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Concept of Causality and Causal Variables
October 28, 2014
2

 

Causality is the relationship between two variables, the first being cause and the second being effect. There are two types of causality relationship between these variable, bidirectional causality and unidirectional causality. The relationship between these two variables should be either unidirectional or bidirectional.

 

Unidirectional causality & bidirectional causality:

 

Unidirectional causality & bidirectional causality

 

Cause is an Independent Variable (IV), whereas Effect is a Dependent Variable (DV)

In cause effect relationship, we will always test sufficient condition first, because if sufficient condition is present then this means that necessary condition will automatically has to be present. For example, consider Cause variable as Clouds and effect variable as rain. The effect variable, that is, rain will also be known as sufficient condition. So in this model we will check if rain is present or not. If suppose we can see that rain is present, then it is automatically necessary that there must be cloud present due to which rain occurred, therefore the presence of cloud is known as necessary condition. We can conclude that if sufficient condition is true, then automatically necessary condition has to be true.

In bidirectional causality, Cause variable causes effect variable, however, at the same time effect variable also causes Cause variable. This means both reactions can take place simultaneously.

 

Unidirectional Causality and Granger Causality Test:

 

Condition for unidirectional cause variables are:

  1. Both variables should be in time series.
  2. Both variable should have shocks (non-Stationarity).
  3. In both time series, shocks should be fixed by the help of 1st or 2nd
  4. AR – process should be present.
  5. GARCH should be significant, that is, volatility should also be present.
  6. Number of variable should be equal to two.

The test used to check unidirectional causality is known as Granger Causality Test.

 

Bidirectional causality and Cross-correlation Test:

 

Condition for bidirectional cause variables are:

  1. Both variables should be in time series.
  2. Both variable should have shocks (non-Stationarity).
  3. In both time series, shocks should be fixed by the help of 1st or 2nd
  4. AR – process should be present.
  5. GARCH should be significant, that is, volatility should also be present.
  6. Number of variable should be equal to two.
  7. There should be a significant unidirectional causality between both time series variables.

 

The test used to check bidirectional causality is known as Cross-correlation Test.

 

 

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Volatility models by ARCH & GARCH econometrics using eviews
October 27, 2014
1
Output of Investigating volatility by GARCH & ARCH in EViews

 

Volatility models by ARCH & GARCH econometrics using eviews

 

In order to investigate Volatility models by ARCH & GARCH econometrics using Eviews you need to follow below steps in Eview 6:

 

1. Open Eviews6.exe

2. Click on File -> Open -> Foreign Data as Workfile…

Investigating volatility by GARCH & ARCH in EViews

 

3. Open the data file “broadband_1 “ by selecting through the path C:\Program Files\SPSSInc\Statistics17\Samples\English

4. Select MARKET_1 and MARKET_2 -> OK

 

Volatility models by ARCH & GARCH econometrics using eviews

 

5. Click on Quick -> Estimate Equation…

 

econometrics using eviews

 

6. Select ARCH (Auto Regressive Conditional Hetroskedasticity in methods tab.

 

arch econometrics

 

7. Write the dependent variable first and then than the independent variable and then click OK:

 

volatility models

 

8. You will see the following output:

 

Volatility models by ARCH & GARCH econometrics using eviews

 

The variance equation table shows that MARKET_2 is not taking any part in the model, which is the property of a catalyst, and therefore the independent variable (MARKET_2) is acting as a catalyst. Since the probability of MARKET_2 is equal to 0, that is P<0.05, therefore condition or catalyst is significantly present in the model.

Now coming to variance equation table, the probability of RESID(-1)^2 [ARCH Term] is equal to 0.25, that is P>0.05, therefore volatility cannot be predicted by ARCH term as its probability is insignificant. However, the probability of the [GARCH Term] GARCH(-1) is equals to 0, that is P<0.05, therefore GARCH term is significantly predicting volatility in this model.

Now we will check significance of the whole model with the help of R-Squared or Adjusted R-Squared. If there is only one independent variable in the model then R-Squared is used and if more than one independent variable, then we use Adjusted R-Squared. In this case, since we have more than one independent variable, therefore we will consider Adjusted R-Squared.

The value of Adjusted R-Squared is equal to 0.884 * 100 = 88.4%. This means that this model is 88.4% healthy.

Lastly, the output model is generated by 3 methods:

  1. Akaike info criterion                   75089605376214
  2. Schwarz criterion                         89051902457694
  3. Hannan-Quinn criter.                 805510273286

The maximum value in the above methods is 15.89, therefore the report or result which is being generated above is represented by Schwarz criterion.

