Excerpt

## CONTENTS

PART 1

COVERED AND UNCOVERED INEREST RATE PARITY

Introduction

Literarute review

Data

Summary Statistics

Random Walk

Unit Root and Stationary Tests

Covered Interest Rate Parity

1. Linear Tests

2. Non Linear Tests

2.1. Threshold Autoregressive (TAR)Model

2.2. Smoothing Transition Autoregressive (STAR) Models

Uncovered Interest Rate Parity

Vector Error-Equilibrium Correction Model (VECM)

Impulse Responses

Threshold Vector error correction model

Dynamic OLS (DOLS)

Conclusions

PART 2

PURCHASING POWER PARITY HYPOTHESIS

Introduction

Literature Review

Data

Summary Statistics

Unit Root and Stationary Tests

Power Purchasing Parity tests

1. Linear Tests

2. Cointegration Tests

3. Panel Unit Root Tests

4. Long span Tests

5. Non- Linear Tests

5.1. Threshold Autoregressive Model (TAR)

5.2. Smoothing Transition Autoregressive Models (STAR)

5.3 Markov two-regime switching model

Conclusions

References

Appendix

## 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 *Ft – St*, where Fama found a first-order autoregressive process. Regressing *Ft – St+1* on *Ft – St* Fama finds that *β 1* is positive in contrary to the negative *β 2* which is generated by *St+1* regressing on *Ft – St*.

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 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 semi-strong 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

illustration not visible in this excerpt

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 *St+k – St* on *rt - r*t*, where *rt* 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, 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 *Ft, 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 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.

(Ft,3-St)/St = α + β (it - it* )/ (1- it* ) + εt

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 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

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## Random walk

In this part we examine if spot rates *St* follows random walk or not. We estimate the following regression and we test about autocorrelation and heteroskedasticity in residuals.

*St = μ + et* (1)

, where *St* are spot rates returns, which are defined as *St = log(st)-log(st-1), μ* is constant and *et* 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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* statistical significant in α=0.05

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

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**Table 4.** ARCH-LM test

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Next we examine the random walk of the following form

*St = μ + St-1 + et* (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.

*St = St-1 + et* (3)

**Table 5.** OLS estimation of equation (2)

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* statistical significant in α=0.01

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

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**Table 7.** ARCH-LM test

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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 *St* follows the random walk.

**Table 8.** OLS estimation of equation (1)

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* statistical significant in α=0.01

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

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**Table 10.** ARCH-LM test

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## 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:

*st = α + δ t + φ st-1 + ε t* (4)

and the hypotheses we test are:

H0: φ=1, δ=0 => st ~ Ι(0) with drift

against the alternative

H1: |φ|<1 => st ~ Ι(1) with deterministic time trend

The problem with Dickey-Fuller test is that is valid when time series, and in our case variable *st*, is characterized by an AR(1) process with white noise error. So *st* might has a more complicate dynamic structure to be captured by a simple AR(1). The Phillips-Perron test is:

*Δ st = α΄ Dt + φ st-1 + ε t* (5)

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

H0: φ=1 , time series is not stationary.

The KPSS statistic is given by

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And we test the hypotheses

H0: time series is stationary against the alternative

H1: time series is not stationary

**Table 11.** Unit root tests: Spot and forward rates

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We must notice that we obtain logarithms of spot rates, so variables *st* and *ft* 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 *St* is *I(1)* as 3-monthly and 6-monthly forward rates are *I(1)* too *.* Also from table 12 we conclude that interest rates *it* and *it** 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

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## 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

*set+k - st = fkt- st => set+k = fkt* (7)

The next step is to obtain the difference between *set+k - fkt =0* and apply *t-statistic* to test the hypotheses:

H0: μ=0 against the alternative H1: μ≠0 , where *μ* is the mean.

We apply the relation (7) for forward rates of three and six months, so *set+k*, will be *set+3* and *set+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 = (it*/ it)*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 *set+k - fkt* with the zero line for *k* =3 and *k* =6 respectively.

**Table 13**. Test hypothesis set+k - fkt =0

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**Figure 1.** Deviations from covered interest parity

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(a) (b)

The second test we apply is the following regression:

*fkt- st = α + β (it-it*) + ut* (9)

, where *it* is the interest rate in the domestic country, in our case Switzerland, and *it** 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 restrictions are requirements for the CIP hypothesis to be valid. In figure 2 we present the interest rates for both countries.

**Figure 2. a)** it, it* with three months maturity and b) and it, it* with six months maturity

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(a) (b)

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 fkt for k=3,6 and it, it* with 3 and 6 months

maturity

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*denotes that it’s statistical significant in α=0.01, ** statistical significant in α=0.05 , t-statistics in parentheses

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 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

illustration not visible in this excerpt

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 three-monthly maturity of interest rates.

