Dr Güven Demirel
BUSM014
Quantitative Research Methods
Revision Lecture
2020/2021 Semester A
Module Outline
2
1. Introduction & Structuring Research
2. Univariate Descriptive Statistics
3. Bivariate Descriptive Statistics, Statistical Graphs & Tables
4. Distributions, LLN & CLT, Confidence Intervals
5. Experiments, Comparing Groups, Testing Hypotheses
6. Testing Hypotheses (ctd)
7. Reading week (no lectures/labs)
8. Regression Analysis – Introduction
9. Regression Analysis – Inference
10. Regression Analysis – Further Aspects
11. Review
12. Linking theory and empirics, developing hypotheses
Revision Lecture Outline
• Recap of Week 9
• Model Specification
• Functional Forms
• Dummy Variables
• Applications
3
Hypothesis Testing
Statistical Inference - Overview
5
• Point estimation: “Best guess” numerical value for an unknown
population parameter (sample mean, sample standard deviation,
standard error, …)
• Confidence interval: Interval estimate for an unknown population
parameter (CI for mean, CI for difference between group means)
• Hypothesis testing: Specify hypotheses for YES / NO questions (e.g.
“Does mean income differ between men & women?”) and use
sample data to test.
Steps of Hypothesis Testing
6
1. Given the hypothesis you intend to test (typically the alternative
hypothesis 1), determine its null-hypothesis (0). In other words,
what is the no-effect version of your hypothesis.
2. Choose the significance level .
3. Compute the probability value (p-value) of observing the
signal you have. To calculate the p-value we need a test statistic.
4. Compare the p-value to the significance level, if it is smaller reject
0. Otherwise, fail to reject 0.
t-Statistic for Hypothesis Testing on Population Mean
7
• Null hypothesis 0: E = ,0
• Alternative hypothesis : E ≠ ,0
• Test statistics are in the general form


• For the hypothesis tests on means and the differences between
means, we use the t-statistic.
o Signal: ത − ,0
o Noise: /
• The population standard deviation is typically unknown, hence we
instead use − =
ഥ−,
ഥ
• ത =


: standard error of the sample mean
Significance Level and Critical Value
8
• Significance level (): the probability with which you are willing to reject the null the
hypothesis when it is true.
= 0 0
• Assuming the null hypothesis, t-statistic: =
ത−,0
ത
~ 0,1 for large n. Therefore
= >
• : the critical value of the t-statistic for which
more extreme values are observed with probability

• For = 0.05, = 1.96 (obtained from the
standard normal distribution for large n.)
• But for small n, CLT does not apply.
• For small samples, if the population is
distributed normally, we can use the t-dist.
Null Hypothesis Rejection Rule
9
• p-value (probability value): the probability of observing the given
or a more extreme signal, given that the null hypothesis 0 is true.
− = > 0
• For given = 0.05, statistical testing on the conditions about the
observed
o Reject 0 if
≥ . , which is equivalent to
− ≤
o Fail to reject otherwise
Two-Sided versus One-sided t-test
10
• Two sided t-test: The alternative
hypothesis does not have a sense of
directionality, we have no idea whether
larger or smaller.
: E ≠ ,0
rejection region rejection region
• One sided t-test: The alternative
hypothesis specifies the direction, i.e.
smaller or larger.
: E < ,0
: > ,
p-value
and
based on
only one
side
Hypothesis Tests for Comparing Population Means
11
0: E − E = 0
: E − E ≠ 0; E − E > 0; E − E < 0
• Paired samples t-test: The same people/ firms are observed before and after the
treatment and the equality of the populations means are tested (e.g. before –
after comparison of the treatment group).
• Independent samples t-test: The two groups tested are drawn independently
and the equality of the populations means are tested (e.g. checking for balance).
• In a paired test, there are n paired observations. The difference between the
pairs is the data, where the standard error uses the sample standard deviation of
the difference data.
• In independent samples, we compare the mean values of the variable in both
groups, where the standard error uses a weighted pooled sample variance.
Testing for statistical independence
12
: Categorical variables X & Y are statistically independent.
: There is an association between X & Y
• For each cell for row i (R rows)& column j (C columns), we have:
o Observed Count: ,
o Expected Count: , =
×

