Noise created through the interactions of arbitrageurs and n

Noise created through the interactions of arbitrageurs and noise traders in the FOREX market. An analysis based on ARCH, GARCH, TGARCH and EGARCH models.

Dr Michel Zaki Guirguis 16/07/2016

Bournemouth University1

Institute of Business and LawFern BarrowPoole, BH12 5BB, UKTel:0030-210-9841550Mobile:0030-6982044429E-mail: [email protected]

Biographical notes

I hold a PhD in Finance from Bournemouth University in the U.K. I have worked for several multinational companies including JP Morgan Chase and Interamerican Insurance and Investment Company in Greece. Through seminars, I learned how to manage and select the right mutual funds according to various clients needs. I supported and assisted the team in terms of six-sigma project and accounts reconciliation. Application of six-sigma project in JP Morgan Chase in terms of statistical analysis is important to improve the efficiency of the department. Professor Philip Hardwick and I have published a chapter in a book entitled “International Insurance and Financial Markets: Global Dynamics and Local Contingencies”, edited by Cummins and Venard at Wharton Business School (University of Pennsylvania in US). I am working on several papers that focus on the Financial Services Sector.

1 I have left from Bournemouth University since 2006. The permanent address of the author’s is, 94, Terpsichoris road, Palaio – Faliro, Post Code: 17562, Athens – Greece.

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Abstract

In this article, we have tested the volatility of the returns of the spot exchange rate of EURO/USD, the returns of a real exchange rate index and the money supply,(M1), for changing conditional variances. Autoregressive Conditional Heteroskedastic models (ARCH), Generalized Autoregressive Conditional Heteroskedastic models, (GARCH), Threshold GARCH,(TGARCH) and exponential, (EGARCH) models take into account the non-linearity that arises in financial time series. In this article, we are not testing the regressive relationship between exchange rate volatility and macroeconomic indicators such as the money supply, interest rates, and real GDP. We have checked the volatility clusters for a long period of time that arises in the financial times series of returns or the fact that large and small values occur persistently in clusters. In other words, we have concluded that negative shocks implied a higher next period conditional variance than positive shocks of the same extent. The asymmetry terms are highly statistically significant by using the family of autoregressive models. We have found that there is a significant positive and negative volatility clusters, which confirms the ARCH and GARCH, TGARCH and EGARCH effect on the time series of the EURO/USD spot rate, based on the F-statistic, the Lagrange Multiplier, (LM), the leverage effect, the Jarque – Bera normality tests. The data that we have used are monthly returns starting from 01/01/2000 to 01/01/2013, which total to 156 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbols of the series are H.10 and H.6.

Keywords: ARCH, GARCH, TGARCH, EGARCH models, EURO/USD spot rate, real exchange rate index, money supply, (M1).

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Introduction

This article has focused on modeling the returns of the EURO/USD spot rate. The returns of a real exchange rate index and the money supply,(M1), by using Autoregressive Conditional Heteroskedastic models, Generalized Conditional Heteroskedastic models, TGARCH and EGARCH models. By using E-Views the models will be tested to validate the hypotheses that will be formulated.

Models, which include a changing conditional variance, are called autoregressive conditional heteroskedastic models. Engle (1982) suggested this type of model for estimating time series with conditional heteroskedasticity. Since then, different types of ARCH models appeared in the literature with different specifications such as mean reversion, mean dependency and variance dependency. For example, GARCH models capture all the history of shocks in a series in a changing variance environment, TGARCH and EGARCH models allow for asymmetric shock to volatility. EGARCH model was calibrated by Nelson in (1991). The advantage of the EGARCH model is that the variance will be positive even if the parameters are negative.

Evaluation of the performance of the volatility models will be based on the function Forecast that is used in E-views. Indicators such as root mean square error, (RMSE), mean absolute error, (MAE), Theil inequality coefficient, bias proportion and variance proportion and the covariance proportion will be used to choose the best model that minimize the error term. It is important to take into consideration important macroeconomic announcements and indicators that affect the forecasting model. For example, the US recession in 2008 was the result of a substantial drop in prices of investment products.

The rest of the paper is organized as follows. Section 1 describes the limits of arbitrage. Section 2 describes the methodology and the data. Section 3 is an analysis of statistical and econometrical tests and Section 4 summarizes and concludes.

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1. The Limits of Arbitrage

The Purchasing Power Parity theorem can be viewed as the proposition that the law of one price will operate internationally. According to the law of one price, identical goods will sell for the same price between countries with the same currency. Price differentials persist because of market imperfections. Arbitrageurs faced with a price differential will buy in a low price market and sell in the higher price market and so make a riskless profit. For example, when the EURO/USD spot rate or the price of one currency in terms of another is appreciated relative to its fundamental value as a result from irrational investors, then it represents a bad buy. Arbitrageurs or rational investors would open a short position to this expensive currency and simultaneously purchase the same or very similar currency at spot rate in another market at a lower price to hedge their risks. The effect of this selling by arbitrageurs is to bring the price of the overpriced currency down to its fundamental value. The same principle applies for an undervalued or depreciated currency. To earn profit arbitrageurs would buy underpriced currency and open a short position to an overpriced to hedge their risk. They prevent with this technique underpricing to be substantial or long lasting.

In real world, arbitrage is limited. Arbitrageurs are risk averse and their horizon is finite. In the case of a currency at spot rate, an arbitrageur would buy the underlying real exchange rate index and sell the currency at spot rate, as it is more volatile. The problem that arbitrageurs will be facing is that discounts are persistent for long period due to volatility clustering as a result of bad news such as the US recession in 2008. Thus, their investment strategy will result in a loss, as the discount will persist since the time the arbitrage trade was done. On the other hand, noise traders would buy the appreciated currency at a spot rate and through sentiment or noise would affect substantially the supply and demand. Their behaviour adds a risk that is not priced and therefore eliminates arbitrageurs investment positions. Thus, arbitrage in FOREX market becomes unprofitable. Arbitrageurs will not open or close investment positions that violate the law of one price. Their concerns focus on irrational traders that may cause mispricing to widen further.

The interaction of these two categories of investors may help to explain the variation of the discount in the currency market. For arbitrageurs to buy funds that are characterized by a constant discount would be costly and, therefore not always profitable. In particular, if the currency at a spot rate trade at a discount and then at a premium, an apparent arbitrage profit can be realised by selling the currency index above the risk free rate. However, if the discount stays relatively constant over the investment horizon, the arbitrageurs make no profit. This active interaction can be measured quantitatively by using as proxies the percentage change of EURO/USD exchange rate, and the logarithmic difference of the EURO/USD exchange rate or a real exchange rate index, which is constructed from a nominal exchange rate, a Euro Harmonised Index of Consumer Prices, (HICP), and a US Price index.

