A Research Examination of Covered-Uncovered Interest Rate Parity and the Purchase Power Parity (PPP) hypothesis: Applications in MATLAB, RATS and EVIEWS

CONTENTS

  PART 1  2

  COVERED AND UNCOVERED INEREST RATE PARITY  2

Introduction..........................................................................................................................   2

  Literarute review  2

  Data.  6

  Summary Statistics.  6

  Random Walk  7

  Unit Root and Stationary Tests  9

  Covered Interest Rate Parity  11

1.  Linear Tests  11

2.  Non Linear Tests  15

2.1.  Threshold Autoregressive (TAR)Model  15

2.2.  Smoothing Transition Autoregressive (STAR) Models  18

  Uncovered Interest Rate Parity.  22

Vector  Error-Equilibrium Correction Model (VECM)  25

Impulse  Responses  31

  Threshold Vector error correction model.  33

  Dynamic OLS (DOLS)  34

  Conclusions.  36

  PART 2 .  37

  PURCHASING POWER PARITY HYPOTHESIS.  37

Introduction.   37

  Literature Review.  38

  Data  43

  Summary Statistics  43

  Unit Root and Stationary Tests  46

  Power Purchasing Parity tests  52

1.  Linear Tests  52

2.  Cointegration Tests  65

3.  Panel Unit Root Tests  72

4.  Long span Tests  76

5.  Non-Linear Tests  78

5.1.  Threshold Autoregressive Model (TAR)  78

5.2.  Smoothing Transition Autoregressive Models (STAR)  82

5.3  Markov two-regime switching model.  89

  Conclusions.  91

  References.  92

Appendix.   96

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PART 1

COVERED AND UNCOVERED INTEREST RATE PARITY

Introduction

In this part we examine the covered and uncovered interest parity between US dollar and Swiss Franc. First we present simple summary statistics for the spot and forward rates, as the mean, kurtosis, skewness and standard deviation. We examine if the spot returns follow random walk with and without drift. Then we apply unit root tests for the above series, to test about stationarity, as we apply unit root in the returns of spot and forward rates. In next section we present the covered interest parity hypothesis and we apply some tests to examine its validation, as deviations from covered interest parity, regression analysis, threshold autoregression and exponential transition autoregression. Then we present the uncovered interest parity and, as in the case of covered interest parity, we apply some tests to examine if it’s valid. We apply Johansen cointegration tests between spot and forward rates, but also between forward premia and interest rates differentials and we test if there is a cointegration equation and we estimate the vector error correction model. After this procedure we present the impulse responses. Next we test if there is a threshold cointegration relation between the above variables. Finally in the last section we apply a dynamic OLS (DOLS) estimation with Newey-West HAC standard errors.

Literature review

Fama ( 1984) obtained spot exchange rates and one-monthly forward rates for nine major currencies, including Belgian Franc (BFR), Canadian Dollar (CAD), French Frank (FFR), Italian Lira (ITL), the Japanese Yen (JPY), Swiss Frank (SFR), British Pound (GBP), Netherlands Guilder (NGL) and German Mark (DEM). He finds that according to standard deviation the current spot rate is a better predictor of the future spot rate than the current forward rate. Also he finds that autocorrelations of changes in spot rates are close to zero. The same situation is observed for the premium, but not for the forecast error, which is F t - S t , where Fama found a first-order autoregressive process. Regressing F t - S t+1 on F t - S t Fama finds that β 1 is positive in contrary to the negative β 2 which is generated by S t+1 regressing on F t - S t.