Chi-Square statistic χ2 = σ=1
σ=1
,−,
2
,
p-value
Regression Analysis
Simple Regression Model
14
• Population Model: simple regression model
= 0 + 1 +
Example: Return to education
= 0 + 1 +
• We refer to as the dependent variable (or regressand, explained variable, response variable,
target variable).
• In the simple and multiple regression models, is continuous.
• We refer to as the independent variable (or regressor, explanatory variable, covariate, feature).
• Independent variable can be continuous or categorical (assume continuous, unless otherwise
stated).
• The variable is called the error term or disturbance.
• 0: intercept parameter
• 1: slope parameter
Simple Regression Estimators
15
• The regression model is for the population and the parameters 0 and 1 are unknown.
• Hence the error term is unknown.
• We have a sample
, : = 1,… ,
• We use data to estimate the parameters 0 and 1.
= መ0 + መ1 + ො
෡: estimator of the intercept parameter 0
෡: estimator of the slope parameter 1
Sample regression function (regression line)
ො = መ0 + መ1
ෝ is the predicted / fitted value of for observation
ෝ = − ො: residual
Ordinary Least Squares (OLS) Estimators
16
• Many alternative rules can be used to estimate the population parameters.
Which estimator should we use?
• Criteria: Minimise the Sum of Squared Residuals

෡,෡
෍
=

ෝ
=෍
=1

− ො
2 =෍
=1

− መ0 − መ1
2
Ordinary Least Squares (OLS) Estimators – Simple Regression
17
• The Sum of Squared Residuals = σ=1
ො
2 = σ=1
− መ0 − መ1
2
is minimised
when the partial derivatives of with respect to መ0 and መ1 are zero.
• OLS slope estimator:
መ1 =
σ=1
− ҧ − ത
σ=1
− ҧ 2
=
෢ ,

2
• The slope is the ratio of the sample covariance between the independent and the
dependent variables and the sample variance of the independent variable.
• OLS intercept estimator:
መ0 = ത − መ1 ҧ
• Consequence: The sample average ҧ, ത is always on the OLS regression line.
Multiple Linear Regression Model
18
• There are multiple explanatory variables that can affect the dependent variable.
• Example: Gender, Age, Tenure, Ethnicity affect Wage alongside Education. If our
focus is on Education, we call rest of the explanatory variables: control variables.
• Model : wage = 0 + 1 + 2 + 2 +
Multiple linear regression model:
= 0 + 11 + 22 +⋯+ +
• 0: intercept parameter
• 1, … , : slope parameters
• 1, … , : explanatory variables
• : dependent variable
• : error / disturbance
Interpretation of Parameters
19
• Multiple linear regression model:
= 0 + 11 + 22 +⋯+ +
• Each slope coefficient provides the effect of its variable, holding other explanatory
variables constant (restricted by variables in the model).
• Model: = 0 + 1 + 2 + 3 +
o 1 is the (expected) effect of one additional year of education, given tenure and
gender are not changing.
∆ = 1∆
o 2 is the effect of one additional year worked in the company, given education and
gender are not changing.
o 3 is the mean difference between the wages women and men, given education and
tenure are not changing.
Ordinary Least Squares (OLS) Estimators – Multiple Regression
20
• The Sum of Squared Residuals = σ=1
ො
2 = σ=1
− መ0 − መ11 −⋯− መ
2
is
minimised when the partial derivatives of with respect to ෡ , ෡, …, ෡ are zero.
• OLS slope estimator:
መ =
σ=1
Ƹ
σ=1
Ƹ
2
where Ƹ are the residuals of the auxiliary regression of on all other explanatory variables.
• Interpretation: መ is the effect of the part of the independent variable that is not
correlated with the rest of the explanatory variables.
• Partialling-out result: መ is called the partial effect of on , which enables the ceteris
paribus interpretation (i.e. while holding other factors constant).
Goodness of Fit
21
• The goodness of fit of the model quantifies how well the model predicts the dependent
variable. (R-squared), also called coefficient of determination, provides the goodness-
of-fit:
2 =