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2.Methodology and data description

The methodologies that will be used are autoregressive volatility models. Heteroskedasticity is a violation of the OLS assumption that the variance error term is constant for all observations. This leads to biased estimates of the standard errors, inefficient estimates of the coefficients and incorrect confidence intervals and statistical tests. In other words, we are checking if the variance of the errors is not constant and is creating volatility clusters. Autoregressive models have been studied by various researchers such as: Alexander,(2003), Brooks, (2002), Nelson, (1991), Bollerslev, (1986), Bollerslev, Chou, and Kroner, (1992), Bollerslev, Engle and Nelson, (1994), Bollerslev, Jeffrey, and Wooldridge, (1992), Ding, Granger and Engle, (1993), Engle, (1982), Engle and Bollerslev, (1986), Engle, Lilien and Robins, (1987), Glosten, Jaganathan and Runkle, (1993), Schwert, (1989), Taylor, (1986) and Zakoian, (1994).

The mean equation of the ARCH (1,5) is as follows:

(1)

Where:

rt: is the expected return of the EURO/USD expressed in percentage and logarithmic terms. It is the return of real exchange rate index or the adjusted money supply, M1.

The conditional variance equation is estimated by regressing the squared residuals on a constant and p lags:

(2)

Where:

The null hypothesis that there is no ARCH effect of order p is H0: The alternative hypothesis is that H1:

The mean equation of the GARCH(1,1), TGARCH(1,1) and EGARCH(1,1) is as follows:

(3)

Where:

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yt is the dependent variable and in our case is the percentage or logarithmic return of the spot rate EURO/USD. It is the return of the real exchange rate index and the money supply, M1.

is a constant.

The variance equation of GARCH(1,1) is as follows:

(4)

Where:

According to Brooks, (2002, p.456), the log –likelihood function, (LIF) of the GARCH model that will be maximised is calculated as follows:

(5)

According to Brooks, (2002, p.470), the conditional variance equation of the EGARCH model is as follows:

(6)

The variance equation of the TGARCH (1,1) could be written in a simpler form as follows:

(7)

Where:

The hypotheses that we are going to formulate and test are as follows:

The null hypothesis, H0, states that by using autoregressive volatility models, returns of the EURO/USD, the real exchange rate index and the money supply, (M1) are constant and similarly distributed over time.

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The alternative hypothesis, H1, states that by using autoregressive volatility models, returns of the EURO/USD, the real exchange rate index and the money supply, (M1) are not constant and similarly distributed over time.

Descriptive statistics will be displayed and to test for non-normality the Jarque – Bera statistic is analysed. We check for stationarity of the series by applying the Augmented Dickey – Fuller’s stationarity test, (ADF). Finally, we apply different types of Autoregressive Heteroskedastic models to test for persistence of volatility clusters.

The data that we will use are monthly returns starting from 01/01/2000 to 01/01/2013 of the EURO/USD spot rate, which total to 156 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbols, H.10 and H.6. The Harmonised Index of Consumer Prices, (HICP), is published by the European Commission, Eurostat and it is available from the statistical department of the European Central Bank. It covers monetary expenditure on final consumption by resident and non – resident households of the Euro Area and it is seasonally adjusted. The US consumer price index was obtained from the Bureau of Labor Statistics. The nominal exchange rate index is a nominal Broad Dollar Index, which according to the Federal Reserve System is a weighted average of foreign exchange values of the U.S. dollar against the currencies of a large group of major U.S. trading partners. The weights of the index change over time and are based on annual trade data from US export shares and from US and foreign import shares. According to Loretan, Mazda and Subramanian, (2005), the index is comprised from twenty-six currencies. It was designed in such a way to capture weight changes of countries of the European Union in order to show the Dollar-Euro exchange rates on trade competition between the United States and the Euro area. Multiplying the Nominal Board Dollar Index by the Harmonised Index of Consumer Prices and dividing this by the US Price Index, we get the real exchange rate index. According to the Federal Reserve Statistical Release, the seasonally adjusted money supply, (M1), consists of currency outside the U.S Treasury, Federal Reserve Banks, the vaults of depository institutions, traveller’s checks of nonblank issuers, demand deposits at commercial banks less cash items in the process of collection and Federal Reserve Float, other checkable deposits, credit union share draft accounts, and demand deposits at thrift institutions.

The return financial series are calculated by taking the log difference of the real exchange index and by using the log difference and the monthly price change returns expressed as percentages of the monthly prices of the EURO/USD. The reason of using these three different methods is to show logarithms and percentages in spotting volatility changes.

The logarithmic formula that we have used is:

(8)

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Where: Rt is the monthly return for month t, Pt is the closing price for month t, and Pt-1

is the previous closing price for month t-1.

The monthly price change returns expressed as percentages is given by the formula

x 100 (9)

Where: Rt is the monthly return for month t, P1 is the price of a financial asset at the end of period 1, and P0 is the price of a financial asset at the beginning of the period.

Figure 1 shows the fluctuations of the percentage returns of the spot rate EURO/USD.

Percentage Return of EUR/USD

-10,00

-8,00-6,00

-4,00-2,00

0,002,00

4,006,00

8,00

2000

-02

2000

-11

2001

-08

2002

-05

2003

-02

2003

-11

2004

-08

2005

-05

2006

-02

2006

-11

2007

-08

2008

-05

2009

-02

2009

-11

2010

-08

2011

-05

2012

-02

2012

-11

Source: Author’s calculation based on monthly data obtained from the US Federal Reserve Statistical Release Department.

According to Figure 1, there was a continuous discount /premium fluctuations of the EUR against the USD. The greatest discount of the spot rate was recorded in October 2008 and the value was -7.50%. This was due to the US recession that started in 2008. There was a persistent negative volatility clusters that resulted to this decrease. A possible explanation is that noise traders have bought in the wrong price due to speculation. For example, when the price of EURO/USD was 1.56 in June of 2008, then there was a continuous drop that reached the price of 1.27 in November of 2008. Another explanation is limited rationality due to the ignorance of the manipulation of thoughts under the Orthodox Approach, therefore, they created a deep discount of the currency. On the other hand, arbitrageurs could not open a short position of a currency option at spot rate, as it was mispriced and volatile. It was away from the fundamental value of the equilibrium or from the average value. They did not know at that time how deep the discount will be and for how long it will last. Arbitrage in the FOREX market at that time becomes unprofitable. Arbitrageurs will not open or close investment positions that violate the law of one price and the discount of the exchange

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rate widen further. In addition, the US recession that took place in 2008 has resulted to deepen the discount, as the volume of seller’s was significantly higher than the volume of buyer’s.

3. Statistical and econometrical tests

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The following Figures and Tables display descriptive statistics, the normality tests of the returns calculated by taking the monthly log difference and the monthly price change returns expressed as percentages of the EURO/USD spot rate, the real exchange rate index and the seasonally adjusted money supply, (M1).