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Fama notes that the negativity of β is owned to the covariation between the risk premium and the expected rate of depreciation. McMillan (2005) tests the forward exchange rate unbiasedness hypothesis (FRUH) applying a panel cointegration test between five major currencies - the Canadian Dollar, French Franc, German Deutschemark, Japanese Yen, and UK Sterling - relative to the US Dollar and he finds that, overall, results support cointegration but not the FRUH. Hai et al. (1997) using the Kalman filter and obtaining maximum likelihood estimates of trend-VARMA(1,1) for log spot and forward dollar prices of the pound, the franc, and the yen they found that log spot and forward exchange rates appear to be cointegrated with cointegration vector (1, -1). Finally they generate simulation distributions of the slope-coefficient estimators and their asymptotic t-ratios and they found that the vector of asymptotic t-ratios, which was drawn from the empirical null distribution is rejected at the 10% level only for the yen. Murfin and Ormerod (1983) examine the relationship between spot and forward rates and the market efficiency using weekly data on the New York foreign exchange market for 1973-1980. The null hypothesis that the forward exchange rate is a unbiased estimate of the future spot exchange rate is rejected for six out of sixteen currencies against the US dollar using OLS estimation or with other words they examinee the restrictions that α=0 and β=1. But when they apply Zellner seemingly unrelated estimations, the efficiency of the estimations are significantly improved and the above two restrictions are rejected for twelve currencies. Goodhart et al. (1997) apply the fully modified OLS procedure on monthly spot and one-month forward exchange rates for the Canadian dollar, French franc, Deutschmark, British pound, Swiss franc and Japanese yen, during January

1975 to December 1987. Wald-statistic for testing the hypothesis that α=0 is

significant at the 1% level for Germany, Britain and Switzerland and at 5% level for France and Japan, but was not found significant for Canada. For the hypothesis that β=1, Wald-statistic is significantly different from unity for the British pound and the Swiss franc at the 1% level and for the French franc, the deutschmark and the Japanese yen at the 5% and not significant for the Canadian dollar. However using the t-statistic β is significantly different than unity at 10% level. So according to these results the unbiasedness hypothesis is rejected for all currencies and marginally rejected for the Canadian dollar. Klein (1990) examines for the case of West Germany the assumption that if the forward rates implicit in the term structure are market expectations of the future spot rates, then forward rates should follow a martingale

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process. He tests the weak and the semi-strong forms for the period January 1975 to December 1986. For the whole period the forward rates are not martingales, but for the period January 1981 to December 1986 are in weak-form efficient. For the semistrong test results are mixed and the semi-strong efficiency was most strongly rejected for the period January 1981 to December 1986.

Sachsida et al. (2001) present the uncovered interest parity (UIP) for Brazil during period January 1984 to October 1998. They consider that expected alterations in the exchange rate should be equal to the interest rate differentials. So they estimate the equation e

They found that UIP hypothesis was accepted only for period January 1990 through June 1994. Carvalho et al. (2004) estimate the same model the uncovered interest parity (UIP) in Argentina, Brazil, Chile and Mexico with monthly data during the period January 1990 through December 2001. They reject the UIP hypothesis for the group of the four countries but they don’t reject the validity of UIP for the group of Argentina, Brazil and Chile for the sub-periods of January 1991 to December 2000, and January 1991 to December 1995 and so they accept the hypothesis that β=1. Rowland (2005) tests UIP for the rate of exchange of the Colombian peso to the US dollar using weekly data for the period January 1994 to August 2002. Between January 1994 and September 1999, exchange rate movements were restricted by a crawling band. In September 1999, the crawling band was dismantled and the exchange rate was allowed to float freely. The UIP hypothesis is tested using 3, 6 and

12-month interest rates. Rowland regress S t+k - S t on r t - r * t , where r t is the domestic interest rate and r * t is the foreign interest rate and the same regression including sovereign spreads using ordinary least squares. The validity if UIP relationship increases with the term of the investment, while the results of the three-month are weak , the UIP hypothesis receive significant support from the twelve-month results. Rowland considers that the main reason why the UIP relationship holds during this period might be the significant macroeconomic transition at this time which Columbia underwent. In Froot’s (1990) paper, 75 published studies are surveyed whose the large majority reject the hypothesis that slope parameter β in the equation should equal unity. Chaboud and Wright (2003) examine the UIP hypothesis for four currencies,

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Japanese Yen, German Mark/Euro, Swiss Franc and Pound Sterling relative to the US Dollar and the obtain 5-minutes data from 1988 to 2002. In their regressions using daily data, the slope coefficients or equivalently β’s range between -3.45 and -4.86 except for the Pound where β is equal with -1.07. Using the same regression , but with

