LECTURE 5
Introduction to Econometrics
Hypothesis testing
October 20, 2017
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ON TODAY’S LECTURE
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ON TODAY’S LECTURE
We are going to discuss how hypotheses about coefficients
can be tested in regression models
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ON TODAY’S LECTURE
We are going to discuss how hypotheses about coefficients
can be tested in regression models
We will explain what significance of coefficients means
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ON TODAY’S LECTURE
We are going to discuss how hypotheses about coefficients
can be tested in regression models
We will explain what significance of coefficients means
We will learn how to read regression output
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ON TODAY’S LECTURE
We are going to discuss how hypotheses about coefficients
can be tested in regression models
We will explain what significance of coefficients means
We will learn how to read regression output
Readings for this week:
Studenmund, Chapter 5.1 - 5.4
Wooldridge, Chapter 4
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QUESTIONS WE ASK
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QUESTIONS WE ASK
What conclusions can we draw from our regression?
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QUESTIONS WE ASK
What conclusions can we draw from our regression?
What can we learn about the real world from a sample?
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QUESTIONS WE ASK
What conclusions can we draw from our regression?
What can we learn about the real world from a sample?
Is it likely that our results could have been obtained by
chance?
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QUESTIONS WE ASK
What conclusions can we draw from our regression?
What can we learn about the real world from a sample?
Is it likely that our results could have been obtained by
chance?
If our theory is correct, what are the odds that this
particular outcome would have been observed?
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HYPOTHESIS TESTING
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HYPOTHESIS TESTING
We cannot prove that a given hypothesis is “correct” using
hypothesis testing
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HYPOTHESIS TESTING
We cannot prove that a given hypothesis is “correct” using
hypothesis testing
All that can be done is to state that a particular sample
conforms to a particular hypothesis
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HYPOTHESIS TESTING
We cannot prove that a given hypothesis is “correct” using
hypothesis testing
All that can be done is to state that a particular sample
conforms to a particular hypothesis
We can often reject a given hypothesis with a certain
degree of confidence
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HYPOTHESIS TESTING
We cannot prove that a given hypothesis is “correct” using
hypothesis testing
All that can be done is to state that a particular sample
conforms to a particular hypothesis
We can often reject a given hypothesis with a certain
degree of confidence
In such a case, we conclude that it is very unlikely the
sample result would have been observed if the
hypothesized theory were correct
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NULL AND ALTERNATIVE HYPOTHESES
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NULL AND ALTERNATIVE HYPOTHESES
First step in hypothesis testing: state explicitly the
hypothesis to be tested
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NULL AND ALTERNATIVE HYPOTHESES
First step in hypothesis testing: state explicitly the
hypothesis to be tested
Null hypothesis: statement of the range of values of the
regression coefficient that would be expected to occur if
the researcher’s theory were not correct
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NULL AND ALTERNATIVE HYPOTHESES
First step in hypothesis testing: state explicitly the
hypothesis to be tested
Null hypothesis: statement of the range of values of the
regression coefficient that would be expected to occur if
the researcher’s theory were not correct
Alternative hypothesis: specification of the range of values of
the coefficient that would be expected to occur if the
researcher’s theory were correct
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NULL AND ALTERNATIVE HYPOTHESES
First step in hypothesis testing: state explicitly the
hypothesis to be tested
Null hypothesis: statement of the range of values of the
regression coefficient that would be expected to occur if
the researcher’s theory were not correct
Alternative hypothesis: specification of the range of values of
the coefficient that would be expected to occur if the
researcher’s theory were correct
In other words: we define the null hypothesis as the result
we do not expect
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NULL AND ALTERNATIVE HYPOTHESES
Notation:
H0 . . . null hypothesis
HA . . . alternative hypothesis
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NULL AND ALTERNATIVE HYPOTHESES
Notation:
H0 . . . null hypothesis
HA . . . alternative hypothesis
Examples:
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NULL AND ALTERNATIVE HYPOTHESES
Notation:
H0 . . . null hypothesis
HA . . . alternative hypothesis
Examples:
One-sided test
H0 : β ≤ 0
HA : β > 0
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NULL AND ALTERNATIVE HYPOTHESES
Notation:
H0 . . . null hypothesis
HA . . . alternative hypothesis
Examples:
One-sided test
H0 : β ≤ 0
HA : β > 0
Two-sided test
H0 : β = 0
HA : β = 0
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TYPE I AND TYPE II ERRORS
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
There are two types of errors we can make
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
There are two types of errors we can make
Type I : We reject a true null hypothesis
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
There are two types of errors we can make
Type I : We reject a true null hypothesis
Type II : We do not reject a false null hypothesis
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
There are two types of errors we can make
Type I : We reject a true null hypothesis
Type II : We do not reject a false null hypothesis
Example:
H0 : β = 0
HA : β = 0
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
There are two types of errors we can make
Type I : We reject a true null hypothesis
Type II : We do not reject a false null hypothesis
Example:
H0 : β = 0
HA : β = 0
Type I error: it holds that β = 0, we conclude that β = 0
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TYPE I AND TYPE II ERRORS
It would be unrealistic to think that conclusions drawn
from regression analysis will always be right
There are two types of errors we can make
Type I : We reject a true null hypothesis
Type II : We do not reject a false null hypothesis
Example:
H0 : β = 0
HA : β = 0
Type I error: it holds that β = 0, we conclude that β = 0
Type II error: it holds that β = 0, we conclude that β = 0