Table 1 and Figure 2 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

0

5

10

15

20

25

30

-0.10 -0.05 0.00 0.05 0.10

Series: LRSample 2 156Observations 155

Mean 0.001926Median 0.002103Maximum 0.121730Minimum -0.110616Std. Dev. 0.029050Skewness 0.058767Kurtosis 5.263322

Jarque-Bera 33.17285Probability 0.000000

Source: Author’s calculation based on E-views software.Significant at 5% significant level.

We state the hypotheses as follows:

H0: The monthly log difference returns of the EURO / USD spot rate are normally distributed.

H1: The monthly log difference returns of the EURO / USD spot rate are not normally distributed.

According to Table 1 and Figure 2, the Jarque – Bera statistic is 33.17, which is very significant at the 5% significance level. The joint test of the null hypothesis that sample skewness equals 0 and sample kurtosis equals 3 is rejected. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic and slightly positively skewed. The EURO/USD spot rate time series is non-normal. The mean is 0.001926 and the dispersion around the mean is 0.029050. There is great dispersion from the actual data.

Table 2 and Figure 3 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the monthly price change returns

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expressed as percentages of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

0

5

10

15

20

25

30

-8 -6 -4 -2 0 2 4 6 8 10 12

Series: RSample 2 156Observations 155





H0: The monthly price change returns expressed as percentages of the EURO/USD spot rate are normally distributed.

H1:. The monthly price change returns expressed as percentages of the EURO/USD spot rate are not normally distributed.

According to Table 2 and Figure 3, the Jarque – Bera statistic is 24.03, which is very significant at the 5% significance level. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic, as the value 4.65 is greater than 3 and slightly positively skewed. The mean is 0.264046 and the dispersion is 2.736006. This implies that there is a high dispersion from the actual data.

Table 3 and Figure 4 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the monthly price change returns

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expressed as percentages of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

0

5

10

15

20

25

30

-2 0 2 4 6 8

Series: RISample 2 156Observations 155

Mean -0.112015Median -0.247006Maximum 7.576207Minimum -3.151097Std. Dev. 1.394967Skewness 1.249082Kurtosis 8.183455




H0: The monthly price change returns expressed as percentages of the real exchange rate index EURO/USD spot rate are normally distributed.

H1:. The monthly price change returns expressed as percentages of the real exchange rate index EURO/USD spot rate are not normally distributed.

According to Table 3 and Figure 4, the Jarque – Bera statistic is 213.83, which is very significant at the 5% significance level. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic, as the value 8.18 is greater than 3 and slightly positively skewed. The mean is -0.112 and the dispersion is 1.395. This implies that there is a high dispersion from the actual data.

Table 4 and Figure 5 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the monthly price change logarithmic returns of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

12

0

5

10

15

20

25

30

-0.02 0.00 0.02 0.04 0.06

Series: LRISample 2 156Observations 155

Mean -0.001217Median -0.002473Maximum 0.073029Minimum -0.032018Std. Dev. 0.013853Skewness 1.136833Kurtosis 7.571941




H0: The monthly log difference returns of the real exchange rate index of the EURO / USD spot rate are normally distributed.

H1: The monthly log difference returns of the real exchange rate index of the EURO / USD spot rate are not normally distributed.

According to Table 4 and Figure 5, the Jarque – Bera statistic is 168.38, which is very significant at the 5% significance level. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic, as the value 7.57 is greater than 3 and slightly positively skewed. The mean is -0.0012 and the dispersion is 0.014.

Table 5 and Figure 6 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the monthly price change returns expressed as percentages of the money supply, (M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

13

0

5

10

15

20

25

-2.50 -1.25 0.00 1.25 2.50 3.75 5.00 6.25

Series: RM1Sample 2 156Observations 155





H0: The monthly price change returns expressed as percentages of the money supply, (M1), are normally distributed.

H1: The monthly price change returns expressed as percentages of the money supply, (M1), are not normally distributed.

According to Table 5 and Figure 6, the Jarque – Bera statistic is 469.29, which is very significant at the 5% significance level. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic, as the value 10.75 is greater than 3 and slightly positively skewed. The mean is 0.510 and the dispersion is 1.12. There is dispersion from the actual data.

Table 6 and Figure 7 display Jarque - Bera normality test and descriptive statistics, which are calculated by taking the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

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0

5

10

15

20

25

30

-0.025 0.000 0.025 0.050

Series: LRM1Sample 2 156Observations 155





H0: The monthly price change logarithmic returns of the money supply, (M1),are normally distributed.

H1: The monthly price change logarithmic returns of the money supply, (M1), are not normally distributed.

According to Table 6 and Figure 7, the Jarque – Bera statistic is 421.05, which is very significant at the 5% significance level. The probability of the p- value 0.0000 is very significant, so we can reject H0 of normality. The distribution is leptokurtic, as the value 10.35 is greater than 3 and slightly positively skewed. The mean is 0.005 and the dispersion is 0.011.

The following Tables show the ADF tests of the of the returns calculated by taking the monthly log difference and the monthly price change returns expressed as percentages of the EURO/USD spot rate, the real exchange rate index and the money supply, (M1).

Table 7 shows the ADF test of the monthly log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

ADF Test Statistic -5.174100 1% Critical Value* -3.4749 5% Critical Value -2.8807 10% Critical Value -2.5769

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*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(LRADF)Method: Least SquaresDate: 08/29/13 Time: 21:06Sample(adjusted): 7 156Included observations: 150 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. LRADF(-1) -0.788353 0.152365 -5.174100 0.0000

D(LRADF(-1)) 0.103821 0.136607 0.760000 0.4485D(LRADF(-2)) -0.060637 0.121588 -0.498708 0.6187D(LRADF(-3)) 0.005706 0.100240 0.056927 0.9547D(LRADF(-4)) 0.002791 0.084773 0.032924 0.9738

C 0.001244 0.002056 0.605349 0.5459R-squared 0.386791 Mean dependent var -0.000242Adjusted R-squared 0.365499 S.D. dependent var 0.031352S.E. of regression 0.024974 Akaike info criterion -4.502820Sum squared resid 0.089810 Schwarz criterion -4.382395Log likelihood 343.7115 F-statistic 18.16601Durbin-Watson stat 1.941085 Prob(F-statistic) 0.000000

Source: Author’s calculation based on E-views software.

For a level of significance of one per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4749. According to Table 7 and to the sample evidence, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -5.174, which is smaller than the critical values, (-3.4749, -2.8807, -2.5769). In other words, the log difference of the EURO/USD is a stationary series.

Table 8 shows the ADF test of the monthly price change returns expressed as percentages of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.



