An Estimation of Mexican Peso Yields with Foreign and. Domestic Factors.

Transcription

1 An Estimation of Mexican Peso Yields with Foreign and Domestic Factors. Pablo Javier Klein 9/7/22 Abstract This study is an estimation of the (daily) yields of Peso-denominated Negotiable Certificates of Deposit (NCD s) dependent on local latent factors and observable factors (the Mexican rates of inflation and growth), a foreign latent factor extracted from Eurodollar yields, and the Federal Reserve s Target rate as announced at FOMC meetings. No-Arbitrage constraints derived in the exponential-affine factor framework of Vasicek (977) are always rejected by the Peso yields. The latent factors derived form the Eurodollar and Peso yields predict the log of the change in the exchange rate substantially better than interest rates. Introduction. This study was submitted as part of my doctoral dissertation at UC Berkeley. I acknowledge invaluable help from Roger Craine (my disseration advisor), as well as financial support from Mexico s National Council for Science and Technology (CONACYT). All errors are my responsibility.

2 This study was motivated by a revival of interest in the Forward Discount Puzzle in international currency markets. Mexico is the emerging market which has been for the longest time under a floating exchange rate regime, and therefore makes an interesting case to contrast with the experience of the main currencies. A linear regression of the variation of the Mexican Peso/US dollar exchange rate on the lagged interest rate differential reinforces the evidence for the FDP, as the Peso-dollar interest differential gives incorrectly signed forecasts of future changes in the exchange rate, independently of the maturity chosen for the interest rates and the frequency of the data. This study looks at forecasts of changes in the Peso-US dollar exchange rate based on latent factors extracted from the Peso and Eurodollar yield curves by means of a Kalman filter. The yields of Mexican Peso-denominated Negotiable Certificates of Deposit (NCD s) are observed with a daily frequency for three maturities, I allow them to depend on domestic and foreign factors. Additionally, I impose No-Arbitrage constraints on the coefficients of the yields, derived from the affine factor framework of Vasicek (977); the different levels of restriction estimated correspond to zero, fixed and time-varying risk premia. Notable examples of the literature that looks for explanations of the FDP along the lines of time-varying risk premia include Backus, Foresi and Telmer (996), Bansal (997), Brandt and Santa Clara (2) and Dewachter and Maes (22) 2. The first Mexico floated its currency on December 22, 994; since February, 996, its currency arrangement can be classified as a managed float. The only developing countries with older managed or free floating currencies are: Albania, Haiti, Kenya and Madagascar. (for a comprehensive classification of exchange regimes, see Reinhart and Rogoff (22)) 2 For other studies using the affine factor framework, seeahn (995), Amin and Jarrow(99), Bakshi and Chen (995), Frachot (994), Nielsen and Saa-Requejo (993) and Saa-Requejo (994). 2

3 example derives theoretical affine factor specifications that replicate the Forward Discount anomaly in a two country model, and find that, in order to be consistent with other stylized facts, at least one factor must be common to both countries interest rates, and the effect of the factors on interest rates and the pricing kernel must be asymmetric 3. Bansal (997) found significant non-linearities in the relationship between interest rates and depreciations for the US dollar and the German Mark; in particular, the Forward Discount bias only occurred in periods when the dollar paid a positive interest premium over these currencies; to replicate this stylized fact, the factor diffusion process must considerably depart from the affine framework. Brandt and Santa Clara (2) derive and jointly estimate the interest rate processes for two currencies (US dollar and German Mark) and the degree of excess volatility due to market incompleteness 4. Dewachter and Maes (22) apply the multifactor-affine term structure framework to jointly estimate bond rates and the exchange rate process as the ratio of two countries Stochastic Discount Factors. They found that a model of the British pound/dollar joint bond markets with two local factors and a common factor is able to partially explain the UIP and the home bias puzzles. Parallel to these studies, a body of literature focused on perfecting the techniques for the estimation of term structure models with affine factors, in particular: Piazzesi (2) developed a Simulated Maximum Likelihood (SML) based method for the 3 Specifically, that the factor attached to interest rates in country A have a stronger effect on the volatility of country B s pricing kernel, and vice versa. 4 Brandt and Santa Clara (2) s main empirical finds are (i) the risk premia associated with interest rates is very small compared to the premia associated with other independent risks (pure currency risks), (ii) the premia tend to vary with the interest rates, and (iii) a large fraction of exchange rate volatility (typically around 8%) is not priced by the market. 3

