2.1 Covered Interest Parity

Economic theory predicts a number of stable relationships among variables through time. Of par- ticular interest in the field of international finance is the CIP. This relationship predicts that under market efficiency, the forward premium (given by the difference between the forward exchange rate and the spot exchange rate) must be equal to the interest rate differential among the assets in two countries (plus some transaction costs or minimum risk premium). Formally, let S_t be the spot exchange rate of USD per 1 MXN and let Ftk be the related forward contract agreed at period t with maturity of k months. Throughout the paper, the followed convention is that the investor resides in the U.S. More generally, he can always borrow in USD and is looking to invest in Mexico. Thus, the U.S. and Mexico are the domestic and the foreign country, respectively. Also, let ius,tk and Fimex,tk be the interest rate on the sovereign bond with maturity of k months issued in domestic currency by the U.S. and Mexico, respectively. Note that by taking ius,tk to compute the CIP, it is implicitly assumed that the traders have access to loans at this rate. This may be true for those at large banks. Assuming individuals are rational and absence of financial frictions and costs, the CIP may be stated as

When deviations from the CIP occur, the economic theory predicts that demand and supply forces will reestablish (2.1) through arbitrage. This is, in theory inequalities provide risk-less profitable opportunities in a friction-less set-up. An example of the latter is as follows. Assume

If an investor in the home country borrows B USD for k months at rate ius,tk , he can buy B/St MXN. hen, he can lend his B/St MXN at a rate imex,tk . At the same time, he signs a forward contract where he promises to deliver 1+imex,tB/St MXN at the end of k months in exchange for FtkB1+imex,tk/St USD. After this sequence of transactions, the investor has in his hand FtkB1+imex,tk/St USD and he owns B1+ius,tk USD. If (2.2) holds, then he can pocket a profit of FtkB1+imex,tk/St-B1+ius,tk USD

The arbitrage process ensures that (2.1) holds, even if there are deviations in the short run. Empirical tests of the CIP advanced by ¹⁹⁷³) or Peel and Taylor (²⁰⁰²) are based on the following approximation to expression (2.1)

where Pk is a factor that adjusts to annual terms the forward premium and c accounts for possible transaction costs and minimum risk premium. Since interest here lies in deviations from the CIP, these are denoted by δtk , and the forward premium by Φtk=PkFtk-StSt . That is

Figure 1 above shows relationship (2.4) for k = 1.

2.2 Funding Liquidity Measures

Among the key money-market indicators for “liquidity events” is the LIBOR-OIS spread, LOIS. It has been used as a market indicator of interbank liquidity conditions by central bankers and market participants as explained by Bernanke (²⁰¹⁵), Hull (²⁰¹⁵), Paulson (²⁰¹⁰), and Thornton (²⁰⁰⁹). In the international finance literature, LOIS has served as a proxy for credit risk. It has a relatively simple interpretation: when LOIS increases, the funding liquidity decreases. A description of how both the LIBOR and the OIS are constructed is found in chapter 3 of Smith (²⁰¹⁰). Thus, in the absence of negative financial events, the spread between the two rates is small. In times of financial distress, however, loans signed in a LIBOR contract are riskier than overnight loans, causing LOIS

to increase. Since the aim of this paper is to obtain a measure of the effects from liquidity shocks on the CIP, LOIS for the U.S. banking sector is used. To compare the effects of said shocks with those born in Europe, LOIS for the European banking sector which is the analogue measure is included in the analysis.²

4.1 Specification

Lag-Length

The first step in specifying the VECM is to choose the number of lags, p, used in estimation. Table 2 displays the Bayesian Information Criterion (BIC), the Hannan-Quinn Criterion (H-Q), and the Lagrange Multiplier Test (LM) for autocorrelation for values of p = 1, ..., 15. From Table 2 it is possible to conclude that p = 11 will imply that residuals are free from autocorrelation, since the LM test fail to reject the null hypothesis of no-autocorrelation at the 5% level. Even though p = 11 may seem large, Kilian and Lütkepohl (²⁰¹⁷) document that if the true lag-length parameter is unknown, an over-parameterised VAR may yield more accurate IRFs. They also discuss why the Ljung-Box Q test may be biased when I(1) series are involved, hence it is not reported here. In this paper, over-parametrization should not be a concern, provided the sample size is 722 observations.

