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1 An Investigation into Regression Model using EVIEWS Prepared by: Sayed Hossain Lecturer for Economics Multimedia University Personal website: www.sayedhossain.comwww.sayedhossain.com Email: sayed.hossain@yahoo.comsayed.hossain@yahoo.com

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2 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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3 (Assumption no. 1) Regression line must be fitted to data strongly (Goodness of Data Fit) *** Guideline : R 2 => 60 percent (0.60) is better

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4 Goodness of Data Fit Data must be fitted reasonable well. That is value of R 2 should be reasonable high, more than 60 percent. Higher the R 2 better the fitted data. www.sayedhossain.com

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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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6 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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7 For Example>> Variables: We have four variables, Y, X 1, X 2 X 3 Here Y is dependent and X 1, X 2 X 3 are independent Population regression model Y = Bo + B 1 X 1 + B 2 X 2 + B 3 X 3 + u Sample regression model Y = bo + b 1 X 1 + b 2 X 2 + b 3 X 3 + 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 b 1, b 2 and b 3 are estimators of big B 1, B 2 and B 3 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.

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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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9 Setting hypothesis for t –test : An example Null Hypothesis: Bo=0 Alternative hypothesis: Bo≠0 Null hypothesis : B 1 =0 Alternative hypothesis: B 1 ≠0 Null Hypothesis : B 2 =0 Alternative hypothesis: B 2 ≠0 Null Hypothesis : B 3 =0 Alternative hypothesis: B 3 ≠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.

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Hypothesis Setting Null hypothesis : B 1 =0 Alternative hypothesis: B 1 ≠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 : B 1 =0 Alternative hypothesis: B 1 <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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11 (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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12 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 (X 1, X 2 X 3 ) jointly can influence dependent variable, that is Y here.

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13 Joint hypothesis setting Null hypothesis Ho: B 1 =B 2 =B 3 =0 Alternative H 1 : 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 www.sayedhossain.com www.sayedhossain.com

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14 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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15 (Assumption no. 4) The sign of the coefficients should follow economic theory or expectation or experiences of others (literature review) or intuition. www.sayedhossain.com

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Residual Analysis

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17 (Assumption no. 5) No serial or auto-correlation in the residual (u). ** Breusch-Godfrey serial correlation LM test : BG test

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18 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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19 How serial correlation can be formed in the model? Incorrect model specification, omitted variables, incorrect functional form, incorrectly transformed data.

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20 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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21 Hypothesis setting Null hypothesis H o : no serial correlation (no correlation between residuals (ui and uj)) Alternative hypothesis H 1 : serial correlation (correlation between residuals (ui and uj )

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22 (Assumption no. 6) The variance of the residual (u) is constant (Homoscedasticity) *** Breusch-Pegan-Godfrey Test

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23 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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24 How the heteroscedasticity may form? –Incorrect model specification, –Incorrectly transformed data,

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25 Hypothesis setting for heteroscedasticity –Null hypothesis H o : Homoscedasticity (the variance of residual (u) is constant) –Alternative hypothesis H 1 : Heteroscedasticity (the variance of residual (u) is not constant )

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26 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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27 (Assumption no. 7) Residuals (u ) should be normally distributed ** Jarque Bera statistics

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28 Setting the hypothesis: Null hypothesis H o : Normal distribution (the residual (u) follows a normal distribution) Alternative hypothesis H 1 : 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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29 An Emperical Model Development www.sayedhossain.com

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30 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 + B 1 X 1 + B 2 X 2 + B 3 X 3 + u Sample regression line Y = bo+ b 1 X 1 + b 2 X 2 +b 3 X 3 + e

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DATA Sample size is 35 taken from population

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DATA obsRESIDX1X2X3YYF 10.41716717001.2200001.20.782833 2-0.2792612001.03180000.650.929257 3-0.1783321001.2190000.60.778327 40.231419937.51151631.20.968581 5-0.332787343.30.97210000.50.832781 60.139639837.90.88153291.21.060361 7-0.0174616480.911614111.017457 8-0.14573739.11.2218760.650.795733 90.48088221000.89171151.51.019118 10-0.0297274.60.23234001.51.529701 11-0.327562310.87161270.751.077562 120.0161131879.10.941768810.983887 13-0.3463119410.99173400.60.946315 140.4857552317.60.87210001.51.014245 150.972181471.40.931600021.027819 16-0.227576780.79163210.91.127572 17-0.26857632.90.93180270.60.868503 18-0.41902510.10.93180230.61.019018 19-0.4259630.60.93156340.61.0259 200.07663215001.031788610.923368