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Augmented Dickey-Fuller Test EquationDependent Variable: D(R)Method: Least SquaresDate: 08/29/13 Time: 21:10Sample(adjusted): 7 156Included observations: 150 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. R(-1) -0.787192 0.148362 -5.305907 0.0000

D(R(-1)) 0.156261 0.132578 1.178639 0.2405D(R(-2)) -0.039551 0.117663 -0.336135 0.7373D(R(-3)) 0.024774 0.096464 0.256819 0.7977D(R(-4)) 0.022032 0.081138 0.271533 0.7864

C 0.186365 0.198732 0.937774 0.3499R-squared 0.374599 Mean dependent var -0.018177Adjusted R-squared 0.352883 S.D. dependent var 2.984189S.E. of regression 2.400588 Akaike info criterion 4.628482Sum squared resid 829.8461 Schwarz criterion 4.748907Log likelihood -341.1362 F-statistic 17.25043Durbin-Watson stat 1.915142 Prob(F-statistic) 0.000000


According to Table 8, for a level of significance of one, five and ten per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4749, -2.8807 and -2.5769. According to Table 8, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -5.3059, which is smaller than the critical values, (-3.4749, -2.8807, -2.5769).

Table 9 shows the ADF test of the monthly price change returns expressed as percentages of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.



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Augmented Dickey-Fuller Test EquationDependent Variable: D(RI)Method: Least SquaresDate: 08/29/13 Time: 21:14Sample(adjusted): 7 156Included observations: 150 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. RI(-1) -0.632279 0.121834 -5.189667 0.0000

D(RI(-1)) 0.162286 0.116740 1.390153 0.1666D(RI(-2)) -0.053449 0.107302 -0.498113 0.6192D(RI(-3)) 0.041463 0.090990 0.455690 0.6493D(RI(-4)) 0.103748 0.081750 1.269091 0.2065

C -0.075319 0.102557 -0.734411 0.4639R-squared 0.322246 Mean dependent var 0.006414Adjusted R-squared 0.298713 S.D. dependent var 1.484616S.E. of regression 1.243260 Akaike info criterion 3.312529Sum squared resid 222.5802 Schwarz criterion 3.432954Log likelihood -242.4397 F-statistic 13.69332Durbin-Watson stat 1.963099 Prob(F-statistic) 0.000000


According to Table 9, for a level of significance of one, five and ten per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4749, -2.8807 and -2.5769. According to Table 9, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -5.1896, which is smaller than the critical values, (-3.4749, -2.8807, -2.5769). The financial series is stationary.

Table 10 shows the ADF test of the monthly price change logarithmic returns of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.



Augmented Dickey-Fuller Test Equation

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Dependent Variable: D(LRI)Method: Least SquaresDate: 08/29/13 Time: 21:16Sample(adjusted): 7 156Included observations: 150 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. LRI(-1) -0.634481 0.122217 -5.191416 0.0000

D(LRI(-1)) 0.162972 0.117002 1.392896 0.1658D(LRI(-2)) -0.054733 0.107501 -0.509136 0.6114D(LRI(-3)) 0.040962 0.091061 0.449829 0.6535D(LRI(-4)) 0.103337 0.081763 1.263863 0.2083

C -0.000818 0.001021 -0.800921 0.4245R-squared 0.323913 Mean dependent var 6.46E-05Adjusted R-squared 0.300438 S.D. dependent var 0.014772S.E. of regression 0.012355 Akaike info criterion -5.910326Sum squared resid 0.021981 Schwarz criterion -5.789901Log likelihood 449.2744 F-statistic 13.79810Durbin-Watson stat 1.962371 Prob(F-statistic) 0.000000


According to Table 10, for a level of significance of one, five and ten per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4749, -2.8807 and -2.5769. According to Table 10, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -5.1914, which is smaller than the critical values, (-3.4749, -2.8807, -2.5769). The financial series is stationary. There is no autocorrelation problem.

Table 11 shows the ADF test of the monthly price change returns expressed as percentages of the money supply, (M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.



Augmented Dickey-Fuller Test EquationDependent Variable: D(RM1)

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Method: Least SquaresDate: 08/29/13 Time: 21:17Sample(adjusted): 7 156Included observations: 150 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. RM1(-1) -0.620345 0.153878 -4.031418 0.0001

D(RM1(-1)) -0.395701 0.146042 -2.709504 0.0076D(RM1(-2)) -0.330809 0.139741 -2.367305 0.0192D(RM1(-3)) -0.065970 0.118525 -0.556586 0.5787D(RM1(-4)) -0.078728 0.083432 -0.943620 0.3469

C 0.344142 0.118797 2.896885 0.0044R-squared 0.540999 Mean dependent var 0.012460Adjusted R-squared 0.525061 S.D. dependent var 1.578738S.E. of regression 1.087999 Akaike info criterion 3.045736Sum squared resid 170.4590 Schwarz criterion 3.166162Log likelihood -222.4302 F-statistic 33.94496Durbin-Watson stat 2.033260 Prob(F-statistic) 0.000000


According to Table 11, for a level of significance of one, five and ten per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4749, -2.8807 and -2.5769. According to Table 11, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -4.0314, which is smaller than the critical values, (-3.4749, -2.8807, -2.5769). The financial series is stationary and according to the Durbin – Watson statistic, there is no autocorrelation problem.

Table 12 shows the ADF test of the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.



Augmented Dickey-Fuller Test EquationDependent Variable: D(LRM1)Method: Least Squares

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Date: 08/29/13 Time: 21:21Sample(adjusted): 7 156Included observations: 150 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. LRM1(-1) -0.620514 0.154175 -4.024746 0.0001

D(LRM1(-1)) -0.398240 0.146290 -2.722262 0.0073D(LRM1(-2)) -0.333638 0.139952 -2.383948 0.0184D(LRM1(-3)) -0.069664 0.118664 -0.587069 0.5581D(LRM1(-4)) -0.080583 0.083419 -0.966014 0.3357

C 0.003396 0.001172 2.898037 0.0043R-squared 0.542042 Mean dependent var 0.000124Adjusted R-squared 0.526141 S.D. dependent var 0.015575S.E. of regression 0.010721 Akaike info criterion -6.193963Sum squared resid 0.016553 Schwarz criterion -6.073538Log likelihood 470.5472 F-statistic 34.08792Durbin-Watson stat 2.034731 Prob(F-statistic) 0.000000


According to Table 12, for a level of significance of one, five and ten per cent, the critical value of the t-statistic from Dickey-Fuller’s table is -3.4749, -2.8807 and -2.5769. According to Table 12, we can reject the null hypothesis namely the existence of a unit root with one, five and ten per cent significance level. The ADF test statistic is -4.0247, which is smaller than the critical values, (-3.4749, -2.8807, -2.5769). The financial series is stationary and according to the Durbin – Watson statistic, there is no autocorrelation problem.