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TYPE I AND TYPE II ERRORS
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TYPE I AND TYPE II ERRORS
Example:
H0 : The defendant is innocent
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TYPE I AND TYPE II ERRORS
Example:
H0 : The defendant is innocent
HA : The defendant is guilty
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TYPE I AND TYPE II ERRORS
Example:
H0 : The defendant is innocent
HA : The defendant is guilty
Type I error = Sending an innocent person to jail
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TYPE I AND TYPE II ERRORS
Example:
H0 : The defendant is innocent
HA : The defendant is guilty
Type I error = Sending an innocent person to jail
Type II error = Freeing a guilty person
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TYPE I AND TYPE II ERRORS
Example:
H0 : The defendant is innocent
HA : The defendant is guilty
Type I error = Sending an innocent person to jail
Type II error = Freeing a guilty person
Obviously, lowering the probability of Type I error means
increasing the probability of Type II error
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TYPE I AND TYPE II ERRORS
Example:
H0 : The defendant is innocent
HA : The defendant is guilty
Type I error = Sending an innocent person to jail
Type II error = Freeing a guilty person
Obviously, lowering the probability of Type I error means
increasing the probability of Type II error
In hypothesis testing, we focus on Type I error and we
ensure that its probability is not unreasonably large
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DECISION RULE
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DECISION RULE
1. Calculate sample statistic
2. Compare sample statistic with the critical value (from the
statistical tables)
The critical value divides the range of possible values of
the statistic into two regions: acceptance region and rejection
region
If the sample statistic falls into the rejection region, we
reject H0
If the sample statistic falls into the acceptance region, we do
not reject H0
The idea is that if the value of the coefficient does not
support H0, the sample statistic should fall into the
rejection region
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ONE-SIDED REJECTION REGION
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ONE-SIDED REJECTION REGION
H0 : β ≤ 0 vs HA : β > 0
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ONE-SIDED REJECTION REGION
H0 : β ≤ 0 vs HA : β > 0
Distribution of β:
Acceptance region Rejection region
Probability of Type I error
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TWO-SIDED REJECTION REGION
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TWO-SIDED REJECTION REGION
H0 : β = 0 vs HA : β = 0
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TWO-SIDED REJECTION REGION
H0 : β = 0 vs HA : β = 0
Distribution of β:
Acceptance region Rejection regionRejection region
Probability of Type I error
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THE t-TEST
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THE t-TEST
We use t-test to test hypothesis about individual regression
slope coefficients
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THE t-TEST
We use t-test to test hypothesis about individual regression
slope coefficients
Test of more than one coefficient at a time (joint
hypotheses) are typically done with the F-test (see next
lecture)
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THE t-TEST
We use t-test to test hypothesis about individual regression
slope coefficients
Test of more than one coefficient at a time (joint
hypotheses) are typically done with the F-test (see next
lecture)
The t-test is appropriate to use when the stochastic error
term is normally distributed and when the variance of that
distribution is unknown
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THE t-TEST
We use t-test to test hypothesis about individual regression
slope coefficients
Test of more than one coefficient at a time (joint
hypotheses) are typically done with the F-test (see next
lecture)
The t-test is appropriate to use when the stochastic error
term is normally distributed and when the variance of that
distribution is unknown
These are the usual assumptions in regression analyses
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THE t-TEST
We use t-test to test hypothesis about individual regression
slope coefficients
Test of more than one coefficient at a time (joint
hypotheses) are typically done with the F-test (see next
lecture)
The t-test is appropriate to use when the stochastic error
term is normally distributed and when the variance of that
distribution is unknown
These are the usual assumptions in regression analyses
The t-test accounts for differences in the units of
measurement of the variables
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THE t-TEST
Consider the model
y = β0 + β1x1 + β2x2 + ε
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THE t-TEST
Consider the model
y = β0 + β1x1 + β2x2 + ε
Suppose we want to test (b is some constant)
H0 : β1 = b vs HA : β1 = b
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THE t-TEST
Consider the model
y = β0 + β1x1 + β2x2 + ε
Suppose we want to test (b is some constant)
H0 : β1 = b vs HA : β1 = b
We know that
β1 ∼ N β1, Var(β1) ⇒
β1 − β1
Var(β1)
∼ N(0, 1)
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THE t-TEST
Problem: Var(β1) depends on the variance of error term σ2,
which is unobservable and therefore unknown
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THE t-TEST
Problem: Var(β1) depends on the variance of error term σ2,
which is unobservable and therefore unknown
It has to be estimated as
ˆσ2
:= s2
=
e e
n − k
,
k is the number of regression coefficients (here k = 3)
e is the vector of residuals
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THE t-TEST
Problem: Var(β1) depends on the variance of error term σ2,
which is unobservable and therefore unknown
It has to be estimated as
ˆσ2
:= s2
=
e e
n − k
,
k is the number of regression coefficients (here k = 3)
e is the vector of residuals
We denote standard error of β1 (sample counterpart of
standard deviation σβ1
) as s.e. β1
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THE t-TEST
We define the t-statistic
t :=
β1 − β1
s.e. β1
∼ tn−k
where β1 is the estimated coefficient and β1 is the value of
the coefficient that is stated in our hypothesis
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THE t-TEST
We define the t-statistic
t :=
β1 − β1
s.e. β1
∼ tn−k
where β1 is the estimated coefficient and β1 is the value of
the coefficient that is stated in our hypothesis
This statistic depends only on the estimate β1, our
hypothesis about β1, and it has a known distribution
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TWO-SIDED t-TEST
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TWO-SIDED t-TEST
Our hypothesis is
H0 : β1 = b vs HA : β1 = b
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TWO-SIDED t-TEST
Our hypothesis is
H0 : β1 = b vs HA : β1 = b
Hence, our t-statistic is
t =
β1 − b
s.e. β1
