Canada: Selected Issues
Author: International Monetary Fund
Publication Date:
18 Jun 1996
eISBN:
9781451806830
Language:
English
Keywords:
ISCR; CR; saving rate; Canada; inflation rate; rate; FEER estimate; equilibrium current account; FEER misalignment estimate; trade elasticity; Marshall-Lerner condition; Exchange rates; Output gap; Real exchange rates
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This Selected Issues paper addresses some of the key policy and economic challenges facing the Canadian economy. The paper presents a new approach to predicting the business cycle in the context of the Canadian economy. This approach uses a range of parametric and nonparametric tests to gauge the ability of various indicators to predict turning points in the business cycle. The paper also presents a model that links the inflation rate to the business cycle and the rates of change in the exchange rate and in unit labor costs.

Abstract

This Selected Issues paper addresses some of the key policy and economic challenges facing the Canadian economy. The paper presents a new approach to predicting the business cycle in the context of the Canadian economy. This approach uses a range of parametric and nonparametric tests to gauge the ability of various indicators to predict turning points in the business cycle. The paper also presents a model that links the inflation rate to the business cycle and the rates of change in the exchange rate and in unit labor costs.

IV. Uncertainty in the Fundamental Equilibrium Exchange Rate 1/

1. Introduction

The assessment of “misalignment” in exchange rates has been the focus of much recent interest. 2/ This interest has been spurred by the seemingly excessive variation of exchange rates relative to fundamentals, concerns about apparent misalignments among major currencies, and the desire to calculate parities for the Exchange Rate Mechanism of the European Monetary System. 3/ One way to gauge the degree of misalignment is by calculating the Fundamental Equilibrium Exchange Rate (FEER)--the exchange rate compatible with full employment and external balance. 4/ Recent studies have constructed FEERs using models ranging from simple partial-equilibrium trade models to full-blown macroeconomic models. 5/

Estimated FEERs depend on estimated parameters, and so like other estimates based on data, are subject to sampling uncertainty. This means that the estimates should not be taken too literally--or as Clark et al. (1994) state, “it is clearly more realistic to think of ranges rather than of point estimates in the assessment of exchange rates” (p. 20). Some earlier work has performed sensitivity analysis on FEERs, but sensitivity analysis does not link the inherent uncertainty in estimated parameters to uncertainty in FEER estimates. 6/ A confidence interval would be more useful, as it would quantify the estimation error and permit the user to decide whether the current exchange rate is a significant distance from equilibrium.

This paper calculates the sampling distribution of the FEER for Canada based on a simple model of trade. 7/ The analysis focuses on the uncertainty introduced by employing estimated trade elasticities. The estimated FEER shows a substantial range of variation that primarily reflects such uncertainty. The results suggest that FEER estimates should be treated with caution, as they may offer a fairly imprecise measure of misalignment.

2. The Fundamental Equilibrium Exchange Rate (FEER)

This paper follows the methodology outlined by Clark et al. (1994) and used by, inter alia, Chandler and Laidler (1995) in studying the equilibrium exchange rate for Canada. The three steps in calculating the FEER are sketched below.

The first step is to calculate the underlying current account, CAu. This is done by adjusting the actual current account, CAa, for foreign and domestic output gaps and the effects of changes in lagged real exchange rates, i.e.

CAuY=CAaY+αxXYYGAPf−αmMYYGAP−[βXXY+(βm−1)MY]*(R−RLAG)(1)

where

RLAG=Θ0R+Θ1R−1+Θ2R−2Θ0+  Θ1+Θ2;Θ0+Θ1+Θ2=1.

αx and αm denote the elasticities of real exports and real imports with respect to foreign and domestic income; βx and βm as denote the elasticities of real exports and real imports with respect to their relative prices; X, M, and Y denote nominal exports, imports and output; YGAPf denotes the export-weighted foreign output gap and YGAP denotes the domestic output gap (both measured in differences from potential in logarithms, where a positive gap indicates output below potential); R denotes the logarithm of the real exchange rate (in this methodology, the multilateral real effective exchange rate); and RLAG is a weighted average of current and past exchange rates.