Finally, we can conclude that in this model GARCH term has predicted volatility, with the condition MARKET_2 being the catalyst.

 

Now we can conclude that GARCH is the model which measures volatility and the econometric tool which is used to gauge volatility is known as ARCH.

At this point, we can now conclude the investigation of Volatility models by ARCH & GARCH econometrics using Eviews

 

Exponential GARCH or e-GARCH

 

Now we are clear on tfe investigation of Volatility models by ARCH & GARCH econometrics using Eviews, so we can also discuss some more complex forms of GARCH.

If multiple volatility is connected together in a linear pattern, than the model used to measure volatility is known as Liner GARCH. Whereas, if there are multiple volatility connected together such that it forms an exponential pattern, then the model used to measure volatility is known as Exponential GARCH or e-GARCH. If the trend of the volatility changes in a way like bubble where t -> 0, then this is called Component GARCH.

 

Exponential GARCH or e-GARCH

 

Simple GARCH model is expressed as:

 

Simple GARCH model

 

Whereas, E-GARCH is expressed as:

 

E-GARCH model

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ARCH models & generalized autoregressive conditional heteroskedasticity
October 27, 2014
0
GARCH Model

 

Concept of volatility

 

If the water changes to gas then this is called volatility. Similarly, if data has random stages such as Brownian motion than this also means that data is volatile.

The variation in data is mostly due to internal factors. The external factor does not change the volatility or does not bring volatility, but it may work as a catalyst to expose the data.

 

ARCH models & generalized autoregressive conditional heteroskedasticity (GARCH)

 

ARCH models & generalized autoregressive conditional heteroskedasticity can be best explain by below model:

 

Volatility= ∝ + ARCH Term+ GARCH Term

 

Volatility is gauged by the model known as GARCH and the econometrical tool used is known as ARCH.

ARCH models & generalized autoregressive conditional heteroskedasticity model can be mathematically be expressed as:

 

arch models & generalized autoregressive conditional heteroskedasticity

 

Whereas, GARCH is also known as volatility.

 

 

 

Heteroskedasticity Vs Homoscedasticity:

 

To understand ARCH models & generalized autoregressive conditional heteroskedasticity, you first need to understand the concept of Heteroskedasticity and Homoscedasticity.

Consider below model:

X= ∝ + β Y+ET

Suppose for the above model, we have different values of x and y with respect to the change in years, whereas ET remains same as mention in below table:

 

homoskedasticity observations

 

Considering the above model and the table we can now define Homo / Heteroskedasticity as:

When independent variable explains or predicts the dependent variable and at every time “ET” or error term remains same, then this is called homoskedasticity, where as if “ET” is different as shown in below table, then this is called heteroskedasticity.

 

Heteroskedasticity observations

 

The formula for the moment is:  Moment= (∑(x-AM)^n )/N

 

Where, AM = Arithmetic mean, N = total number of observation, n = nth Moment about mean

If n = 2, then this equation turns to Variance.

Therefore the mathematical expression or the formula for Homo / Heteroskedasticity can be further derived from the variance formula as mentioned below:

 

Heteroskedasticity and Homoscedasticity model

 

Where ET = Error term and AM (ET) = Arithmetic mean of the error term.

If H = 0, then this means homoskedasticity is present, else Heteroskedasticity.

 

ARCH (Auto regressive conditional heteroskedasticity / GARCH (Generalized Auto regressive conditional heteroskedasticity) :

 

The term ARCH stands for Auto Regressive conditional heteroskedasticity. Up till now we are clear on all of the terms meaning in ARCH except for conditional heteroskedasticity. If heteroskedasticity is due to any condition that is due to any catalyst involved then this kind of heteroskedasticity will be termed as conditional heteroskedasticity. Therefore we can summarize, that if AR process exists in a model and side by side heteroskedasticity exists due to any catalyst present then this will be known as ARCH.

Remember: if homoskedacity is present then volatility cannot exist, whereas on the other hand if heteroskedasticity is there, then volatility may exist.

 

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Autoregressive distributed lag model by Vector autoregression
October 23, 2014
0
Investigating ARLD model through Vector auto regressive

Autoregressive distributed lag model

 

Autoregressive distributed lag model by Vector autoregression

 

The above model contains ARDL (autoregressive distributed lag model) in addition to VAR / vector autoregression because of both variable, independent and dependent.

The above model is also another form of ARDL model (autoregressive distributed lag model) because AR process is also their and similarly Lag distribution of the dependent variable is there as well.

