2.1. UIP and relationship between the exchange return and interest rate diffe- rentials

The UIP is an equilibrium condition for exchange market and has been widely studied by several resear- chers. The UIP indicates that the expected depreciation of exchange rate [E(e_t+k)−e_t]/e_t equals to difference between the local it and foreign interest rate i_t∗, where E(e_t+k) is the expected exchange rate for the t + k period and et is the spot exchange rate. In this way, the exchange rate would be in equilibrium if the interest rate differential fully explains the exchange return. Usually, this relationship has been tested through the following regression model:

According to this specification, UIP would be valid if the model parameters (1) are α = 0, β₁ = 1 and ε_t is a non-autocorrelated residue. However, an extensive empirical evidence has shown that the UIP is not met. This empirical literature has pointed out that the UIP prediction is biased because the interest rate differential only explains a fraction of the exchange returns (^{Mussa, 1984}; ^{Frenkel, 1981}; ^{Froot and Thaler, 1990}). Even ^{Froot and Thaler (1990)} and ^{Froot (1990)} argue that the most common result is an interest rate differential that affects negatively the subsequent direction of exchange rate, i.e. 1<0. This result is usually known as forward premium puzzle or forward discount bias, and would be common in foreign exchange markets of developed countries (^{Fama, 1984}; ^{Mussa, 1984}; ^{Hodrick, 1987}; ^{Froot and Frankel, 1989}; ^{Froot, 1990}; ^{Lewis, 1995}; ^{Engel, 1996}; ^{Bacchetta and Van Wincoop, 2010}; ^{Olmo and Pilbeam, 2011}; ^{Bacchetta, 2013}; ^{Bhatti, 2014}). Other studies that analyzed the prior periods to Bretton Woods, where lower exchange rate volatility was observed, found similar results to forward premium puzzle (^{McFarland, McMahon and Ngama, 1994}; ^{Phillips, McFarland and McMahon, 1996}; ^{Choudhry, 2013}). ^{Omer et al. (2013)} warn dependence between currencies explains these results and would be closely related to the existence of negative interest rate differentials.

More recent researches provides a different vision and that would validate the UIP. Initial studies on this subject found that 1 is close to 1 when the estimation is based on long-term interest rates (^{Chinn and Meredith, 2004}, ²⁰⁰⁵; ^{Bekaert, Wei and Xing, 2007}). ^{Herger (2018)} analyzed the currencies of France, Netherlands, Belgium and Germany and demonstrated UIP was a valid condition while exchange rates were related to investments in bills of exchange during the gold standard period. However, the actual debate on this line was deepened to a greater extent due to a more detailed analysis about interest rate differentials behavior. ^{Aggarwal (2013)} studied the exchange markets of Japan, Australia and the United States between 1992 and 2005 and found favorable evidence for UIP. This research argues that higher interest rate differentials lead to a risk premium that explain better the exchange rate behavior. According to ^{Yung (2017)}, this premium of each exchange market, would contribute to explaining more than half of the exchange rate variations. ^{Lothian and Wu (2011)} support this vision using an extensive sample of years and countries. Their results validate UIP, especially in periods when interest rate differentials are high. The authors add that 1 becomes negative otherwise. Other empirical studies also support this view (^{Chaboud and Wright, 2005}; ^{Lambelet and Mihailov, 2005}; ^{Sarno, Valente and Leon, 2006}; ^{Baillie and Kilic, 2006}; ^{Lothian, Pownall, and Koedijk, 2013}, ^{Ismailov and Rossi, 2018}). However, ^{Bansal and Dahlquist (2000)} and ^{Frankel and Poonawala (2010)} warn that the interest rate differential has a different behavior between developed and emerging countries. Their works reveal that there is less bias for UIP in emerging countries, while the forward discount bias would be observed more frequently in developed markets. ^{Li, Ghoshray and Morley (2012)} and ^{Lothian (2016)} add that the higher premium adjustment is more evident in emerging markets, where indeed interest rate differentials are comparatively higher than developed markets. ^{Nunes and Piloiu (2017}) indicate higher interest differentials reflects higher sistematic risk in exchange markets.

The debate described by the empirical literature validate UIP in countries that experience higher exchange returns and interest rate differentials, leaving this theory as a mechanism relates significant variations of these variables. This fact is observed mainly in emerging markets. While the forward discount bias is associated with low interest rate differentials, which are common in developed countries. These conditions allow us to argue that the relationship between the exchange rate returns and interest rate differentials has a non-linear shape, where the UIP bias decreases as such spreads increase. Even, ^{Aggarwal (2013)} warns that UIP could be valid with significant and higher variations both positive and negative. Therefore, we propose two research hypotheses regarding the relationship between exchange returns and interest rate differentials through UIP. So:

3.1. Data

The research’ data was extracted from the World Developing Indicators (WDI) of the World Bank. The information was organized in a panel data for 83 countries between 1980 and 2015. Table 1 shows the variables description.

