LECTURE 7
Introduction to Econometrics
Omitted & Irrelevant Variables
Hieu Nguyen
Fall semester, 2024
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SPECIFICATION OF A REGRESSION
►We discussed the specification of a regression equation
►
►Specification consists of choosing:
►
1.correct independent variables
2.correct functional form
3.correct form of the stochastic error term
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►A specification error occurs if any of these choices is wrong
►
►In lecture 6, we discussed the correct functional form. Now we will learn how to deal with the
other two in today’s and the following two lectures.

ON TODAY’S LECTURE
►We will talk about the problem of not adding relevant independent variables or adding irrelevant
independent variables
►
►We will learn that
•Omitting a relevant variable brings bias to our estimates of the other coefficients
•
•Including an irrelevant variable increase the variance of our estimates of the other coefficients
•
•Since in real estimation, it is often hard to judge whether or not to include a variable, we need
economic theory and statistical tools to decide
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OMITTING RELEVANT VARIABLES
e We omit a variable when we
forget to include it
do not have data for it
e This misspecification results in
not having the coefficient for this variable
biasing estimated coefficients of other variables in the  equation −→ omitted variable bias
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OMITTED VARIABLES
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OMITTED VARIABLES
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OMITTED VARIABLES
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OMITTED VARIABLES
•Example: estimating the price of chicken meat in the US
Yt        . . . per capita chicken consumption
PCt . . . price of chicken
PBt . . . price of beef
YDt . . . per capita disposable  income
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OMITTED VARIABLES
•When we omit price of beef:
, n = 44
R2 = 0.895
•Compare to the true model:
R2 = 0.986 , n = 44
•We observe positive bias in the coefficient of PC (was it  expected?)
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OMITTED VARIABLES
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OMITTED VARIABLES
•In reality, we usually do not have the true model to  compare with
Because we do not know what the true model is
Because we do not have data for some important variable
•We can often recognize the bias if we obtain some  unexpected results
•We can prevent omitting variables by relying on the theory
•If we cannot prevent omitting variables, we can at least  determine in what way this biases our
estimates
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IRRELEVANT VARIABLES
•A second type of specification error is including a variable  that does not belong to the model
•
•This    misspecification
•
•does not cause bias
•but it increases the variances of the estimated coefficients of  the included variables
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Why variance increases on estimated coefficients when we include irrelevant variable. Explanation
https://stats.stackexchange.com/questions/193229/why-does-the-parameter-variance-change-when-contro
l-variables-are-added-to-a-reg

IRRELEVANT VARIABLES
•True model:
yi = βxi + ui
(1)
(2)
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IRRELEVANT VARIABLES


SUMMARY OF THE THEORY
•Bias – efficiency  trade-off:
Omitted variable
Irrelevant variable
Bias
Yes*
No
Variance
Decreases *
Increases*
* As long as we have correlation between x and z
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FOUR IMPORTANT SPECIFICATION CRITERIA
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Does a variable belong to the equation?
1.Theory: Is the variable’s place in the equation  unambiguous and theoretically sound? Does
intuition tells  you it should be included?
2.
2.t-test: Is the variable’s estimated coefficient significant in  the expected direction?
3.
3.R2: Does the overall fit of the equation improve (enough)  when the variable is added to the
equation?
4.
4.Bias: Do other variables’ coefficients change significantly  when the variable is added to the
equation?

FOUR IMPORTANT SPECIFICATION CRITERIA
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•If all conditions hold, the variable belongs in the equation
•If none of them holds, the variable is irrelevant and can be  safely excluded
•If the criteria give contradictory answers, most importance  should be attributed to theoretical
justification
•
•
•Therefore, if theory (intuition) says that variable belongs to  the equation, we include it (even
though its coefficients  might be insignificant!).

EXAMPLE FOR SPECIFICATION CRITERIA
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EXAMPLE FOR SPECIFICATION CRITERIA


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EXAMPLE FOR SPECIFICATION CRITERIA


THE DANGER OVERSPECIFICATION
•”If you just torture the data long enough, they will confess.”
•
•If too many specifications are tried:
•
•The final result may have the desired properties only by chance
•The statistical significance of the result is overestimated because the estimations of the
previous regressions are ignored.
•
•How to solve this issue:
•
•Keep the number of try of regressions low
•Focus on theory (very important)
•Save all regression you tried
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SPECIFICATION TEST
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RESET TYPE I
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RESET TYPE II


SUMMARY
•Omitting a relevant variable brings bias to our estimates of the other coefficients
•
•Including an irrelevant variable increase the variance of our estimates of the other coefficients
•
•Since in real estimation, it is often hard to judge whether or not to include a variable, we need
economic theory and statistical tools to decide
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