Once the underlying current account has been calculated, the second step is to specify the equilibrium or desired current account, CAe. For example, CAe could be the current account consistent with a desirable ratio of net foreign assets to GDP, or the current account consistent with a particular model of saving and investment.

The third and last step is to consider the adjustment of the underlying current account to the equilibrium current account. This implies that the real exchange rate adjusts to its equilibrium value, Re: 1/

CAuY−CAeY=[βxXY+(βm−1)MY](Re−R),

or

Re−R=CAuY−CAeY[βxXY+(βm−1)MY].(2)

(Re-R) is the estimated degree of misalignment, which is the focus of this study.

3. The FEER as a statistical estimate

It is obvious that the estimated degree of misalignment is a function of unknown quantities, in particular of trade elasticities and the equilibrium current account. In practice, the equilibrium current account is replaced by an estimate based on judgement, and trade elasticities are replaced by statistical estimates. Thus the FEER is also a statistical estimate, and like other statistical estimates has a sampling distribution. If known, that sampling distribution could be used for formal hypothesis testing. For example, it could be used to construct a test that the estimated degree of misalignment is not significantly different from zero.

A closed-form distribution for the FEER that could be applied in a sample of any size would be ideal. At best, however, one might be able to derive an analytic approximation to the distribution of the FEER that is accurate in large samples. For example, given that the elasticities are normally distributed in large samples, the PEER estimate of misalignment will be normally distributed in large samples as well. 2/ However, it is hard to say when a sample is large enough for the approximation to be a good one. For the data used in this paper, some tests (not shown) imply that the approximation is poor--that the estimated FEER is not normally distributed. 3/

Since an analytic approximation to the distribution of the PEER is likely to be poor, the simulations proceeded by a different route, namely, by simulating repeated samples. That is, elasticities were drawn at random from specified distributions, and then used to calculate the PEER, just as would be done in an actual random sample. This process was repeated 10,000 times, generating elasticities at random 10,000 times and calculating 10,000 FEERs, under various assumptions about the equilibrium current account and other aspects of the model. The 10,000 FEW estimates for each case then yielded a picture of the sampling distribution of the PEER. More details on how this was done are in the following section.

4. Simulating the sampling distribution of the FEER

Calculating the FEER misalignment estimate required trade elasticities, a value for the equilibrium current account, and miscellaneous data such as imports, exports, and GDP. The source for each of these is described in turn.

Elasticities were generated at random by two methods. In the first method, the elasticities reflected the range of estimates typically encountered in the literature. Elasticities were drawn at random from uniform distributions, where the minimum and maximum of each distribution were fixed at the minimum and maximum elasticities for Canada shown in the survey of empirical trade models by Goldstein and Khan (1985). These ranges are shown in the tabulation below. The estimated elasticities from the recent study of the PEER for Canada by Chandler and Laidler (1995) fall in these ranges.

Ranges for Trade Elasticities 1/

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As a second method, a bootstrap methodology was used: elasticities were drawn at random from an empirically-determined distribution of estimates. 1/ This empirical distribution was created by simulating how trade elasticities would vary as they were re-estimated in repeated samples. The resulting FEER estimates reflect the sampling variability in estimated elasticities.

Three values of the equilibrium current account were specified: a surplus of one percent of GDP, balance (no deficit or surplus), and a deficit of one percent of GDP. The trade balance for 1994 that stabilizes the ratio of net international liabilities to GDP is estimated at about 1.5 percent of GDP. 2/ Hence, a range of (-1, +1) for the desired current account seems reasonable, and includes levels of the current account that would put external debt on a declining path as a percentage of GDP, which is one of the objectives of the Canadian authorities. 3/

Exports, imports, GDP, the current account, and the domestic output gap (all for 1995) were taken from the World Economic Outlook database. 4/ Estimates made by the Research Department were used for the foreign output gap and the weights on lagged exchange rates (θ0, θ1, and θ2).

