Med Selim Econometrie

8/8/2019 Med Selim Econometrie

1/14

FINANCIAL

ECONOMETRICS Course Work

Malvin Moyo | Sean Anderson |Dominic calus | Mihajlo Tomic | Mohamed Selim Sta

11/10/2010


2/14

1

Table of Contents1 Question 1 ............................................................................................................................................. 2

2 Question 2: ............................................................................................................................................ 5

3 Question 3 ............................................................................................................................................. 7

4 Question 4 ........................................................................................................................................... 10

5 Question 5 ........................................................................................................................................... 10

6 Question 6 ........................................................................................................................................... 11

7 Question 7 ........................................................................................................................................... 12


3/14

2

1 Question 1ii.

The following are the graphs illustrating the evolution over time of the explanatory variables that weare going to use during this analysis:

It is clear from the graphs above that the financial time series are not stationary. In order totransform those to stationary series we take the logarithmic difference.

-2

-1

0

1

2

3

4

5

01 02 03 04 05 06 07 08 09 10

Term spread

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

01 02 03 04 05 06 07 08 09 10

FX

0

20

40

60

80

100

120

140

01 02 03 04 05 06 07 08 09 10

OIL


4/14

3

A common pattern in the 4 graphs above is the 2007-2008 financial meltdowns that inflicted a greatdeal of damage to the stationarity of financial time series.

After running the augmented dickey fuller test we can observe that only the term spread variable isnot stationary over time. The result of the test is shown below.

Null Hypothesis: D(TS_RET) has a unit rootExogenous: ConstantLag Length: 2 (Automatic based on SIC, MAXLAG=12)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -14.54805 0.0000Test critical values: 1% level -3.487550

5% level -2.88650910% level -2.580163

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

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(TS_RET,2)Method: Least Squares

-10

-5

0

5

10

15

20

25

01 02 03 04 05 06 07 08 09 10

Term Spread Return Graph

-.15

-.10

-.05

.00

.05

.10

01 02 03 04 05 06 07 08 09 10

FTSE_RET

-.12

-.08

-.04

.00

.04

.08

.12

01 02 03 04 05 06 07 08 09 10

FX_RET

-.5

-.4

-.3

-.2

-.1

.0

.1

.2

.3

01 02 03 04 05 06 07 08 09 10

Crude Return Series


5/14

4

Date: 11/23/10 Time: 05:58Sample (adjusted): 2001M03 2010M10Included observations: 116 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(TS_RET(-1)) -2.999698 0.206192 -14.54805 0.0000D(TS_RET(-1),2) 1.195992 0.151261 7.906807 0.0000D(TS_RET(-2),2) 0.510118 0.081269 6.276926 0.0000

C -0.002945 0.209956 -0.014029 0.9888

R-squared 0.823852 Mean dependent var 0.001276Adjusted R-squared 0.819134 S.D. dependent var 5.317154S.E. of regression 2.261296 Akaike info criterion 4.503627Sum squared resid 572.7075 Schwarz criterion 4.598579Log likelihood -257.2104 Hannan-Quinn criter. 4.542172F-statistic 174.6102 Durbin-Watson stat 2.171746Prob(F-statistic) 0.000000

iii.In order to observe distributional properties of the portfolio and market excess returns, we carried out a

Normality test. Our results in EViews are shown below;

Figure 2

As we can observe, the mean and median are not equal and the p-value is less than the critical 5% value, thus we

do not assume that the distribution is normally distributed. As the skewness is -0.878741 and the kurtosis is

greater than 3, the distribution is negatively skewed, confirming non-normality. To check this, we also observe

that the Jarque-Bera is not too high and significant at the 1% level, thus we can suggest the distribution is not

normal.

0

2

4

6

8

10

12

14

16

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Series: Portfolio excess returnSample 2000M10 2010M10Observations 120

Mean 0.005759Median 0.011285Maximum 0.101791Minimum -0.196917Std. Dev. 0.049575Skewness -0.878741Kurtosis 4.652985

Jarque-Bera 29.10550Probabil ity 0 .000000


6/14

5

Figure 3Figure 3 shows the Market (FTSE 100) Excess Returns distribution. In similar fashion to our observation for the

portfolio excess return, the mean and median are not equal and the p-value is less than the critical 5% level. The

higher the Jarque-Bera, the more normal the distribution and it is not so high. In addition, the Kurtosis is also

greater than 3, which shows that the distribution is not normal, but negatively skewed. The Jarque-Bera statistic

is insignificant at any level.

