LECTURE 5
Introduction to Econometrics
Hypothesis testing & Goodness of fit
October 20, 2017
1 / 25
ON THE PREVIOUS LECTURE
2 / 25
ON THE PREVIOUS LECTURE
We discussed the principle of hypothesis testing
2 / 25
ON THE PREVIOUS LECTURE
We discussed the principle of hypothesis testing
Type I and Type II errors
2 / 25
ON THE PREVIOUS LECTURE
We discussed the principle of hypothesis testing
Type I and Type II errors
Critical value and rejection region
2 / 25
ON THE PREVIOUS LECTURE
We discussed the principle of hypothesis testing
Type I and Type II errors
Critical value and rejection region
We derived the t-statistic t = β−β
s.e.(β)
2 / 25
ON THE PREVIOUS LECTURE
We discussed the principle of hypothesis testing
Type I and Type II errors
Critical value and rejection region
We derived the t-statistic t = β−β
s.e.(β)
We defined the concept of the p-value
2 / 25
ON THE PREVIOUS LECTURE
We discussed the principle of hypothesis testing
Type I and Type II errors
Critical value and rejection region
We derived the t-statistic t = β−β
s.e.(β)
We defined the concept of the p-value
We explained what significance of a coefficient means
2 / 25
ON THE PREVIOUS LECTURE
We studied the impact of years of education on wages:
3 / 25
ON THE PREVIOUS LECTURE
We studied the impact of years of education on wages:














3 / 25
ON TODAY’S LECTURE
4 / 25
ON TODAY’S LECTURE
We will explain how multiple hypotheses are tested in a
regression model
4 / 25
ON TODAY’S LECTURE
We will explain how multiple hypotheses are tested in a
regression model
We will define the notion of the overall significance of a
regression
4 / 25
ON TODAY’S LECTURE
We will explain how multiple hypotheses are tested in a
regression model
We will define the notion of the overall significance of a
regression
We will introduce a measure of the goodness of fit of a
regression (R2)
4 / 25
ON TODAY’S LECTURE
We will explain how multiple hypotheses are tested in a
regression model
We will define the notion of the overall significance of a
regression
We will introduce a measure of the goodness of fit of a
regression (R2)
Readings for this week:
Studenmund, Chapters 5.5 & 2.4
Wooldridge, Chapters 4 & 3
4 / 25
TESTING MULTIPLE HYPOTHESES
Suppose we have a model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
5 / 25
TESTING MULTIPLE HYPOTHESES
Suppose we have a model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
Suppose we want to test multiple linear hypotheses in this
model
5 / 25
TESTING MULTIPLE HYPOTHESES
Suppose we have a model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
Suppose we want to test multiple linear hypotheses in this
model
For example, we want to see if the following restrictions on
coefficients hold jointly:
β1 + β2 = 1 and β3 = 0
5 / 25
TESTING MULTIPLE HYPOTHESES
Suppose we have a model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
Suppose we want to test multiple linear hypotheses in this
model
For example, we want to see if the following restrictions on
coefficients hold jointly:
β1 + β2 = 1 and β3 = 0
We cannot use a t-test in this case (t-test can be used only
for one hypothesis at a time)
5 / 25
TESTING MULTIPLE HYPOTHESES
Suppose we have a model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
Suppose we want to test multiple linear hypotheses in this
model
For example, we want to see if the following restrictions on
coefficients hold jointly:
β1 + β2 = 1 and β3 = 0
We cannot use a t-test in this case (t-test can be used only
for one hypothesis at a time)
We will use an F-test
5 / 25
RESTRICTED VS. UNRESTRICTED MODEL
6 / 25
RESTRICTED VS. UNRESTRICTED MODEL
We can reformulate the model by plugging the restrictions
as if they were true (model under H0)
6 / 25
RESTRICTED VS. UNRESTRICTED MODEL
We can reformulate the model by plugging the restrictions
as if they were true (model under H0)
We call this model restricted model as opposed to the
unrestricted model
6 / 25
RESTRICTED VS. UNRESTRICTED MODEL
We can reformulate the model by plugging the restrictions
as if they were true (model under H0)
We call this model restricted model as opposed to the
unrestricted model
The unrestricted model is
