applied financial econometrics

Time Series, Volatility & Value-at-Risk Modeling & Causality Analysis

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Assignment:

TIME SERIES, VOLATILITY & VALUE-AT-RISK MODELLING & CAUSALITY

ANALYSIS

Author: Andreas Poulopoulos Student’s Number: l7110166 Course: Advanced Econometrics Supervisor: Bekiros D. Stelios Department: Accounting & Finance University: Athens University Business and Economics “One of the Great Rules of Economics According to John Green If you are rich, you have to be an idiot not to stay rich. And if you are poor, you have to be really smart to get rich.”

John Green


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TABLE OF CONTENTS SECTION I: TIME SERIES ANALYSIS ......................................................................................... 3

Part I ...................................................................................................................................... 3

PART II.................................................................................................................................. 21

Part III .................................................................................................................................. 40

SECTION II VOLATILITY MODELING & VALUE-at-RISK.......................................................... 48

PART I................................................................................................................................... 48

Part II ................................................................................................................................... 74

SECTION III: Causality Analysis ............................................................................................ 80

Part I .................................................................................................................................... 80

PART II.................................................................................................................................. 84

References ........................................................................................................................... 92

APPENDIX 1.......................................................................................................................... 93

APPENDIX 2.......................................................................................................................... 96

APPENDIX 3.......................................................................................................................... 98


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SECTION I: TIME SERIES ANALYSIS

PART I

a) Create the plots of the data. Determine whether the currency price series ARE NON-Stationary. Apply the augmented Dickey-Fuller (ADF) test with up to 12 lags with a constant BUT NO Trend in the test equation. Use the Schwartz criterion to determine the optimum lag length in the ADF test (default).Can the null Hypothesis of unit root in the price series be rejected?

Few word about the data: We are going to use daily “closing” prices for Euro (EUR/USD) and Japanese Yen (USD/JPY). There are total of 1522 daily observations running from 01/01/2008 to 30/10/2013. From now on the term “price” indicates the log-price series. Choosing logarithmic or linear price scale depends on the trader’s trading style. There is no significant difference. The only noticeable thing is that the prices are squeezed down, so it is easier to handle our data.

The Figure 1 and Figure 2 depicts our daily observations. As you can see from the scale it is not the prices (we defined before the prices as the log-price series), the reason we demonstrate these graphs is to show the scale difference.

1.1

1.2

1.3

1.4

1.5

1.6

1.7

I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV

2008 2009 2010 2011 2012 2013

EUR/USD

75

80

85

90

95

100

105

110

115


2008 2009 2010 2011 2012 2013

USD/JPY

FIGURE 1

FIGURE2


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It is easily noticed the difference between the 2 group of graphs. The difference is better understood if you notice the JPY scale. In Figure 2 the scale start from 75-115 when in the Figure 4 this values has been “decreased” squeezed down to 4.3-4.8.

FIGURE 5 EUR PRICE, JPY PRICE

The Figure 5 illustrates the two currencies, how do they move through the time. As we can see their motion is directly depending from the time. This information from the graph, indicates that our data are probably non-stationary. In order to prove that we have to run some unit roots tests to determine whether is non-stationary or not.

.1

.2

.3

.4

.5

4.3

4.4

4.5

4.6

4.7

4.8


2008 2009 2010 2011 2012 2013

PEUR PJPY

.16

.20

.24

.28

.32

.36

.40

.44

.48


2008 2009 2010 2011 2012 2013

PEUR

4.3

4.4

4.5

4.6

4.7

4.8


2008 2009 2010 2011 2012 2013

PJPY

FIGURE 3 FIGURE 4


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It would be a great mistake if we analyze our data in this primal stage, without having computed the Unit Root tests. To explain why, if you generate a regression with non-stationary time series data you may have a big R-square, but the whole regression is completely useless and has not any economic importance. So the next step is to apply the Augmented Dickey-Fuller test to our data in order to determine if our data are stationary or not-stationary. Augmented Dickey Fuller (ADF) test. Dickey-Fuller have proposed three specifications for the unit root test. In our project we are going to use the one with a constant but no trend. 𝛥𝑌𝑡 = 𝛾𝑌𝑡−𝑖 + 𝛼 + ∑ 𝛽𝑖 𝛥𝛶𝑡−𝑖 + 휀𝑖 (1.1)

𝛥𝑌𝑡 = (𝑝 − 1)𝑌𝑡−𝑖 + 𝛼 + ∑ 𝛽𝑖 𝛥𝛶𝑡−𝑖 + 휀𝑖 (1.2)

Null Hypothesis: p=1 or γ=0, we can conclude that our data are non-stationary process. Alternative Hypothesis: p≠0 or γ≠0 if we reject the Null Hypothesis then we can conclude that the series is stationary.

Null Hypothesis: PEUR has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)

t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.321407 0.1653

Test critical values: 1% level -3.434443 5% level -2.863235 10% level -2.567720

*MacKinnon (1996) one-sided p-values.

TABLE 1.1 PEUR ADF TEST

*peur= log-price eur/usd *pjpy= log-price usd/jpy

Null Hypothesis: PJPY has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)


Test critical values: 1% level -3.434443 5% level -2.863235 10% level -2.567720 *MACKINNON (1996) ONE-SIDED P-VALUES.

TABLE 1.2 PJPY ADF TEST


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The results of the ADF test for both currencies conclude with our indications from the Figure 5 , that both series EUR/USD and JPY/USD are non-stationary and we have to take the first differences in order to make them stationary but we have also test that. Interpreting the Table 1.1 and 1.2 Table 1.1 For the PEUR the Null Hypothesis is not rejected because p-value is 16.53% which >> 5% and 1% significance levels. Given that, PEUR series is non-stationary because it has a Unit Root. Table 1.2 The same explanation as above. For the PJPY the Null Hypothesis is not rejected because p-value is 26.01% >> 5% and 1% significance levels. At this point we accept the Null Hypothesis which implies that there is unit root and consequently the PJPY series is non-stationary. Furthermore, in order to cross check our results about the Unit Root Tests we are going to apply some additional Unit Root tests. The tests that we are going to perform are the Phillips-Peron and Kwiatkowski-Phillips-Schmidt-Shin (KPSS).

b) Instead of the ADF test, run the Phillips-Perron and KPSS tests as above. Explain the difference- if any- between testing with the ADF or PP and testing with the KPSS. Compare the results to the ADF test.

Phillips-Peron Unit Root Test. Dickey-Fuller tests assume that the residuals do not auto-correlated and that they have constant variance. In this section Phillips-Peron developed two tests statistics for the unit root without the Dickey-Fuller’s strictly conditions, for the residuals’ distribution. Hence, Phillips-Peron suggest two new statistics 𝛧𝛼 and 𝛧𝑡. These statistics are modified Dickey-Fuller statistics, so that, the auto-correlation do not affect their asymptotic distribution.

𝑍𝛼 = 𝛵(�̂� − 1) − (𝑠2 − 𝑠𝑢2)(2𝑇−2 ∑ 𝑌𝑡−1

2 )−1𝑇𝑖=1 (1.3 )

𝑍𝑡 =𝑠𝑢

𝑠𝑡𝛼 −

1

2(𝑠2 − 𝑠𝑢

2)(𝑠2𝑇−2 ∑ 𝑌𝑡−12 )−

1

2𝑇𝑡=1 (1.4)


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Null Hypothesis: PEUR has a unit root Exogenous: Constant Bandwidth: 11 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.* Phillips-Perron test statistic -2.390856 0.1445

Test critical values: 1% level -3.434443 5% level -2.863235 10% level -2.567720 *MacKinnon (1996) one-sided p-values.

TABLE 1.3 PEUR PP TEST

Null Hypothesis: PJPY has a unit root Exogenous: Constant Bandwidth: 9 (Newey-West automatic) using Bartlett kernel



TABLE 1.4 PJPY PP TEST

Interpret Tables 1.3 and 1.4 Table 1.3: Represent the Phillip-Peron (PP) test for the PEUR. The Null Hypothesis is not rejected because p-value is 14.45% >> 5% and 1% significance levels. If we cannot reject the Null Hypothesis we have to accept it, and that means that with the PP unit root test the PEUR series are non-stationary. Table 1.4: This table also demonstrate the PP unit root test for the PJPY. The Null Hypothesis also here cannot be rejected because probability is 28.92% >> 5% and 1% significance levels. As a result PJPY series is non-stationary. The last kind of Unit Root test that we are going to perform is KPSS (Kwiatkowski-Phillips-Schmidt-Shin). At this point, we have to note that KPSS Null Hypothesis is defined with different way. Here the Null Hypothesis, determine that the series IS STATIONARY, that means series DO NOT have a UNIT ROOT.


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Kwiatkowski-Phillips-Schmidt-Shin suggest an LM criterion in order to check the Null Hypothesis.

𝐿𝑀𝐾𝑃𝑆𝑆 = ∑ 𝑆𝑡

2𝛵𝑡=1

�̂�𝑢2 (1.5)

Null Hypothesis: PEUR is stationary

Exogenous: Constant

Bandwidth: 31 (Newey-West automatic) using Bartlett kernel LM-Stat.

Kwiatkowski-Phillips-Schmidt-Shin test statistic 1.633802

Asymptotic critical values*: 1% level 0.739000

5% level 0.463000

10% level 0.347000

*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)

TABLE 1.5 PEUR KPSS TEST

Null Hypothesis: PJPY is stationary Exogenous: Constant Bandwidth: 31 (Newey-West automatic) using Bartlett kernel

LM-Stat. Kwiatkowski-Phillips-Schmidt-Shin test statistic 2.065102

Asymptotic critical values*: 1% level 0.739000 5% level 0.463000 10% level 0.347000 *Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)

TABLE 1.6 PJPY KPSS TEST

Interpret Tables 1.5 and 1.6 Table 1.5: Checking now the stationarity of PEUR with a different way. The Null Hypothesis now implies that PEUR series is stationary. In the table the Null Hypothesis is rejected because LM stat. is 1.6333> 5% and 1% critical values. If the Null Hypothesis is rejected we choose the Alternative that PEUR is non-stationary.


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Table 1.6: Same with the previous explanation. The LM stat. for the PJPY series is 2.065102> that 5% and 1% critical values. That means, Null Hypothesis is rejected, that the series is stationary and we accept the Alternative. In other words PJPY is also non-stationary. At this stage we have completed all the diagnostic tests for the existence of Unit Root. In order to overcome the stationarity obstacle, we will take the first differences and we will check again for the existence of unit root. If at the first differences there is not unit root then we can go further to our analysis, otherwise we will have to take the second differences and then to perform again the test for unit root and so on.


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c) If you find that the levels of the series are non-stationary, create the logarithmic first differences series (log-retuns) and repeat the analysis a) and b) on the returns. What is the results?

Consider a situation where the value of a time series at 𝑡,𝑦𝑡, is a linear function of the last p

values of exogenous terms, denoted by 휀𝑡.

𝑦𝑡 = 𝛼1𝑦𝑡−1 + 𝛼2𝑦𝑡−2 + ⋯ + 𝛼𝑝𝑦𝑡−𝑝 + 휀𝑡 (1.6)

The expressions of type (1.6) are called difference equations. The first differences are defined are the difference between the present value of the dependent variable minus the same dependent variable the previous period.

𝛥𝑦𝑡 = 𝑦𝑡 − 𝑦𝑡−1 (1.7)

Null Hypothesis: REUR has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)



TABLE 1.7 REUR ADF TEST

Null Hypothesis: RJPY has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)



TABLE 1.8 PJPY ADF TEST

*REUR = log-returns eur/usd. *RJPY = log-returns usd/jpy.


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Interpret Tables 1.7 and 1.8 Table 1.7: After taking the first differences, the Null Hypothesis of the ADF test is rejected, because probability is 0.000. We accept the Alternative, so REUR series is no longer non-stationary because it has not unit root. Table 1.8: Same results also for the RJPY series. The Null Hypothesis is rejected, because probability is 0.000, given that the RJPY series is stationary.

Null Hypothesis: REUR has a unit root Exogenous: Constant Bandwidth: 9 (Newey-West automatic) using Bartlett kernel



TABLE 1.9 REUR PP TEST

Null Hypothesis: RJPY has a unit root Exogenous: Constant Bandwidth: 10 (Newey-West automatic) using Bartlett kernel



TABLE 1.10 PJPY PP TEST


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Interpreting Tables 1.9 and 1.10 Table 1.9: Philips-Peron proves that the REUR series is stationary, because with probability 0.000 we can reject the Null Hypothesis and to accept the Alternative Hypothesis. Table 1.10 The same output has for the RJPY the Philips-Peron test. The Null Hypothesis of unit root is rejected because the probability is 0.000 and our series is stationary.

Null Hypothesis: REUR is stationary

Exogenous: Constant



Asymptotic critical values*: 1% level 0.739000

5% level 0.463000

10% level 0.347000


TABLE 1.11 REUR KPSS TEST

Null Hypothesis: RJPY is stationary Exogenous: Constant Bandwidth: 10 (Newey-West automatic) using Bartlett kernel



TABLE 1.12 RJPY KPSS TEST

Interpreting Tables 1.11 and 1.12 Table 1.11 The results of the KPSS for the REUR series are inconclusive, because the Null Hypothesis is not rejected at the 1% significant level, so at this level the REUR series is stationary. For the level 5% and 10% the Null Hypothesis is rejected. We will take in account the 1% significant level so REUR is stationary.


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Table 1.12 The KPSS’s test result are the same for the RJPY series. The Null Hypothesis of stationarity is not rejected only at 1% significance level when at 5% and 10% the Null Hypothesis is rejected.

d) If the levels of the series are non-stationary, test for cointergration between them using the Engle-granger approach on a regression estimation. Would you have expected the series cointergrate? Why or why not? What would this tell you about their long-term relationship?

e) Perform diagnostic analysis on the residuals of the regression equation. Firstly plot

the residuals. Employ the ADF test on the residuals series assuming that up to 12 lags are permitted, and that a constant but not a trend is included in the regression on the level prices. What is the result? Is the Null Hypothesis of unit root rejected? Are the two series cointergrated or not?

We have the variables Y, 𝑋1 𝑋2, … , 𝑋𝑘 and we assume that there is long-turn relationship between them.

𝑌𝑡 = 𝑎𝑜 + 𝑎1𝑋1𝑡 + 𝑎2𝑋2𝑡 + ⋯ + 𝑎𝑘𝑋𝑘𝑡 + 𝑢𝑡 (1.8) We make an assumption that all variables are integrated I(1), so they will be cointergrated if their linear combination:

𝑢𝑡 = 𝑌𝑡 − 𝑎0 − 𝑎1𝑋1𝑡 − 𝑎2𝑋2𝑡 − ⋯ − 𝑎𝑘𝑋𝑘𝑡 (1.9)

Is integrated I(0), meaning that 𝑢𝑡 is a stationary series. The (1.8) equation that we have wrote before can be estimated with the OLS method and it is referred as cointergrating regression or static regression. The test for cointegration existence is relying on the residuals behavior, which have been produce from the OLS method. So if the residuals:

�̂�𝑡 = 𝑌𝑡 − �̂�0 − �̂�1𝑋1𝑡 − �̂�2𝑋2𝑡 − ⋯ − �̂�𝑘𝑋𝑘𝑡 (1.10) is a stationary series that means there is a long-run relationship between the variables. The next task for this project is to run a regression model with the levels of the series. We will run the following regressions 1) PEUR = C (1) + C (2)*PJPY 2) PJPY = C (1) + C (2)*PEUR From the previous diagnostic tests we found that the PEUR (EUR/USD log-prices) and the PJPY (JPY/USD) are non-stationary, which simple means that we cannot run a regression model, but if we do it we will be a spurious regression with no economic importance. BUT, if their residuals are I (0) (stationary), that means the variables of the models (1),(2) are cointergrated or they have long run relationship or equilibrium relationship between them.