5-minutes data from 16:30 to 21:00 New York Time the slope coefficients for Swiss

Franc, Mark/Euro, Pound Sterling and Yen were 0.79, 1.02, 1.44 and -1.00 respectively. So they reject the UIP hypothesis for three out of four currencies. Turnovsky and Ball (1983) test the covered interest parity and they use quarterly and monthly data of Australian and US three-month interest rates, the Australian and US exchange rate and the corresponding three-month forward exchange rate between Australian and US dollars. The chosen period is September 1974 to December 1981. They found that the CIP hypothesis is accepted at the 5% level for the quarterly data, but not for the monthly data. Also they estimate an additional regression where the lagged forward rate F t, t-3 is included as additional repressor. This is the speculative efficiency hypothesis, which is accepted at the 1% level and not at 5% for the quarterly data, while is accepted at 5% level for the monthly data. Balke and Wohar (1998) test the CIP using threshold autoregression model with OLS and the nonlinear TAR/TARCH model. They consider that the presence of the transaction cost implies that deviations from the covered interest parity might be modeled as threshold autoregression. They use daily data for USA and UK from January 1974 to

September 1983. They confirm that market imperfections such the transaction costs and the standard deviations from CIP ignoring the neutral zone are statistically significant. So the profit opportunities from covered interest parity are not only small, but they have short duration. Finally using and the impulse response function implied by the threshold autoregression, they found that deviations from covered interest parity that are outside the transaction costs band are less persistent than those inside the band. Kui-yin and PO-ming (1994) used daily data of spot and forward swap US dollars rates versus Hong Kong dollars as well as Hong Kong dollar and US dollar interbank bid and offered rates with maturity of one , three and six months. They calculate the one-way and two-way arbitrage according to specific inequalities and they found there are few unexploited profit opportunities for the covered interest arbitrage. A great deal of profit opportunities was persisted for the one-way arbitrage. Also they found that larger profit returns exist for the long term contracts which had lower swap costs between the swap market and the Hong Kong and the US dollar

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interbank market. Cody (1990) used daily data on French franc versus five currencies- Deutschemark, Japanese yen , pound sterling , Swiss franc and U.S. dollar

- and uses the following model to test the CIP hypothesis.

^{* *} (F t,3 -S t )/S t = α + β (i t - i t )/ (1- i t ) + ε t

and he tests the hypothesis if α=0 and β=1 . Using OLS estimation and chi-square statistic the covered interest parity hypothesis is rejected for three out of four currencies. Tol and Wolff (2005) apply a multivariate threshold VECM of spot and forward exchange rates that allows for different forms of equilibrium reversion in each of the cointegrating residual series and they found the superiority of multivariate VECM against the linear VECM.

Data

We use monthly data in our estimations. We estimate for all models after 1973, where the Bretton Woods system collapsed and a system of floating currencies was established after that period. Also depending on the availability of all data, as the spot rates, forward rates and the domestic and foreign interest rates, the chosen period is from January 1982 through August 2008.

The data for our analysis have been obtained by website www.econstat.com and the website of the National Bank of Switzerland, which is www.snb.ch. We obtained

3-monthly and 6- monthly forward rates and the respective monthly maturity of

treasury bills interest rates. The domestic country is Switzerland and the foreign country is U.S.A.

Summary statistics

Descriptive statistics for the spot and three-monthly and six-monthly forward exchange rates returns are reported in table 1. We observe that in all cases negative mean returns are observed, but one might say that are very close to zero. Also in both three rates returns negative skewness is presented, but kurtosis is very close to three, as is defined by the normal distribution. Based on the Jarque-Bera statistics the hypothesis of normality for spot and forward exchange rates is not rejected. So from

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these results we consider that OLS estimation could be the appropriate method for, so GARCH models are not necessary as they could be if we had obtained daily data.