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
6 / 25
RESTRICTED VS. UNRESTRICTED MODEL
We can reformulate the model by plugging the restrictions
as if they were true (model under H0)
We call this model restricted model as opposed to the
unrestricted model
The unrestricted model is
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
We derive (on the lecture) the restricted model:
6 / 25
RESTRICTED VS. UNRESTRICTED MODEL
We can reformulate the model by plugging the restrictions
as if they were true (model under H0)
We call this model restricted model as opposed to the
unrestricted model
The unrestricted model is
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
We derive (on the lecture) the restricted model:
y∗
i = β0 + β1x∗
i + εi ,
where y∗
i = yi − xi2 and x∗
i = xi1 − xi2
6 / 25
IDEA OF THE F-TEST
7 / 25
IDEA OF THE F-TEST
If the restrictions are true, then the restricted model fits the
data in the same way as the unrestricted model
7 / 25
IDEA OF THE F-TEST
If the restrictions are true, then the restricted model fits the
data in the same way as the unrestricted model
residuals are nearly the same
7 / 25
IDEA OF THE F-TEST
If the restrictions are true, then the restricted model fits the
data in the same way as the unrestricted model
residuals are nearly the same
If the restrictions are false, then the restricted model fits the
data poorly
7 / 25
IDEA OF THE F-TEST
If the restrictions are true, then the restricted model fits the
data in the same way as the unrestricted model
residuals are nearly the same
If the restrictions are false, then the restricted model fits the
data poorly
residuals from the restricted model are much larger than
those from the unrestricted model
7 / 25
IDEA OF THE F-TEST
If the restrictions are true, then the restricted model fits the
data in the same way as the unrestricted model
residuals are nearly the same
If the restrictions are false, then the restricted model fits the
data poorly
residuals from the restricted model are much larger than
those from the unrestricted model
The idea is thus to compare the residuals from the two
models
7 / 25
IDEA OF THE F-TEST
8 / 25
IDEA OF THE F-TEST
How to compare residuals in the two models?
8 / 25
IDEA OF THE F-TEST
How to compare residuals in the two models?
Calculate the sum of squared residuals in the two models
Test if the difference between the two sums is equal to zero
(statistically)
H0: the difference is zero (residuals in the two models are
the same, restrictions hold)
HA: the difference is positive (residuals in the restricted
model are bigger, restrictions do not hold)
8 / 25
IDEA OF THE F-TEST
How to compare residuals in the two models?
Calculate the sum of squared residuals in the two models
Test if the difference between the two sums is equal to zero
(statistically)
H0: the difference is zero (residuals in the two models are
the same, restrictions hold)
HA: the difference is positive (residuals in the restricted
model are bigger, restrictions do not hold)
Sum of squared residuals
SSE =
n
i=1
(yi − yi)2
=
n
i=1
e2
i
8 / 25
F-TEST
9 / 25
F-TEST
The test statistic is defined as
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k ,
9 / 25
F-TEST
The test statistic is defined as
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k ,
where:
SSER . . . sum of squared residuals from the restricted model
9 / 25
F-TEST
The test statistic is defined as
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k ,
where:
SSER . . . sum of squared residuals from the restricted model
SSEU . . . sum of squared residuals from the unrestricted model
9 / 25
F-TEST
The test statistic is defined as
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k ,
where:
SSER . . . sum of squared residuals from the restricted model
SSEU . . . sum of squared residuals from the unrestricted model
J . . . number of restrictions
9 / 25
F-TEST
The test statistic is defined as
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k ,
where:
SSER . . . sum of squared residuals from the restricted model
SSEU . . . sum of squared residuals from the unrestricted model
J . . . number of restrictions
n . . . number of observations
