an elementary investigation into regression model using eviews 5.1
TRANSCRIPT
1
An Investigation into Regression Model using
EVIEWS
Module One
Prepared by:
Sayed Hossain
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Seven assumptions about a good regression model
1. Regression line must be fitted to data strongly.
2. Most of the independent variables should be individually significant to explain dependent variable
3. Independent variables should be jointly significant to influence or explain dependent variable.
4. The sign of the coefficients should follow economic theory or expectation or experiences or intuition.
5. No serial or auto-correlation in the residual (u)
6. The variance of the residual (u) should be constant meaning that homoscedasticity
7. The residual (u) should be normally distributed.
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(Assumption no. 1)
Regression line must be fitted to data strongly
(Goodness of Data Fit)
***
Guideline : R2 => 60 percent (0.60) is better
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Goodness of Data Fit
• Data must be fitted reasonable well.
• That is value of R2 should be reasonable high, more than 60 percent.
• Higher the R2 better the fitted data.
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5
(Assumption no. 2)
Most of the independent variables should
be individually individually significant
**
t- test
t –test is done to know whether each and every
independent variable (X1, X2 and X3 etc here) is
individually significant or not to influence the
dependent variable, that is Y here.
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Individual significance of the variable
• Most of the independent variables should be individually
significant.
• This matter can be checked using t test.
• If the p-value of t statistics is less than 5 percent (0.05) we can reject the null and accept alternative hypothesis.
• If we can reject the null hypothesis, it means that particular independent variable is significant to influence dependent variable in the population.
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For Example>>
Variables: We have four variables, Y, X1, X2 X3 Here Y is dependent and X1, X2 X3 are independent
Population regression model
Y = Bo + B1X1+ B2X2 + B3X3 + u
Sample regression model
Y = bo + b1X1+ b2X2 + b3X3 + e
Here, sample regression line is a estimator of population regression line. Our target is to estimate population regression line (which is almost impposible or time and money consuming to estimate) from sample regression line. For example, small b1, b2 and b3 are estimators of big B1, B2 and B3
Here, u is the residual for population regression line while e is the residual for sample regression line. e is the estimator of u. We want to know the nature of u from e.
Tips
If the sample collection is done as per the
statistical guideline (several random procedures)
then sample regression line can be a
representative of population regression line.
Our target is to estimate the population
regression line from a sample regression line.
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Setting hypothesis for t –test : An example
Null Hypothesis: Bo=0
Alternative hypothesis: Bo≠0
Null hypothesis : B1=0
Alternative hypothesis: B1≠0
Null Hypothesis : B2=0
Alternative hypothesis: B2≠0
Null Hypothesis : B3=0
Alternative hypothesis: B3 ≠0
Hypothesis setting is always done for population, not for sample. That
is why we have taken all big B (from population regression line) but
not small b from sample regression line.
Hypothesis Setting Null hypothesis : B1=0
Alternative hypothesis: B1≠0
• Since the direction of alternative hypothesisis is ≠, meaning that we assume that there exists a relationship between independent variable (X1 should be here) with dependent variable (Y here) in the population. But it can not say whether the relationship is negative or positive. This direction ≠ is a two tail hypothesis.
Null hypothesis : B1=0
Alternative hypothesis: B1<0
• But if we set hypothesis as above, then we assume that in the population, there exists a negative relationship between X1 and Y as the direction in alternative hypothesis is <. It requires one tail test.
`
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(Assumption no. 3)
Joint Significace
Independent variables should be jointly significant
to explain dependent variable
**
F- test
ANOVA
(Analysis of Variance)
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Joint significance
• Independent variables should be jointly
significant to explain Y. This can be checked using F-test.
• If the p-value of F statistic is less than 5 percent (0.05) we can reject the null and accept alternative hypothesis.
• If we can reject null hypothesis, it means that all the independent variables (X1, X2 X3 ) jointly can influence dependent variable, that is Y here.
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Joint hypothesis setting
Null hypothesis Ho: B1=B2=B3=0
Alternative H1: Not all B’s are simultaneously equal to
zero
Here Bo is dropped as it is not associated with any
variable.
Here also taken all big B
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Few things
• Residual ( u or e) = Actual Y – estimated (fitted)
Y
• Residual, error term, disturbance term all are
same meaning.
• Serial correlation and auto-correlation are same
meaning.