Table 1. Descriptive statistics

Deviation Bera

Spot returns -0.00273 -0.00185 0.08853 -0.08398 0.02925 -0.17451 3.25455 3.226 0.199 Three-monthly forward -0.00274 -0.00138 0.08991 -0.08338 0.02914 -0.17061 3.26602 3.237 0.198 returns

Six-monthly

forward -0.00274 -0.00074 0.09018 -0.08362 0.02904 -0.17294 3.24468 3.104 0.211 returns

Random walk

In this part we examine if spot rates S t follows random walk or not. We estimate the following regression and we test about autocorrelation and heteroskedasticity in residuals. S t = μ + e t (1)

, where S t are spot rates returns, which are defined as S t = log(s t )-log(s t-1 ), μ is constant and e t express the residuals. The estimation results of equation (1) are presented in table 2. In table 3 we present the correlogram of squared residuals .According on these tables there is no autocorrelation. In table 4 we present the ARCH-LM test results and we conclude that there is no heteroskedasticity. We decided to chose two lags for the ARCH-LM test.

Table 2. OLS estimation of equation (1)

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Table 3. Correlogram of squared residuals for equation (1)

AC 0.006 0.048 0.077 -0.029 0.004 0.051 0.028 0.077 0.052 0.051 0.025 -0.051 PAC 0.006 0.048 0.077 -0.032 -0.003 0.048 0.032 0.072 0.042 0.043 0.012 -0.061 Q-Stat 0.0160 0.9957 35.049 38.476 38.551 49.494 52.792 78.068 89.525 10.048 10.323 11.450 Prob 0.899 0.608 0.320 0.427 0.570 0.550 0.626 0.453 0.442 0.436 0.502 0.491 LAGS 13 14 15 16 17 18 19 20 21 22 23 24 AC -0.048 -0.044 0.001 0.069 -0.030 0.036 0.042 0.027 -0.021 -0.013 -0.029 -0.094 PAC -0.058 -0.048 0.006 0.068 -0.038 0.022 0.037 0.046 -0.017 -0.012 -0.021 -0.100 Q-Stat 12.443 13.289 13.290 15.366 15.756 16.307 17.074 17.386 17.575 17.655 18.039 21.987 Prob 0.492 0.504 0.580 0.498 0.541 0.571 0.585 0.628 0.676 0.726 0.755 0.580

Next we examine the random walk of the following form

S t = μ + S t-1 + e t (2)

Based on the table 6 there is autocorrelation for the first eight lags, but not for the rest

of them. According to table 6 we accept the null hypothesis of no autocorrelation.

Also from table 7 we conclude that there isn’t heteroskedasticity. The last model we

examine is the following random walk with no constant.

S t = S t-1 + e t (3)

Table 5. OLS estimation of equation (2)

Variable Coefficient t-statistic Probability

Table 6. Correlogram of squared residuals for equation (2)

LAGS 1 2 3 4 5 6 7 8 9 10 11 12

AC -0.051 0.044 0.065 -0.041 0.061 0.069 0.015 0.006 0.039 0.059 -0.003 -0.033 PAC -0.051 0.041 0.070 -0.036 0.052 0.074 0.023 -0.008 0.033 0.064 -0.007 -0.052 Q-Stat 10.790 18.753 36.678 43.569 59.406 79.241 80.237 80.374 86.774 10.144 10.149 10.603 Prob 0.299 0.392 0.300 0.360 0.312 0.244 0.331 0.430 0.468 0.428 0.517 0.563 LAGS 13 14 15 16 17 18 19 20 21 22 23 24 AC -0.033 -0.029 0.040 0.067 -0.025 0.022 0.022 0.020 0.048 -0.018 -0.035 -0.079 PAC -0.045 -0.029 0.034 0.068 -0.017 0.016 0.027 0.028 0.037 -0.018 -0.039 -0.095 Q-Stat 11.074 11.437 12.140 14.110 14.372 14.582 14.785 14.957 15.949 16.089 16.638 19.421

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The results for equation (3) are exactly the same with that of model (2), so we

conclude that there isn’t neither autocorrelation or heteroskedasticity in the

residuals, so the general conclusion from the above results is that S t follows the

random walk.