• is the ratio of the explained variation = σ=1
ො − ത
2 to the total variation
= σ=1
− ത
2 in the dependent variable. : actual value, ො: predicted value, ത:
sample mean.
• 0 ≤ 2 ≤ 1.
• Having a low or high R-squared are not per-se bad/good. In social science low R-squared
values are common as there are many unobserved factors and true randomness.
• 2 never decreases when new variables are added (under no missing data), which will give
us criteria on whether a group of control variables should be added to the model.
Properties of OLS Estimators
Unbiasedness of OLS Est. for Multiple Regression
23
MLR Assumptions:
1. Linearity in parameters: In the population model
= 0 + 11 + 22 +⋯+ +
2. Random sampling: We have a random sample of size n
1, 2, … , , : = 1,… ,
3. No perfect collinearity: In the sample, none of the variables is constant and there are
no exact linear relationships between the independent variables. They can be
correlated but not perfectly.
4. Zero conditional mean: The error has an expected value of zero given any value of
the explanatory variable E |1, 2, … , = 0.
If the MLR-Assumptions 1-4 hold, then OLS estimators are unbiased:
E መ0 = 0, E መ1 = 1, E መ2 = 2,…,E መ =
• For a random sample, the main concern is the last assumption.
Homoskedasticity Assumption
24
Additional assumption:
5. Homoskedasticity: The error has the same variance given any value of the
explanatory variables Var |1, 2, … , =
2.
homoskedastic heteroskedastic
MLR Assumptions, Unbiasedness, and Efficiency of OLS
25
Unbiasedness of OLS: If MLR-Assumptions 1-4 (Linearity in parameters; Random
sampling; No perfect collinearity; Zero conditional mean) hold, then OLS
estimators are unbiased:
• E መ0 = 0,E መ1 = 1, E መ2 = 2,…,E መ =
Efficiency of OLS: If Gauss-Markov Assumptions (MLR 1-4 + Homoskedasticity) hold,
then the OLS estimator መ for is the Best Linear Unbiased Estimator (BLUE).
• Best: the estimator with the least variance
• Linear: ෨ = σ=1

Variance of OLS Estimator
26
If MLR-Assumptions 1-5 hold, then the variance of the OLS estimator መ is
෡ =


.
• Error variance, : Higher noise in the equation, more difficult to estimate partial
effects.
o The only way to reduce it is to add more explanatory variables.
• Total sample variation in , = σ=1
− ҧ
2
= − 1
2 : More sample
variation in , the more precise the estimate becomes.
o The higher the sample variance
, the easier to estimate.
o The larger the sample size , the more precise the estimate becomes.
• Variance Inflation Factor, = ൗ

−
: The higher the VIF of , the less precise
the estimate becomes.
Variance Inflation Factor and Multi-Collinearity
27
If MLR-Assumptions 1-5 hold, then the variance of the OLS estimator መ is
෡ =


.
• Variance Inflation Factor, = ൗ

−
: The higher the VIF of , the less precise the
estimate becomes.
• Here
is the R-squared of the following auxiliary regression
on 1, 2, … , −1, +1, … ,
o The higher the proportion of the variation in that can be explained by the variation
in the other variables (
), the less precise the estimate is. Hence, if the variable is
linearly related with other variables, the harder it becomes to estimate its partial
effect. This high VIFj is called multi-collinearity.
o Multi-collinearity does not violate the perfect collinearity assumption, which is when

2 = 1 but makes estimates imprecise.
OLS Standard Error
28
Under Gauss-Markov Assumptions ෡ =


