solution of the model represents an equilibrium value for the real exchange rate consistent with both the internal and the external equilibria of the economy.
Assuming that there are two countries in the world, each producing a tradable good (T) and a nontradable good (N), the REER (q) in logarithmic terms can be defined as
q = s + p – p*, (1)
where p and p* are the domestic and the foreign consumer price indices (CPI), respectively, and s is the nominal exchange rate. For each country, the CPI, which is formed by prices of domestic and foreign tradable goods and nontradable goods, can be expressed as follows:
where the αs determine the share of each good in the consumer price index. Substituting these expressions into (1), we obtain
where, for simplicity, the weights of nontradable goods for the two countries are assumed to be the same, and the lack of perfect substitution between tradable goods between different countries is also considered. From (4) it is clear that the exchange rate is determined by two different components: the evolution of relative prices of domestic to foreign tradable goods, which reflects the external dimension of the economy:
and the behavior of nontradable goods relative to tradable goods across countries, which relates the internal dimension of the economy:
Thus, the equilibrium exchange rate
The external equilibrium. The external balance clears the tradable goods market, and is characterized by the achievement of a desired stock of net foreign assets. The evolution of the current account balance, which determines adjustments to the equilibrium, leads to an accumulation of net foreign assets. The current account balance (ca) is defined as the trade balance (x) plus the net income received or paid by residents (r*) on foreign asset holdings (nfa) expressed in real terms:
ca = x + r* nfa. (7)
The trade balance depends on the evolution of the external real exchange rate,51 namely,
x = –γqx, (8)
and, following Mussa (1984), the current account adjusts to the difference between the current and the desired level of net foreign assets, so that a current account surplus would reflect a net foreign asset position below the desired level
In the long run,
The internal equilibrium. The evolution of the internal real exchange rate is determined by the different behavior of sectoral relative prices between countries, which in turn are related to the evolution of sector productivity. Starting from the productivity hypothesis, it can be shown that
where the ys are the average sectoral productivities. Neglecting constant terms, it follows that the equilibrium internal exchange rate can be expressed as follows:
Putting together the external and internal equilibria concepts produces the equation for the equilibrium REER:
where v is the speed of adjustment of net foreign assets to changes in relative prices; (k – k*) is the difference between measures of relative sector productivity at home and abroad (where k = yT – yN and
The empirical model. The theoretical model has identified two main determinants of the real exchange rate (q) in the long run: the stock of net foreign assets (nfa) and the relative sectoral prices between countries (n), and could be rewritten in the following form by factoring nfa:
In this form the equilibrium REER is a function of three variables: nfa, the difference between measures of relative sector productivity at home and abroad, and demand shocks. Abstracting from demand shocks and using price differentials in lieu of the relative sector productivity differential at home and abroad, we obtain our empirical model:
qt = β0 + β1nfat + β2nt + ut
Since our main objective is to compute the equilibrium exchange rate as a function of its fundamentals, we have to, first, establish the existence of a long-run relationship among the variables, and, second, we have to compute the equilibrium levels of the determinants nfa and n. In order to determine the existence of a long-run relationship among variables (i.e., to test for cointegration), we use the Johansen procedure for cointegration. To establish the equilibrium level of the REER, we assume that qt fluctuates around its long-term value, but is not permanently at that value. Moreover, in order to derive the equilibrium exchange rate, we also allow for the possibility of nfat and nt deviating from their long-run values. From an empirical point of view, the three variables in the system are decomposed into transitory [
Bearing in mind that a unique decomposition between permanent and transitory components does not exist, we consider the decomposition suggested by Gonzalo and Granger (1995), based on the assumption that shocks to the transitory component (i.e., our estimate of the misalignment) do not affect the permanent component (i.e., our estimate of the equilibrium).52
They derive a decomposition in which the transitory component does not Granger-cause the permanent component in the long run and in which the permanent component is a linear combination of contemporaneous observed variables. In other words, the first restriction implies that a change in the transitory component today will not affect the long-run values of the variables. The second restriction makes the permanent component observable and assumes that the contemporaneous observations contain all the necessary information to extract the permanent component. The decomposition is done using the identification implicit in the cointegration of the series. In particular, if cointegration exists among a number of variables, then the vector will have a common, or factor, decomposition. Gonzalo and Granger demonstrate that the common factor can be estimated if it is assumed to be a linear combination of the series under analysis and if it is further assumed that the residuals from this model do not have a permanent effect on the original series. The former assumption makes the common factor observable, while the second permits identification.
