2.1. A quick review about Exchange rate and UIP

Exchange market has been an analysis focus for many researchers. An important part of the researches has centered the study in the exchange market equilibrium, mainly through UIP. The UIP indicates that the exchange rate expected depreciation [((e _t+k )-e _t ]/e _t is adjusted according to differential between the local i _t and foreign i _t ^* interest rate, where E(e _t+k ) is the expected exchange rate and e _t is the spot exchange rate. This relationship has been tested empirically through this regression:

Under this specification, short-term exchange market equilibrium will be fulfilled if the interest rate differential fully explains the exchange rate return. That is, α=0, β₁=1 and ε _t is a non-autocorrelated residue. Empirical researches have shown a persistent lack of consensus and most of them have demonstrated contrary results for UIP. This literature has indicated that UIP prediction is biased, so that the interest rate differential only explains a fraction of the exchange rate return (^{Frenkel, 1981}; ^{Mussa, 1984}). Indeed, most studies have found that α=0, although often β₁<0, fact that reveals that the interest rate differential less predicts the subsequent direction of the exchange rate. ^{Froot and Thaler (1990)} summarized 75 empirical studies and found very few cases where β₁> 0. Most studies showed β₁<0 with -0.88 average. In this sense, diverse studies have supported this empirical finding, called forward premium puzzle or forward discount bias, and which would be common in developed countries foreign exchange markets (^{Fama, 1984}; ^{Mussa, 1984}; ^{Hodrick, 1987}; ^{Froot, 1990}; ^{Lewis, 1995}; ^{Engel, 1996}; ^{Olmo and Pilbeam, 2011}; ^{Bhatti, 2014}). Even other studies that analyzed periods prior to Bretton Woods, characterized by lower exchange rate volatility, found results similar to forward premium puzzle (^{McFarland, McMahon and Ngama, 1994}; ^{Phillips, McFarland and McMahon, 1996}; ^{Choudhry, 2013}).

Other researches have found favorable results to UIP under very specific conditions such as the use of long-term interest rates or high interest rate differentials (^{Chinn and Meredith, 2004}, ²⁰⁰⁵; ^{Chaboud and Wright, 2005}; ^{Lambelet and Mihailov, 2005}; ^{Sarno, Valente and Leon, 2006}; ^{Baillie and Kilic, 2006}; ^{Bekaert, Wei and Xing, 2007}; ^{Lothian and Wu, 2011}; ^{Lothian, Pownall and Koedijk, 2013}; ^{Lothian, 2016}). Under these conditions, the empirical evidence show that the bias would be lower in emerging markets, while forward discount bias would be concentrated in developed markets (^{Bansal and Dahlquist, 2000}; ^{Frankel and Poonawala, 2010}).

2.2. Exchange risk premium in the foreign exchange market

Various theories have explained the persistent UIP deviation; being one of them the risk premium existence. ^{Frankel (1982)} indicates that the risk premium is a function of prediction error variance and exchange rate movements. Fact supported by ^{Froot and Frankel (1989)} and ^{Mark and Wu (1998)}. In this line, ^{Domowitz and Hakkio (1985)} elaborated a model for UIP that extends the ^{Lucas (1982)} and ^{Hodrick and Srivastava (1984)} models, and that incorporates a time-varying risk premium associated to exchange rate volatility σ _t . Following ^{Engle (1982)}, ^{Engle, Lilien and Robins (1987)} and ^{Bollerslev (1990)}, this relationship has been tested empirically through GARCH models for:

Under this specification, short-term equilibrium for exchange rate will be fulfilled if the interest rate differential fully explains the exchange rate return and there is no risk premium. That is, α=0, β₁=1, β₂=0 and ε _t is a non-autocorrelated residue. If there is a risk premium, ^{Frankel and Chinn (1993)} and ^{Cavaglia, Verschoor and Wolff (1994)} point out that risk premium would have the capacity to correct the UIP deviation. That is, α=0 and β₁=1. Empirically, the existence, nature, and capability of risk premium to correct the UIP deviation has been the focus of debate.