The canonical VAR analysis no longer requires the residuals to satisfy Normality and Ho- moskedasticity. On the one hand, Normality is no longer a necessary condition to estimate through Maximum Likelihood, since reliance on Quasi-Maximum Likelihood techniques is enough. On the other hand, Heteroskedasticity is no longer an issue in estimating a VAR according to Juselius (²⁰⁰⁶).

Rank Test

One of the appealing features of model (4.1) is that it allows to test whether the equilibrium relationships predicted by economic theory are satisfied by the data. In particular, when each of the equations (β ^′ X _t−1 )j is I(0), the long-run relationships are satisfied. This is, a subset of elements of X _t are cointegrated. Note that j = 1, 2.., r < n. Another appealing feature of the model (4.1) is that it allows to impose restrictions on the matrix β. Moreover, if the number of restrictions in each row β'j is larger than r-1 , these are testable. The latter means that determining r , the cointegrating rank, is a necessary condition for undertaking inference on the cointegrating relationships and the n − r common trends. Johansen (¹⁹⁸⁸) tests is a popular way to select r.⁵

Table 3 contains the tests for several possible values of r denoted r0. In this case, Johansen’s Trace Test suggests r = 2. Estimation of β^ yields:⁶

Inspection of (4.4) reveals that estimates for the main variables of CIP are similar across cointegration relationships, hence imposing r=2 may be redundant. A further criterion to choose r is discussed in Martin et al. (²⁰¹³). In particular, they suggest that a valid alternative to Johansen’s test is selecting the restricted model with the least BIC. The last column in Table 3 provides said criterion. Indeed, it suggests r=1 . The estimation of β^ with r=1 yields:

Expression (4.5) presents the cointegrating relationship.⁷ At this point it is not possible to interpret the estimates. It is possible, however, to discuss the stationarity of the cointegrating relationship. Figure 6 presents β^'Xt and suggests that it is stationary, as desired. Unit root tests are shown in Table 4, confirming the latter. In particular, the Augmented Dickey Fuller and Phillips-Perron tests are included, without a mean or a trend. The Table also shows a t-test statistic for the sample mean of β^'Xt being zero, justifying the exclusion of any deterministic terms in the unit root tests. Results show unambiguous support for the absence of a unit root confirming the existence of one cointegrating relationship.

Figure 6 Cointegrating Relationship β^'Xt. 

4.2 Testing the Theoretical Relationship

Assuming that a Data Generating Process can be modelled by a VECM(p) entails the a priori conjecture that level relationships among the elements of Xt are stationary. Hence, economic theory outlined in Section 2 is used to justify the only cointegration relationship: the CIP. Economic theory predicts a stable relationship among its components, which in econometric terms means that δt as defined in expression (2.4) is stationary. Let the parameters to be estimated of the CIP, be given by β=β1,β2,β3,β4,β5,c' . If the data obey the CIP in equilibrium, then the estimate β^ will satisfy:

Results displayed in expression (4.5) somewhat satisfy conditions 1 and 2, however, condition 3 is far from being observed. The VECM(p) allows to test whether 3 can be imposed. Let β~R be the restricted estimate of β , for the set of restrictions R . The first set of restrictions to test, R=A , is defined as

Imposing the restriction set (4.6a)-(4.6d), the following estimates are obtained:

The conducted test to assess whether the restrictions imposed in (4.7) are valid suggest they are not. Table 5 presents the results of the test which rejects the null hypothesis of β~A being valid.

Since the aim is to test whether changes in funding liquidity are affecting the CIP, define the restrictions set B as a subset of A where B is composed only by (4.6a) and (4.6b). That is, define β~B as the restricted estimate of β , where β~1B=-β~2B and β~1B=β~3B are imposed and both β~4B and β~5B are left unrestricted. Estimating the model under the restriction set B yields:

As shown in Table 5, the Null Hypothesis of the restriction set B being valid is not rejected at a 10% level. This means that there is a stationary equilibrium relationship that includes the funding liquidity measures and has a mean different from zero. Once the validity of the relationship is established, a test to determine whether δt-1B is stationary ensues. Figure 7 suggests that δt-1B is stationary. The latter is confirmed by the results of the unit root tests included in Table 6. This is, δtB , as efined in expression (4.8), is indeed stationary, hence cointegrated. With δtB in hand, an economic interpretation for the econometric results obtained so far can be provided.

Figure 7 Cointegrating Relationship β^B'Xt=δtB. 