These observations are not surprising as research study suggests that return series data are not normally

distributed.

2 Question 2:

Dependent Variable: PORT_RETMethod: Least SquaresDate: 11/23/10 Time: 06:24Sample (adjusted): 2000M11 2010M10Included observations: 120 after adjustments


C 0.006219 0.002043 3.043936 0.0029FTSE_RET 0.953420 0.047689 19.99255 0.0000OIL_RET 0.076620 0.020932 3.660356 0.0004FX_RET -0.098087 0.082026 -1.195806 0.2342TS_RET -4.74E-05 0.000988 -0.047980 0.9618

R-squared 0.805679 Mean dependent var 0.005759Adjusted R-squared 0.798920 S.D. dependent var 0.049575S.E. of regression 0.022231 Akaike info criterion -4.733920Sum squared resid 0.056833 Schwarz criterion -4.617775Log likelihood 289.0352 Hannan-Quinn criter. -4.686753F-statistic 119.2011 Durbin-Watson stat 2.335443Prob(F-statistic) 0.000000

Examination of the R-squared and the adjusted R-squared values suggests that a good proportion of the variabilityof excess returns in the portfolio is explained by the independent variables, at approximately 80% level. As the p-

0

4

8

12

16

20

-0.10 -0.05 -0.00 0.05

Series:Port Excess ReturnSample 2000M10 2010M10Observations 120

Mean -0.001052

Median 0.006456Maximum 0.083000Minimum -0.139546Std. Dev. 0.044360Skewness -0.738131Kurtosis 3.549304

Jarque-Bera 12.40542Probability 0.002024


7/14

6

values for the Market Excess Returns (FTSE_RET) and Brent Crude Oil returns (OIL_RET) are less than the

critical 5% level, it suggest they are the insignificant terms in the regression and the null hypothesis for

transformed FTSE_RET and OIL_RET into unexpected changes would be rejected. The intercept, and returns

on the FX and TS suggests these are the significant and our assumption is made on the 1% critical value test.

The Durbin-Watson statistic result, which is a measure of autocorrelation of the first order, is around 2, which

shows that there is no evidence of positive or negative serial correlation. The coefficient estimate of 0.953420 for

the excess returns of the FTSE 100 means that if the previous increases by one unit, the dependent variable will

be expected to increase by 0.953420, everything else being equal. With a S.E. of regression of 0.022231, we can

assume certainty of our model, as the values of our coefficients are accurate without much variation and as the fit

of the line to the actual data is close, with the error terms concentrated around the line.

After we identified the insignificant variables as Excess Returns on the FTSE 100 and the Brent Crude Oil

Returns, we dropped both variables from the model and run a new model. As can be seen from our output, our

regression output for the model actually got worse. The R2 decreased drastically .However, the DW statistic is

still around 2 and the p-value now significant at the 10% level.

ependent Variable: PORT_RETMethod: Least SquaresDate: 11/23/10 Time: 07:04

Sample (adjusted): 2000M11 2010M10Included observations: 120 after adjustments


C 0.006362 0.004488 1.417484 0.1590FX_RET 0.114546 0.168054 0.681606 0.4968TS_RET -0.004483 0.002125 -2.109570 0.0370

R-squared 0.039015 Mean dependent var 0.005759Adjusted R-squared 0.022588 S.D. dependent var 0.049575S.E. of regression 0.049012 Akaike info criterion -3.168806Sum squared resid 0.281059 Schwarz criterion -3.099118

Log likelihood 193.1283 Hannan-Quinn criter. -3.140505F-statistic 2.375020 Durbin-Watson stat 1.965776Prob(F-statistic) 0.097483

Wald TestThe Wald statistic is a measurement of how close the unrestricted estimates come to satisfying the restrictionsunder the Ho: null hypothesis. We want to check if our is statistically different from 1. If the restrictions areindeed true, then the unrestricted estimates should come close to satisfying the restrictions. WE will use the Chi-square version of the test as we have a large sample size. The p-value of 0 indicates that the of our portfolio(PER) is statistically significantly different from 1.


