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APPLIED ECONOMICS RESEARCH CENTRE
2013-14
ECONOMETRICS 2
TERM PAPER
SAFIA ASLAM
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CONTENTS
Chapter8
Multiple Regression Analysis: The Problem of Inference
Rest r i ct ed Least Squares: Testing Linear Equality RestrictionsWa ld Test
Testing For Structural or Parameter Stability of Regression Models
O The Chow Test
Computer Application using EViews
Chapter10
Multicollinearity: What Happens If The Regressors Are Correlated?
The Nature of Multicollinearity
Sources of Multicollinearity
Practical Consequences of Multicollinearity
Detection of Multicollinearity
O High R2 but
Few Significant T-Ratios
O High Pair-wise Correlations among Regressors
O Examination of Partial Correlations
O Auxiliary Regressions
O Tolerance and Variance Inflation Factor
O Remedial Measures
O Do Nothing Or
O Follow Some Rules Of Thumb.
O A Priori Information
O Combining Cross-Sectional and Time Series Data
O Dropping a Variable(S) and Specification Bias
O Transformation of Variables
O First Difference Form
O Ratio Transformation
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Ad di ti on al or New Data
O Computer Application using EViews
Chapter12
Autocorrelation: What Happens If the Error Terms Are Correlated?
The Nature of Autocorrelation
Sources of Autocorrelation
O Inertia.
O Specification Bias: Excluded Variables Case
O Specification Bias: Incorrect Functional Form
O Cobweb Phenomenon
O Lags.
O Manipulation Of Data
O Data TransformationPractical Consequences of Autocorrelation
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2
i
Detection of Autocorrelation Gr ap hi ca l Method
The Runs Test DurbinWatson D Test
A General Test of Autocorrelation: The BreuschGodfrey (Bg) Test Remedial Measures of Autocorrelation
NeweyWest Method. Genera l i zed Least-Square (G|LS) Method.
Correcting For (Pure) Autocorrelation: The Method of Generalized LeastSquares (GLS)
O When Is Known:
O When Is Not Known:The First-Difference Method:
O BerenbluttWebb Test,
Based On DurbinWatson D Statistic:
Estimated From The Residuals:
Theil-Nagar Estimate Based On D Statistic:
Estimating : The CochraneOrcutt (CO) Iterative
Procedure: Estimating : Durbins Two-Step Method:
The Durbin h Statistic
Computer Application using EViews
Chapter11
Heteroscedasticity: What Happens If the Error Variance Is Non constant?
The Nature of Heteroscedasticity
Sources of Heteroscedasticity
Error-Learning Models
Data Collecting Techniques Improve Is Likely To Decrease.
Outliers.
Other Sources of Heteroscedasticity:
O Incorrect Data Transformation (E.G., Ratio or First Difference
Transformations)
O Incorrect Functional Form (E.G., Linear Versus LogLinear Models).
Practical Consequences of Heteroscedasticity
Detection of Heteroscedasticity
Informal Methods
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Nature Of The Problem
Graphical Method
Formal Methods
O Park Test
O Glejser Test
O Goldfeld-Quandt Test
Whites General Heteroscedasticity Test Koenker Bassett (Kb) Test
Remedial Measures
1. When i2
is knownThe method of weighted least square
2. When i2is not known
Whites heteroscedasticity-Consistent Variances and Standard Errors
Plausible Assumptions about Heteroscedasticity Pattern.
Assumption 1: The Error Variance Is Proportional To Xi2
Assumption 2: The Error Variance Is Proportional ToXi. The Square Root
Transformation
Assumption 3: The Error Variance Is Proportional To the Square Of The
Mean Value OfY.
Assumption 4: A Log Transformation
3. Computer Application using EViews
Chapters18 To 20
Simultaneous Regression Models
1. The Nature Of Simultaneous-Equation Models2. The Identification Problem3. Rules For Identification
The Order Condition Of Identifiability
The Rank Condition Of Identifiability Hausman Specification Test
The Method Of Indirect Least Squares (ILS): A Just Identified Equation
The Method Of Two-Stage Least Squares (2SLS): An Over identified
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Equation
The Method Of Three-Stage Least Squares (3SLS) Using EViews
The Granger Test
Computer Application using EViews
Chapter9
Dummy Variable Regression Models
The Nature of Dummy Variable
Caution In the Use of Dummy VariablesAnalysis of Variance (ANOVA) Model
Analysis of Covariance (ANCOVA) Models.
Computer Application using EView
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CHAPTER 8
MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF
INFERENCE
RESTRICTED LEAST SQUARES: TESTING LINEAR EQUALITY RESTRICTIONS
WALD Test: Wald test is used to test the validity of the linear restriction imposed on the
parameters. There are occasions where economic theory may suggest that the coefficients in a
regression model satisfy some linear equality restrictions.
For instance, consider the CobbDouglas production function
Y=KLe
Where Yi = output, Li = labor input, and Ki = capital input.
Written in log form, the equation becomes
LnYi = 1 + 2 ln Ki + 3 ln Li + i (8.1)
Now if there are constant returns to scale (equi proportional change in output for an
equiproportional change in the inputs), economic theory would suggest that
2 + 3 = 1
This is an example of a linear equality restriction.
TESTING LINEAR RESTRICTIONS IN EVIEWS:
Step1: First Regress the Model (8.1) equation.
Quickestimate equationwrite: log(Y) C log(K) log(L)
Dependent Variable: LOG(Y)
Method: Least Squares
Date: 08/21/13 Time: 13:41
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Sample: 1960 2012
Included observations: 53
VariableCoefficien
t Std. Error t-Statistic Prob.
C 10.94926 0.420685 26.02720 0.0000
LOG(K) 3.734688 0.138653 26.93548 0.0000
LOG(L) -1.271933 0.214719 -5.923718 0.0000
R-squared 0.996592 Mean dependent var 18.36495
Adjusted R-squared 0.996456 S.D. dependent var 0.432456
S.E. of regression 0.025746 Akaike info criterion -4.426138
Sum squared resid 0.033143 Schwarz criterion -4.314612
Log likelihood 120.2927 F-statistic 7310.653
Durbin-Watson stat 0.054050 Prob(F-statistic) 0.000000
Step2: First Regress the Model (8.1) equationRegression window
ViewCoefficient TestsWald-Coefficients Restrictions: Write: C(2) + C(3) = 1
Wald Test:
Equation: Untitled
Test Statistic Value df Probability
F-statistic 319.2578 (1, 50) 0.0000
Chi-square 319.2578 1 0.0000
Null Hypothesis Summary:
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Normalized Restriction (= 0) Value Std. Err.
-1 + C(2) + C(3) 1.462755 0.081865
Restrictions are linear in coefficients.
WALD TEST MANUALLY (USING EVIEWS):
Unrestricted model: lnYi = 0 + 2 ln Ki + 3 ln Li + i
STEP 1:
First Regress the (8.1) equation.
Quickestimate equationwrite: log(Y) C log(K) log(L)
And Obtain RSSUR
Dependent Variable: LOG(Y)
Method: Least Squares
Date: 08/21/13 Time: 13:41
Sample: 1960 2012
Included observations: 53
VariableCoefficien
t Std. Error t-Statistic Prob.
C 10.94926 0.420685 26.02720 0.0000
LOG(K) 3.734688 0.138653 26.93548 0.0000
LOG(L) -1.271933 0.214719 -5.923718 0.0000
R-squared 0.996592 Mean dependent var 18.36495
Adjusted R-squared 0.996456 S.D. dependent var 0.432456
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S.E. of regression 0.025746 Akaike info criterion -4.426138
Sum squared resid 0.033143 Schwarz criterion -4.314612
Log likelihood 120.2927 F-statistic 7310.653
Durbin-Watson stat 0.054050 Prob(F-statistic) 0.000000
Restricted model: ln (Yi /Li) = 0 + 2 ln (Ki / Li) + i
STEP 2:
First Regress the (8.2) equation.
Quickestimate equationwrite: log(Y/L) C log(K/L)
And Obtain RSSR
Dependent Variable: LOG(Y/L)
Method: Least Squares
Date: 08/21/13 Time: 13:48
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 18.42422 0.119150 154.6300 0.0000
LOG(K/L) 5.936631 0.170981 34.72092 0.0000
R-squared 0.959412 Mean dependent var 14.30042
Adjusted R-squared 0.958617 S.D. dependent var 0.340546
S.E. of regression 0.069277 Akaike info criterion -2.464402
Sum squared resid 0.244764 Schwarz criterion -2.390052
Log likelihood 67.30666 F-statistic 1205.542
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Durbin-Watson stat 0.035469 Prob(F-statistic) 0.000000
STEP 3:
Apply the F Test of RSS Version.