Analytically, consider a 3×1 vector xt = [qt, nfat, nt]′, which under the null hypothesis of one cointegration vector admits the following representation:
Δxt = A1Δxt–1 + … + Ap–1Δxt–p + Πxt–p + et, (17)
where et is a vector white noise process with zero mean and variance Σ and Π is a 3 × 3 matrix, whose rank will determine the number of cointegration vectors. If cointegration exists, Π is not full rank (r < 3, with r = 1 in our case) and can be written as the product of two rectangular matrices, Π = αβ′, where β is the matrix whose columns are the linearly independent cointegrating vectors and α is the factor-loading matrix, indicating the speed with which the system responds to last period’s deviation from the equilibrium level of the exchange rate. Next, one can always define the orthogonal complements α⊥ and β⊥ as the eigenvectors associated with the unit eigenvalues of the matrices (I – α(α′α)–1α′) and (I – β(β′β)–1β′), respectively. The matrix α⊥ is formed by the vectors defining the space of the common stochastic trends, and therefore should be informative about the key “driving” variable(s) in each of the systems, while β⊥ gives the loadings associated with the series that are driven by the common trends. Notice that α′⊥α = 0 and β′⊥β = 0. If the vector x is of reduced rank, r, Gonzalo and Granger have demonstrated that the elements of x can be explained in terms of a smaller number of (3 – r) of I(1) variables called common factors, ft, plus some I(0) components, the transitory elements,
The identification of the common factors may be achieved in the following way. If it is assumed that the common factors are linear combinations of the variables xt,
ft = B1xt, (19)
and if A1ft and
ft = α⊥xt. (20)
This identification of the common factors allows one to obtain the following permanent-transitory decomposition of xt:
xt = β⊥(α′⊥β⊥)-1α⊥xt α(β′α)-1β′xt, (21)
where the permanent and the transitory components are captured by the terms β⊥(α′⊥β⊥)–1α⊥xt and α(β′α)–1β′xt, respectively. Gonzalo and Granger show that the transitory components defined in this way will not have any effect on the long-run values of the variables captured by the permanent components. The identification of the permanent component with the equilibrium implies that
from which the estimation of the equilibrium exchange rate and its deviation follow directly.
B. Econometric Methodology
To understand the link between the concept of equilibrium and those of integration and cointegration, we start from the PPP hypothesis, which implies a constant value for the equilibrium exchange rate. In practice, this does not mean that the real exchange rate is expected to be always at its equilibrium level, but that on average it is. Econometrically speaking, the PPP hypothesis implies a stationary process, or a process integrated of order zero I(0), for the real exchange rate. If the latter is integrated of order one I(1) (i.e., it contains a unit root), it will not revert to its mean (i.e., no constant equilibrium can be defined) and the PPP hypothesis can be rejected. However, if the equilibrium is thought of as a time-varying one, the real exchange rate will fluctuate around this time-varying equilibrium. The latter will be characterized by the long-run cointegration relationship (i.e., its coefficients) if the variables are cointegrated, which means that a stationary combination among these variables exists. Thus the presence of a cointegration relationship implies the existence of a time-varying equilibrium exchange rate.
To infer the stationary characteristics of the series under analysis, we use panel integration and cointegration techniques, since standard unit root and cointegration methods are known to have low power if applied to short-length time series. Given the short period of available data for these countries (1994-2002), an alternative to increase the power of the tests is to add the cross-sectional dimension to the analysis. In the literature, Im, Pesaran, and Shin (1995) and Pedroni (1996), among others, developed unit roots and cointegration statistics that, under quite general conditions, have more power than standard time-series tests. In addition, the tests by Im, Pesaran, and Shin (IPS) and Pedroni allow for heterogeneity in the dynamics of each of the cross-section units in the panel. This implies that under the hypothesis of a unit root in either the series of interest or the residuals of the cointegration regression, the dynamics of each cross-section unit may differ. Under the alternative hypothesis of no unit root, there are no homogeneity restrictions. This flexibility makes the use of these tests particularly suitable to our framework, where the coefficients of the long-run equilibrium and the short-run dynamics are likely to differ across countries.
Im, Pesaran, and Shin’s statistic (t-IPS) tests the null hypothesis of a unit root in a panel. The test is based on the average of the standard ADFt statistics obtained from individual tests. Under the null hypothesis of a unit root, the panel unit root test is distributed as a standard normal. For analyzing the cointegration properties, we follow Pedroni, who proposes several panel cointegration tests. Two are used in this exercise, the Group PP (GPP) and the Group t (Gt). The former is computed on the basis of the individual Phillips-Perron statistics applied on the residuals of each cointegration regression, while the Gt is calculated on the basis of the individual ADFt statistics applied on the same residuals. In both cases, the panel cointegration tests are asymptotically normal.