The exchange return (EXRET), measured by the annual percentage change of the exchange rate, is the dependent variable of this paper. The exchange rate is quantified as the value of US dollar in terms of the local currency. This measure is widely used by several international studies (^{Fama, 1984}; ^{Domowitz and Hakkio, 1985}; ^{Baillie and Bollerslev, 1990a}; ^{Lewis, 1995}; ^{Engel, 2016}). The interest rate differential (DIF) is measured by the difference between the 30-days interbank rates in country i and the US rate. Both variables are used to specify the UIP, theory that will be used for exchange rates valuation.

The analysis also includes dummy variables that adopt the value 1 in the years of the Asian (1997-1998), subprime (2008-2009) and European (2012-2013) crises. These variables allow controlling extreme events over foreign exchange market.

3.2. Econometric methodology

In this section we present the econometric models used in this research. Preliminarily, we will evaluate the UIP validity using the model (1), which will be estimated by a fixed-effects panel data regression:

Where EXRET_it is the exchange return of the country i in the period t, which is controlled over the interbank interest rate differential (DIF_it). Additionally, η_i represent the individual fixed effects, η_t are the temporal effects and ε_it is a random disturbance. According to (1) and (2) the exchange market will be in equilibrium if α = 0, β₁ = 1 and ε_t is a non-autocorrelated residue. However, if η_i and η_t are significant, the UIP condition is not valid because would exist idyiosincratic and temporal unobservable factors that explain exchange returns behavior.

To evaluate the hypothesis H1, where we test a possible non-linear effect of the interest rate differential on exchange returns, we will use the following regression:

Where EXRET_it is the exchange return of the country i in the period t, which is controlled over the interbank interest rate differential (DIF_it). We note that DIFit2 is the quadratic value for the interest rate differential, where β₂ is the parameter that captures its possible non-linear effect on exchange return. In addition, we have that η_i represent the individual fixed effects, η_t are the temporal effects and ε_it is a random disturbance.

To evaluate hypothesis H2 we will estimate the regression (2) through two complementary ways. First, we will use a quartiles regression (QR), where the exchange returns are classified by quartiles to visualize their relationship with the interest rate differentials. Our interest is to analyze this relationship on their extreme quartiles (Q1 and Q4). Second, we will use a pooled data regression (POLS) where we analyze the effect of the absolute value of interest rate differentials on the absolute value of exchange returns. In this case, we have increasingly ordered the absolute values of the interest rate differentials by quartiles. In this way, ^{Aggarwal (2013)} indicates that the 1 increase shows the lower UIP bias according to exchange rate returns or interest rate differentials experience higher changes. In all these models we will test the hypothesis H₀: α = 0 and β₁ = 1, which validates the UIP (see UIP1 in Table 3, UIP2 in Table 4 and UIP3 in Table 5). If this hypothesis is not support, we will observe the UIP bias, which will reduce if β₁ is not negative or close to 1.

Finally, models (2) and (3) include dummy variables to capture the effects of Asian, subprime and Eu- ropean crises on exchange rate returns. All these models were estimated through robust variance to control heteroskedasticity patterns.

4.1. Descriptive and correlational analysis

Table 2 presents the descriptive statistics, where the sample of countries was divided according to income level. On average, the exchange return is 19.46 %, which shows that most of the exchange fluctuations would be concentrated in quartiles 3 and 4. In addition, there is a clear heterogeneity between countries. Low and low-middle income countries exhibit exchange returns of 38.16 % and 24.20 % respectively, while upper-middle and high-income countries have returns that are below of sample average. This pattern for exchange returns is also seen in the quartiles for each type of country, where are visible the lower exchange returns from high-income countries and higher returns from low-income countries. Interest rate differentials have a similar pattern to the exchange returns. In fact, it is observed that its correlation with foreign exchange returns is positive and significant, which would according to UIP. The average interest rates differential is 17.56 %, where the low-income countries show values above the average, while the low-middle (7.36 %), upper-middle (5.26 %) and high (-0.02 %) income countries are below the average. Likewise, interest rate differentials are concentrated in quartiles 3 and 4, which also indicates that countries experience high spreads episodes. An idea that is also displayed in the quartile 4 values for each type of country.