F = { (RSSRRSSUR) /M} / {RSSUR/ (N-K)}
And compare it with F critical Values at m and (n-k) degrees of freedom;
if | FCal | > | FCritical | then Restriction is invalid and vice versa
F = {0.244764 0.033143 / 1} / {0.033143 / (53-3)}
F = 319.69
The above example shows that F test is significant 1% level of significance.
TESTING FOR STRUCTURAL OR PARAMETER STABILITY OF REGRESSIONMODELS: THE CHOW TEST
When we use a regression model involving time series data, it may happen that there is a
structural change in the relationship between the regressand Y and the regressors. By structural
change, we mean that the values of the parameters of the model do not remain the same
through the entire time period.
USING E VIEWS:
Step 1: First Regress the Model (8.3) equation (with n = 34);
Quickestimate equationwrite: Sav C Yd
Dependent Variable: SAV
Method: Least Squares
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Date: 08/21/13 Time: 14:08
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -15.73656 1.331322 -11.82025 0.0000
YD 0.771388 0.022667 34.03118 0.0000
R-squared 0.957821 Mean dependent var 29.38251
Adjusted R-squared 0.956994 S.D. dependent var 4.246259
S.E. of regression 0.880589 Akaike info criterion 2.620554
Sum squared resid 39.54728 Schwarz criterion 2.694904
Log likelihood -67.44468 F-statistic 1158.121
Durbin-Watson stat 0.024164 Prob(F-statistic) 0.000000
Step 2: First Regress the Model (8.3) equation. Regression windowViewStability TestsChow
Breakpoint Test:
Write Breakpoint Year in the Box: 1980
Chow Breakpoint Test: 1980
F-statistic 303.2648 Probability 0.000000
Log likelihood ratio 137.4620 Probability 0.000000
The above example shows that F test is significant 1% level of significance
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MODEL
DETERMINANTS OF LIFE EXPECTANCY IN PAKISTAN:
Y=B1+B2X1+B3X2++B4X3+B5X4+B6X5
X1: POPULATION
X2: GDP
X3: UNEMPLOYEMENT
X4: URBAN POPULATION
X5: HEALTH EXPENDITURE
REGRESSION RESULTS:
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 13:57
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
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S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
Log likelihood -7.334900 F-statistic 3456.958
Durbin-Watson stat 0.196325 Prob(F-statistic) 0.000000
CHAPTER 10
MULTICOLLINEARITY: WHAT HAPPENS IF THE
REGRESSORS ARE CORRELATED?
THE NATURE OF MULTICOLLINEARITY
The term multicollinearity is due to Ragnar Frisch. Originally it meant the existence of a
perfect, or exact, linear relationship among some or all explanatory variables of aregression model.
X1 + X2 . + Xk = 0 (10.1.1)
X1 + X2 . + Xk + i = 0 (10.1.2)
Why does the classical linear regression model assume that there is no multicollinearity among
the Xs? The reasoning is this: If multicollinearity is perfect in the sense of (10.1.1), the
regression coefficients of the X variables are indeterminate and their standard errors are
infinite. If multicollinearity is less than perfect, as in (10.1.2), the regression coefficients,
although determinate, possess large standard errors (in relation to the coefficients themselves),
which means the coefficients cannot be estimated with great precision or accuracy.
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SOURCES OF MULTICOLLINEARITY
There are several sources of multicollinearity. As Montgomery and Peck note, multicollinearity
may be due to the following factors;
1. The data collection method employed, for example, sampling over a limited range of the valuestaken by the regressors in the population
2. Constraints on the model or in the population being sampled. For example, in the regressionof electricity consumption on income (X2) and house size (X3) there is a physical constraint in
the population in that families with higher incomes generally have larger homes than families
with lower incomes.
3. Model specification, for example, is adding polynomial terms to a regression model, especiallywhen the range of the X variable is small.
4. An over-determined model. This happens when the model has more explanatory variables thanthe number of observations. This could happen in medical research where there may be a smallnumber of patients about whom information is collected on a large number of variables.
An additional reason for multicollinearity, especially in time series data, may be that theregressors included in the model share a common trend, that is, they all increase or decrease
over time.
PRACTICAL CONSEQUENCES OF MULTICOLLINEARITY
In cases of near or high multicollinearity, one is likely to encounter the following consequences:
a) Although BLUE, the OLS estimators have large variances and co variances, making precise estimationdifficult.
b) Because of consequence 1, the confidence intervals tend to be much wider, leading to the acceptance of thezero null hypothesis (i.e., the true population coefficient is zero) more readily.
c) Also because of consequence 1, the t-ratio of one or more coefficients tends to be statistically insignificant.d) Although the t-ratio of one or more coefficients is statistically insignificant, R2, the overall measure of
goodness of fit, can be very high.
e) The OLS estimators and their standard errors can be sensitive to small changes in the data.
DOING REGREESION USING EVIEWS FOR MULTICOLLINEARITY
DETECTION METHODS:
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1. HIGH R2 BUT FEW SIGNIFICANT t- RATIO:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
STEP:
First regress the above equation.
Open the File containing dataQuickestimate equationwrite: Y C X1 X2 X3 X4 X5
And check the R2 and t-ratios.
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 13:57
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
Log likelihood -7.334900 F-statistic 3456.958
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Durbin-Watson stat 0.196325 Prob(F-statistic) 0.000000
The above regression results shows that R2
= 0.997288(high).X1, X2, X3, X4 are significant while X5 is insignificant.
There may be multicollinearity here.
1. HIGH PAIR WISE CORRELATION AMONG REGRESSORS:
STEP:
First open the File containing data QuickGroup StatisticsCorrelation
Write: X1 X2 X3 X4 X5
X1 X2 X3 X4 X5
X1 1.000000 0.898710 0.900580 0.988885 -0.071599
X2 0.898710 1.000000 0.713399 0.863979 -0.222654
X3 0.900580 0.713399 1.000000 0.902906 -0.159905
X4 0.988885 0.863979 0.902906 1.000000 -0.053507
X5 -0.071599 -0.222654 -0.159905 -0.053507 1.000000
The above correlation matrix shows that X1 and X4 are highly correlated.
2. EXAMINATION OF PARTIAL CORRELATION:
STEP:
First regress the equations. Open the File containing data Quickestimate
equationwrite: Y C X1, Y C X2, Y C X3, Y C X4, Y C X5 individually and compare their r2 with R2 of overall
regression.
REGRESS Y ON X1:
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Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 14:02
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 46.13325 0.681312 67.71239 0.0000
X1 1.20E-07 6.11E-09 19.57344 0.0000
R-squared 0.882521 Mean dependent var 58.49076
R2
0.997288
REGRESS Y ON X2:
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 14:03
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 54.61037 0.659574 82.79642 0.0000
X2 7.42E-11 8.68E-12 8.554942 0.0000
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R-squared 0.589329 Mean dependent var 58.49076
R2 0.997288
REGRESS Y ON X3:
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 14:03
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 47.82694 0.889198 53.78662 0.0000
X3 3.221181 0.245909 13.09909 0.0000
R-squared 0.770875 Mean dependent var 58.49076
R2
0.997288
REGRESS Y ON X4:
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 14:03
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
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C 22.00695 1.082998 20.32039 0.0000
X4 1.241685 0.036487 34.03118 0.0000
R-squared 0.957821 Mean dependent var 58.49076
R2
0.997288
REGRESS Y ON X5:
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 14:04
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 58.94157 5.968745 9.875036 0.0000
X5 -0.146728 1.927387 -0.076128 0.9396
R-squared 0.000114 Mean dependent var 58.49076
R2
0.997288
The above regression shows data had no multicollinearity
3. AUXILLARY REGRESSIONS:
STEP:
First regress the equations. Open the File containing data Quickestimate equationwrite: X2 C X3 X4,
write: X3 C X2 X4, write: X4 C X2 X3 individually and compare the r2 with R2.