9 / 25
F-TEST
The test statistic is defined as
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k ,
where:
SSER . . . sum of squared residuals from the restricted model
SSEU . . . sum of squared residuals from the unrestricted model
J . . . number of restrictions
n . . . number of observations
k . . . number of estimated coefficients (including intercept)
9 / 25
F-TEST
Rejection region :
p-value = 5%
Distribution FJ,n-k
5%
FJ,n-k,0.95
10 / 25
EXAMPLE
11 / 25
EXAMPLE
We had the model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
11 / 25
EXAMPLE
We had the model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
We wanted to test
H0 :
β1 + β2 = 1
β3 = 0
vs. HA :
β1 + β2 = 1
β3 = 0
11 / 25
EXAMPLE
We had the model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
We wanted to test
H0 :
β1 + β2 = 1
β3 = 0
vs. HA :
β1 + β2 = 1
β3 = 0
Under H0, we obtained the restricted model
y∗
i = β0 + β1x∗
i + εi ,
where y∗
i = yi − xi2 and x∗
i = xi1 − xi2
11 / 25
EXAMPLE
We run the regression on the unrestricted model, we
obtain SSEU
12 / 25
EXAMPLE
We run the regression on the unrestricted model, we
obtain SSEU
We run the regression on the restricted model, we obtain
SSER
12 / 25
EXAMPLE
We run the regression on the unrestricted model, we
obtain SSEU
We run the regression on the restricted model, we obtain
SSER
We have k = 4 and J = 2
12 / 25
EXAMPLE
We run the regression on the unrestricted model, we
obtain SSEU
We run the regression on the restricted model, we obtain
SSER
We have k = 4 and J = 2
We construct the F-statistic F = (SSER−SSEU)/2
SSEU/(n−4)
12 / 25
EXAMPLE
We run the regression on the unrestricted model, we
obtain SSEU
We run the regression on the restricted model, we obtain
SSER
We have k = 4 and J = 2
We construct the F-statistic F = (SSER−SSEU)/2
SSEU/(n−4)
We find the critical value of the F distribution with 2 and
n − 4 degrees of freedom at the 95% confidence level
12 / 25
EXAMPLE
We run the regression on the unrestricted model, we
obtain SSEU
We run the regression on the restricted model, we obtain
SSER
We have k = 4 and J = 2
We construct the F-statistic F = (SSER−SSEU)/2
SSEU/(n−4)
We find the critical value of the F distribution with 2 and
n − 4 degrees of freedom at the 95% confidence level
If F > F2,n−4,0.95, we reject the null hypothesis
we reject that the restrictions hold jointly
12 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
13 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
Usually, we are interested in knowing if the model has
some explanatory power, i.e. if the independent variables
indeed “explain” the dependent variable
13 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
Usually, we are interested in knowing if the model has
some explanatory power, i.e. if the independent variables
indeed “explain” the dependent variable
We test this using the F-test of the joint significance of all
(k − 1) slope coefficients:
H0 :



β1 = 0
β2 = 0
...
βk−1 = 0
vs. HA :



βj = 0
for at least one j = 1, . . . , k − 1
13 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
Unrestricted model:
yi = β0 + β1xi1 + β2xi2 + . . . + βk−1xik−1 + εi
14 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
Unrestricted model:
yi = β0 + β1xi1 + β2xi2 + . . . + βk−1xik−1 + εi
Restricted model:
yi = β0 + εi
14 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
Unrestricted model:
yi = β0 + β1xi1 + β2xi2 + . . . + βk−1xik−1 + εi
Restricted model:
yi = β0 + εi
F-statistic:
F =
(SSER − SSEU)/(k − 1)
SSEU/(n − k)
∼ Fk−1,n−k
14 / 25
OVERALL SIGNIFICANCE OF THE REGRESSION
Unrestricted model:
yi = β0 + β1xi1 + β2xi2 + . . . + βk−1xik−1 + εi
Restricted model:
yi = β0 + εi
F-statistic:
F =
(SSER − SSEU)/(k − 1)
SSEU/(n − k)
∼ Fk−1,n−k
Number of restrictions = k − 1
This F-statistic and the corresponding p-value are part of
the regression output
14 / 25
EXAMPLE














15 / 25
GOODNESS OF FIT MEASURE
16 / 25
GOODNESS OF FIT MEASURE
We know that education and experience have a significant
influence on wages
16 / 25
GOODNESS OF FIT MEASURE
We know that education and experience have a significant
influence on wages
But how important are they in determining wages?
