hapter I – The Simple Linear Regression Model
Section I – Basic Assumptions and Counter-Examples of Assumptions
Definition
According to Malinvaud (1970):
“Econometrics uses mathematical tools and statistical inference to test economic theories.”
Econometrics aims to estimate relationships between economic phenomena whose
existence is asserted by economic theory. It makes it possible to identify both the direction
and the strength of the relationships between variables.
Econometrics also allows us to explain and forecast the evolution of economic variables
(the dependent variable) in order to facilitate decision-making.
Basic Framework
Yt=b+aXt+Ut t=1,2,…,n
Where:
Yt : endogenous variable or variable to be explained
Xt: exogenous variable or explanatory variable
Ut: error term, which contains omitted variables that may affect Y other than X
a and b: real and unknown parameters to be estimated using the observations yt and xt
Assumption 1: The Model Is Correctly Specified
A model is said to be misspecified when there is an incorrect definition of the functional
relationship between the explanatory variable and the dependent variable, a poor choice of
explanatory variables, the omission of relevant variables, or an inappropriate definition of
variables (index, level, variation, etc.) that does not correspond to the problem under study.
Therefore, the chosen explanatory variable must be the most appropriate one, without
omitting other relevant explanatory variables.
Example
Household consumption is explained by household income and the prices of goods.
Counter-example
, It is incorrect to explain the quantity of shirts sold by the price of cereals, as this represents a
misspecification of the model.
A misspecification may also arise from tautological relationships. Two types of tautology
can be identified: direct and indirect.
Direct tautology
Example:
CA=f(RT) Given that CA=RT=Q×P, the variability of the dependent variable is explained by
itself. In this case, the model is misspecified.
Indirect tautology
Example:
I=f(GDP) Given that: GDP=C+I+X−M
investment is included by definition in GDP. Therefore, this situation leads to a problem of
endogeneity.
Assumption 2: The variables Yi and Xi are numerical quantities
observed without measurement error
Y is a random variable due to the introduction of the error term Ut .
E(Ut)=0 for any Xt(t=1,2,…,n)
The values Xi are observed without error, and the error term has a zero mean that is identical
for all observations:
E(Ui)=0 ∀Xi(t=1,…,n)
The assumption that the expected value of the error term is zero, i.e. E(ut)=0 means that the
error term does not exhibit a systematic positive or negative tendency. The errors represent
only the variation not explained by the explanatory variables, without any directional bias:
E(ut∣Xt)=0 (exogeneity)
Example
Section I – Basic Assumptions and Counter-Examples of Assumptions
Definition
According to Malinvaud (1970):
“Econometrics uses mathematical tools and statistical inference to test economic theories.”
Econometrics aims to estimate relationships between economic phenomena whose
existence is asserted by economic theory. It makes it possible to identify both the direction
and the strength of the relationships between variables.
Econometrics also allows us to explain and forecast the evolution of economic variables
(the dependent variable) in order to facilitate decision-making.
Basic Framework
Yt=b+aXt+Ut t=1,2,…,n
Where:
Yt : endogenous variable or variable to be explained
Xt: exogenous variable or explanatory variable
Ut: error term, which contains omitted variables that may affect Y other than X
a and b: real and unknown parameters to be estimated using the observations yt and xt
Assumption 1: The Model Is Correctly Specified
A model is said to be misspecified when there is an incorrect definition of the functional
relationship between the explanatory variable and the dependent variable, a poor choice of
explanatory variables, the omission of relevant variables, or an inappropriate definition of
variables (index, level, variation, etc.) that does not correspond to the problem under study.
Therefore, the chosen explanatory variable must be the most appropriate one, without
omitting other relevant explanatory variables.
Example
Household consumption is explained by household income and the prices of goods.
Counter-example
, It is incorrect to explain the quantity of shirts sold by the price of cereals, as this represents a
misspecification of the model.
A misspecification may also arise from tautological relationships. Two types of tautology
can be identified: direct and indirect.
Direct tautology
Example:
CA=f(RT) Given that CA=RT=Q×P, the variability of the dependent variable is explained by
itself. In this case, the model is misspecified.
Indirect tautology
Example:
I=f(GDP) Given that: GDP=C+I+X−M
investment is included by definition in GDP. Therefore, this situation leads to a problem of
endogeneity.
Assumption 2: The variables Yi and Xi are numerical quantities
observed without measurement error
Y is a random variable due to the introduction of the error term Ut .
E(Ut)=0 for any Xt(t=1,2,…,n)
The values Xi are observed without error, and the error term has a zero mean that is identical
for all observations:
E(Ui)=0 ∀Xi(t=1,…,n)
The assumption that the expected value of the error term is zero, i.e. E(ut)=0 means that the
error term does not exhibit a systematic positive or negative tendency. The errors represent
only the variation not explained by the explanatory variables, without any directional bias:
E(ut∣Xt)=0 (exogeneity)
Example
