## _wp07287 ## Source details **Canonical URL:** [_wp07287](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07287.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07287.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07287.pdf.json) --- ### Purpose and contribution - Applies the panel cointegration method of Larsson and Lyhagen (2007) to test purchasing power parity (PPP) for the G7 during the float period January 1973 through December 1998. - Provides tests and estimates robust to: - cross-sectional dependence of error terms, - choice of normalization (which variable on left-hand side), - possibility of more than one cointegrating vector. - Contrasts Larsson and Lyhagen maximum-likelihood results with Pedroni’s panel cointegration tests and the FMOLS and DOLS estimators to assess practical importance of methodological differences. ### Data and basic model - Model: s_{i,t} = μ_i + β_i p_{i,t} + ε_{i,t}, where s_{i,t} is log bilateral nominal exchange rate, p_{i,t} is log aggregate price ratio (CPI) relative to base country. - Strong PPP: β_i = 1, ∀i. - Countries: Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States (United States as base). - Sample period: January 1973 through December 1998. - Panel dimensions: T = 312 and N = 6. ### Methodology — Larsson and Lyhagen maximum likelihood - Framework: panel vector error correction model (panel-VECM) generalizing Johansen (1988). - Key features: - Tests for number r of cointegrating vectors (more than one possible). - No arbitrary normalization choice; unified result. - Allows substantial short-run dependence and cross-sectional dependence while cointegrating relations are group-specific. - Model (compact): ΔY_t = μ + Π Y_{t-1} + Σ_{k=1}^{m-1} Γ_k ΔY_{t-k} + ε_t, with Π assumed reduced rank Π = A B′ and ε_t ~ N(0, Σ). - Restrictions: constant allowed in cointegrating relations but no drift in the system (A′_⊥ μ = 0). - Testing sequence: - Test whether Π has reduced rank (existence and number r of cointegrating relations). - Conditional on cointegration, test homogeneity of cointegrating vectors across groups. - If homogeneous, test whether common vector = [1; -1] (strong PPP). ### Methodology — Pedroni’s tests, FMOLS and DOLS - Pedroni’s residual-based cointegration tests estimate country-specific cointegrating regressions and test unit roots in residuals ^u_{i,t}; the null is no cointegration and the alternative can allow heterogeneous cointegrating vectors. - FMOLS: non-parametric correction for endogeneity and serial correlation using ^u_{i,t} and Δy_{2,i,t}; permits standard normal inference. - DOLS: parametric correction via leads and lags of Δy_{2,i,t}: y_{1,i,t} = μ_i + β_i′ y_{2,i,t} + Σ_{s=-s_i}^{s_i} ψ_{i,s} Δy_{2,i,t-s} + v_{i,t}. - Between-dimension estimators pool information across cross-sections; point estimates interpreted as mean values for cointegrating vectors. - Limitations: - FMOLS and DOLS rely on cross-sectional independence of errors. - Residual-based methods cannot test for more than one cointegrating relation. - Results can depend on normalization (choice of dependent variable). ### Empirical findings — Larsson and Lyhagen panel cointegration (bootstrap) - Number of bootstrap replicates = 399. - Null of no cointegration rejected in favour of panel cointegration with one cointegrating vector at the five-percent level. - Noted substantive difference between asymptotic and bootstrapped critical values: using asymptotic critical values would indicate two cointegrating vectors; bootstrap indicates one. - Homogeneity of cointegrating vectors: - Null of homogenous cointegrating vectors forcefully rejected. - Observed likelihood ratio: 73:66 - Asymptotic 5% critical value: 11:07 - Bootstrapped 5% critical value: 39:46 - Block diagonality test: - Likelihood ratio: 75.06 - Asymptotic 5% critical value: 43.77 - Bootstrapped 5% critical value: 99.86 - Sample period: January 1973 to December 1998. ### Parameter estimates of normalized cointegrating vectors (bootstrapped s.e., 399 replicates) - Cointegrating vector normalized such that ii = [1; i]0. All estimates significantly different from -1 at the 5 percent level. - Individual estimates and standard errors: - UK: -3.66 0.22 - France: -2.39 0.15 - Italy: -1.36 0.05 - Germany: 2.43 0.18 - Japan: 5.44 0.32 - Canada: -0.22 0.27 - Interpretation: - Individual estimates are quite far from the theoretical -1. - UK, France, Italy: parameter estimates and standard errors roughly similar. - Germany and Japan: parameters have the wrong sign and are significantly so. - Canada: parameter close to zero and not significant. - Block diagonality assumption could account for divergent estimates, but likelihood-ratio test with bootstrapped critical values does not reject block diagonality. ### Pedroni panel cointegration tests (sensitivity to time dummies and normalization) - Ignoring time specific effects: null of no cointegration not rejected for any test regardless of dependent variable. - Including time dummies increases power; out of fourteen tests (seven for each dependent variable), ten reject the null and find support for some PPP relationship. - Table 3 test-statistics (sample January 1973 to December 1998), where * denotes 5 percent significance: - Normalization s_i,t = _i + p_i,t + ε_i,t and p_i,t = _i + s_i,t + ε_i,t (time dummies Yes/No): - Panel v: 3.84* 1.02 -1.73 -2.45 - Panel rho: -2.30* 0.18 -2.23* 0.33 - Panel PP: -1.77* 0.34 -1.83* -0.07 - Panel ADF: -1.55 0.48 -1.78* -0.07 - Group rho: -1.90* 0.61 -1.34 1.40 - Group PP: -1.82* 0.53 -2.07* 0.79 - Group ADF: -1.60 0.67 -1.83* 0.75 - Findings: - Results consistent between normalizations only when no time dummies are included (no cointegration rejected). - With time dummies included, only Panel rho, Panel PP and Group PP reach the same conclusions under both normalizations. - Choice of dependent variable and cointegration test can generate different conclusions. ### FMOLS and DOLS between-dimension estimation (exchange-rate normalization) - Sample: January 1973 to December 1998. * Denotes 5 percent significance. t-statistics test H0: = 1 in parentheses. - Panel estimates (exchange-rate normalization): - DOLS Panel : 0.97 (5.76) with Time dummies Yes; 1.37 (6.39) with Time dummies No. - FMOLS Panel : 0.92 (3.48) with Time dummies Yes; 1.28 (3.48) with Time dummies No. - Individual country estimates (DOLS / FMOLS, Time dummies Yes / No) and t-statistics: - UK: 0.73* 0.68* 0.78* 0.68* ( -3.00, -3.15, -2.19, -2.23 ) - France: 2.10* 1.87* 2.11* 2.12* ( 10.09, 4.26, 10.18, 7.16 ) - Italy: 1.01 1.02 1.03 1.16* ( 0.22, 0.37, 0.92, 2.20 ) - Japan: 1.68* 1.87* 1.69* 1.89* ( 10.98, 6.49, 11.22, 7.12 ) - Germany: 0.74* 0.94 0.65* 1.05 ( -7.20, -0.43, -4.35, 0.33 ) - Canada: -0.77* 1.31 -0.44* 1.32 ( -2.56, 0.99, -1.69, 1.07 ) - Interpretation: - Strong PPP rejected in all cases. - Panel average of cointegrating vectors close to one for FMOLS and DOLS when time dummies are used, but this reflects averaging over heterogeneous cointegrating vectors rather than support for strong PPP. - Only Italy approximates strong PPP reasonably well in individual vectors. ### FMOLS and DOLS estimation (price-index-ratio normalization) - Panel estimates: - DOLS Panel : 0.65 ( -40.60 ) with Time dummies Yes; 0.51 ( -27.72 ) with Time dummies No. - FMOLS Panel : 0.64 ( -36.48 ) with Time dummies Yes; 0.51 ( -26.52 ) with Time dummies No. - Individual country estimates (DOLS / FMOLS, Time dummies Yes / No) and t-statistics: - UK: 0.78* 0.70* 0.77* 0.68* ( -2.36, -3.05, -2.67, -3.20 ) - France: 0.42* 0.34* 0.44* 0.35* ( -26.82, -15.68, -31.20, -16.65 ) - Italy: 0.97 0.88* 0.98 0.88* ( -0.99, -2.16, -0.71, -2.09 ) - Japan: 0.56* 0.42* 0.56* 0.42* ( -21.69, -20.42, -24.51, -22.37 ) - Germany: 1.18* 0.54* 1.18* 0.54* ( 3.18, -5.72, 3.19, -5.91 ) - Canada: -0.04* 0.20* -0.03* 0.21* ( -40.70, -17.92, -43.54, -17.67 ) - Findings: - Null of strong PPP forcefully rejected for all specifications under this normalization. - Strong PPP appears an even worse approximation under price-ratio normalization; coefficients on the exchange rate are very far from one. - Estimates of when exchange rate is dependent variable are often far from the inverse of that when price ratio is dependent variable. ### Overall conclusions and implications - Across methodologies (Larsson and Lyhagen (2007), Pedroni (1995), FMOLS, DOLS) the consistent picture is: - Strong PPP hypothesis is forcefully rejected. - Weak PPP with heterogeneous cointegrating vectors is supported. - Strong PPP does not seem to be an acceptable approximation of observed data in