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PEUR = C (1) + C (2)*PJPY PEUR = -0.790171393293 + 0.244893104548*PJPY

Dependent Variable: PEUR

Method: Least Squares Date: 06/23/15 Time: 01:37 Sample: 1/01/2008 10/30/2013 Included observations: 1522

Variable Coefficient Std. Error t-Statistic Prob. C -0.790171 0.063857 -12.37402 0.0000

PJPY 0.244893 0.014205 17.24010 0.0000 R-squared 0.163558 Mean dependent var 0.310437

Adjusted R-squared 0.163008 S.D. dependent var 0.063327 S.E. of regression 0.057936 Akaike info criterion -2.857638 Sum squared resid 5.102023 Schwarz criterion -2.850637 Log likelihood 2176.663 Hannan-Quinn criter. -2.855032

F-statistic 297.2212 Durbin-Watson stat 0.016692 Prob(F-statistic) 0.000000

TABLE 1.13 REGRESSION, DEP.VAR. PEUR PJPY = C (1) + C (2)*PEUR

PJPY = 4.28690660042 + 0.667875336093*PEUR

Dependent Variable: PJPY

Method: Least Squares Date: 06/23/15 Time: 01:37 Sample: 1/01/2008 10/30/2013 Included observations: 1522

Variable Coefficient Std. Error t-Statistic Prob. C 4.286907 0.012274 349.2751 0.0000

PEUR 0.667875 0.038740 17.24010 0.0000 R-squared 0.163558 Mean dependent var 4.494240



TABLE 1.14 REGRESSION DEP.VAR. PJPY

From the Tables 1.13 and 1.14 we can see, we have signs for cointergration because on both regression models the Durbin-Watson stat is << R-squared. Table 1.13 R-square 0.163558 >> Durbin-Watson stat. 0.016692.


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Table 1.14 R-square 0.163558 >> Durbin-Watson stat. 0.009191. Now, if we take a closer look at the Figure 5 we cannot say for sure that these 2 series are cointergrating. If we see a small part of the picture from the start of 2008 until end of third quarter of 2008 we can argue that they moving “together” almost. But for the rest periods we cannot claim that there is a relation between them. From this point of I wouldn’t have expected to be cointergrated. But if their residuals for the both model are stationary then the 2 series have a long-term relationship.

FIGURE 1.6 PEUR REGRESSION RESIDS

-.15

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

PEUR_PJPY_RESID

-.20

-.15

-.10

-.05

.00

.05

.10

.15

.20


2008 2009 2010 2011 2012 2013

PJPY_PEUR_RESID

FIGURE 1.7 PJPY REGRESSION RESIDS


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Null Hypothesis: PEUR_PJPY_RESID has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)



TABLE 1.15 PEUR REGRESSION RESIDS ADF TEST

Null Hypothesis: PJPY_PEUR_RESID has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)



TABLE 1.16 PJPY REGRESSION RESIDS ADF TEST

Interpreting Tables 1.15 and 1.16 Table 1.15 In this table we check if there is Unit Root on the residuals which have been derived from regression (1) where the dependent variable is PEUR and the independent is PJPY. The ADF test do not reject the Null Hypothesis, because probability is 11.09% higher than 10%, 5% and 1%. We have to mention here that the p-value is marginally higher that 10% significance level, the distance between them is only 1%. Table 1.16 Checking also the residuals from the regression, number 2 where the dependent variable is PJPY and the independent is PEUR, the ADF test do not reject the Null Hypothesis because the probability is 20% higher that all the significance levels. In sum, the residuals are not stationary, so the two variables are not cointergrated, so the regressions that we have ran above are useless. The regression that their variables are not stationary, we called them spurious and they are economic insignificant.


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f) Based on the results of the previous question e), a pure first difference equation model or an error correction model (ECM) is appropriate in order to capture the long-run relationship between the series as well as the short-run relationship? Then, estimate the ECM or the pure first difference equation model based on the results of question e) and describe the cointergration relationship based on the Engle-Granger approach, or the short-run relationship between the variables.

In the previous question e) we found that there is no long-run or equilibrium relationship between the two variables, because their residuals were not stationary, so the regression where spurious. For that reason we will perform a pure first differences model in order to capture their between relationship. REUR = C (1) + C (2)*RJPY REUR = -4.61437217621e-05 - 0.0751016048227*RJPY

Dependent Variable: REUR

Method: Least Squares Date: 06/23/15 Time: 01:48 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments

Variable Coefficient Std. Error t-Statistic Prob. C -4.61E-05 0.000182 -0.253649 0.7998

RJPY -0.075102 0.024328 -3.087065 0.0021 R-squared 0.006235 Mean dependent var -3.99E-05



TABLE 1.17 REUR REGRESSION MODEL

From the Table 1.17, we can see now the reverse results in contrast with the spurious regression that we have ran before. Durbin-Watson is higher than R-square (1.988255 >> 0.006235)

The independent variable’s RJPY coefficient is statistically significance because the provability is very low 0.0021. At this part we can assume that there is relationship between the two variables. Explaining that, if we increase the RJPY by one unit the REUR will be decreased by -0.075102. The constant term is insignificant because the probability 0.7998 is greater than 5% and 1% significance levels, but we will not ignore it because we do not want a regression where the starting point is (0,0).


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Tests for the Residuals of REUR regression model:

Heteroskedasticity Test: Breusch-Pagan-Godfrey

F-statistic 5.105592 Prob. F(1,1519) 0.0240

Obs*R-squared 5.095189 Prob. Chi-Square(1) 0.0240 Scaled explained SS 9.641376 Prob. Chi-Square(1) 0.0019

Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 06/24/15 Time: 10:57 Sample: 1/02/2008 10/30/2013 Included observations: 1521

Variable Coefficient Std. Error t-Statistic Prob. C 5.02E-05 2.51E-06 20.01502 0.0000

RJPY -0.000758 0.000335 -2.259556 0.0240 R-squared 0.003350 Mean dependent var 5.03E-05

Adjusted R-squared 0.002694 S.D. dependent var 9.79E-05 S.E. of regression 9.78E-05 Akaike info criterion -15.62572 Sum squared resid 1.45E-05 Schwarz criterion -15.61872 Log likelihood 11885.36 Hannan-Quinn criter. -15.62311 F-statistic 5.105592 Durbin-Watson stat 1.751319 Prob(F-statistic) 0.023990

TABLE 1.18 BREUSCH-PAGAN TEST ON REUR RESIDUALS

Running the Breusch-Pagan Heteroskedasticity test on residuals we found that at 1% level significance do not reject the Null Hypothesis which declares that residuals have homoscedasticity.

FIGURE 1.8 REUR REGRESSION MODEL RESIDUALS HISTOGRAM

The Null Hypothesis of normality is rejected and our residuals are distributed non-normal. This may affect a little bit the statistics controls.

0

40

80

120

160

200

240

280

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

Series: Residuals

Sample 1/02/2008 10/30/2013

Observations 1521

Mean -3.37e-19

Median 0.000175

Maximum 0.032609

Minimum -0.029076

Std. Dev. 0.007092

Skewness 0.080175

Kurtosis 4.794474

Jarque-Bera 205.7058

Probability 0.000000


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RJPY = C (1) + C (2)*REUR RJPY = -8.69998857623e-05 - 0.0830172409075*REUR

Dependent Variable: RJPY Method: Least Squares Date: 06/23/15 Time: 01:49 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments


REUR -0.083017 0.026892 -3.087065 0.0021 R-squared 0.006235 Mean dependent var -8.37E-05



TABLE 1.18 RJPY REGRESSION MODEL

From the Table 1.18, we have the same also evidence with the previous Table 1.17. Durbin-Watson is higher than R-square (2.097429 >> 0.006235). This outcome implies that our regression is not spurious. It explain the relationship between RJPY and REUR this time.

The independent variable’s REUR coefficient is statistically significance because the provability is very low 0.0021. At this part we can assume that there is relationship between the two variables. Explaining that, if we increase the REUR by one unit the RJPY will be decreased by -0.083017. The constant term is insignificant because the probability 0.6493 is greater than 5% and 1% significance levels, but we will not ignore it because we do not want a regression where the starting point is (0,0).


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Tests for the Residuals of REUR regression model:

Heteroskedasticity Test: Breusch-Pagan-Godfrey

F-statistic 0.039356 Prob. F(1,1519) 0.8428

Obs*R-squared 0.039407 Prob. Chi-Square(1) 0.8426 Scaled explained SS 0.137772 Prob. Chi-Square(1) 0.7105

Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 06/24/15 Time: 10:56 Sample: 1/02/2008 10/30/2013 Included observations: 1521


REUR -0.000105 0.000531 -0.198384 0.8428 R-squared 0.000026 Mean dependent var 5.56E-05

Adjusted R-squared -0.000632 S.D. dependent var 0.000147 S.E. of regression 0.000147 Akaike info criterion -14.80809 Sum squared resid 3.29E-05 Schwarz criterion -14.80108 Log likelihood 11263.55 Hannan-Quinn criter. -14.80548 F-statistic 0.039356 Durbin-Watson stat 1.788285 Prob(F-statistic) 0.842771

FIGURE 1.19 RJPY BREUSCH-PAGAN RESIDUALS TEST

It is easily noticed that also here the residuals are homoscedastic because the test cannot reject the Null Hypothesis the probabilities are high enough. 0.8428.

FIGURE 1.9 RJPY REGRESSION RESIDUALS HISTOGRAM

The residuals also here are not normal distributed, because their probability is 0.000 which means that the Null hypothesis of normality is rejected.

0

50

100

150

200

250

300

-0.025 0.000 0.025 0.050

Series: Residuals

Sample 1/02/2008 10/30/2013

Observations 1521

Mean -5.53e-19

Median -0.000124

Maximum 0.054097

Minimum -0.040219

Std. Dev. 0.007456

Skewness 0.001634

Kurtosis 8.010689




21

PART II

a) Perform an examination of the autocorrelation and partial autocorrelation functions for up to 12 lags. What do the ACF and PACF function plots and Ljung-Box test tell you?

REUR series

Date: 06/19/15 Time: 01:47 Sample: 1/01/2008 10/30/2013 Included observations: 1521

Autocorrelation Partial Correlation AC PAC Q-Stat Prob | | | | 1 0.009 0.009 0.1116 0.738

| | | | 2 -0.018 -0.018 0.6215 0.733 | | | | 3 -0.006 -0.006 0.6844 0.877 | | | | 4 0.015 0.015 1.0460 0.903 | | | | 5 -0.000 -0.001 1.0461 0.959 | | | | 6 0.030 0.031 2.4238 0.877 | | | | 7 0.024 0.024 3.3340 0.852 | | | | 8 0.015 0.015 3.6740 0.885 | | | | 9 -0.051 -0.050 7.6213 0.573 | | | | 10 -0.001 0.000 7.6221 0.666 | | | | 11 0.032 0.030 9.2176 0.602 | | | | 12 0.002 -0.000 9.2249 0.684

TABLE 1.20 REUR AUTOCORRELATION FUNCTION

RJPY series


Autocorrelation Partial Correlation AC PAC Q-Stat Prob | | | | 1 -0.048 -0.048 3.4808 0.062

| | | | 2 -0.035 -0.037 5.3331 0.069 | | | | 3 -0.013 -0.016 5.5868 0.134 | | | | 4 -0.030 -0.033 7.0035 0.136 | | | | 5 0.000 -0.004 7.0036 0.220 *| | *| | 6 -0.067 -0.070 13.832 0.032 | | | | 7 0.059 0.052 19.217 0.008 | | | | 8 0.011 0.010 19.410 0.013 | | | | 9 -0.005 -0.002 19.447 0.022 | | | | 10 0.051 0.050 23.491 0.009 | | | | 11 -0.043 -0.035 26.345 0.006 | | | | 12 0.040 0.037 28.801 0.004

TABLE 1.21 RJPY AUTOCORRELATION FUNCTION


22

Interpret Table 1.20 and 1.21 Table 1.20: That we can notice here is that all coefficient of ACF and PACF are statistically insignificant. Moreover the values of the coefficients are very low almost close to zero. In other words that means that between them there is no correlation. From this point we can assume with no safety that probably REUR series is an ARMA (0, 0). Table 1.21: In contrast with the REUR series, RJPY series ACF and PACF are insignificant for the first 5 lags because their probability is higher than 10%. It is worthy to mention that the coefficient for the lag 6 are significant, and if we continue to then next lags, the probability is less than 5% which means that the series has autocorrelation. Also here the coefficients values are very low close to zero. The results here are inconclusive. The Ljung-Box test which is a modified approach, which came from the Box-Pierce test, it tests jointly the significance of autocorrelation coefficients through the residuals control.

Null Hypothesis: 𝜌1 = 𝜌2 = ⋯ = 𝜌𝑚 = 0 Ljung-Box statistic is defined as:

𝑄𝐿𝐵 = 𝑇(𝑇 + 2) ∑�̂�𝑠

2

𝑇−𝑠𝑚𝑠=1 (1.11)

Testing now with Ljung-Box statistic for the Table 1.20 the last lag number 12 the 𝑄𝐿𝐵 Do not reject the Null Hypothesis because the probability is 68.4% is high enough. That means the REUR series has no serial autocorrelation.

For the Table 1.21 the results are different. The 𝑄𝐿𝐵 reject the Null Hypothesis and accept

the alternative that the RJPY series has autocorrelation. This happens because at lag 12 the

probability of 𝑄𝐿𝐵 is less that 1% and that implies the rejection of the Null Hypothesis.


23

b) Suppose that ARMA models from order (0, 0) to (2, 2) are plausible for the two currency return series. Use the information criteria AIC and SBIC for each ARMA model order from (0, 0) to (2, 2). Which models do the criteria select i.e., which model for the AIC and which for the SBIC for each return series (four models in total)? Compare the results of the information criteria to the results from ACF and PACF question a)

Representing the equations for AR (p) MA (p) and ARMA (p, q) AR (p)

𝑌𝑡 = 𝛼0 + 𝛼1𝑌𝑡−1 + 𝛼2𝑌𝑡−2 + ⋯ + 𝛼𝑝𝑌𝑡−𝑝 + 휀𝑡 (1.12)

MA (q)

𝑌𝑡 = 𝜇 + 휀𝑡 + 𝜃1휀𝑡−1 + 𝜃2휀𝑡−2 + ⋯ + 𝜃𝑝휀𝑡−𝑞 (1.13)

ARMA (p, q)

𝑌𝑡 = 𝛿 + 𝛼1𝑌𝑡−1 + 𝛼2𝑌𝑡−2 + ⋯ + 𝛼𝑝𝑌𝑡−𝑝 + 휀𝑡 + 𝜃1휀𝑡−1 + 𝜃2휀𝑡−2 +

⋯ + 𝜃𝑝휀𝑡−𝑞 (1.14)

The following equations are the ARMA (0, 0) to (2, 2) that we have estimated.