 

Investigating ARLD (autoregressive distributed lag model model) through VAR (vector autoregression) in EViews:

In order to investigate ARLD model by the help of vector autoregression in Eviews, you need to follow bellow steps:

1. Open Eviews6.exe

2. Click on File -> Open -> Foreign Data as Workfile…

 

Investigating ARLD model through Vector auto regressive

 

3. Open the data file “broadband_1 “ by selecting through the path C:\Program Files\SPSSInc\Statistics17\Samples\English

4. Select MARKET_1 and MARKET_2 -> OK

 

Investigating ARLD model through Vector auto regressive

 

5. Click on Quick -> Estimate VAR…

 

Investigating ARLD model through Vector auto regressive

 

6. First write the dependent variable and then independent variable. In our case, MARKET_1 is dependent variable, whereas MARKET_2 is independent variable, then Click OK:

 

Investigating ARLD model through Vector auto regressive

 

7. The output shows a table with t-statistic value and coefficients. Since we have selected MARKET_1 as our dependent variable, therefore we will use the value in MARKET_1 column.

Investigating ARLD model through Vector auto regressive

 

Mathematically, the above vector autoregression model can be expressed as:

 

Autoregressive distributed lag model by Vector autoregression

 

From the above expression of β1 we can define that, if previous lag of M1 increases by 1, then the current lag of M1 will increase by 1.206. We define β2, β3 & β4 in the similar manner.

To be significant the value of t-stats should be greater than 1.5, therefore from the above model we can see that M1 (t-1) is significantly predicting M1 (t) but M1 (t-2) is not significant. Since the dependent variable is significantly being predicted by one of its lag, therefore AR process executes. On the other hand, the t-stats of M2 (t-1) & M2 (t-2) both are insignificant; therefore LD process cannot be proved in this model.

Now we will check significance of the whole model with the help of R-Squared or Adjusted R-Squared. If there is only one independent variable in the model then R-Squared is used and if more than one independent variable, then we use Adjusted R-Squared. In this case, since we have more than one independent variable, therefore we will consider Adjusted R-Squared.

The value of Adjusted R-Squared is equal to 0.998 * 100 = 98.8%. This means that this model is 98.8% healthy.

Now to check whether the above value of -Squared or Adjusted R-Squared is significant or not we will consider F-statistic value.

F-statistic or F > 3.84 is significant

Therefore in the above table F = 10926.4 are insignificant.

Hence we can conclude that this model explains 98.9% which is proved by F statistics.

Lastly, the output model is generated by 2 methods:

  1. Akaike AIC 81048631603165
  2. Schwarz SC 98811071349254

The maximum value in the above methods is 11.988, therefore the report or result which is being generated above is represented by Schwarz criterion.

Finally, we can conclude that this model is an ARLD model, but only AR process has been proved in the model.

 

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Auto Regreser (AR) process and Auto Regressive distributed lag (ARDL) Model
October 23, 2014
0
auto regressive distributed lag model or ARDL model

The prediction of dependent variable by the predictors (independent Variable) is called regression.

 

regression

 

If the previous lags of a variable is distributed in a model such that it is predicting the dependent variable than this is called lag distribution model. Whereas, if the dependent variable is being predicted by its own previous dependent variable than this model is known as auto regressive distributed lag model or ARDL model.

Example of such model are mention below:

 

auto regressive distributed lag model or ARDL model

 

Property of ARDL (Auto regressive distributed Lag) Model:

Two property of ARDL must be there:

  1. Lags are present in the model.
  2. AR process should execute.

 

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Johansen cointegration test
October 23, 2014
2
Output of Investigation of Johansen Cointegration by EViews

Cointegration

 

Before jumping to johansen cointegration test you first need to understand Cointegration.

If two dependent variables are associated with each other, that is, shows the same trend and side by side their independent variable are dissimilar then this kind of association is known as Co-integration.

