LECTURE 5
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Introduction to Econometrics
Multiple Regression Analysis:
Hypothesis Testing II
Hieu Nguyen
Fall semester, 2024

REVISION: the Classical Linear Model (CLM) Assumptions
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1.Linearity: the regression model is linear in the parameters  (coefficients)
2.Random sampling: the data is a random sample drawn  from the population and each data point
follows the  population equation
3.No perfect collinearity: the values of explanatory variables  are not all the same and no
explanatory variable is a  perfect linear function of any other explanatory variable(s)
4.Zero conditional mean: values of explanatory variables  must contain no information about the
mean of the  unobserved factors - explanatory variables are  uncorrelated with the error term
5.Homoskedasticity: the error term has a constant variance
6.Normality of the error term: the error term is normally  distributed

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e We continue our discussion on how hypotheses about coefficients can be tested in regression
models
e We will explain what significance of coefficients mean
e We will learn how to read regression output
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e Readings:
             Wooldridge Chapter 4;
              Studenmund Chapter 5.1-5.4
TODAY’S LECTURE

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e    „Statistically significant“ variables in a regression
§If a regression coefficient is different from zero in a two-sided test, the corresponding variable
is said to be „statistically significant“
§If the number of degrees of freedom is large enough so that the normal approximation applies, the
following rules of thumb apply:
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„statistically significant at 10 % level“
„statistically significant at 5 % level“
„statistically significant at 1 % level“
INFERENCE: The t Test

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e    Guidelines for discussing economic and statistical significance
§If a variable is statistically significant, discuss the magnitude of the coefficient to get an
idea of its economic or practical importance
§The fact that a coefficient is statistically significant does not necessarily mean it is
economically or practically significant!
§If a variable is statistically and economically important but has the „wrong“ sign, the regression
model might be misspecified
§If a variable is statistically insignificant at the usual levels (10%, 5%, 1%), one may think of
dropping it from the regression
§If the sample size is small, effects might be imprecisely estimated so that the case for dropping
insignificant variables is less strong
INFERENCE: The t Test

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e  Testing more general hypotheses about a regression coefficient
e  Null hypothesis
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e   t-statistic
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e   The test works exactly as before, except that the hypothesized value is substracted from the
estimate when forming the statistic
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Hypothesized value of the coefficient
INFERENCE: The t Test

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e  Example: Campus crime and enrollment
§An interesting hypothesis is whether crime increases by one percent   if enrollment is increased
by one percent
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The hypothesis is rejected at the 5% level
Estimate is different from one but is this difference statistically significant?
INFERENCE: The t Test

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e   Computing p-values for t-tests
§If the significance level is made smaller and smaller, there will be a point where the null
hypothesis cannot be rejected anymore
§The reason is that, by lowering the significance level, one wants to avoid more and more to make
the error of rejecting a correct H0
§The smallest significance level at which the null hypothesis is still rejected, is called the
p-value of the hypothesis test
§A small p-value is evidence against the null hypothesis because one would reject the null
hypothesis even at small significance levels
§A large p-value is evidence in favor of the null hypothesis
§P-values are more informative than tests at fixed significance levels
INFERENCE: The t Test

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e   How the p-value is computed (here: two-sided test)
The p-value is the significance level at which one is indifferent between rejecting and not
rejecting the null hypothesis.
In the two-sided case, the p-value is thus the probability that the t-distributed variable takes on
a larger absolute value than the realized value of the test statistic, e.g.:
From this, it is clear that a null hypothesis is rejected if and only if the corresponding p-value
is smaller than the significance level.
For example, for a significance level of 5% the t-statistic would not lie in the rejection region.
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value of test statistic
These would be the critical values for a 5% significance level
INFERENCE: The t Test

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Critical value of
two-sided test
e   Confidence intervals
e    Simple manipulation of the result in Theorem 4.2 implies that
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e   Interpretation of the confidence interval
§The bounds of the interval are random
§In repeated samples, the interval that is constructed in the above way will cover the population
regression coefficient in 95% of the cases
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Lower bound of the
Confidence interval
Upper bound of the
Confidence interval
Confidence level
INFERENCE: Confidence Intervals

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e    Confidence intervals for typical confidence levels
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e    Relationship between confidence intervals and hypotheses tests
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 reject                         in favor of
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Use rules of thumb
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INFERENCE: Confidence Intervals

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e    Example: Model of firms‘ R&D expenditures
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Spending on R&D
Annual sales
Profits as percentage of sales
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The effect of sales on R&D is relatively precisely estimated as the interval is narrow. Moreover,
the effect is significantly different from zero because zero is outside the interval.
This effect is imprecisely estimated as the in-
terval is very wide. It is not even statistically
significant because zero lies in the interval.
(0.0128 )
0.0217
(0.0128 )
INFERENCE: Confidence Intervals

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e    Example: Return to education at 2 year vs. at 4 year colleges
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Years of education at 2 year colleges
Years of education at 4 year colleges
Test                                against                              .
A possible test statistic would be:
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The difference between the estimates is normalized by the estimated standard deviation of the
difference. The null hypothesis would have to be rejected if the statistic is „too negative“ to
believe that the true difference between the parameters is equal to zero.
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INFERENCE: Testing Hypotheses About a Linear Combination of Parameters

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e    Impossible to compute with standard regression output because
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e    Alternative method
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Usually not available in regression output
Define                          and test                       against                     .
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a new regressor (= total years of college)
Insert into original regression
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INFERENCE: Testing Hypotheses About a Linear Combination of Parameters