ARMA Models for REUR

REUR_USD = C(1) REUR_USD = -3.98583969865e-05

Dependent Variable: REUR_USD Method: Least Squares Date: 06/19/15 Time: 01:55 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments

Variable Coefficient Std. Error t-Statistic Prob. C -3.99E-05 0.000182 -0.218500 0.8271 R-squared 0.000000 Mean dependent var -3.99E-05

Adjusted R-squared 0.000000 S.D. dependent var 0.007114

S.E. of regression 0.007114 Akaike info criterion -7.052763

Sum squared resid 0.076932 Schwarz criterion -7.049261 Log likelihood 5364.627 Hannan-Quinn criter. -7.051459 Durbin-Watson stat 1.981881

TABLE 1.22 REUR ARMA (0, 0)


24

REUR_USD = C(1) + [AR(1)=C(2)] REUR_USD = -4.56829888873e-05 + [AR(1)=0.00855673464111]

Dependent Variable: REUR_USD Method: Least Squares Date: 06/24/15 Time: 15:46 Sample (adjusted): 1/03/2008 10/30/2013 Included observations: 1520 after adjustments Convergence achieved after 2 iterations


AR(1) 0.008557 0.025653 0.333562 0.7388 R-squared 0.000073 Mean dependent var -4.56E-05

Adjusted R-squared -0.000585 S.D. dependent var 0.007113


Sum squared resid 0.076849 Schwarz criterion -7.044855 Log likelihood 5361.416 Hannan-Quinn criter. -7.049254 F-statistic 0.111263 Durbin-Watson stat 1.999974 Prob(F-statistic) 0.738756

Inverted AR Roots .01


REUR_USD = C(1) + [AR(1)=C(2),AR(2)=C(3) REUR_USD = -4.64459079444e-05 + [AR(1)=0.00855414440284,AR(2)=-0.0183611266713]

Dependent Variable: REUR_USD Method: Least Squares Date: 06/19/15 Time: 01:57 Sample (adjusted): 1/04/2008 10/30/2013 Included observations: 1519 after adjustments Convergence achieved after 3 iterations


AR(1) 0.008554 0.025679 0.333122 0.7391 AR(2) -0.018361 0.025667 -0.715356 0.4745

R-squared 0.000408 Mean dependent var -4.66E-05




Inverted AR Roots .00+.14i .00-.14i

TABLE 1.24 REUR ARMA (2,0)


25

REUR_USD = C(1) + [AR(1)=C(2),MA(1)=C(3)] REUR_USD = -4.57072483802e-05 + [AR(1)=0.0121038671185,MA(1)=-0.0036176158463]

Dependent Variable: REUR_USD Method: Least Squares Date: 06/19/15 Time: 01:53 Sample (adjusted): 1/03/2008 10/30/2013 Included observations: 1520 after adjustments Convergence achieved after 16 iterations MA Backcast: 1/02/2008


AR(1) 0.012104 0.782544 0.015467 0.9877 MA(1) -0.003618 0.782989 -0.004620 0.9963






Inverted MA Roots .00

TABLE 1.25 REUR ARMA (1, 1) REUR_USD = C(1) + [AR(1)=C(2),MA(1)=C(3),MA(2)=C(4)] REUR_USD = -4.64928308861e-05 + [AR(1)=0.132237221792,MA(1)=-0.123782846334,MA(2)=-0.0189499470392]

Dependent Variable: REUR_USD Method: Least Squares Date: 06/19/15 Time: 02:01 Sample (adjusted): 1/03/2008 10/30/2013 Included observations: 1520 after adjustments Convergence achieved after 26 iterations MA Backcast: 1/01/2008 1/02/2008


AR(1) 0.132237 0.690835 0.191417 0.8482 MA(1) -0.123783 0.691073 -0.179117 0.8579 MA(2) -0.018950 0.026032 -0.727958 0.4668


Adjusted R-squared -0.001543 S.D. dependent var 0.007113 S.E. of regression 0.007119 Akaike info criterion -7.049594 Sum squared resid 0.076822 Schwarz criterion -7.035577 Log likelihood 5361.691 Hannan-Quinn criter. -7.044375 F-statistic 0.219810 Durbin-Watson stat 1.999940 Prob(F-statistic) 0.882685


Inverted MA Roots .21 -.09



26

REUR_USD = C(1) + [AR(1)=C(2),AR(2)=C(3),MA(1)=C(4)] REUR_USD = -4.65566074203e-05 + [AR(1)=0.123051086687,AR(2)=-0.0195896889671,MA(1)=-0.114540768038]



AR(1) 0.123051 1.306596 0.094177 0.9250 AR(2) -0.019590 0.026999 -0.725571 0.4682 MA(1) -0.114541 1.306837 -0.087647 0.9302



Inverted AR Roots .06+.13i .06-.13i




27

REUR_USD = C(1) + [AR(1)=C(2),AR(2)=C(3),MA(1)=C(4),MA(2)=C(5)] REUR_USD = -4.78882692696e-05 + [AR(1)=1.38478847919,AR(2)=-0.988895255425,MA(1)=-1.39405481743,MA(2)=0.996747543413]



AR(1) 1.384788 0.004080 339.4317 0.0000 AR(2) -0.988895 0.004226 -233.9856 0.0000 MA(1) -1.394055 0.001977 -705.2120 0.0000 MA(2) 0.996748 0.001873 532.1130 0.0000


Adjusted R-squared 0.003399 S.D. dependent var 0.007115 S.E. of regression 0.007103 Akaike info criterion -7.053252 Sum squared resid 0.076390 Schwarz criterion -7.035722 Log likelihood 5361.945 Hannan-Quinn criter. -7.046726 F-statistic 2.294377 Durbin-Watson stat 1.973178 Prob(F-statistic) 0.057320

Inverted AR Roots .69-.71i .69+.71i

Inverted MA Roots .70+.71i .70-.71i



28

REUR_USD = C(1) + [MA(1)=C(2)] REUR_USD = -3.98112777815e-05 + [MA(1)=0.00889897904922]



MA(1) 0.008899 0.025657 0.346845 0.7288 R-squared 0.000076 Mean dependent var -3.99E-05


Inverted MA Roots -.01


REUR_USD = C(1) + [MA(1)=C(2),MA(2)=C(3)] REUR_USD = -3.98925760421e-05 + [MA(1)=0.00849177292224,MA(2)=-0.0176450667197]



MA(1) 0.008492 0.025663 0.330897 0.7408 MA(2) -0.017645 0.025664 -0.687550 0.4918






29

ARMA MODELS for the RJPY RUSD_JPY = C(1) RUSD_JPY = -7.09744938706e-05

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:47 Sample: 1/03/2008 10/30/2013 Included observations: 1520

Variable Coefficient Std. Error t-Statistic Prob. C -7.10E-05 0.000191 -0.370635 0.7110 R-squared 0.000000 Mean dependent var -7.10E-05

Adjusted R-squared 0.000000 S.D. dependent var 0.007466 S.E. of regression 0.007466 Akaike info criterion -6.956304 Sum squared resid 0.084667 Schwarz criterion -6.952800 Log likelihood 5287.791 Hannan-Quinn criter. -6.955000 Durbin-Watson stat 2.096746

TABLE 1.31 RJPY ARMA (0, 0) RUSD_JPY = C(1) + [AR(1)=C(2)] RUSD_JPY = -7.16559684958e-05 + [AR(1)=-0.0477972276001]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:48 Sample: 1/03/2008 10/30/2013 Included observations: 1520 Convergence achieved after 3 iterations


AR(1) -0.047797 0.025582 -1.868379 0.0619 R-squared 0.002294 Mean dependent var -7.10E-05


Inverted AR Roots -.05

TABLE 1.32 RJPY ARMA (1, 0)


30

RUSD_JPY = C(1) + [AR(1)=C(2),AR(2)=C(3)] RUSD_JPY = -7.04836113029e-05 + [AR(1)=-0.0502679323386,AR(2)=-0.0372799772109]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:49 Sample (adjusted): 1/04/2008 10/30/2013 Included observations: 1519 after adjustments Convergence achieved after 3 iterations


AR(1) -0.050268 0.025666 -1.958541 0.0503 AR(2) -0.037280 0.025614 -1.455468 0.1457



Inverted AR Roots -.03+.19i -.03-.19i


RUSD_JPY = C(1) + [AR(1)=C(2),MA(1)=C(3)] RUSD_JPY = -6.35103386702e-05 + [AR(1)=0.601343417629,MA(1)=-0.659196702125]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:50 Sample: 1/03/2008 10/30/2013 Included observations: 1520 Convergence achieved after 7 iterations MA Backcast: 1/02/2008


AR(1) 0.601343 0.183218 3.282128 0.0011 MA(1) -0.659197 0.172661 -3.817876 0.0001







31

RUSD_JPY = C(1) + [AR(1)=C(2),MA(1)=C(3),MA(2)=C(4)] RUSD_JPY = -6.47377856051e-05 + [AR(1)=0.493389114595,MA(1)=-0.546306726364,MA(2)=-0.0168829996338]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:50 Sample: 1/03/2008 10/30/2013 Included observations: 1520 Convergence achieved after 10 iterations MA Backcast: 1/01/2008 1/02/2008


AR(1) 0.493389 0.270545 1.823687 0.0684 MA(1) -0.546307 0.271536 -2.011911 0.0444 MA(2) -0.016883 0.034623 -0.487618 0.6259







32

RUSD_JPY = C(1) + [AR(1)=C(2),AR(2)=C(3),MA(1)=C(4)] RUSD_JPY = -6.58768983536e-05 + [AR(1)=0.60019879704,AR(2)=-0.00808551734886,MA(1)=-0.652774607452]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:51 Sample (adjusted): 1/04/2008 10/30/2013 Included observations: 1519 after adjustments Convergence achieved after 15 iterations MA Backcast: 1/03/2008


AR(1) 0.600199 0.266216 2.254559 0.0243 AR(2) -0.008086 0.035419 -0.228281 0.8195 MA(1) -0.652775 0.264958 -2.463695 0.0139



Inverted AR Roots .59 .01




33

RUSD_JPY = C(1) + [AR(1)=C(2),AR(2)=C(3),MA(1)=C(4),MA(2)=C(5)] RUSD_JPY = -6.41767590284e-05 + [AR(1)=-0.403862711222,AR(2)=0.501765492758,MA(1)=0.358222640078,MA(2)=-0.579521212547]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/19/15 Time: 02:52 Sample (adjusted): 1/04/2008 10/30/2013 Included observations: 1519 after adjustments Convergence achieved after 28 iterations MA Backcast: 1/02/2008 1/03/2008


AR(1) -0.403863 0.189277 -2.133718 0.0330 AR(2) 0.501765 0.185977 2.697992 0.0071 MA(1) 0.358223 0.179496 1.995717 0.0461 MA(2) -0.579521 0.177604 -3.262999 0.0011



Inverted AR Roots .53 -.94




34

RUSD_JPY = C(1) + [MA(1)=C(2)] RUSD_JPY = -8.31163821567e-05 + [MA(1)=-0.0521545604963

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/23/15 Time: 21:57 Sample: 1/02/2008 10/30/2013 Included observations: 1521 Convergence achieved after 6 iterations MA Backcast: 1/01/2008


MA(1) -0.052155 0.025625 -2.035323 0.0420 R-squared 0.002490 Mean dependent var -8.37E-05



TABLE 1.38 RJPY ARMA (0, 1) RUSD_JPY = C(1) + [MA(1)=C(2),MA(2)=C(3)] RUSD_JPY = -8.27333215652e-05 + [MA(1)=-0.0516546634749,MA(2)=-0.0398063993234]

Dependent Variable: RUSD_JPY Method: Least Squares Date: 06/23/15 Time: 21:58 Sample: 1/02/2008 10/30/2013 Included observations: 1521 Convergence achieved after 6 iterations MA Backcast: 12/31/2007 1/01/2008


MA(1) -0.051655 0.025652 -2.013642 0.0442 MA(2) -0.039806 0.025652 -1.551761 0.1209






35

We have calculated all the possible combination for ARMA models (0,0) to (2,2) for both series REUR and RJPY. We are going to use the information criteria AIC and SBIC in order to choose from the two currencies the best models depend on that criteria, (total four models).

AIC = ln∑ 𝑢2

𝑇+

2𝑘′

𝑇 (1.15 )

SBIC= ln∑ �̂�2

𝑇+

𝑘′

𝑇𝑙𝑛𝑇 (1.16 )

Where,

∑ �̂�2 = 𝑠𝑢𝑚 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠

𝑇 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

𝑘′ = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 (𝑝 + 𝑞 + 1)

Schwarz Information Criterion for REUR ARMA MODELS

AR / MA 0 1 2

0 -7.049.261 -7.044.520 -7.040.023

1 -7.044.855 -7.040.035 -7.035.577

2 -7.039.732 -7.034.924 -7.035.722

TABLE 1.41 SBIC REUR

Akaike Information Criterion for REUR ARMA MODELS

AR / MA 0 1 2

0 -7.052.763 -7.051.525 -7.050.530

1 -7.051.864 -7.050.548 -7.049.594

2 -7.050.250 -7.048.948 -7.053.252 TABLE 1.40 AIC REUR


36

Akaike Information Criterion for RJPY ARMA MODELS

AR / MA 0 1 2

0 -6.956.304 -6.953.735 -6.953.875

1 -6.957.286 -6.958.506 -6.957.321

2 -6.956.809 -6.957.067 -6.959.728

TABLE 1.42 AIC RJPY

Schwarz Information Criterion for RJPY ARMA MODELS

AR / MA 0 1 2

0 -6.952.800 -6.946.730 -6.943.368

1 -6.950.277 -6.947.993 -6.943.304

2 -6.946.290 -6.943.043 -6.942.197

TABLE 1.43 SBIC RJPY

The Final Four models for REUR and RJPY chosen by the criteria mentioned above are Comparing the results of the information criteria to the results from ACF and PACF of question a) we cannot explain the results. From the table of ACF and PACF for the REUR we found that there is no autocorrelation which SBIC proves it, because it selected the AR (0), but AIC did not have such an information instead chose the an ARMA (2, 2). For the RJPY the autocorrelation table shows that there is autocorrelation after 6th lag. Instead of that the AIC chose ARMA (2, 2) as the best model and SBIC AR(0). In sum, we cannot rely on the ACF and PACF information because the divergence is very significant.

(*The red boxes were the best models but there is a restriction not to choose them and to choose the next “best” model).

REUR RJPY

AIC ARMA (2,2) ARMA(2,2)

SIBC AR(1) AR(1) TABLE 1.44 SELECTED MODEL


37

c) For the four models selected above, the inverses of the AR and MA roots of the

characteristic equation can be used to check whether the process implied by the model is stationary and invertible. What is the result in this case?

Inverse Roots of AR/MA Polynomial(s)

Specification: REUR_USD C AR(1)

Date: 06/24/15 Time: 18:35

Sample: 1/01/2008 10/30/2013

Included observations: 1520 AR Root(s) Modulus 0.008557 0.008557 No root lies outside the unit circle.

ARMA Model is stationary.

The REUR ARMA (0, 1) and the RJPY ARMA (0, 1) from the figures are stationary procedures and they do not have unit root because the roots lies outside the unit circle.


Specification: RUSD_JPY C AR(1)

Date: 06/24/15 Time: 18:37

Sample: 1/03/2008 10/30/2013

Included observations: 1520 AR Root(s) Modulus Cycle -0.047797 0.047797 No root lies outside the unit circle.

ARMA Model is stationary.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

AR

ro

ots


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

AR

ro

ots


FIGURE 1.10 REUR ARMA (1, 0) UNIT ROOTS FIGURE 1.11 RJPY ARMA (1, 0) UNIT ROOTS


38

Furthermore REUR ARMA(2, 2) and RJY ARMA (2, 2) are stationary because their unit roots do not lies outside the unit root test and also the MA parts of the ARMA procedure is invertible.

Inverse Roots of AR/MA Polynomial(s) Specification: RUSD_JPY C AR(1) AR(2) MA(1) MA(2)

Date: 06/24/15 Time: 18:45

Sample: 1/03/2008 10/30/2013

Included observations: 1519 AR Root(s) Modulus -0.938506 0.938506

0.534643 0.534643 No root lies outside the unit circle.

ARMA Model is stationary. MA Root(s) Modulus -0.961161 0.961161


ARMA Model is invertible.

Inverse Roots of AR/MA Polynomial(s) Specification: RUSD_JPY C AR(1) AR(2) MA(1) MA(2)

Date: 06/24/15 Time: 18:47

Sample: 1/03/2008 10/30/2013

Included observations: 1519 AR Root(s) Modulus -0.938506 0.938506


ARMA model is stationary. MA Root(s) Modulus -0.961161 0.961161


ARMA Model is invertible.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

AR roots

MA roots


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

AR roots

MA roots


FIGURE 1.12 REUR ARMA (2, 2) FIGURE 1.13 RJPY ARMA (2, 2)


39

d) For each currency series, estimate one model that you fell most appropriate given

the results that you found from the previous two questions (two models total). The previous models are all stationary and inverted. We will choose the best model based on the Akaike Information Criterion. For the REUR series we will choose the ARMA (2, 2) because Akaike Information Criterion is the lower comparing with the ARMA (1 ,0).

For the RJPY series we will choose the ARMA (2, 2) because Akaike Information Criterion is the lower comparing with the ARMA (1 ,0).

Note: If we have chosen the SBIC as the final criterion for the both series the correct model would have been the ARMA (1, 0).

REUR AIC

ARMA (2,2) -7.053.252

ARMA (1, 0) -7.051.864

TABLE 1.45 ARMA (2,2) SELECTED

RJPY AIC

ARMA (2,2) -6.959.728

ARMA (1, 0) -6.957.286

TABLE 1.46 ARMA (2,2) SELECTED


40

PART III Use the estimated models from Part II to forecast future values of each of the return series. Considering that the model selected was estimated using observations from 02//01/2008 to 10/07/2013 (1441 daily observations), leave out 80 reaming observations ( for the period 11/07/2013 to 30/10/2013) to construct forecasts and to test forecast accuracy.

a) Calculate a sequence of one-step-ahead forecasts, rolling the sample forward one observation after each forecast, in order to use actual rather than forecasted values for lagged dependent variables. Produce the forecast screenshot in Eviews.