the sample January 1973 to December 1998. - Potential mechanisms for failure of strong PPP include: - Measurement errors, - Differences in price indices between countries, - Transportation costs, - Differential productivity shocks. - Implications for macroeconomic theory: - If failure is due to measurement errors, models imposing strong PPP may be misleading when taken to data. - If failure stems from omitted fundamental mechanisms, this points to deeper theoretical shortcomings. - Methodological recommendation: - When panel length and dimension permit, Larsson and Lyhagen's (2007) maximum likelihood approach is preferred because it is robust to cross-sectional dependence and does not rely on a particular normalization. - Further research: - More research on the causes of failure of the strong PPP hypothesis is warranted. *Source: _wp07287 (IMF working paper PDF).* ### 1. Introduction......................................................................................................... ### _wp07287 - 1. Introduction ### Purpose and contribution - Applies the recently developed panel cointegration method of Larsson and Lyhagen (2007) to test purchasing power parity (PPP) for the G7 during the recent float period. - Provides tests and estimates that are robust to: - cross-sectional dependence of error terms, - choice of normalization (which variable on left-hand side), - and allows testing for more than one cointegrating vector. - Contrasts maximum-likelihood results with Pedroni’s panel cointegration tests and the FMOLS and DOLS estimators to assess the practical importance of methodological differences. ### PPP concepts and hypotheses - Long-run PPP implies nominal exchange rates and aggregate price ratios are cointegrated; strong PPP: cointegrating vector = [1; -1]. - Weak PPP allows: - homogeneous cointegrating vectors across countries but different from [1; -1], or - heterogeneous cointegrating vectors across countries. - Under weak PPP, the real exchange rate (residual to a [1; -1] vector) will not be stationary (i.e., will not return to a time-invariant equilibrium). ### Data and basic model - Model: s_{i,t} = μ_i + β_i p_{i,t} + ε_{i,t}, where s_{i,t} is log bilateral nominal exchange rate, p_{i,t} is log aggregate price ratio (CPI) relative to base country; strong PPP if β_i = 1, ∀i. - Countries: Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States (United States as base). - Sample period: January 1973 through December 1998 (after Bretton Woods transition until EMU irrevocable fixings). - Panel dimensions: T = 312 and N = 6. - Data shown in Figures 1 and 2 (log exchange rates and log CPI ratios). ### Methodology — Larsson and Lyhagen maximum likelihood - Framework: panel vector error correction model (panel-VECM) generalizing Johansen (1988). - Key features: - Allows testing for number of cointegrating vectors (more than one possible). - No arbitrary normalization choice; one unified result. - Permits substantial short-run dependence and cross-sectional dependence via unrestricted parts of the model while cointegrating relations are group-specific. - Model structure (compact): ΔY_t = μ + Π Y_{t-1} + Σ_{k=1}^{m-1} Γ_k ΔY_{t-k} + ε_t, with Π assumed reduced rank Π = A B′ and normality ε_t ~ N(0, Σ). - Restrictions: constant allowed in cointegrating relations but no drift in the system (A′_⊥ μ = 0). - Testing sequence: - Test whether Π has reduced rank (existence and number r of cointegrating relations). - Conditional on cointegration, test whether cointegrating vectors are homogeneous across groups. - If homogeneous, test whether common vector = [1; -1] (strong PPP). ### Methodology — Pedroni’s tests, FMOLS and DOLS - Pedroni’s cointegration tests: - Residual-based approach: estimate country-specific cointegrating regressions y_{1,i,t} = μ_i + β_i′ y_{2,i,t} + u_{i,t}, then test unit roots in residuals ^u_{i,t}. - Null of no cointegration can allow heterogeneous cointegrating vectors under the alternative. - Between-dimension FMOLS and DOLS estimators: - Correct endogeneity and serial correlation to permit standard normal inference. - FMOLS: non-parametric correction using ^u_{i,t} and Δy_{2,i,t}. - DOLS: parametric correction via leads and lags of Δy_{2,i,t}: y_{1,i,t} = μ_i + β_i′ y_{2,i,t} + Σ_{s=-s_i}^{s_i} ψ_{i,s} Δy_{2,i,t-s} + v_{i,t}. - Between-dimension estimators pool information across cross-sections; point estimates can be interpreted as mean values for cointegrating vectors. - Limitations noted: - FMOLS and DOLS rely on cross-sectional independence of errors. - Residual-based methods cannot test for more than one cointegrating relation. - Results can depend on normalization (choice of dependent variable) for residual-based methods. ### Empirical findings (preview) - Using Larsson and Lyhagen’s method with bootstrap: - Number of bootstrap replicates = 399. - Test supports one cointegrating vector at the five-percent level. - Noted: substantive difference between asymptotic critical values and bootstrapped critical values; if asymptotic critical values were used, two cointegrating vectors would be indicated. - Overall inference across methods: - Strong PPP is forcefully rejected in favor of weak PPP with heterogeneous cointegrating vectors. - Strong PPP does not seem to be an acceptable approximation of observed data. - The comparison indicates FMOLS and DOLS shortcomings matter in practice because they affect inference, although all methods point in the same general direction. ### Paper organization - Section 2: Basic model and data. - Section 3: Larsson and Lyhagen panel-cointegration framework and discussion of Pedroni’s tests and between-dimension FMOLS and DOLS. - Section 4: Results. - Section 5: Conclusions. - Appendix and References follow; Tables and Figures (including Tables 1–5 and Figures 1–2) present detailed empirical results and data plots. *Source: _wp07287 - 1. Introduction (IMF working paper PDF).* ### conclusion, which clearly would have been both surprising and unsatisfying. ### _wp07287 - conclusion, which clearly would have been both surprising and unsatisfying. ### Panel cointegration and homogeneity tests (Larsson and Lyhagen methodology) - Null of no cointegration rejected in favour of panel cointegration with one cointegrating vector. - Null of homogenous cointegrating vectors forcefully rejected: - Observed likelihood ratio: 73:66 - Asymptotic 5% critical value: 11:07 - Bootstrapped 5% critical value: 39:46 - Block diagonality test: - Likelihood ratio: 75.06 - Asymptotic 5% critical value: 43.77 - Bootstrapped 5% critical value: 99.86 - Sample period: January 1973 to December 1998. ### Parameter estimates of normalized cointegrating vectors (bootstrapped standard errors) - Cointegrating vector normalized such that ii = [1; i]0. Standard errors bootstrapped using 399 replicates. All estimates significantly different from -1 at the 5 percent level. - Individual estimates and s.e.: - UK: -3.66 0.22 - France: -2.39 0.15 - Italy: -1.36 0.05 - Germany: 2.43 0.18 - Japan: 5.44 0.32 - Canada: -0.22 0.27 - Interpretation: - Individual estimates are quite far from the theoretical -1. - UK, France, Italy: parameter estimates and standard errors roughly similar. - Germany and Japan: parameters have the wrong sign and are significantly so. - Canada: parameter close to zero and not significant. - Block diagonality assumption could be a reason for divergent estimates, but likelihood-ratio test with bootstrapped critical values does not reject block diagonality. ### Comparison with other studies - Jacobson et al. (2002) using Larsson and Lyhagen method on a subsample find support for weak PPP with homogenous cointegrating vectors near strong PPP expectations; current paper finds that result not very robust. ### Pedroni panel cointegration tests (sensitivity to time dummies and normalization) - Ignoring time specific effects: null of no cointegration not rejected for any test regardless of dependent variable. - Cross-sectional dependencies plausibly deflate test size and reduce power. - Including time dummies increases power; out of fourteen tests (seven for each dependent variable), ten reject the null hypothesis and find support for some form of PPP relationship. - Minor difference between panel and group versions of tests. - Table 3 test-statistics (sample January 1973 to December 1998), where * denotes 5 percent significance: - Normalization s_i,t = _i + p_i,t + ε_i,t (time dummies Yes/No) and p_i,t = _i + s_i,t + ε_i,t (time dummies Yes/No): - Panel v: 3.84* 1.02 -1.73 -2.45 - Panel rho: -2.30* 0.18 -2.23* 0.33 - Panel PP: -1.77* 