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(Assumption no. 4)
The sign of the coefficients should follow
economic theory or expectation or experiences
of others (literature review) or intuition.
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Residual Analysis
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(Assumption no. 5)
No serial or auto-correlation in the residual (u).
** Breusch-Godfrey serial correlation LM test : BG test
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Serial correlation
• Serial correlation is a statistical term used to the
describe the situation when the residual is
correlated with lagged values of itself.
• In other words, If residuals are correlated, we
call this situation serial correlation which is not
desirable.
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How serial correlation can be formed in the
model?
• Incorrect model specification,
• omitted variables,
• incorrect functional form,
• incorrectly transformed data.
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Detection of serial correlation
• Many ways we can detect the existence of serial
correlation in the model.
• An approach of detecting serial correlation is
Breusch-Godfrey serial correlation LM test : BG
test
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Hypothesis setting
Null hypothesis Ho: no serial correlation (no correlation between residuals (ui and uj))
Alternative hypothesis H1: serial correlation
(correlation between residuals (ui and uj )
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(Assumption no. 6)
The variance of the residual (u) is
constant (Homoscedasticity)
***
Breusch-Pegan-Godfrey Test
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• Heteroscedasticity is a term used to the
describe the situation when the variance of the
residuals from a model is not constant.
• When the variance of the residuals is constant,
we call it homoscedasticity. Homoscedasticity
is desirable.
• If residuals do not have constant variance, we
call it hetersocedasticty, which is not desirable.
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How the heteroscedasticity may form?
– Incorrect model specification,
– Incorrectly transformed data,
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Hypothesis setting for heteroscedasticity
– Null hypothesis Ho: Homoscedasticity (the
variance of residual (u) is constant)
– Alternative hypothesis H1 : Heteroscedasticity
(the variance of residual (u) is not constant )
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Detection of heteroscedasticity
• There are many test involed to detect
heteroscedasticity.
• One of them is Bruesch-Pegan-Godfrey test
which we will employ here.
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(Assumption no. 7)
Residuals (u ) should be normally distributed
**
Jarque Bera statistics
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Setting the hypothesis:
• Null hypothesis Ho : Normal distribution (the residual (u) follows a normal distribution)
• Alternative hypothesis H1: Not normal distribution (the residual (u) follows not normal distribution)
Detecting residual normality:
• Histogram-Normality test (Perform Jarque-Bera Statistic).
• If the p-value of Jarque-Bera statistics is less than 5 percent (0.05) we can reject null and accept the alternative, that is residuals (u) are not normally distributed.
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Our hypothetical model
Variables:
We have four variables, Y, X1, X2 X3 Here Y is dependent and X1, X2 and X3 are independent
Population regression model
Y = Bo + B1X1+ B2X2 + B3X3 + u
Sample regression line
Y = bo+ b1X1+ b2X2+b3X3 + e
DATA
Sample size is 35 taken from
population
DATA
obs RESID X1 X2 X3 Y YF
1 0.417167 1700 1.2 20000 1.2 0.782833
2 -0.27926 1200 1.03 18000 0.65 0.929257
3 -0.17833 2100 1.2 19000 0.6 0.778327
4 0.231419 937.5 1 15163 1.2 0.968581
5 -0.33278 7343.3 0.97 21000 0.5 0.832781
6 0.139639 837.9 0.88 15329 1.2 1.060361
7 -0.01746 1648 0.91 16141 1 1.017457
8 -0.14573 739.1 1.2 21876 0.65 0.795733
9 0.480882 2100 0.89 17115 1.5 1.019118
10 -0.0297 274.6 0.23 23400 1.5 1.529701
11 -0.32756 231 0.87 16127 0.75 1.077562
12 0.016113 1879.1 0.94 17688 1 0.983887