Table 8. OLS estimation of equation (1)

Variable Coefficient t-statistic Probability

^S t-1 0.306604 6.551 0.00000

* statistical significant in α=0.01

Table 9. Correlogram of squared residuals for equation (2)

LAGS 1 2 3 4 5 6 7 8 9 10 11 12 AC -0.051 0.044 0.065 -0.041 0.061 0.069 0.015 0.006 0.039 0.059 -0.003 -0.033 PAC -0.051 0.041 0.070 -0.036 0.052 0.074 0.023 -0.008 0.033 0.064 -0.007 -0.052 Q-Stat 10.790 18.753 36.678 43.569 59.406 79.241 80.237 80.374 86.774 10.144 10.149 10.603 Prob 0.299 0.392 0.300 0.360 0.312 0.244 0.331 0.430 0.468 0.428 0.517 0.563 LAGS 13 14 15 16 17 18 19 20 21 22 23 24 AC -0.033 -0.029 0.040 0.067 -0.025 0.022 0.022 0.020 0.048 -0.018 -0.035 -0.079 PAC -0.045 -0.029 0.034 0.068 -0.017 0.016 0.027 0.028 0.037 -0.018 -0.039 -0.095 Q-Stat 11.074 11.437 12.140 14.110 14.372 14.582 14.785 14.957 15.949 16.089 16.638 19.421

Unit root and stationary tests

We examine if spot rates are stationary in levels or in differences as we examine if

the returns of the above variable is stationary. We apply three unit root tests,

Augmented Dickey-Fuller, Phillips-Perron and KPSS tests. The simplest version is

the random walk with drift as the equation (2) and without drift as equation (3). We

test equation (2), but we include also the trend. So the model becomes the following:

s t = α + δt + φs t-1 + ε t (4)

and the hypotheses we test are:

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H 0 : φ=1, δ=0 => s t ~ Ι(0) with drift against the alternative H 1 : |φ|<1 => s t ~ Ι(1) with deterministic time trend

The problem with Dickey-Fuller test is that is valid when time series, and in our case variable s t, is characterized by an AR(1) process with white noise error. So s t might has a more complicate dynamic structure to be captured by a simple AR(1). The Phillips-Perron test is: Δs t = α΄D t + φs t-1 + ε t (5)

, where ε t ~I(0) and we test the null hypothesis

H 0 : φ=1 , time series is not stationary. The KPSS statistic is given by

, where

And we test the hypotheses

H 0 : time series is stationary against the alternative H 1 : time series is not stationary

Table 11. Unit root tests: Spot and forward rates

^s t ^{in levels -2.979998 -2.948672 0.165047 -14.87725 -14.66071 0.081483 s} t ^{in first differences f} t 3-monthly in -3.003810 -3.030816 0.156224 levels -3.98 -3.98 0.216

^{1% level 1% level 1% level f} t 3-monthly in -3.42 -3.42 0.146 first -14.8520 5% level -14.67625 5% level 0.78938 5% level -3.13 -3.13 0.119 differences

^{10% level 10% level 10% level -3.031696 -3.064812 0.149161 f} t ^{6-monthly in levels f} t 6-monthly in first -14.78465 -14.61649 0.77420 differences

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We must notice that we obtain logarithms of spot rates, so variables s t and f t are in

logarithm values. We conclude, from table 11, that spot exchange rates are not

stationary in levels but they are in first differences, so variable S t is I(1) as 3-monthly

and 6-monthly forward rates are I(1) too. Also from table 12 we conclude that interest

rates i t and i t * are I(1) for 3 and 6 months of maturity.