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REGRESS X1 ON X2 X3 X4 X5:
Dependent Variable: X1
Method: Least Squares
Date: 08/19/13 Time: 14:06
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -1.22E+08 11339153 -10.78366 0.0000
X2 0.000186 2.70E-05 6.900419 0.0000
X3 4481648. 1162156. 3.856322 0.0003
X4 6366302. 583130.7 10.91745 0.0000
X5 4542256. 1906829. 2.382100 0.0212
R-squared 0.989170 Mean dependent var 1.03E+08
R2
0.997288
REGRESS X2 ON X1 X3 X4 X5:
Dependent Variable: X2
Method: Least Squares
Date: 08/19/13 Time: 14:07
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
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C 1.84E+11 7.50E+10 2.451252 0.0179
X1 2677.158 387.9703 6.900419 0.0000
X3 -2.16E+10 3.97E+09 -5.426820 0.0000
X4 -8.45E+09 3.94E+09 -2.141991 0.0373
X5 -2.88E+10 6.42E+09 -4.489499 0.0000
R-squared 0.910163 Mean dependent var 5.23E+10
R2
0.997288
REGRESS X3 ON X1 X2 X4 X5:
Dependent Variable: X3
Method: Least Squares
Date: 08/19/13 Time: 14:07
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 1.445271 2.266943 0.637542 0.5268
X1 5.28E-08 1.37E-08 3.856322 0.0003
X2 -1.76E-11 3.25E-12 -5.426820 0.0000
X4 -0.011486 0.118091 -0.097265 0.9229
X5 -0.757352 0.189557 -3.995372 0.0002
R-squared 0.894134 Mean dependent var 3.310530
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R2
0.997288
REGRESS X4 ON X1 X2 X3 X5:
Dependent Variable: X4
Method: Least Squares
Date: 08/19/13 Time: 14:08
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 18.57003 0.745923 24.89536 0.0000
X1 1.12E-07 1.03E-08 10.91745 0.0000
X2 -1.03E-11 4.82E-12 -2.141991 0.0373
X3 -0.017156 0.176382 -0.097265 0.9229
X5 -0.051514 0.267322 -0.192705 0.8480
R-squared 0.981090 Mean dependent var 29.38251
R2
0.997288
REGRESS X5 ON X1 X2 X3 X4:
Dependent Variable: X5
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Method: Least Squares
Date: 08/19/13 Time: 14:08
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 2.736547 1.448764 1.888884 0.0650
X1 2.33E-08 9.77E-09 2.382100 0.0212
X2 -1.03E-11 2.29E-12 -4.489499 0.0000
X3 -0.329525 0.082477 -3.995372 0.0002
X4 -0.015007 0.077873 -0.192705 0.8480
R-squared 0.351577 Mean dependent var 3.072447
R2
0.997288
The above regression shows data had no multicollinearity.
REMEDIAL MEASURES:
A. DROPPING A VARIABLE(S) AND SPECIFICATION BIAS :
STEP:
First Regress the (10.1) equation without X1 (by assuming it causes Multicollinearity).
Open the File containing dataQuickestimate equationwrite: Y C X2 X4
And check the R2 and t-ratios
Dependent Variable: Y
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Method: Least Squares
Date: 08/19/13 Time: 14:12
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 9.097915 1.536073 5.922841 0.0000
X2 -3.84E-11 3.65E-12 -10.50413 0.0000
X3 -0.751619 0.157433 -4.774216 0.0000
X4 1.907816 0.078995 24.15124 0.0000
X5 -0.706191 0.258311 -2.733880 0.0087
R-squared 0.987744 Mean dependent var 58.49076
Adjusted R-squared 0.986722 S.D. dependent var 5.387357
S.E. of regression 0.620775 Akaike info criterion 1.973891
Sum squared resid 18.49733 Schwarz criterion 2.159768
Log likelihood -47.30811 F-statistic 967.0998
Durbin-Watson stat 0.340734 Prob(F-statistic) 0.000000
After dropping X1, regression shows that multicollinearity has been removed.
B. TRANSFORMATION OF VARIABLE:
I m transforming model into lag form.
Dependent Variable: D(Y)
Method: Least Squares
Date: 08/19/13 Time: 14:18
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Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 1.080539 0.091881 11.76017 0.0000
D(X1) -2.20E-07 1.15E-08 -19.15005 0.0000
D(X2) -4.13E-12 1.19E-12 -3.461294 0.0012
D(X3) 0.002617 0.020385 0.128386 0.8984
D(X4) -0.482934 0.278996 -1.730969 0.0902
D(X5) 0.009456 0.021666 0.436466 0.6645
R-squared 0.922727 Mean dependent var 0.362255
Adjusted R-squared 0.914328 S.D. dependent var 0.179894
S.E. of regression 0.052655 Akaike info criterion -2.941961
Sum squared resid 0.127535 Schwarz criterion -2.716817
Log likelihood 82.49098 F-statistic 109.8591
Durbin-Watson stat 1.059673 Prob(F-statistic) 0.000000
C. ADDITION OR NEW DATA.
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CHAPTER
12
AUTOCORRELATION: WHAT HAPPENS IFTHE ERROR
TERMS ARE CORRELATED?
THE NATURE OF AUTOCORRELATION
The term autocorrelation may be defined as correlation between members of series ofobservations ordered in time [as in time series data] or space [as in cross-sectional data].In theregression context, the classical linear regression model assumes that such autocorrelation
does not exist in the disturbances Ui.
Tintner defines autocorrelation as lag correlation of a given series with itself, lagged by anumber of time units, whereas he reserves the term serial correlation to lag correlationbetween two different series.
SOURCES OF AUTOCORRELATION
1. Inertia. A salient feature of most economic time series is inertia, or sluggishness. Asis well known, time series such as GNP, price indexes, production, employment, and
unemployment exhibit (business) cycles.
2. Specification Bias: Excluded Variables Case3. Specification Bias: Incorrect Functional Form4. Cobweb Phenomenon
5. Lags.6. Manipulation of Data: Another source of manipulation is interpolation or
extrapolation of data.
7. Data Transformation
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PRACTICAL CONSEQUENCES OF AUTOCORRELATION
In the presence of autocorrelation the usual OLS estimators, although linear, unbiased, and
asymptotically (i.e., in large samples) normally distributed, are no longer minimum varianceamong all linear unbiased estimators. In short, they are not efficient relative to other linear and
unbiased estimators. Put differently, they may not be BLUE. As a result, the usual, t, F, and may not be valid.
DOING REGREESION USING EVIEWS FOR AUTOCORRELATION
DETECTION METHOD:
1. GRAPHICAL METHOD:
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
X C Y
Step 2: obtain the Residuals
From the estimated equation result window Procmake residuals series(name) ok.
Step 3: Plot these residuals and see the pattern for Autocorrelation.
From step2QuickGraphScatter plot
Write: r1(-1) r1
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It is clear from the Graph that the residuals Rt and Rt-1 are serially correlated and positively correlated
2. RUNS TEST:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1 X2 X3 X4 X5
Step 2: obtain the Residuals
From the estimated equation result window Procmake residuals series (name) ok.
Step 3: check the Runs by looking at changing sign of residuals, then find the interval by calculating mean variance
the check whether Runs obtained lies in interval or not for clarity regarding Autocorrelation
(- - - - - - -, + + + + + + + + + + , - - - - - - - - - - - - - -,+, - - -, + + + + + + + + + +, - - - - -, + + )
R=8
N1=23
N2=29
MEAN(R) = 2N1N2 /N +1
= 1334/52+1
= 25.653
VARIANCE () = 2N1N2 (2N1N2N) / N2
(N1)
= 1334*1282/2704*51
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= 1710188/137904
= 12.4012
If the null hypothesis of randomness is sustainable, following the properties of the normal
distribution, we should expect that
PROB [E(R)-1.96 R E(R) +1.96]
PROB [25.653-1.96(12.4012) 8 25.653+1.96(12.4012)]
PROB [1.3466 8 49.959]
Obviously, this interval includes 8. So we can accept the null hypothesis that residuals do not containautocorrelation.
3. DURBIN WATSON d TEST:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X2X3 X4 X5
In the regression results there is Durbin-Watson dstatistics
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 13:57
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
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C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
Log likelihood -7.334900 F-statistic 3456.958
Durbin-Watson stat 0.196325 Prob(F-statistic) 0.000000
4. A GENERAL TEST OF AUTOCORRELATION:THE BREUSCHGODFREY (BG) TEST
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equationequation
Specification window
Write:
Y C X2X3 X4 X5
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Step 2: from step 1, in estimated equation windowViewResidual Tests
Serial Correlation LM Test
Breusch-Godfrey Serial Correlation LM Test:
F-statistic 28.96111 Probability 0.000000
Obs*R-squared 42.88204 Probability 0.000000
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Date: 08/19/13 Time: 14:44
Pre sample missing value lagged residuals set to zero.