C. The Data
The REERs analyzed in this section are for the Estonian kroon, the Latvian lats, and the Lithuanian litas. The time period under consideration is 1994Q1–2002Q1 and the data are quarterly.53 The analysis has been conducted using the following variables:
- Real Exchange Rate (qt): the multilateral CPI-based REER of the currency of the domestic economy relative to its trading partner countries is used. The variable is expressed in logarithms. The series used are the ones published by the Information Notice System (INS).
- Net Foreign Assets (nfat) positions for each country are calculated by adding up the current account balances. The initial stock of net foreign assets is 1999Q3, from the International Investment Position data provided by the country authorities. The net foreign assets position is then normalized by the GDP in order to adjust for the size of the economy.
- Relative Prices of Nontradable to Tradable Goods (nt) are defined as the ratio of the domestic consumer price index (CPI) to the producer price index (PPI) relative to the corresponding weighted average of partner country ratios, using the same weights as the ones applied to qr The variable is expressed in logarithms.
D. Econometric Results
Testing for the existence of time-varying equilibrium exchange rates
Panel integration and cointegration techniques are used to infer the long-run property of the series and thereby test whether the PPP hypothesis holds and, if it does not, whether a time-varying equilibrium exchange rate exists for the three countries. For comprehensiveness, the results of time-series unit root (ADF) and cointegration (Johansen, 1988) tests are also presented.
Table A1 shows the results of the unit root tests. Both the panel and the individual unit root tests indicate that the hypothesis that the variables are integrated of order one cannot be rejected, suggesting the presence of a unit root in all three variables for all three countries. Thus, there is evidence that the PPP hypothesis does not hold for any of the Baltic countries.
|Panel integration (t – IPS)||–0.18||2.76||–0.15|
|Time-series integration (ADF)|
5 percent critical values: for t–IPS, – 1.69 and for ADF, – 2.95.
5 percent critical values: for t–IPS, – 1.69 and for ADF, – 2.95.
The results of the cointegration analysis are presented in Table A2. Although the table shows some disparity in the results, with the panel cointegration test Gt strongly rejecting the null hypothesis of no cointegration while the GPP test does not confirm the same result, considering the evidence of the time-series cointegration tests overall, we can infer the presence of cointegration for the three countries.
|Pedroni’s GPP||Pedroni’s Gt|
Critical value of panel tests at 5 percent is – 1.69.
Table A3 displays the cointegration vectors for the three countries together with some diagnostic statistics on the residuals of the cointegration regression. [The parameters associated with relative prices (n) are, as expected, systematically very close to one. Contrary to the empirical evidence for many other countries, however, the net foreign assets position enters into the long-run relationship with a negative sign for all three countries. Given the similarities of these three countries, this behavior could be attributed to the fact that, for the period under consideration, the current account deficit was financed by increased demand for these countries’ assets, since they started with very small liability positions at the beginning of the period and, with high productivity growth, relatively cheap labor costs, and a stable macroeconomic environment, offered good opportunities for foreign investors.
|Stationarity tests (c.v. 5.99)||28.92||27.40||19.88|
|Exclusion tests (c.v. 3.84)||19.79||13.99||27.25|
|Stationarity tests (c.v. 5.99)||188.20||207.50||211.90|
|Exclusion tests (c.v. 3.84)||56.70||66.50||90.60|
|Stationarity tests (c.v. 5.99)||56.37||56.80||54.49|
|Exclusion tests (c.v. 3.84)||37.67||45.04||48.30|
Equilibrium exchange rates
The estimation results for each of the Baltic countries are discussed in detail below and illustrated in Figures A1 and 17 in the main text. Figure A1 displays the historic series of the REERs and their determinants, while Figure 17 in the main text reports the estimated equilibrium real exchange rates. The left-hand panels of Figure 17 in the main text display the actual and the estimated equilibrium exchange rates, and the panels on the right-hand side present the deviations from equilibrium along with the computed 95 percent standard error bands. Positive values of misalignments indicate overvaluation of the REER with respect to its equilibrium. From Figure 17 in the main text it can be seen that in Estonia and Lithuania, REERs have fluctuated around their equilibrium rates within a range of about plus or minus 15 percent. Moreover, the recent pattern has been similar, with both countries experiencing overvaluation following the 1998 Russia crisis, although the overvaluation was more prolonged in the case of Lithuania, reflecting the sharp appreciation of the U.S. dollar, and hence the litas, against the euro during 1999–2000. By early 2002, however, both the kroon and the litas were modestly undervalued, by about 5–6 percent, with respect to their equilibrium levels. The accuracy of the results for Latvia, however, is very low (as can be seen from the wide confidence bands in Figure 17 in the main text) and it is not possible to say with any confidence whether the lats was under- or overvalued throughout the period under observation.
Figure A1.Real Exchange Rates and Their Determinants
Source: IMF staff estimates.
