## _wp04106

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---

### I. INTRODUCTION — framing and main argument
- Framework:
  - The intertemporal approach to the current account views net accumulation of foreign assets as smoothing consumption intertemporally in response to idiosyncratic income shocks; under simplifying assumptions the current account equals the present discounted value of expected future net income declines.
- Empirical testing tradition and tension:
  - Researchers apply the Campbell-Shiller methodology (estimate unrestricted VARs on income changes and the current account and construct model-predicted series) and commonly find the actual current account series are positively correlated with, but much more volatile than, model-predicted series — an “excess volatility” result.
  - Many papers try to generate extra predicted volatility via model augmentations (examples cited in the source).
- Main argument of the paper:
  - The Campbell-Shiller methodology can be misleading under near-singularity conditions typically generated by persistence in current account data.
  - Under near-singularity, small sample estimation errors can produce disproportionately large effects on the predicted series because the K vector is a non-linear function involving the inverse of M ≡ I − (1/(1+r))C.
  - Posterior distributions of predicted series, correlation coefficients, and variance ratios will be very wide; the Delta method approximation to Var(K) can fail in short samples, making the usual Wald test invalid in the singularity region.
  - The valid alternative: an F-test based on regressions of Rt on the information set It−1 (where Rt = CAYt − CAYt−1 − ∆Yt) does not rely on a bad linear approximation and yields correct inference under Gaussian residuals.

### II. TESTING THE PRESENT VALUE MODEL — methodology and failure modes
- Model and testing setup (as in the source):
  - Present value model under quadratic utility and constant real return r predicts CA_it as a discounted sum of expected future net income.
  - Joint DGP of ΔY and CA assumed to be an unrestricted l order VAR; construct X_t and companion matrix C; model-predicted current account p_t CA = K X_t with K = A (1 − (1/(1+r)) C)^{-1}.
  - Testing equality of p̂_t CA and CA_t is equivalent to testing whether K equals the 2l vector T (zeros except a 1 at the (l+1)th element).
  - Wald statistic: W = (K̂ − T)' [J V J']^{-1} (K̂ − T) with asymptotic χ^2_{2l} distribution under usual regularity (Delta method used to approximate Var(K) as J V J′).
- Problems under near-singularity (analytical and empirical implications):
  - If M has an eigenvalue close to zero (i.e., C has an eigenvalue close to 1+r), small estimation errors in VAR parameters can cause very large deviations in M^{-1} and thus in K and the model-predicted series.
  - Delta method failure and Wald distortions:
    - False rejection: Delta method can give a variance-covariance that is “too small,” making W “too large,” leading to rejection when likelihood support near the null is actually larger.
    - False acceptance: Delta method can give a variance-covariance that is “too large,” making W “too small,” leading to acceptance when likelihood support near the null is actually smaller.
  - Practical prevalence: near-singularity is described as “very common” in practice for current account VARs because of persistence (small coefficients on lagged income changes and high coefficients on lagged CA).

