Polytechnic University of the Philippines Sta. Mesa, Manila ECON 3023 ECONOMETRICS GROUP PROJECT Submitted By: GROUP 8 Camacho, Irwin Dave De Ramos, Liezel Gonzales, Divina Oliver, Ralph Laurence B.S. ECONOMICS 3-1 Submitted To: Prof. Alberto Guillo Vice President for Administration, PUP
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Assumptions of Simple and Multiple Linear Regression Model
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Polytechnic University of the Philippines
Sta. Mesa, Manila
ECON 3023 ECONOMETRICS
GROUP PROJECT
Submitted By:
GROUP 8
Camacho, Irwin Dave
De Ramos, Liezel
Gonzales, Divina
Oliver, Ralph Laurence
B.S. ECONOMICS 3-1
Submitted To:
Prof. Alberto Guillo
Vice President for Administration, PUP
ASSUMPTIONS OF SIMPLE AND
MULTIPLE REGRESSION MODEL
ASSUMPTION 1: LINEAR REGRESSION MODEL
Departures from/ Violations of assumptions (both Two-Variable Linear Model and
Multiple Variable Linear Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as discussed in a research output published in a professional research
journal (At least one for each
violation)
Nonlinearity in Parameters
Scatterplot
Plot of observed versus predicted values
Plot of residuals versus predicted values -Better than the observed-versus-predicted plot for this purpose, because it eliminates the visual distraction of a sloping pattern
Chow Test
The normal equations for nonlinear regression have the unknowns (the B’s) both on the left and right-hand sides of the equations.
As a consequence, we cannot obtain explicit solutions of the unknowns in terms of the known quantities.
Wrong regressors
Changing parameters
Perform a curvilinear transformation
If there appears to be
a quadratic pattern to
the residuals,
polynomial
transformation of
degree 2 should be
tried.
Log transformation
Locally Weighted
Scatterplot Smoothing
(LOWESS)
Does High Public Debt Consistently Stifle Economic Growth? A Critique of Reinhart and Rogoff
Authors & Institutional Affiliations: -Thomas Herndon -Michael Ash -Robert Pollin -Political Economy Research Institute, -University of Massachusetts Amherst
Publication/Journal Title: -Working Paper Series
Date of Publication,
Volume No. and Year: -April 2013, Number 322
Violation:
-Nonlinear relationship between the variables
How was it detected: -Scatterplot
Remedial Measure: -LOWESS
ASSUMPTION 2: X VALUES ARE FIXED IN REPEATED SAMPLING
Departures from/ Violations of assumptions (both Two-Variable Linear Model and
Multiple Variable Linear Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as discussed in a research output published in a professional research
journal (At least one for each
violation)
Stochastic regressor(s)
Endogeneity
Graphically examining
the cumulative sum of
the recursive residuals
Durbin-Wu-Hausman Specification Test
Johansen Test
Engle-Granger two-step method
Simultaneous
equation or
simultaneity problem
The estimates of the
slope and intercept
will be biased
Use of Proxy variable
Instrumental Variable (IV) Regression
Clever Sample Selection.
Drop the polluted observations of X that covary with the disturbance
Instrumental Variables or Control Variables.
In each observation, drop the polluted component of X or control for the polluted component of X.
Full Information Methods.
Model the covariation of errors across the equations.
Foreign Direct Investment Inflows and Economic Growth in Ghana
Author: -Baba Insah
Publication/Journal Title: -International Journal of Economic Practices and Theories
Date of Publication, Volume No. and Year: -April 2013, Vol.3, No.2
Violation: -Stochastic regressors
How was it detected:
-Johansen Test
Remedial Measure: -Engle-Granger two-step methodology for error correction was employed -Error Correction was utilized.
ASSUMPTION 3: ZERO MEAN VALUE OF DISTURBANCE ui
Departures from/ Violations of assumptions (both Two-Variable Linear Model and
Multiple Variable Linear Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as discussed in a research output published in a professional research
journal (At least one for each
violation)
Nonzero mean of ui
Plot of residuals
against the
predictor(s)
if there are more than
a couple of
predictors, at least
against fitted values
Cobb-Douglas Function
The coefficient
estimate gets biased
down
Randomization of the measurement order
Randomization can
effectively convert
systematic
measurement errors
into additional
random process error.
While adding to the
random error of the
process is undesirable,
this will provide the
best possible
information from the
data about the
regression function.
