Covered Interest Parity

Equation (4) represents a CIP relationship between CETES and PAGAFES. Figure 1 shows the evolution of (1 + i_t) and (1 + i_t*)(F_t/S_t) over the period January 1987 to July 1990. The figure shows that deviations from covered interest arbitrage were small for most of the observations, with notable exceptions occurring between November 1987 and March 1988. That period, however, coincided with the beginning of the current stabilization program in Mexico, when a series of structural reforms were announced, a major devaluation took place, and the authorities implemented a price-wage pact between the Government, labor, and business, with the aim of controlling prices. These developments undoubtedly increased uncertainties about the future course of the economy, which, combined with some official intervention in domestic financial markets, might have prevented covered arbitrage from holding.

The deviations from CIP are presented in Table 1. As the table shows, 71 percent of the deviations are less than 0.5 percentage point, and 86 percent of the deviations are less than 1 percentage point. By comparison, the average bid-ask spreads of the exchange rate in the controlled and the free markets during the period were 0.95 percent and 2.1 percent, respectively. Besides being small, the deviations from interest parity have a mean very close to zero and are uncorrected at all lags. Indeed, the results of a Q-test for serial correlation indicate that the null hypothesis that the series is white noise cannot be rejected at the 1 percent significance level.

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CETES vs. PAGAFES (Adjusted)

Citation: IMF Staff Papers 1991, 003; 10.5089/9781451930801.024.A007

Note: CETES (Certificados de la Tesorería); PAGAFES (Pagares de la Tesorería de la Federación). The unit on the y-axis is 1 + i. where i is the interest rate per month.^aAdjusted for the premium in the forward market.

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These results indicate that, with the exception of a short period, when uncertainties in the economy were unusually large, deviations from interest rate parity have been small and random occurrences. This evidence seems to provide strong support for CIP.⁸

Table 1.

Analysis of the Deviations from Interest Rate Parity

(Series: (1 + i_t) − (1 +i_t)(F_t/S_t)

^aThe null hypothesis of no serial correlation is rejected at the 1 percent level only if the value of x² is greater than the critical value of 21.7.

Table 1.

Analysis of the Deviations from Interest Rate Parity

(Series: (1 + i_t) − (1 +i_t)(F_t/S_t)

Item	Result
Number of observations	43
Mean	0.0004
Standard deviation	0.010
Proportion of deviations (in percent)
Less than 0.5 percentage point	0.71
Less than 100 percentage points	0.86
Q-test
Degrees of freedom	9
x2-statistica	7.88

^aThe null hypothesis of no serial correlation is rejected at the 1 percent level only if the value of x² is greater than the critical value of 21.7.

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Table 1.

Analysis of the Deviations from Interest Rate Parity

(Series: (1 + i_t) − (1 +i_t)(F_t/S_t)

Item	Result
Number of observations	43
Mean	0.0004
Standard deviation	0.010
Proportion of deviations (in percent)
Less than 0.5 percentage point	0.71
Less than 100 percentage points	0.86
Q-test
Degrees of freedom	9
x2-statistica	7.88

^aThe null hypothesis of no serial correlation is rejected at the 1 percent level only if the value of x² is greater than the critical value of 21.7.

Uncovered Interest Parity

The uncovered version of interest parity between CETES and PAGAFES is represented by equation (2).

The empirical literature on UIP has broadly tested equation (2) (see, for example, Cumby and Obstfeld (1980) and Lizondo (1983a)). However, since E_t(S_t+1) is an unobservable variable, the empirical tests have been joint tests of equation (2) and two hypotheses of expectations behavior: (1) the rational expectations hypothesis that states that E_t(S_t+1) is a mathematical conditional expectation, based on the true probability distribution underlying the behavior of the exchange rate; and (2) the hypothesis that the foreign exchange market is “weakly” efficient, in that expectations of the future exchange rate incorporate all information contained in past forecast errors of the exchange rate. These joint hypotheses imply that the forecast error

should have zero mean and be serially uncorrelated.