### III. EMPIRICAL ILLUSTRATION AND RESULTS — diagnostics and numeric findings
- Data and samples:
  - Countries: Belgium, Canada, Denmark, Sweden, and the United Kingdom.
  - Frequencies: Annual and quarterly data for each country (country sample periods listed in the Data appendix).
  - Estimation conventions: Means removed from CA and ΔY; real interest rates set to 4 percent (annual) and 1 percent (quarterly); VAR lag length selected by the Akaike information criterion.
- Key empirical findings (summary of diagnostics and numerical results):
  - Test discrepancies:
    - In four out of ten cases the Wald test and the F-test lead to opposite inferences at traditional confidence levels.
    - The Wald statistic can suggest levels of significance that are up to eighty-eight percentage points different from their actual level as given by the F-test.
    - The average distortion across the ten country samples equals thirty three percentage points.
    - The Wald test always rejects the model with quarterly data while the F-test accepts it for two out of five countries (Belgium and Canada).
  - Posterior dispersion of K and predicted series:
    - For each country sample 10,000 draws are generated from the multivariate Normal distribution given by the estimated VAR parameters and their variance-covariance (5,000 draws for the UK quarterly data).
    - The posterior variance of the (l+1)th coefficient of the K vector is very large in all samples; even when the F-test accepts the model the coefficient often has high probability of being far from its theoretical value and can be negative.
    - The 25.th and 97.5.th posterior percentile lines of the model-predicted series are almost without exception very wide compared to the actual series (percentile lines computed for each of the K draws).
  - Variance ratio distributions:
    - For each draw i construct predicted series i,pt CA = K_i X_t and compute variance ratio = Var(i,pt CA)/Var(actual CA).
    - Variance ratios are often very dispersed as a direct consequence of dispersion in K in the singularity region.
    - Even when the model is consistent with the data the predicted current account can be much more or much less volatile than the actual.
      - Example: model strongly consistent with Belgian annual data yet there is a 40 percent probability that the predicted current account is over four times as volatile as the actual.
      - There is also a 20 percent probability that the predicted current account displays less than a fourth of the volatility of the actual series.
    - The probability that the predicted series is more volatile than the actual averages 44 percent over the 10 samples, ranging from 11 percent for Belgian quarterly data to over 97 percent for Swedish quarterly data.
  - Correlation distributions:
    - Posterior distributions of in-sample correlation between actual and predicted series are very wide, reflecting K dispersion.
    - Correlations can place substantial probability mass near both 1 and −1; distributions often cluster around one and minus one under near-singularity and one lag when one K coefficient dominates.
    - Example probabilities reported in the text:
      - For Belgian and Danish annual data the valid F-test accepts the model, yet there is over 45 percent probability that the correlation lies between 1− and 09−..
      - For Swedish annual data there is a 37 percent probability that the correlation will exceed 095. even though the test has rejected the model.
- Table 1 — selected numeric entries (one-minus-cdf figures for Wald and F tests; a star denotes model acceptance by the F-test at the 5 percent level):
  - Annual data
    - Belgium 1953–1998: Number of Lags 1; Eigenvalue 1.01; Wald test 96.9%; F-test 36.7%*
    - Canada 1948–2002: Number of Lags 1; Eigenvalue 0.61+0.2i; Wald test 27.8%; F-test 0.6%
    - Denmark 1966–2002: Number of Lags 1; Eigenvalue 0.96; Wald test 94.9%; F-test 20%*
    - Sweden 1950–2002: Number of Lags 1; Eigenvalue 0.93; Wald test 89.4%; F-test 0.8%
    - United Kingdom 1948–2002: Number of Lags 2; Eigenvalue 0.67; Wald test 2.8%; F-test 0.1%
  - Quarterly data
    - Belgium 1980–1998: Number of Lags 3; Eigenvalue 0.98; Wald test 0.7%; F-test 43.7%*
    - Canada 1948–2002: Number of Lags 8; Eigenvalue 0.94; Wald test 2%; F-test 38.1%*
    - Denmark 1988–2002: Number of Lags 4; Eigenvalue 0.93; Wald test 0%; F-test 0%
    - Sweden 1990–2002: Number of Lags 4; Eigenvalue 0.97; Wald test 0%; F-test 0%
    - United Kingdom 1955–2002: Number of Lags 4; Eigenvalue 0.94; Wald test 0%; F-test 0%
  - Notes: The figures correspond to one minus the cdf of the test statistic; the number of lags was selected by the Akaike criterion.
- Graphical diagnostics (as described, figures referenced in the source):
  - Figures illustrate false rejection/acceptance phenomena in the singularity region, ex-post posterior distributions of K coefficients (Figures 4–5), actual vs predicted series with percentile bands (Figures 6–9), posterior distributions of variance ratios (Figures 10–11) and correlation coefficients (Figures 12–13).
  - Percentile lines used: 25th and 97 5th. percentiles for annual panels; 2.5th and 97.5th percentiles for quarterly panels (as annotated in the source).

### IV. CONCLUSION — implications and recommendations
- Methodological implications:
  - Inference based on Campbell-Shiller VAR-based present value testing is fragile under near-singularity, a condition frequently met in current account data because of persistence.
  - The Delta method-based Wald test can deliver misleading conclusions in short samples (both false rejections and false acceptances).
  - The F-test described (regress Rt on It−1 and perform an F-test on joint nullity) is recommended as the valid formal test because it does not rely on the Delta method linear approximation and is valid under Gaussianity regardless of unit roots.
- Empirical implications:
  - Claims of excess current account volatility (or its absence) based on direct comparison of actual and VAR-predicted series should be treated with caution given the wide posterior uncertainty documented.
  - Researchers should systematically construct ex-post posterior distributions of the predicted series, variance ratios, and correlation coefficients rather than base inference solely on single-sample predicted paths.
- Broader relevance:
  - The concerns extend to other applications of the Campbell-Shiller methodology (for example, expectations tests for the term structure of interest rates) where bivariate VARs and present value constructs are employed.
- Practical guidance:
  - Use the F-test rather than the traditional Wald test for formal testing.
  - Report posterior dispersion (percentile bands), variance ratio distributions, and correlation distributions to reflect estimation uncertainty in the singularity region.
  - Recognize that given high variance of cross-equation parameters in the singularity region, useful inference about the relationship between data and model predictions will often be difficult.