Dependent Variable: REUR_USD

Method: Least Squares Date: 06/24/15 Time: 20:00 Sample (adjusted): 1/04/2008 7/10/2013 Included observations: 1439 after adjustments Convergence achieved after 20 iterations MA Backcast: 1/02/2008 1/03/2008


AR(1) 0.744482 0.003678 202.3975 0.0000 AR(2) -0.991695 0.003558 -278.7192 0.0000 MA(1) -0.737843 0.002025 -364.3906 0.0000 MA(2) 0.994653 0.001717 579.3973 0.0000



Inverted AR Roots .37-.92i .37+.92i

Inverted MA Roots .37+.93i .37-.93i

TABLE 1.47 REUR ARMA (2, 2) IN SAMPLE


41

FIGURE 1.14 REUR ARMA (2, 2) FORECAST

Dependent Variable: RUSD_JPY

Method: Least Squares Date: 06/24/15 Time: 20:05 Sample (adjusted): 1/04/2008 7/10/2013 Included observations: 1439 after adjustments Convergence achieved after 42 iterations MA Backcast: 1/02/2008 1/03/2008


AR(1) -0.446956 0.215095 -2.077945 0.0379 AR(2) 0.464586 0.211048 2.201324 0.0279 MA(1) 0.402205 0.205785 1.954487 0.0508 MA(2) -0.538645 0.203392 -2.648306 0.0082



Inverted AR Roots .49 -.94


TABLE 1.48 RJPY ARMA (2, 2) IN SAMPLE

-.016

-.012

-.008

-.004

.000

.004

.008

.012

.016

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

REUR_USDF ± 2 S.E.

Forecast: REUR_USDF

Actual: REUR_USD

Forecast sample: 7/11/2013 10/30/2013

Included observations: 80

Root Mean Squared Error 0.003776

Mean Absolute Error 0.002812

Mean Abs. Percent Error 104.6104

Theil Inequality Coefficient 0.967634

Bias Proportion 0.044183

Variance Proportion 0.900423

Covariance Proportion 0.055393


42

FIGURE 1.15 RJPY ARMA (2, 2) FORECAST

b) What is the square root of mean error (RMSE), the MAE and the MAPE? Create a comparative Table for each currency.

Here we write down all the criteria for the forecast evaluation. Root Mean Square Error

𝑅𝑀𝑆𝐸 = √1

𝑀 ∑ (𝑌𝑡

𝑓− 𝑌𝑡

𝛼)2𝑀𝑡=1 (1.17)

𝑌𝑡

𝑓= forecasted value

𝑌𝑡𝛼= observed value

M = number of periods

Mean Absolute Error

𝑀𝐴𝐸 =1

𝑀∑ |𝑌𝑡

𝑓− 𝑌𝑡

𝛼|𝑀𝑡=1 (1.18)

Mean Absolute Percentage Error

𝑀𝐴𝑃𝐸 = 1

𝑀∑ |

𝑌𝑡𝑓

−𝑌𝑡𝛼

𝑌𝑡𝛼 |𝑀

𝑡=1 (1.19)

-.020

-.015

-.010

-.005

.000

.005

.010

.015

.020

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

RUSD_JPYF ± 2 S.E.

Forecast: RUSD_JPYF

Actual: RUSD_JPY

Forecast sample: 7/11/2013 10/30/2013










43

Theil’s Inequality Coefficient

𝑈 = √

1

𝑀 ∑ (𝑌𝑡

𝑓−𝑌𝑡

𝛼)2𝑀𝑡=1

√1

𝑀 ∑ (𝑌𝑡

𝛼)2𝑀𝑡=1

(1.20)

Theil’s Inequality coefficient is independent from the unit of measurement and that is why the most appropriate to compare models’ predictability power.

If the forecasted values are the same with the real values, the price of U is ZERO

The forecasts are very bad when the U>1.

When U=1 the all forecasts are ZERO.

In the case of U=1, in our model we are not using prices but variations of the prices. So that means the predicted variation are zero and as a result the have the continuation of the present situation. Analyzing a bit more the Theil’s Inequality Coefficient a new equation is coming up.

𝑈2 =(�̅�𝑓−�̅�𝛼)2

𝛢+

(𝜎𝑓−𝜎𝛼)2

𝛢+

2(1−𝜌)𝜎𝑓𝜎𝛼

𝛢 (1.21)

Where,

𝑌𝑡𝑓̅̅̅̅

= mean forecasted values

𝑌𝑡𝛼̅̅ ̅̅ = mean observed values

𝜎𝑓 = standard deviation of 𝑌𝑡𝑓

𝜎𝛼= standard deviation of 𝑌𝑡𝛼

𝜌 = correlation coefficient of 𝑌𝑡𝑓

and 𝑌𝑡𝛼.

The first term of the (1. ) equation (�̅�𝑓−�̅�𝛼)2

𝛢 is the bias proportion. The second term

(𝜎𝑓−𝜎𝛼)2

𝛢 is variance proportion and the last term is

2(1−𝜌)𝜎𝑓𝜎𝛼

𝛢 the autocorrelation and

it can be consider as a measure of incomplete covariation. The first two terms, bias proportion and variance proportion is the systematic errors that must be avoided. The third term is non-systematic which cannot be avoided.


44

ARMA (2, 2)

REUR RJPY

RMSE 0,003776 0,00647

MAPE 104,6104 101,1219

MAE 0,002812 0,00514

TABLE 1.49 ARMA (2,2) RMSE, MAPE, MAE

Briefly, both REUR and RJPY has very low RMSE, which is means that the actual values are close to the forecast values. But the Theil’s Inequality Coefficient is close to 1 which means the results continue the present situation. So the model’s predictability power is really low and we cannot rely on it. Both REUR and RJPY have a Theil greater than 0.90. REURS Theil’s Bias Proportion is extremely high in comparison to the RJPY. We have referred that we must avoid this because Bias Proportion is the systematic source of error. Both REUR and RJPY have a great Variance Proportion, that we must also avoid that kind error because is systematic. In sum, the ARMA (2, 2) is not a good model in order to forecast the two currencies.

ARMA (2, 2)

REUR RJPY

Theil’s Inequality Coefficient 0,967634 0,912072

Bias Proportion 0,044183 0,000405

Variance Proportion 0,900423 0,873446

Covariance Proportion 0,055393 0,12615

TABLE 1.50 ARMA (2,2) THEIL’S COEFFICIENT


45

c) Compare the forecasting accuracy of your chosen ARMA model to that of an arbitrary ARMA (1, 1) and a random walk with the drift in the price levels.

FIGURE 1.16 REUR ARMA (0 , 0) FORECAST

For both currencies ARMA’s (0, 0) prediction power “does not exist”. ARMA (0, 0) cannot give us reliable forecasts, as we can also see from the graph the blue line which is the forecast line is the same with axis x’x. Comparing to this ARMA (2, 2) is better.

-.015

-.010

-.005

.000

.005

.010

.015

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

REUR_USDF_00 ± 2 S.E.

Forecast: REUR_USDF_00

Actual: REUR_USD

Forecast sample: 7/11/2013 10/30/2013







Variance Proportion NA

Covariance Proportion NA

-.016

-.012

-.008

-.004

.000

.004

.008

.012

.016

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

RUSD_JPYF4 ± 2 S.E.

Forecast: RUSD_JPYF4

Actual: RUSD_JPY

Forecast sample: 7/11/2013 10/30/2013







Variance Proportion NA

Covariance Proportion NA

FIGURE 1.17 RJPY ARMA (0 , 0) FORECAST


46

FIGURE 1.18 REUR ARMA (1 , 1) FORECAST

The ARMA (2,2) in contrast with the ARMA (1,1) for the REUR_USD is that the Theil’s Inequality Coefficient is lower than the ARMA (1, 1), from this point the ARMA (2, 2) is slighter better than ARMA (1, 1). As the errors concerns there is not a noticeable differences between them. So ARMA still remains the best Model comparing to ARMA (0, 0) and ARMA (1, 1), but still it is not a good model for making forecasts.

-.015

-.010

-.005

.000

.005

.010

.015

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

REUR_USDFARMA11 ± 2 S.E.

Forecast: REUR_USDFARMA11

Actual: REUR_USD

Forecast sample: 7/11/2013 10/30/2013









REUR_USD ARMA(2,2) ARMA(1,1)




Covariance Proportion 0,055393 0,015277 TABLE 1.51 ARMA(2,2)-ARMA(0,0) COMPARISON REUR

REUR_USD

ARMA(2, 2) ARMA(1,1)

RMSE 0,003776 0,003779

MAPE 104,6104 101,8828

MAE 0,002812 0,002816

TABLE 1.52 ARMA(2,2)-ARMA(0,0) COMPARISON REUR


47

For the RJPY series the “best” still remains the ARMA (2, 2) because Theil’s Inequality is lower than the ARMA’s (1, 1). And the errors don’t have big differences. Also comparing the blue lines form the forecast model ARMA’s (2, 2) is more vivid than the ARMA’s (1, 1). In conclusion, the “best” forecast model for both currencies among the ARMA(1 ,1) and ARMA(0, 0) and ARMA(2 ,2), is the ARMA(2, 2). Despite, that ARMA (2, 2) is the best does it means that is also an good forecast model. It forecast power is significant low and we cannot use for predictions.

RJPY_USD

ARMA (2, 2) ARMA(1, 1)




Covariance Proportion 0,12615 0,104453

TABLE 1.53 ARMA(2,2)-ARMA(1,1) COMPARISON RJPY

RJPY_USD

ARMA(2, 2) ARMA(1, 1)

RMSE 0,00647 0,006510

MAPE 101,1219 103,3668

MAE 0,005140 0,005201

TABLE 1.54 ARMA(2,2)-ARMA(0,0) COMPARISON RJPY

-.020

-.015

-.010

-.005

.000

.005

.010

.015

.020

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

RUSD_JPYF_ARMA11 ± 2 S.E.

Forecast: RUSD_JPYF_ARMA11

Actual: RUSD_JPY

Forecast sample: 7/11/2013 10/30/2013









FIGURE 1.19 RJPY ARMA (1 , 1) FORECAST


48

SECTION II VOLATILITY MODELING & VALUE-AT-RISK

PART I

A) Before estimating a GARCH-type model, test for ARCH effects in stock returns. Compute the Engle test for ARCH effects to make sure that this class of models is appropriate for the data. Estimate an ARMA (1, 1) model and then test or the presence of ARCH in the residuals. Use five lags for the tests and comment on the Engle ARCH test (F- and x^2 version) for the presence of ARCH in each of the stock market returns.

ARCH rank (p) MODEL

𝜎𝑡2 = 𝛼0 + 𝛼1𝑢𝑡−1

2 + 𝛼2𝑢𝑡−22 + ⋯ + 𝛼𝑝𝑢𝑡−𝑝

2 (2.1)

Now the function (2.1) can be generalized, thus the conditional variance 𝜎𝑡2 to be an

additional a function of itself with time lag. To be more specific: GARCH rank (p) MODEL

𝜎𝑡2 = 𝛼0 + 𝛼1𝑢𝑡−1

2 + 𝛼2𝑢𝑡−22 + ⋯ + 𝛼𝑝𝑢𝑡−𝑝

2 + 𝛾1𝜎𝑡−12 + ⋯ + 𝛾𝑞𝜎𝑡−𝑞

2 (2.2) The function (2.2) is known as Generalized Autoregressive Conditional Heteroskedasticity or GARCH model. So here we have to estimate a GARCH-type model for our data. Few words for our data We are going to use daily “closing” prices of the DAX/German Stock Index and the NIKKEI225/Japanese index series. There are total of 1523 observations running from 31/12/2007 to 30/10/2013. DAX_30= Deutcher Aktien-Indice as we know is a German Stock Index. It is traded on the Frankfurt Stock Exchange which is the biggest stock exchange in Germany. DAX measures the development of the 30 largest and best-performing companies on the German equities market and represents around 80% of the market capitalization in Germany. NIKKEI 225 It is a price-weighted index consisting of 225 prominent stocks on the Tokyo Stock Exchange. The Nikkei has been calculated since 1950 and its direction is considered an indicator of the state of the Japanese economy.


49

Conditions In order to run GARCH-type model and generally all the ARCH family models, time series data must fulfill some mandatory conditions.

1) Firstly, our data must be stationary, which means that the mean and the variance have to not change through the time, in other words to be constant values.

2) Secondly, all ARCH family models assume that the variance of the residuals is related to the size of previous periods’ residuals, giving rise to Volatility Clustering. This means that low volatility tend to be followed by periods of low volatility for a prolonged time. As well as, periods of high volatility tend to be followed by periods of high volatility for a prolonged time. This suggest that residuals or error terms is conditionally Heteroskedasticity and it can be represented by ARCH and GARCH model.

Performing our first steps for our analysis, we have to determine whether our data is stationary or non-stationary. In order to prove that, we are going to perform unit root tests, and specially we will use Augmented Dickey Fuller (ADF), Phillips-Perron (PP) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests.

4,000

6,000

8,000

10,000

12,000

14,000

2008 2009 2010 2011 2012 2013

DAX 30

6,000

8,000

10,000

12,000

14,000

16,000

2008 2009 2010 2011 2012 2013

NIKKEI225

FIGURE 2.1: DAX, NIKKEI DAILY “CLOSING” PRICES


50

It is easily noticeable from the Figure 1 that we have suspicions for non-stationarity in our daily “closing” prices observations. In order to cast out our doubt we will perform the tests that we mentioned before. DAX Unit Root Tests.

Null Hypothesis: DAX_30 has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -1.761352 0.4001


TABLE 2.1 DAX ADF TEST

Null Hypothesis: DAX_30 has a unit root

Exogenous: Constant

Bandwidth: 1 (Newey-West automatic) using Bartlett kernel Adj. t-Stat Prob.*

Phillips-Perron test statistic -1.766306 0.3976

Test critical values: 1% level -3.434440 5% level -2.863233 10% level -2.567720


TABLE 2.2 DAX PP TEST

Null Hypothesis: DAX_30 is stationary

Exogenous: Constant



Asymptotic critical values*: 1% level 0.739000 5% level 0.463000 10% level 0.347000


TABLE 2.3 DAX KPSS TEST 1


51

NIKKEI Unit Root Tests.

Null Hypothesis: NIKKEI225 has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -2.114693 0.2390 Test critical values: 1% level -3.434440

5% level -2.863233 10% level -2.567720 *MacKinnon (1996) one-sided p-values.

TABLE 2.4 NIKKEI ADF TEST

Null Hypothesis: NIKKEI225 has a unit root Exogenous: Constant Bandwidth: 1 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*



TABLE 2.5 NIKKEI PP TEST

Null Hypothesis: NIKKEI225 is stationary

Exogenous: Constant



Asymptotic critical values*: 1% level 0.739000 5% level 0.463000 10% level 0.347000


TABLE 2.6 NIKKEI KPSS TEST


52

Briefly, commenting on unit test outputs, for the DAX daily prices we cannot reject the Null Hypothesis, that the series has a unit root because in Table 2.1 and 2.2 probability is higher than 10% and at KPSS test Table 2.3 LM is 1.154971 which is greater that the critical values so the Null Hypothesis that DAX series is stationary is rejected. In other words DAX series is non-stationary and we will take the log-returns to make it stationary. As far Nikkei concerns, ADF test in Table 2.4 and PP test in Table 2.5 cannot reject the Null Hypothesis because 0.2390> 0,10 and 0.2527>0,10 but from the KPSS test we have a “paradox”. LM-stat is 0,581879 which in 5% and 10% significant level Null Hypothesis rejected but in 1% does not rejected. In order to continue our analysis we will assume a 5%sgnificant level so both series DAX and NIKKEI are non-stationary and we will examine the log returns.

4,000

6,000

8,000

10,000

12,000

14,000

16,000


2008 2009 2010 2011 2012 2013

NIKKEI225 DAX 30

FIGURE 2.2 NIKKEI, DAX PRICES


53

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

RDAX

-.15

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

RNIKKEI

Figure 2.3 represents DAX, NIKKEI log-returns, which now are referred as RDAX and RNIKKEI. As we can see from the graphs the mean now is not changing through the time, but instead is moving up and down from the zero. With the log-returns our data fulfill the first condition to be stationary and we will show also the unit roots tests which confirm this state.

RDAX

Mean 3.33E-05

Median 0.000221

Maximum 0.123697

Minimum -0.096010

Std. Dev. 0.020002

Skewness 0.020196

Kurtosis 7.381147



Sum 0.050700

Sum Sq. Dev. 0.608494 Observations 1522

RNIKKEI

Mean -3.55E-05

Median 0.000000

Maximum 0.132346

Minimum -0.121110

Std. Dev. 0.017669

Skewness -0.577337

Kurtosis 10.98250



Sum -0.054050

Sum Sq. Dev. 0.474863

Observations 1522

FIGURE 2.3 DAX, NIKKEI LOG-RETUNS


54

RDAX Unit Roots Tests.