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e    Estimation results
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e    This method works always for single linear hypotheses
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Total years of college
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Hypothesis is rejected at 10% level but not at 5% level
INFERENCE: Testing Hypotheses About a Linear Combination of Parameters

TESTING MULTIPLE HYPOTHESES
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e Suppose we have a model
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
e Suppose we want to test multiple linear hypotheses in this  model
e For example, we want to see if the following restrictions on  coefficients hold jointly:
β1  + β2 = 1 and β3  = 0
e We cannot use a t-test in this case (t-test can be used only  for one hypothesis at a time)
e We will use an F-test

RESTRICTED VS. UNRESTRICTED MODEL
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e We can reformulate the model by plugging the restrictions  as if they were true (model under H0)
e We call this model restricted model as opposed to the
unrestricted model
e The unrestricted model is
yi = β0 + β1xi1 + β2xi2 + β3xi3 + εi
e Restricted model can be derived to have the following  form:
y∗i = β0 + β1x∗i       + εi ,
where   y∗i = yi − xi2    and   x∗i = xi1 − xi2

IDEA OF THE F-TEST
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e 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
e 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
e The idea is thus to compare the residuals from the two  models

IDEA OF THE F-TEST
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e How to compare residuals in the two models?
vCalculate the sum of squared residuals in the two models
vTest if the difference between the two sums is equal to zero  (statistically)
vH0: the difference is zero (residuals in the two models are  the same, restrictions hold)
vHA: the difference is positive (residuals in the restricted  model are bigger, restrictions do not
hold)
e Sum of squared residuals

F-TEST
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e The test statistic is defined as
F =
(SSRr − SSRur)/q  SSRur/(n − k − 1)
∼ F
q,n−k−1
,
. . . sum of squared residuals from the restricted model
. . . sum of squared residuals from the unrestricted model
where:
SSRr
SSRur  q
. . . number of restrictions
n
. . . number of observations
k
. . . number of estimated coefficients

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e    Testing multiple linear restrictions: The F-test
e    Testing exclusion restrictions
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Years in
the league
Average number of
games per year
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Salary of major lea-
gue baseball player
Batting average
Home runs per year
Runs batted in per year
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against
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Test whether performance measures have no effect/can be exluded from regression.
INFERENCE: The F Test

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e    Estimation of the unrestricted model
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None of these variabels are statistically significant when tested individually
Idea: How would the model fit be if these variables were dropped from the regression?
INFERENCE: The F Test

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e    Estimation of the restricted model
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e    Test statistic
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The sum of squared residuals necessarily increases, but is the increase statistically significant?
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The relative increase of the sum of squared residuals when going from
H1 to H0 follows a F-distribution (if
the null hypothesis H0 is correct)
 Number of restrictions
INFERENCE: The F Test

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e    Rejection rule
A F-distributed variable only takes on positive values. This corresponds to the fact that the sum
of squared residuals can only increase if one moves from H1 to H0.
Choose the critical value so that the null hypo-thesis is rejected in, for example, 5% of the
cases, although it is true.
INFERENCE: The F Test

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e    Test decision in example
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e    Discussion
§The three variables are „jointly significant“
§They were not significant when tested individually
§The likely reason is multicollinearity between them
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Number of restrictions to be tested
Degrees of freedom in
the unrestricted model
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The null hypothesis is overwhel-mingly rejected (even at very small significance levels).
INFERENCE: The F Test

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e    Test of overall significance of a regression
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e    The test of overall significance is reported in most regression packages; the null hypothesis
is usually overwhelmingly rejected
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The null hypothesis states that the explanatory variables are not useful at all in explaining the
dependent variable
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Restricted model
(regression on constant)
INFERENCE: The F Test

GOODNESS OF FIT MEASURE
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e We know that education and experience have a significant  influence on wages
e But how important are they in determining wages?
e How much of difference in wages between people is  explained by differences in education and in
experience?
e How well variation in the independent variable(s) explains  variation in the dependent variable?
e This are the questions answered by the goodness of fit  measure - R2

TOTAL AND EXPLAINED VARIATION
e Total variation in the dependent variable:
e Predicted value of the dependent variable = part that is  explained by independent variables:
(case of regression line - for simplicity of notation)
e Explained variation in the dependent variable:
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GOODNESS OF FIT - R2
e Denote:
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e Define the measure of the goodness of fit:
R2 = SSE = Explained variation in y  SST Total variation in y

GOODNESS OF FIT - R2
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e  In all models: 0 ≤ R2 ≤ 1
e 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
e Higher R2 means better fit of the regression model (not  necessarily a better model!)

DECOMPOSING THE VARIANCE
e For models with intercept, R2 can be rewritten using the  decomposition of variance.
e Variance decomposition:
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VARIANCE DECOMPOSITION AND R2
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e Variance decomposition: SST = SSE + SSR
e Intuition: total variation can be divided between the  explained variation and the unexplained
variation
residual ei (unexplained part)
e We can rewrite R2:
2
R =
=
SSE SST − SSR SST
SST
= 1 −
SSR SST

ADJUSTED R2
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e The sum of squared residuals (SSR) decreases when  additional explanatory variables are
introduced in the  model, whereas total sum of squares (SST) remains the  same
2 SSR
SST
R   = 1 − increases if we add explanatory variables
 Models with more variables automatically have better fit.
e To deal with this problem, we define the adjusted R2:
R2
adj
= 1 −
   SSR
 n−k−1
SST
n−1
.≤ R2
(k is the number of coefficients)
e This measure introduces a “punishment” for including more  explanatory variables

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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e If all conditions hold, the variable belongs in the equation
e If none of them holds, the variable is irrelevant and can be  safely excluded
e 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!).