*Source: _wp04106 - References..............................................................................................................*

### References..............................................................................................................

### _wp04106 - References..............................................................................................................

### I. INTRODUCTION
- Framework:
  - The intertemporal approach to the current account (Sachs (1981); surveyed in Obstfeld and Rogoff (1995)) views net accumulation of foreign assets as smoothing consumption intertemporally in response to idiosyncratic income shocks.
  - Under simplifying assumptions, the current account equals the present discounted value of expected future net income declines.
- Empirical testing tradition:
  - Following Sheffrin and Woo (1990) and using Campbell (1987) and Campbell and Shiller (1987) insights, researchers estimate unrestricted VARs on income changes and the current account to construct model-predicted series and compare them to actual series.
  - Many studies find the model is rejected: actual current account series are positively correlated with, but much more volatile than, model-predicted series. This “excess volatility” result has been interpreted as evidence against full international capital mobility (Feldstein and Horioka).
  - Papers attempting to augment the model to generate extra predicted volatility include Bergin and Sheffrin (2000), Gruber (2004), and Nason and Rogers (2003).
- Subperiod instability problem:
  - Example: Figure 1 (Actual (-) and Predicted (--) Series: Subsample Estimation) shows Canada: good fit in first subperiod, strong excess volatility in second; UK: almost opposite pattern.
  - This subperiod variation is problematic for explanations that generate extra volatility uniformly (e.g., nonseparable preferences, world interest rate shocks).
- Main argument of the paper:
  - The Campbell-Shiller methodology can be misleading under near-singularity conditions typically generated by persistence in current account data.
  - Under near-singularity, small sample estimation errors can produce disproportionately large effects on the predicted series.
  - Posterior distributions of predicted series, correlation coefficients, and variance ratios will be very wide, undermining inference based on direct comparison of actual and predicted series.
  - The traditional Wald test is not valid in the singularity region because the Delta method linear approximation to the variance-covariance matrix becomes imprecise in short samples; the Wald test can thus lead to false rejection or false acceptance relative to a valid linear F-test.
- Empirical illustration and key empirical claims:
  - Sample: Belgium, Canada, Denmark, Sweden, and the UK (same five countries discussed in Obstfeld and Rogoff (1996)).
  - Results summary:
    - Likelihoods differ sharply between the Wald and F-tests; in four out of ten cases the Wald test gives the wrong inference at traditional confidence levels (rejects where the F-test accepts and vice versa).
    - Posterior distributions of model-predicted series (constructed by drawing from the multivariate normal of VAR parameter estimates and variance-covariance and reestimating predicted series) show that the 25.th and 97.5.th posterior percentile lines are almost without exception very wide compared to actual series.
    - Posterior distributions of the variance ratio are very dispersed and indicate substantial probability that the actual series is both several times more and several times less volatile than the model-predicted series.
    - Posterior distributions of the correlation coefficient are very dispersed and show substantial probability that the correlation is close to both 1 and -1.
    - These dispersion findings occur regardless of whether the F-test accepts or rejects the model.
- Prior warnings and contribution:
  - Kasa (2004) warned that plausible linear income processes could undermine the approach; this paper’s criticism is broader, based on near-singularity conditions that are easy to verify and commonly met in the data.
- Relevance to other literature:
  - The issues apply to other applications of the Campbell-Shiller methodology (e.g., expectations theory of the term structure of interest rates; Campbell and Shiller (1987, 1991); Hardouvelis (1994)).