Using additional
information
Redesigning the
measurement system
Consumption of
Tobacco and
Alcoholic Beverages
Among Spanish
Consumers
Authors &
Institutional
Affiliations:
-Anna-Lena Beutel
- Stefan Minner
-Department of
Business
Administration,
University of Vienna,
Austria
to eliminate the
systematic errors
Reformulating the
problem to obtain the
needed information
the other way
Publication/Journal
Title:
-International Journal
of Production
Economics
Date of Publication &
Volume No:
2011
Violation:
-non zero mean of
error
How was it detected:
-Using plot of
disturbance
Remedial Measures
-Used service level
model which performs
better for the target
fill rate than for in-
stock probabilities
ASSUMPTION 4: HOMOSCEDASTICITY OR EQUAL VARIANCE OF 𝒖𝒊
Departures from/ Violations
of assumptions (both Two-
Variable Linear Model and
Multiple Variable Linear
Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as
discussed in a research
output published in a
professional research
journal
(At least one for each
violation)
Heteroscedasticity
Scatterplot
look for any pattern:
No heteroscadasticity
Not Linear and nature is unknown
Linear increase and
presence of heteroscadasticity
Heteroscadasticity
with quadratic relationship
Quadratic relationship
Park Test
Breush-Pagan / Cook-Weisberg Test for Heteroscedasticity
White General Test
for Heteroscedasticity
Goldfeld-Quandt Test
Gives equal weight to
all observations
Standard errors are
biased
This in turn leads to
bias in test statistics
and confidence
intervals
Respectify the Model/
Transform the
Variable
Use Robust Standard
Error (also referred as
Huber/White
estimators or
sandwich estimators
of variance)
Use Weighted Least
Square
Meta-analysis of
alcohol price and
income elasticities –
with corrections for
publication bias
Authors &
Institutional
Affiliations:
-Jon P. Nelson
Publication/Journal
Title:
-Nelson Health
Economics Review
Date of Publication &
Volume No:
2013, 3:17
Violation:
-Heteroscedasticity
How was it detected:
-Scatterplot; a
“funnel-shaped” plot
was detected
Remedial Measures:
-Dividing equation (1)
by the standard error
to yield
where ti is the t-
statistic for the i-th
observation, 1/Sei is
its precision, and vi is
an error term
corrected for
heteroscedasticity.
ASSUMPTION 5: NO AUTOCORRELATION BETWEEN THE DISTURBANCES
Departures from/ Violations
of assumptions (both Two-
Variable Linear Model and
Multiple Variable Linear
Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as
discussed in a research
output published in a
professional research
journal
(At least one for each
violation)
Autocorrelated Disturbances
Time sequence plot
Residuals plots
The Runs Test
The Durbin Watson
Test
The Breusch – Godfrey
the OLS estimator is unbiased
the OLS estimator is inefficient; that is, it is not BLUE
The estimated variances and covariances of the OLS estimates are biased and inconsistent
If there is positive autocorrelation, and if the value of a right-hand side variable grows over time, then the estimate of the standard error of the coefficient estimate of this variable will be
Cochrane-Orcutt
estimator
Hildreth-Lu estimator
AUTOREG Procedure
An Examination of
Socioeconomic
Determinants of
Average Body Mass
Indices in Rwanda
Authors &
Institutional
Affiliations:
-Edward Mutandwa
-College of Forest
Resources, Mississippi
State University, USA
Publication/Journal
Title:
-Open Obesity Journal
too low and hence the t-statistic too high
Hypothesis tests are not valid.
Date of Publication &
Volume No:
-2015, 7, 1-9
Violation:
-Positive
autocorrelation
How was it detected:
-By using Durbin-
Watson test value,
obtained 0.93,
indicating positive
autocorrelation
(p<0.05)
Remedial Measures:
-By using “proc
autoreg”
ASSUMPTION 6: ZERO COVARIANCE BETWEEN 𝒖𝒊AND 𝑿𝒊
Departures from/ Violations of assumptions (both Two-Variable Linear Model and
Multiple Variable Linear Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as discussed in a research output published in a professional research
journal (At least one for each
violation)
Nonzero covariance between disturbances and regressor
Graphically examining
the cumulative sum of
the recursive residuals
Durbin-Wu-Hausman test for endogeneity
OLS estimator will be
both biased and
inconsistent
Endogeneity
Clever Sample Selection.
Drop the polluted observations of X that covary with the disturbance
Instrumental Variables or Control Variables.
In each observation, drop the polluted component of X or control for the polluted component of X.
Full Information Methods.
Model the covariation of errors across the equations.
Publication/Journal Title: -The Geneva Papers on Risk and Insurance
Date of Publication &
Volume No: October 2004, Vol.29 No.4
Violation: -Multicollinearity
How was it detected:
-An OLS regression of the inverse Mill’s ratio ƛ on the explanatory variables results in an R2 of 0.9897, suggesting almost perfect linearity
Remedial Measures:
-Multicollinearity is at least mitigated by employing a two-part model in addition to the Heckman model. The two-part model separates the selection part from the equation that explains the level of HCE. Thus, the correlation between the selection term ƛ and the age variables as a source of multicollinearity is eliminated.
ASSUMPTION 11: NORMALITY OF DISTURBANCES
Departures from/ Violations of assumptions (both Two-Variable Linear Model and
Multiple Variable Linear Model)
Graphical Approach
Statistical Approach
Effect/s of the violation in
the model
Remedial Measures
Sample of violation with
remedial measures as discussed in a research output published in a professional research
journal (At least one for each
violation)
Non-normality of disturbances
Skew -non-symmetricality -one tail longer than the other
Kurtosis -too flat or too peaked -kurtosed
Outliers -individual cases which are far from the distribution
Histograms
Boxplots
P-P Plots
Anderson-Darling test
of normality
Jarque-Bera test of
normality
Correlation Test
Obtain correlation between observed residuals and expected values under normality
Compare correlation with critical value
Reject the null
hypothesis of normal
errors if the
correlation falls below
the table value
Shapiro-Wilk Test
In finite samples,
without the normality
assumption the usual
t and F statistics may
not follow the t and F
distributions.
Skew biases the mean, in
direction of skew
Kurtosis standard deviation is
biased -and hence standard errors, and significance tests