4.3 Economics of the Estimation

Expression (4.8) should be interpreted as an equilibrium relationship among the 5 elements of X _t . In particular, said expression may be written as

where ν _t is a mean zero stationary random variable as implied by Table 6.

The relationship between δ _t and LOIS _us,t is negative, which can be rationalised as follows: (1) An increase in the LOIS _us,t spread signals an increase of the perception of credit risk, thus a decrease in funding liquidity in the U.S. (2) The increase in LOIS _us,t will cause δ _t to decrease in one of three ways: (a) USD appreciates if there is a flight to quality, diminishing the forward premium Φ _t ; or (b) i _us,t increases since individuals demand less treasuries due to mature in the short run; or (c) both.

The relationship between δ _t and LOIS _eur,t is positive since: (1) An increase in LOIS _eur,t signals a decrease in funding liquidity in Europe. (2) There is a flight-to-quality to the U.S. assets. This causes: (a) an appreciation of the USD, thus Φ _t increases (closer to zero); or (b) i _us,t decreases; or (c) both.

Finally the constant term represents the minimum profit investors demand to trade MXN and Mexican Treasuries. It is worth noting that despite the similar nature of changes in funding liquidity conditions, whether these are originated in the U.S. or Europe matters. In particular, the two sources of funding liquidity shocks have opposite sign on δ _t .

4.4 Weak Exogeneity and the Common Trends Representation

Weak Exogeneity

A further advantage provided by the model (4.1) is that it allows to test for “weakly exogenous variables”. As defined by Juselius (^{2006, pp. 193}), a variable x_j,t ∈ {X_t} is said to be weakly exogenous if it affects other variables in the system X_t while it is not affected by them. This is, if the row j of matrix α contains only zeros. Since α is a n × r matrix, and r = 1 in this paper, then a weak exogeneity test is equivalent to testing for each of the 5 rows of α for being different from zero. To this end, a simple t − test statistic is not suitable since the comparison is made across 5 models. In particular, a Likelihood-Ratio statistic is required. Table 7 displays the weak exogeneity test for the variables in X_t.

Results from the test allow to conclude that i _mex,t , LOIS _us,t and LOIS _eur,t are weakly ex- ogenous. Note, however, that the p-value for i _us,t implies that this variable adjusts towards the cointegrating relationship. There is, however, a plausible rationalisation for this seemingly counter- intuitive result. In particular, note that (4.9) contains LOIS _us,t and note that funding liquidity shortages may be strongly related to an increase of demand of the safest asset in the market (i.e. the U.S. treasuries). The latter, in turn, should cause yields to change. A similar argument may apply when the change in funding liquidity conditions is born in Europe. This may explain why within the estimated system, i _us,t is not weakly exogenous. The latter is confirmed by the restricted estimation and a test for the validity of the restrictions embedded in α˜, contained in Table 8:

The restricted estimate for α in equation (4.10), implies that Φ _t and i _us,t are the only variables of the system that adjust in response to deviations from the cointegration relationship β ^′ X _t−1 . Moreover, the estimates imply that it takes slightly more than 3 and 7.5 weeks for deviations from the previous period equilibrium relationship to dissipate for Φ _t and i _us,t , respectively. This is, suppose there is a stationary one-time shock to δ ^B , seemingly creating arbitrage opportunities.

Their effect on Φ _t will take 3 or 7.5 weeks to dissipate entirely.

Common Trends Representation

The VECM also allows writing (4.1) in its common trends representation or Moving Average (MA) form, as proved in the Granger Representation Theorem. This allows both the estimation of the n − r common trends in the system X _t and a very intuitive interpretation. The MA form is written as

where β _⊥ is the orthogonal complement of β, defined as the (n−r)×r matrix satisfying β ^′ β = 0 and similarly for α _⊥.⁸ The first component of equation (4.11) deserves special attention. In particular, note that the product C∑s=1tεs is β~⊥α'⊥∑s=1tεs using (4.12). It is possible to interpret β~⊥ as a measure of the importance (weight) that each of the common trends α'⊥∑s=1tεs has on Xt .

The n × n matrix C defined in (4.12) is known as the “long-run impact matrix” and it is interpreted in two effect of c _ij on x _i . Row-wise, the element c _ij means if statistically significant, means the cumulated effect ∑s=1tεj,s has a relevant effect of cij on xi . Row-wise, the element cij means that xi may be permanently influenced by ∑s=1tεj,s in some measure cij . The matrices Cj* are the current and previous “one-time" effects of each element of εt on the system Xt and X0 is its initial value.

