This document discusses statistical inference for multiple regression analysis. It begins by recapping the assumptions of the classical linear model (CLM), which are used to derive the sampling distributions of the OLS estimators. It then explains that the sampling distributions are needed to test hypotheses about the population parameters. Under the CLM assumptions, specifically the additional normality assumption, the sampling distributions are normal. This allows hypotheses about a single parameter to be tested using the t-distribution. The document provides an example of testing the null hypothesis that a parameter equals zero against a one-sided alternative hypothesis.
This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.
This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
Data Science - Part XII - Ridge Regression, LASSO, and Elastic NetsDerek Kane
This lecture provides an overview of some modern regression techniques including a discussion of the bias variance tradeoff for regression errors and the topic of shrinkage estimators. This leads into an overview of ridge regression, LASSO, and elastic nets. These topics will be discussed in detail and we will go through the calibration/diagnostics and then conclude with a practical example highlighting the techniques.
Logistic regression is used to obtain odds ratio in the presence of more than one explanatory variable. The procedure is quite similar to multiple linear regression, with the exception that the response variable is binomial. The result is the impact of each variable on the odds ratio of the observed event of interest.
Descriptive Statistics Formula Sheet Sample Populatio.docxsimonithomas47935
Descriptive Statistics Formula Sheet
Sample Population
Characteristic statistic Parameter
raw scores x, y, . . . . . X, Y, . . . . .
mean (central tendency) M =
∑ x
n
μ =
∑ X
N
range (interval/ratio data) highest minus lowest value highest minus lowest value
deviation (distance from mean) Deviation = (x − M ) Deviation = (X − μ )
average deviation (average
distance from mean)
∑(x − M )
n
= 0
∑(X − μ )
N
sum of the squares (SS)
(computational formula) SS = ∑ x
2 −
(∑ x)2
n
SS = ∑ X2 −
(∑ X)2
N
variance ( average deviation2 or
standard deviation
2
)
(computational formula)
s2 =
∑ x2 −
(∑ x)2
n
n − 1
=
SS
df
σ2 =
∑ X2 −
(∑ X)2
N
N
standard deviation (average
deviation or distance from mean)
(computational formula) s =
√∑ x
2 −
(∑ x)2
n
n − 1
σ =
√∑ X
2 −
(∑ X)2
N
N
Z scores (standard scores)
mean = 0
standard deviation = ± 1.0
Z =
x − M
s
=
deviation
stand. dev.
X = M + Zs
Z =
X − μ
σ
X = μ + Zσ
Area Under the Normal Curve -1s to +1s = 68.3%
-2s to +2s = 95.4%
-3s to +3s = 99.7%
Using Z Score Table for Normal Distribution
(Note: see graph and table in A-23)
for percentiles (proportion or %) below X
for positive Z scores – use body column
for negative Z scores – use tail column
for proportions or percentage above X
for positive Z scores – use tail column
for negative Z scores – use body column
to discover percentage / proportion between two X values
1. Convert each X to Z score
2. Find appropriate area (body or tail) for each Z score
3. Subtract or add areas as appropriate
4. Change area to % (area × 100 = %)
Regression lines
(central tendency line for all
points; used for predictions
only) formula uses raw
scores
b = slope
a = y-intercept
y = bx + a
(plug in x
to predict y)
b =
∑ xy −
(∑ x)(∑ y)
n
∑ x2 −
(∑ x)2
n
a = My - bMx
where My is mean of y
and Mx is mean of x
SEest (measures accuracy of predictions; same properties as standard deviation)
Pearson Correlation Coefficient
(used to measure relationship;
uses Z scores)
r =
∑ xy−
(∑ x)(∑ y)
n
√(∑ x2−
(∑ x)2
n
)(∑ y2−
(∑ y)2
n
)
r =
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦
r
2
= estimate or % of accuracy of predictions
PSYC 2317 Mark W. Tengler, M.S.
Assignment #9
Hypothesis Testing
9.1 Briefly explain in your own words the advantage of using an alpha level (α) = .01
versus an α = .05. In general, what is the disadvantage of using a smaller alpha
level?
9.2 Discuss in your own words the errors that can be made in hypothesis testing.
a. What is a type I error? Why might it occur?
b. What is a type II error? How does it happen?
9.3 The term error is used in two different ways in the context of a hypothesis test.
First, there is the concept of sta
For this assignment, use the aschooltest.sav dataset.The dMerrileeDelvalle969
For this assignment, use the aschooltest.sav dataset.