4 estimation of the yield curve derived from LIBOR-quality interest rate swaps; this method had the advantage of solving the likelihood for a non-gaussian diffusion process for two latent factors (based on CIR (985)) with the Fed. Target rate added as a first observable factor following a jump process. Ang and Piazzesi (2) compared the predictive power of the Vasicek (977) model with a simple unconstrained VAR for monthly rates on five maturities of US Treasuries, and found that the restricted model had better out of sample performance than the VAR. Duffee and Stanton (2) compared the performance of different estimation techniques with simulated data and concluded that a simple Kalman filter-based estimation produced the best results, even when it was inconsistent with the specification of the true model 5. This study uses daily data on the interest rates paid by bank-issued Negotiable Certificates of Deposit denominated in Pesos, as published by Banco de México in the official gazette; they are averages for the banking sector. The time frame of this study is the relatively stable period from 996 to 2 6. During those years, Mexican interest and inflation rates gradually decreased from more than 3% to single digit levels and the economy grew steadily at rates above 5%, except for the second half of 998 (see Figures 4-9). Most of the variation in the levels of Eurodollar yields can be explained by the Fed Target, which is used as a first (observed) factor. The same is 5 The techniques compared by Duffie and Stanbton (2) are: (i) a combination of EMM with Seminonparametric (SNP) auxiliary model approximating the non-gaussian process of the factors (ii) EMM with a linearized Kalman Filer as auxiliary model (iii) direct estimation of the linearized Kalman filter. 6 During 995, as the Mexican crisis unfolded, the exchange rate was extremely volatile and interest rates remained above 4%. For a study on the behavior of exchange rate volatility in this period, see Werner (997). 4

5 true for the Peso yields and the Mexican rate of inflation. Expressing the ED rates as spreads over the target and the Peso rates as spreads over Mexican inflation allows the estimation of high frequency processes with a faster rate of mean-reversion than the unmodified interest rates. This procedure also avoids the problem of negative interest rates associated with the Vasicek (977) framework. For given starting values of the coefficients of the interest rates, a Kalman filter procedure extracts the ED and Peso latent factors and the likelihood of the sample. An iterative optimization algorithm is then used to obtain Maximum Likelihood estimators of the yield coefficients and the autoregressive coefficients of the latent factors. I estimate different versions of the model with two Peso factors and with one Peso factor, with free coefficients and with No-Arbitrage constraints. The term structure model used to derive no-arbitrage constraints is an extension of the exponential-affine factor framework of Vasicek (977). This choice makes the yields depend linearly on the latent and observable factors and the conditional distribution of the latent factors gaussian. Since I don t observe a reliable one day interest rate for an equivalent asset 7, the coefficients of the 28 day interest rate of Peso-denominated NCD s are fixed at the values obtained in the unconstrained estimation, with the underlying assumption that these are consistent with the unobserved instant rate under no- arbitrage. Under this assumption, it is possible to derive constraints on the 9 day and 82 day Peso NCD s 8. Different specifications of the pricing kernel for Peso 7 Ang and Piazzesi (2) use monthly data, they estimate the process for the one month ED rate as the instant rate for the derivation of the yield curve. 8 For the details of this derivation, see the Appendix. 5

6 assets are tested: variable prices of risk, constant prices of risk and zero prices of risk (Perfect Expectations Hypothesis). The constrained models are always rejected by the data in favor of a specification with free coefficients in the yield equations 9. Eurodollar and Peso interest rates are strongly negatively correlated; this fact can be attributed to the Federal Reserve s policy of changing the target in response to events in international markets; in particular its lowering of the rates after the financial crises in Russia, Argentina and Brazil in 998. The Fed. Target and the Eurodollar latent factor (extracted from the difference between Eurodollar yields and the Target) have a negative effect on the Peso rates, however the effect of the Target on the Peso rates was not statistically significant; probably because the Eurodollar latent factor is already a good predictor of future changes in the Target. A change in the Eurodollar latent factor had a stronger effect on the longest yields, which are more volatile than the shorter ones; this is confirmed by the positive relation between the slope and the level of the Eurodollar yield curve. This behavior is very different from the Peso yields, which are characterized by a negative correlation between the slope and the level and higher volatility for the shorter yields. The difference in behavior between the Eurodollar and Peso yield curves can be attributed to the high persistence in the Federal Reserve s policy; which implements its medium run objective in small consecutive changes. 9 This rejection could be caused by the functional form of the pricing kernel: in the most complex case, the price of risk is an affine function of the Eurodollar and Peso latent factors. To estimate a more complex functional form would require more than three observable yields. See Piazzesi (2) for an econometric estimate of the persistence in the Fed s policy. 6

7 In addition to depending on the rate of inflation with a coefficient of one, Peso yields are significantly negatively correlated with Mexico s rate of growth in Economic Activity, indicating that Mexico s Central Bank abstained from using monetary policy as a tool for economic stabilization during the period of study. The estimated price of risk under No-Arbitrage was consistent with the average positive slope of the Peso yield curve. With a flexible price of risk dependent on the foreign (eurodollar) and domestic (Peso) latent factor, the foreign factor has a stronger influence than the domestic factor on the price of risk associated with Peso interest rates. Both the foreign and domestic factors have a positive effect on the slope of the Peso yield curve (corresponding to an increase in the price of risk), even though they have opposite effects on the level of Peso rates. After estimating the different constrained and unconstrained yield specifications, I measure the power of lagged interest rates and yield factors as predictors of the log change in the exchange rate. The standard result that the interest rate differential between two currencies produces incorrectly signed forecasts of future changes in the exchange rates is confirmed in the sample for all maturities. This effect is due to the behavior of the latent factors associated with the Peso. An increase in the level of Peso interest rates raises lowers Eurodollar interest rates and causes an appreciation of next period s exchange rate, in violation with the theory of Uncovered Interest Parity (UIP). The Eurodollar and Peso latent factors, extracted from the spreads (over This violation of UIP was first documented by Fama (984); there is an extensive literature that proposes explanations for this puzzle. 7