For the present model, restricted estimates of the long-run impact matrix yield the following common trend representation.⁹ Recall that Σ is a full matrix as described in (4.3), Table 9 shows the corresponding correlations.

The MA form is the representation of each variable as the sum of the history of previous shocks. Although correlated, these shocks are useful in explaining changes in each element of system Xt . Here, the restricted estimate C~ is used to construct the estimated MA expressions. Let Ck,⋅j* be the kth row of matrix Cj* . Omitting stationary and terms not different from zero statistically, the MA representation for Φt is given by

In line with the weak exogeneity test, equation (4.13) shows all the stochastic trends have a non-negligible effect on Φt . LOISeur,t is present through the correlation between εLOIS,us and εLOIS,eur which is relatively high, as shown in Table 9.

For i _us,t , equation (4.14) shows its non weakly exogenous status. In particular, it is affected by shocks on the forward premium

The results of the estimation for i _mex,t displayed in (4.15) suggests some de-coupling of the monetary policy for the period. A case for the presence of at least a second stochastic trend could be made easily in order to explain changes in the Mexican Treasury yield. This issue deserves a deeper study, since the correlations between i _mex,t and the rest of the variables is not high enough to justify some second-order effect from a different variable in the system. These second-order effects may be present since the reduced form errors ε _t are contemporaneously related

Finally, the MA forms of LOIS _us,t and LOIS _eur,t in (4.16) and (4.17) show these variables behaving as two weakly exogenous elements, with a high correlation among them

So far, evidence suggests unambiguous support for the relevance of both LOIS _us,t and LOIS _eur,t in explaining the behaviour of the variables that define δ _t . The previous analysis comes a long way in separating effects from each shock into the behaviour of each element of X _t . There is a shortcoming,however, and that is related to the covariance matrix of the reduced form errors, Σ contained in Table 9, not being diagonal. Shocks between LOIS _us,t and LOIS _eur,t cannot be distinguished clearly. To accomplish this, a structural analysis is undertaken in the next section.

4.5 Structural Analysis

The Structural VECM(p) can be readily derived from expressions (4.1) or (4.11). In order to relate the reduced-form errors vector ε _t to the “structural” errors vector u _t , a Cholesky decomposition is used. This identification strategy relies on the assumption of some causal direction among the variables. Thus, it is assumed that i _us,t is independent from the rest of the variables in X _t and affects LOIS _us,t . In turn, LOIS _eur,t is affected by shocks to LOIS _us,t . Also, i _mex,t is subject to changes in all non-domestic variables. Finally, Φ _t responds to shocks to all previous variables. The selected order obeys the economic logic behind the variables in the system. That is, structurally, the i _us,t follows its own shocks; the banking system of the U.S. is larger than the European one, however interconnected; the European banking sector is independent of the structural shocks to i _mex,t , given that Mexico is a small open economy.¹⁰

Impulse Response Analysis

The model VECM(11) in (4.1) is used to estimate the IRFs, imposing the identified restriction set (4.8). The responses of both Φ _t and i _mex,t are displayed in Figure 8. An increase in i _mex,t yields a negative response of Φ _t , reflecting an MXN appreciation. Also, more stringent funding liquidity conditions given by increases in either LOIS _us,t or LOIS _eur,t yield a positive response of Φ _t . This is, an appreciation of the USD. The right column shows that increases in Φ _t , given by an appreciation of the MXN, yield a negative response of i _mex,t . Moreover, increases in either i _us,t or LOIS _eur,t imply an increase in i _mex,t . Note that this is the expected monetary policy reaction to more stringent funding conditions, in a broad sense.

The response of the rest of the variables to the shocks is shown in Figure 9. As suggested by the weak exogeneity analysis, i _us,t responds positively to increases in Φ _t and to no other shock. More- over, LOIS _us,t responds negatively to shocks to i _us,t and LOIS _eur,t . Finally, LOIS _eur,t responds negatively to increases in both Φ _t and i _us,t , while responding positively to shocks to LOIS _us,t .