Null Hypothesis: RDAX has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=12)



TABLE 2.7 RDAX ADF TEST

Null Hypothesis: RDAX has a unit root

Exogenous: Constant

Bandwidth: 5 (Newey-West automatic) using Bartlett kernel Adj. t-Stat Prob.*


Test critical values: 1% level -3.434443

5% level -2.863235

10% level -2.567720


TABLE 2.8 RDAX PP TEST

Null Hypothesis: RDAX is stationary Exogenous: Constant Bandwidth: 5 (Newey-West automatic) using Bartlett kernel

LM-Stat.



TABLE 2.9 RDAX KPSS TEST


55

RNIKKEI Unit Root Tests.

Null Hypothesis: RNIKKEI has a unit root

Exogenous: Constant

Lag Length: 0 (Automatic - based on SIC, maxlag=12) t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -39.92947 0.0000

Test critical values: 1% level -3.434443

5% level -2.863235

10% level -2.567720


TABLE 2.10 RNIKKEI ADF TEST

Null Hypothesis: RNIKKEI has a unit root Exogenous: Constant Bandwidth: 9 (Newey-West automatic) using Bartlett kernel



TABLE 2.11 RNIKKEI PP TEST

Null Hypothesis: RNIKKEI is stationary Exogenous: Constant Bandwidth: 9 (Newey-West automatic) using Bartlett kernel



TABLE 2.12 RNIKKEI KPSS TEST


56

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

AR roots

MA roots


FIGURE 2.4 RDAX INVERTED ROOTS

At 5% level significance all the Unit Tests confirm that RDAX and RNIKKEI are stationary. The fact that our data are stationary, gives us the access to go further to our analysis. Before estimating the GARCH type model, we have to check out our data for ARCH effect. In this part of our case we will estimate an ARMA (1, 1). Estimation Equation: RDAX = C(1) + [AR(1)=C(2),MA(1)=C(3) (2.3) RDAX = 3.27387400402e-05 + [AR(1)=-0.266023506183,MA(1)=0.276542661939] (2.4)

Dependent Variable: RDAX

Method: Least Squares Date: 06/20/15 Time: 18:36 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Convergence achieved after 19 iterations White Heteroskedasticity-consistent standard errors & covariance MA Backcast: 1/01/2008

Variable Coefficient Std. Error t-Statistic Prob. C 3.27E-05 0.000518 0.063208 0.9496

AR(1) -0.266024 2.983757 -0.089157 0.9290 MA(1) 0.276543 2.975043 0.092954 0.9260

R-squared 0.000129 Mean dependent var 3.33E-05




TABLE 2.13 RDAX ARMA (1, 1)


57

Table 2.13 represent an ARMA (1, 1) for the RDAX and Figure 2.4 represents the stationarity of this ARMA (1, 1) procedure. The stationarity is proved because ARMA’s roots are inside the inverted circle. What we will do next is to check ARMA’s (1, 1) residuals for ARCH effect. We will test this with 2 ways, first with the residual graph and then with the Engle ARCH test.

From the Figure 2.5 we can see that from the end of the third quarter of 2008 until the second quarter of 2009 we have a high volatility period, from the third quarter of 2009 until end of the second quarter of 2011 residuals represent low volatility at the third quarter of 2011 we have again high volatility until the of the 2011. From this graph we can abstract the information that ARMA (1, 1) of RDAX has an ARCH effect because there is volatility clustering, and our empirical diagnosis will be confirmed from the Engle ARCH test.

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

RDAX Residuals

FIGURE 2.5 RDAX ARMA(1,1) RESIDUALS


58

Heteroskedasticity Test: ARCH F-statistic 44.15565 Prob. F(5,1510) 0.0000

Obs*R-squared 193.3811 Prob. Chi-Square(5) 0.0000

Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 06/20/15 Time: 17:53 Sample (adjusted): 1/09/2008 10/30/2013 Included observations: 1516 after adjustments

Variable Coefficient Std. Error t-Statistic Prob. C 0.000168 2.95E-05 5.682497 0.0000

RESID^2(-1) 0.012925 0.025271 0.511476 0.6091 RESID^2(-2) 0.123735 0.025137 4.922493 0.0000 RESID^2(-3) 0.154155 0.025025 6.159958 0.0000 RESID^2(-4) 0.101913 0.025137 4.054261 0.0001 RESID^2(-5) 0.188896 0.025271 7.474691 0.0000

R-squared 0.127560 Mean dependent var 0.000401


TABLE 2.14 RDAX ARMA (1, 1) ENGLE ARCH TEST

There is ARCH effect in the German stock market returns. This is confirmed by the probability of the F and X-square. The probability is zero for both statistics. That means that the Null Hypothesis of Homoscedasticity is rejected and we choose the Alternative which declares the Heteroskedasticity of the residuals. In sum, our data for the DAX German stock Index fulfill the criteria to run a GARCH type model.


59

Estimation Equation: RNIKKEI = C(1) + [AR(1)=C(2),MA(1)=C(3) (2.5) RNIKKEI = 2.72773511647e-05 + [AR(1)=0.830755466947,MA(1)=-0.855128109923 (2.6)

Dependent Variable: RNIKKEI

Method: Least Squares Date: 06/20/15 Time: 18:38 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Convergence achieved after 16 iterations White Heteroskedasticity-consistent standard errors & covariance MA Backcast: 1/01/2008

Variable Coefficient Std. Error t-Statistic Prob. C 2.73E-05 0.000394 0.069298 0.9448

AR(1) 0.830755 0.166412 4.992154 0.0000 MA(1) -0.855128 0.158338 -5.400662 0.0000





TABLE 2.15 RNIKKEI ARMA (1, 1)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

AR roots

MA roots


FIGURE 2.6 RNIKKEI INVERTED ROOTS


60

Table 2.15 represent an ARMA (1, 1) for the RNIKKEI and Figure 2.6 represents the stationarity of this ARMA (1, 1) procedure. The stationarity is proved because ARMA’s roots are inside the inverted circle.

FIGURE 2.7 RNIKKEI ARMA (1, 1) RESIDUALS

There is also here volatility clustering. From the second quarter of 2009 we have low volatility until the fourth quarter of 2010. Means that low volatility period is followed by low volatility period. Next we will perform the Engle Arch test on the ARMA (1 ,1) RNIKKEI residuals.

-.15

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

RNIKKEI Residuals


61

Heteroskedasticity Test: ARCH F-statistic 137.3536 Prob. F(5,1510) 0.0000

Obs*R-squared 473.9418 Prob. Chi-Square(5) 0.0000

Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 06/20/15 Time: 17:52 Sample (adjusted): 1/09/2008 10/30/2013 Included observations: 1516 after adjustments


RESID^2(-1) 0.050431 0.025722 1.960663 0.0501 RESID^2(-2) 0.393854 0.025177 15.64340 0.0000 RESID^2(-3) 0.018227 0.027126 0.671927 0.5017 RESID^2(-4) 0.211025 0.025174 8.382808 0.0000 RESID^2(-5) 0.027798 0.025720 1.080807 0.2800

R-squared 0.312627 Mean dependent var 0.000311


TABLE 2.16 RNIKKEI ARMA (1, 1) ARCH TEST

There is ARCH effect in the Japanese stock market returns. This is confirmed by the probability of the F and X-square. The probability is zero for both statistics. That means that the Null Hypothesis of Homoscedasticity is rejected and we choose the Alternative which declares the Heteroskedasticity of the residuals. We are now ready to estimate GARCH-type model for both stock returns, because they fulfilled all the conditions that we mentioned before.


62

b) Estimate a GARCH-type model for each of the two series. It is necessary to specify the mean and the variance equations, as well as the estimations technique and sample. In order to estimate a GARCH (1, 1) model use one ARCH and one GARCH term. Leave the default estimation options unchanged.

Dependent Variable: RDAX

Method: ML - ARCH Date: 06/21/15 Time: 00:47 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Convergence achieved after 17 iterations Bollerslev-Wooldridge robust standard errors & covariance MA Backcast: 1/01/2008 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob. C 0.000772 0.000341 2.261692 0.0237

AR(1) 0.700098 0.136366 5.133963 0.0000 MA(1) -0.730855 0.130967 -5.580436 0.0000

Variance Equation C 6.37E-06 3.49E-06 1.824903 0.0680

RESID(-1)^2 0.097691 0.030613 3.191139 0.0014 GARCH(-1) 0.887060 0.034320 25.84668 0.0000

R-squared -0.000473 Mean dependent var 3.33E-05

Adjusted R-squared -0.001791 S.D. dependent var 0.020008 S.E. of regression 0.020026 Akaike info criterion -5.277138 Sum squared resid 0.608782 Schwarz criterion -5.256124 Log likelihood 4019.264 Hannan-Quinn criter. -5.269315 Durbin-Watson stat 1.929954



TABLE 2.17 RDAX GARCH (1, 1)

As we can easily see the table 2.17 depicts the GARCH-type model for the RDAX (we have defined that RDAX is the log returns of DAX German Index stock prices).


63

The Table consists two kind of equations, the Mean and Variance equation.

Mean Equation

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000772 0.000341 2.261692 0.0237

AR(1) 0.700098 0.136366 5.133963 0.0000 MA(1) -0.730855 0.130967 -5.580436 0.0000

TABLE 2.18 RDAX GARCH (1, 1) MEAN EQ.

RDAX = 0.000772077941062 + [AR (1) =0.70009792746, MA (1) =-0.730855189197 (2.7) The Mean Equation of the Table 2.18 has three variables, C, AR (1), and MA (1). All these variables are significant because probability is less than 5%. From the mean equation we derived the residuals and we have estimated the variance equation of residuals.


RESID(-1)^2 0.097691 0.030613 3.191139 0.0014 GARCH(-1) 0.887060 0.034320 25.84668 0.0000

TABLE 2.19 RDAX GARCH (1, 1) VARIANCE EQ. GARCH = 6.36609338872e-06 + 0.0976910615361*RESID (-1) ^2 + (2.8) 0.887059881654*GARCH (-1) The variance equation, has three variables, C, RESID (-1) ^2 and GARCH (-1). Only the constant term is insignificant. The other two variables RESID (-1) ^2 which is the ARCH part of the GARCH model and the GARCH (-1) are significant. The GARCH dependent variable of the equation (2.8) is the volatility of the DAX log-returns. The volatility of DAX log returns can be explained by the significant variables RESID (-1) ^2 and GARCH (-1), which are the shocks which influence the DAX.


64

Dependent Variable: RNIKKEI

Method: ML - ARCH Date: 06/20/15 Time: 18:39 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Convergence achieved after 12 iterations Bollerslev-Wooldridge robust standard errors & covariance MA Backcast: 1/01/2008 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)


AR(1) -0.460705 0.486582 -0.946817 0.3437 MA(1) 0.422688 0.494797 0.854265 0.3930


RESID(-1)^2 0.113330 0.031221 3.629907 0.0003 GARCH(-1) 0.861106 0.030247 28.46915 0.0000

R-squared -0.000953 Mean dependent var -3.55E-05




TABLE 2.20 RNIKKEI GARCH (1, 1)


65

Mean Equation

RNIKKEI = 0.000472682918 + [AR(1)=-0.460704595196,MA(1)=0.422688040307 (2.9)

The Mean Equation of the Table 2.18 has three variables, C, AR (1), and MA (1). All these variables are not significant because probability is higher than 5%. From the mean equation we derived the residuals and we have estimated the variance equation of residuals.

GARCH = 7.14562151202e-06 + 0.113329578779*RESID(-1)^2 + (2.10) 0.861105580633*GARCH(-1) The variance equation, has three variables, C, RESID (-1) ^2 and GARCH (-1). All the variables are significant. The two variables RESID (-1) ^2 which is the ARCH part of the GARCH model and the GARCH (-1) are also significant. The GARCH depended variable of the equation (2.10) is the volatility of the NIKKEI log-returns. The volatility of NIKKEI log returns can be explained by the significant variables RESID (-1) ^2 and GARCH (-1), which are the shocks which influence the NIKKEI. The estimation method that we use for both stock market returns is ML-ARCH (Marquardt). For the Error distribution we used Normal (Gaussian). Our sample start from the 12/31/2007 – 10/30/2013 (Eviews default) but here our adjusted sample starts from 01/02/2008- 10/30/2013. From the 1523 initially observation now we have 1521. This issue occurred from the fact that we have used logarithmic first differences, and that is why we lose the 2 observations. It should be also referred that for the GARCH estimation we used from the tab option the coefficient covariance to Heteroskedasticity consistent covariance (Bollerslev-Wooldridge).


AR(1) -0.460705 0.486582 -0.946817 0.3437 MA(1) 0.422688 0.494797 0.854265 0.3930

TABLE 2.21 RNIKKEI GARCH (1, 1) MEAN EQ.


RESID(-1)^2 0.113330 0.031221 3.629907 0.0003 GARCH(-1) 0.861106 0.030247 28.46915 0.0000

TABLE 2.22 RNIKKEI GARCH (1, 1) VARIANCE EQ.


66

c) In order to perform diagnostic testing first produce “Actual”, “Fitted” and “Residuals graphs”. Then produce the GARCH graph. For the residuals testing generate the Correlogram-Q statistics and the Histogram-Normality Test. Comment on these results. RDAX

FIGURE 2.8 RDAX GARCH (1, 1) RESIDUAL, ACTUAL, FITTED

-.10

-.05

.00

.05

.10

.15

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

Residual Actual Fitted

.0000

.0004

.0008

.0012

.0016

.0020

.0024

.0028

.0032


2008 2009 2010 2011 2012 2013

Conditional variance


67

FIGURE 2.9 RDAX GARCH (1, 1) GRAPH

Date: 06/20/15 Time: 23:06 Sample: 1/02/2008 10/30/2013 Included observations: 1521 Q-statistic probabilities adjusted for 2 ARMA term(s)

Autocorrelation Partial Correlation AC PAC Q-Stat Prob | | | | 1 0.030 0.030 1.3834

| | | | 2 0.009 0.008 1.5175 | | | | 3 -0.008 -0.009 1.6174 0.203 | | | | 4 0.027 0.028 2.7524 0.253 | | | | 5 -0.006 -0.007 2.8020 0.423 | | | | 6 0.010 0.010 2.9524 0.566 | | | | 7 0.015 0.015 3.2814 0.657 | | | | 8 0.003 0.001 3.2954 0.771 | | | | 9 -0.010 -0.009 3.4359 0.842 | | | | 10 -0.011 -0.011 3.6142 0.890 | | | | 11 0.032 0.033 5.2146 0.815 | | | | 12 0.009 0.007 5.3493 0.867

TABLE 2.23 RDAX GARCH (1, 1) CORRELOGRAM- Q

FIGURE 2.10 RDAX GARCH (1, 1) HISTOGRAM-NORMALITY TEST

0

50

100

150

200

250

300

350

-6 -5 -4 -3 -2 -1 0 1 2 3 4

Series: Standardized Residuals

Sample 1/02/2008 10/30/2013

Observations 1521

Mean -0.048307

Median -0.025059

Maximum 3.798798

Minimum -6.034618

Std. Dev. 0.999546

Skewness -0.329956

Kurtosis 4.478023




68

FIGURE 2.11 RDAX GARCH (1, 1) VARIANCE AND DAX SERIES

From figure 2.8 and 2.9 we can notice that the GARCH (1, 1) model can explain quite enough the high and also the low volatility of the residuals. It is easy to notice that in the periods where residuals have high volatility the GARCH model depicts high spikes. To be more specific, at the third quarter of 2008 until first quarter of 2009 we can see that GARCH graph hits high variances and also residuals has high volatility. Furthermore, from Figure 2.11 we compare the GARCH (1, 1) variances with the DAX_30 daily closing observation. Periods of high volatility, for instance the period from the fourth quarter of 2008 until the first quarter of 2009, we have high variance and at the same time the DAX’s price is falling. That high volatility is “blurring” the investors’ horizon, and for that reason, the investors are acting more behaviorally than rationally, which cause the market to fall rapidly. Comparing that with the period at 2009 until first two quarters of 2011 where the volatility is really low the German stock market has a “bull” character, means that investors are sure about the market and the do not afraid to invest. In addition, the Table 2.23 which is the Correlogram-Q, the probability for all the lags are higher than 5% which means that the Null Hypothesis is not rejected and we can say that there is no serial correlation on residuals, fact that make our GARCH model efficient. However, if we take a closer look at the residuals we will notice that are not normally distributed and this is not a good sign. We can support our argument because the p-value is ZERO, which means that we have to reject the Null Hypothesis for the Residual’s normality. The weakness of Normal Distribution (Gaussian) is the non-normality of the residuals, but many suggest that non-normality on residuals may not be that serious problem as ESTIMATORS are still consistent. It would be good to know if the non-normality residuals problem is solving by using another distribution such as Student’s t, Generalized Error (GED), but on the current project is not a question that need to be answered.