8 the target and inflation, respectively) produce substantially better forecasts of next day s Peso/dollar exchange rate than the interest rates, although their explanatory power remains below one percent. The main findings of this paper are: () Peso-denominated NCD s reject no arbitrage constraints for the chosen specifications for the price of risk. (2) The different behavior of the ED and Peso yield curves is consistent with the high persistence in the Federal Reserve s policy and the relative neutrality in the policy of Mexico s Central Bank (3) The Forward Discount Puzzle (FDP) is confirmed for the Eurodollar/Peso exchange rate and is robust to different maturities and frequencies. This is mainly due to the effect of the Peso latent factor on interest and exchange rates. (4) The latent factors extracted from the spreads between yields and a level driving variable (the Fed. Target for ED and the Mexican rate of inflation for Pesos) predict exchange rates substantially better than the raw yields. The following section describes the data in more detail, in particular the statistical properties and the behavior of the yields over time; as well as the behavior of the level, slope and curvature of the Ed and Peso yield curves. The data The dataset consists of interest rates for Peso-denominated negotiable Certificates of Deposit (NCD s) with 28, 9 and 82 day maturities (PAG28, PAG9 and PAG82) and the daily change in the log of the Peso-dollar exchange rate (DLTC). These 8

9 figures are averages of the Mexican banking sector, published every business day in Mexico s Official Gazette. Eurodollar rates for similar maturities are available from the web site of the Federal Reserve Bank of Chicago (EDM, ED3M, ED6M). Monthly figures for the Mexican rate of inflation (DINPC), and the growth in Mexico s monthly Global Economic Activity index (DIGAE) are published by Mexico s institute for Statistics, Geography and Informatics (INEGI). The Federal Target rate is announced periodically after FOMC s scheduled meetings. Table presents summary statistics for all the series. The one moth Eurodollar rate averages 5.55 percent over the sample, with the 3 month and 6 month rates paying average premia of.9 and.7 percent points respectively. The 28 day Peso rate is 6.5 on average, although its range of variation is larger than for eurodollars (7.5%- 4.% versus 4.75%-6.7%), the 9 day and 82 day rates pay premia of.3 and.32 basis points over the Peso one month rate, on average. At a glance, the most striking differences between Peso and eurodollar rates are: () the Peso yield curve looks less linear than the ED yield curve: the average yields for the 9 and 82 day rates are extremely close, and those two maturities move virtually together during the first part of the sample. (2) while for the Peso, the short rates are the most volatile, for the ED rates the opposite is true: the longer yields appear to have more volatility than the shorter yields. This last stylized fact could be explained by the high persistence in the level of ED rates, which is determined by the Federal Reserve. 9

10 Table Summary Statistics Mean Std. Dev. Minimum Maximum Sum Variance Skewness Kurtosis Levels EDM ED3M ED6M PAG PAG PAG DLTC TARGET DINPC DIGAE Spreads EDM - TARGET ED3M - EDM ED6M - EDM PAG28 - DINPC PAG9 - PAG PAG82 - PAG Notes: EDM: ED3M: ED6M: PAG28: PAG9: PAG82: DLTC: TARGET: DINPC: DIGAE: 3 day ED rate 9 day ED rate 8 day ED rate 28 day Peso rate 9 day Peso rate 82 day Peso rate Change in the log XR Fed Target rate Mex. Inflation Mex. Production

11 Figure shows interest paid by Eurodollar-denominated bonds with 3, 9 and 8 day maturities, and Figure 3 shows the Fed Target rate. The levels of Eurodollar rates follow the Target very closely; Figure 2 shows eurodollar interest spreads of long bonds over the one month rate. Both spreads (9-3 day and 8-3 day) appear to be strongly correlated, and positively correlated with the level of interest rates. Figure 4 shows the levels of interest rates on Peso-denominated NCD s for maturities of 28, 9 and 82 days. Figure 6 shows Mexico s monthly inflation rate 2. This appears to be the most important factor affecting the level of Peso interest rates in the medium to long run. 996 was a year of high and volatile interest rates as the Mexican economy was still struggling in the aftermath of the Peso crisis, burdened by the huge non-recoverable loan portfolio of its banking sector, a severe adjustment program, and relatively high inflation as the real exchange rate realigned towards its long term equilibrium level. From 996 to 2, interest rates gradually decreased from their initial high level in the 3% range towards single digit values. The most significant interruption to this trend was the period of relatively high interest rates in the fall of 998 and the spring of 999, caused by contagion effects from other emerging markets, as well as the continuing problems of Mexico s banking sector, as well as the need for fiscal reform caused by the loss of government revenue from oil exports. The and 9-28 day spreads (Figure 5) appear negatively correlated with the level of the interest rates, the opposite to what is observed with Eurodollar spreads. During 996, as interest rates were above 25 %, longer maturities often 2 Mexico s inflation (DINPC) and growth (DIGAE) rates were seasonally adjusted and converted to daily figures by extrapolation.

