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Title

lmcol: OLS Multicollinearity Diagnostic Tests

Table of Contents

Syntax

Description

Options

References

*** Examples

Authors

Syntax

lmcol depvar indepvars [if] [in] , [ noconstant coll ]

Options

noconstant Exclude Constant Term from Equation

coll keep collinear variables; default is removing collinear variables.

Description

lmcol computes OLS Multicollinearity Diagnostic Tests

1- VIF: variance inflation factors for independent variables
2- Eigenvalues (Eigenval)
3- Condition Index (C_Index)
4- Condition Number (C_Number)
5- R2 between each independent variable with other independent variables (R2_xi,X)

* Correlation Matrix
* Multicollinearity Diagnostic Criteria
* Farrar-Glauber Multicollinearity Tests
Ho: No Multicollinearity - Ha: Multicollinearity
* (1) Farrar-Glauber Multicollinearity Chi2-Test
* (2) Farrar-Glauber Multicollinearity F-Test
* (3) Farrar-Glauber Multicollinearity t-Test
* Multicollinearity Ranges
* Determinant of |X'X|
* Theil R2 Multicollinearity Effect:
- Gleason-Staelin Q0
- Heo Range Q1

R2, R2 Adjusted, and F-Test, are obtained from 4 ways:

1- (Buse 1973) R2.

2- Raw Moments R2.

3- squared correlation between predicted (Yh) and observed dependent variable (Y).

4- Ratio of variance between predicted (Yh) and observed dependent variable (Y).

- Adjusted R2: R2_a=1-(1-R2)*(N-1)/(N-K-1).

- F-Test=R2/(1-R2)*(N-K-1)/(K).

- Multicollinearity Detection:

1. A high F statistic or R2 leads to reject the joint hypothesis that all of the coefficients are zero, but individual t-statistics are low.

2. High simple correlation coefficients are sufficient but not necessary for multicollinearity.

3. One can compute condition number. That is, the ratio of largest to smallest root of the matrix x'x. This may not always be useful as the standard errors of the estimates depend on the ratios of elements of characteristic vectors to the roots.

- Multicollinearity Remediation:

1. Use prior information or restrictions on the coefficients. One clever way to do this was developed by Theil and Goldberger. See tgmixed, and Theil(1971, pp 347-352).

2. Use additional data sources. This does not mean more of the same. It means pooling cross section and time series.

3. Transform the data. For example, inversion or differencing.

4. Use a principal components estimator. This involves using a weighted average of the regressors, rather than all of the regressors.

5. Another alternative regression technique is ridge regression. This involves putting extra weight on the main diagonal of X'X.

6. Dropping troublesome RHS variables. This begs the question of specification error.

References

D. Belsley (1991)
"Conditioning Diagnostics, Collinearity and Weak Data in Regression",
John Wiley & Sons, Inc., New York, USA.

D. Belsley, E. Kuh, and R. Welsch (1980)
"Regression Diagnostics: Identifying Influential Data and Sources of Collinearity",
John Wiley & Sons, Inc., New York, USA.

Damodar Gujarati (1995)
"Basic Econometrics"
3rd Edition, McGraw Hill, New York, USA.

Evagelia, Mitsaki (2011)
"Ridge Regression Analysis of Collinear Data",
http://www.stat-athens.aueb.gr/~jpan/diatrives/Mitsaki/chapter2.pdf

Farrar, D. and Glauber, R. (1976)
"Multicollinearity in Regression Analysis: the Problem Revisited",
Review of Economics and Statistics, 49; 92-107.

Greene, William (1993)
"Econometric Analysis",
2nd ed., Macmillan Publishing Company Inc., New York, USA; 616-618.

Greene, William (2007)
"Econometric Analysis",
6th ed., Upper Saddle River, NJ: Prentice-Hall; 387-388.

Griffiths, W., R. Carter Hill & George Judge (1993)
"Learning and Practicing Econometrics",
John Wiley & Sons, Inc., New York, USA; 602-606.

Judge, Georege, R. Carter Hill, William . E. Griffiths, Helmut Lutkepohl, & Tsoung-Chao Lee (1988)
"Introduction To The Theory And Practice Of Econometrics",
2nd ed., John Wiley & Sons, Inc., New York, USA.

Judge, Georege, W. E. Griffiths, R. Carter Hill, Helmut Lutkepohl, & Tsoung-Chao Lee(1985)
"The Theory and Practice of Econometrics",
2nd ed., John Wiley & Sons, Inc., New York, USA; 615.

Maddala, G. (1992)
"Introduction to Econometrics",
2nd ed., Macmillan Publishing Company, New York, USA; 358-366.

Marquardt D.W. (1970)
"Generalized Inverses, Ridge Regression, Biased Linear Estimation, and Nonlinear Estimation",
Technometrics, 12; 591-612.

Rencher, Alvin C. (1998)
"Multivariate Statistical Inference and Applications",
John Wiley & Sons, Inc., New York, USA; 21-22.

Theil, Henri (1971)
"Principles of Econometrics",
John Wiley & Sons, Inc., New York, USA.

William E. Griffiths, R. Carter Hill and George G. Judge (1993)
"Learning and Practicing Econometrics",
John Wiley & Sons, Inc., New York, USA.