16 / 25
GOODNESS OF FIT MEASURE
We know that education and experience have a significant
influence on wages
But how important are they in determining wages?
How much of difference in wages between people is
explained by differences in education and in experience?
16 / 25
GOODNESS OF FIT MEASURE
We know that education and experience have a significant
influence on wages
But how important are they in determining wages?
How much of difference in wages between people is
explained by differences in education and in experience?
How well variation in the independent variable(s) explains
variation in the dependent variable?
16 / 25
GOODNESS OF FIT MEASURE
We know that education and experience have a significant
influence on wages
But how important are they in determining wages?
How much of difference in wages between people is
explained by differences in education and in experience?
How well variation in the independent variable(s) explains
variation in the dependent variable?
This are the questions answered by the goodness of fit
measure - R2
16 / 25
TOTAL AND EXPLAINED VARIATION
17 / 25
TOTAL AND EXPLAINED VARIATION
Total variation in the dependent variable:
n
i=1
(yi − yn)2
17 / 25
TOTAL AND EXPLAINED VARIATION
Total variation in the dependent variable:
n
i=1
(yi − yn)2
Predicted value of the dependent variable = part that is
explained by independent variables:
yi = β0 + β1xi
(case of regression line - for simplicity of notation)
17 / 25
TOTAL AND EXPLAINED VARIATION
Total variation in the dependent variable:
n
i=1
(yi − yn)2
Predicted value of the dependent variable = part that is
explained by independent variables:
yi = β0 + β1xi
(case of regression line - for simplicity of notation)
Explained variation in the dependent variable:
n
i=1
(yi − yn)2
17 / 25
GOODNESS OF FIT - R2
18 / 25
GOODNESS OF FIT - R2
Denote:
SST =
n
i=1
yi − yn
2
. . . Total Sum of Squares
SSR =
n
i=1
(yi − yn)2
. . . Regression Sum of Squares
18 / 25
GOODNESS OF FIT - R2
Denote:
SST =
n
i=1
yi − yn
2
. . . Total Sum of Squares
SSR =
n
i=1
(yi − yn)2
. . . Regression Sum of Squares
Define the measure of the goodness of fit:
R2
=
SSR
SST
=
Explained variation in y
Total variation in y
18 / 25
GOODNESS OF FIT - R2
19 / 25
GOODNESS OF FIT - R2
In all models: 0 ≤ R2 ≤ 1
19 / 25
GOODNESS OF FIT - R2
In all models: 0 ≤ R2 ≤ 1
R2 tells us what percentage of the total variation in the
dependent variable is explained by the variation in the
independent variable(s)
R2
= 0.3 means that the independent variables can explain
30% of the variation in the dependent variable
19 / 25
GOODNESS OF FIT - R2
In all models: 0 ≤ R2 ≤ 1
R2 tells us what percentage of the total variation in the
dependent variable is explained by the variation in the
independent variable(s)
R2
= 0.3 means that the independent variables can explain
30% of the variation in the dependent variable
Higher R2 means better fit of the regression model (not
necessarily a better model!)
19 / 25
DECOMPOSING THE VARIANCE
20 / 25
DECOMPOSING THE VARIANCE
For models with intercept, R2 can be rewritten using the
decomposition of variance.
Variance decomposition:
n
i=1
yi − yn
2
=
n
i=1
(yi − yn)2
+
n
i=1
e2
i
20 / 25
DECOMPOSING THE VARIANCE
For models with intercept, R2 can be rewritten using the
decomposition of variance.
Variance decomposition:
n
i=1
yi − yn
2
=
n
i=1
(yi − yn)2
+
n
i=1
e2
i
SST =
n
i=1
yi − yn
2
. . . Total Sum of Squares
20 / 25
DECOMPOSING THE VARIANCE
For models with intercept, R2 can be rewritten using the
decomposition of variance.
Variance decomposition:
n
i=1
yi − yn
2
=
n
i=1
(yi − yn)2
+
n
i=1
e2
i
SST =
n
i=1
yi − yn
2
. . . Total Sum of Squares
SSR =
n
i=1
(yi − yn)2
. . . Regression Sum of Squares
20 / 25
DECOMPOSING THE VARIANCE
For models with intercept, R2 can be rewritten using the
decomposition of variance.