### II. TESTING THE PRESENT VALUE MODEL
- Model and basic testing setup:
  - Under quadratic utility, constant real return on a single internationally traded bond, and subjective discount factor equal to the inverse of the (gross) return, the present value model (PVM) predicts the current account as:
    - Equation (1): CA_it = (1/(1+r)) * sum_{i=0}^{∞} [E_t (Y_{t+i} - ... )]  (expression as in source).
    - Definitions: r is the (constant) real interest rate; Y is income net of government spending and investment (referred to as “income” in the paper).
  - Assume the joint data generating process of income changes (ΔY) and the current account (CA) is an unrestricted l order VAR.
    - Construct 2l column vector X_t = [ΔY_{t},...,ΔY_{t-l+1}, CA_t,...,CA_{t-l+1}]'.
    - Companion form: X_t = C X_{t-1} + u_t with u_t zero-mean homoskedastic errors.
    - k-step-ahead forecast: E_{t+k} ΔY_{t+k} = A C^k X_t where A = [1 0 0 ... 0].
  - Model-predicted current account at time t:
    - p_t CA = K X_t, where K = A (1 - (1/(1+r)) C)^{-1} (matrix expression as in source).
    - The estimate p̂_t CA is obtained by replacing C with its empirical estimate Ĉ.
  - Testing equality of p̂_t CA and actual CA_t is equivalent to testing whether K equals the 2l vector T (all zeros except a 1 at the (l+1)th element).
  - Variance-covariance of K is approximated by J V J′ using the Delta method (V = variance-covariance of VAR parameters; J = Jacobian of K).
  - Wald statistic:
    - W = (K̂ - T)' [J V J']^{-1} (K̂ - T) has asymptotic chi-square distribution with 2l degrees of freedom. (Equation (3) as in source.)
- Problems under near-singularity:
  - Analytical sensitivity:
    - p_t CA is a linear function of the inverse of M ≡ I - (1/(1+r)) C.
    - If M has an eigenvalue close to zero (i.e., C has an eigenvalue close to 1+r), small estimation errors in VAR parameters can cause very large deviations in M^{-1} and thus in the model-predicted series.
    - Consequence: posterior distributions of predicted series, correlation coefficients, and variance ratios will be very wide.
  - Delta method failure and size/power distortions:
    - In short samples near singularity, the Delta method poorly approximates Var(K), distorting the Wald statistic W relative to its asymptotic distribution.
    - Two types of errors illustrated:
      - False rejection: Delta method yields a variance-covariance that is “too small,” making W “too large,” leading to rejection of the model when the likelihood support near the null is actually larger.
      - False acceptance: Delta method yields a variance-covariance that is “too large,” making W “too small,” leading to acceptance of the model when the likelihood support near the null is actually smaller.
    - Figures illustrating these phenomena: Figure 2 (False Rejection in the Singularity Region) and Figure 3 (False Acceptance in the Singularity Region).
  - Practical prevalence:
    - The near-singularity issue is characterized as “very common” in practice; Table 1 (Estimation Results) reports estimated eigenvalues of the VAR companion matrix for each country sample, with these eigenvalues noted as being above (text cut off in source).

### III. EMPIRICAL ILLUSTRATION (summary of approach and empirical diagnostics)
- Data and samples:
  - Countries: Belgium, Canada, Denmark, Sweden, and the UK.
  - Both annual and quarterly data are used; numerous figures present actual vs. predicted series and posterior distributions:
    - Figures include: Actual (-) and Predicted (--) Series: Subsample Estimation (Figure 1); Posterior Distributions of the (l+1)th Coefficient of the K Vector: Annual Data (Figure 4) and Quarterly Data (Figure 5); Actual vs Predicted series for Annual and Quarterly Data (Figures 6–9); Posterior Distributions of the Variance Ratio: Annual and Quarterly Data (Figures 10–11); Posterior Distributions of the Correlation Coefficient: Annual and Quarterly Data (Figures 12–13).
  - Data Appendix referenced.
- Key empirical findings (as summarized in the text):
  - Test discrepancies: In four out of ten cases the Wald test and the F-test lead to opposite inferences at traditional confidence levels.
  - Posterior dispersion: The 25.th and 97.5.th posterior percentile lines of the model-predicted series are almost without exception very wide compared to the actual series.
  - Variance ratio and correlation distributions: Posterior distributions are highly dispersed, indicating substantial probability mass for both much higher and much lower volatility of the actual series relative to the model prediction, and correlations concentrated near both 1 and -1.
  - These dispersion patterns arise irrespective of F-test outcomes, undermining claims of robust excess current account volatility.