The joint hypothesis containing UIP and the hypothesis of weak market efficiency was tested for Mexico using nonoverlapping monthly observations covering the period January 1987–July 1990. Before these results are discussed, it should be pointed out that over this period, the exchange rate was not freely floating, but was managed by the authorities and followed three different regimes: (1) from January 1987 to December 1987 the exchange rate was depreciated by an unspecified amount each day; (2) from January 1988 to December 1988 the exchange rate was fixed except for a small change in February; and (3) from January 1989 to May 1990 the exchange rate was depreciated by an announced 1 peso per U.S. dollar a day.⁹ Recent studies have analyzed the problems involved in testing the U1P hypothesis in the presence of intervention in exchange rate markets. In particular. Krasker (1980) and Lizondo (1983b) have shown that in the presence of a small and positive probability of a devaluation, an efficient exchange rate market will imply that the expected value of the future spot rate will reflect the probability of that event. However, as long as the devaluation does not take place, the expectation of the future spot rate will consistently overestimate the realized future spot rate. As a result, the forecast error in the exchange market will show a positive bias,¹⁰ but this will not be sufficient to reject the joint hypothesis that UIP holds and that the market is weakly efficient.

Table 2 presents an analysis of the forecast error, ∈_t. The most important result is that although the Q-test indicates that at the 1 percent significance level we cannot reject the hypothesis that the errors are uncorrelated, the mean of the forecast error is positive.¹¹ These results imply that S_t[(1 + i_t)/(l + i_t*)] overstates the future spot rate. Based on our previous discussion, however, the existence of a positive mean in the forecast errors does not allow us to accept or reject the hypothesis that UIP holds under conditions of weak market efficiency, since the tests for UIP are not appropriate in the context of a peso problem. However, the lack of autocorrelation in the forecast errors seems to indicate that the interest rate differentials between CETES and PAGAFES incorporated all the information available to generate predictions of the future exchange rate in conditions where the peso problem prevailed, namely in a situation where the exchange rate was not allowed to float freely and there was always a small probability of a devaluation.¹² Indeed, the differential between the spread in the interest rates on CETES and PAGAFES and the preannounced depreciation of the exchange rate was greatest in the early part of 1988 following the transition to a fixed exchange rate regime, which can be taken as an indication of a lack of full credibility in the exchange rate policy (Figure 2). However, as the authorities persisted in their policies and financial conditions improved, the differential has tended to decline.

Table 2.

Analysis of the Forecast Error

$\left({ϵ}_t=S_t\left[\left(1+i_t\right)/\left(1+i_t^{*}\right)\right]-S_{t+1}\right)$

^aThe critical value for the test with 9 degrees of freedom at the 1 percent level is 21.7.

Table 2.

Analysis of the Forecast Error

$\left({ϵ}_t=S_t\left[\left(1+i_t\right)/\left(1+i_t^{*}\right)\right]-S_{t+1}\right)$

Item	Result
Number of observations	43
Mean	25.762
Standard deviation	68.580
g-test
Degrees of freedom	9
x2-statistica	1.83

^aThe critical value for the test with 9 degrees of freedom at the 1 percent level is 21.7.

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Table 2.

Analysis of the Forecast Error

$\left({ϵ}_t=S_t\left[\left(1+i_t\right)/\left(1+i_t^{*}\right)\right]-S_{t+1}\right)$

Item	Result
Number of observations	43
Mean	25.762
Standard deviation	68.580
g-test
Degrees of freedom	9
x2-statistica	1.83

^aThe critical value for the test with 9 degrees of freedom at the 1 percent level is 21.7.

Implicit Yield in the Secondary Market for Mexico’;s External Debt

The implicit yield to maturity for Mexico’;s external debt was obtained from the observed secondary market price, P_t (from Solomon Brothers data),¹⁶ and the application of the following present value formula:

where $i_t^{sm}$ represents the implicit annual yield to maturity evident in the secondary market price for Mexico’;s external debt in period t. The face value, FV, is set at 100, since the discounts quoted in the secondary markets apply to $100 worth of contractual debt; the contractual coupon payment, C_t, is the interest rate on six-month LIBOR plus the interest rate spread paid by Mexico (assumed to be 13/16 percent over the entire period); and the average maturity, n, was assumed to be equal to 20 years. A monthly series for $i_t^{sm}$ was constructed covering the period August 1986–July 1990.