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DATA obsRESIDX1X2X3YYF 21-0.373499491618.31.1165370.50.873499 220.1837993472009.80.96176551.150.966201 230.1958325071562.40.96231001.150.954167 24-0.4613870712000.88131300.61.061387 250.3095779681310312051310.690422 26-0.210732043739.60.92174090.750.960732 27-0.083511573241.2145250.750.833512 28-0.020608542385.80.891520711.020609 290.145776441698.50.93154091.151.004224 30-0.060006495440.871890011.060006 31-0.505102041769.10.45176770.851.355102 320.87037022510650.65150922.11.22963 330.274774344803.10.98180141.250.975226 34-0.14967571616.71289880.750.899676 350.0627321492101.2217860.870.807268 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)

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Regression Output

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35 Dependent Variable: Y35 ObservationMethod: Least Squares Included observations: 98 VariableCoefficientStd. Errort-StatisticProb. C1.8000.48363.720.0008 X1-2.11E-052.58E-05-0.8200.4183 X2-0.75270.3319-2.2670.0305 X3-3.95E-062.08E-05-0.1890.8509 R-squared0.1684Mean dependent var0.9834 Adjusted R-squared0.087S.D. dependent var0.3912 S.E. of regression0.3736Akaike info criterion0.9762 Sum squared resid4.328Schwarz criterion1.15 Log likelihood-13.08F-statistic2.093 Durbin-Watson stat2.184Prob(F-statistic)0.1213 Regression output

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Few things t- statistics= Coeffient / standard error t-statistics (absolute value) and p values always move in opposite direction

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Output Actual Y, Fitted Y, Residual and its plotting obsActualFittedResidualResidual Plot 11.20.7828329910.417167009|. |.* | 20.650.92925722-0.27925722|.* |. | 30.60.778327375-0.178327375|. * |. | 41.20.968581150.23141885|. | *. | 50.50.8327808-0.3327808| * |. | 61.21.0603605490.139639451|. | *. | 711.017457055-0.017457055|. *. | 80.650.79573323-0.14573323|. * |. | 91.51.0191181630.480881837|. |. * | 101.51.529701243-0.029701243|. *. | 110.751.077562408-0.327562408| * |. | 1210.9838870190.016112981|. *. | 130.60.946314864-0.346314864| * |. | 141.51.0142449390.485755061|. |. * | 1521.0278191050.972180895|. |. * 160.91.127572088-0.227572088|.* |. | 170.60.868503447-0.268503447|.* |. | 180.61.019018495-0.419018495| *. |. | 190.61.025899595-0.425899595| *. |. | 2010.9233683040.076631696|. |*. |

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obsActualFittedResidualResidual Plot 210.50.873499486-0.373499486| * |. | 221.150.9662006530.183799347|. | *. | 231.150.9541674930.195832507|. | *. | 240.61.061387074-0.461387074| *. |. | 2510.6904220320.309577968|. | * | 260.750.960732042-0.210732042|. * |. | 270.750.833511567-0.083511567|. *|. | 2811.020608541-0.020608541|. *. | 291.151.004223560.14577644|. | *. | 3011.060006494-0.060006494|. *|. | 310.851.355102042-0.505102042| *. |. | 322.11.2296297750.870370225|. |. * | 331.250.9752256560.274774344|. | *. | 340.750.899675696-0.149675696|. * |. | 350.870.8072678510.062732149|. |*. | Output Actual Y, Fitted Y, Residual and its plotting

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Actual Y, Fitted Y and Residual

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Sample residual

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41 (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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42 (Assumption no. 2) Joint Hypothesis : F statistics F statistics: 2.093 and Prob 0.1213 Null hypothesis Ho: B 1 =B 2 =B 3 =0 Alternative H 1 : 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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43 Assumption No. 3 Independent variable significance For X1, p-value : 0.4183 Null Hypothesis: B 1 =0 Alternative hypothesis: B 1 ≠0 Since the p-value is more than 5 percent (0.05) we can not reject null and meaning we accept null meaning B 1 =0. In other words, X1 can not influence Y in the population. For X2, p-value: 0.0305 (3.05 percent) Null Hypothesis: B 2 =0 Alternative hypothesis: B 2 ≠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.

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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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45 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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46 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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47 Assumption no. 7 Residual (u) Normality Test 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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48 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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References Essentials of Econometrics by Damodar Gujarati, McGraw Hill Publication. Basic Econometrics by Damodar Gujarati, McGraw Hill Publication. An Introduction to Econometrics by Cheng Ming Yu, Sayed Hossain and Law Siong Hook. McGraw Hill Publication.Sayed Hossain

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50 Prepared by: Sayed Hossain Lecturer for Economics Multimedia University, Malaysia Peronal website: www.sayedhossain.comwww.sayedhossain.com Email: sayed.hossain@yahoo.comsayed.hossain@yahoo.com Year: 2009 Use the information of this website on your own risk. This website shall not be responsible for any loss or expense suffered in connection with the use of this website. PLEASE COMMENT IN MY GUESTBOOK AT: www.sayedhossain.com www.sayedhossain.com

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