8/14

7

Wald Test:Equation: EQ02

Test Statistic Value df Probability

F-statistic 49014.57 (1, 117) 0.0000

Chi-square 49014.57 1 0.0000

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

-1 + C(1) -0.993638 0.004488

Restrictions are linear in coefficients.

Diagnostic Tests on our Model

3 Question 3Residuals Normality Test

Histogram Normality test, check for normality of the residuals series. If the distribution is normal, then the mean

and median should be observed as more or less equal. Also, the distribution should not be skewed and as such

have a coefficient of Kurtosis of 3. The shape of the histogram should be observed as bell-shaped and the Jarque-

Bera stat should be high, in other words not significant. In the case of our observations, the kurtosis is higher than

3, suggesting that our series is negatively skewed. The p-value is less than the critical at the 1% level, thus we

would reject the null H0: normality. However, as our sample size is sufficiently large, we can ignore this rule, as

the violation of normality in this instance would be insignificant. According to the theory of central limit, as the

distribution sample gets sufficiently larger, the statistics observed will follow the appropriate distributions, even

if the error distribution is not normal.

Multicollinearity Test

0

4

8

12

16

20

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Series: RESIDSample 2000M10 2010M10Observations 120

Mean 2.33e-18Median 0.003456Maximum 0.095444

Minimum -0.199436Std. Dev. 0.048599Skewness -0.872480Kurtosis 4.661271



9/14

8

FTSE_RET OIL_RET TS_RET FX_RETFTSE_RET 1.000000OIL_RET 0.204845 1.000000TS_RET -0.193507 -0.103390 1.000000FX_RET 0.050633 0.359992 0.064105 1.000000

A very good method of testing the extent of Multicollinearity is by looking at the matrix of correlations betweenthe independent variables. Our model involves correlation relationships between 4 variables and we can observethat there is no high positive or negative correlation between the variables. We can thus say that Multicollinearitydoes exist, because the correlation between went as high as 36%. (FX variable and Crude Oil)

Hetroskedasticity Test

Heteroskedasticity Test: White

F-statistic 0.992080 Prob. F(14,105) 0.4668Obs*R-squared 14.01889 Prob. Chi-Square(14) 0.4483Scaled explained SS 19.15720 Prob. Chi-Square(14) 0.1590

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 11/23/10 Time: 07:29Sample: 2000M11 2010M10Included observations: 120


C 0.000409 0.000110 3.718084 0.0003

FTSE_RET -0.000636 0.002104 -0.302376 0.7630FTSE_RET^2 0.056588 0.029416 1.923727 0.0571

FTSE_RET*OIL_RET 0.032861 0.024824 1.323764 0.1885FTSE_RET*TS_RET 0.001424 0.005152 0.276472 0.7827FTSE_RET*FX_RET 0.080362 0.099635 0.806563 0.4217

OIL_RET 0.000459 0.000904 0.507198 0.6131OIL_RET^2 -0.008495 0.005492 -1.546910 0.1249

OIL_RET*TS_RET 0.000834 0.000636 1.310327 0.1929OIL_RET*FX_RET -0.033815 0.028683 -1.178928 0.2411

TS_RET 5.04E-05 0.000148 0.340401 0.7342TS_RET^2 6.54E-06 2.10E-05 0.311960 0.7557

TS_RET*FX_RET 0.003675 0.004228 0.869375 0.3866FX_RET 0.000877 0.003365 0.260658 0.7949

FX_RET^2 0.053434 0.090689 0.589198 0.5570

R-squared 0.116824 Mean dependent var 0.000474Adjusted R-squared -0.000933 S.D. dependent var 0.000820S.E. of regression 0.000821 Akaike info criterion -11.25608Sum squared resid 7.07E-05 Schwarz criterion -10.90764Log likelihood 690.3646 Hannan-Quinn criter. -11.11457F-statistic 0.992080 Durbin-Watson stat 2.327283Prob(F-statistic) 0.466760