.000

.001

.002

.003

.004

4,000

6,000

8,000

10,000

12,000

14,000


2008 2009 2010 2011 2012 2013

RDAX_GARCH11_VAR DAX 30


69

RNIKKEI

FIGURE 2.12 RNIKKEI GARCH (1, 1) RESIDUAL, ACTUAL, FITTED

FIGURE 2.13 RNIKKEI GARCH (1, 1) GARCH

-.15

-.10

-.05

.00

.05

.10

.15

-.15

-.10

-.05

.00

.05

.10

.15


2008 2009 2010 2011 2012 2013

Residual Actual Fitted

.000

.001

.002

.003

.004

.005


2008 2009 2010 2011 2012 2013

Conditional variance


70

FIGURE 2.14 RNIKKEI GARCH (1, 1) HISTOGRAM-NORMALITY TEST

0

50

100

150

200

250

-5 -4 -3 -2 -1 0 1 2 3


Sample 1/02/2008 10/30/2013

Observations 1521

Mean -0.034670

Median -0.017904

Maximum 3.185992

Minimum -5.372672

Std. Dev. 1.000870

Skewness -0.399635

Kurtosis 4.149554



Date: 06/20/15 Time: 23:09

Sample: 1/02/2008 10/30/2013

Included observations: 1521 Q-statistic probabilities adjusted for 2 ARMA term(s)

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

| | | | 1 0.013 0.013 0.2770

| | | | 2 0.005 0.005 0.3140

| | | | 3 0.007 0.007 0.3953 0.530

| | | | 4 0.007 0.006 0.4623 0.794

| | | | 5 0.006 0.006 0.5195 0.915

| | | | 6 -0.006 -0.006 0.5719 0.966

| | | | 7 -0.000 -0.000 0.5721 0.989

| | | | 8 0.016 0.016 0.9648 0.987

| | | | 9 0.004 0.004 0.9884 0.995

| | | | 10 0.030 0.030 2.3953 0.966

| | | | 11 -0.028 -0.029 3.5826 0.937

| | | | 12 -0.004 -0.004 3.6061 0.963

TABLE 2.24 RNIKKEI GARCH (1, 1) CORRELOGRAM-Q


71

FIGURE 2.15 RNIKKEI GARCH (1, 1) VARIANCE AND NIKKEI SERIES

From figure 2.12 and 2.13 we can notice that the GARCH (1, 1) model can explain quite enough the high and also the low volatility of the residuals. It is easy to notice that in the periods where residuals have high volatility the GARCH model depicts high spikes. To be more specific, at the start of fourth quarter of 2008 until the start of the first quarter of 2009 we can see that GARCH graph hits high variances and also residuals has high volatility. Furthermore, from Figure 2.15 we compare the GARCH (1, 1) variances with the NIKKEI 225 daily closing observation. Periods of high volatility, for instance the period from the start of fourth quarter of 2008 until the first quarter of 2009, we have high variance and at the same time the NIKKEI’s price is falling. That high volatility is “blurring” the investors’ horizon, and for that reason, the investors are acting more behaviorally than rationally, which cause the market to fall rapidly. In addition, the Table 2.24 which is the Correlogram-Q, the probability for all the lags are higher than 5% which means that the Null Hypothesis is not rejected and we can say that there is no serial correlation on residuals, fact that make our GARCH model efficient. However, if we take a closer look at the residuals we will notice that are not normally distributed and this is not a good sign. We can support our argument because the p-value is ZERO, which means that we have to reject the Null Hypothesis for the Residual’s normality.

.000

.001

.002

.003

.004

.005

6,000

8,000

10,000

12,000

14,000

16,000


2008 2009 2010 2011 2012 2013

RNIKKEI_GARCH_VAR NIKKEI225


72

d) Check the coefficients on both lagged squared residual and lagged conditional variance terms in the conditional variance equation. Are they statistically significant? Also – as is typical of GARCH model estimates – is the sum of the coefficients on the lagged squared error and lagged conditional variance very close to unity? What is the value of the variance intercept term? RDAX GARCH (1, 1) Dependent Variable: RDAX Method: ML - ARCH (Marquardt) - Normal distribution Date: 06/22/15 Time: 18:28 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Convergence achieved after 17 iterations Bollerslev-Wooldridge robust standard errors & covariance MA Backcast: 1/01/2008 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)


AR(1) 0.700098 0.136366 5.133963 0.0000 MA(1) -0.730855 0.130967 -5.580436 0.0000


RESID(-1)^2 0.097691 0.030613 3.191139 0.0014 GARCH(-1) 0.887060 0.034320 25.84668 0.0000





TABLE 2.17 RDAX GARCH (1, 1)

We mentioned before that the coefficient of GARCH (1, 1) are statistically significant because the p-value is less than the 5% so both RESID (-1) ^2 and GARCH (-1) their coefficients are significant, meaning that both influence the dependent variable which is the GARCH. Moreover if we sum up the RESID (-1) ^2 and GARCH (-1) coefficients the result is 0.097691+0.887060= 0.984751. Their sum is close to unity but it is not 1. That means that our GARCH is stationary but it is really on the edge of being a random walk model. The value of variance intercept term is 6.37E-06 but is 5% level significance the C term is not

statistically significant.


73

RNIKKEI GARCH (1, 1)

Dependent Variable: RNIKKEI Method: ML - ARCH Date: 06/20/15 Time: 18:39 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Convergence achieved after 12 iterations Bollerslev-Wooldridge robust standard errors & covariance MA Backcast: 1/01/2008 Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)


AR(1) -0.460705 0.486582 -0.946817 0.3437 MA(1) 0.422688 0.494797 0.854265 0.3930

Variance Equation

C 7.15E-06 2.22E-06 3.221053 0.0013 RESID(-1)^2 0.113330 0.031221 3.629907 0.0003 GARCH(-1) 0.861106 0.030247 28.46915 0.0000





TABLE 2.20 RNIKKEI GARCH (1, 1)

For the RNIKKEI GARCH (1, 1), the coefficient of GARCH (1, 1) are statistically significant because the p-value is less than the 5% so both RESID (-1) ^2 and GARCH (-1) their coefficients are significant, meaning that both influence the dependent variable which is the GARCH. Moreover, if we sum up the RESID (-1) ^2 and GARCH (-1) coefficients the result is 0.113330+0.861106= 0.974436. Their sum is close to unity but it is not 1. That means that our GARCH is stationary but it is really on the edge of being a random walk model. The value of variance intercept term is 7.15E-06 and is statistically significance in 5% Significance.


74

PART II The Value-at-Risk (VaR) of a portfolio is defined as the value (return) such that the probability that the loss one the portfolio over the given time horizon exceeds this value is the given probability level (assuming normally distributed markets).

a) Consider naively that each time series comprises one portfolio by itself, i.e., two investors have taken long position on each of the stock returns. Thus, for each stock time series calculate the Value-at-Risk (VaR) on the last day of in-sample (t) and 5-days-ahead (t+5) of the out of sample, using 5% and 1% probability level. The standard deviation σt should be derived from the previously estimated GARCH model, as well as from the historical method.

RDAX GARCH (1, 1) IN SAMPLE

Dependent Variable: RDAX Method: ML - ARCH Date: 06/21/15 Time: 15:51

Sample (adjusted): 1/02/2008 10/23/2013

Included observations: 1516 after adjustments Convergence achieved after 19 iterations Bollerslev-Wooldridge robust standard errors & covariance MA Backcast: 1/01/2008 Presample variance: backcast (parameter = 0.7)

GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob. C 0.000748 0.000343 2.182408 0.0291

AR(1) 0.708762 0.128818 5.502032 0.0000 MA(1) -0.739699 0.123497 -5.989591 0.0000


RESID(-1)^2 0.097374 0.030831 3.158330 0.0016 GARCH(-1) 0.886506 0.034972 25.34927 0.0000





TABLE 2.25 RDAX GARCH(1, 1) IN SAMPLE


75

FIGURE 2.16 RDAX GARCH (1, 1) FORECAST

Obs RDAX_GARCH11_FORECAST_Var RDAX_GARCH11_Var

10/24/2013 0.0001186350656552773 0.000116604640553715

10/25/2013 0.0001180985966627276 0.0001160173056033685

10/28/2013 0.0001113661198423097 0.0001092926528578422

10/29/2013 0.0001056890147372009 0.0001036326400486436

10/30/2013 0.0001013039608499178 99219234077,366

TABLE 2.26 RDAX GARCH(1, 1) FORECAST AND ACTUAL VARIANCES

-.03

-.02

-.01

.00

.01

.02

.03

24 25 28 29 30

2013m10

RDAXF ± 2 S.E.

Forecast: RDAXF

Actual: RDAX

Forecast sample: 10/24/2013 10/30/...









.000100

.000104

.000108

.000112

.000116

.000120

24 25 28 29 30

2013m10

Forecast of Variance


76

RNIKKEI GARCH (1, 1) IN SAMPLE

Dependent Variable: RNIKKEI Method: ML - ARCH Date: 06/21/15 Time: 17:19

Sample (adjusted): 1/02/2008 10/23/2013

Included observations: 1516 after adjustments Convergence achieved after 12 iterations Bollerslev-Wooldridge robust standard errors & covariance MA Backcast: 1/01/2008 Presample variance: backcast (parameter = 0.7)

GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob. C 0.000478 0.000345 1.382687 0.1668

AR(1) -0.388642 0.624786 -0.622041 0.5339 MA(1) 0.354174 0.632578 0.559889 0.5756


RESID(-1)^2 0.113324 0.031230 3.628667 0.0003 GARCH(-1) 0.861490 0.030173 28.55129 0.0000





TABLE 2.27 RNIKKEI GARCH(1, 1) IN SAMPLE

Obs RNIKKEI_GARCH_FORECATS Var RNIKKEI_GARCH_Var

10/24/2013 0.0001409316707569713 0.0001417627812576701

10/25/2013 0.0001294625538938838 0.0001302351957900088

10/28/2013 0.0002073194842154681 0.0002075535079426517

10/29/2013 0.0002310910120174029 0.0002306611158701568

10/30/2013 0.0002081230681137855 0.0002075945479159629

TABLE 2.28 RNIKKEI GARCH(1, 1) FORECAST AND ACTUAL VARIANCES


77

FIGURE 2.17 RNIKKEI GARCH (1, 1) FORECAST

Time Std.Dev(DAX) Std.Dev(NIKKEI) VaR_95%_(DAX) VaR_95%_(NIKKEI)

24/10/2013 0,010891973 0,011871465 17.971,75 19.587,92

25/10/2013 0,010867318 0,011378161 17.931,07 18.773,97

28/10/2013 0,010553015 0,014398593 17.412,47 23.757,68

29/10/2013 0,010280516 0,015201678 16.962,85 25.082,77

30/10/2013 0,010064987 0,014426471 16.607,23 23.803,68

TABLE 2.29 RDAX, RNIKKEI FORECASTED VAR 95% 5 DAYS

Time Std.Dev(DAX) Std.Dev(NIKKEI) VaR_99%_(DAX) VaR_99%_(NIKKEI)

24/10/2013 0,010891973 0,011871465 25.378,30 27.660,51

25/10/2013 0,010867318 0,011378161 25.320,85 26.511,12

28/10/2013 0,010553015 0,014398593 24.588,52 33.548,72

29/10/2013 0,010280516 0,015201678 23.953,60 35.419,91

30/10/2013 0,010064987 0,014426471 23.451,42 33.613,68

TABLE 2.30 RDAX, RNIKKEI FORECASTED VAR 99% 5DAYS

-.04

-.02

.00

.02

.04

24 25 28 29 30

2013m10

RNIKKEIFF ± 2 S.E.

Forecast: RNIKKEIFF

Actual: RNIKKEI

Forecast sample: 10/24/2013 10/30/...









.00012

.00014

.00016

.00018

.00020

.00022

.00024

24 25 28 29 30

2013m10

Forecast of Variance


78

Total VaR for 5 forecasted periods

Std. Dev(DAX) Std.Dev(NIKKEI) VaR_95%_(DAX) VaR_95%_(NIKKEI)

0,023560407 0,030280816 38.874,67 49963,3464


0,023560407 0,030280816 54.895,75 70554,30128

Historic Method


0,044725832 0,0441725 73797,6228 72884,625


0,044725832 0,0441725 104211,1886 102921,925

b) Compare the Value at Risk of both series on that day, in all cases from. Infer the riskiness of the positions in the German and Japanese stock market for the two investors.

Taking into account the 5 days ahead forecast VaR at 95% and 99%, (the variances has been extracted from the estimated GARCH (1, 1)), the DAX is less risky than the NIKKEI. VaR 95% Garch(1,1) Variances The maximum loss for the investors taking a long position for DAX and NIKKEI is 38.874,67 euro and 49.963,3464 respectively with a 5% probability the loss to be greater than the maximum which was referred. At this point NIKKEI is more risky. VaR 99% Garch(1,1) Variances The maximum loss for the investors taking a long position for DAX and NIKKEI is 54.895,75 euro and 70.554,30128 euro respectively with a 1% probability the loss to be greater than the maximum which was referred. At this point NIKKEI again is riskier than DAX. Nevertheless, if we focus our attention to the Historical Method VaR the outcome is not the same.


79

VaR 95% Historical Method Relying now on the historical method the DAX Value-at-Risk is 73.797,6228 euro when the NIKKEI’s Value-at-Risk is calculated at 72.884,625 less than the DAX. Meaning that there is a 5% the loss to be greater than the calculated VaR. In this particular situation DAX is riskier than the NIKKEI. But can we rely on this outcome? VaR 99% Historical Method Checking also the VaR at 99% significance level, NIKKEI’s VaR is still lower than the DAX’s because 102.921,25 is lower than 104.211,1886. In conclusion, if one of the investor chooses to invest based on the historical Method VaR and the other chooses with the GARCH (1,1) variances, one of the two will definitely loss a big portion of his portfolio value. And probably will be the first one because his analysis is based on an average variance when the other’s is based on a daily variance fluctuations. The variance is not standard, but it is changing through the time so we must take into consideration the variance’s changes in order to predict and to invest without having any loss.

*The VaR calculation = 𝑉𝑎𝑅 = 𝑎𝜎𝑊𝑜 a = the significance level 1% = 2,33 and 5% = 1,65. σ = standard deviation Wo= portfolio value, here we assuming a portfolio 1.000.000. euro


80

SECTION III: CAUSALITY ANALYSIS

PART I

a) Construct a bivariate VAR (p) model for the EUR/USD – USD/JPY pair. Determine the optimal lag length based on two multivariate information criteria, one of which should be Akaike. If you will be invited to specify the maximum number of lags, arbitrary select 10. Show the results in a Table.