Table 13 shows the Autoregressive Conditional Heteroskedastic test, (ARCH 1,5) of the monthly price change returns expressed as percentages of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

ARCH Test:F-statistic 3.226441 Probability 0.008692Obs*R-squared 15.07856 Probability 0.010032

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 08/29/13 Time: 21:30Sample(adjusted): 12 156

21

Included observations: 145 after adjusting endpointsVariable Coefficient Std. Error t-Statistic Prob.

C 3.493528 1.122152 3.113241 0.0022RESID^2(-1) -0.047330 0.083426 -0.567322 0.5714RESID^2(-2) 0.264626 0.083468 3.170402 0.0019RESID^2(-3) 0.022804 0.086419 0.263874 0.7923RESID^2(-4) -0.047776 0.083513 -0.572074 0.5682RESID^2(-5) 0.168377 0.083204 2.023667 0.0449

R-squared 0.103990 Mean dependent var 5.488024Adjusted R-squared 0.071759 S.D. dependent var 8.601273S.E. of regression 8.286917 Akaike info criterion 7.107732Sum squared resid 9545.546 Schwarz criterion 7.230907Log likelihood -509.3106 F-statistic 3.226441Durbin-Watson stat 1.932654 Prob(F-statistic) 0.008692


The hypotheses that were formulated and have been tested are as follows:

H0: There is no ARCH effect that manifests in the time series.H1: There is ARCH effect that manifests in the time series.

The presence of ARCH (1,5) in the residuals is calculated by regressing the squared residuals on a constant and p lags. According to Table 13, both the F-statistic and the Lagrange Multiplier, LM- statistics are significant, thus, the sample evidence suggests that we can reject the null hypothesis. ARCH effect is present in the monthly price change returns expressed as percentages of the EURO/USD spot rate. Specifically, the F-statistic is 3.23 with a significant p-value of 0.008 and the LM – statistic is 15.08 with a significant p-value of 0.01.

Table 14 displays the forecasting results of the Autoregressive Conditional Heteroskedastic test, (ARCH 1,5) of the monthly price change returns expressed as percentages of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Forecast: RFActual: RSample: 7 156Include observations: 150

Root Mean Squared Error 2.537298Mean Absolute Error 2.022152Mean Abs. Percent Error 119.3570Theil Inequality Coefficient 0.887742

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Bias Proportion 0.000003 Variance Proportion 0.831106 Covariance Proportion 0.168891Source: Author’s calculation based on E-views software.

According to Table 14, the root mean square error, (RMSE), and mean absolute error, (MAE), have a quite high value. Theil inequality coefficient should be between zero and one. In our case, it is 0.89, which shows that the model is not best fit. The closer is this value to zero indicates that the model is best fit. The bias proportion is very low and the value of the covariance proportion is 0.17.

Figure 8 shows the residuals graph of the monthly price change returns expressed as percentages of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

-8

-6

-4

-2

0

2

4

6

8

20 40 60 80 100 120 140

R Residuals


According to Figure 8, we can see that there is a significant positive and negative volatility clusters depicted by the illustrated arrows, which confirms the ARCH effect on the time series of the EURO/USD based on the Lagrange Multiplier, (LM). This figure shows that the mean and variance are not constant.Table 15 shows the Autoregressive Conditional Heteroskedastic test, (ARCH 1,5) of the monthly price change returns expressed as percentages of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.


Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 08/29/13 Time: 21:47Sample(adjusted): 12 156Included observations: 145 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C 1.433009 0.444772 3.221896 0.0016

RESID^2(-1) -0.029243 0.084813 -0.344794 0.7308

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RESID^2(-2) 0.159969 0.084785 1.886747 0.0613RESID^2(-3) -0.039499 0.085800 -0.460360 0.6460RESID^2(-4) -0.039366 0.084757 -0.464457 0.6430RESID^2(-5) -0.006334 0.084785 -0.074710 0.9406

R-squared 0.028782 Mean dependent var 1.501948Adjusted R-squared -0.006154 S.D. dependent var 4.236078S.E. of regression 4.249092 Akaike info criterion 5.771787Sum squared resid 2509.615 Schwarz criterion 5.894962Log likelihood -412.4545 F-statistic 0.823842Durbin-Watson stat 1.999938 Prob(F-statistic) 0.534702


According to Table 15, both the F-statistic and the Lagrange Multiplier, LM- statistics are not significant. ARCH effect is not present in the monthly price change of the real exchange rate index of the EURO/USD spot rate. Specifically, the F-statistic is 0.82 with a non-significant p-value of 0.53 and the LM – statistic is 4.17 with a non-significant p-value of 0.52. A possible explanation is that the nominal Broad Index contains twenty-six currencies in relation to the Euro area. The international currencies are related to the economy of Canada, China, Euro, Mexico, Japan, United Kingdom, Korea, Taiwan, Malaysia, Brazil, Thailand, Singapore, India, Israel, Switzerland, Sweden, Philippines, Indonesia, Hong Kong, Australia, Russia, Colombia, Chile, Venezuela, Argentina, and Saudi Arabia. The weights are related to the share of US imports and exports from the countries of the economies mentioned. It is not focused only on the Euro economies. Another explanation is that the ARCH (1,5) model is not the best fit to show persistence of index return volatility as the TGARCH and EGARCH that we will analyse later.

Table 16 shows the Autoregressive Conditional Heteroskedastic test, (ARCH 1,5) of the monthly price change returns expressed as percentages of the money supply, (M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.


Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 08/29/13 Time: 21:59Sample(adjusted): 12 156Included observations: 145 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C 0.758395 0.317073 2.391865 0.0181

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RESID^2(-1) 0.231510 0.084830 2.729112 0.0072RESID^2(-2) 0.002622 0.087065 0.030116 0.9760RESID^2(-3) 0.152293 0.086090 1.768998 0.0791RESID^2(-4) -0.019860 0.087042 -0.228163 0.8199RESID^2(-5) -0.015490 0.084784 -0.182703 0.8553

R-squared 0.080673 Mean dependent var 1.167099Adjusted R-squared 0.047604 S.D. dependent var 3.350365S.E. of regression 3.269647 Akaike info criterion 5.247740Sum squared resid 1485.993 Schwarz criterion 5.370915Log likelihood -374.4612 F-statistic 2.439524Durbin-Watson stat 2.000853 Prob(F-statistic) 0.037389


According to Table 16, both the F-statistic and the Lagrange Multiplier, LM- statistics are significant. ARCH effect is present in the monthly price change returns expressed as percentages of the money supply, (M1). Specifically, the F-statistic is 2.44 with a significant p-value of 0.04 and the LM – statistic is 11.70 with a significant p-value of 0.04.

Figure 9 shows the residuals graph of the monthly price change returns expressed as percentages of the money supply, (M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

-4

-2

0

2

4

6

20 40 60 80 100 120 140

RM1 Residuals


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According to Figure 9, we can see that there is a significant positive and negative volatility clusters, depicted by the illustrated arrows, which confirms the ARCH effect on the time series of the money supply based on the Lagrange Multiplier, (LM). This figure shows that the mean and variance are not constant.