Table 12.Unit root tests: Domestic and foreign interest rates with maturity of 3 and 6 months

Variables Test Critical values Test Critical values Test Critical values ADF PP KPSS t-statistic t-statistic t-statistic

^{-2.680624 -3.190921 3-monthly i} t in 0.182528

levels

^{3-monthly i} t in

first -17.36022 -17.68232 0.077427 differences

^{-2.632137 -2.943607 6-monthly i} t in 0.167161

levels

^{6-monthly i} t in -15.93596 -16.27941 first 0.066691

differences

^{-3.787258 -3.13 -3.246644 -3.13 0.118727 0.119 3-monthly i} t*

10% level in levels 10% level 10% level

^{-13.00650 -13.94888 3-monthly i} t* in 0.051335

first differences

^{-3.987898 -3.374537 6-monthly i} t* 0.131189

in levels

^{6-monthly i} t* in

first -12.61888 -13.59231 0.055167 differences

Covered interest parity

1. Linear Tests

In this section we apply various tests to confirm the validity of the covered interest

parity (CIP). The first test we apply is the following

s e t+k - s t = f k t - s t => s e t+k = f k (7) t

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The next step is to obtain the difference between s e t+k - f k t =0 and apply t-statistic to test the hypotheses:

H 0 : μ=0 against the alternative H 1 : μ≠0 , where μ is the mean. We apply the relation (7) for forward rates of three and six months, so s e t+k , will be s e t+3 and s e t+6 respectively. The expected spot rate differs from current primarily by the extent to which inflation expectations in the two currencies differ. The expected spot rates can be calculated as: ^* Expected spot rates = (i t / i t )*spot rates (8)

The results are presented in table 12 and we reject the null hypothesis , which means that relation (7) is not valid, so we reject the covered interest parity. Also in figure 1.a and 1.b we present the difference s e t+k - f k t with the zero line for k=3 and k=6 respectively.

The second test we apply is the following regression:

f k t - s t = α + β (i t -i t *) + u t (9)

, where i t is the interest rate in the domestic country, in our case Switzerland, and i t * is the interest rate in the foreign country, which in our case is U.S.A. We apply equation (9) for both forward rates of three and six month and interest rates of 3 and 6 months maturity respectively. We expect to obtain α=0 and β=1, and these

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restrictions are requirements for the CIP hypothesis to be valid. In figure 2 we present

the interest rates for both countries.

Figure 2. ^{a) i} t, ⁱ t ^{* with three months maturity and b) and i} t, ⁱ t * with six months maturity

In table 14 we present the OLS estimation results for equation (9) obtaining the

forward rates of 3-month and 6-month as the interest rates with 3 and 6 months

maturity. The intercept is statistically insignificant for 6 month interest rates, but is

significant and negative in the case of 3-monthly maturity interest rate.

Table 14. OLS estimation results of equation (8) with f k t ^{for k=3,6 and i} t, ⁱ t * with 3 and 6 months

maturity

f ^{k 1.382 -0.049 Lag 1} t ^{for k=3 and i} t, ⁱ t *

with three months

maturity

f ^{k Lag 1} t ^{for k=6 and i} t, ⁱ t *

-1.393 0.006 F-statistic = 4223.298 with six months

(-43.878)* (0.355) Probability= 0.0000 maturity

We observe that in the estimation for k=3 β is statistically significant but not equal

with unity, as for k=6 is significant but has a negative sign. Finally we apply ARCH-

LM test to test for heteroskedasticity and in all cases we reject null hypothesis, so we

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have ARCH effects. We could apply also and asymmetric GARCH models as TGARCH and EGARCH but the coefficient indicating the leverage effects, is not statistically significant in both models. This is happening and in the following regressions that we estimate in the next parts of the paper. We don’t show the results of the asymmetric GARCH models.

GARCH models characterize the conditional distribution of ε t by imposing serial dependence on the conditional variance of the innovations.GARCH (1,1) , which process is much resemblance to the general ARMA process but it permits a more parsimonious description in many situations (Bollerslev, 1986). The general GARCH (p,q) model is

In table 15 we present the GARCH (1,1) estimations of equation (9) as in the case of OLS. GARCH(1) is statistically significant only in the case with k=3 and threemonthly maturity of interest rates.

