Group 8: “Could you please explain more
about the conditional expected value?”
Yes! Let’s go over Conditional Expectations.
E[Y|X]
This means: suppose we know the value of the variable X.
Then, what is the expected value of variable Y?
PROPERTIES
If X and Y are independent, then E[Y|X]=E[Y] (and E[X|Y]=E[X])
If Y is simply a function of X, then, it’s simple:
𝐸 𝑓 𝑋 𝑋 =𝑓 𝑋
Eg. If Y= 3X2+2. What is E[Y|X]?
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Group 8: “Could you please explain more
about the conditional expected value?”
All the properties of Expected Value we learned also apply to
Conditional Expected Value.
So, we can use these properties (linearity, constant) to solve:
E[Y|X] for: 𝑌 = 𝐵0 + 𝐵1 𝑋 + 𝑈
Now, a practice exercise in your groups.
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The Simple
Regression Model
Recall: OLS estimators are random variables
The estimated regression coefficients are random variables
because they are calculated from a random sample
Data is random and depends on particular sample that has been drawn
The question is what the estimators will estimate on average
and how large their variability in repeated samples is
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The Simple
Regression Model
Standard assumptions for the linear regression model
Assumption SLR.1 (Linear in parameters)
In the population, the true relationship
between y and x is linear
Assumption SLR.2 (Random sampling)
The data is a random sample
drawn from the population
Each data point therefore follows the population
equation. Specifcially, 𝑦𝑖 only depends on 𝑥𝑖 and 𝑢𝑖 ,
rather than depending also the x, u, or y of other
observations in the sample.
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The Simple
Regression Model
Assumptions for the linear regression model (cont.)
Assumption SLR.3 (Sample variation in explanatory variable)
The values of the explanatory variables are not all
the same (otherwise, it is impossible to study how
variation in x corresponds with variation in y!)
Assumption SLR.4 (Zero conditional mean)
The value of the explanatory variable must
contain no information about the mean of
the unobserved factors
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The Simple
Regression Model
Theorem 2.1 (Unbiasedness of OLS)
Understanding unbiasedness
Remember, the estimated coefficients may be smaller or larger that the true
parameters, depending on our sample (which results from one random draw).
However, on average, they will be equal to the values that charac-terize the
true relationship between y and x in the population
"On average" means if sampling was repeated, i.e. if drawing the random
sample and doing the estimation was repeated many times
Most of the time, we only have one random sample, so we have no way of
knowing how near or far, above or below, our estimates are from true values.
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The Simple
Regression Model
Wages and education
Population regression function:
Hourly wage in dollars
Years of education
Fitted regression or Sample Regression Function
Intercept
𝜷𝟎
Slope
𝜷𝟏
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The Simple
Regression Model
This implies that each additional year of education increases
average wage by a constant amount, 54 cents.
Based on this model, one year of middle school has the same
effect on predicted wage as:
One year of high school
One year of college
One year of graduate school
Do you think this is true, on average?
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The Simple
Regression Model
Incorporating nonlinearities: Semi-logarithmic form
Regression of log wages on years of eduction
Natural logarithm of wage as Y, instead of wage itself
This changes the interpretation of the regression coefficient:
Recall that in the standard linear model:
Each 1-unit increase in x changes y by B1.
Now,
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The Simple
Regression Model
Incorporating nonlinearities: Semi-logarithmic form
Regression of log wages on years of eduction
Natural logarithm of wage
This changes the interpretation of the regression coefficient:
Proportional change of wage
… if years of education
are increased by one year
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The Simple
Regression Model
See p. 713-715 for more explanation
The wage increases by 8.3 % for
every additional year of education
(= return to education)
For example:
Introduces nonlinearity of a particular form:
Growth rate of wage is estimated as 8.3 %
per year of education
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The Simple
Regression Model
Incorporating nonlinearities: Log-log form
CEO salary and firm sales
Natural logarithm of CEO salary
Natural logarithm of his/her firm‘s sales
This changes the interpretation of the regression coefficient:
Percentage change of salary
… if sales increase by 1 %
Logarithmic changes are
always percentage changes
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The Simple
Regression Model
CEO salary and firm sales: fitted regression
For example:
+ 1 % sales → + 0.257 % salary
The log-log form postulates a constant elasticity model,
whereas the semi-log form assumes a semi-elasticity model
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The Simple
Regression Model
Remember, the estimates we get of
are random
variables themselves! Every random sample we draw will give
us a different “draw” of them.
Fitted regression line
(depends on sample)
Unknown population regression line
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