Table 17 displays the GARCH(1,1) model of the log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRMethod: ML – ARCHDate: 08/29/13 Time: 22:19Sample: 2 156Included observations: 155Convergence not achieved after 100 iterationsBollerslev – Wooldridge robust standard errors & covariance

Coefficient Std. Error z-Statistic Prob. C 0.015717 0.002572 6.109826 0.0000

Variance EquationC 0.000168 0.000152 1.108455 0.2677

ARCH(1) 2.085080 0.342634 6.085448 0.0000GARCH(1) 0.249840 0.123063 2.030175 0.0423

R-squared -0.013871 Mean dependent var 0.005768Adjusted R-squared -0.034014 S.D. dependent var 0.084749S.E. of regression 0.086178 Akaike info criterion -3.292222

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Sum squared resid 1.121432 Schwarz criterion -3.213682Log likelihood 259.1472 Durbin-Watson stat 2.089705


According to Table 17, the z-statistic, the coefficients and the p- values of the variance equation of the log difference of the EURO/USD spot rate are very significant. The log likelihood is maximised at the value of 259.15. This means that the conditional variance is highly persistent. As we saw earlier, there are positive and negative volatility clusters that show continuous large fluctuations of prices for a specified period of time.

Table 18 displays the forecasting results of the Autoregressive Conditional Heteroskedastic test, (GARCH 1,1) of the log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Forecast:LRFActual: LRSample: 2 156Include observations: 155

Root Mean Squared Error 2.561989Mean Absolute Error 2.052641Mean Abs. Percent Error 133.1345Theil Inequality Coefficient 0.876579 Bias Proportion 0.003819 Variance Proportion 0.996181 Covariance Proportion 0.000000Source: Author’s calculation based on E-views software.

According to Table 18, the root mean square error, (RMSE), and mean absolute error, (MAE), have a quite high value of 2.56 and 2.05 respectively. Theil inequality is 0.88. The bias proportion is very low, the variance proportion is close to 1 and the value of the covariance proportion is close to zero.

Table 19 shows the GARCH(1,1) model of the monthly logarithmic price changes returns of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRIMethod: ML – ARCHDate: 08/29/13 Time: 22:25Sample: 2 156Included observations: 155Convergence achieved after 27 iterationsBollerslev – Wooldridge robust standard errors & covariance

Coefficient Std. Error z-Statistic Prob. C -0.002084 0.001157 -1.800632 0.0718

Variance EquationC 0.000148 4.65E-05 0.191372 0.8214

ARCH(1) 0.304367 0.105764 0.877792 0.9340GARCH(1) -0.110706 0.263050 -0.420855 0.6739

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R-squared -0.003944 Mean dependent var -0.001217Adjusted R-squared -0.023889 S.D. dependent var 0.013853S.E. of regression 0.014017 Akaike info criterion -5.790814Sum squared resid 0.029669 Schwarz criterion -5.712274Log likelihood 452.7881 Durbin-Watson stat 1.155106


According to Table 19, the z-statistic, the coefficients and the p- values of the variance equation are not statistically significant. The conditional variance is not persistent. The shortcoming of this model will be overcome by using the TGARCH and EGARCH models.

Table 20 shows the GARCH(1,1) model of the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRM1Method: ML – ARCHDate: 08/29/13 Time: 22:31Sample: 2 156Included observations: 155Convergence achieved after 23 iterationsBollerslev – Wooldridge robust standard errors & covariance


Variance EquationC 1.88E-05 8.09E-06 2.325239 0.0201

ARCH(1) 0.750577 0.135706 5.530921 0.0000GARCH(1) 0.343081 0.096494 3.555465 0.0004

R-squared -0.073073 Mean dependent var 0.005029Adjusted R-squared -0.094393 S.D. dependent var 0.011009S.E. of regression 0.011516 Akaike info criterion -6.397606Sum squared resid 0.020027 Schwarz criterion -6.319066Log likelihood 499.8145 Durbin-Watson stat 1.824653


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According to Table 20, the z-statistic, the coefficients and the p- values of the variance equation of the log difference of the monthly price change logarithmic returns of the money supply,(M1) are very significant. The log likelihood is maximised at the value of 499.8145. The sum of the coefficients on the lagged squared error and lagged conditional variance is very close to unity. This means that the conditional variance is highly persistent. The variance intercept C is very small.

Table 21 displays the Threshold TGARCH(1,1) model of the log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRMethod: ML - ARCHDate: 08/30/13 Time: 08:49Sample: 2 156Included observations: 155Convergence achieved after 202 iterationsBollerslev – Wooldridge robust standard errors & covariance


Variance EquationC 0.000412 0.000125 3.283895 0.0010

ARCH(1) -0.050032 0.150550 -0.332325 0.7396(RESID<0)*ARCH(1) 4.816924 1.028003 4.685711 0.0000

GARCH(1) 0.181124 0.093066 1.946198 0.0516R-squared -0.000017 Mean dependent var 0.005768Adjusted R-squared -0.026684 S.D. dependent var 0.084749S.E. of regression 0.085872 Akaike info criterion -3.469584Sum squared resid 1.106108 Schwarz criterion -3.371409Log likelihood 273.8927 Durbin-Watson stat 2.118657


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According to Table 21, the asymmetry terms, (RESID<0)*ARCH(1), has a coefficient of 4.817 and a z-statistic of 4.69, which is very statistically significant at the 5 % significance level. Positive shocks related to good news shocks implied a higher next period conditional variance than negative shocks related to bad news of the same extent. The convergence of the maximum – likelihood occurred after 202 iterations and not after the default maximum of 100 iterations.

The mean and variance equation (3) and (7) will be used in combination with Table 21 to show the coefficients .

Table 22 shows the estimation of the coefficientsCoefficients Coefficients

Estimatedz-statistic Probability

0.005424 2.092748* 0.0364

0.000408 3.283895* 0.0010-0.050032 -0.332325 0.7396

4.816924 4.685711* 0.00000.181124 1.946198* 0.0516

Source: Author’s calculation based on E-views software.* Significant at 5% significant level.

According to Table 22, all the coefficients are significant, except the coefficient that captures the ARCH effect. Specifically, the constant of the mean equation has a value of 0.005 and it is significantly positive at 5% significance level. The coefficient of the constant of the variance equation is statistically significant with a z- statistic of 3.28. The , which explains the ARCH effect is not significant. The leverage effect is very significant with a z-statistic of 4.69, which implies that there is leverage effect and persistent asymmetry in the time series as justified by the Jarque Bera test. Finally, the coefficient of the variance equation, which measures the GARCH effect, is positively significant at the 5% significance level. The persistent asymmetry is a strong evidence of the US recession in relation to the interactions of noise traders and arbitrageurs. The noise traders by buying in the wrong price they have created persistent fluctuations or volatility clusters of the price of the EURO/USD spot rate.