Various studies argue that the risk premium existence does not correct the UIP deviation. ^{Domowitz and Hakkio (1985)}, analyzing the currencies of Germany, France, Japan, Switzerland, and United Kingdom through ARCH-in-Mean models, finding evidence that support the time-varying risk premium existence for Japan and United Kingdom markets. However, the UIP bias was only partially reduced. ^{Tai (2001)} finds similar evidence for Asian markets, ruling out UIP compliance. Despite the advantages of GARCH models for modeling exchange rate volatility, other studies have showed similar results (^{Baillie and Bollerslev, 1990a}, ^1990b; ^{Forsberg and Bollerslev, 2002}; ^{Olmo and Pilbeam, 2011}; ^{Aysun and Lee, 2014}; ^{Engel, 2016}). Another relatively minor part of empirical evidence has found that risk premium inclusion corrects the UIP deviation. ^{Aggarwal (2013)}, in an empirical study for Japan, Australia and United States foreign exchange markets found favorable evidence for UIP. The author argues that the individual risk premium of each market is attributable to high interest rate differentials episodes. ^{Li, Ghoshray and Morley (2012)} corroborate this view and add that the risk premium adjustment would be more evident in emerging markets, where interest rate differentials are greater than developed markets. According to ^{Yung (2017)}, the risk premium of each exchange market, would explain more than half of the exchange rate changes. The Latin American markets have these qualities and for this reason we formulate this hypothesis:

H1: The exchange risk premium in each market corrects the UIP bias.

The exchange risk premium could not only have each market characteristics, but also regional. The greater international trade between same region countries, common financial development policies and the greater interdependence degree among the countries would favor the regional valuation for exchange risk premium (^{He, 2017}). ^{Bollerslev (1990)}, in an empirical analysis for the German mark, Italian lira, Swiss franc, French franc and pound sterling showed that the monetary unification increased the exchange returns correlations. This fact increased the risk premium for all these markets due to higher co-movements between exchange markets. Even crises periods increased the foreign exchange markets dependence (^{Yang, Kolari and Min, 2003}; ^{Assidenou, 2011}). The Latin American markets are characterized by low segmentation degree and have similar idiosyncratic qualities that make it difficult, on the one hand, the diversification possibilities in the region for investors, and on the other, they force them to value financial assets through a regional vision (^{Mellado and Escobari, 2015}). ^{Mellado and García (2014)} argue that in Latin American markets these qualities would be supported by high correlations between markets. When risk premium is valued regionally, the UIP itself would not be valid because those co-movements between markets would explain the exchange rate return, in addition to interest rate differentials. The literature for Latin American markets is almost nonexistent in this matter, and for that reason we formulate this hypothesis:

H2: The exchange risk premium is valued regionally in the Latin American foreign exchange markets.

2.3. Effect of MILA on Latin-American exchange markets

The economic or financial integration processes developed by different countries would have significant effects on macroeconomic and financial system, mainly on markets financial liberalization (^{Francis, Hasan and Hunter, 2002}; ^{Seerattan and Birchwood, 2004}). Several empirical studies have evaluated the effects of these processes. ^{Fratzscher (2002)} states that the economic and financial integration process developed in Europe helped to mitigate the markets uncertainty and positioned them better in relation to the United States. For Asian markets, ^{Shin and Sohn (2006)} argue that, although the integration degree of the region is lower than that European markets, the main effects of regional integration are visible in larger price movements.

The stock and exchange markets relationship is narrow. However, the empirical evidence that has investigated the effects of stock markets integration on foreign exchange markets is still scarce. ^{Glick and Rose (1999)} affirm that capital flows movements, inherent to these integration processes, could affect the foreign currency portfolios investors’ positions. This fact, according to ^{Beine (2004)} and ^{Tai (2007)}, would increase the financial contagion probability due to the higher exchange markets dependence. Conclusion that is also supported by ^{Syllignakis and Kouretas (2011)} and ^{Celik (2012)}. ^{Bollerslev (1990)} analyzed the effect of these processes on exchange markets of Europe. Their results indicated that the monetary unification increased the exchange returns correlations. Along the same lines, ^{Fratzscher (2002)} argues that the equity markets integration deepened the effect of economic integration in Europe, generating a significant reduction of exchange volatility. Similar results were found by ^{De Brouwer (1997)} and ^{Janor, Ali and Shaharudin (2007)} for the Asian markets. These authors add that the exchange rate parity would be closely related to the countries financial openness. More recently, other studies reveal that both economic and financial integration processes deepen dependence on foreign exchange markets in both developed (^{Reboredo et al., 2021}) and emerging (^{Aftab et al., 2020}) markets. This would be seen in higher levels of correlation and spillover effects. Even ^{Malik and Umar (2019)} warn that this interdependence could be affected by the interaction between the foreign exchange, stock and commodity markets.