The simulations account for variability in elasticities and judgement about the equilibrium current account. However, no attempt was made to account for uncertainty in the measure of the output gap, nor for likely developments in fiscal policy or demographics. As such, the estimates presented below understate the uncertainty in FEER estimates.

5. Results

This section presents the results of the simulations. Four sets of simulations are presented below. The first set of simulations proceeded as described above. The second set of simulations imposed the Marshall-Lerner condition on the simulated elasticities. The third set of simulations added randomness in the equilibrium current account, and the final set of simulations examined the sampling distribution in the underlying current account.

a. The Sampling distribution of the FEER

Chart IV-1 displays the simulated joint distribution of the estimated degree of misalignment (Re-R) and the estimated underlying current account (CAu), while Table IV-1 shows some statistics on the distribution of (Re-R). 1/ Each simulation shows an overvalued real effective exchange rate on average. The median is higher than the mean in every case, implying positive skewness, and thus measuring significance in the usual way (a point estimate two standard deviations from zero) may not yield accurate inferences. Another measure of uncertainty, one that does not rely on the symmetry of the underlying distribution, is the percentile range. For example, the range between the 95th and 5th percentiles can be thought of as a 90 percent confidence interval for the estimate.

CHART IV-1
CHART IV-1

CANADA JOINT DISTRIBUTION OF ESTIMATED DEGREE OF MISALIGNMENT AND ESTIMATED UNDERLYING CURRENT ACCOUNT

Citation: IMF Staff Country Reports 1996, 038; 10.5089/9781451806830.002.A004

Table IV-1.

Canada: Estimates of the Degree of Real Misalignment

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The bounds of the 90 percent confidence intervals (denoted by q05 and q95) are shown in the last two columns of Table IV-1. The confidence intervals imply that even when using the uniform distribution for elasticities (cases A-C), the uncertainty in the estimated degree of misalignment is not negligible. When using the empirical distribution for elasticities (cases D-F), it is quite large indeed. For the case with an equilibrium current account deficit of 1 percent of GDP, with 90 percent confidence, the degree of required depreciation is between 18 percent and about 1 percent (using uniform elasticities) or between 67 percent and -45 percent (using empirically-distributed elasticities). 2/ In either case, discerning the degree of misalignment is difficult.

b. Sampling distribution of the PEER under the Marshall-Lerner condition

The experiments described in the previous section did not impose the Marshall-Lerner condition that a real appreciation worsens the current account. Although nothing in theory guarantees that the condition must hold, it is sufficiently well-entrenched as a stylized fact of empirical trade models that it is worthwhile to consider its implications. Hence, the experiments A above were repeated, but using combinations of βx* and βm* for which β^x+β^m was greater than unity.

Table IV-2 and Chart IV-2 display the results of the simulations with the Marshall-Lerner condition imposed (for ease of comparison with the previous experiments, the corresponding results for the unrestricted case are reproduced from Table IV-1). Imposing the Marshall-Lerner condition shifts the distribution of the estimated degree of misalignment to the left, as evidenced by the smaller (more negative) means and medians for the distributions. This is not surprising, as the Marshall-Lerner condition means that only depreciation (not appreciation) can close the gap between the underlying and equilibrium current accounts. Accordingly, the confidence intervals for the bootstrapped cases no longer bracket zero--indeed, unless the estimated underlying current account is quite volatile, they cannot bracket zero. Finally, while imposing the Marshall-Lerner condition makes the confidence intervals smaller, it does not make them small. The narrowest confidence interval brackets 20 percent overvaluation to about 2 percent overvaluation, while the widest confidence interval brackets 230 percent overvaluation to 17 percent overvaluation.