13 -0.34631 1941 0.99 17340 0.6 0.946315
14 0.485755 2317.6 0.87 21000 1.5 1.014245
15 0.972181 471.4 0.93 16000 2 1.027819
16 -0.22757 678 0.79 16321 0.9 1.127572
17 -0.2685 7632.9 0.93 18027 0.6 0.868503
18 -0.41902 510.1 0.93 18023 0.6 1.019018
19 -0.4259 630.6 0.93 15634 0.6 1.0259
20 0.076632 1500 1.03 17886 1 0.923368
DATA obs RESID X1 X2 X3 Y YF
21 -0.37349949 1618.3 1.1 16537 0.5 0.873499
22 0.183799347 2009.8 0.96 17655 1.15 0.966201
23 0.195832507 1562.4 0.96 23100 1.15 0.954167
24 -0.46138707 1200 0.88 13130 0.6 1.061387
25 0.309577968 13103 1 20513 1 0.690422
26 -0.21073204 3739.6 0.92 17409 0.75 0.960732
27 -0.08351157 324 1.2 14525 0.75 0.833512
28 -0.02060854 2385.8 0.89 15207 1 1.020609
29 0.14577644 1698.5 0.93 15409 1.15 1.004224
30 -0.06000649 544 0.87 18900 1 1.060006
31 -0.50510204 1769.1 0.45 17677 0.85 1.355102
32 0.870370225 1065 0.65 15092 2.1 1.22963
33 0.274774344 803.1 0.98 18014 1.25 0.975226
34 -0.1496757 1616.7 1 28988 0.75 0.899676
35 0.062732149 210 1.2 21786 0.87 0.807268
Y, X1, X2 and X3 are actual sample data collected from population
YF= Estimated, forecasted or predicted Y
RESID (e) = Residuals of the sample regression line that is, e=Actual Y – Predicted Y (fitted Y)
Regression Output
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Dependent Variable: Y 35 Observation Method: Least Squares
Included observations: 98
Variable Coefficient Std. Error t-Statistic Prob.
C 1.800 0.4836 3.72 0.0008
X1 -2.11E-05 2.58E-05 -0.820 0.4183
X2 -0.7527 0.3319 -2.267 0.0305
X3 -3.95E-06 2.08E-05 -0.189 0.8509
R-squared 0.1684 Mean dependent var 0.9834
Adjusted R-squared 0.087 S.D. dependent var 0.3912
S.E. of regression 0.3736 Akaike info criterion 0.9762
Sum squared resid 4.328 Schwarz criterion 1.15
Log likelihood -13.08 F-statistic 2.093
Durbin-Watson stat 2.184 Prob(F-statistic) 0.1213
Regression output
Few things
• t- statistics= Coeffient / standard error
• t-statistics (absolute value) and p values always
move in opposite direction
Output Actual Y, Fitted Y, Residual and its plotting
obs Actual Fitted Residual Residual Plot
1 1.2 0.782832991 0.417167009 | . | .* |
2 0.65 0.92925722 -0.27925722 | .* | . |
3 0.6 0.778327375 -0.178327375 | . * | . |
4 1.2 0.96858115 0.23141885 | . | *. |
5 0.5 0.8327808 -0.3327808 | * | . |
6 1.2 1.060360549 0.139639451 | . | * . |
7 1 1.017457055 -0.017457055 | . * . |
8 0.65 0.79573323 -0.14573323 | . * | . |
9 1.5 1.019118163 0.480881837 | . | . * |
10 1.5 1.529701243 -0.029701243 | . * . |
11 0.75 1.077562408 -0.327562408 | * | . |
12 1 0.983887019 0.016112981 | . * . |
13 0.6 0.946314864 -0.346314864 | * | . |
14 1.5 1.014244939 0.485755061 | . | . * |
15 2 1.027819105 0.972180895 | . | . *
16 0.9 1.127572088 -0.227572088 | .* | . |
17 0.6 0.868503447 -0.268503447 | .* | . |
18 0.6 1.019018495 -0.419018495 | *. | . |
19 0.6 1.025899595 -0.425899595 | *. | . |
20 1 0.923368304 0.076631696 | . |* . |
obs Actual Fitted Residual Residual Plot
21 0.5 0.873499486 -0.373499486 | * | . |
22 1.15 0.966200653 0.183799347 | . | * . |
23 1.15 0.954167493 0.195832507 | . | * . |
24 0.6 1.061387074 -0.461387074 | *. | . |
25 1 0.690422032 0.309577968 | . | * |
26 0.75 0.960732042 -0.210732042 | . * | . |
27 0.75 0.833511567 -0.083511567 | . *| . |
28 1 1.020608541 -0.020608541 | . * . |
29 1.15 1.00422356 0.14577644 | . | * . |
30 1 1.060006494 -0.060006494 | . *| . |
31 0.85 1.355102042 -0.505102042 | * . | . |
32 2.1 1.229629775 0.870370225 | . | . * |
33 1.25 0.975225656 0.274774344 | . | *. |
34 0.75 0.899675696 -0.149675696 | . * | . |
35 0.87 0.807267851 0.062732149 | . |* . |
Output Actual Y, Fitted Y, Residual and its plotting
Actual Y, Fitted Y and Residual
-0.8
-0.4
0.0
0.4
0.8
1.2
0.4
0.8
1.2
1.6
2.0
2.4
5 10 15 20 25 30 35
Residual Actual Fitted
Sample residual
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
5 10 15 20 25 30 35
Y Residuals
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(Assumption no. 1)
Goodness of Fit Data
R-square: 0.1684
• It means that 16.84 percent variation in Y can
be explained jointly by three independent
variables such as X1, x2 and X3. The rest
83.16 percent variation in Y can be explained
by residuals or other variables other than X1 X2
and X3.