In Latin America, MILA began its operations in May 2011 through virtual financial integration between Chilean, Colombian and Peruvian markets, to which Mexico would later join. Despite integration, each stock markets continues to operate independently. Currently, MILA has become the second largest stock market in the region, second only to the Brazilian stock market. Some studies have argued that MILA has reported benefits in terms of stock market activity, liquidity, and depth (^{Castro and Marín, 2014}; ^{Lizarzaburu, Burneo, Galindo and Berggrun, 2015}; ^{Cardona, Gutiérrez and Agudelo, 2017}). Even other research has shown that MILA implementation has generated support for economic activity during crisis periods (^{Asness, Israelov and Liew, 2011}).

Despite of the above, its effects on exchange markets of the region have been scarcely investigated. ^{Mellado and García (2014)}, in an empirical work carried out for MILA initial markets (Chile, Colombia and Peru) found that the exchange returns are significantly and dynamically correlated to their long-term average. This, according to ^{Heston, Geert and Wessels (1995)}, reflects the greater dependence between these markets. In any case, ^{Mellado y García (2014)} indicate that MILA implementation did not have major effects on exchange returns dynamic correlations, except for co-movements between Chilean and Peruvian markets, where MILA generated a reduction. According to these authors, this reduction would allow to diversify the exchange risk in these markets. But this research did not analyze the MILA effect on exchange risk premium.

Thus, the higher dependence between Latin American markets observed by ^{Chen, Firth and Meng (2002)}, added to its scant segmentation degree that mitigates the potential diversification benefit, would mean that the exchange returns, and risk premium of the region's exchange markets would experience co-movements. Given that MILA has strengthened the financial market integration in the region, it would be expected that such co-movements will be accentuated in the exchange markets, mainly among the MILA’ markets. In this area, the evidence is scarce and motivates us to formulate this hypothesis:

H3: MILA positively affects the correlations between the exchange returns of Latin American markets.

H4: MILA positively affects the correlations between the exchange risk premiums of Latin American markets.

3.1. Sample data

The research data were extracted from Bloomberg database. The information corresponds to monthly time series between January 1997 and December 2021 for Brazil, Chile, Colombia, Mexico and Peru markets. Each time series contains 300 observations. We use monthly time series to attenuate the effects of volatility on the parametric convergence of the multivariate models. Table 1 shows the variables.

The exchange return (EXRET), measured by exchange rate monthly percentage change, is the dependent variable. The exchange rate quantifies the US dollar value in terms local currency. This measure is widely used by several international studies (^{Fama, 1984}; ^{Domowitz and Hakkio, 1985}; ^{Baillie and Bollerslev, 1990a}; ^{Lewis, 1995}). While that, interest rate differential (DIF) is measured as the difference between the 30-day interbank rate of country i and the United States rate. Both variables are used to specify the UIP, theory that will be used for valuing exchange rates in the short-term.

The foreign exchange risk premium (PREM) is measured by the conditional standard deviation of foreign exchange returns. This measurement is based on prediction obtained from a GARCH-in-Mean(1,1) model where the conditional mean equation corresponds to ARMA(1,1) specification for exchange returns. To control systematic factors associated with economic and financial crises we define three dummy variables, which adopt the value 1 for the Asian (ASIA), Subprime (SUB) crises periods, and the Covid-19 pandemic respectively; and 0 otherwise.

Finally, we define the MILA variable as a dummy variable that adopts value 1 since May 2011, the date on which MILA started to operate.

3.2. Econometric methodology

In this section we present the econometric models used in this research. For a UIP preliminary analysis, we estimate the following regression by OLS:

Where EXRET _t is the exchange return in period t, which is controlled on interest rates differential (DIF _t ). Moreover, ε _t is a random disturbance. According to (1) and (3) the exchange market will be in equilibrium if α=0 and β₁=1.