Table IV-2.

Canada: Sensitivity of Estimates of the Degree of Real Misalignment to Imposition of the Marshall-Lerner Condition 1/

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The corresponding cases from Table IV-1 are reproduced here for ease of comparison.

CHART IV-2
CHART IV-2

CANADA JOINT DISTRIBUTION OF ESTIMATED DEGREE OF MISALIGNMENT AND ESTIMATED UNDERLYING CURRENT ACCOUNT, WITH MARSHALL-LERNER CONDITION IMPOSED

Citation: IMF Staff Country Reports 1996, 038; 10.5089/9781451806830.002.A004

c. Sampling distribution of the FEU when CAe is random

As noted, these experiments understate the true degree of uncertainty, by ignoring the uncertainty in estimates of output gaps, for example. There is also judgmental uncertainty about the desired current account. It would be useful to quantify this uncertainty by treating the desired current account as a random variable rather than a parameter. Hence, this uncertainty was simulated by drawing equilibrium current accounts at random from a uniform distribution with bounds equal to the maximum and minimum of the values used above (-1 and +1). The Marshall-Lerner condition is imposed for these simulations.

The simulations using a random equilibrium current account are presented in Table IV-3. Since the average equilibrium current account is zero in these simulations, they correspond to Cases B and E in Table IV-2 where the equilibrium current account is fixed at zero, and so are labeled Case B’ and Case E’ in the table. The means, medians and confidence intervals are scarcely affected by randomness in the equilibrium current account. The effect of uncertainty in the judgmental estimate of the equilibrium current account seems small relative to the effect of uncertainty in the elasticity estimates.

Table IV-3.

Canada: Sensitivity of Estimates of the Degree of Real Misalignment to Uncertainty about Equilibrium Current Account 1/

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For all cases, the Marshall-Lerner condition is imposed. The corresponding cases (B and E) from Table IV-2 are reproduced here for ease of comparison.

d. Sampling distribution of the underlying current account

It is also useful to isolate the degree of uncertainty in the underlying current account. Table IV-4 presents descriptive statistics for the distribution of the underlying current account (since CAe is not used in estimating the underlying current account, there are only two cases, one for each type of distribution for the elasticities). Without the Marshall-Lerner condition imposed, the 90 percent confidence interval for the estimated underlying current account is about (-2.9, -1.7) for the case with uniformly-distributed elasticities and about (-2.4, -1.9) for the case with bootstrapped elasticities, in both cases well on the deficit side. In each case, there is a significant distance between the estimated underlying current account and the equilibrium current accounts specified above; for example, a deficit of one percent of GDP lies above these confidence intervals. This means that the estimated misalignment is insignificant not because the estimated current-account adjustment is insignificant, but because the FEER’s dependence on estimated elasticities makes it imprecise.

Table IV-4.

Canada: Estimates of the Underlying Current Account

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6. Conclusions

Since the FEER is a function of both sample data and judgement, the point estimate that emerges from the procedure should be treated with caution. The results described above suggest that the uncertainty about the point estimate is very large, even when the degree of implied adjustment in the current account is significant. Most of this uncertainty comes from the use of estimated trade elasticities. Accounting for judgmental uncertainty in the specification of the equilibrium current account only slightly magnifies the variability of the FEER estimates.

The variability in the FEER estimates for Canada is large enough that it is often impossible to tell whether the real exchange rate is overvalued or undervalued with any precision. As a result, the usefulness of misalignment indicators based on such trade models seems questionable. Even given the relative simplicity of the model relative to some others used to produce FEERs, large apparent misalignments can be small relative to sampling uncertainty.