Table 15. GARCH (1,1) estimation results of equation (8) with f k t ^{for k=3,6 and i} t, ⁱ t * with 3 and 6

months maturity

variance equation

f k t ^{for k=3 and i} t,

^{(3.465)* i} t * with three

months maturity

f ^{k 2.81e-0.5 1.320 -0.0768 -0.0039 -1.228} t ^{for k=6 and i t, (2.585)** (6.540)* (-2.765)** (-6.087)* (-659.34)* i} t * with six months maturity

So the main conclusion is the we reject the covered interest rate parity with the specific tests, as we found a negative sign of parameter β, in the case of k=6, while we find the same parameter statistically significant different from zero. In figure 3 we present the OLS recursive β coefficients for equation (9) for k=3 ,6, where we confirm the rejection of covered interest parity, that parameter β it’s not stable through the time period we examine. In the next part we apply non-linear tests to examine if the

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covered interest parity can be characterized and interpreted by non-linear time-series models.

Figure 3. OLS recursive estimates of β’s of equation (8) for a) k=3 and b) k=6

2. Non-linear Tests

2.1 Threshold Autoregressive (TAR) models

The first non-linear model we estimate is the threshold autoregressive model. We obtain the variable h t = f k t - s t - (i t -i t *) and we apply an AR(1)-TAR and one threshold is estimated using a grid-search procedure based on the quantiles of the distribution of h t . We estimate TAR models in winRATS 6.0 (Enders, 2003). A two-regime AR(1) model is defined as:

, where α and β are the coefficients, i is the order of TAR, r is the threshold value and It is the Heaviside indicator function

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First we should apply a simple test to examine if h t is non-linear or not, based on a test proposed by Terasvirta, Lin and Granger (1993), We estimate the following regression

β β + = h h

o t 1

So for p=1 and d=1 will be for example

γ γ γ β β ^{3 2} + + + + = h h h h h h h h − − − − − − − ^{1 1 31 1 1 21 1 1 11 1 1} t t t t t t t o t (13)

So we estimate relation (12) and we compare the F-statistics with the F-values from critical tables, where we examine the following hypotheses. H 0 : No TAR non-linearity against the H 1 : TAR non-linearity. The results are presented in table 16.

Table 16. Non-linearity test results

Constant

-0.008296

(-0.966215) F-statistic 2493.878

t-statistics in parentheses, * statistical significant in α=0.05, * *statistical significant in α=0.10

Even if F-statistic is much higher then the critical F- critical value the coefficients γ 11 and γ 21 are statistically insignificant so there is a weak evidence of TAR non-linearity. We estimate two TAR models one for 3 and one for 6 months. The programming routine procedure is presented in appendix and the estimation results in table 17 and in figure 4 the residuals sum of squares and the threshold value are presented.

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Table 17. AR(1)-TAR estimation results for k=3

^h t ^{< r h} t ≥r

^α 0 ^α 1 ^β 0 ^β 1

-0.0882 0.1476 -1.856 -1.7144 0.170936

(0.1773) (0.250) (0.7302) (0.7343)

[-0.49783] [0.59041] [-2.541]** [-2.334]**

Figure 4. Residuals sums of squares and therhold for TAR estimation and for k=3

Threshold

From table 18 we observe that the coefficients in the first regime are statistically

insignificant, so h t follows a random walk. In the second regime both coefficients are

significant. The estimated threshold coefficient is r = -0.21425. In table 18 we present

the AR(1) results for k=6 and we have the same conclusion as previous.

Table 18. AR(1)-TAR estimation results for k=6

^h t ^{< r h} t ≥r

^α 0 ^α 1 ^β 0 ^β 1

-0.1073 0.2398 -1.6916 -1.681 0.130572

(0.1763) (0.3422) (0.7272) (0.8023)

[-0.6090] [0.7006] [-2.3259]** [-2.0952]**

Figure 5 presents once again the residuals sum of squares and the threshold value

which is r = -0.19303.