As the first analysis model, we used the GARCH-in-Mean(1,1) model proposed by ^{Engle, Lilien and Robins (1987)} on UIP condition. The purpose is to determine the particular risk premium presence to each exchange market. The conditional mean equation is:

Where EXRET _t is the exchange return in period t, which is controlled on interest rates differential (DIF _t ) and conditional standard deviation of exchange returns (σ _t ). This last regressor is the GARCH-in-Mean component, while ε _t is a random disturbance. According to (4), the exchange market will be in short-term equilibrium if α=0, β₁=1 and β₂=0. That is, there is no risk premium (α=0 and β₂=0) and that the interest rates differential fully explains the exchange return (β₁=1). Note that if α≠0 the risk premium is constant, whereas if β₂≠0 the risk premium is time-varying. Both facts would support the risk premium existence in the exchange market. Additionally, the equation for conditional variance of exchange returns is:

Where σ² _t-1 is the GARCH(1) component represented by lag of the conditional variance in t-1, while ε² _t-1 is the ARCH(1) component measured by quadratic residue in t-1. Note that the coefficients γ₀, γ₁ and γ₂ are non-negative. If γ₁ and γ₂ were not significant, then the conditional variance of exchange returns would be homoscedastic and equal to γ₀. In addition, the variance persistence coefficient is equivalent to (γ₁ + γ₂), a factor that the closer it is to 1, more persistent are the variance temporal moves.

Second, we estimate a multivariate GARCH model with dynamic conditional correlation (DCC-MGARCH) proposed by ^{Engle (2002)} and ^{Tse and Tsui (2002)}. The purpose is to determine if exchange risk premium is valued regionally. The model is:

The model (6) is the UIP multivariate representation, where the dependent variable is the exchange return (EXRET _t ) at time t. Moreover, A₀ is the constants vector, B₁ is the coefficients matrix for interest rate differentials (DIF _t ) in period t and Θ₁ is the coefficients matrix associated with risk premiums (PREM _t ). Finally, ε _t is the errors vector. The advantage of model (6) over (4) is that it allows evaluating whether the risk premium affects the exchange rate return when co-movements between exchange markets are included. If Θ₁ is significant, then risk premium would be valued regionally in Latin American markets.

Thirdly, to estimates and predict the dynamics correlation for exchange risk premiums, we estimated a DCC-MGARCH(1,1) model:

The model (7) is a VAR(1) process for exchange risk premium. Where PREM _t is the risk premium matrix for period t and PREM _t-1 is the risk premium matrix lagged in t-1. In addition, Φ₀ and Φ₁ are matrices that group the constants and lags coefficients of the VAR(1) process, respectively. Finally, ε _t is the errors vector.

For multivariate (6) and (7) models, we define σ _ijt as the conditional covariance between exchange markets i and j in period t, which is modeled by a GARCH(1,1) process:

Where ε _it is the residue of the market i in period t and σ_ijt-1 is the conditional covariance between exchange markets i and j in period t-1. Note that when i=j the conditional variance is modeled. The parameters λ₁ and λ₂ are coefficients to be estimated and represent the correlations dynamic adjustment, such that 0<(λ₁+λ₂)<1. So, we can rewrite (8) in its multivariate form MGARCH(1,1):

Where Σ_t the conditional variances and covariances matrix, and E _t-1 is the vector of lagged residuals in t-1. The conditional dynamic correlation model proposed by ^{Engle (2002)} spectrally decomposes the matrix Σ_t as follow:

Where D _t is the standard deviation matrix such as D _t D _t ’=diag(Σ_t) and Γ _t the correlation matrix whose approximation we will denote by quasi-correlations matrix Q _t . Thus, dynamic quasi-correlations can follow this process:

Being ξ_t the standardized residuals vector, and ξ_t=ε_it /σ_it . Note that Λ₁ and Λ₂ are the dynamic quasi-correlations adjustment parameters to their long-term averages. If Λ₁ and Λ₂ are non-significant parameters, then these correlations will be constants.

Finally, we estimate the following AR(1) process for determine the MILA effects on exchange markets:

Where y _t is the dependent variable measured by exchange returns, risk premiums and their dynamic correlations. These dynamic correlations were predicted from (6) model for exchange returns and from (7) model for risk premium. MILA is a dummy variable that adopts value 1 since May 2011 and 0 otherwise, and therefore (_MILA measures the effect of this financial integration process on y _t . Moreover, we use a t-test to evaluate the means differences for these variables before and after MILA. All estimations include dummy variables for Asian, Subprime and Covid-19 periods. These variables allow us to control the effects of the crisis periods and the Covid-19 pandemic on exchange returns, exchange volatilities, and dynamic correlations.