12 carried a negative premium over the 28 day Peso NCD s. In subsequent years, as interest rates converged to a level below %, term premia became consistently positive, around percent for the day spread and a third of one percent for the 9-28 day spread. Table 2 shows the results of Augmented Dickey-Fuller (ADF) unit root tests on the levels of eurodollar and Peso-denominated interest rates and spreads, as well as the Fed Target, the Mexican inflation and growth rates. For virtually all variables, with the exception of 28 and 9 day interest rates for Pesos at the 9% confidence level, the series are unable to reject the null hypothesis of a unit root. This surprising fact is probably caused by the estimation of processes with very high persistence over a relatively short time interval. This high persistence has been shown to cause a bias in the autocorrelation coefficients of the underlying factors 3. Unit root tests on the spreads between the ED rates and the target and the Peso rates and the Mexican inflation rate respectively reject a unit root at the 95% confidence level (the only exception is the 6 month ED rate). Expressing the yields as spreads over the Fed target and over inflation for Eurodollars and Pesos respectively reduces the possibility of estimation problems resulting from the high persistence in the underlying observable factors (that is the Fed target and the Mexican rate of inflation). The one month spreads for eurodollars and Pesos are pictured in Figures and. Finally, at the bottom of Table 2 are the results of unit root tests on spreads of longer maturities over the one month rates, which always reject unit roots. 3 As shown by Duffee and Stanton (2) 2

13 Table 2 ADF Unit Root Tests Test Stat. P- Value Nr. Of lags Levels EDM ED3M ED6M PAG *.7468 PAG * PAG TARGET DINPC DIGAE Spreads EDM - TARGET **.6 22 ED3M - TARGET ** ED6M - TARGET PAG28 DINPC **.98 3 PAG9 DINPC ** PAG82 - DINPC ** ED3M - EDM **. 22 ED6M - EDM * PAG9 - PAG **. 6 PAG82 - PAG ** ** Rejects the null hypothesis of a unit root with 95% confidence * Rejects the null hypothesis of a unit root with 9% confidence The traditional factors of Litterman and Sheinkman (993) were estimated with linear combinations of the yields 4. The correlations between these traditional factors for both currencies are shown on Table 3. For the Peso rates, the level is negatively correlated with the slope and with the curvature, suggesting that investors consider Peso interest rates to be mean-reverting and their volatility to increase with the level. For Eurodollars, the levels are also negatively correlated with the curvature, but they 4 The level, slope and curvature are estimated as the following linear combinations of the yields: LEV=(Y + Y2 + Y3)/3, SLO= Y3 Y, CUR= Y 2Y2 +Y3. where Y Y2 Y3 are the one, three and six month rates, respectively. 3

14 are positively correlated with the slope, suggesting a higher degree of persistence in the underlying process. The ED level is also strongly negatively correlated with the Peso level, although this result may be a spurious consequence of the slow mean reversion in the rates and the short sample of study. Table 3 Traditional Factor Correlations Eurodollars Pesos Level Slope Curvature Level Slope Curvature Eurodollars Level. Slope.523. Curvature Pesos Level Slope Curvature Figure 9 shows the growth in Mexico s Index of Global Economic Activity (DIGAE). This provides a month to month measure of economic growth in Mexico. The Mexican Economy grew at a relatively high rate (from 6 to %) from 996 to 998 and in the years 999-2, with virtually zero growth during 998 and the start of a recession in early 2. The fast growth before 998 occurred as the Mexican economy recovered from the severe contraction of 995. The stagnation in 998 can be explained by the factors mentioned previously (banking debt, fiscal reform, oil prices), while the slowdown in 2 is related the same trend in the US economy. Between January 996 and January 2, the exchange rate varied between 7.3 and.6 Pesos to the dollar and it was not statistically distinguishable from a random 4

15 walk with drift for all time horizons (see Figures 7 and 8). The average annualized rate of depreciation in this period was around 5%. The largest depreciation (+2% over a couple of weeks) occurred during the Russian crisis in September 998, but most of this jump reverted subsequently. Figure 3 Day eurodollar rate minus the Fed. Target Figure 28 day Peso rate minus inflation /3/2 /3/ /3/2 /3/2-6. 5