### IV. CONCLUSION (implications)
- Methodological implication:
  - Inference based on Campbell-Shiller VAR-based present value testing is fragile under near-singularity, a condition frequently met in current account data due to persistence.
  - Standard Delta method-based Wald tests can produce misleading conclusions (both false rejections and false acceptances) in short samples.
- Empirical implication:
  - Claims of excess current account volatility (or its absence) based on direct comparison of actual and VAR-predicted series should be treated with caution given the wide posterior uncertainty documented.
- Broader relevance:
  - These concerns extend to other applications of the Campbell-Shiller methodology (e.g., term structure expectations tests) where bivariate VARs and present value constructs are employed.

*Source: _wp04106 - References..............................................................................................................*

### 09. for all

### 09. for all

### C. A Valid Test
- Define Rt = CAYt − CAYt−1 − ∆Yt and let It−1 be the information set containing all the values of ...,∆Yt−1, CAt−1, CAYt−1, as well as the previous values of any other variable. Equation (1) implies that E(Rt | It−1) = 0.
- One can regress Rt on It−1 with the appropriate number of lags and perform a simple F-test on the joint nullity of the coefficients of all the regressors in the information set.
- The F-test is fully valid because it does not rely on a (bad) linear approximation and, under Gaussianity of residuals, the likelihood’s shape is Gaussian regardless of unit roots.
- From a Bayesian perspective the usual OLS confidence intervals are correct under a flat prior; the Delta method can distort a true posterior probability interval.
- The F-test can be used to “measure” distortions in the standard Wald test by comparing posterior probabilities that the model is true as given by the Wald and the F tests.
- The F-test has been used in the literature but much less frequently than the Wald test.

### III. EMPIRICAL RESULTS

#### Data and general issues
- Five small open economies studied: Belgium, Canada, Denmark, Sweden, and the United Kingdom.
- Data used at both annual and quarterly frequencies.
- Eigenvalues of the variance-covariance matrix show near-singularity is a real issue in most country samples.
- Near-singularity is generated by current account persistence; VAR estimations show small coefficients on lagged income changes and high coefficients on the lagged current account.
- In the region of singularity, small differences in estimated companion matrix coefficients translate into very large differences in the coefficients of the K vector.

#### A. Tests of the Model
- Table 1 reports likelihood of Wald and F tests expressed as one minus the cdf of the test statistic.
- The distortion associated with the Wald test can be very large:
  - The Wald statistic suggests levels of significance that are up to eighty-eight percentage points different from their actual level as given by the F-test.
  - The average distortion across the ten country samples equals thirty three percentage points.
  - In four out of ten cases the Wald test leads to the wrong inference at the traditional 95 percent level of confidence.
- The Wald test always rejects the model with quarterly data while the F-test accepts it for two out of five countries (Belgium and Canada).
- These results illustrate how misleading the Wald test commonly used in the literature can be.

#### B. The K Vector
- The K vector is a non-linear function of the parameters of the companion matrix of the VAR.
- Near-singularity region: variance of the K vector coefficients becomes very large as ĉ$ approaches zero.
- For each country sample, 10,000 draws are generated from the multivariate Normal distribution given by the estimated VAR parameters and their associated variance-covariance matrix (5,000 draws for the UK quarterly data for computational reasons).
- For each draw the associated K vector is computed, yielding a Bayesian posterior distribution of the K vector representative of estimation uncertainty.
- Figures 4 and 5 show the ex-post distribution of the 1lth+ coefficient of K (supposed to equal one under the null):
  - For all country samples, the variance of the coefficient is very large.
  - Even when the model is consistent with the data as determined by the F-test, there is a high probability that the coefficient will be far from its theoretical value; the coefficient can easily be negative.
- Since by construction ptt, CA = K X, the large posterior variance of the K vector has strong implications for the predicted current account path, its variance, and its correlation with actual data.

#### C. Graphical Analysis
- Literature comparisons of actual and predicted current account paths (e.g., UK first oil shock, Japan second oil shock) can be misleading when not accounting for K dispersion.
- For each of the 10,000 K draws the predicted series is recalculated; plotted percentile lines correspond to the 25th and 97 5th. percentiles of the distribution at each point in time.
- Percentile lines are typically very wide compared with the actual series, often dramatically so.
- Example: actual deficits in the UK following the first oil shock exceed the single-sample model prediction, but particularly in quarterly data the percentile lines easily encompass those deficits, so the conclusion that the model underestimated the deficits is unwarranted.