Variable Coefficient Std. Error t-Statistic Prob.
C -0.511900 0.658397 -0.777494 0.4413
X1 -5.69E-09 4.60E-09 -1.238405 0.2226
X2 1.71E-12 1.19E-12 1.433777 0.1592
X3 0.035050 0.040369 0.868241 0.3903
X4 0.027397 0.034699 0.789556 0.4343
X5 0.030173 0.062743 0.480893 0.6331
RESID(-1) 0.772369 0.154615 4.995436 0.0000
RESID(-2) 0.109866 0.201820 0.544377 0.5891
RESID(-3) 0.067084 0.201378 0.333126 0.7407
RESID(-4) -0.015117 0.211764 -0.071384 0.9434
RESID(-5) 0.029055 0.218522 0.132961 0.8949
RESID(-6) -0.316267 0.171752 -1.841416 0.0728
R-squared 0.809095 Mean dependent var -2.50E-15
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Adjusted R-squared 0.757877 S.D. dependent var 0.280545
S.E. of regression 0.138045 Akaike info criterion -0.926361
Sum squared resid 0.781315 Schwarz criterion -0.480257
Log likelihood 36.54857 F-statistic 15.79697
Durbin-Watson stat 1.489135 Prob(F-statistic) 0.000000
REMEDIAL MEASURES:
1. NEWEYWEST CONSISTENT STANDARD ERRORS METHOD:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step: Regress the Model
Open the File containing dataQuickestimate equationequation
Specification window
Write:
Y C X2X3
In equation specification windowoptionsestimation option windowtick
Heteroscedasticity-consistent covariance NeweyWestclick ok
Dependent Variable: Y
Method: Least Squares
Date: 08/21/13 Time: 11:22
Sample: 1960 2012
Included observations: 53
Newey-West HAC Standard Errors & Covariance (lag truncation=3)
Variable Coefficient Std. Error t-Statistic Prob.
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C -5.519635 2.390986 -2.308518 0.0254
X1 -1.20E-07 1.43E-08 -8.372859 0.0000
X2 -1.61E-11 2.39E-12 -6.749603 0.0000
X3 -0.215865 0.081656 -2.643598 0.0111
X4 2.668870 0.133068 20.05638 0.0000
X5 -0.163192 0.136790 -1.193012 0.2389
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
Log likelihood -7.334900 F-statistic 3456.958
Durbin-Watson stat 0.196325 Prob(F-statistic) 0.000000
Only Std. Error of coefficients are corrected for autocorrelation
2. GENERALIZED LEAST SQUARES (GLS) METHOD:
When is known: We use that rho to estimate the generalized difference equation through OLSthen these results will be reliable and dont have autocorrelation problem.
When is Unknown:
based on DurbinWatson d Statistic:
1 d/2
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
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Write:
Y C X1 X2X3 X4 X5
And take Durbin Watson statistics and calculate rho (= p).
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 13:57
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
Log likelihood -7.334900 F-statistic 3456.958
Durbin-Watson stat 0.196325 Prob(F-statistic) 0.000000
1 0.196325/2
0.9018
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Step 2: Regress the equation
Open the File containing dataQuickestimate equation
Write: Y- 0.9018*d(Y) 1-0.9018 X1-0.9018*d(X1) X5-0.9018*(X5)
Dependent Variable: Y-0.9018*D(Y)
Method: Least Squares
Date: 08/21/13 Time: 11:30
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
1-0.9018 -54.11289 13.94341 -3.880893 0.0003
X1-0.9018*D(X1) -1.17E-07 9.85E-09 -11.87465 0.0000
X2-0.9018*D(X2) -1.71E-11 2.88E-12 -5.928562 0.0000
X3-0.9018*D(X3) -0.235757 0.089920 -2.621862 0.0118
X4-0.9018*D(X4) 2.659116 0.071015 37.44442 0.0000
X5-0.9018*D(X5) -0.185270 0.137451 -1.347904 0.1843
R-squared 0.997261 Mean dependent var 58.39192
Adjusted R-squared 0.996964 S.D. dependent var 5.332351
S.E. of regression 0.293827 Akaike info criterion 0.496515
Sum squared resid 3.971373 Schwarz criterion 0.721658
Log likelihood -6.909378 Durbin-Watson stat 0.187082
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estimated from the Residuals:
t= t-1 + t
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1 X2X3 X4 X5
Obtain the Residuals, from the estimated equation result window Procmake
Residuals series (name) ok
Step 2: Regress the residual on it lag to get rho to transform model in GDE
Write:
R1R1 (-1)
Dependent Variable: R1
Method: Least Squares
Date: 08/21/13 Time: 11:32
Sample (adjusted): 1962 2012
Included observations: 51 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
D(R1) 0.356889 0.311125 1.147091 0.2568
R-squared 0.024186 Mean dependent var 0.010390
Adjusted R-squared 0.024186 S.D. dependent var 0.271481
S.E. of regression 0.268178 Akaike info criterion 0.225078
Sum squared resid 3.595959 Schwarz criterion 0.262957
Log likelihood -4.739484 Durbin-Watson stat 0.133401
Step 3: Regress the equation
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Open the File containing dataQuickestimate equation
Write: Y- 0.3568*d(Y) 1-0.3568 X1-0.3568*d(X1) X5-0.3568*(X5)
Dependent Variable: Y-.3568*D(Y)
Method: Least Squares
Date: 08/21/13 Time: 11:35
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
1-.3568 -7.471330 2.116689 -3.529725 0.0010
X1-.3568*D(X1) -1.14E-07 9.48E-09 -12.06953 0.0000
X2-.3568*D(X2) -1.71E-11 2.64E-12 -6.461472 0.0000
X3-.3568*D(X3) -0.247995 0.090371 -2.744203 0.0086
X4-.3568*D(X4) 2.635871 0.070181 37.55831 0.0000
X5-.3568*D(X5) -0.198586 0.146473 -1.355786 0.1818
R-squared 0.997367 Mean dependent var 58.58935
Adjusted R-squared 0.997081 S.D. dependent var 5.237607
S.E. of regression 0.282978 Akaike info criterion 0.421274
Sum squared resid 3.683530 Schwarz criterion 0.646418
Log likelihood -4.953129 Durbin-Watson stat 0.146685
from Durbin two steps method:
Step 1: Regress the equation given suggested by Durbin for his two step methods
Open the File containing dataQuickestimate equation
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write: Y C X1 d(X1) X2 d(X2) X3 d(X3) X4 d(X4) X5 d(X5) d(Y)
The coefficient of Yt-1 is estimated rho (=P)
Dependent Variable: Y
Method: Least Squares
Date: 08/21/13 Time: 11:39
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -17.98627 3.065724 -5.866892 0.0000
X1 -1.47E-07 1.22E-08 -12.00310 0.0000
D(X1) -3.78E-07 2.41E-07 -1.570401 0.1242
X2 -2.07E-11 2.75E-12 -7.517914 0.0000
D(X2) 4.08E-12 8.47E-12 0.481797 0.6326
X3 -0.213495 0.091318 -2.337915 0.0245
D(X3) 0.176647 0.106567 1.657618 0.1052
X4 3.177549 0.142231 22.34072 0.0000
D(X4) -0.389698 1.727353 -0.225604 0.8227
X5 -0.037026 0.130887 -0.282886 0.7787
D(X5) 0.074601 0.113507 0.657232 0.5148
D(Y) 3.333366 1.153424 2.889976 0.0062
R-squared 0.998433 Mean dependent var 58.71860
Adjusted R-squared 0.998002 S.D. dependent var 5.175647
S.E. of regression 0.231364 Akaike info criterion 0.109524
Sum squared resid 2.141171 Schwarz criterion 0.559812
Log likelihood 9.152363 F-statistic 2316.511
Durbin-Watson stat 0.453925 Prob(F-statistic) 0.000000
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= 3.3333
Step 2: Regress the equation (12.4)
Open the File containing dataQuickestimate equation
Write: Y- 3.3333*d(Y) 1-3.333 X1-3.3333*d(X1) X5-3.3333*(X5)
Dependent Variable: Y-3.333*D(Y)
Method: Least Squares
Date: 08/21/13 Time: 11:43
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
1-3.333 3.158922 0.687286 4.596224 0.0000
X1-3.333*D(X1) -1.40E-07 1.09E-08 -12.87266 0.0000
X2-3.333*D(X2) -9.24E-12 3.84E-12 -2.405610 0.0202
X3-3.333*D(X3) -0.060601 0.055325 -1.095376 0.2791
X4-3.333*D(X4) 2.759547 0.086447 31.92176 0.0000
X5-3.333*D(X5) -0.032140 0.061097 -0.526048 0.6014
R-squared 0.994743 Mean dependent var 57.51121
Adjusted R-squared 0.994172 S.D. dependent var 5.756341
S.E. of regression 0.439444 Akaike info criterion 1.301554
Sum squared resid 8.883108 Schwarz criterion 1.526698
Log likelihood -27.84041 Durbin-Watson stat 0.369519
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BerenbluttWebb test g-statistics for estimated =1:
To test the hypothesis that 1. The test statistic they use is called the g-statistic, which
is defined as follows:
g = et2 / t
2
Step 1: Regress the equation
Open the File containing dataQuickestimate equation
Write: Y C X1 X2 X3 X4 X5
Obtain residuals, t2
Step 2: Regress the equation First difference
Open the File containing dataQuickestimate equation
Write: D(Y) C D(X10 D(X2) D (X3) D(X4) D(X5)
Obtain residuals, et2
Dependent Variable: Y
Method: Least Squares
Date: 08/21/13 Time: 16:27
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
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Dependent Variable: D(Y)
Method: Least Squares
Date: 08/21/13 Time: 16:29
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 1.080539 0.091881 11.76017 0.0000
D(X1) -2.20E-07 1.15E-08 -19.15005 0.0000
D(X2) -4.13E-12 1.19E-12 -3.461294 0.0012
D(X3) 0.002617 0.020385 0.128386 0.8984
D(X4) -0.482934 0.278996 -1.730969 0.0902
D(X5) 0.009456 0.021666 0.436466 0.6645
R-squared 0.922727 Mean dependent var 0.362255
Adjusted R-squared 0.914328 S.D. dependent var 0.179894
S.E. of regression 0.052655 Akaike info criterion -2.941961
Sum squared resid 0.127535 Schwarz criterion -2.716817
g = 0.0311
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
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We find that dL=1.364 and dU =1.590 (5 percent level)
It is clear that g statistics lies in the 0dL range we can take first difference by assuming =1.