4.1. Data description

Table 2 shows the statistical summary. The exchange returns oscillate on average between 0% and 1%. However, Colombia and Mexico stand out with slightly higher exchange returns than the rest of countries, being Peru the lower.

Interest rate differentials and risk premium series have similar behavior across countries. Brazil, Mexico and Colombia markets have the highest interest rate differentials in relation to the United States, with averages of 14.27%, 9.05% and 7.17%, respectively. Moreover, these same markets exhibit the highest risk premiums. While Chile and Peru markets have lower interest rate differentials and risk premiums. The highest record of exchange volatility occurred during the Subprime crisis and the Covid-19 pandemic.

Table 2 show that the Augmented Dickey-Fuller tests are significant at 1%. So, exchange returns, interest rate differentials and risk premiums are stationary processes. The results of the KPSS test confirm the stationary behavior of the time series. The time series of exchange returns, and exchange risk premium have ARCH structure that it justifies the use of GARCH models. In addition, the correlations between interest rate differentials and risk premium show a possible relationship with exchange returns. We observe that interest rate differentials are positively correlated with the exchange returns, except for Mexico. In addition, the risk premium would be positively associated with exchange rate returns of the Brazilian and Peruvian markets (except the Chilean market).

4.2. Testing the presence and nature of exchange risk premium

In this section we analyze the potential risk premium presence on Latin American exchange markets. Table 3 presents the UIP results, where panel A shows the model (3) results and panel B presents model (4) estimation.

Panel A of Table 3 shows the UIP results through OLS estimations. We observe that constant α=0, which rules out the constant risk premium presence in each exchange markets. Additionally, the coefficient β₁ is not significant, so the interest rate differential (DIF) does not explain the exchange return (EXRET). The UIP1 test that under the null hypothesis indicates the short-term equilibrium for exchange market (H₀: α=0 and β₁=1), is rejected at 1% significance. Even, the Wald test shows that OLS estimation is not significance. So, UIP is not fulfilled, and each exchange market is not at its short-term equilibrium. The UIP non-compliance is supported by several international studies (^{Fama, 1984}; ^{Mussa, 1984}; ^{Froot, 1990}; ^{Engel, 1996}; ^{Choudhry, 2013}). But the observed bias is clearly lower than that exhibited in developed markets, which also be in accordance with international evidence (^{Bansal and Dahlquist, 2000}; ^{Lothian, 2016}).

Panel B shows the model (4) results estimated through GARCH-in-mean specification. The purpose of this UIP specification is to include a time-varying risk premium that potentially corrects the UIP bias. This premium would be generated individually in each exchange markets and would be independent among them. The results do not differ from panel A. The UIP2 test, which under the null hypothesis indicates α=0, β₁=1 and β₂=0, is rejected at 1% for all markets. Even, the risk premium existence and short-term equilibrium are discarded for Latin American exchange markets. Despite of that the Chilean market has a time-varying negative risk premium and Brazil market presents favorable evidence for forward discount bias. This result goes against hypothesis H1, so the risk premium presence in each market capable to correct the UIP bias is discarded (^{Domowitz and Hakkio, 1985}; ^{Frankel and Chinn, 1993}; ^{Cavaglia, Verschoor and Wolff, 1994}; ^{Tai, 2001}; ^{Forsberg and Bollerslev, 2002}; ^{Olmo and Pilbeam, 2011}).

The previous results, although they reveal that there is no individual risk premium in each market (except Chile), do not conclude if this risk premium is regionally valued for Latin American markets. Investors assess risk regionally when markets have a low segmentation and diversification degree, as Latin America (^{De Jong and De Roon, 2005}; ^{Abid, Kaabia and Guesmi, 2014}; ^{Berggrun, Lizarzaburu and Cardona, 2016}). To evaluate the regionally valued risk premium existence, the model (6) was estimated by DCC-MGARCH specification. Table 4 shows its results. This UIP specification considers that the exchange returns are explained by interest differential (DIF), risk premium (PREM) and co-movements between exchange returns. The model includes dummy variables to control the effects of the Asian, Subprime crises, and Covid-19 pandemic.