. But we do not know the error
variance 2, which needs to be estimated from data.
• Under Gauss-Markov Assumptions, the following is an unbiased estimator of the error
variance, i.e. ො2 = 2:
ො2 =

− − 1
=
σ=1
− ො
2
− − 1
ො is called the standard error of regression
• This leads to the standard error of ෡, which is the estimator of the sampling standard
deviation of መ.
o Sampling standard deviation of መ: ෡ =



o Standard error of መ: ෡ =
ෝ


Distribution of OLS Estimators
29
• We have learned the mean and the sampling variation of the OLS estimator መ. The next
question is “What is the sampling distribution of መ?” If we know it, we can test hypotheses
on regression coefficients. For this, we need an additional assumption.
Classical Linear Model (CLM) Assumptions (Gauss-Markov & normality):
(1) Linearity in parameters; (2) Random sampling; (3) No perfect collinearity; (4) Zero
conditional mean; (5) Homoskedasticity
6. Normality: The error is independent of the explanatory variables and is distributed
normally with mean zero and variance 2.
|, … , ~ + +⋯+ ,

Under CLM assumptions, ෡~ , ෡ .Hence, the standardized t-statistic follows a t-
distribution with − − 1 degrees of freedom
=
෡ −
෡
~−−
Testing Hypotheses About a Single Parameter
30
= 0 + 11 + 22 +⋯+ +
Testing whether the variable has a significant effect on the dependent variable, once
accounted for other explanatory variables.
0: = 0
1: < 0 (directional) 1: ≠ 0 (un-directional) 1: > 0 (directional)
• We know that the t-statistic =
෡
෡
is distributed as −−1 under the null hypothesis,
where the real effect is zero. Hence we can proceed the hypothesis testing by calculating
the p-value and comparing by the significance level .
• Remember that we have made CLM assumptions. If the sample is random and the error
term is not correlated with the explanatory variables, the t-statistic follows a standard
normal distribution under large sample sizes even if the population error is not normal.
F-statistic
31
Unrestricted Model
= 0 + 11 +⋯+ −− + −+1−+1 +⋯+ +
q exclusion restrictions:
0: −+1 = 0,… , = 0,
Restricted Model
= 0 + 11 +⋯+ −− +
Fact: R-squared decreases as some explanatory variables are removed from the model,
that is
≥
. Here
2 and
2 the R-squared values of the unrestricted and
restricted models.
Question: Is the decrease in R-squared statistically significant?
Test statistic: F-statistic =

−
/
−
/ −−
Testing Overall Significance of a Regression
32
Unrestricted Model
= 0 + 11 +⋯+ +
2 = 2
k exclusion restrictions:
0: 1 = 0,… , = 0,
Restricted Model: = 0 +
2 = 0
Question: Does the overall model explain any significant variation in the dependent
variable? Are all the variables jointly significant? Is the model plausible at all?
F-statistic: F =
2/
1−2 / −−1
is distributed as ,−−1.
• This is the default F-test reported in Stata.
• If − ≥ , fail to reject 0. There is not enough evidence that the explanatory
variables help to explain the dependent variable jointly. Hence, the model is rejected.
Omitted Variables, Functional Forms,
Dummy Variables, Interactions
Multiple Regression Analysis: Perspective
34
• How many variables should we add? On what basis should variables be
included in an empirical model?
o Model Under-Specification: Leaving out variables that affect the
dependent variable and are also correlated with existing variables lead
to biased OLS estimators (Omitted Variable Bias - OVB) .
o Model Over-Specification: Including irrelevant variables that do not
affect the dependent variable do not lead to bias but lead to higher
variance of the estimators; hence should be avoided.
Omitted Variable Bias
35
• If you leave out variables that affect the dependent variable and are also correlated with
existing variables, this leads to biased OLS estimators (Omitted Variable Bias - OVB).
• True model: = 0 + 11 + 22 +⋯+ +
o OLS estimators: = መ0 + መ11 + መ22 +⋯+ መ
o Unbiasedness: መ =
• Omitted variable:
o OLS estimators: = ෨0 + ෨11 + ෨22 +⋯+ ෨−1−1
= ෩ − = ෩,
where ෩ is the slope coefficient of in the auxiliary regression of the omitted variable
on the rest of the variables. A useful consideration for determining the direction of OVB is
to assume the variables other than are pairwise uncorrelated. Then ~
෣ ,