In modeling the market value of claims of debtor countries, Dooley (1988) has argued that the secondary market price of debt should equal the creditor’;s expected present value of total debt-service payments. This argument, which requires the assumption that creditors are risk neutral, can be formalized as

where E_t is the expectations operator during period t, and $i_t^f$ stands for the risk-free market rate (which in this case is taken as corresponding to the rate on six-month LIBOR prevailing at period t).

A comparison between equations (8) and (9) reveals that the implicit yield obtained from the price in the secondary market depends on the creditor’;s expectations about receiving full payments on the contractual debt. The derivation of a specific formula for the probability of default on Mexico’;s external debt is subject to specific assumptions about the distributional properties of debt-service payments.¹⁷ In what follows, no attempt is made to derive a measure of the probability of default. Instead, by testing the hypothesis that the long-run behavior of the interest rate on PAGAFES can be explained by the behavior of the implicit yield for Mexico’;s external debt, one is also implicitly testing the hypothesis that the probability of default on Mexico’;s external debt also applies to Mexico’;s domestic debt.¹⁸

Test for Cointegration

The first step is to test whether the series on the interest rate on PAGAFES (i_t*) and on the implicit yield from the secondary market $\left(i_t^{sm}\right)$ are I(0); that is, whether the series are stationary. This is done by using the Dickey-Fuller (DF) and the augmented Dickey-Fuller (ADF) tests. In both tests, the null hypothesis is that the series have a unit root, and the alternative hypothesis is that the series are I(0). The ADF test consists in running the following regressions using ordinary least squares:

where the number of lags in each equation (p or q) is selected, such that ω_1,t and ω_2t are white noise. The difference between the DF test and the ADF test is that in the former, γ_1,j = γ_2,j = 0.

In both tests, the test statistic is the ratio of each β_i (i = 1,2) to its corresponding standard error. The null hypothesis is rejected if the β_i is negative and significantly different from zero.¹⁹

The results from the tests, which covered the period August 1986–July 1990, are presented in Table 3. In the case of the ADF test, two lags were needed for ω_1,t to be white noise (that is, p = 1 in equation (10)), whereas only one lag was enough for ω_2,t to be white noise (that is, q = 2 inequation (11)). As shown in the table, the null hypothesis of a unit root cannot be rejected at the 5 percent significance level, indicating that the series on interest rates on PAGAFES and on the implicit yield for Mexico’;s debt are nonstationary processes (Figure 4).

Tabie 3.

Unit Root Test for

i_t* and ${\text{i}}_{\text{t}}^{\text{sm}}$

Note: DF is the Dickey-Fuller test; ADF is the augmented Dickey-Fuller test. The numbers in parentheses indicate the number of lags sufficient for ω_{i, t} (i = 1,2) to be white noise.

Tabie 3.

Unit Root Test for

i_t* and ${\text{i}}_{\text{t}}^{\text{sm}}$

Variable	DF	ADF
it*	-0.86	-0.71 (2)
$i_t^{sm}$	0.40	0.36(1)
$\Delta i_t^{sm}$	-6.40	-4.7(1)
Critical value at
5 percent level	-2.56	-2.9

Note: DF is the Dickey-Fuller test; ADF is the augmented Dickey-Fuller test. The numbers in parentheses indicate the number of lags sufficient for ω_{i, t} (i = 1,2) to be white noise.

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Tabie 3.

Unit Root Test for

i_t* and ${\text{i}}_{\text{t}}^{\text{sm}}$

Variable	DF	ADF
it*	-0.86	-0.71 (2)
$i_t^{sm}$	0.40	0.36(1)
$\Delta i_t^{sm}$	-6.40	-4.7(1)
Critical value at
5 percent level	-2.56	-2.9

Note: DF is the Dickey-Fuller test; ADF is the augmented Dickey-Fuller test. The numbers in parentheses indicate the number of lags sufficient for ω_{i, t} (i = 1,2) to be white noise.