CHAPTER 11
HETEROSCEDASTICITY: WHAT HAPPENS IF THE ERROR
VARIANCE IS NONCONSTANT?
THE NATURE OF HETEROSCEDASTICITY
A critical assumption of the classical linear regression model is that the disturbances i have all
the same variance, 2 if this assumption is not satisfied, there is heteroscedasticity. Hence,there is heteroscedasticity.
E i = i2
Notice the subscript ofi2,
which reminds us that the conditional variances
of i(= conditional variances ofYi) are no longer constant
SOURCES OF HETEROSCEDASTICITY
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1. Following the error-learning models, as people learn, their errors of behavior
become smaller over time. In this case, i2is expected to decrease.
2. As data collecting techniques improve, i2
is likely to decrease.
3. Heteroscedasticity can also arise as a result of the presence ofoutliers. An outlying
observation, or outlier, is an observation that is much different (either very small or
very large) in relation to the observations in the sample.
4. Another source of heteroscedasticity arises from violating Assumption 9 of CLRM,
namely, that the regression model is correctly specified.
5. Another source of heteroscedasticity is skewness in the distribution of one or more
regressors included in the model.
6. Other sources of heteroscedasticity: As David Hendry notes, heteroscedasticity can
also arise because of (1) incorrect data transformation (e.g., ratio or first difference
transformations) and (2) incorrect functional form (e.g., linear versus loglinear models).
PRACTICAL CONSEQUENCES OF HETEROSCEDASTICITY
In the presence of heteroscedasticity the usual OLS estimators, although linear, unbiased, and
asymptotically (i.e., in large samples) normally distributed, are no longer minimum variance
among all linear unbiased estimators. In short, they are not efficient relative to other linear and
unbiased estimators. Put differently, they may not be BLUE. As a result, the usual, t, F, and may not be valid.
DOING REGREESION USING EVIEWS FOR HETEROSCEDASTICITY
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DETECTION OF HETEROSCEDASTICITY:
1. GRAPHICAL METHOD:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write: Y C X1 X2 X3 X4 X5
Step 2: obtain the Residuals
From the estimated equation result window Procmake residuals series(name) ok.
Step 3: Plot these residuals squares and X1, X2, X3, X4, X5 & individually and see the pattern for
Hetroscedasticity. Y
From step2QuickGraphScatter plot
Write:
X1 R1^2, X2 R1^2, X3 R1^2, X4 R1^2, X5 R1^2
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It is clear from the graph that there is such pattern seen that causes hetroscedasticity.
2. FORMAL METHODS:
a. PARK TEST:
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MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1X2
And obtain the Residuals from the estimated equation result window Procmake
Residuals series (name) ok.
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 13:57
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
R-squared 0.997288 Mean dependent var 58.49076
Adjusted R-squared 0.997000 S.D. dependent var 5.387357
S.E. of regression 0.295091 Akaike info criterion 0.503204
Sum squared resid 4.092694 Schwarz criterion 0.726256
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Log likelihood -7.334900 F-statistic 3456.958
Durbin-Watson stat 0.196325 Prob(F-statistic) 0.000000
Lni =1++ 2lnXi+i
Step 2: Regress the Equation (11.1) suggested by Park.
Open the File containing dataQuickestimate equation
Write:
log(r1^2) c log(x1)
log(r1^2) c log(x2)
log(r1^2) c log(x3)
log(r1^2) c log(x4)
log(r1^2) c log(x5)
And check the significance of coefficient of explanatory variable
Dependent Variable: LOG(R1^2)
Method: Least Squares
Date: 08/21/13 Time: 12:07
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 39.65786 10.25252 3.868109 0.0003
LOG(X1) -2.361010 0.557693 -4.233531 0.0001
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Dependent Variable: LOG(R1^2)
Method: Least Squares
Date: 08/21/13 Time: 12:07
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 17.17790 5.018112 3.423181 0.0012
LOG(X2) -0.866555 0.207702 -4.172101 0.0001
Dependent Variable: LOG(R1^2)
Method: Least Squares
Date: 08/21/13 Time: 12:07
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -1.586677 0.639137 -2.482532 0.0164
LOG(X3) -1.935249 0.532505 -3.634238 0.0007
Dependent Variable: LOG(R1^2)
Method: Least Squares
Date: 08/21/13 Time: 12:08
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
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Variable Coefficient Std. Error t-Statistic Prob.
C 19.37869 5.598606 3.461342 0.0011
LOG(X4) -6.848214 1.657294 -4.132166 0.0001
Dependent Variable: LOG(R1^2)
Method: Least Squares
Date: 08/21/13 Time: 12:08
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -5.142800 2.376507 -2.164016 0.0353
LOG(X5) 1.264550 2.120873 0.596240 0.5537
If2s turns out to be statistically significant at 1%, it would suggest that heteroscedasticity is present
in the data in the each case.
b. GLESJER TEST:
MODEL: Yi = 1 + 2X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
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Write:
Y C X1X2
And obtain the Residuals from the estimated equation result window Procmake
Residuals series (name) ok
= 1+2+Xi+i
Step 2: Regress the Equation (11.1a) suggested by Glesjer.
Open the File containing dataQuickestimate equation
Write:
abs(r1) c x1
abs(r1) c x2
abs(r1) c x3
abs(r1) c x4
abs(r1) c x5
And check the significance of coefficient of explanatory variable
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:16
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.462368 0.051704 8.942531 0.0000
X1 -2.31E-09 4.60E-10 -5.020498 0.0000
Dependent Variable: ABS(R1)
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Method: Least Squares
Date: 08/21/13 Time: 12:16
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.290397 0.029296 9.912631 0.0000
X2 -1.30E-12 3.82E-13 -3.408431 0.0013
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:16
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.423221 0.050627 8.359624 0.0000
X3 -0.060295 0.013883 -4.343235 0.0001
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:17
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.914931 0.137492 6.654407 0.0000
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X4 -0.023500 0.004612 -5.094806 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:17
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.123370 0.185112 0.666459 0.5082
X5 0.031866 0.059835 0.532554 0.5967
If2s turns out to be statistically significant at 1%, it would suggest that heteroscedasticity is presentin the data in each case.
= 1+2+Xi+ i
Step 2: Regress the Equation (11.1b) suggested by Glesjer.