16 Figure Eurodollar Interest Rates (Percent) 3, 9 and 8 days Figure 2 Eurodollar Interest Spreads Percent points Figure 3 Fed. Target Rate Percent 9 Days 8 Days 3 Days Days /3/2 /3/2 /3/ Days /3/2 /3/2 /3/2 Figure 4 Figure 5 Figure 6 Peso Interest Rates (Percent) 28, 9 and 82 days Peso Interest Spreads Percent points Monthly Inflation Rate Annualized, Percent /3/2 /3/2 Figure 7 Exchange Rate Pesos Per Dollar Figure 8 Daily Depreciation Percent Figure 9 Mexico's Economic Activity Index, Monthly variation Percent, Annualized Days Days /3/ Days 9-28 Days /3/ /3/2 /3/2 /3/2 /3/ /3/2 /3/2 /3/2 /3/

17 Unrestricted Estimation Eurodollar yields spreads over the Federal Reserve s Target are estimated as affine functions of a single factor. Let Y *, Y 2 *, Y 3 * be the yields of eurodollar-denominated bonds with one, three and six month maturities, then, for n = to 3; Y τ = G + G F + u () * * * * nt t n n t n Similarly, Peso yield spreads over inflation, Y *, Y 2 *, Y 3 * are linear functions of the eurodollar factor, two Peso factors, and the exogenous, observable variables. Y π = G + G F + G F + G F + G g + G τ + u (2) nt t n n t n2 2t n3 3t ng t nτ t n Where π t is the rate of inflation, F, F2, F3 are: the eurodollar factor and the two Peso factors respectively, g t is the rate of growth in Mexico s index of Global Economic Activity, τ t is the Fed Target rate, and u n are iid normal disturbances. The latent factors are modeled as standard AR() processes. Note that the Peso latent factor F3 is only included in the estimations with two domestic latent factors. F t = af t + ξ t F2t = a2f2t + ξ2t F3t = a3f3t + a4f2t + ξ 3t (3) 7

18 Where ξ t, ξ 2t, ξ 3t are gaussian disturbances. I estimate the ED and Peso interest rates as spreads over a first observable factor (the Fed Target for ED and the Mexican inflation rate for Pesos). These first factors approximate the variation in the yields over the medium to long run, leaving the short run dynamics to be explained by the remaining latent factors 5. For given parameters, a Kalman filter procedure can be applied to extract a number of latent autoregressive factors and a value for the likelihood of the sample 6. The next step is to obtain Maximum Likelihood estimates of the coefficients by integrating this likelihood evaluation routine in a standard maximization algorithm. I estimate two versions of the unrestricted model: (i) with two factors explaining the Peso yields, (ii) with a single factor for the Peso yields. The general procedure is as follows: (a) I obtain a ML estimate of the coefficients for the ED equations and the single ED latent factor. (b) I use the ED latent factor, together with the target and the change in the index of Global Economic Activity (DIGAE) as exogenous variables for the Peso rate equation, and I obtain ML estimates of the Peso equation coefficients and the Peso latent factors. 5 This is the same as restricting all the coefficients of the target and the inflation rates to be equal to one. A preliminary estimation without these restrictions produced coefficients not statistically different from one. 6 For a detailed description of the Kalman filter technique, see Hamilton (994), chapter3. 8

19 Restricted Estimation In a second estimation, I restrict the coefficients of the Peso yields in (2) to be consistent with the absence of arbitrage in a simple Vasicek (977) affine factor framework. Since we are actually extracting factors from interest spreads that can be negative (see Table ) the usual criticism that the Vasicek specification allows negative interest rates does not apply in this case. The following derivation of noarbitrage conditions is taken directly from Ang and Piazzesi (2). The only difference with their study is the coefficients of the 28 day Peso yield are fixed to be equal to the values obtained in the unrestricted estimation, and the coefficients of the 9 and 82 day yields are then obtained from the 28 day coefficients (see the Appendix for details). This procedure fixes the instant rate at a reasonable level. A LR test of the restricted model versus the unrestricted one is a test of whether the 9 day and 82 day Peso yields are consistent with the absence of arbitrage, given that the 28 day rate coefficients estimated without restrictions are consistent with the absence of arbitrage. The instant interest rate depends on the same variables that affect the yields: rt πt = β+ βf t + β2f2t + βggt + βττt (4) Term structure models use the assumption of no-arbitrage 7 to guarantee the existence of an equivalent martingale measure Q such that the ratio of the prices of any non- 7 See Harrison and Kreps (979) for the theoretical foundations of asset pricing under no arbitrage. 9

20 dividend paying assets follows a Maringale, in particular the ratio of the price of any asset over the risk-free one period zero-coupon bond paying the instant rate. For the purposes of this study, I assume that the Radom-Nykodim derivative of Q, ζ follows the log-normal process: ζ ζ λ λξ 2 2 t+ = t exp t t 2t+ (5) Where ξ 2t + is the disturbance term that drives Peso latent factor, and λ t the price of Peso risk is: λ = λ + λ F + λ F (6) t t 2 2t Given the specification of the risk-neutral measure and the instant interest rate, I define the pricing kernel for all Peso-denominated assets: 2 log( mt+ ) = rt + λt + λξ t 2t+ (7) 2 For the restricted estimation, there is only one domestic latent factor, and the Peso SDF only depends on domestic risk. While a model with two domestic sources of risk produces a superior fit, I have found that including more than one source of risk in the 2