VAR (p) Model is defined :

Endogenous variables: REUR RJPY Exogenous variables: C Date: 06/23/15 Time: 16:29 Sample: 1/01/2008 10/30/2013 Included observations: 1511

Lag LogL LR FPE AIC SC HQ 0 10587.54 NA 2.82e-09* -14.01130* -14.00426* -14.00868*

1 10590.12 5.149357 2.82e-09 -14.00942 -13.98830 -14.00156

2 10592.77 5.276079 2.83e-09 -14.00763 -13.97242 -13.99452 3 10596.15 6.744863 2.83e-09 -14.00682 -13.95753 -13.98846 4 10599.48 6.616227 2.83e-09 -14.00593 -13.94255 -13.98233 5 10600.51 2.031541 2.84e-09 -14.00199 -13.92453 -13.97314 6 10605.50 9.896088* 2.84e-09 -14.00330 -13.91175 -13.96921 7 10608.19 5.324738 2.85e-09 -14.00157 -13.89593 -13.96223 8 10608.73 1.083799 2.86e-09 -13.99700 -13.87728 -13.95242 9 10612.96 8.353647 2.86e-09 -13.99730 -13.86350 -13.94748 10 10616.76 7.488429 2.86e-09 -13.99704 -13.84915 -13.94196 * indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion

TABLE 3.1 VAR LAG SELECTION REUR_RJPY

The optimum lag length if we choose the Akaike and Schwarz is 0. Nevertheless, there is a restriction not to choose the zero lag but the next “best”. So from the Table the next best is lag 1 for both Akaike and Schwarz criterion


81

The VAR (1) Model for REUR and RJPY

Vector Autoregression Estimates Date: 06/23/15 Time: 16:39 Sample (adjusted): 1/03/2008 10/30/2013 Included observations: 1520 after adjustments Standard errors in ( ) & t-statistics in [ ]

REUR RJPY REUR(-1) 0.007772 -0.029969 (0.02574) (0.02698) [ 0.30195] [-1.11092]

RJPY(-1) -0.009455 -0.050047 (0.02448) (0.02566) [-0.38615] [-1.95039]

C -4.61E-05 -7.65E-05 (0.00018) (0.00019) [-0.25270] [-0.39958] R-squared 0.000172 0.003105

Adj. R-squared -0.001147 0.001791 Sum sq. resids 0.076842 0.084404 S.E. equation 0.007117 0.007459 F-statistic 0.130158 2.362757 Log likelihood 5361.491 5290.155 Akaike AIC -7.050646 -6.956783 Schwarz SC -7.040133 -6.946270 Mean dependent -4.56E-05 -7.10E-05 S.D. dependent 0.007113 0.007466

Determinant resid covariance (dof adj.) 2.80E-09

Determinant resid covariance 2.79E-09 Log likelihood 10656.22 Akaike information criterion -14.01345 Schwarz criterion -13.99243

TABLE 3.2 VAR (1) REUR_RJPY, AIC CHOICE


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b) Next, run a Granger causality/Block Exogenity Testing procedure based on the number of lags the criteria selected. For each criterion explain the Granger-Causality results i.e., whether there is weak, strong, or no evidence of lead-lag interaction between the series using 5% and 1% probability level, and why. Produce the table of statistics. Are the results (i) the same for the two criteria and (ii) are they supported by the theory or the can more like be considered just a statistical outcome?

System: VAR_SUSTEM_LAG1 Estimation Method: Least Squares Date: 06/23/15 Time: 16:40 Sample: 1/03/2008 10/30/2013 Included observations: 1520 Total system (balanced) observations 3040

Coefficient Std. Error t-Statistic Prob. C(1) 0.007772 0.025740 0.301945 0.7627

C(2) -0.009455 0.024484 -0.386155 0.6994 C(3) -4.61E-05 0.000183 -0.252703 0.8005 C(4) -0.029969 0.026977 -1.110917 0.2667 C(5) -0.050047 0.025660 -1.950386 0.0512 C(6) -7.65E-05 0.000191 -0.399583 0.6895

Determinant Residuals Covariance 2.79E-09

Equation: REUR = C(1)*REUR(-1) + C(2)*RJPY(-1) + C(3) Observations: 1520

R-squared 0.000172 Mean dependent var -4.56E-05 Adjusted R-squared -0.001147 S.D. dependent var 0.007113 S.E. of regression 0.007117 Sum squared resid 0.076842 Durbin-Watson stat 2.000577

Equation: RJPY = C(4)*REUR(-1) + C(5)*RJPY(-1) + C(6) Observations: 1520

R-squared 0.003105 Mean dependent var -7.10E-05 Adjusted R-squared 0.001791 S.D. dependent var 0.007466 S.E. of regression 0.007459 Sum squared resid 0.084404 Durbin-Watson stat 2.003842

TABLE 3.3 VAR(1) SYSTEM EQUATIONS


83

TABLE 3.4 GRANGER CAUSALITY TEST VAR(1)

Equation: REUR = C(1)*REUR(-1) + C(2)*RJPY(-1) + C(3)

The dependent variable is REUR and the independent RJPY. The Null Hypothesis states that RJPY(-1) does not cause REUR. And in this particular situation the Null Hypothesis cannot be rejected because probability is 69.94% >> 5% and 1% significance levels. So the RJPY does not cause the REUR. Equation: RJPY = C(4)*REUR(-1) + C(5)*RJPY(-1) + C(6) Same logic with the previous one. The dependent variable is now the RJPY and the independent is REUR. The Null Hypothesis states that REUR(-1) does not cause RJPY. And in this particular situation the Null Hypothesis cannot be rejected, because probability is 26.66% >> 5% and 1% significance levels. So the REUR does not cause the RJPY. In conclusion neither REUR nor RJPY cause each other at 5% and 1% probability level. The results are same for both criteria because both criteria suggest lag length number one.

VAR Granger Causality/Block Exogeneity Wald Tests


Dependent variable: REUR Excluded Chi-sq df Prob. RJPY 0.149116 1 0.6994 All 0.149116 1 0.6994

Dependent variable: RJPY Excluded Chi-sq df Prob. REUR 1.234136 1 0.2666 All 1.234136 1 0.2666


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From the SECTION I of this project we proved that there is no a long-term relationship between the EUR and JPY. We cannot find a solid theory explaining if there is any Granger causality between the two currencies for the period 2008-2013. Our VAR(1) model proved that neither EUR/USD nor USD/JPY affect each other. At this point we cannot determine if our results is proven by the theory or it is just an statistical outcome. Although, there is a research, but not for our examination period. The research’s period is between 3/20/1991 and 3/20/2007. Using the same VAR as we have used in our project they have shown that there is indeed Granger’ causality JPY to EUR, something that in our VAR model does not happen. We can explain this difference to the financials shock that occurred from 2008-2013. The shocks generally affects the market a lot, and change their equilibrium state of economies.

PART II

a) Construct a bivariate VAR (p) model for the DAX-NIKKEI indices pair. Determine the optimal lag length based on two multivariate information criteria, one of which should be Akaike. If you will be invited to specify the maximum number of lags, arbitrary select 12. Show the results in a Table.

VAR Lag Order Selection Criteria Endogenous variables: RDAX RNIKKEI Exogenous variables: C Date: 06/23/15 Time: 11:56 Sample: 12/31/2007 10/30/2013 Included observations: 1510

Lag LogL LR FPE AIC SC HQ 0 7789.344 NA 1.14e-07 -10.31436 -10.30732 -10.31174

1 7984.100 388.7392 8.83e-08 -10.56702 -10.54588* -10.55915 2 7997.003 25.72055 8.72e-08 -10.57881 -10.54358 -10.56569* 3 7999.492 4.954650 8.74e-08 -10.57681 -10.52749 -10.55844 4 8003.817 8.597940 8.74e-08 -10.57724 -10.51383 -10.55362

5 8010.608 13.48343* 8.71e-08* -10.58094* -10.50343 -10.55207 6 8012.202 3.161464 8.73e-08 -10.57775 -10.48615 -10.54364 7 8013.186 1.948093 8.77e-08 -10.57376 -10.46806 -10.53440 8 8013.427 0.475781 8.81e-08 -10.56878 -10.44899 -10.52417 9 8014.638 2.391498 8.84e-08 -10.56508 -10.43121 -10.51523 10 8016.805 4.273274 8.87e-08 -10.56266 -10.41469 -10.50755 11 8020.909 8.084090 8.87e-08 -10.56279 -10.40073 -10.50244 12 8021.199 0.570951 8.91e-08 -10.55788 -10.38173 -10.49228 * indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion

TABLE 3.5 VAR LAG SELECTION DAX_NIKKEI


85

Vector Autoregression Estimates Date: 06/23/15 Time: 15:02 Sample (adjusted): 1/02/2008 10/30/2013 Included observations: 1521 after adjustments Standard errors in ( ) & t-statistics in [ ]

RDAX RNIKKEI RDAX(-1) 0.021288 0.424594 (0.02697) (0.02121) [ 0.78928] [ 20.0139]

RNIKKEI(-1) -0.173908

(0.03054) (0.02402) [-1.98375] [-7.24045]

C 3.00E-05 -5.76E-05 (0.00051) (0.00040) [ 0.05845] [-0.14284] R-squared 0.002607 0.209249

Adj. R-squared 0.001293 0.208207 Sum sq. resids 0.606907 0.375498 S.E. equation 0.019995 0.015728 F-statistic 1.984031 200.8472 Log likelihood 3793.850 4158.982 Akaike AIC -4.984681 -5.464802 Schwarz SC -4.974174 -5.454295 Mean dependent 3.33E-05 -3.55E-05 S.D. dependent 0.020008 0.017675

Determinant resid covariance (dof adj.) 8.76E-08

Determinant resid covariance 8.72E-08 Log likelihood 8045.388 Akaike information criterion -10.57119 Schwarz criterion -10.55017

TABLE 3.6 VAR (1) SBIC CHOICE DAX_NIKKEI


86

TABLE 3.7 VAR (5) AIC CHOICE

Vector Autoregression Estimates

Date: 06/23/15 Time: 15:03 Sample (adjusted): 1/08/2008 10/30/2013 Included observations: 1517 after adjustments Standard errors in ( ) & t-statistics in [ ]

RDAX RNIKKEI RDAX(-1) 0.021156 0.433630

(0.02738) (0.02138)

[ 0.77275] [ 20.2864]

RDAX(-2) -0.002593 0.105360

(0.03121) (0.02436)

[-0.08310] [ 4.32428]

RDAX(-3) -0.019983 0.029149

(0.03138) (0.02450)

[-0.63679] [ 1.18970]

RDAX(-4) 0.057569 0.049748

(0.03134) (0.02447)

[ 1.83673] [ 2.03289]

RDAX(-5) -0.044451 0.059655

(0.03088) (0.02411)

[-1.43931] [ 2.47404]

RNIKKEI(-1) -0.060763 -0.232760

(0.03501) (0.02733)

[-1.73574] [-8.51619]

RNIKKEI(-2) -0.068002 -0.071442

(0.03572) (0.02789)

[-1.90394] [-2.56198]

RNIKKEI(-3) -0.019831 -0.070410

(0.03578) (0.02793)

[-0.55433] [-2.52078]

RNIKKEI(-4) -0.022836 -0.019154

(0.03563) (0.02782)

[-0.64090] [-0.68852]

RNIKKEI(-5) -0.002024 -0.004197

(0.03122) (0.02438)

[-0.06483] [-0.17218]

C 4.32E-05 -4.12E-05

(0.00051) (0.00040)

[ 0.08421] [-0.10280]

R-squared 0.012678 0.225989

Adj. R-squared 0.006122 0.220849

Sum sq. resids 0.600601 0.366111

S.E. equation 0.019970 0.015592

F-statistic 1.933852 43.97084

Log likelihood 3789.797 4165.251

Akaike AIC -4.981934 -5.476929

Schwarz SC -4.943326 -5.438320

Mean dependent 4.98E-05 8.18E-08

S.D. dependent 0.020032 0.017664 Determinant resid covariance (dof adj.) 8.55E-08

Determinant resid covariance 8.43E-08

Log likelihood 8050.340

Akaike information criterion -10.58450

Schwarz criterion -10.50728


87

b) Next, run a Granger causality/Block Exogenity Testing procedure based on the

number of lags the criteria selected. For each criterion explain the Granger-Causality results i.e., whether there is weak, strong, or no evidence of lead-lag interaction between the series using 5% and 1% probability level, and why. Produce the table of statistics. Are the results (i) the same for the two criteria and (ii) are they supported by the theory or the can more like be considered just a statistical outcome?

System: VAR_1_SYSTEM Estimation Method: Least Squares Date: 06/23/15 Time: 15:24 Sample: 1/02/2008 10/30/2013 Included observations: 1521 Total system (balanced) observations 3042

Coefficient Std. Error t-Statistic Prob. C(1) 0.021288 0.026971 0.789282 0.4300

C(2) -0.060576 0.030536 -1.983745 0.0474 C(3) 3.00E-05 0.000513 0.058449 0.9534 C(4) 0.424594 0.021215 20.01387 0.0000 C(5) -0.173908 0.024019 -7.240447 0.0000 C(6) -5.76E-05 0.000403 -0.142839 0.8864

Determinant residual covariance 8.72E-08

Equation: RDAX = C(1)*RDAX(-1) + C(2)*RNIKKEI(-1) + C(3) Observations: 1521

R-squared 0.002607 Mean dependent var 3.33E-05 Adjusted R-squared 0.001293 S.D. dependent var 0.020008 S.E. of regression 0.019995 Sum squared resid 0.606907 Durbin-Watson stat 2.006185

Equation: RNIKKEI = C(4)*RDAX(-1) + C(5)*RNIKKEI(-1) + C(6) Observations: 1521

R-squared 0.209249 Mean dependent var -3.55E-05 Adjusted R-squared 0.208207 S.D. dependent var 0.017675 S.E. of regression 0.015728 Sum squared resid 0.375498 Durbin-Watson stat 2.095477

TABLE 3.8 VAR(1) SYSTEM EQUATIONS SBIC


88

VAR Granger Causality/Block Exogeneity Wald Tests Date: 06/23/15 Time: 15:23 Sample: 12/31/2007 10/30/2013 Included observations: 1521

Dependent variable: RDAX Excluded Chi-sq df Prob. RNIKKEI 3.935246 1 0.0473 All 3.935246 1 0.0473

Dependent variable: RNIKKEI Excluded Chi-sq df Prob. RDAX 400.5551 1 0.0000 All 400.5551 1 0.0000

TABLE 3.9 VAR(1) GRANGER CAUSALITY TEST SBIC

Equation: RDAX = C(1)*RDAX(-1) + C(2)*RNIKKEI(-1) + C(3) The dependent variable is RDAX when the independent is RNIKKEI. The SBIC suggest us to compute that equation between RDAX and RNIKKEI. The Null Hypothesis states that RNIKKEI(-1) DOES not cause the RDAX. In 5% level significance the Null Hypothesis is rejected because probability is 4,73%. In this case marginally can the Null Hypothesis be rejected. In sum, at 5% significance level the RNIKKEI causes RDAX. Nonetheless, at 1% level significance we cannot reject the Null Hypothesis rather we accept it and automatically we accept that RNIKKEI (-1) does not cause RDAX. Equation: RNIKKEI = C(4)*RDAX(-1) + C(5)*RNIKKEI(-1) + C(6) In this equation the dependent variable is RNIKKEI and the independent is the RDAX. Again the Null Hypothesis assumes that RDAX(-1) does not cause the RNIKKEI. The probability for this equation is 0,000 so the Null Hypothesis is rejected, meaning that RDAX(-1) indeed causes RNIKKEI at 5% and 1% significance level. At 5% level significance we can support that there is a bi-directional Granger-causality from RNIKKEI to RDAX At 1% level significance there is unidirectional Granger-causality from RDAX to RNIKKEI.