Variance decomposition:
n
i=1
yi − yn
2
=
n
i=1
(yi − yn)2
+
n
i=1
e2
i
SST =
n
i=1
yi − yn
2
. . . Total Sum of Squares
SSR =
n
i=1
(yi − yn)2
. . . Regression Sum of Squares
SSE =
n
i=1
e2
i . . . Sum of Squared Residuals
20 / 25
VARIANCE DECOMPOSITION AND R2
21 / 25
VARIANCE DECOMPOSITION AND R2
Variance decomposition: SST = SSR + SSE
21 / 25
VARIANCE DECOMPOSITION AND R2
Variance decomposition: SST = SSR + SSE
Intuition: total variation can be divided between the
explained variation and the unexplained variation
the true value y is a sum of estimated (explained) y and the
residual ei (unexplained part)
yi = yi + ei
21 / 25
VARIANCE DECOMPOSITION AND R2
Variance decomposition: SST = SSR + SSE
Intuition: total variation can be divided between the
explained variation and the unexplained variation
the true value y is a sum of estimated (explained) y and the
residual ei (unexplained part)
yi = yi + ei
We can rewrite R2:
R2
=
SSR
SST
=
SST − SSE
SST
= 1 −
SSE
SST
21 / 25
ADJUSTED R2
22 / 25
ADJUSTED R2
The sum of squared residuals (SSE) decreases when
additional explanatory variables are introduced in the
model, whereas total sum of squares (SST) remains the
same
22 / 25
ADJUSTED R2
The sum of squared residuals (SSE) decreases when
additional explanatory variables are introduced in the
model, whereas total sum of squares (SST) remains the
same
R2
= 1 − SSE
SST increases if we add explanatory variables
Models with more variables automatically have better fit.
22 / 25
ADJUSTED R2
The sum of squared residuals (SSE) decreases when
additional explanatory variables are introduced in the
model, whereas total sum of squares (SST) remains the
same
R2
= 1 − SSE
SST increases if we add explanatory variables
Models with more variables automatically have better fit.
To deal with this problem, we define the adjusted R2:
R2
adj = 1 −
SSE
n−k
SST
n−1
≤ R2
(k is the number of coefficients including intercept)
22 / 25
ADJUSTED R2
The sum of squared residuals (SSE) decreases when
additional explanatory variables are introduced in the
model, whereas total sum of squares (SST) remains the
same
R2
= 1 − SSE
SST increases if we add explanatory variables
Models with more variables automatically have better fit.
To deal with this problem, we define the adjusted R2:
R2
adj = 1 −
SSE
n−k
SST
n−1
≤ R2
(k is the number of coefficients including intercept)
This measure introduces a “punishment” for including more
explanatory variables
22 / 25
EXAMPLE














23 / 25
F-TEST - REVISITED
24 / 25
F-TEST - REVISITED
Let us recall the F-statistic:
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k
24 / 25
F-TEST - REVISITED
Let us recall the F-statistic:
F =
(SSER − SSEU)/J
SSEU/(n − k)
∼ FJ,n−k
We can use the formula R2 = 1 − SSE
SST to rewrite the
F-statistic in R2 form:
F =
(R2
U − R2
R)/J
(1 − R2
U)/(n − k)
∼ FJ,n−k
We can use this R2 form of F-statistic under the condition
that SSTU = SSTR (the dependent variables in restricted
and unrestricted models are the same)
24 / 25
SUMMARY
25 / 25
SUMMARY
We showed how restrictions are incorporated in regression
models
25 / 25
SUMMARY
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
25 / 25
SUMMARY
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
We defined the notion of the overall significance of a
regression
25 / 25
SUMMARY
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
We defined the notion of the overall significance of a
regression
We introduced the measure or the goodness of fit - R2
25 / 25
SUMMARY
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
We defined the notion of the overall significance of a
regression
We introduced the measure or the goodness of fit - R2
We learned how total variation in the dependent variable
can be decomposed
25 / 25