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TWO-SIDED t-TEST
Our hypothesis is
H0 : β1 = b vs HA : β1 = b
Hence, our t-statistic is
t =
β1 − b
s.e. β1
where β1 is the estimated regression coefficient of β1
b is the constant from our null hypothesis
s.e. β1 is the estimated standard error of β1
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TWO-SIDED t-TEST
How to determine the critical value for this test statistic?
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TWO-SIDED t-TEST
How to determine the critical value for this test statistic?
The critical value is the value that distinguishes the
acceptance region from the rejection region
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TWO-SIDED t-TEST
How to determine the critical value for this test statistic?
The critical value is the value that distinguishes the
acceptance region from the rejection region
1. We set the probability of Type I error
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TWO-SIDED t-TEST
How to determine the critical value for this test statistic?
The critical value is the value that distinguishes the
acceptance region from the rejection region
1. We set the probability of Type I error
Let’s set the Type I. error to 5%
We say the p-value of the test is 5% or that we have a test at
95% confidence level
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TWO-SIDED t-TEST
How to determine the critical value for this test statistic?
The critical value is the value that distinguishes the
acceptance region from the rejection region
1. We set the probability of Type I error
Let’s set the Type I. error to 5%
We say the p-value of the test is 5% or that we have a test at
95% confidence level
2. We find the critical values in the statistical tables: tn−k,0.975
and tn−k,0.025
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TWO-SIDED t-TEST
How to determine the critical value for this test statistic?
The critical value is the value that distinguishes the
acceptance region from the rejection region
1. We set the probability of Type I error
Let’s set the Type I. error to 5%
We say the p-value of the test is 5% or that we have a test at
95% confidence level
2. We find the critical values in the statistical tables: tn−k,0.975
and tn−k,0.025
The critical value depends on the chosen level of Type I
error and n − k
Note that tn−k,0.975 = −tn−k,0.025
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TWO-SIDED t-TEST
Rejection region :
p-value = 5%
Distribution tn-k
2.5 % 2.5 %
tn-k,0.975tn-k,0.025
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TWO-SIDED t-TEST
Rejection region :
p-value = 5%
Distribution tn-k
2.5 % 2.5 %
tn-k,0.975tn-k,0.025
We reject H0 if |t| > tn−k,0.975
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ONE-SIDED t-TEST
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ONE-SIDED t-TEST
Suppose our hypothesis is
H0 : β1 ≤ b vs HA : β1 > b
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ONE-SIDED t-TEST
Suppose our hypothesis is
H0 : β1 ≤ b vs HA : β1 > b
Our t-statistic still is
t =
β1 − b
s.e. β1
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ONE-SIDED t-TEST
Suppose our hypothesis is
H0 : β1 ≤ b vs HA : β1 > b
Our t-statistic still is
t =
β1 − b
s.e. β1
We set the probability of Type I error to 5%
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ONE-SIDED t-TEST
Suppose our hypothesis is
H0 : β1 ≤ b vs HA : β1 > b
Our t-statistic still is
t =
β1 − b
s.e. β1
We set the probability of Type I error to 5%
We compare our statistic to the critical value tn−k,0.95
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ONE-SIDED t-TEST
Rejection region :
p-value = 5%
Distribution tn-k
5%
tn-k,0.95
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ONE-SIDED t-TEST
Rejection region :
p-value = 5%
Distribution tn-k
5%
tn-k,0.95
We reject H0 if t > tn−k,0.95
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SIGNIFICANCE OF THE COEFFICIENT
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SIGNIFICANCE OF THE COEFFICIENT
The most common test performed in regression is
H0 : β = 0 vs HA : β = 0
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SIGNIFICANCE OF THE COEFFICIENT
The most common test performed in regression is
H0 : β = 0 vs HA : β = 0
with the t-statistic
t =
β
s.e. β
∼ tn−k
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SIGNIFICANCE OF THE COEFFICIENT
The most common test performed in regression is
H0 : β = 0 vs HA : β = 0
with the t-statistic
t =
β
s.e. β
∼ tn−k
If we reject H0 : β = 0, we say the coefficient β is
significant
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SIGNIFICANCE OF THE COEFFICIENT
The most common test performed in regression is
H0 : β = 0 vs HA : β = 0
with the t-statistic
t =
β
s.e. β
∼ tn−k
If we reject H0 : β = 0, we say the coefficient β is
significant
This t-statistic is displayed in most regression outputs
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THE p-VALUE
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THE p-VALUE
Classical approach to hypothesis testing: first choose the
significance level, then test the hypothesis at the given
level of significance (e.g. 5%)
However, there is no ”correct” significance level.
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THE p-VALUE
Classical approach to hypothesis testing: first choose the
significance level, then test the hypothesis at the given
level of significance (e.g. 5%)
However, there is no ”correct” significance level.
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THE p-VALUE
Classical approach to hypothesis testing: first choose the
significance level, then test the hypothesis at the given
level of significance (e.g. 5%)
However, there is no ”correct” significance level.
Or we can ask a more informative question:
What is the smallest significance level at which the null
hypothesis would still be rejected?
This level of significance is known as the p-value.
Remember that the significance level describes the
probability of type I. error. The smaller the p-value, the
smaller the probability of rejecting the true null hypothesis
(the bigger the confidence the null hypothesis is indeed
correctly rejected).
The p-value for H0 : β = 0 is displayed in most regression
outputs
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EXAMPLE
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EXAMPLE
Let us study the impact of years of education on wages:
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EXAMPLE
Let us study the impact of years of education on wages:
wage = β0 + β1education + β2experience + ε
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EXAMPLE
Let us study the impact of years of education on wages:
wage = β0 + β1education + β2experience + ε
Output from Gretl:
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EXAMPLE
Let us study the impact of years of education on wages:
wage = β0 + β1education + β2experience + ε
Output from Gretl:
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
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
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
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