.
Model Over-Specification
36
• If we include irrelevant variables that do not affect the dependent variable:
= 0 + 11 + 22 +⋯+ + ,with = 0
• OLS estimators of the model with the irrelevant variable are still unbiased, that is
መ = , = 0,… ,
• The sampling variance of estimators መ =
2

increases if the irrelevant
variable is correlated with .
• If you consider whether adding a new variable, you should contrast
o Omitted Variable Bias
o The sampling variance of estimators (decreases 2 if relevant but may increase )
Linearity in Parameters
37
Linear regression refers to linearity in the parameters
• Simple regression model: = 0 + 1 +
• Multiple regression model: = 0 + 11 + 22 +
• The following are also linear regression models:
o = 0 + 1 log +
o log = 0 + 1
2 +
o = 0 + 11 + 22 + 312 +
• You can transfer the above to the standard form by redefining the variables
accordingly, e.g. z = log .
• For instance, the following are not linear in parameters:
o = 0 + 1 + 3
2 +
o =
1
0+1
+
Logarithmic Variables
38
• Logarithms are commonly used either informed by theory or for skewed variables such
as monetary variables (e.g. wage, GDP) and size (population, number of employees). It
has special interpretations
• Log on level: log = 0 + 11 + ℎ
o 100 × 1 is approximately the percentage change in , if 1 increases by one unit.
• Level on log: y = 0 + 1 log 1 + ℎ
o 1/100 is approximately the level change in , if 1 increases by one percent.
• Log on log: log = 0 + 1 log 1 + ℎ
o 1 is approximately the percentage change in , if 1 increases by one percent. It is
called elasticity (
ൗ∆
Τ∆
).
Interactions of Continuous Variables
39
• If the partial effect depends on the magnitude of another explanatory variable, it is called
an interaction effect.
• The most common form is the multiplication of two levels:
= 0 + 11 + 22 + 312 + ℎ
• Example: Attendance and prior performance impact on exam results, ATTEND dataset of
Wooldridge(2019)
= 0 + 1 + 2 + 3 +
4
2 + 5
2 + 6 × +
: standardized test result for a final exam
: attendance rate; : GPA prior; : ACT test result
• Previous performance can be expected to have a decreasing / increasing marginal effect.
• The attendance can be expected to have a different impact on students with different past
performance.
Dummy Variable Coefficients
40
Example: sex, base group: male
log = 0 + 1 + 2 +
0: intercept for male
0 + 1: intercept for female
2: common slope
Interactions between Dummy & Continuous Variables
41
log = 0 + 1
+ 2 + 3 × +
• 0: intercept for male
• 0 + 1: intercept for female
• 2: slope for male
• 2 + 3: slope for female
Hypotheses for Telecommuting
42
• H1: Are the treatment and control groups identical before the experiment?
Before means = 0 and the difference between control and
treatment group is given by switching on :
/ = 0 + 1 × + 2 × 0 + 3 × 0 +
• H2: Does the control group change in the experiment?
Control group means TreatGroup = 0 and the difference between before and
after is given by switching on :
/ = 0 + 1 × 0 + 2 × + 3 × 0 +
• H3: Does the treatment group change because of the experiment?
Treatment group means = 1 and the difference between before and
after is given by switching on :
/ = 0 + 1 × 1 + 2 × + 3 × +
Hypotheses for Telecommuting
43
• H4: Does the treatment group change net of the control group?
Treatment group change: 2 + 3
Control group change: 2
Difference-in-Differences effect (ATE): 3

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