89

System: VAR_5_SYSTEM

Estimation Method: Least Squares

Date: 06/23/15 Time: 15:39

Sample: 1/08/2008 10/30/2013


Total system (balanced) observations 3034 Coefficient Std. Error t-Statistic Prob. C(1) 0.021156 0.027378 0.772750 0.4397

C(2) -0.002593 0.031207 -0.083104 0.9338

C(3) -0.019983 0.031381 -0.636786 0.5243

C(4) 0.057569 0.031343 1.836728 0.0663

C(5) -0.044451 0.030883 -1.439311 0.1502

C(6) -0.060763 0.035007 -1.735743 0.0827

C(7) -0.068002 0.035716 -1.903945 0.0570

C(8) -0.019831 0.035775 -0.554327 0.5794

C(9) -0.022836 0.035630 -0.640904 0.5216

C(10) -0.002024 0.031224 -0.064828 0.9483

C(11) 4.32E-05 0.000513 0.084213 0.9329

C(12) 0.433630 0.021375 20.28644 0.0000

C(13) 0.105360 0.024365 4.324283 0.0000

C(14) 0.029149 0.024501 1.189702 0.2343

C(15) 0.049748 0.024471 2.032894 0.0422

C(16) 0.059655 0.024112 2.474037 0.0134

C(17) -0.232760 0.027332 -8.516185 0.0000

C(18) -0.071442 0.027886 -2.561982 0.0105

C(19) -0.070410 0.027932 -2.520785 0.0118

C(20) -0.019154 0.027818 -0.688519 0.4912

C(21) -0.004197 0.024378 -0.172181 0.8633

C(22) -4.12E-05 0.000400 -0.102802 0.9181 Determinant residual covariance 8.43E-08

Equation: RDAX = C(1)*RDAX(-1) + C(2)*RDAX(-2) + C(3)*RDAX(-3) + C(4)

*RDAX(-4) + C(5)*RDAX(-5) + C(6)*RNIKKEI(-1) + C(7)*RNIKKEI(-2) +

C(8)*RNIKKEI(-3) + C(9)*RNIKKEI(-4) + C(10)*RNIKKEI(-5) + C(11)

Observations: 1517



S.E. of regression 0.019970 Sum squared resid 0.600601

Durbin-Watson stat 1.998029

Equation: RNIKKEI = C(12)*RDAX(-1) + C(13)*RDAX(-2) + C(14)*RDAX(-3)

+ C(15)*RDAX(-4) + C(16)*RDAX(-5) + C(17)*RNIKKEI(-1) + C(18)

*RNIKKEI(-2) + C(19)*RNIKKEI(-3) + C(20)*RNIKKEI(-4) + C(21)

*RNIKKEI(-5) + C(22)

Observations: 1517



S.E. of regression 0.015592 Sum squared resid 0.366111

Durbin-Watson stat 1.996156

TABLE 3.10 VAR (5) SYSTEM EQUATIONS AIC


90

VAR Granger Causality/Block Exogeneity Wald Tests


Dependent variable: RDAX Excluded Chi-sq df Prob. RNIKKEI 5.727995 5 0.3336 All 5.727995 5 0.3336

Dependent variable: RNIKKEI Excluded Chi-sq df Prob. RDAX 425.1102 5 0.0000 All 425.1102 5 0.0000

TABLE 3.11 VAR (5) GRANGER CAUSALITY AIC

Equation: RDAX = C(1)*RDAX(-1) + C(2)*RDAX(-2) + C(3)*RDAX(-3) + C(4) *RDAX(-4) + C(5)*RDAX(-5) + C(6)*RNIKKEI(-1) + C(7)*RNIKKEI(-2) + C(8)*RNIKKEI(-3) + C(9)*RNIKKEI(-4) + C(10)*RNIKKEI(-5) + C(11) Dependent variable is the RDAX and the independent variables that we want to check for causality are RNIKKEI(-1), RNIKKEI(-2), RNIKKEI(-3), RNIKKEI-4), RNIKKEI(-5). The Null Hypothesis is that all these lags of RNIKKEI jointly cause the RDAX. From the Causality test the probability is 33,36% >> 5% and 1% significance level. In this situation we cannot reject the Null Hypothesis, so RNIKKEI does not cause the RDAX.

Equation: RNIKKEI = C(12)*RDAX(-1) + C(13)*RDAX(-2) + C(14)*RDAX(-3) + C(15)*RDAX(-4) + C(16)*RDAX(-5) + C(17)*RNIKKEI(-1) + C(18) *RNIKKEI(-2) + C(19)*RNIKKEI(-3) + C(20)*RNIKKEI(-4) + C(21) *RNIKKEI(-5) + C(22) The dependent variables in this equation is RNIKKEI and the independent variables that we want to check for causality are RDAX(-1), RDAX(-2), RDAX(-3), RDAX(-4), RDAX(-5). Again the Null Hypothesis is that all these lags of RDAX jointly case the RDAX. The probability that Granger’s test provide is 0.000 and this implies the rejection of Null Hypothesis. So RDAX causes RNIKKEI. On this VAR (5) model there is causality from RDAX to RNIKKEI the reverse behavior is no detect.


91

In conclusion, both criteria SBIC and AIC REJECT the Null Hypothesis when the dependent variable is RNIKKEI, at all significance levels. Moreover when RDAX is the dependent variable both criteria accept the Null Hypothesis at 1% level significance BUT SBIC reject the Null Hypothesis at 5% level significance. Unfortunately our observations are from 2008-2013 and we did not find so far any theory to explain the causality for the indices at this period of time. What we have found is a research for the period 1985-1997. The research found that at period 03/01/88-10/02/89 after the market crash 1987 there is an unidirectional Granger-causality from NIKKEI to DAX, when before that incident there was no Granger’s causality at all between them. In our analysis we have shown that at 5% level significance RNIKKEI influence RDAX (VAR(1)). Before that great incident the research do not refer any causality between the two indices. So, after that shock in the markets the equilibrium point have changed, is what we can assume in order to explain the causality. The research at period 03/05/90 to 12/29/92 still do not find a Granger-Causality from DAX to NIKKEI as our analysis shows. At the last period 01/04/93 to 10/20/97 there is no Granger-Causality from DAX to NIKKEI. What we can assume is that our results cannot be considered as a statistical outcome, and we can explain the unidirectional Granger’s casualty from DAX to NIKKEI by the recent financial crisis in Europe. This shock may have changed the equilibrium point and as a result to have that unidirectional causality from DAX to NIKKEI. But all these are assumption because we lack of evidences. So a possible explanation for the unidirectional Granger’s causality from DAX to NIKKEI, is that it may be created when a great shock similar to 1987 crash markets have taken places creating relationships between the two indices. We have similar shocks at the periods 2008-2013 the Financial Crisis that has affected the whole Europe can be considered as one.


92

REFERENCES ARIMASel (2010) http://www.eviews.com/Addins/addins.shtml Γ.Κ. Χρήστου (2007), Εισαγωγή στην Οικονομετρία, Τόμος Α’ & Β’, Gutenberg. Fabozzi Frank J., Focardi Sergio M., Jasic Teo, Mittnik Stefan, Rachiev Svetlozar T (2007)., “Financial Econometrics” From Basics to Advanced Modeling Techniques, Wiley Finance. Moslem Peymany(2009), A simple Eviews Program for ARMA selection, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1523045 Robert Dornau, Shock Around the Clock- On the Causal Relations Between International Stock Markets, the Strength of Causality and the Intensity of Shock Transmission. An Econometric Analysis. http://ftp.zew.de/pub/zew-docs/dp/dp1398.pdf Stelios D. Bekiros, Cees G.H. Diks (2008), The nonlinear dynamic relationship of exchange

rates: Parametric and nonparametric causality tests, doi:10.1016/j.jmacro.2008.04.001,

Journal of Macroeconomics. Yang Fan (2011), Are international stock market correlated? Comparing NIKKEI, Dow Jones, and DAX in the periods of 1991-2000 and 2001-2010, Jonkoping International Business School. Yoshihiro Kitamura (2012), Informational linkages among the major currencies in the EBS market: Evidence from the spot rates of the Euro, Yen and Swiss franc, pp (17-26),

doi:10.1016/j.japwor.2011.12.003, Japan and World Economy.

http://www.eviews.com/Addins/addins.shtml

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1523045

http://ftp.zew.de/pub/zew-docs/dp/dp1398.pdf

http://dx.doi.org/10.1016/j.jmacro.2008.04.001

http://dx.doi.org/10.1016/j.japwor.2011.12.003


93

APPENDIX 1 Section I / Part III In the opposite case, choosing the Schwarz Criterion for selection, the qualified models are ARMA (1, 0) for both currencies.

Dependent Variable: REUR_USD

Method: Least Squares

Date: 06/23/15 Time: 10:58

Sample (adjusted): 1/03/2008 7/10/2013

Included observations: 1440 after adjustments

Convergence achieved after 2 iterations Variable Coefficient Std. Error t-Statistic Prob. C -8.77E-05 0.000193 -0.454741 0.6494

AR(1) 0.008023 0.026397 0.303921 0.7612 R-squared 0.000064 Mean dependent var -8.77E-05



Sum squared resid 0.075710 Schwarz criterion -7.005270

Log likelihood 5051.067 Hannan-Quinn criter. -7.009860

F-statistic 0.092368 Durbin-Watson stat 1.996944

Prob(F-statistic) 0.761233 Inverted AR Roots .01

TABLE 1 APPENDIX 1 RUER ARMA (1, 0) IN SAMPLE

FIGURE 1 APPENDIX 1 REUR ARMA (1, 0) FORECAST

-.015

-.010

-.005

.000

.005

.010

.015

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

REUR_USDF ± 2 S.E.

Forecast: REUR_USDF

Actual: REUR_USD

Forecast sample: 7/11/2013 10/30/2013










94

Dependent Variable: RUSD_JPY

Method: Least Squares

Date: 06/25/15 Time: 02:33

Sample (adjusted): 1/03/2008 7/10/2013

Included observations: 1440 after adjustments

Convergence achieved after 3 iterations Variable Coefficient Std. Error t-Statistic Prob. C -6.70E-05 0.000189 -0.354920 0.7227

AR(1) -0.048219 0.026315 -1.832389 0.0671 R-squared 0.002330 Mean dependent var -6.68E-05



Sum squared resid 0.081024 Schwarz criterion -6.937425

Log likelihood 5002.219 Hannan-Quinn criter. -6.942015

F-statistic 3.357648 Durbin-Watson stat 2.001622

Prob(F-statistic) 0.067100 Inverted AR Roots -.05

TABLE 2 APPENDIX 1 RJPY ARMA (1, 0) IN SAMPLE

FIGURE APPENDIX 2 RJPY ARMA (1, 0) FORECAST

As we can see the ARMA( 1, 0) has the highest Theil Inequality coefficient so it is not a good prediction model for the RJPY.

-.020

-.016

-.012

-.008

-.004

.000

.004

.008

.012

.016

15 22 29 5 12 19 26 2 9 16 23 30 7 14 21 28

M7 M8 M9 M10

RUSD_JPYF ± 2 S.E.

Forecast: RUSD_JPYF

Actual: RUSD_JPY

Forecast sample: 7/11/2013 10/30/2013









RJPY_USD ARMA (2, 2) ARMA(1, 1) ARMA(1, 0)

Theil’s Inequality Coefficient 0,912072 < 0,932488 < 0,951320

Bias Proportion 0,000405 0,000425 0,000202

Variance Proportion 0,873446 0,895122 0,904453

Covariance Proportion 0,12615 0,104453 0,095346


95

Also here ARMA(1, 0) is not better than the ARMA(2, 2) but is better than the ARMA(1, 1) relying on the Theil’s Inequality coefficient. After all choosing our model with the Akaike information Criterion is proved that it was a good choice.

REUR_USD ARMA(2,2) ARMA(1,1) ARMA(1, 0)

Theil’s Inequality Coefficient 0,967634 < 0,982055 > 0,981887

Bias Proportion 0,044183 0,043905 0,043828

Variance Proportion 0,900423 0,940817 0,939348

Covariance Proportion 0,055393 0,015277 0,016825


96

APPENDIX 2 Section II / Part I In c) question we mention that the histogram of the residuals with the Gaussian distribution is prove that the residuals are non-normal. Will the result change if we change the distributions.

FIGURE 1 APPENDIX 2 RNIKEI GARCH(1,1) STUDENT’S T AND GENERALIZED DISTRIBUTION HISTOGRAM After changing the distributions for the RNIKKEI, again the probability is zero, and that mean our residuals are non-normal.

0

50

100

150

200

250

300

-5 -4 -3 -2 -1 0 1 2 3


Sample 1/02/2008 10/30/2013

Observations 1521

Mean -0.045144

Median -0.026396

Maximum 3.147706

Minimum -5.385535

Std. Dev. 0.999875

Skewness -0.420472

Kurtosis 4.277618



0

50

100

150

200

250

-5 -4 -3 -2 -1 0 1 2 3


Sample 1/02/2008 10/30/2013

Observations 1521

Mean -0.032059

Median -0.013307

Maximum 3.082589

Minimum -5.321982

Std. Dev. 0.995864

Skewness -0.410691

Kurtosis 4.212159




97

FIGURE 2 APPENDIX 2 RDAX GARCH(1,1) STUDENT’S AND GENERALIZED ERROR DISTRIBUTION HISTOGRAM

The results do not change either for the RDAX, the residuals are non-normal because the probability is 0,0000. In sum whatever distribution we choose the residuals are non-normal but this is not so bad if the estimators are consistent.

0

40

80

120

160

200

240

280

320

360

-6 -5 -4 -3 -2 -1 0 1 2 3 4


Sample 1/02/2008 10/30/2013

Observations 1521

Mean -0.044191

Median -0.022016

Maximum 3.827444

Minimum -6.412314

Std. Dev. 1.000140

Skewness -0.356350

Kurtosis 4.718814



0

40

80

120

160

200

240

280

320

360

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4


Sample 1/02/2008 10/30/2013

Observations 1521

Mean -0.050044

Median -0.029093

Maximum 3.796796

Minimum -6.696921

Std. Dev. 0.996385

Skewness -0.394231

Kurtosis 4.938564




98

APPENDIX 3 Code for ARMA selection. 'Reading data wfopen "c:\data.xls" series data=x1 '-------------------------------------------------------------------------------------------------------------------------------------- 'Determining max lag of AR terms (p) and MA terms (q) !maxp=3 !maxq=3 '-------------------------------------------------------------------------------------------------------------------------------------- 'Preparing a matrix for necessary calculations, this matrix will be deleted at the end of code matrix(!maxp+1,!maxq+1) m_sc matrix(!maxp+1,!maxq+1) m_aic '-------------------------------------------------------------------------------------------------------------------------------------- 'Preparing tables to show results table (!maxp+5,!maxq+2) results_sc setline(results_sc,2) setline(results_sc,!maxp+4) results_sc(1,1)="SBC" results_sc(3,2)="---" table (!maxp+5,!maxq+2) results_aic setline(results_aic,2) setline(results_aic,!maxp+4) results_aic(1,1)="AIC" results_aic(3,2)="---" for !i=0 to !maxp results_sc(!i+3,1)="AR("+@str(!i)+")" results_aic(!i+3,1)="AR("+@str(!i)+")" next for !i=0 to !maxq results_sc(1,!i+2)="MA("+@str(!i)+")" results_aic(1,!i+2)="MA("+@str(!i)+")" next '-------------------------------------------------------------------------------------------------------------------------------------- 'Filling first row of tables %1="" for !j=1 to !maxq %1=%1+"ma("+@str(!j)+")" equation eq1.ls data c {%1} results_sc(3,!j+2)=eq1.@sc results_aic(3,!j+2)=eq1.@aic m_sc(1,!j+1)=eq1.@sc m_aic(1,!j+1)=eq1.@aic next '-------------------------------------------------------------------------------------------------------------------------------------- 'Filling first column of tables


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%2="" for !k=1 to !maxp %2=%2+"ar("+@str(!k)+")" equation eq2.ls data c {%2} results_sc(!k+3,2)=eq2.@sc results_aic(!k+3,2)=eq2.@aic m_sc(!k+1,1)=eq2.@sc m_aic(!k+1,1)=eq2.@aic next '-------------------------------------------------------------------------------------------------------------------------------------- 'Filling other parts of tables %3="" for !p=1 to !maxp %3=%3+"ar("+@str(!p)+")" %4="" for !q=1 to !maxq %4=%4+"ma("+@str(!q)+")" %5=%3+%4 equation eq.ls data c {%5} results_sc(!p+3,!q+2)=eq.@sc results_aic(!p+3,!q+2)=eq.@aic m_sc(!p+1,!q+1)=eq.@sc m_aic(!p+1,!q+1)=eq.@aic next next '-------------------------------------------------------------------------------------------------------------------------------------- 'Determining the bestmodel !pq=@max(m_sc) m_sc(1,1)=!pq !pq=@max(m_aic) m_aic(1,1)=!pq !maxpp=!maxp+1 !maxqq=!maxq+1 !pq=@min(m_sc) for !pp=1 to !maxpp for !qq=1 to !maxqq if m_sc(!pp,!qq)=!pq then !finalp=!pp-1 !finalq=!qq-1 endif next next results_sc(!maxp+5,1)="Best Model Is ARMA("+@str(!finalp)+","+@str(!finalq)+")" !pq=@min(m_aic) for !pp=1 to !maxpp for !qq=1 to !maxqq if m_aic(!pp,!qq)=!pq then !finalp=!pp-1 !finalq=!qq-1 endif next next results_aic(!maxp+5,1)="Best Model Is ARMA("+@str(!finalp)+","+@str(!finalq)+")" '-------------------------------------------------------------------------------------------------------------------------------------- 'Deleting excess data delete eq eq1 eq2 m_sc m_aic


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'-------------------------------------------------------------------------------------------------------------------------------------- 'Showing results show results_sc show results_aic

In additions, for the ARMA selection model, also used an Add-In which is called ARIMASEL. The Add-in was abstracted from the http://www.eviews.com/Addins/addins.shtml .

http://www.eviews.com/Addins/addins.shtml

applied financial econometrics

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