Forecast Variance Decomposition Analysis

A forecast variance decomposition analysis is undertaken to complete the analysis of the aver- age effect of shocks to the variables. Having established that funding liquidity shocks have a role in explaining deviations form the CIP, the aim is to quantify the relative importance of changes in LOIS _us,t and LOIS _eur,t . To this end, a VAR model in levels based on the VECM(11) is esti- mated. As discussed by Kilian and Lütkepohl (²⁰¹⁷), the analysis may well be carried out on the VECM(11) even though all variables contained in X _t are I(1). This, however, may entail some loss of information since the model is expressed in differences.

Estimation was carried out in EViews 10.

Figure 8 Impulse Response Functions and ± 2 Standard Errors computed by Monte Carlo Simulations.  

Estimation was carried out in EViews 10.

Figure 9 Impulse Response Functions and ± 2 Standard Errors computed by Monte Carlo Simulations.  

The estimated VAR model in levels is given by

where X _t = (LOIS _us,t , LOIS _eur,t , δ _t )^′, δ _t is defined in (2.4), Θj are n×n parameter matrices, c is a n×1 vector of constant terms, and vt is the error term. The estimation of model (4.18) and inference obtained from it should pose no obstacle, as long as the system is stable. That is, all the roots of the system are within the unit circle. Indeed, the largest implied root from the VAR in levels is 0.974, well below 1.¹¹

Results from the forecast error variance decomposition for δt are summarised in Table 10. The 52 week (one year) horizon serves to find the convergence level of each variance decomposition percentage. The Cholesky ordering is that given in X~t . Results suggest that the relative importance of δt to explain its own forecast errors decreases within the horizon from 75.5% to 44.9%. The relative importance of LOISus,t increases from 21.3% to 40.5% while that of LOISeur,t increases from 3.3% to 14.6%.

Interestingly, the relative importance of LOIS _us,t is statistically the same as that of δ _t in ex- plaining the forecast error variance of δ _t after 15 weeks. The latter may be rationalised by, first, noting that the U.S. banking sector is the largest in the world and, second, the analysis of the CIP involves the USD, the currency in which LOIS _us,t is determined.

A further result worth of attention is that, however small when compared with LOISus,t, changes in LOISeur,t are important in explaining deviations from the CIP. Indeed, in a horizon of 52 weeks, 14.6% of the forecasts error variance may be attributed to this funding liquidity measure. Thus evidence supports that, notwithstanding the apparent weak relationship that may exist between the USD-MXN market and European financial markets, there is a non-negligible link.

Historical Decomposition Analysis

Up to this stage, the structural analysis suggests that on average, changes in funding liquidity conditions are relevant to determine deviations from CIP. The analysed period is characterised by a sequence of financial events, not all taking place simultaneously. Hence, an Historical Decomposition Analysis may shed light on the relative importance of each measure of funding liquidity at each date. Figure 10 displays the results. In particular, the solid line is the relative importance of each variable on the behaviour of δ _t while the shade corresponds to de-mean δ _t itself.

Shaded area corresponds to δ _t , solid line corresponds to the relative importance of each variable explaining δ _t . Estimation was carried out in EViews 10.

Figure 10 Historical Decomposition of δ _t computed by Monte Carlo Simulations.  

Some conclusions emerge. Prior to the onset of the financial crisis in summer 2007, most of the dynamics of δ _t were explained by itself (i.e. changes in Φ _t , i _us,t , and i _mex,t ). After August 2007, the funding liquidity measures are able to explain several episodes of stark deviations from CIP. In particular LOIS _us,t is able to account for the observed large deviations from early 2008 to mid 2009, including of course those registered in the fourth quarter of 2008 when Lehman failed, AIG was bailed-out, and TARP was approved by the U.S.’ Congress. It then explains a large share of δ _t from mid 2010 to mid 2011. Finally, beginning in 2012, possibly related to new regulation concerning funding of commercial banks, the funding liquidity measure re-gained importance.

The role of LOIS _eur,t is reflected mainly in four episodes. First, from mid 2007 to mid 2008, a period where, as documented by Ivashina et al. (²⁰¹⁵), European banks experienced marked USD liquidity constraints. Second, from mid 2010 to early 2011, when several southern European economies were on the brink of entering into a sovereign debt crisis. Third, throughout 2011 when the sovereign debt crisis was at its peak, particularly in Greece. The role of LOIS _eur,t started to diminish after the European Central Bank (ECB) stated that “Within our mandate, the ECB is ready to do whatever it takes to preserve the euro.” in mid 2012.¹² Finally, towards the end of the sample in 2018, this funding liquidity measure is able to account for deviations from the CIP.