For all cases, the correlations between exchange returns are significant, which indicate the higher integration degree between these markets. We observe that the returns correlations between MILA markets -Chile, Colombia, Mexico and Peru- are positive, while the correlations between these markets and Brazil are negative and not significant in general. This result shows a diversification possibility between Brazilian and MILA exchange markets, while among the MILA markets diversification benefit is more limited. About these correlations, we note that the quasicorrelations adjustment parameters, λ₁ and λ₂, are significant. So, the exchange returns correlations between Latin American markets are dynamically adjusted, which validate the DCC-MGARCH model. The LR test that evaluates the significance for λ₁ and λ₂ also supports this conclusion.

The conditional mean equation results show that UIP condition is not met for these exchange markets. However, we observed that there is a significant time-varying risk premium valued regionally in Latin American markets. This fact supports hypothesis H2. Even Table 5 that shows the model (7) results, also support this conclusion. We add that risk premium correlates significantly between the Latin American markets. The VAR(1) component for conditional mean equation indicates also the temporal dependence between risk premiums. These results confirm hypothesis H2 from another perspective, which also agrees with ^{Mellado and García (2014)}.

4.3. Effect of MILA on foreign exchange markets

In this section we analyze the effects of MILA on Latin American exchange markets. Table 6 shows the t-test that quantifies the mean difference significance through MILA implementation and model (12) results. It should be noted that the dynamic correlations for exchange returns and risk premiums were predicted from the DCC-MGARCH models (6) and (7), respectively. In addition, the idea of analyzing the exchange markets of Latin America through Brazil, which does not belong to MILA; and Chile, Colombia, Mexico and Peru, which do belong, is to visualize the potential complementary effect (positive co-movements) or substitute (negative co-movements) of exchange returns and risk between these markets.

The results indicate that MILA did not has significant effects on exchange returns of Brazil, Chile, Colombia, Mexico and Peru markets. However, MILA had significant effects on dynamic correlations of exchange returns between these markets. At this point, we observe two interesting results. First, the dynamic correlations of exchange returns between the markets that belong to MILA increased significantly. This result shows that exchange markets integration degree has increased due to MILA. This fact supports hypothesis H3 and the view of the various authors regarding the effects of these integration processes on price moves (^{Bollerslev, 1990}; ^{Shin and Sohn, 2006}). However, this finding shows that the exchange diversification opportunities within MILA markets are more limited. For these reasons, it is considered that Latin American markets have low segmentation degree and investors would value their risk regionally. Second, the dynamics correlations of exchange returns between each MILA market and Brazil were significantly reduced. This fact contradicts hypothesis H3. In general, these correlations were negatives, and as a MILA’s result, these correlations became little more negative. This result indicates that the Brazilian market is a relevant diversification source for international investors in Latin America. The Covid-19 pandemic increased the dynamic correlations, but it did not affect the previous results.

Regarding risk premium, the results are mixed after MILA and before Covid-19. On one hand, the MILA implementation significantly increased the exchange risk premium for Colombia (0.58%) and Mexico (0.14%), while in Chile it reduced it by 0.39%. In Brazil and Peru it did not have significant effects. Including the Covid-19 period, all exchange risk premium figures increased. On the other hand, the MILA’s effects on dynamic correlations of exchange risk premium between each market were significant. The increase on dynamic correlations between the exchange risk premium of Brazil and the MILA markets stands out (except for Peru market). We observe a similar situation between Colombia, Mexico and Peru markets. These facts corroborate that exchange risk premium has regional qualities in Latin America. So, the increase on dynamic correlations of risk premium between these markets supports the hypothesis H4. However, the dynamic correlations of risk premium between Chile and the MILA markets are negative and due to MILA they decreased a little (except for Mexico). Fact that is against hypothesis H4. From a regional perspective, any risk increase in Latin America would generates capital flows movements to Chile, a country characterized by low risk. In fact, these movements generate a lower exchange rate return, consistent with the negative individual risk premium in this market and the Chilean peso strengthening. A similar pattern was observed when it included the Covid-19 period.

We observe that the interest rate differentials decreased significantly, except for Chilean market. However, they increased after Covid-19 period due to the inflationary pressures. In general, these results are conditioned by monetary policy direction implemented for these countries in the last two decades because inflation and economic crisis scenarios.