#### D. Variance Analysis
- Literature has emphasized that actual current account series are typically more volatile than model predictions.
- For each draw i from the multivariate Normal, the predicted current account i,pt CA = K_i X_t is constructed for all t; for each draw the variance ratio is the intertemporal variance of i,pt CA over t divided by the intertemporal variance of the actual series.
- Variance ratios are often very dispersed as a direct consequence of dispersion in the K vector in the singularity region.
- Even when the model is consistent with the data the predicted current account can be much more or much less volatile than the actual.
  - Example: model strongly consistent with Belgian annual data yet there is a 40 percent probability that the predicted current account is over four times as volatile as the actual.
  - There is also a 20 percent probability that the predicted current account displays less than a fourth of the volatility of the actual series.
- The probability that the predicted series is more volatile than the actual averages 44 percent over the 10 samples, ranging from 11 percent for Belgian quarterly data to over 97 percent for Swedish quarterly data.
- One cannot conclude that the current account is less volatile than model predictions either.

#### E. Correlation Analysis
- Some authors claim the model has value because correlation between actual and predicted series tends to be high.
- Figures 12 and 13 show posterior distributions of in-sample correlation between actual and predicted series; these distributions are very wide reflecting dispersion in the K vector.
- Correlation values are not indicative of the model’s statistical validity.
  - For Belgian and Danish annual data the valid F-test accepts the model, yet there is over 45 percent probability that the correlation lies between 1− and 09−..
  - For Swedish annual data there is a 37 percent probability that the correlation will exceed 095. even though the test has rejected the model.
- Distributions often cluster around one and minus one; under near-singularity and one lag, if k̂2 is positive (negative) and very large relative to k̂1 then p̂CA is mostly driven by k̂2 CA and the correlation will tend to one (minus one).

### IV. CONCLUDING REMARKS
- The Campbell-Shiller methodology can be misleading when used on present value models of the current account.
- The paper casts doubt on results in the large empirical literature that uses similar techniques, including extensions to the present value model.
- Statistical analysis does not support any conclusion of excess current account volatility.
- The F-test, and not the traditional Wald test, should be used to formally test the model.
- Inference should not be limited to the sample estimate of the predicted current account path; it is essential to systematically construct ex-post distributions of the predicted series and associated correlation coefficients and variance ratios.
- Given the potentially high variance of cross-equation parameters in the singularity region, researchers will often find it difficult to make useful inferences about the relationship between the data and the model’s predictions.