Table 23 displays the forecasting results of the Threshold TGARCH(1,1) model of the log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Forecast: LRFActual: LRSample: 2 156Include observations: 155Root Mean Squared Error 0.084476Mean Absolute Error 0.034504Mean Absolute Percentage Error 158.8749Theil Inequality Coefficient 0.937840 Bias Proportion 0.000019

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Variance Proportion 0.999981 Covariance Proportion 0.000000


According to Table 23, the root mean square error, (RMSE), and mean absolute error, (MAE), have a low value of 0.08 and 0.03 respectively. Theil inequality coefficient is 0.94. The bias proportion is very low, the variance proportion is close to unity and the value of the covariance proportion is zero. The TGARCH(1,1) is better fit than the GARCH(1,1) model as it minimizes the error term.

Table 24 shows the EGARCH(1,1) model of the log difference of the EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRMethod: ML - ARCHDate: 08/30/13 Time: 09:07Sample: 2 156Included observations: 155Convergence achieved after 140 iterationsBollerslev – Wooldridge robust standard errors & covariance

Coefficient

Std. Error z-Statistic Prob.

C 0.003402 0.003005 1.132103 0.2576 Variance Equation

C -1.110689 0.422041 -2.631709 0.0085|RES|/SQR[GARCH]

(1)0.072616 0.080169 0.905787 0.3650

RES/SQR[GARCH](1) -0.585463 0.129181 -4.532102 0.0000EGARCH(1) 0.834988 0.063650 13.11846 0.0000



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According to Table 24, the asymmetry terms, RES/SQR[GARCH](1), has a coefficient of -0.59 and a z-statistic of -4.53, which is very statistically significant at the 5 % significance level. EGARCH lagged one period has a z – statistic of 13.12 and it is statistically significant at the 5% significance level. Negative shocks due to bad news implied a higher next period conditional variance than positive shocks due to good news of the same extent. The convergence of the maximum – likelihood occurred after 140 iterations and not after the default maximum of 100 iterations.

Table 25 displays the Threshold TGARCH(1,1) model of the monthly logarithmic price changes returns of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRIMethod: ML - ARCHDate: 08/30/13 Time: 07:48Sample: 2 156Included observations: 155Convergence achieved after 19 iterationsBollerslev – Wooldridge robust standard errors & covariance


Variance EquationC 4.94E-05 1.25E-05 3.945854 0.0001

ARCH(1) 0.178337 0.049053 3.635581 0.0003(RESID<0)*ARCH(1) -0.332512 0.071757 -4.633840 0.0000

GARCH(1) 0.674920 0.099815 6.761702 0.0000R-squared -0.004829 Mean dependent var -0.001217Adjusted R-squared -0.031624 S.D. dependent var 0.013853S.E. of regression 0.014070 Akaike info criterion -5.828039Sum squared resid 0.029695 Schwarz criterion -5.729864Log likelihood 456.6730 Durbin-Watson stat 1.154088


According to Table 25, the asymmetry terms, (RESID<0)*ARCH(1), has a coefficient of -0.33 and a z-statistic of -4.63, which is very statistically significant at the 5 %

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significance level. Negative shocks implied a higher next period conditional variance than positive shocks of the same extent. The convergence of the maximum – likelihood occurred after 19 iterations and not after the default maximum of 100 iterations. The TGARCH(1,1) is a better fit than the ARCH(1,5) and the GARCH(1,1), as it captures the persistence of the conditional variance




-0.002176 -2.118055* 0.0342

4.94E-05 3.945854* 0.00010.178337 3.635581* 0.0003

-0.332512 -4.633840* 0.00000.674920 6.761702* 0.0000


According to Table 26, all the coefficients are statistically significant at the 5% significance level. In more detail, the constant of the mean equation has a coefficient value of -0.002 and a z – statistic of -2.12 and it is significant at 5% significance level. The coefficient of the constant of the variance equation is statistically significant with a z- statistic of 3.95. The , which explains the ARCH effect is significant. It has a coefficient value of 0.18 and a z – statistic of 3.64. The leverage effect is very significant with a z-statistic of -4.63, which implies that bad news have created negative shocks and persistent asymmetry in the time series as justified by the Jarque Bera test. Finally, the coefficient of the variance equation, which measures the GARCH effect, is positively and statistically significant at the 5% significance level with a coefficient of 0.67 and a z –statistic of 6.76.

Table 27 shows the forecasting results of the Threshold TGARCH(1,1) model of the monthly logarithmic price changes returns of the real exchange rate index

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EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Forecast: LRIFActual: LRISample: 2 156Include observations: 155Root Mean Squared Error 0.013841Mean Absolute Error 0.010524Mean Absolute Percentage Error 112.8868Theil Inequality Coefficient 0.863052 Bias Proportion 0.004806 Variance Proportion 0.995194 Covariance Proportion 0.000000


According to Table 27, the root mean square error, (RMSE), and mean absolute error, (MAE), have a low value of 0.014 and 0.01 respectively. Theil inequality coefficient is 0.86. The bias proportion is very low and the value of the covariance proportion is zero. The variance proportion is close to unity.

Table 28 shows the EGARCH(1,1) model of the monthly logarithmic price changes returns of the real exchange rate index EURO/USD spot rate. The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRIMethod: ML - ARCHDate: 08/30/13 Time: 14:01Sample: 2 156Included observations: 155Convergence achieved after 121 iterationsBollerslev-Wooldrige robust standard errors & covariance


Variance EquationC -3.539898 1.835148 -1.928944 0.0537

|RES|/SQR[GARCH](1)

0.098512 0.314307 0.313426 0.7540

RES/SQR[GARCH](1) 0.255693 0.149505 1.710260 0.0872EGARCH(1) 0.601989 0.215120 2.798389 0.0051

R-squared -0.005791 Mean dependent var -0.001217Adjusted R-squared -0.032612 S.D. dependent var 0.013853S.E. of regression 0.014077 Akaike info criterion -5.782312Sum squared resid 0.029724 Schwarz criterion -5.684137Log likelihood 453.1292 Durbin-Watson stat 1.152984


According to Table 28, the asymmetry terms, RES/SQR[GARCH](1), has a coefficient of 0.26 and a z-statistic of 1.71, which is not statistically significant at the 5 % significance level. EGARCH lagged one period has a z – statistic of 2.80 and it is statistically significant at the 5% significance level. The convergence of the maximum

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– likelihood occurred after 121 iterations and not after the default maximum of 100 iterations.