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(Assumption no. 2)
Joint Hypothesis : F statistics
F statistics: 2.093 and Prob 0.1213
Null hypothesis Ho: B1=B2=B3=0 Alternative H1: Not all B’s are simultaneously equal to zero
Since the p-value is more tha than 5 percent (here 12.13 percent), we can not reject null. In other words, it means that all the independent variables (here X1 X2 and X3) can not jointly explain or influence Y in the population.
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Assumption No. 3 Independent variable significance
• For X1, p-value : 0.4183
Null Hypothesis: B1=0
Alternative hypothesis: B1≠0
Since the p-value is more than 5 percent (0.05) we can not reject null and meaning we accept null meaning B1=0. In other words, X1 can not influence Y in the population.
• For X2, p-value: 0.0305 (3.05 percent)
Null Hypothesis: B2=0
Alternative hypothesis: B2≠0
Since p-value (0.03035) is less than 5 percent meaning that we can reject null and accept alternative hypothesis. It means that variable X2 can influence variable Y in the population but what direction we can not say as alternative hypothesis is ≠.
• For X3, p-value: 0.8509. So X3 is not significant to explain Y.
Assumption No. 4
Sign of the coefficients
Our sample model:
Y=bo+b1x1+b2x2+b3x3+e
Sign we expected after estimation as follows:
Y=bo - b1x1 + b2x2 - b3x3
Decision : The outcome did not match with our expectation.
So assumption 4 is violated.
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Breusch-Godfrey Serial Correlation LM Test:
F-statistic 1.01 Prob. F(2,29) 0.3751
Obs*R-squared 2.288 Prob. Chi-Square(2) 0.3185
Null hypothesis : No serial correlation in the residuals
(u)
Alternative: There is serial correlation in the residuals
(u)
Since the p-value ( 0.3185) of Obs*R-squared is
more than 5 percent (p>0.05), we can not reject null
hypothesis meaning that residuals (u) are not
serially correlated which is desirable.
Assumption no 5
SERIAL OR AUTOCORRELATION
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Assumption no. 6 Heteroscedasticy Test
F-statistic 1.84 Probability 0.3316
Obs*R-squared 3.600 Probability 0.3080
Breusch-Pegan-Godfrey test (B-P-G Test)
Null Hypothesis: Residuals (u) are Homoscedastic
Alternative: Residuals (u) are Hetroscedastic
The p-value of Obs*R-squared shows that we can not
reject null. So residuals do have constant variance which
is desirable meaning that residuals are homoscedastic.
B-P-G test normally done for large sample
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Assumption no. 7 Residual (u) Normality Test
0
1
2
3
4
5
6
-0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
Series: Residuals
Sample 1 35
Observations 35
Mean 1.15e-16
Median -0.029701
Maximum 0.972181
Minimum -0.505102
Std. Dev. 0.356788
Skewness 0.880996
Kurtosis 3.508042
Jarque-Bera 4.903965
Probability 0.086123
Null Hypothesis: residuals (u) are normally distribution
Alternative: Not normally distributed
Jarque Berra statistics is 4.903 and the corresponding p value is
0.08612. Since p vaue is more than 5 percent we accept null meaning
that population residual (u) is normally distrbuted which fulfills the
assumption of a good regression line.
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Evaluation of our model on the basis of
assumptions
1. R-square is very low ( Bad sign)
2. There is no serial correlation (Good sign)
3. Independent variables are not jointly can influence Y (Bad sign)
4. Signs are not as expected (Bad sign)
5. Only X2 variable is significant out of three (Bad sign).
6. Heteroscedasticity problem is not there (Good sign)
7. Residuals are normally distributed (Good sign)
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