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Figure 5. Residuals sums of squares and therhold for TAR estimation and for k=6

Threshold

Generally we observe that the coefficients in the nonlinear regime are statistically significant, while in the linear regime aren’t significant. This means that the specific time-series we examine can be interpreted by a Threshold Autoregressive model. So in both cases, where k=3 and k=6, we observe that h t follows a random walk process as also we reject the hypothesis that r=0, confirming the results of those in the study of Peel and Taylor (2002), but our results are not consistent with those of the same study, where the authors accept that r=0.05, so we reject the Keynes-Einzig conjecture hypothesis. So we conclude that there is an arbitrage opportunity for the case we study.

2.2 Smoothing Transition Autoregressive (STAR) models

The next non-linear tests for the covered interest parity test include the smoothing transition autoregressive (STAR) models and we consider two STAR functions the logistic and the exponential, which are defined by the relations (14.a) and (14.b) respectively.

, where γ>0.

So LSTAR and ESTAR models are:

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p p − = h

1 ( t

, where ) ( d h I − can be either LSTAR or ESTAR function.

t

In order to specify the d we estimate (16) for various values of d and p we choose d=1,…6 and for p=1…3 For h t with both interest rates of three and six months maturity we choose p=3 , where the minimum P-value is obtained.

If we reject the linearity we test the following sequences of hypotheses (Terasvirta and Anderson, 1992)

If we reject the (17) hypothesis then we choose LSTAR model. If (17) is accepted and (18) is rejected than ESTAR model is selected. Finally accepting (17) and (18) and rejecting (19) we choose LSTAR model. In tables 19 and 20 we present the linearity tests for k=3 and k=6 respectively. In both cases we reject linearity hypothesis for d=5 and in α=0.05, while in the first case, where k=3, we reject linearity test also for d=4 , but at α=0.10 statistically significance level, while also Pvalue is higher, so as we mentioned previously we choose these of values of d, where the minimum P-value is observed. In table 21 we present the results of the specification test between ESTAR and LSTAR, where in the both cases for k=3,6 we reject hypothesis (17) so consequently we accept LSTAR model.

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Table 21. ^{F-statistics Specification of the nonlinear model} Delay F-statistic F-statistic F-statistic Type of

(d)

5

5

P-values in parentheses

We applied the grid search with both Gauss-Newton and Levenberg-Marquarft algorithms and nonlinear lest squares and in both cases and for both k=3,6 the values of parameters γ and c are 3.26 and 1.95 respectively. Also we choose p=3, because for this value the minimum P-value is obtained. In tables 22 and 23 we present the results of AR(3)-LSTAR-OLS for k=3 and k=6 respectively. We observe that in both cases the R 2 adj. is high, as we reject the hypothesis of autocorrelation and ARCH effects, based on LBQ and ARCH-LM values. Also we observe that the F-statistic is very high in both cases but the t-statistics are very low, except from coefficient α 0 , concluding that all the remained coefficients are statistically insignificant, LSTAR models probably doesn’t not explain efficient the covered interest parity.

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Table 22. AR(3)-LSTAR OLS estimation results for k=3

α 1 α 2 α 3 α 0

Coefficients in linear (middle)regime

β 0 β 1 β 2 β 3

Coefficients in nonlinear (outer) regime

*denotes significance in 0.01 level , standard errors in parentheses, t-statistics in brackets, p-values in {} AIC and SBC refer respectively to Akaike and Schwarz information criteria,, LL is the Log Likelihood, LBQ ² is the Ljung-Box test on squared standardized residuals with 12 lags, ARCH-LM denotes Lagrange Multiplier test for ARCH effects with 5 lags.

Table 23. AR(3)-LSTAR OLS estimation results for k=6

α 0 α 1 α 2 α 3

Coefficients in linear (middle)regime

β 0 β 1 β 2 β 3

Coefficients in nonlinear (outer) regime

*denotes significance in 0.01 level , standard errors in parentheses, t-statistics in brackets, p-values in {} AIC and SBC refer respectively to Akaike and Schwarz information criteria,, LL is the Log Likelihood, LBQ ² is the Ljung-Box test on squared standardized residuals with 12 lags, ARCH-LM denotes Lagrange Multiplier test for ARCH effects with 5 lags.

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