Table 29 shows the Threshold TGARCH(1,1) model of the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRM1Method: ML - ARCHDate: 08/30/13 Time: 14:10Sample: 2 156Included observations: 155Convergence achieved after 33 iterationsBollerslev – Wooldridge robust standard errors & covariance


Variance Equation

C 1.73E-05 5.60E-06 3.087955 0.0020ARCH(1) 1.323377 0.564845 2.342905 0.0191

(RESID<0)*ARCH(1) -1.292823 0.529829 -2.440073 0.0147GARCH(1) 0.352020 0.080655 4.364504 0.0000



According to Table 29, the asymmetry terms, (RESID<0)*ARCH(1), has a coefficient of -1.29 and a z-statistic of -2.44, which is very statistically significant at the 5 % significance level. Negative shocks as a result of bad news implied a higher next period conditional variance than positive shocks of the same extent. The convergence

35

of the maximum – likelihood occurred after 33 iterations and not after the default maximum of 100 iterations.




0.002612 5.190765* 0.0000

1.73E-05 3.087955* 0.00201.323377 2.342905* 0.0191

-1.292823 -2.440073* 0.01470.352020 4.364504* 0.0000


According to Table 30, all the coefficients are statistically significant at the 5% significance level. In more detail, the constant of the mean equation has a coefficient value of 0.002 and a z – statistic of 5.19 and it is significant at 5% significance level. The coefficient of the constant of the variance equation is statistically significant with a z- statistic of 3.09. The ,which explains the ARCH effect is significant. It has a coefficient value of 1.32 and a z – statistic of 2.34. The leverage effect is very significant with a z-statistic of -2.44, which implies that there is leverage effect and persistent asymmetry in the time series. Finally, the coefficient of the variance equation, which measures the GARCH effect, is positively and statistically significant at the 5% significance level with a coefficient of 0.35 and a z –statistic of 4.36.

Table 31 displays the forecasting results of the Threshold TGARCH(1,1) model of the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Forecast: LRM1FActual: LRM1Sample: 2 156Include observations: 155Root Mean Squared Error 0.011236Mean Absolute Error 0.007166Mean Absolute Percentage Error 173.1389Theil Inequality Coefficient 0.765256 Bias Proportion 0.046277 Variance Proportion 0.953723 Covariance Proportion 0.000000


According to Table 31, the root mean square error, (RMSE), and mean absolute error, (MAE), have a low value of 0.011 and 0.007 respectively. Theil inequality coefficient is 0.77. The bias proportion is very low and the value of the covariance proportion is zero.

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Table 32 shows the EGARCH(1,1) model of the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Dependent Variable: LRM1Method: ML - ARCHDate: 08/30/13 Time: 14:14Sample: 2 156Included observations: 155Convergence achieved after 48 iterationsBollerslev – Wooldridge robust standard errors & covariance


Variance EquationC -3.290249 0.672555 -4.892161 0.0000

|RES|/SQR[GARCH](1) 0.466821 0.221377 2.108714 0.0350RES/SQR[GARCH](1) 0.488197 0.128355 3.803499 0.0001

EGARCH(1) 0.691518 0.073026 9.469479 0.0000R-squared -0.033099 Mean dependent var 0.005029Adjusted R-squared -0.060648 S.D. dependent var 0.011009S.E. of regression 0.011338 Akaike info criterion -6.498933Sum squared resid 0.019281 Schwarz criterion -6.400758Log likelihood 508.6673 Durbin-Watson stat 1.895256


According to Table 32, the asymmetry terms, RES/SQR[GARCH](1), has a coefficient of 0.49 and a z-statistic of 3.80, which is statistically significant at the 5 % significance level. EGARCH lagged one period has a z – statistic of 9.47 and it is statistically significant at the 5% significance level. The convergence of the maximum – likelihood occurred after 48 iterations and not after the default maximum of 100 iterations.

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Table 33 displays the forecasting results of the EGARCH(1,1) model of the monthly price change logarithmic returns of the money supply,(M1). The dataset has covered the period starting from 01/01/2000 to 01/01/2013.

Forecast: LRM1FActual: LRM1Sample: 2 156Include observations: 155Root Mean Squared Error 0.011153Mean Absolute Error 0.007103Mean Absolute Percentage Error 189.8736Theil Inequality Coefficient 0.738445 Bias Proportion 0.032039 Variance Proportion 0.967961 Covariance Proportion 0.000000


According to Table 33, the root mean square error, (RMSE), and mean absolute error, (MAE), have a low value of 0.011 and 0.007 respectively. Theil inequality coefficient is 0.74. The bias proportion is very low and the value of the covariance proportion is zero. The EGARCH(1,1) is better fit than the TGARCH(1,1) model as it minimizes the error term because the Theil inequality coefficient has a lower value.

4. Summarizes and Concludes

In this article, we have attempted to explain, illustrate, test and analyze the returns by using Autoregressive Conditional Heteroskedastic models, Generalized Conditional Heteroskedastic models, Threshold Generalised Conditional Heteroskedastic models and Exponential Generalised Conditional Heteroskedastic models. We have tested the volatility of the returns of the spot exchange rate of EURO/USD, the returns of a real exchange rate index and the money supply,(M1), for changing conditional variances. The data that we have used are monthly returns starting from 01/01/2000 to 01/01/2013, which total to 156 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbols of the series are H.10 and H.6.

We have concluded that the conditional variance is highly persistent. There is a significant positive and negative volatility clusters, which confirms the ARCH and GARCH, TGARCH and EGARCH effect on the time series of the EURO/USD spot rate, based on the F-statistic, the Lagrange Multiplier, (LM), the leverage effect, the Jarque – Bera normality tests. Thus, we reject the null hypothesis that the returns of the EURO/USD spot rate, the returns of the real exchange rate index and the returns of the money supply are constant and similarly distributed over time.

Evaluation of the performance of the volatility models was done by using the Forecast Function for each class of Autoregressive Conditional Heteroskedastic models. Our evidence shows that TGARCH and EGARCH models are better fit than ARCH (1,5) and GARCH (1,1) for the real exchange rate index of EURO/USD to show asymmetry, variance persistence and volatility clusters. For the rest series namely

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percentage and logarithmic returns of EURO/USD spot rate and returns of money supply all the autoregressive models showed strong evidence of asymmetry and conditional variance persistence.

Noise created through the interaction of these two categories of investors may help to explain the volatility clusters or conditional variation persistence in the currency market. This active interaction has been measured quantitatively by using as proxies the percentage change of EURO/USD exchange rate, and the logarithmic difference of the EURO/USD exchange rate or a real exchange rate index, which is constructed from a nominal exchange rate, a Euro Harmonised Index of Consumer Prices, (HICP), and a US Price index. Our findings support the evidence from the above models that asymmetry in the time series of the spot rate of exchange rate is created from the noise traders that are buying or selling in the wrong price. Their actions are accompanied by macroeconomic shocks such as changes in the money supply or recession that was created in the US in 2008. The statistically significant values of the leverage effect indicate that negative shocks implied a higher next period conditional variance than positive shocks of the same extent.

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Noise created through the interactions of arbitrageurs and n

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