*Source: _wp04106 - 09. for all*

### REFERENCES

### REFERENCES

### Bibliographic citations
- Agénor, P.R., C. Bismut, P. Cashin, and C.J. McDermott, 1999, “Consumption Smoothing and the Current Account: Evidence from France,” 1970–1996, Journal of International Money and Finance, Vol. 18, pp. 1–12.
- Bergin, P.R., and S.M. Sheffrin, 2000, “Interest Rates, Exchange rates, Present Value Models of the Current Account,” Economic Journal, Vol. 110, pp. 535–58.
- Callen, T., and P. Cashin, 2002, “Capital Controls, Capital Flows and External Crises: Evidence from India,” Journal of International Trade and Economic Development, Vol. 11, pp. 77–98.
- Campbell, J.Y., 1987, “Does Saving Anticipate Declining Labor Income? An Alternative Test of the Permanent Income Hypothesis,” Econometrica, Vol. 55, pp. 1249–73.
- Campbell, J.Y., and R. Shiller, 1987, “Cointegration and Tests of Present Value Models,” Journal of Political Economy, Vol. 93, pp. 1062–88.
- Campbell, J.Y., 1991, “Yield Spreads and Interest Rate Movements: A Bird’s Eye View,” Review of Economic Studies, Vol. 58, pp. 495–514.
- Cashin, P., and C.J. McDermott, 1998a, “Are Australia’s Current Account Deficits Excessive?” The Economic Record, Vol. 74, pp. 346–61.
- Cashin, P., and C.J. McDermott, 1998b, “International Capital Flows and National Creditworthiness: Do the Fundamental Things Apply As Time Goes By?” IMF Working Paper 98/172, (Washington: International Monetary Fund).
- Ghosh, A.R., 1995, “International Capital Mobility Amongst the Major Industrialized Countries: Too Little or Too Much?” Economic Journal, Vol. 105, pp. 107–28.
- Ghosh, A.R., and J.D. Ostry, 1995, “The Current Account in Developing Countries: A Perspective from the Consumption-Smoothing Approach,” World Bank Economic Review, Vol. 9, pp. 305–33.
- Gruber, J., 2004, “A Present Value Test of Habits and the Current Account,” Journal of Monetary Economics, forthcoming.
- Hall, V.B., K. Kim, and R.A. Buckle, 2001, “New Zealand’s Current Account Deficit: Analysis Based on the Intertemporal Optimisation Approach,” New Zealand’s Treasury Working Paper 01/02.
- Hardouvelis, G.A., 1994, “The Term Structure Spread and Future Changes in Long and Short Rates in the G7 Countries: Is There a Puzzle?” Journal of Monetary Economics, Vol. 33, pp. 255–83.
- Kasa, K., 2004, “Testing Present Value Models of the Current Account: A Cautionary Note,” Journal of International Money and Finance, forthcoming.
- Nason, J., and J.H. Rogers, 2003, “The Present Value Model of the Current Account has been Rejected: Round up the Usual Suspects,” International Finance Discussion Paper 2003–760 (Washington: Board of Governors of the Federal Reserve System).
- Obstfeld, M., and K. Rogoff, 1995, “The Intertemporal Approach to the Current Account”, in Handbook of International Economics, Vol. 3, ed. by G. Grossman, and K. Rogoff (Amsterdam: North-Holland), pp. 1731–99.
- Obstfeld, M., and K. Rogoff, 1996, Foundations of International Macroeconomics (Cambridge: MIT Press).
- Ostry, J.D., 1997, “Current Account Imbalances in ASEAN Countries: Are They a Problem?” IMF Working Paper 97/51 (Washington: International Monetary Fund).
- Otto, G., 1992, “Testing a Present Value Model of the Current Account: Evidence from the U.S. and Canadian Time Series,” Journal of International Money and Finance, Vol. 11, pp. 414–30.
- Sachs, J., 1981, “The Current Account and Macroeconomic Adjustment in the 1970’s,” Brookings Papers on Economic Activity, Brookings Institution, Vol. 12, pp. 201–68.
- Sheffrin, S.M., and W.T. Woo, 1990, “Present Value Tests of an Intertemporal Model of the Current Account,” Journal of International Economics, Vol. 29, pp. 237–253.
- Sims, C.A., and T. Zha, 1995, “Error Bands for Impulse Responses,” Federal Reserve Bank of Atlanta Working Paper 95/6.

### Data appendix — definitions, data sources, and samples
- Data source: International Financial Statistics of the International Monetary Fund.
- Variable definitions (expressed in real, per capita terms):
  - Yt = GDPt - Gt - It (Net output)
  - CAt = GNPt - Ct - It - Gt (Current account)
- Corresponding IFS series:
  - GNP: gross national income (line 99a)
  - G: government consumption (line 91f)
  - I: sum of private gross fixed capital formation (line 93e) and increase/decrease in stocks (line 93i)
  - C: household consumption (line 96f)
  - GDP: gross domestic product (line 99b)
- Conversion to real, per capita terms uses:
  - GDP volume in 1995 or 1996 terms (line 99b)
  - population (line 99z)
- Country sample periods (longest available sample in IFS):
  - Belgium: Annual Data 1953–1998; Quarterly Data 1980:1–1998:4
  - Canada: Annual Data 1948–2002; Quarterly Data 1948:1–2002:4
  - Denmark: Annual Data 1966–2002; Quarterly Data 1988:1–2002:4
  - Sweden: Annual Data 1950–2002; Quarterly Data 1990:1–2002:4
  - United Kingdom: Annual Data 1948–2002; Quarterly Data 1955:1–2002:4
- Data treatment and estimation conventions:
  - Means removed from the current account and from the first difference in net output (testing dynamic restrictions).
  - Annual and quarterly real interest rates set to 4 percent and 1 percent respectively.
  - VAR lag length selected by the Akaike information criterion.
  - Results robust to changes in number of lags or in the value of the real interest rate.
- Robustness check:
  - Estimated a consumption-tilting parameter when the discount factor is not equal to the inverse of the gross real interest rate (as in Ghosh (1995)).
  - Estimated consumption-tilting parameters are usually close to one.
  - Resulting series display similar properties and results remain robust.