21 SDF leads to non-identification of the prices of risk 8. I estimate three levels of restriction: (i) flexible prices of risk, where I allow the ED latent factor to affect the price of risk associated with the Peso factor (ii) fixed prices of risk and (iii) zero prices of risk, equivalent to the Perfect Expectations Hypothesis. The 28 day yield equation is then fixed at the unrestricted estimate. Y π = G + G F + G F + G g + G τ + u (8) t t t 2 2t g t τ t Then the 9 and 82 yield equations are derived from the 28 day coefficients and the restricted parameters. (see the Appendix for details). The coefficients obtained this way are then fed to the Kalman filter to derive the Peso latent factor and the likelihood. Maximum likelihood estimates are then obtained for the parameters λ, λ, λ 2 and a 2. The restricted long yields, therefore, have the following specification: (For n = 2,3) ( ) ( ) ( ) Y π = G θ + G θ F + G θ F + G g + G τ + u (9) nt t n n t n2 2t g t τ t n 8 The intercept of the price of risk λ and the implied volatility of the Peso factor fully determine two yield intercepts. Since the first (one month) yield coefficient is restricted, maximum likelihood yields full identification. A more complex pricing kernel specification cannot be estimated unless we had more than three yields of the same type of instrument. See the Appendix for a more detailed explanation of this problem. 2

22 The Results Table 4 presents the coefficients of the ED and Peso yields for all maturities for the two unrestricted estimations (three factors and two factors) and for the restricted estimation with variable, fixed and zero prices of risk respectively. Figures 2-2 show the yield impulse responses to standard deviation shocks in the factors for the unrestricted estimation with three factors (one ED, two Peso). Figures 2-26 show the yield impulse responses for the unrestricted estimation with two factors. In the three factor, unrestricted case, the ED factor F has a positive effect on ED yields and a negative effect on Peso yields. In the ED case, the long yields are affected the most, while for the Pesos, a shock in F affects more the 9 day yield. This response is consistent with the stylized facts previously mentioned: (i) the slope of the ED yield curve increases with shocks in the level (ii) the longer yields show more volatility (iii) the ED and Peso interest rates are negatively correlated. An explanation for the different behavior of ED yields could be the persistence of US monetary policy, when a change in the economic outlook prompts the Federal Reserve to raise its desired interest rate in the long run, the Fed Target rate is increased towards that objective in gradual steps at each FOMC meeting. Factor F incorporates expected future changes in the Fed. Target rate. An explanation for the negative correlation between ED and Peso rates might be the strong linkage between US and Mexican economies and the counter-cyclical nature of the Federal Reserve s 22

23 policy: when both economies are in a recessive trend, Peso rates tend to be high, and the Fed lowers its target to stimulate the US economy. In the three factor specification, F2 and F3 are related to the Peso yields. They don t affect ED yields by construction. Factor F2 is a curvature factor and factor F3 is a level and slope factor by construction. The effect of F2 is hump-shaped and negative, it reaches a minimum after 3.5 years. F3 is the most persistent of all three factors: its has a half life of nearly six years, while factors F and F2 have a half life of about.5 years. The two factor case has very similar results: a very persistent Peso factor, a less persistent ED factor affecting more the long rates. The only difference with the two factor case is the positive effect of the ED factor (F) on the 82 Peso rate. All unrestricted and restricted cases produce the same signs for the effects of the latent factors and the exogenous variables on the yields (factor F2 in the two factor case corresponds to factor F3 in the three factor case ). The change in the index of global economic activity (DIGAE) has a significant negative effect on all Peso rates, suggesting that Peso interest rates are counter-cyclical as the Mexican monetary authority follows a neutral policy. The Fed target has a negative effect on Peso rates, but the magnitude of the effect is not statistically significant. Since future moves in the Target are anticipated and incorporated in F, the remaining explanatory power of TARGET on Peso yields, after including F is, not surprisingly, very low. 23

24 Table 4 Interest Rate Coefficients Unrestricted Estimation Restricted Estimates Three Factors Two Factors Time-VaryingFixed Zero Maturity Estimate Std Error Estimate Std Error Risk Prices Risk Prices Risk Prices Eurodollar Rates: a. **.. **. month C.... F.8 **..8 **. 3 months C F.22 **..22 **. 6 months C -. *.6 -. *.6 F.45 **.2.45 **.2 Peso Rates a2. **.. **.... a3. **. a4.. month C 2.59 ** ** F ** F **..76 ** F3.62 **.4 DY -.2 ** ** TARGET * months C 2.74 ** ** F -.34 ** * F **..75 ** F3.45 **.2 DY -.22 ** ** TARGET months C.32 ** ** F ** F **.8.7 ** F3.6 **.2 DY -.2 ** ** TARGET * Significant with 9% confidence ** Significant with 95% confidence