The Estonian kroon
On the basis of the cointegration results (Table A2), there is evidence of one significant cointegration vector for the system regarding the Estonian kroon. As reported in Table A3, the cointegration relationship, normalized on the exchange rate, produces the following relationship:
qt = –.018β1nfat + 1.0β2nt.
The adjustment (or α loading matrix) associated with the cointegration vector is reported in Table A4. The significantly negative α coefficient in the exchange rate equation indicates that the exchange rate moves to close the gap of a disequilibrium by approximately 50 percent every five quarters, or that most of the adjustment to a shock to the real exchange rate will be offset after two and a half years. Moreover, the significantly positive α coefficient in the Balassa-Samuelson variable equation suggests that the Balassa-Samuelson variable moves to close the gap of a disequilibrium at approximately the same pace.
|– 0.12||– 0.09||0.10|
As the variables in the system are integrated of order one and there exists one cointegration relationship implied by the cointegration vector, one can infer that there must be two common trends. Tables A5 and A6, which report α and β orthogonal components, respectively, should read as follows: in Table A5, the row headings are the common trends, while the column headings show the contributions of the variables to the trends. Looking across rows, the cell with the largest absolute value indicates that the shock to the variable in the row heading makes the largest contribution to the common trend. In Table A6, the column headings indicate the weights attached to the common trends, and the rows show how the weights are distributed among the variables. Focusing on a row, one can see which trend has the largest effect on a particular variable.
|β⊥1||– 0.70||0.34||– 0.63|
|β⊥2||– 0.16||– 0.93||– 0.33|
The results in Table A5 indicate that the first common trend appears to correspond to unanticipated shocks to the net foreign assets position, while the second trend is driven by the real exchange rate. Table A6 indicates that the real exchange rate and the Balassa-Samuelson variable appear to be driven by the first common trend, and the net foreign assets position by the second trend.
Additional information on the driving variables in the systems may be obtained looking at the long-run impact matrix (Table A7), which measures the combined effect of the α and β orthogonal components and indicates if a shock to a particular variable has a permanent effect on the other variables in the system. Table A7 indicates that shocks to the Balassa-Samuelson variable have a significant cumulative impact on the real exchange rate, while the cumulative effect of a shock to the net foreign assets position does not seem to be significant.
|Shock to q||Shock to nfa||Shock to n|
The Latvian lats
There is evidence of one cointegration vector for the lats, which, once normalized for the exchange rate, produces the following relationship:
qt = –.015β1nfat + 0.99β2nt.
The estimated α coefficient in Table A8 indicates that the exchange rate adjusts to close the gap of a disequilibrium, but the speed of adjustment is slower than in Estonia, with adjustment completed within four and a half years. The Balassa-Samuelson variable, however, moves to close the gap of a disequilibrium much faster, in approximately six months.
|α||– 0.05||– 0.05||0.42|
As shown in Table A9, there is evidence that the first common trend is mainly driven by the net foreign assets position, while the second one corresponds to unanticipated shocks to the exchange rate. Based on the results in Table A10, the net foreign assets position appears to be driven by the first common trend, and the exchange rate and the relative price differentials by the second trend.
Finally, from the estimated long-run impact matrix in Table A11, one can learn that shocks to the net foreign assets position have a significant cumulative effect on the real exchange rate, while the cumulative effect of a shock to the Balassa-Samuelson term does not appear to be significant.
|Shock to q||Shock to nfa||Shock to n|
|qt||0.12||– 0.43||– 0.04|
|nt||0.19||– 0.26||– 0.01|
The Lithuanian litas
For the system regarding the real exchange rate for the Lithuanian litas, there is also evidence of a cointegration vector, which produces the following relationship, once normalized for the exchange rate:
qt = –.018β1nfat + 1.02β2nt.
Similar to the finding for the other two countries, the real exchange rate adjusts to close the gap of a disequilibrium (Table A12). The speed of adjustment is faster than in the other cases and suggests an adjustment within two years.
|– 0.16||– 0.03||0.06|
The results in Table A13 indicate that both the first and second common trends appear to correspond to unanticipated shocks to the Balassa-Samuelson term. Table A14 shows that the variable that adjusts most to the first trend is the net foreign assets position, while the REER and the Balassa-Samuelson term seem to be mostly affected by the second trend.
|α⊥2||– 0.38||0.42||– 0.82|
The estimated long-run impact matrix for the Lithuanian litas is reported in Table A15. As for the Estonian kroon system, shocks to the Balassa-Samuelson term have a significant cumulative effect on the real exchange rate, while the cumulative effect of a shock to the net foreign assets position does not seem to be significant.
|Shock to q||Shock to nfa||Shock to n|
|nfat||– 0.41||1.39||– 0.40|