Open the File containing dataQuickestimate equation
Write:
abs(r1) c x1^0.5
abs(r1) c x2^0.5
abs(r1) c x3^0.5
abs(r1) c x4^0.5
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abs(r1) c x5^0.5
And check the significance of coefficient of explanatory variable
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:18
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.697759 0.093902 7.430750 0.0000
X1^.5 -4.76E-05 9.19E-06 -5.181553 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:18
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.383647 0.041604 9.221475 0.0000
X2^.5 -8.01E-07 1.80E-07 -4.443441 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:18
Sample (adjusted): 1961 2012
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Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.633219 0.090155 7.023649 0.0000
X3^.5 -0.230573 0.049249 -4.681793 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:18
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 1.599442 0.269902 5.926016 0.0000
X4^.5 -0.254295 0.049674 -5.119270 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:19
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.004676 0.366416 0.012761 0.9899
X5^.5 0.123829 0.209160 0.592029 0.5565
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If2s turns out to be statistically significant at 1%, it would suggest that heteroscedasticity is present
in the data in each case.
= 1+2+1/Xi+ i
Step 2: Regress the Equation (11.1c) suggested by Glesjer.
Open the File containing dataQuickestimate equation
Write:
abs(r1) c 1/x1
abs(r1) c 1/x2
abs(r1) c 1/x3
abs(r1) c 1/x4
abs(r1) c 1/x5
And check the significance of coefficient of explanatory variable
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:20
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -0.010629 0.048537 -0.218995 0.8275
1/X1 20352326 3925383. 5.184800 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
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Date: 08/21/13 Time: 12:20
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.135861 0.026436 5.139347 0.0000
1/X2 1.42E+09 2.98E+08 4.761296 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:20
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.009848 0.044574 0.220930 0.8260
1/X3 0.577436 0.110386 5.231046 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:20
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -0.451516 0.132770 -3.400733 0.0013
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1/X4 19.46190 3.801832 5.119086 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:21
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.357066 0.180871 1.974150 0.0539
1/X5 -0.410269 0.541472 -0.757693 0.4522
If2s turns out to be statistically significant at 1%, it would suggest that heteroscedasticity is presentin the data in each case.
= 1+2+1/Xi+ i
Step 2: Regress the Equation (11.1d) suggested by Glesjer.
Open the File containing dataQuickestimate equation
Write:
abs(r1) c 1/x1^0.5
abs(r1) c 1/x2^0.5
abs(r1) c 1/x3^0.5
abs(r1) c 1/x4^0.5
abs(r1) c 1/x5^0.5
And check the significance of coefficient of explanatory variable
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Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:22
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -0.245656 0.090575 -2.712173 0.0091
1/X1^0.5 4471.569 848.7235 5.268582 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:22
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.047878 0.037971 1.260883 0.2132
1/X2^0.5 25715.18 4894.805 5.253565 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:22
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Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -0.199286 0.083843 -2.376886 0.0213
1/X3^0.5 0.713478 0.138597 5.147859 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/21/13 Time: 12:22
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -1.135887 0.265139 -4.284128 0.0001
1/X4^0.5 7.318108 1.426138 5.131418 0.0000
Dependent Variable: ABS(R1)
Method: Least Squares
Date: 08/22/13 Time: 13:12
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.475984 0.362154 1.314313 0.1947
1/X5^0.5 -0.443667 0.629237 -0.705088 0.4840
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If2s turns out to be statistically significant at 1%, it would suggest that heteroscedasticity is present
in the data in each case
c. WHITES GENERAL HETEROSCEDASTICITY TEST:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1X2X3 X4 X5
Step 2: applying Whites General Hetroscedasticity Test
In estimated equation in step1viewResiduals Tests
White Heteroskedasticity Test (cross terms)
or
White Heteroskedasticity Test (no cross term)
White Heteroskedasticity Test: no cross term
F-statistic 5.982381 Probability 0.000014
Obs*R-squared 31.13871 Probability 0.000557
As n*R2 (=31.13871) > the critical chi-square (=3.94) value at the 5% level of significance, the
conclusion is that there is heteroscedasticity
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White Heteroskedasticity Test: cross term
F-statistic 3.782162 Probability 0.000405
Obs*R-squared 37.24425 Probability 0.010937
As n*R2 (=37.24425) > the critical chi-square (=10.85) value at the 5% level of significance, theconclusion is that there is heteroscedasticity
d. KOENKERBASSETT (KB) TEST:
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1X2X3 X4 X5
I2= 1+2I
2+i
Step 2: Regress the (11.6a) equation
Open the File containing dataQuickestimate equation
Write: in
R1^2 C (Y_CAP)^2
Where I are the estimated values from the model (11.6). The null hypothesis is that 2= 0. If this is not rejected,then one could conclude that there is no heteroscedasticity.
The null hypothesis can be tested by the usual ttest or theFtest,
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Dependent Variable: R1^2
Method: Least Squares
Date: 08/21/13 Time: 12:44
Sample (adjusted): 1961 2012
Included observations: 52 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.414590 0.072768 5.697453 0.0000
(Y_CAP)^2 -9.74E-05 2.07E-05 -4.714133 0.0000
R-squared 0.307700 Mean dependent var 0.076373
Adjusted R-squared 0.293854 S.D. dependent var 0.104297
S.E. of regression 0.087643 Akaike info criterion -1.993383
Sum squared resid 0.384066 Schwarz criterion -1.918336
Log likelihood 53.82797 F-statistic 22.22305
Durbin-Watson stat 0.407608 Prob(F-statistic) 0.000020
REMEDIAL MEASURES:
WHEN i2
IS KNOWN: THE METHOD OF WEIGHTED LEAST SQUARES:
Ifi2
is known, the most straightforward method of correcting heteroscedasticity is by
Means of weighted least squares, for the estimators thus obtained are BLUEYi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Yi/ = 1 (1/)+ 2 X1i /+ 3 X2i/+ 4 X3i/ +/+6X5i/+i WEIGHTED LEAST SQUARE
Where iare the standard deviations
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WHEN i2
IS NOT KNOWN: Whites Heteroscedasticity-Consistent Variances and Standard Errors.
MODEL: Yi = 1 + 2 X1i + 3 X2i + 4 X3i +5X4i + 6X5i +i
Step 1: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1X2X3 X4 X5
Dependent Variable: Y
Method: Least Squares
Date: 08/19/13 Time: 13:57
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.350874 -4.085972 0.0002
X1 -1.20E-07 9.29E-09 -12.86161 0.0000
X2 -1.61E-11 2.45E-12 -6.580235 0.0000
X3 -0.215865 0.085649 -2.520339 0.0152
X4 2.668870 0.070082 38.08228 0.0000
X5 -0.163192 0.129846 -1.256813 0.2150
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Step 2: Regress the Model
Open the File containing dataQuickestimate equation
Write:
Y C X1X2X3 X4 X5
In equation specification windowoptionsestimation option windowtick
Heteroscedasticity-consistent covariance Whitesclick ok
Dependent Variable: Y
Method: Least Squares
Date: 08/22/13 Time: 14:14
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C -5.519635 1.416856 -3.895694 0.0003
X1 -1.20E-07 8.29E-09 -14.42540 0.0000
X2 -1.61E-11 1.64E-12 -9.811624 0.0000
X3 -0.215865 0.056197 -3.841198 0.0004
X4 2.668870 0.078266 34.09986 0.0000
X5 -0.163192 0.106943 -1.525967 0.1337
As we can see that in case of heteroscedasticity the OLS standard errors of slope coefficient
of X1 X2 X3 are over estimated and X4 X5 are under estimated. And intercept was under
estimated. After applying Whites Hetroscedasticity-Consistent Variances and Standard Errorsremedial procedure they were now corrected for hetroscedasticity.