The dataset consists of Reading, Writing, Math, Science, and Social Studies test scores for 200 students. Demographic data include gender, race, SES, school type, and program type.
Instructions:
Work with the aschooltest.sav datafile and respond to the following questions in a few sentences. Please submit your SPSS output either in your assignment or separately.
1. Identify an Independent and Dependent Variable (of your choice) and develop a hypothesis about what you expect to find. (
note: the IV is a grouping variable, which means it needs to have more than 2 categories and the DV is continuous)
2. Run Assumption tests for Normality and initial Homogeneity of Variance. What are your results?
3. Run the one-way ANOVA with the Levene test & Tukey post hoc test.
a. What are the results of the Levene test? What does this mean?
b. What are the results of the one-way ANOVA (use notation)? What does it mean?
c. Are post hoc tests necessary? If so, what are the results of those analyses?
4. How do your analyses address your hypotheses?
Is concentration of single parent families associated with reading scores?
Using the AECF state data, the regression below measures the effect of the state's percentage of single parent families on the percentage of 4th graders with below basic reading scores.
%belowbasicread = β0 + β1x%SPF + u
Stata Output
1) Please write out the regression equation using the coefficients in the table
2) Please provide an interpretation of the coefficient for SPF
3) How does the model fit?
4) What is the NULL hypothesis for a T test about a regression coefficient?
5) What is the ALTERNATE hypothesis for a T test about a regression coefficient?
6) Look at the p value for the coefficient SPF.
a) Report the p value
b) How many stars would it get if we used our standard convention?
* p ≤ .1 ** p ≤ .05 *** p ≤ .01
image1.png
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Tw ...
Data categories are groupings of data with common characteristics or features. They are useful for managing the data because certain data may be treated differently based on their classification. Understanding the relationship and dependency between the different categories can help direct data quality effort
Regression Analysis presentation by Al Arizmendez and Cathryn LottierAl Arizmendez
We present an overview of regression analysis, theoretical construct, then provide a graphic representation before performing multiple regression analysis step by step using SPSS (audio files accompany the tutorial).
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
Data Science - Part XII - Ridge Regression, LASSO, and Elastic NetsDerek Kane
This lecture provides an overview of some modern regression techniques including a discussion of the bias variance tradeoff for regression errors and the topic of shrinkage estimators. This leads into an overview of ridge regression, LASSO, and elastic nets. These topics will be discussed in detail and we will go through the calibration/diagnostics and then conclude with a practical example highlighting the techniques.
Logistic regression is used to obtain odds ratio in the presence of more than one explanatory variable. The procedure is quite similar to multiple linear regression, with the exception that the response variable is binomial. The result is the impact of each variable on the odds ratio of the observed event of interest.
Descriptive Statistics Formula Sheet Sample Populatio.docxsimonithomas47935
Descriptive Statistics Formula Sheet
Sample Population
Characteristic statistic Parameter
raw scores x, y, . . . . . X, Y, . . . . .
mean (central tendency) M =
∑ x
n
μ =
∑ X
N
range (interval/ratio data) highest minus lowest value highest minus lowest value
deviation (distance from mean) Deviation = (x − M ) Deviation = (X − μ )
average deviation (average
distance from mean)
∑(x − M )
n
= 0
∑(X − μ )
N
sum of the squares (SS)
(computational formula) SS = ∑ x
2 −
(∑ x)2
n
SS = ∑ X2 −
(∑ X)2
N
variance ( average deviation2 or
standard deviation
2
)
(computational formula)
s2 =
∑ x2 −
(∑ x)2
n
n − 1
=
SS
df
σ2 =
∑ X2 −
(∑ X)2
N
N
standard deviation (average
deviation or distance from mean)
(computational formula) s =
√∑ x
2 −
(∑ x)2
n
n − 1
σ =
√∑ X
2 −
(∑ X)2
N
N
Z scores (standard scores)
mean = 0
standard deviation = ± 1.0
Z =
x − M
s
=
deviation
stand. dev.