APPENDIX Construction of Empirical Distribution for Trade Elasticities

In constructing a simulated distribution for the PEER, trade elasticities were “bootstrapped” or resampled from empirical distributions for (α^x, β^x) and (α^m, β^m). The bootstrap technique constructs an estimate of the distribution of some statistic of interest by repeatedly sampling subsets of the data--in effect, treating the sample as if it were the population. The estimated distribution can then be used to perform inference. Bootstrapping is valuable when little is known about the distribution of a statistic in small samples. 1/

The first step in creating the empirical distribution for elasticities was to estimate representative elasticities. This required data on exports and imports at constant prices, the relative prices of imports and exports, and foreign and domestic income at constant prices. The relative price of exports was measured by the ratio of the Canadian export price deflator (converted to U.S. dollar terms) to export-weighted partner-country export prices in U.S. dollar terms. The relative price of imports was measured by the ratio of the import price deflator to the consumer price index. Domestic income was measured by real GDP, and foreign income was measured by an index of export-weighted partner-country real GDP. All data were quarterly, and the sample covered 1971:I to 1995:II.

Since the series showed evidence of unit-root nonstationarity and cointegration, the Stock and Watson (1993) estimator was used to estimate the elasticities. Starting with 8 leads and 8 lags in the short-run adjustment mechanism, insignificant short-run coefficients were trimmed to yield a parsimonious model that passed tests for autoregressive conditional heteroskedasticity, a structural break at mid-sample, and tests for autocorrelation, after a correction for first-order autoregressive (AR(1)) disturbances.

The rest of the procedure followed Li (1994). The sample of import elasticities was generated as follows (the export elasticities were generated in an analogous fashion). Denote by M the dependent variable (imports in constant-dollar terms), and denote by Z a vector containing the independent variables (the relative price of imports and domestic income at constant prices). 2/ Residuals (e^t), estimated coefficients (δ^,π^), and an estimated AR(1) coefficient ρ^ were calculated by estimating the model

Mt = μ + Σj=pp δj ΔZt+j + π Zt + et, et = ρet−1 + ut, ut i.i.d.(A1)

with P > 0 > p (P leads and p lags). The series st = ΔZt was calculated as well.

Next, drawing at random with replacement from the estimated disturbances (u^t)t=1T and from the shocks (st)t=1T synthetic series et* = ρ^ et−1* + u^t and Zt* = Zt−1*+ st were constructed. These were used in turn to construct Mt* by replacing (δ,π) with (δ^,π^), et with e^t, and ρ with ρ^ in equation (A1). The parameters (δ,π) were then re-estimated from the synthetic data M* and Z*, yielding estimates (δ*,π*) from fitting

Mt* = μ* + Σj=pP δj* ΔZt+j* + π* Zt* + et*,

again with an AR(1) correction. This process was repeated 1000 times for each equation, yielding 1000 simulated estimates of αx, βx, αm, and βm.

These estimates were sampled at random for each of the 10,000 trials where Re - R was calculated. The samples were drawn in matched pairs: that is, if the 100th estimated αx and the 200th estimated αm were chosen for a particular estimate of Re - R, then the 100th estimated βx and the 200th estimated βm were chosen as well. This scheme preserved the dependence between α^x and β^x and the dependence between α^m and β^m. Chart IV-3 summarizes the results of the bootstrapping experiment, showing the distributions of the estimated import and export elasticities.

CHART IV-3
CHART IV-3

CANADA DISTRIBUTION OF SIMULATED IMPORT AND EXPORT ELASTICITIES

Citation: IMF Staff Country Reports 1996, 038; 10.5089/9781451806830.002.A004

References

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1/

Prepared by Charles Kramer.

2/

The notion of “misalignments” in market-determined exchange rates is controversial. The paper does not address this issue, but rather focuses on the methodological issue of whether a particular model of the equilibrium exchange rate can produce estimates that are precise enough to be useful.

3/

See for example Clark et al. (1994).

4/

Sometimes also referred to as a Desired Equilibrium Exchange Rate (DEER) so as to emphasize its normative nature.