- Note on Belgium: For Belgium we cut the sample in 1998 as there is a break in the data following the adoption of the euro in 1999.

### Table 1 — Estimation results (selected numeric entries preserved exactly)
- Table header labels include: Number of Lags; Eigenvalue; Wald test; F-test.
- Annual data
  - Belgium 1953–1998: Number of Lags 1; Eigenvalue 1.01; Wald test 96.9%; F-test 36.7%*
  - Canada 1948–2002: Number of Lags 1; Eigenvalue 0.61+0.2i; Wald test 27.8%; F-test 0.6%
  - Denmark 1966–2002: Number of Lags 1; Eigenvalue 0.96; Wald test 94.9%; F-test 20%*
  - Sweden 1950–2002: Number of Lags 1; Eigenvalue 0.93; Wald test 89.4%; F-test 0.8%
  - United Kingdom 1948–2002: Number of Lags 2; Eigenvalue 0.67; Wald test 2.8%; F-test 0.1%
- Quarterly data
  - Belgium 1980–1998: Number of Lags 3; Eigenvalue 0.98; Wald test 0.7%; F-test 43.7%*
  - Canada 1948–2002: Number of Lags 8; Eigenvalue 0.94; Wald test 2%; F-test 38.1%*
  - Denmark 1988–2002: Number of Lags 4; Eigenvalue 0.93; Wald test 0%; F-test 0%
  - Sweden 1990–2002: Number of Lags 4; Eigenvalue 0.97; Wald test 0%; F-test 0%
  - United Kingdom 1955–2002: Number of Lags 4; Eigenvalue 0.94; Wald test 0%; F-test 0%
- Notes for Table 1:
  - The figures for the tests correspond to one minus the cdf of the test statistic.
  - A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - For each sample the number of lags was selected using the Akaike criterion.

### Figures and graphical notes (appendix)
- Figures referenced:
  - Figure 1. Actual (-) and Predicted (--) Series: Subsample Estimation
  - Figure 2. False Rejection in the Singularity Region
    - Notation and comment: c0 is the true value of the parameter, ĉ the empirical estimate, and [ĉ,LR] the probability interval around ĉ. The Delta method yields the probability interval [, ,K̂LR δδ ] around K̂ rather than the correct [K̂,LR].
  - Figure 3. False Acceptance in the Singularity Region
    - Notation and comment: c0 is the true value of the parameter, ĉ the empirical estimate, and [ĉ,LR] the probability interval around ĉ. The Delta method yields the probability interval [, ,K̂LR δδ ] around K̂ which includes 0K even though [ĉ,LR] excludes 0c.
  - Figure 4. Posterior Distributions of the (1)th l+Coefficient of the K Vector: Annual Data
    - Note: A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 5. Posterior Distributions of the (1)th l+Coefficient of the K Vector: Quaterly Data
    - Note: A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 6. Actual (-) and Predicted (--) Series: Annual Data
    - Note: A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 7. Actual (-), Predicted (--), and Percentile Lines (Bold): Annual Data
    - Example panel labels and notes: Belgium 1953 - 1998*; Canada 1948 - 2002; Denmark 1966 - 2002*; Sweden 1950 - 2002; United Kingdom 1948 - 2002.
    - Note: Bold lines correspond to the 25th. and 97 5th. percentiles of the posterior distribution of the predicted series. A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 8. Actual (-) and Predicted (--) Series: Quarterly Data
    - Note: A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 9. Actual (-), Predicted (--), and Percentile Lines (Bold): Quarterly Data
    - Notes: Bold lines correspond to the 2.5th and 97.5th percentiles of the posterior distribution of the predicted series. A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 10. Posterior Distributions of the Variance Ratio: Annual Data
    - Notes: The variance ratio is expressed as the variance of the predicted series over that of the actual. A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 11. Posterior Distributions of the Variance Ratio: Quarterly Data
    - Notes: The variance ratio is expressed as the variance of the predicted series over that of the actual. A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 12. Posterior Distributions of the Correlation Coefficient: Annual Data
    - Note: A star denotes model acceptance by the F-test at the 5 percent level of confidence.
  - Figure 13. Posterior Distirbutions of the Correlation Coefficient: Quarterly Data
    - Note: A star denotes model acceptance by the F-test at the 5 percent level of confidence.

*Italic source attribution.*

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