Examples

clear all

sysuse lmcol.dta , clear

db lmcol

lmcol y x1 x2 x3

. clear all
. sysuse lmcol.dta , clear
. lmcol y x1 x2 x3

==============================================================================
* Ordinary Least Squares (OLS)
==============================================================================
y = x1 + x2 + x3
------------------------------------------------------------------------------
Sample Size = 17
Wald Test = 253.9319 | P-Value > Chi2(3) = 0.0000
F-Test = 84.6440 | P-Value > F(3 , 13) = 0.0000
(Buse 1973) R2 = 0.9513 | Raw Moments R2 = 0.9986
(Buse 1973) R2 Adj = 0.9401 | Raw Moments R2 Adj = 0.9983
Root MSE (Sigma) = 5.7724 | Log Likelihood Function = -51.6441
------------------------------------------------------------------------------
- R2h= 0.9513 R2h Adj= 0.9401 F-Test = 84.64 P-Value > F(3 , 13) 0.0000
- R2v= 0.9513 R2v Adj= 0.9401 F-Test = 84.64 P-Value > F(3 , 13) 0.0000
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | 1.060841 .2769969 3.83 0.002 .4624256 1.659257
x2 | -1.397391 .2321721 -6.02 0.000 -1.898969 -.895814
x3 | -.0034456 .0514889 -0.07 0.948 -.1146807 .1077894
_cons | 132.2612 36.46863 3.63 0.003 53.47554 211.0469
------------------------------------------------------------------------------

==============================================================================
*** Multicollinearity Diagnostic Tests
==============================================================================

* Correlation Matrix
(obs=17)

| x1 x2 x3
-------------+---------------------------
x1 | 1.0000
x2 | 0.1788 1.0000
x3 | -0.1832 -0.9296 1.0000

* Multicollinearity Diagnostic Criteria
+-------------------------------------------------------------------------------+
| Var | Eigenval | C_Number | C_Index | VIF | 1/VIF | R2_xi,X |
|-------+-----------+-----------+-----------+-----------+-----------+-----------|
| x1 | 1.9954 | 1.0000 | 1.0000 | 1.0353 | 0.9659 | 0.0341 |
| x2 | 0.9342 | 2.1361 | 1.4615 | 7.3632 | 0.1358 | 0.8642 |
| x3 | 0.0704 | 28.3396 | 5.3235 | 7.3753 | 0.1356 | 0.8644 |
+-------------------------------------------------------------------------------+

* Farrar-Glauber Multicollinearity Tests
Ho: No Multicollinearity - Ha: Multicollinearity
--------------------------------------------------

* (1) Farrar-Glauber Multicollinearity Chi2-Test:
Chi2 Test = 28.7675 P-Value > Chi2(3) 0.0000

* (2) Farrar-Glauber Multicollinearity F-Test:
+--------------------------------------------------------+
| Variable | F_Test | DF1 | DF2 | P_Value |
|------------+----------+----------+----------+----------|
| x1 | 0.247 | 14.000 | 3.000 | 0.971 |
| x2 | 44.543 | 14.000 | 3.000 | 0.005 |
| x3 | 44.627 | 14.000 | 3.000 | 0.005 |
+--------------------------------------------------------+

* (3) Farrar-Glauber Multicollinearity t-Test:
+-------------------------------------+
| Variable | x1 | x2 | x3 |
|----------+--------+--------+--------|
| x1 | . | | |
| x2 | 0.680 | . | |
| x3 | -0.697 | -9.435 | . |
+-------------------------------------+

* Determinant of |X'X|:
|X'X| = 0 Multicollinearity - |X'X| = 1 No Multicollinearity
Determinant of |X'X|: (0 < 0.1313 < 1)
---------------------------------------------------------------

* Theil R2 Multicollinearity Effect:
R2 = 0 No Multicollinearity - R2 = 1 Multicollinearity
- Theil R2: (0 < 0.7606 < 1)
---------------------------------------------------------------

* Multicollinearity Range:
Q = 0 No Multicollinearity - Q = 1 Multicollinearity
- Gleason-Staelin Q0: (0 < 0.5567 < 1)
1- Heo Range Q1: (0 < 0.8356 < 1)
2- Heo Range Q2: (0 < 0.8098 < 1)
3- Heo Range Q3: (0 < 0.6377 < 1)
4- Heo Range Q4: (0 < 0.5425 < 1)
5- Heo Range Q5: (0 < 0.8880 < 1)
6- Heo Range Q6: (0 < 0.5876 < 1)
------------------------------------------------------------------------------

Authors

- Emad Abd Elmessih Shehata
Professor (PhD Economics)
Agricultural Research Center - Agricultural Economics Research Institute - Egypt
Email: emadstat@hotmail.com
WebPage: http://emadstat.110mb.com/stata.htm
WebPage at IDEAS: http://ideas.repec.org/f/psh494.html
WebPage at EconPapers: http://econpapers.repec.org/RAS/psh494.htm

- Sahra Khaleel A. Mickaiel
Professor (PhD Economics)
Cairo University - Faculty of Agriculture - Department of Economics - Egypt
Email: sahra_atta@hotmail.com
WebPage: http://sahraecon.110mb.com/stata.htm
WebPage at IDEAS: http://ideas.repec.org/f/pmi520.html
WebPage at EconPapers: http://econpapers.repec.org/RAS/pmi520.htm

LMCOL Citation

Shehata, Emad Abd Elmessih & Sahra Khaleel A. Mickaiel (2012)

LMCOL: "OLS Multicollinearity Diagnostic Tests"

Online Help:

* OLS Multicollinearity Tests:
lmcol OLS Multicollinearity Diagnostic Tests
fgtest Farrar-Glauber Multicollinearity Chi2, F, t Tests
theilr2 Theil R2 Multicollinearity Effect