5/

Examples include Goldman Sachs (1995), Chandler and Laidler (1995), Church (1992), and the papers in Williamson (1994). Other approaches to assessing movements in real exchange rates, including purchasing power parity, terms of trade, and productivity trends (Balassa-Samuelson), are discussed in Amano and van Norden (1995), Ronald MacDonald (1995), Asea and Gorden (1994), and Isard and Symansky (1995).

6/

Some studies of FEERs that contain sensitivity analysis are discussed in Clark et al. (1994).

7/

Though the data employed are for Canada, nothing about the appropriateness of the exchange rate for the Canadian dollar is implied by or should be inferred from what follows.

1/

For the sake of simplicity, this expression, like the previous one, is a comparative statics definition that ignores hysteresis effects (see Bayoumi et al. (1994)).

2/

This follows from the so-called ‘delta method’ of approximation. See Serfling (1980).

3/

Normality tests (see Epps and Pulley (1983)) rejected the null of normality at the one percent level for the simulated values of the FEERs presented later. Test results are available on request.

1/

See Goldstein and Khan (1985), Tables 4.1 and 4.3.

1/

The Appendix briefly discusses the bootstrap methodology and explains its application to the problem at hand. See Davidson and MacKinnon (1993), Chapter 21, for a more detailed discussion of bootstrapping and simulation in econometrics.

2/

See Sheila MacDonald (1995).

3/

See The Economic and Fiscal Update, Department of Finance, Canada, December 6, 1995.

4/

Complete data for 1995 did not become available until after the simulations were performed, so some of the data employed were based on staff estimates. Some of the experiments were later repeated with complete data for 1995, and the results were qualitatively the same as the ones shown here--the FEER estimates showed a wide range of uncertainty.

1/

In some random draws, the denominator of the expression for the PEER (equation (2)) was very small. The subsequent large outliers in the results unduly influenced the descriptive statistics. To remedy this problem, 2.5 percent of the distribution of the FUR was trimmed from each tail (trimming the distributions of the elasticities did not reduce the number of outliers). This trimming means that the statistics will understate the true degree of imprecision in the PEER estimator, however.

2/

Since the Marshall-Lerner condition is not imposed at this stage, some of the random draws might imply that an apperciation is needed to reduce the current account deficit to its equilibrium level. The implications of imposing the Marshall-Lerner condition are explored in the next section.

1/

See Davidson and MacKinnon (1993), Chapter 21, for a discussion of bootstrapping and related topics.

2/

M and Z were expressed in logarithms.

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Cover IMF Staff Country Reports
Canada: Selected Issues
Author: International Monetary Fund
Volume/Issue:
Volume 1996: Issue 038
Publisher:
International Monetary Fund
ISBN:
9781451806830
ISSN:
1934-7685
Pages:
168
DOI:
https://doi.org/10.5089/9781451806830.002
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    CANADA JOINT DISTRIBUTION OF ESTIMATED DEGREE OF MISALIGNMENT AND ESTIMATED UNDERLYING CURRENT ACCOUNT

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    CANADA JOINT DISTRIBUTION OF ESTIMATED DEGREE OF MISALIGNMENT AND ESTIMATED UNDERLYING CURRENT ACCOUNT, WITH MARSHALL-LERNER CONDITION IMPOSED

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    CANADA DISTRIBUTION OF SIMULATED IMPORT AND EXPORT ELASTICITIES

CANADA JOINT DISTRIBUTION OF ESTIMATED DEGREE OF MISALIGNMENT AND ESTIMATED UNDERLYING CURRENT ACCOUNT

CANADA JOINT DISTRIBUTION OF ESTIMATED DEGREE OF MISALIGNMENT AND ESTIMATED UNDERLYING CURRENT ACCOUNT, WITH MARSHALL-LERNER CONDITION IMPOSED

CANADA DISTRIBUTION OF SIMULATED IMPORT AND EXPORT ELASTICITIES