Since the test statistics for i_t* are negative, it can be concluded that if follows an I(l) process. However, since the test statistics for $i_t^{sm}$ are positive, it is necessary to test for the stationarity of the first difference; that is, $\Delta i_t^{sm}$. Indeed, the test statistics for $\Delta i_t^{sm}$ are negative and significant, implying that the original series is I(l). The result that i_t* and $i_t^{sm}$ are I(1) is in line with the hypothesis of rational expectations and market efficiency.

As discussed above, given the characteristics of PAGAFES and commercial bank claims on Mexico traded in the secondary market, it is reasonable to expect divergence in the behavior of the variables in the short run; however, we would expect the variables to move together in the long run. To test if the two variables are cointegrated, we first need to form the cointegration equation:

where, as mentioned before, z_t should be an I(0) process if i_t* and $i_t^{sm}$ are cointegrated. In this sense, z_t measures the extent to which the system deviates from the long-run relationship between i_t* and $i_t^{sm}$. If z_t is stationary, such relationship will hold in the long run.

An estimate of/I. the long-run coefficient relating i_t* and $i_t^{sm}$, is derived from the following vector autoregression (VAR) model:

The numbers in parenthesis are t-values.

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Interest Rate on PAGAFES vs. Yield to Maturity

(In percent per year)

Citation: IMF Staff Papers 1991, 003; 10.5089/9781451930801.024.A007

Note: PAGAFES (Pagares de la Tesorería de la Federación).

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Using equation (13) to obtain the long-run relationship between i_t* and $i_t^{sm}$, the estimated form of equation (11) is

where i_t* is the estimated value of i_t*, and z_t = i_t* − i_t*.

To test whether z_t is an I(0) process, we again used the DF and ADF tests. In addition, we also analyzed the Durbin-Watson of the cointegration equation (CRDW). In the context of the present exercise, this test consists in obtaining the DW statistic by running z_t against a constant. The null hypothesis that z_t has a unit root will be rejected if the CRDW is significantly above zero.

The results from the tests, which are presented in Table 4, are somewhat mixed. The CRDW test rejects the hypothesis of a unit root in the residuals of the cointegration equation at the 5 percent significance level and therefore supports the hypothesis that i_t* and $i_t^{sm}$ are cointegrated. However, neither the DF nor the ADF supports the hypothesis of cointegration at the 5 percent significance level. The DF statistics are significant at the 10 percent level, but the ADF statistics are on the borderline.

Since our results are not conclusive, we further investigate whether i_t* and $i_t^{sm}$ are cointegrated by testing for the existence of a generating mechanism having what is called an error-correcting form:²⁰

Table 4.

Tests for Cointegration Between

i_t* and ${\text{i}}_{\text{t}}^{\text{sm}}$

Note: DF is the Dickey-Fuller test; ADF is the augmented Dickey-Fuller test; and CRDW is the Durbin-Watson of the cointegration equation.

^aThe literature reports values ranging from -2.89 (Schwert (1988)) to -3.17 (Hall and Henry (1988)).

^bHall and Henry (1988) report a value of -2.84, which probably corresponds to the upper end of the range.

Table 4.

Tests for Cointegration Between

i_t* and ${\text{i}}_{\text{t}}^{\text{sm}}$

			Critical Values
			5 percent	10 percent
Test	Statistic	Number of Lags	level	level
DF	-2.25	—	-2.56	-2.18
ADF	-2.77	1	-2.93a	…b
CRDW	0.460	—	0.397	0.322

Note: DF is the Dickey-Fuller test; ADF is the augmented Dickey-Fuller test; and CRDW is the Durbin-Watson of the cointegration equation.

^aThe literature reports values ranging from -2.89 (Schwert (1988)) to -3.17 (Hall and Henry (1988)).

^bHall and Henry (1988) report a value of -2.84, which probably corresponds to the upper end of the range.

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Table 4.