 

Johansen Cointegration Test

 

To investigate Johansen cointegration by using EViews, you need to follow below steps:

 

1. Open Eviews6.exe

2. Click on File -> Open -> Foreign Data as Workfile…

 

Johansen cointegration test in Eviews

 

3. Open the data file “broadband_1 “ by selecting through the path
C:\Program Files\SPSSInc\Statistics17\Samples\English

4. Select MARKET_1 and MARKET_2 -> OK

 

Johansen cointegration test in Eviews

 

5. Click on Quick -> Group Statistics -> Cointegration Test

Johansen cointegration test in Eviews

 

6. Write down the two variables in which you want to check co-integration and click OK twice

 

Johansen cointegration test in Eviews

 

Johansen cointegration test in Eviews

 

7. Following output for johansen cointegration test will be visible:

Johansen cointegration test in Eviews

 

We will only study the following table in order to interpret the result:

 

Johansen cointegration test in Eviews Output

 

From the above Johansen cointegration test by using EViews, we can conclude following results:

 

Since Max-Eigen Stats (8.836) < Critical value at 5% (14.07) and 1% (18.63), therefore the co-integration does not exists in the current lag (None). Similarly, Max-Eigen Stats (0.014) < Critical value at 5% (3.76) and 1% (6.65), therefore co-integration does not exists in previous lags as well (At most 1).

 

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Seemingly unrelated regression model (SURE)
October 23, 2014
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seemingly unrelated estimation

Before jumping to SURE or seemingly unrelated regression model, you need to understand following important terms:

 

  • If two variables are associated with each other than this is called correlation.
  • If two observation of a same variable are associated with each other than this is known as auto correlation.
  • If in two variables auto correlation exist and these two variables are associated with each other, then cross correlation exists between the two variables.
  • If two observation of independent variable are associated with each other than this is called Co-linear.

 

Seemingly unrelated regression model

 

Finally, if two observation of dependent variable are associated with each other than this is called SURE or seemingly unrelated regression model.

 

seemingly unrelated regression model

 

As per the above two equations, two dependent variable (CB & JB) has same independent variable (Z), then the two dependent variables are seemingly unrelated to each other. Therefore, this type of association of Seemingly unrelated equations variables is called SURE and the model is known as seemingly unrelated regression model.

 

Johenson Cointegration

 

On the other hand, as described in the above model, if two dependent variables are associated with each other, that is, shows the same trend and side by side their independent variable are dissimilar then this kind of association is known as Co-integration. Co-integration can be investigated through Johenson Cointegration test.

If we have to investigate trend of more then 1 equations and at the same time their independent variable are dissimilar, then the tool used is known as Co-integration. Whereas, if we want to investigate trend of one equation only, then the econometric tool deployed is known as Hodrick Prescott filter.

 

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Relationship / Association between variables
October 23, 2014
0
Relationship - Association between variables

Relationship tells the association between two variables. There are four types of relationships:

Deterministic Relationship (DR): Output and the input are always pre-determined and it has the same execution at all times. This means that its output will always remain the same that is independent of time and place.

Stochastic Relationship (SR): The output in such scenarios may differ from place to pace or from time to time.

Linear Relationship: if the relationship / association between two variables are shown by 1 equation is called linear relationship.

Curvilinear Relationship: if the relationship / association between two variables are shown by more than 1 equations is called Curvilinear relationship.

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Test for non-stationarity by Autocorrelation in SPSS
October 23, 2014
0
Output of Investigation of Shocks by Auto correlation in SPSS

 

Test for non-stationarity

 

In order to investigate shock or test for non-stationarity by the help of Autocorrelation, you need to follow below steps:

1. Open SPSS 17.0

 

2. Click on file -> open -> Data

test for non-stationarity in SPSS

 

3. Open the data file “Broadband 1 “ by selecting through the path
C:\Program Files\SPSSInc\Statistics17\Samples\English

 

4. Remember, MARKET_1 is a time series as the data is expressed in months. Now click on Analyze -> Forecasting            -> Autocorrelations

test for non-stationarity in SPSS

 

5. Select Subscriber for Market_1, click OK

test for non-stationarity in SPSS

 

6. Output window will open and following figure will be displayed. This is the figure for dice down pattern which also               proves that shocks are present in this data:

Test for non-stationarity output

 

Hence, the test for non-stationarity of the above model proves that shocks are present in the given set of data.

 

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Developing PAC (Partial Auto correlation) to investigate non-stationarity (Shock)
October 23, 2014
0
Partial Auto correlation to investigate Shock

If two consecutive observations are associated with each other is called autocorrelation. PAC investigates autocorrelation as well as shocks in a data but it cannot tell whether the data series has a permanent or a temporary shocks.

Partial Auto correlation to investigate Shock

 

If output (PAC) of 1st lag is positive and the 2nd lag is “0” or negative, then this is called cutoff and when cut off is present then this means shocks are there. For example graph A.

If the output (PAC) is like a decreasing lags this is called dice down, then this will also tell that there is shocks present in the data series. For example graph B.

Output other than the above two example will indicate that there is no shocks present in the data series.

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