25 Figures 2-2 Impulse Responses (Unrestricted, Three Factor) I One Standard deviation shock in F Figure 2 Figure 3 Figure 4 Impulse Response Impulse Response Impulse Response Latent Factors Eurodollar Rates Peso Rates days days days days 8 days.2 F days II One Standard deviation shock in F2 Figure 5 Figure 6 Figure 7 Impulse Response Impulse Response Impulse Response Latent Factors Eurodollar Rates Peso Rates F days days -.5. F days III One Standard deviation shock in F3 Figure 8 Figure 9 Figure 2 Impulse Response Impulse Response Impulse Response Latent Factors Eurodollar Rates Peso Rates F days 9 days 82 days

26 Figure 2 Impulse Response F Figures 2-26 Impulse Responses (Unrestricted, Two Factor) I. One standard deviation shock in F 9 days 3 days Figure 22 Impulse Response Eurodollar Rates 8 days Figure 23 Impulse Response Peso Rates -. 9 days days 82 days II. One standard deviation shock in F2 Figure 24 Figure 25 Figure 26 Impulse Response Impulse Response Impulse Response F2 Eurodollar Rates Peso Rates days 28 days

27 Table 5 presents LR tests for the different models estimated, table 6 shows the fit of the levels and spreads for the ED and Peso interest rates for the unrestricted estimations and for the Peso levels and spreads for the models with no-arbitrage constraints. Figures 27-35, 36-4 and show the observed and predicted levels and spreads for the unrestricted and restricted estimations respectively. The results clearly indicate that the unrestricted model with two factors (one Peso factor) is rejected in favor of a model with three factors (two Peso factors). In particular, the three factor model replicates the behavior of the rate spreads in a much better way than the two factor model (see Figures ). Similarly, all the models with no arbitrage constraints are rejected by the data, and the more constrained models are always rejected. The levels of the yields are predicted with an R-squared above 97% for all models, which is not surprising given the relatively small term premia and the high persistence in all rates. It is really when we look at the fit of the spreads themselves that we can differentiate the models: the least restrictive specification explains 6% of the 9-28 day Peso spread and 97% of the day Peso spread. On the other side, the most restrictive model (no arbitrage constraints with zero risk prices=peh) has a negative figure for the r-squared of the 9-28 day spread and only explains 4.65% of the day spread. 27

28 Table 5 Likelihood Ratio (LR) Test Statistics Restrictions Value H: True model has no restrictions (3 factors) H: True model has no restrictions (2 factors) ** H: Time-varying RP ** H: Constant RP ** H: Zero RP ** H: True model has no restrictions (2 factors) H: Time-varying RP ** H: Constant RP ** H: Zero RP ** H: True model has time-varynig RP H: Constant RP ** H: Zero RP ** H: True model has nonzero RP H: Zero RP 4.8 ** * Rejects H with 9% confidence ** Rejects H with 95% confidence Table 6 Fit of the Models (R-squared, %) Unrestricted Unrestricted Time-Varying Constant Zero Three Factor Two Factor Risk Prices Risk Prices Risk Prices Levels EDM ED3M ED6M PAG PAG PAG Spreads ED3M - EDM ED6M - EDM PAG9 - PAG PAG82 - PAG

29 /3/2 /3/2 /3/2 Figures Fit (Unrestricted, Two Factors) Interest Rates (levels)* Figure 28 Fit (Unrestricted) 9 day Eurodollars Figure 27 Figure 29 Fit (Unrestricted) 3 day Eurodollars Fit (Unrestricted) 8 day Eurodollars /3/2 /3/2 /3/2 /3/2 /3/2 Figure 3 Figure 3 Figure 32 Fit (Unrestricted, Three Factors) Fit (Unrestricted, Three Factors) Fit (Unrestricted, Three Factors) 28 day Pesos 9 day Pesos 82 day Pesos /3/2 /3/2 /3/2 /3/2 /3/2 Figure 33 Fit (Unrestricted, Two Factors) 28 day Pesos Figure 34 Fit (Unrestricted, Two Factors) 9 day Pesos Figure 35 Fit (Unrestricted, Two Factors) 82 day Pesos /3/2 /3/2 /3/2 /3/2 /3/2 * Dark lines represent actual values, light lines represent fitted values

30 Figures 36-4 Fit (Unrestricted) Interest Spreads* Figure 36 Fit (Unrestricted) 9-3 day spread, Eurodollars Figure 37 Fit (Unrestricted) 8-3 day spread, Eurodollars /3/2 /3/ /3/2 /3/ Figure 38 Figure 39 Fit (Unrestricted, Three Factors) Fit (Unrestricted, Three Factors) 9-28 day spread, Pesos day spread, Pesos /3/2 /3/ /3/2 /3/ Figure 4 Fit (Unrestricted, Two Factors) 9-28 day spread, Pesos Figure 4 Fit (Unrestricted, Two Factors) day spread, Pesos /3/2 /3/ /3/2 /3/2 -.5 * Dark lines represent actual values, light lines represent fitted values -3