Tests for Cointegration Between

i_t* and ${\text{i}}_{\text{t}}^{\text{sm}}$

			Critical Values
			5 percent	10 percent
Test	Statistic	Number of Lags	level	level
DF	-2.25	—	-2.56	-2.18
ADF	-2.77	1	-2.93a	…b
CRDW	0.460	—	0.397	0.322

Note: DF is the Dickey-Fuller test; ADF is the augmented Dickey-Fuller test; and CRDW is the Durbin-Watson of the cointegration equation.

^aThe literature reports values ranging from -2.89 (Schwert (1988)) to -3.17 (Hall and Henry (1988)).

^bHall and Henry (1988) report a value of -2.84, which probably corresponds to the upper end of the range.

where dB is a finite polynomial in the lag operator B, and v_t is white noise.

Equation (14) is, therefore, a standard error-correction model, which includes the lagged residuals of the cointegration regression. The variables i_t* and $i_t^{sm}$ will be cointegrated if ρ₁, the coefficient of the lagged error term, is significantly negative. This is so because equation (15) indicates that the amount and direction of a change in i_t* will take into account the size and sign of the previous deviation from equilibrium z_t-1 If the variables are cointegrated, z_t is stationary and is therefore inclined to move toward its mean (which is zero), and hence, the equilibrium relationship (14) tends to be restored.

Table 5 presents the results from estimating a simple error-correction model that includes the lagged error term, z_t-1. Since ρ₁ is significant at the 5 percent level, this implies that there exists an error-correlation mechanism that tends to restore the long-run equilibrium relationship between i_t* and $i_t^{sm}$. Therefore, this test supports the hypothesis that i_t* and $i_t^{sm}$ are cointegrated.

Finally, as mentioned above, an important result of cointegration analysis is that if two variables are I(1) and cointegrated, there must be Granger causality in at least one direction. Our conjecture is that, Mexico being a small open economy, the implicit yield for Mexico’;s external debt causes (in Granger terms) the domestic interest rates on PAGAFES. Since the Granger is an F-test, it is applicable only to stationary variables; therefore, the test is applied to the first differences of i_t* and $i_t^{sm}$. Table 6 reports the results from this test. As is shown, when Δi_t* is the dependent variable, the F-value is significant at the 5 percent level, supporting the hypothesis that $i_t^{sm}$ causes (in Granger terms) i_t*. Moreover, the test rejects the hypothesis that i_t* causes (in Granger terms) $i_t^{sm}$, since the F-vaiue was found to be not significant at the 5 percent level when $\Delta i_t^{sm}$ was treated as the dependent variable.

Table 5.

Error-Correction Model

(Dependent variable: Δi_t*)

Note: Only the lagged variables that were found to be significant are included; R² is the coefficient of determination; and DW is the D urbi n-Watson statistic,

Table 5.

Error-Correction Model

(Dependent variable: Δi_t*)

Variable	Coefficient	t-value
Constant	0.232	0.284
Z t-1	-0.224	-2.362
Δi*t-1	0.333	2.182
$\Delta i_{t-2}^{sm}$	-0.984	2.068
$\Delta i_{t-3}^{sm}$	0.981	1.878
$\Delta i_{t-4}^{sm}$	0.923	1.791
Memorandum items:
R2 = 0.378
DW = 1.90

Note: Only the lagged variables that were found to be significant are included; R² is the coefficient of determination; and DW is the D urbi n-Watson statistic,

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Table 5.

Error-Correction Model

(Dependent variable: Δi_t*)

Variable	Coefficient	t-value
Constant	0.232	0.284
Z t-1	-0.224	-2.362
Δi*t-1	0.333	2.182
$\Delta i_{t-2}^{sm}$	-0.984	2.068
$\Delta i_{t-3}^{sm}$	0.981	1.878
$\Delta i_{t-4}^{sm}$	0.923	1.791
Memorandum items:
R2 = 0.378
DW = 1.90

Note: Only the lagged variables that were found to be significant are included; R² is the coefficient of determination; and DW is the D urbi n-Watson statistic,

On balance, the evidence seems to support the hypothesis that i_t* and $i_t^{sm}$ are cointegrated. Nevertheless, the tests are not very robust and further investigation would be useful as more data become available.