Plausible Assumptions about Heteroscedasticity Pattern :
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Assumption 1: The error variance is proportional to XI2:
E(i2)=2Xi
2
We may transform by dividing the original model through byXi
Transforming using X1:
Yi = 1(1/X1) + 2 + 3 X2i/X1 + 4 X3i/X1 + +6X5i/X1 +i TRANSFORMED MODEL
Step 1: Regress the equation (11.7a)
Open the File containing dataQuickestimate equationequation specification window
Write: Y/X1 C 1/X1 X2/X1 X3/X1 X4/X1 X5/X1
Dependent Variable: Y/X1
Method: Least Squares
Date: 08/22/13 Time: 14:31
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C -1.35E-07 1.30E-08 -10.36097 0.0000
1/X1 -8.939349 1.641135 -5.447055 0.0000
X2/X1 -1.65E-11 3.01E-12 -5.480641 0.0000
X3/X1 -0.360089 0.097640 -3.687939 0.0006
X4/X1 2.859217 0.091651 31.19696 0.0000
X5/X1 -0.206358 0.123891 -1.665644 0.1024
R-squared 0.999641 Mean dependent var 6.51E-07
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Adjusted R-squared 0.999603 S.D. dependent var 2.18E-07
S.E. of regression 4.34E-09 Akaike info criterion -35.56657
Sum squared resid 8.85E-16 Schwarz criterion -35.34352
Log likelihood 948.5141 F-statistic 26205.57
Durbin-Watson stat 0.188567 Prob(F-statistic) 0.000000
Similarly for X2 X3 X4 and X5:
Dependent Variable: Y/X2
Method: Least Squares
Date: 08/22/13 Time: 14:33
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C -1.60E-11 1.20E-11 -1.337315 0.1876
X1/X2 -1.80E-07 2.77E-08 -6.486804 0.0000
1/X2 -15.75433 2.283040 -6.900594 0.0000
X3/X2 -0.400503 0.187234 -2.139050 0.0377
X4/X2 3.241575 0.140842 23.01573 0.0000
X5/X2 -0.141550 0.103679 -1.365271 0.1787
R-squared 0.999966 Mean dependent var 3.39E-09
Adjusted R-squared 0.999962 S.D. dependent var 3.44E-09
S.E. of regression 2.12E-11 Akaike info criterion -46.21324
Sum squared resid 2.11E-20 Schwarz criterion -45.99019
Log likelihood 1230.651 F-statistic 274036.8
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Durbin-Watson stat 0.158014 Prob(F-statistic) 0.000000
Dependent Variable: Y/X3
Method: Least Squares
Date: 08/22/13 Time: 14:34
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C -0.308845 0.117205 -2.635089 0.0114
X1/X3 -1.49E-07 1.46E-08 -10.18503 0.0000
X2/X3 -1.40E-11 3.08E-12 -4.564873 0.0000
1/X3 -10.39333 1.716490 -6.054987 0.0000
X4/X3 2.948408 0.095104 31.00182 0.0000
X5/X3 -0.215436 0.120129 -1.793374 0.0793
R-squared 0.999735 Mean dependent var 20.99458
Adjusted R-squared 0.999707 S.D. dependent var 8.062351
S.E. of regression 0.137977 Akaike info criterion -1.017182
Sum squared resid 0.894776 Schwarz criterion -0.794130
Log likelihood 32.95532 F-statistic 35499.75
Durbin-Watson stat 0.192584 Prob(F-statistic) 0.000000
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Dependent Variable: Y/X4
Method: Least Squares
Date: 08/22/13 Time: 14:35
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C 2.734611 0.080073 34.15149 0.0000
X1/X4 -1.25E-07 9.49E-09 -13.13093 0.0000
X2/X4 -1.64E-11 1.99E-12 -8.252539 0.0000
X3/X4 -0.255739 0.068636 -3.726028 0.0005
1/X4 -6.702676 1.430325 -4.686120 0.0000
X5/X4 -0.189060 0.115224 -1.640812 0.1075
R-squared 0.990734 Mean dependent var 2.006037
Adjusted R-squared 0.989748 S.D. dependent var 0.111929
S.E. of regression 0.011333 Akaike info criterion -6.015897
Sum squared resid 0.006037 Schwarz criterion -5.792845
Log likelihood 165.4213 F-statistic 1005.009
Durbin-Watson stat 0.185140 Prob(F-statistic) 0.000000
Dependent Variable: Y/X5
Method: Least Squares
Date: 08/22/13 Time: 14:35
Sample: 1960 2012
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Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C -0.107515 0.114543 -0.938644 0.3527
X1/X5 -1.19E-07 7.82E-09 -15.21492 0.0000
X2/X5 -1.56E-11 1.60E-12 -9.744259 0.0000
X3/X5 -0.213828 0.057516 -3.717711 0.0005
X4/X5 2.657934 0.074921 35.47637 0.0000
1/X5 -5.458239 1.338748 -4.077122 0.0002
R-squared 0.999139 Mean dependent var 19.34730
Adjusted R-squared 0.999047 S.D. dependent var 3.065848
S.E. of regression 0.094632 Akaike info criterion -1.771378
Sum squared resid 0.420893 Schwarz criterion -1.548326
Log likelihood 52.94151 F-statistic 10906.54
Durbin-Watson stat 0.202398 Prob(F-statistic) 0.000000
Notice that in the transformed regression the intercept term 2 is the slope coefficient in the
original equation and the slope coefficient 1 is the intercept term in the original model.
Therefore, to get back to the original model we shall have to multiply the estimated
transformed model byXi.
Assumption 2: The error variance is proportional toXi. The square root transformation:
E (i2) =2Xi
Transforming using X1:
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Yi /X1= 1 (1/X1) + 2X1 + 3 X2i/X1 + 4 X3i/X1 + +6X5i/X1 +i TRANSFORMED
Step 1: Regress the equation (11.8a)
Open the File containing dataQuickestimate equationequation specification window
Write:
Y/x1^0.5 c 1/x1^0.5 x2/x1^0.5 x3/x1^0.5 x4/x1^0.5 x5/x1^0.5
Dependent Variable: Y/X1^0.5
Method: Least Squares
Date: 08/22/13 Time: 14:42
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C -0.003423 0.000293 -11.69658 0.0000
1/X1^0.5 -3.483622 1.212010 -2.874252 0.0061
X2/X1^0.5 -2.95E-11 1.99E-12 -14.88087 0.0000
X3/X1^0.5 -0.356717 0.062820 -5.678385 0.0000
X4/X1^0.5 3.378221 0.131543 25.68143 0.0000
X5/X1^0.5 -0.166805 0.123406 -1.351681 0.1829
R-squared 0.998082 Mean dependent var 0.006034
Adjusted R-squared 0.997878 S.D. dependent var 0.000748
S.E. of regression 3.45E-05 Akaike info criterion -17.60661
Sum squared resid 5.58E-08 Schwarz criterion -17.38355
Log likelihood 472.5751 F-statistic 4892.212
Durbin-Watson stat 0.287508 Prob(F-statistic) 0.000000
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Similarly do for X2 X3 X4 and X5.
Note an important feature of the transformed model: It has no intercept term.
Therefore, one will have to use the regression-through-the-origin model to
Estimate 1 and 2. Having run (11.8), one can get back to the original model
Simply by multiplying transformed model by Xi.
Assumption 3: The error variance is proportional to the square of the mean value ofY.
E (i2)=2E(Yi)
2
Equation postulates that the variance of i2 is proportional to the square of the expected
Value ofY.
Therefore, if we transform the original equation as follows:
Yi /E (Yi) = 1 (1/ E (Yi)) + 2X1/E (Yi) +3 X2i/ E (Yi) + 4 X3i/ E (Yi) + +6X5i/ E (Yi)+i
The transformation (11.9) is, however, in operational because E(Yi) depends on 1 and 2,
which are unknown. Of course, we know, which is an estimator ofE(Yi)
Transforming using: i
Yi /i = 1 (1/ i) + 2X1i/ i+ 3 X2i/ i + 4 X3i/i + +6X5i/ i + i
Step: Regress the equation (11.8a)
Open the File containing dataQuickestimate equationequation specifications window
Write: Y/Y_CAP 1/Y_CAP X1/Y_CAP X2/Y_CAP X3/Y_CAP X4/Y_CAP X5/Y_CAP
Dependent Variable: Y/Y_CAP
Method: Least Squares
Date: 08/22/13 Time: 14:55
Sample: 1960 2012
Included observations: 53
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White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
1/Y_CAP -6.465594 1.412492 -4.577437 0.0000
X1/Y_CAP -1.24E-07 9.12E-09 -13.58257 0.0000
X2/Y_CAP -1.62E-11 1.85E-12 -8.803458 0.0000
X3/Y_CAP -0.237086 0.063967 -3.706393 0.0006
X4/Y_CAP 2.721034 0.078917 34.47970 0.0000
X5/Y_CAP -0.182424 0.112227 -1.625497 0.1107
R-squared 0.013722 Mean dependent var 0.999988
Adjusted R-squared -0.091201 S.D. dependent var 0.005202
S.E. of regression 0.005434 Akaike info criterion -7.486055
Sum squared resid 0.001388 Schwarz criterion -7.263003
Log likelihood 204.3804 Durbin-Watson stat 0.186775
Assumption 4: A log transformation such as:
lnYi = 1 + 2 lnXi+i
very often reduces heteroscedasticity.