0.34 -1.83* -0.07 - Panel ADF: -1.55 0.48 -1.78* -0.07 - Group rho: -1.90* 0.61 -1.34 1.40 - Group PP: -1.82* 0.53 -2.07* 0.79 - Group ADF: -1.60 0.67 -1.83* 0.75 - Findings: - Results are consistent between normalizations only when no time dummies are included (no cointegration rejected). - With time dummies included, only Panel rho, Panel PP and Group PP reach the same conclusions under both normalizations. - Choice of dependent variable and cointegration test can generate different conclusions. ### FMOLS and DOLS between-dimension estimation (normalization on exchange rate) - Sample: January 1973 to December 1998. * Denotes 5 percent significance. t-statistics test H0: = 1 in parentheses. - Panel estimates (exchange-rate normalization): - DOLS Panel : 0.97 (5.76) with Time dummies Yes; 1.37 (6.39) with Time dummies No. - FMOLS Panel : 0.92 (3.48) with Time dummies Yes; 1.28 (3.48) with Time dummies No. - Individual country estimates (DOLS / FMOLS, Time dummies Yes / No): - UK: 0.73* 0.68* 0.78* 0.68* (t-stats: -3.00, -3.15, -2.19, -2.23) - France: 2.10* 1.87* 2.11* 2.12* (10.09, 4.26, 10.18, 7.16) - Italy: 1.01 1.02 1.03 1.16* (0.22, 0.37, 0.92, 2.20) - Japan: 1.68* 1.87* 1.69* 1.89* (10.98, 6.49, 11.22, 7.12) - Germany: 0.74* 0.94 0.65* 1.05 (-7.20, -0.43, -4.35, 0.33) - Canada: -0.77* 1.31 -0.44* 1.32 (-2.56, 0.99, -1.69, 1.07) - Interpretation: - Strong PPP rejected in all cases. - Point estimate of the average of cointegrating vectors is close to one for both FMOLS and DOLS when time dummies are used, but this does not imply support for strong PPP — it reflects averaging over heterogeneous cointegrating vectors. - Only Italy appears to approximate strong PPP reasonably well in individual vectors. ### FMOLS and DOLS estimation (normalization on price index ratio) - Panel estimates: - DOLS Panel : 0.65 ( -40.60 ) with Time dummies Yes; 0.51 ( -27.72 ) with Time dummies No. - FMOLS Panel : 0.64 ( -36.48 ) with Time dummies Yes; 0.51 ( -26.52 ) with Time dummies No. - Individual country estimates (DOLS / FMOLS, Time dummies Yes / No): - UK: 0.78* 0.70* 0.77* 0.68* (-2.36, -3.05, -2.67, -3.20) - France: 0.42* 0.34* 0.44* 0.35* (-26.82, -15.68, -31.20, -16.65) - Italy: 0.97 0.88* 0.98 0.88* (-0.99, -2.16, -0.71, -2.09) - Japan: 0.56* 0.42* 0.56* 0.42* (-21.69, -20.42, -24.51, -22.37) - Germany: 1.18* 0.54* 1.18* 0.54* (3.18, -5.72, 3.19, -5.91) - Canada: -0.04* 0.20* -0.03* 0.21* (-40.70, -17.92, -43.54, -17.67) - Findings: - Null of strong PPP forcefully rejected for all specifications under this normalization. - Strong PPP appears an even worse approximation under price-ratio normalization; coefficients on the exchange rate are very far from one. - Estimates of when exchange rate is dependent variable are often far from the inverse of that when price ratio is dependent variable (only asymptotically should they be inverses). ### Overall conclusions and implications - Across methodologies (Larsson and Lyhagen (2007), Pedroni (1995), FMOLS, DOLS) the overall picture is consistent: - Strong PPP hypothesis is forcefully rejected. - Weak PPP with heterogeneous cointegrating vectors is supported. - Strong PPP does not seem to be an acceptable approximation of the observed data in the sample (January 1973 to December 1998). - Potential mechanisms for failure of strong PPP include: - Measurement errors, - Differences in price indices between countries, - Transportation costs, - Differential productivity shocks. - Implications for macroeconomic theory depend on underlying causes: - If failure is due to measurement errors, models imposing strong PPP may be misleading when taken to data. - If failure stems from omitted fundamental mechanisms, this points to deeper theoretical shortcomings. - Methodological recommendation: - When panel length and dimension permit, Larsson and Lyhagen's (2007) maximum likelihood based approach is preferred because it is robust to cross-sectional dependence and does not rely on a particular normalization. - Further research: - More research on the causes of failure of the strong PPP hypothesis is clearly warranted. *Source: _wp07287 - conclusion, which clearly would have been both surprising and unsatisfying. (Sample January 1973 to December 1998)* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07287.pdf_