This table shows the variable of year 2008 that we will examine starting with (y) which is dependent variable and (x) is independent variable . As we mentioned before The stock price index of the market is our dependent variable (y) and the stock price index of Commercial Bank of Dubai is our independent variable (x). Also we explained that if any change happened in stock price index of Commercial Bank of Dubai will lead to changes in stock price index of the market which means that it’s the changes when the independent variable changes.
To explain the relationship between the stock price index of Commercial Bank of Dubai and stock price index of the market, we drew a graph showing their performance of the year 2008.As we can notice they almost have the same shape indicating that they are moving with the same fluctuating in the value stock. And we found that there is a positive relationship between them.
Estimating the regression Equation :
In , regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a and one or more . The mathematical model of this relationship between variable is usually called regression equation. The regression equation contains estimates of unknown regression parameters .These parameters are usually estimated by ordinary last squares (OLS) method using a given data set . This method aims to obtain optimal solution for each parameter in the liner regression to minimize the error sum of squares (SSE) .
Using E-views to apply econometric software by importing the data above .The following will show the estimation result:
Then we are interesting in defining the simple regression equation which is :
yt=b1+b2 xt+et
where , C=b1 and X=b2 . if Xt = 0 then yt = 482.8090 , c.p. And If xt increase by 1 unites then yt will increase by 2136.569 unites ,c.p. Therefore , the regression equation will be after using E-views and conducting the result from above :
yt=469.8173+2127.691 xt+et
After that we estimate the regression equation conducting diagnostic test to figure out the relationship between the dependent variable and independent variable throw 4 assumption and if it is statistically significant or not .Starting with testing the following hypothesis :
1. Ho: b1=0 H1: b1≠0
if the probability of the parameter is 0.0035 then we reject the null hypothesis because it is lower than the significant level (α=0.05).which we conclude that b1≠0 and it is statistically significant with respect to dependent variable . and it would be unrestricted model if it was under the alternative hypothesis .
2. Ho: b2=0 H1: b2≠0
Since the probability of the parameter is 0.0000 then we strongly reject the null hypothesis because it is lower than the significant level (α=0.05).
Now estimating the coefficient of determination which denoted by R2 .Going back to the above result the R2 will equal 0.744859. This values means that more than 74.5% of variation in the dependent variable yt is explained by the regression model and the rest is explained by the error .
In addition to contracting a regression equation . Now we should check weather the assumption are fulfilled or not . And by using E-views the result will be shown below:
Firstly: Test for No-Misspecification:
Recall to what we took in the course that the most important assumption for a good model is that the expected value of the error term is equal to zero ( E(et)=0). Because this required to obtain unbiased estimated parameters. This can be tested by RESET test as suggested by Ramsey which is mentioned by the following steps:
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1. Estimate the regression model (yt=b1+b2 xt+et ) and obtain t , 2t , 3t and 4t . After that estimate the following regression :
yt=b1+b2 xt+c1 2t + c23t + c3 4t +ut
- 2. Test the following null hypoythisis of no-mispicification against the alternative hypothesis of misspecification by using the F-test :
Ho: c1= c2 = c3 =0 H1: c1≠0 , c2 ≠0 or c3 ≠0
No -Misspecification Misspecification
E(et)=0 E(et) ≠0
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3. Then test via F-test using these results : propapility =0.0000 , r=3 restrictions , k=5 prameters and T= 262 observation :
F=(SSER- SSEU )/3
SSEU /(262-5)
Note that :
SSER is the residual sum of squares in the equation (yt=b1+b2 xt+et ) = e2t
SSEU is the residual sum of squares in the equation (yt=b1+b2 xt+c1 2t +c23t + c3 4t +ut) =u2t
Since the probability of the F-test = 0.0000 is lower than the significant level (α=0.05).) then we reject null hypothesis at that significant level . Then we accept H1 of misspecification which indicate that the estimated parameter is not unbiased (biased) . So if the null hypnosis is rejected we need to find remedy to solve the problem .