The White's test is a test of the null hypothesis H0: no Hetroskedasticity against Hetroskedasticity of some

unknown general form, in other words that there is homoscedasticity. In EViews, the test statistic is computed by


10/14

9

an auxiliary regression, where we regress the squared residuals on all possible (non-redundant) cross products of

the regression. In this instance, we reject the null hypothesis of no Hetroskedasticity at the 1% significance level

because we obtain a p-value of 0.466760, which is more than the critical. As such, we can assume that there is no

evidence of Hetroskedasticity, meaning the variation of the errors is not constant. The absence of

Hetroskedasticity in our model means we can use an OLS model, hence, any inference and conclusions made

could be assumed to be correct.

Bellow you can observe a scatter plot of the error terms. It is clear that there is no trend in the graph which

confirms the absence of Hetroskedasticity.

Autocorrelation Test

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 1.063464 Prob. F(4,111) 0.3781Obs*R-squared 4.429028 Prob. Chi-Square(4) 0.3510

Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 11/23/10 Time: 07:40Sample: 2000M11 2010M10Included observations: 120Presample missing value lagged residuals set to zero.


C 3.01E-06 0.002006 0.001498 0.9988FTSE_RET -0.004666 0.046954 -0.099383 0.9210

FX_RET 0.038649 0.083670 0.461921 0.6450OIL_RET -0.002432 0.021077 -0.115410 0.9083TS_RET 7.38E-06 0.000986 0.007483 0.9940

RESID(-1) -0.182567 0.095943 -1.902863 0.0596RESID(-2) 0.025933 0.096978 0.267406 0.7897RESID(-3) 0.030002 0.097573 0.307484 0.7591RESID(-4) -0.050625 0.096326 -0.525559 0.6002

R-squared 0.036909 Mean dependent var 1.45E-18Adjusted R-squared 0.003410 S.D. dependent var 0.021854

-.08

-.06

-.04

-.02

.00

.02

.04

.06

.08

01 02 03 04 05 06 07 08 09 10

RESID


11/14

10

S.E. of regression 0.021816 Akaike info criterion -4.704861Sum squared resid 0.054735 Schwarz criterion -4.495799Log likelihood 291.2916 Hannan-Quinn criter. -4.619960F-statistic 0.550893 Durbin-Watson stat 1.979638Prob(F-statistic) 0.815755

The Breusch-Godfrey autocorrelation test was carried out to test for serial correlation for higher order ARMAerrors. The serial correlation test is carried out to analyze whether the errors are statistically independent anduncorrelated of each other. The null hypothesis assumes that there is no serial correlation of independentvariables and if this assumption is broken, the OLS is again biased, giving misleading values. After conducting theBreusch-Godfrey test, we got a p-value of 0.815755 and thus came to the conclusion to not reject the nullhypothesis at 5% significance level. We used 4 lags and as such, under the null hypothesis the current error is notrelated to any of its other 4 previous values. Due to the evidence of no autocorrelation, there is no need forremedy.

Misspecification Remedies to the Model

4 Question 4Whilst carrying out our diagnostic tests, we encountered only few problems such as a partial multicolinearity between two of our variables. We have explored different independent variable combinations and have witnessed better error term normalitywhen introducing two dummy variables. The first takes the value 1 at the .com bubble in 2000 and the secondshadows the financial crisis of July 2007. Also, omitting the Crude oil independent variable has shown to be beneficial for the normality of the error terms. Below is the histogram graph:

The P-statistic increased dramatically from 0.6% to 1.4% in the residual normality test which is a sign that ourremedy had improved the model.