31 Figures Fit (Restricted) Interest Spreads* I. Variable Risk Prices Figure 42 Fit (Restricted) 9-28 day spread, Pesos Figure 43 Fit (Restricted) day spread, Pesos /3/2 /3/ /3/2 /3/ II. Fixed Risk Prices Figure 44 Figure 45 Fit (Restricted) Fit (Restricted) 9-28 day spread, Pesos day spread, Pesos /3/2 /3/ /3/2 /3/ II. Zero Risk Prices Figure 46 Figure 47 Fit (Restricted) Fit (Restricted) 9-28 day spread, Pesos day spread, Pesos /3/2 /3/ /3/2 /3/ * Dark lines represent actual values, light lines represent fitted values

32 Table 7 shows the prices of risk obtained from the restricted estimations, The sign of the estimated price of risk in the fixed case is consistent with the average positive slope of the yield curve. When the prices of risk depend on the latent factors, the ED has a particularly strong effect on the price of risk related to the Peso factor. A positive shock in both factors increases the slope of the Peso yield curve, although they have opposite effects on the level. Table 7 Prices of Risk Time-Varying Fixed Zero Parameter Risk Prices Risk Prices Risk Prices λ ** (3.5689) (3.436) λ ** (.28) λ ** (.5499) * Significant with 9% confidence ** Significant with 95% confidence Table 8 shows the correlations between the latent factors (F, F2, F3) and the traditional factors (as derived in the data section) for the three factor case and the two factor case. The results confirm, again, that F is positively correlated with the ED level and slope, negatively correlated with the Peso level, and positively correlated with the Peso slope. The Peso level factor (F3 in the three factor estimation, F2 in the two factor estimation) is positively correlated with the Peso level and negatively correlated with the Peso slope and the ED level and slope. Contrary to the results 32

33 from Table 4 and the impulse responses (Figures 2-2), factor F2 has a very similar effect to factor F3 in the three factor case. Table 8 Traditional and latent factor correlations Unconstrained Estimations Three Factor Estimation Two Factor Estimation F F2 F3 F F2 Lat. Factors F.. F F Eurodollars Level Slope Curvature Pesos Level Slope Curvature Table 9 presents the results of linear regressions of the change in the log of the exchange rate on the one month Peso-dollar interest rate differential, on the one month interest rates and on the latent factors obtained from the restricted and unrestricted estimations. There is a negative relationship between the Peso-dollar interest rate differential and next period s Peso/dollar exchange rate. 9 This confirms the Fama (984) result for the Mexican Peso/dollar exchange rate at the daily horizon. When the interest rates of one month instruments in both currencies are used instead 9 This result is robust to the maturity of the instruments and the frequency of the data. 33

34 of the differential (case 2), we can see that the negative correlation is due to the Peso component. This result is confirmed when we replace the interest rates with the factors. Recall that factor F ( the ED factor) increases ED rates and decreases Peso rates. In all cases, factor F causes next period s Peso/dollar rate to appreciate. Factors F2 and F3 (the Peso factors) cause the Peso interest rates to increase and next period s Peso/dollar exchange rate to appreciate, which is opposite to the predictions of Uncovered Interest Parity (UIP). The explanatory power in all cases is extremely low (under %), using the factors as explanatory variables significantly improves the fit of the log change in the Peso/dollar exchange rate. 34

35 Table 9 Regression Results Dependent Variable: Change in the log of the Exchange Rate Eq. Constant PAG28(-)-EDM(-) PAG28(-) EDM(-) F3(-) F2(-) F(-) R- Squared (%) (7.328) (.5747) ( ) (.685) (9.657) Unrestricted: Three Factors ** **.9628 (7.549) (3.22) (2.7686) (9.3239) Unrestricted: Two Factors ** (5.662) (.3283) (8.798) Variable Risk Prices ** (5.7293) (.3266) (8.93) Fixed Risk Prices ** (6.66) (.3339) (8.2326) Zero Risk Prices ** (5.7947) (.32) (8.235) * Significant with 9% confidence ** Significant with 95% confidence

36 Conclusions The yields of Mexican Peso-denominated Negotiable Certificates of Deposit (NCD s) are estimated as a function of local latent factors and a foreign factor extracted from Eurodollar yields and exogenous, observable factors: the Mexican monthly inflation and global economic index growth rates and the Fed target. I found that the level of Mexican rates in the medium to long run is mostly explained by the domestic rate of inflation, which can be used as a first observable level factor. The Peso latent factors, extracted with a Kalman filter procedure, capture the high frequency dynamics of the level and the slope of Mexico s yield curve. I found that two latent factors produce a superior fit of the term spreads of Mexican NCD s, and that models based on No Arbitrage constraints derived in the affine factor framework of Vasicek (977) are always rejected by the data. The Peso/dollar exchange rate is negatively correlated with the lag of the Peso-dollar interest differential. This effect is caused by swings in the factor extracted from Peso rates, which causes the next period s Peso/dollar exchange rate to appreciate when Peso interest rates increase. The ED and Peso latent factors explain next period s exchange rate significantly better than the one month interest rates. 36

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