Remedy :
By using E-views we will repeat the same step and start again. Just to solve the problem we need to add one or more variables for example :
yt=b1+b2 xt+ b3 zt +et
Going throw the result below . we would notice that the F-test has increase to 0.4555 which make it higher than significant level (α=0.05).) then we don’t reject null hypothesis at that significant level And by that we could have unbiased estimated model by assuming No –misspecification
Second :Test For No-Autocorrelation :
Another assumption for a good model that the value of error terms should not be dependent on each other (Cov (et ,et-s ) =0) .By using Bruesch-Godfery Social correlation LM test will indicate if the estimated parameter is efficient or not :
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1. Estimate the regression model (yt=b1+b2 xt+et ) and obtain After that estimate the following regression :
=b1+b2 xt+p1 + wt
- 2. Test the following null hypoythisis of no- Autocorrelation of order one
against the alternative hypothesis of misspecification by using the F-test :
Ho: p1= 0 H1: p1≠0
No -Autocorrelation of degree 1 Autocorrelation of degree 1
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3. Then test via F-test using these results : probability =0.0000 , r=1 restrictions , k=3 parameters and T= 262 observation :
F=(SSER- SSEU )/1
SSEU /(262-3)
Note that :
SSER is the residual sum of squares in the equation (yt=b1+b2 xt+et ) = e2t
SSEU is the residual sum of squares in the equation (=b1+b2 xt+p1 + wt
) =w2t
Since the probability of the F-test = 0.0000 is lower than the significant level (α=0.05).) then we reject null hypothesis at that significant level . It means that there is autocorrelation and the estimated parameter is not efficient .One Potential solution is to do remedy .
Remedy :
It has been shown that the null hypotheses has been rejected and the model is not efficient .so we need to do remedy to fix the problem by making the model dynamic :
yt=b1+b2 xt+ b3 xt-1 + b4 yt-1 +et
looking at the figure below we could see that F-test (0.1064 ) is higher than the significant level (α=0.05).) and by that we will not reject the null hypothesis .So the model can remove autocorrelation of degree one and the estimated parameter would be efficient :
yt=-0.524322+11.28928 xt+ -14.22664 xt-1 + 0.001366 yt-1 +et
Third :Test for Homoscedasticity :
Homoscedasticity is required in addition to non autocorrelation in order to make the efficient estimated parameter . One of the testing is Brysesch-Pagen test which indicate if we have constant variance otherwise if the variance of the error term is not constant across the observation then Heteroscedasticity occurs .According to th B-P test :
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1. Estimate the regression model (yt=b1+b2 xt+et ) and obtain and After that estimate the following regression :
=ho+h1 xt+ v
- 2. Test the following null hypothesis of Homoscedasticity against the alternative hypothesis of Heteroscedasticity by using the F-test :
Ho: h1= 0 H1: h1≠0
Homoscedasticity Heteroscedasticity
(constant variance) (Not constant variance )
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3. Then test via F-test using these results : probability =0.0000 , r=1restrictions , k=2 parameters and T= 262 observation :
F=(SSER- SSEU )/1
SSEU /(262-2)
Note that :
SSER is the residual sum of squares in the equation (yt=b1+b2 xt+et ) = w2t
SSEU is the residual sum of squares in the equation (=ho+h1 xt+ v) =v2t
if the probability of the F-test = 0.0000 then we reject null hypothesis because it is lower than the significant level (α=0.05).)..and when null hypothesis is rejected it means there is Heteroscedasticity and the estimated parameter is not efficient . One Potential solution is to do remedy .
Remedy :
One of the way to make the model sufficient is to make use of one of the solution and these are : robust stander errors and weighted last squares in the estimations . When we used the E-views to repeat the steps we used weighted last squares to have new t-test and to make the model more efficient . And the probability of the t-test = 0.0988 which is higher than l the significant level (α=0.05).).. which means there is Homoscedasticity.
Fourth :Test For Normality :
This assumption is important because the distribution that are used to test different null hypothesis are based on this assumption . One testing we can use for this assumption is Jarque-Bera test .
Therefore , the null hypothesis of normality against the alternative hypothesis of non normality is :
Ho: S=(K-3)= 0 H1: S≠ 0 or (K-3) ≠ 0
Normality Non-Normality
The null hypothesis of normality is rejected if the probability of JB –test is lower than the significant level . the best solution for the problem is using logarithm if possible , remove outlines via dummy variables , allowing parameters instability or using alternative methods like boot simulation .
References