23 / 1
CONFIDENCE INTERVAL
24 / 1
CONFIDENCE INTERVAL
A 95% confidence interval of β is an interval centered
around β such that β falls into this interval with
probability 95%
24 / 1
CONFIDENCE INTERVAL
A 95% confidence interval of β is an interval centered
around β such that β falls into this interval with
probability 95%
P β − c < β < β + c =
= P

−
c
s.e. β
<
β − β
s.e. β
<
c
s.e. β

 = 0.95
24 / 1
CONFIDENCE INTERVAL
A 95% confidence interval of β is an interval centered
around β such that β falls into this interval with
probability 95%
P β − c < β < β + c =
= P

−
c
s.e. β
<
β − β
s.e. β
<
c
s.e. β

 = 0.95
Since β−β
s.e.(β)
∼ tn−k, we derive the confidence interval:
β ± tn−k,0.975 · s.e. β
24 / 1
CONFIDENCE INTERVAL
Output from Gretl (wage regression):
25 / 1
CONFIDENCE INTERVAL
Output from Gretl (wage regression):














25 / 1
CONFIDENCE INTERVAL
Output from Gretl (wage regression):














Confidence interval for coefficient on education:
β ± tn−k,0.975 · s.e. β = 0.644 ± 1.960 · 0.054
β ∈ [0.538; 0.750] with 95% probability
25 / 1
SUMMARY
26 / 1
SUMMARY
We discussed the principle of hypothesis testing
26 / 1
SUMMARY
We discussed the principle of hypothesis testing
We derived the t-statistic
26 / 1
SUMMARY
We discussed the principle of hypothesis testing
We derived the t-statistic
We defined the concept of the p-value
26 / 1
SUMMARY
We discussed the principle of hypothesis testing
We derived the t-statistic
We defined the concept of the p-value
We explained what significance of a coefficient means
26 / 1
SUMMARY
We discussed the principle of hypothesis testing
We derived the t-statistic
We defined the concept of the p-value
We explained what significance of a coefficient means
We observed a regression output on an example
26 / 1