**Table 15**. GARCH (1,1) estimation results of equation (8) with fkt for k=3,6 and it, it* with 3 and 6

months maturity

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*denotes that it’s statistical significant in α=0.01, ** statistical significant in α=0.05 , z-statistics in parentheses

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 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

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### 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 ht= fkt- st - (it-it*) 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 *ht.* We estimate TAR models in winRATS 6.0 (Enders, 2003). A two-regime AR(1) model is defined as:

illustration not visible in this excerpt

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

*It* = {[illustration not visible in this excerpt]

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

illustration not visible in this excerpt

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

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So we estimate relation (12) and we compare the *F-statistics* with the *F-values* from critical tables, where we examine the following hypotheses.

H0: No TAR non-linearity against the H1: TAR non-linearity. The results are presented in table 16.

**Table 16**. Non-linearity test results

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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.

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

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*Standard errors in parentheses, t-statistics in brackets

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

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From table 18 we observe that the coefficients in the first regime are statistically insignificant, so *ht* 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

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*Standard errors in parentheses, t-statistics in brackets

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

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.

illustration not visible in this excerpt

, where γ>0.

So LSTAR and ESTAR models are:

illustration not visible in this excerpt

, where [illustration not visible in this excerpt] can be either LSTAR or ESTAR function.

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 *ht* with both interest rates of three and six months maturity we choose *p=3*, where the minimum *P-value* is obtained.

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

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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 *P-value* 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.

**Table 19**. F-statistics Specification of the nonlinear model for interest rates of three

months maturity

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p-values in parenthesis

**Table 20**. F-statistics Specification of the nonlinear model for interest rates of six

months maturity

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p-values in parenthesis

**Table 21**. F-statistics Specification of the nonlinear model

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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 *R2adj.* 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.

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

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*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, LBQ2 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

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*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, LBQ2 is the Ljung-Box test on squared standardized residuals with 12 lags, ARCH-LM denotes Lagrange Multiplier test for ARCH effects with 5 lags.

*Uncovered interest parity*

The first test we apply for the UIP is the following regression

*Δ st+k = α + β (fkt- st) + nt+k* (20)

In table 24 we present the results of equation (20). For both estimations with *k=3, 6*, *β* coefficient hasn’t the expected sign, as we expected to be equal with 1 and we found it equal with -1. Also intercept is statistically significant and negative. In the same table we present the ARCH-LM test results and we conclude that there is heteroscedasticity. Finally in tables 25 and 26 we see the correlogram of squared residuals and equation (20) for *k* =3 and *k* =6 respectively. So from these results we reject the uncovered interest parity.

**Table 24**. OLS estimation results of equation (20) with fkt for k=3,6

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* denotes significance in 0.01 level , t-statistics in parentheses

**Table 25.** Correlogram of squared residuals for equation (20) and fkt for k=3

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**Table 26.** Correlogram of squared residuals for equation (21) and fkt for k=6

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**Table 27**. ARCH (1) estimation results of equation (21) with fkt for k=3,6

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* denotes significance in 0.01 level , z-statistics in parentheses

In table 27 we show the results of the ARCH(1) estimation, while GARCH (1) coefficient isn’t statistically significant. Once again the sign for coefficient *β* has been found negative resulting to reject the uncovered interest parity.

The second test we apply is the following

*st+k = α + β fkt + nt+k* (21)

In table 28 we present the OLS results of equation (21). The situation is very different with that of tables 24 and 27 and equation (20), as in this case *β* has the correct sign Furthermore if we test the hypothesis H0: *β* =1 against the alternative H0: *β* ≠1, with Wald test then we reject the null hypothesis for k=3 and 6. The test hypothesis results are presented in table 29. We conclude that we reject the uncovered interest party hypothesis. Furthermore as we observe in the table 28 and the ARCH-LM test results we conclude that there is heteroskedasticity. Also in tables 30 and 31 we present the correlograms of squared residuals and we conclude that there is autocorrelation in residuals. In table 32 we present the ARCH(1) estimation for equation (21), as GARCH(1), is insignificant. Once again we reject the hypothesis that *β* =1 in both cases, where *k=3* and *k=6.*

**Table 28**. OLS estimation results of equation (17) with fkt for k=3,6

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* denotes significance in 0.01 level , t-statistics in parentheses

**[...]**

- Quote paper
- Eleftherios Giovanis (Author), 2008, A Research Examination of Covered-Uncovered Interest Rate Parity and the Purchase Power Parity (PPP) hypothesis: Applications in MATLAB, RATS and EVIEWS, Munich, GRIN Verlag, https://www.grin.com/document/142520

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