It should be noted that for estimation process, both exchange rate returns and interest rate differentials are stationary variables. The unit root test is significant at 1 % in all cases.

4.2. Non-linear effect of interest rate differentials on exchange return

In this section we present the econometric analysis results. Table 3 presents the model (2) results, which was estimated through pooled (POLS) and fixed-effects panel data (FE) regressions. Wald and F-test corres- pond to global significance tests for these models. In the latter case, the Hausman test reports the significance for the individual and temporal fixed-effects on regression (2) estimation. So, it result support the statistical relevance of the fixed-effects estimator. This model presents a preliminary UIP analysis. Its results are not originals because UIP is not met. The F-test denoted by UIP1, that indicates under null hypothesis H₀: α = 0 and β₁ = 1, is rejected in all cases. Even these findings coincide with fixed-effects existence because idiosyncra- tic factors also explain the exchange returns. This result is consistent with other empirical studies (^{Hodrick, 1987}; ^{Froot, 1990}; ^{McFarland, McMahon and Ngama, 1994}; ^{Lewis, 1995}; ^{Engel, 1996}; ^{Phillips, McFarland and McMahon, 1996}; ^{Bacchetta and Van Wincoop, 2010}; ^{Olmo and Pilbeam, 2011}; ^{Choudhry, 2013}; ^{Bhatti, 2014}). However, we note that UIP bias is relatively lower in low-middle income countries and increases in high-income countries. This result also agrees with other studies (^{Bansal and Dahlquist, 2000}; ^{Frankel and Poonawala, 2010}; ^{Li, Ghoshray and Morley, 2012}; ^{Lothian, 2016}, ^{Nunes and Piloiu, 2017}).

Table 4 presents the model (3) results through pooled (POLS) and fixed-effects (FE) regression. The Hausman test corroborates the fixed-effects estimator is better than random-effects estimator. The similar way, the F-test denoted by UIP2, that indicates under null hypothesis H0: =0 and 1=1, is rejected in all cases and support that UIP condition is not met. Despite this, FE estimator is appropriate econometric tool to capture the difference in the exchange returns and interest rate differentials between countries. We observe that the 2 parameter is positive and significant. This fact indicates that the interest rate differential (DIF) has a non-linear effect on exchange return (EXRET), which validates our hypothesis H1. The non-linearity shape suggests that for lower interest rate differentials, the UIP is not met and the forward discount bias predominates over exchange rates behavior. In high-income countries we observe the higher forward discount bias predominance, as these countries present low exchange returns and interest rate differentials. On the other hand, for higher interest rate differentials, we observe a positive relationship between differentials and exchange returns. In any case, there are threshold values that separate these effects. Critical values described in Table 4 were obtained by maximizing the exchange returns of the model (3) with respect to DIF variable, only when the parameters 1 and 2 are significant. In this way, critical values are obtained from the expression -(1/22). In low-middle and upper-middle income countries, where this bias is lower, the threshold values for interest rate differentials fluctuate between 38.43 % and 40.62 % according to estimation. The threshold values existence for interest rate differentials would allow policymakers, such as central banks and ministries, to identify the possible effects of monetary policy on exchange rates and economy. Even for investors, it provides guidelines that would allow them to analyze the effects of interest rate differentials changes on their investments in currencies.

Table 5 shows the model (2) results through pooled data and quartiles regressions. So, these regressions would show the relationship between Q1 (lower variations for exchange returns) and Q4 quartiles (higher variations for exchange returns) with interest differential. In addition, pooled data regressions evaluate the effect of absolute value of the interest rate differentials on absolute value of the exchange returns. The regres- sions have been classified in quartiles (Q4 and Q1), ordered by absolute values of interest rate differentials. The idea is to observe the UIP behavior according to magnitude of variations of these variables, regardless of the sign of change. Once again, our results support that UIP is not met. UIP3 test that evaluates the null hypothesis α = 0 and β₁ = 1 is rejected at 1 % in each pooled data and quartile regression. But, according the positive values of β₁, we confirm that UIP bias is lower. The pooled data regression for quartile 1, which groups together the lowest absolute values of the interest rate differentials, presents a positive and not signi- ficant parameter equal to 0.0348; while the regression for quartile 4, which groups the highest absolute values of the interest rate differentials, has a significant parameter equal to 0.2682. The regressions by quartiles, which are grouped in quartiles according to the values of exchange returns, show a similar result. The UIP bias is reduced as the exchange returns and interest differentials show higher variations. These results validate hypothesis H2, which corroborates the trend to reduce UIP bias when its fundamentals undergo significant variations.