Log Transforming:
lnYi = 1 + 2 lnX1i + 3 lnX2i + 4 lnX3i + 5lnX4i+6lnX5i+i
Step: Regress the equation (11.8c)
Open the File containing dataQuickestimate equationequation specifications window
Write: log(y) c log(x1) log(x2) log(x3) log(x4) log(x5)
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Dependent Variable: LOG(Y)
Method: Least Squares
Date: 08/22/13 Time: 15:00
Sample: 1960 2012
Included observations: 53
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C 5.341832 0.443418 12.04694 0.0000
LOG(X1) -0.397072 0.050292 -7.895290 0.0000
LOG(X2) -0.040926 0.010557 -3.876748 0.0003
LOG(X3) 0.020821 0.008782 2.370920 0.0219
LOG(X4) 2.058282 0.171616 11.99351 0.0000
LOG(X5) 0.037678 0.012541 3.004484 0.0043
R-squared 0.988388 Mean dependent var 4.064502
Adjusted R-squared 0.987152 S.D. dependent var 0.095517
S.E. of regression 0.010827 Akaike info criterion -6.107355
Sum squared resid 0.005509 Schwarz criterion -5.884303
Log likelihood 167.8449 F-statistic 800.0900
Durbin-Watson stat 0.393330 Prob(F-statistic) 0.000000
This result arises because log transformation compresses the scales in which the variables
are measured, thereby reducing a tenfold difference between two values to a twofold
difference.
To conclude our discussion of the remedial measures, we reemphasize that all the
transformations discussed previously are ad hoc; we are essentially speculating
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about the nature ofi2.
Which of the transformations discussed previously will
work will depend on the nature of the problem and the severity of
heteroscedasticity.
CHAPTERS 18 TO 20
SIMULTANEOUS REGRESSION MODELS
THE NATURE OF SIMULTANEOUS-EQUATION MODELS
In contrast to single-equation models, in simultaneous-equation models more than onedependent, or endogenous, variable is involved, necessitating as many equations as the
number of endogenous variables. A unique feature of simultaneous-equation models is that the
endogenous variable (i.e., regressand) in one equation may appear as an explanatory variable
(i.e., regressor) in another equation of the system.
Y1i = 10 + 12Y2i + 11 X1i + u1i
Y2i = 20 + 21Y1i + 21 X1i + u2i
Where Y1 and Y2 are mutually dependent, or endogenous, variables and X1 is an exogenous
variable and where u1 andu2 are the stochastic disturbance terms, the variables Y1 and Y2 areboth stochastic.
Therefore, unless it can be shown that the stochastic explanatory variable Y2 in (18.1) is
distributed independently of u1 and the stochastic explanatory variable Y1in (18.2) is
distributed independently of u2, application of the classical OLS to these equations individually
will lead to inconsistent estimates.
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As a consequence, such an endogenous explanatory variable becomes stochastic and is usually
correlated with the disturbance term of the equation in which it appears as an explanatory
variable.
In this situation the classical OLS method may not be applied because the estimators thus
obtained are not consistent, that is, they do not converge to their true population values no
matter how large the sample size. Since simultaneous-equation models are used frequently,
especially in econometric models, alternative estimating techniques have been developed by
various authors.
DOING REGREESION USING EVIEWS SIMULTENENOUS MODELS
MODEL: Yi = 1 + 2 X1i + 3 X2i +4X4i + 5X5i +1i
X1i= 1+2Yi+3X2i+4X3i+2i
Where Yi and X1iare mutually dependent, or endogenous, variables X2i, X3i, X4i and X5i
are an exogenous variable and where 1i and2i are the stochastic disturbance terms, the
variables YiandX1iare both stochastic. Therefore, unless it can be shown that the
stochastic explanatory variable Yi is distributed independently of1i
and the stochastic explanatory variableX1i is distributed independently
of2i, application of the classical OLS to these equations individually will lead to
inconsistent estimates.
THE IDENTIFICATION PROBLEM
RULES FOR IDENTIFICATION
THE ORDER CONDITION OF IDENTIFIABILITY
A necessary (but not sufficient) condition of identification, known as the order condition,
May be stated in two different but equivalent ways as follows (the necessary as well as
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Sufficient condition of identification will be presented shortly):
MK G - 1
M= numbers of variables in the simultaneous model
iK= numbers of variables in the specific equation
G = numbers of endogenous variables or equations
FOR EQUATION 1:
65 2 1
1 1
FOR EQUATION 2:
64 2-1
2 1
Hence equation 1 is just identified and equation 2 is over identified. As we know equation 1 is
just identified, is estimated using ILS and 2SLS and equation 2 is over identified, is estimated
using 2SLS method.
THE RANK CONDITION FOR IDENTIFIABILITY:
Here we can not apply the rank condition there are only two simultaneous equations only
HAUSMAN SPECIFICATION TEST
It is also called test of endogeneity or simultaneity problem
EVI EWS:
Step 1: Regress the equation 1. Run Yi onX1 X2 X3 X4 X5
Open the File containing dataQuickestimate equation
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Write:
Y C X1 X2 X4 X5
And obtain the Residuals from the estimated equation result window Procmake
Residuals series (name) ok
Dependent Variable: Y
Method: Least Squares
Date: 09/04/13 Time: 12:58
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C 9.097915 1.536073 5.922841 0.0000
X2 -3.84E-11 3.65E-12 -10.50413 0.0000
X3 -0.751619 0.157433 -4.774216 0.0000
X4 1.907816 0.078995 24.15124 0.0000
X5 -0.706191 0.258311 -2.733880 0.0087
R-squared 0.987744 Mean dependent var 58.49076
Adjusted R-squared 0.986722 S.D. dependent var 5.387357
S.E. of regression 0.620775 Akaike info criterion 1.973891
Sum squared resid 18.49733 Schwarz criterion 2.159768
Log likelihood -47.30811 F-statistic 967.0998
Durbin-Watson stat 0.340734 Prob(F-statistic) 0.000000
Step 2: Regress the equation 2 using residual of step 1 as explanatory variable
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Open the File containing dataQuickestimate equation
Write:
X1 C Y X2 X3 R1
Now check the significance of this residual coefficient if it exceeds critical then we can say both the equation are
simultaneous
Dependent Variable: X1
Method: Least Squares
Date: 09/04/13 Time: 13:01
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -1.53E+08 10008321 -15.31155 0.0000
Y 3794853. 203298.8 18.66638 0.0000
X2 0.000291 1.31E-05 22.24633 0.0000
X3 5855713. 671827.1 8.716101 0.0000
RESID01 -10309118 797193.2 -12.93177 0.0000
R-squared 0.994332 Mean dependent var 1.03E+08
Adjusted R-squared 0.993859 S.D. dependent var 42305993
S.E. of regression 3315248. Akaike info criterion 32.95555
Sum squared resid 5.28E+14 Schwarz criterion 33.14143
Log likelihood -868.3221 F-statistic 2104.972
Durbin-Watson stat 0.634807 Prob(F-statistic) 0.000000
As residual is significant 1% level of significance; w can conclude that both equation are
simultaneous
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Similarly for equation 2 Repeat step 1 now.
Step 2: Regress the equation 1 using residual of step 1 as explanatory variable
Open the File containing dataQuickestimate equation
Write:
y c x1 x2 x4 x5 r1
Now check the significance of this residual coefficient if it exceeds critical then we can say both the equation are
simultaneous
Dependent Variable: Y
Method: Least Squares
Date: 09/04/13 Time: 13:05
Sample: 1960 2012
Included observations: 53
Variable Coefficient Std. Error t-Statistic Prob.
C -0.811293 1.899331 -0.427146 0.6712
X1 -9.78E-08 1.21E-08 -8.112440 0.0000
X2 -1.70E-11 2.24E-12 -7.562896 0.0000
X4 2.397581 0.101523 23.61622 0.0000
X5 -0.049479 0.107277 -0.461224 0.6468
RESID01 0.362655 0.101890 3.559284 0.0009
R-squared 0.997575 Mean dependent var 58.49076
Adjusted R-squared 0.997317 S.D. dependent var 5.387357
S.E. of regression 0.279035 Akaike info criterion 0.391313
Sum squared resid 3.659450