5 Question 5



C 0.021388 0.004911 4.355431 0.0000TSSQ -0.000106 0.000105 -1.013161 0.3131OILSQ -0.189713 0.240181 -0.789874 0.4312

0

4

8

12

16

20

-0.050 -0.025 -0.000 0.025 0.050

Series: ResidualsSample 2000M11 2010M10Observations 120

Mean 9.54e-19Median -0.001188Maximum 0.067206Minimum -0.065667Std. Dev. 0.022823Skewness 0.441043Kurtosis 3.961133



12/14

11

FXSQ 3.396208 3.501980 0.969797 0.3342FTSESQ -7.889109 1.279905 -6.163822 0.0000

R-squared 0.279410 Mean dependent var 0.005759Adjusted R-squared 0.254346 S.D. dependent var 0.049575S.E. of regression 0.042809 Akaike info criterion -3.423361

Sum squared resid 0.210751 Schwarz criterion -3.307216Log likelihood 210.4017 Hannan-Quinn criter. -3.376194F-statistic 11.14787 Durbin-Watson stat 2.156670Prob(F-statistic) 0.000000

We augmented our model by including the squares of the factor shocks, which do not take the direction intoaccount, as the squared number will always be positive. By squaring it, we found the magnitude of the factorshocks and not the direction of the movement.By continuing our augmentation of the model, we indeed found our model not to improve, when we excludedsome of the variables. As such, we decided that the model which gave us the best statistics at this point was withindependent variables; squared FTSE100 returns, squared crude oil log return, squared GBP/Dollar log returnand squared term spread.Chow Breakpoint Test6 Question 6

From the residuals graph above it can be seen that there are several significant outliers within the range from

2007 until 2008. In order to test for breakpoints we are going to run a Chow breakpoint test and decided to

choose September 2007 and December 2008 as breakpoints.

Chow Breakpoint Test: 2007M06 2008M12Null Hypothesis: No breaks at specified breakpointsVarying regressors: All equation variablesEquation Sample: 2000M11 2010M10

F-statistic 1.565860 Prob. F(10,105) 0.1270Log likelihood ratio 16.68056 Prob. Chi-Square(10) 0.0817Wald Statistic 15.65860 Prob. Chi-Square(10) 0.1098

-.10

-.05

.00

.05

.10

-.2

-.1

.0

.1

.2

01 02 03 04 05 06 07 08 09 10

Residual Actual Fitted


13/14

12

The results of the test show us that the chosen breakpoints are significant at the 5% significance level. Therefore,

we reject the null hypothesis of no breakpoints and conclude that some events caused fluctuations within our

portfolio returns. The conclusion is that we need to add dummy variables into the model and check if our model

will improve.



FTSE_RET 0.951119 0.049080 19.37891 0.0000OIL_RET 0.082671 0.021475 3.849694 0.0002

FX_RET -0.084597 0.084987 -0.995414 0.3216TS_RET 0.000235 0.001015 0.231198 0.8176DUM1 0.006491 0.004292 1.512371 0.1332

R-squared 0.824117 Mean dependent var 0.005759Adjusted R-squared 0.846956 S.D. dependent var 0.049575S.E. of regression 0.022882 Akaike info criterion -4.676126Sum squared resid 0.060214 Schwarz criterion -4.559980Log likelihood 285.5675 Hannan-Quinn criter. -4.628959Durbin-Watson stat 2.216209

The model above includes a dummy variable for the period from September 1998 until the end of the series. Aswe can see our model improved slightly, as the R2 and the Adjusted R2 increased above 0.84, a value that is

already considered as an excellent goodness of fit.

Examination of the Best Model

7 Question 7Although we are fully aware that this is not a perfect model, as for instance the Durbin-Watson statistic is

worrying, we believe that this model is able to predict excess returns.

By implementing a second order integration and a dummy variable we have succeeded to bring the residual to anear normality status as the below graph can show:


14/14

13

Skewness is closer to zero and kurtosis is the closest we could get to 3. Another factor that informs us on the

quality of the model is the AIC and BIC criterions. In fact we have managed to decrease the criterions throughout

the models.

0

4

8

12

16

20

-0.06 -0.04 -0.02 -0.00 0.02 0.04 0.06

Series: ResidualsSample 2000M11 2010M10Observations 120

Mean 0.004542Median 0.002523Maximum 0.065961Minimum -0.061159Std. Dev. 0.022027Skewness 0.339612Kurtosis 3.946519


Documents

Med Selim Econometrie