X = M + Zs
Z =
X − μ
σ
X = μ + Zσ
Area Under the Normal Curve -1s to +1s = 68.3%
-2s to +2s = 95.4%
-3s to +3s = 99.7%
Using Z Score Table for Normal Distribution
(Note: see graph and table in A-23)
for percentiles (proportion or %) below X
for positive Z scores – use body column
for negative Z scores – use tail column
for proportions or percentage above X
for positive Z scores – use tail column
for negative Z scores – use body column
to discover percentage / proportion between two X values
1. Convert each X to Z score
2. Find appropriate area (body or tail) for each Z score
3. Subtract or add areas as appropriate
4. Change area to % (area × 100 = %)
Regression lines
(central tendency line for all
points; used for predictions
only) formula uses raw
scores
b = slope
a = y-intercept
y = bx + a
(plug in x
to predict y)
b =
∑ xy −
(∑ x)(∑ y)
n
∑ x2 −
(∑ x)2
n
a = My - bMx
where My is mean of y
and Mx is mean of x
SEest (measures accuracy of predictions; same properties as standard deviation)
Pearson Correlation Coefficient
(used to measure relationship;
uses Z scores)
r =
∑ xy−
(∑ x)(∑ y)
n
√(∑ x2−
(∑ x)2
n
)(∑ y2−
(∑ y)2
n
)
r =
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦
r
2
= estimate or % of accuracy of predictions
PSYC 2317 Mark W. Tengler, M.S.
Assignment #9
Hypothesis Testing
9.1 Briefly explain in your own words the advantage of using an alpha level (α) = .01
versus an α = .05. In general, what is the disadvantage of using a smaller alpha
level?
9.2 Discuss in your own words the errors that can be made in hypothesis testing.
a. What is a type I error? Why might it occur?
b. What is a type II error? How does it happen?
9.3 The term error is used in two different ways in the context of a hypothesis test.
First, there is the concept of sta
For this assignment, use the aschooltest.sav dataset.The dMerrileeDelvalle969
For this assignment, use the aschooltest.sav dataset.
The dataset consists of Reading, Writing, Math, Science, and Social Studies test scores for 200 students. Demographic data include gender, race, SES, school type, and program type.
Instructions:
Work with the aschooltest.sav datafile and respond to the following questions in a few sentences. Please submit your SPSS output either in your assignment or separately.
1. Identify an Independent and Dependent Variable (of your choice) and develop a hypothesis about what you expect to find. (
note: the IV is a grouping variable, which means it needs to have more than 2 categories and the DV is continuous)
2. Run Assumption tests for Normality and initial Homogeneity of Variance. What are your results?
3. Run the one-way ANOVA with the Levene test & Tukey post hoc test.
a. What are the results of the Levene test? What does this mean?
b. What are the results of the one-way ANOVA (use notation)? What does it mean?
c. Are post hoc tests necessary? If so, what are the results of those analyses?
4. How do your analyses address your hypotheses?
Is concentration of single parent families associated with reading scores?
Using the AECF state data, the regression below measures the effect of the state's percentage of single parent families on the percentage of 4th graders with below basic reading scores.
%belowbasicread = β0 + β1x%SPF + u
Stata Output
1) Please write out the regression equation using the coefficients in the table
2) Please provide an interpretation of the coefficient for SPF
3) How does the model fit?
4) What is the NULL hypothesis for a T test about a regression coefficient?
5) What is the ALTERNATE hypothesis for a T test about a regression coefficient?
6) Look at the p value for the coefficient SPF.
a) Report the p value
b) How many stars would it get if we used our standard convention?
* p ≤ .1 ** p ≤ .05 *** p ≤ .01
image1.png
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Tw ...
Data categories are groupings of data with common characteristics or features. They are useful for managing the data because certain data may be treated differently based on their classification. Understanding the relationship and dependency between the different categories can help direct data quality effort
Regression Analysis presentation by Al Arizmendez and Cathryn LottierAl Arizmendez
We present an overview of regression analysis, theoretical construct, then provide a graphic representation before performing multiple regression analysis step by step using SPSS (audio files accompany the tutorial).
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
What website can I sell pi coins securely.DOT TECH
Currently there are no website or exchange that allow buying or selling of pi coins..
But you can still easily sell pi coins, by reselling it to exchanges/crypto whales interested in holding thousands of pi coins before the mainnet launch.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and resell to these crypto whales and holders of pi..
This is because pi network is not doing any pre-sale. The only way exchanges can get pi is by buying from miners and pi merchants stands in between the miners and the exchanges.
How can I sell my pi coins?
Selling pi coins is really easy, but first you need to migrate to mainnet wallet before you can do that. I will leave the telegram contact of my personal pi merchant to trade with.
Tele-gram.
@Pi_vendor_247
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Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
Even tho Pi network is not listed on any exchange yet.
Buying/Selling or investing in pi network coins is highly possible through the help of vendors. You can buy from vendors[ buy directly from the pi network miners and resell it]. I will leave the telegram contact of my personal vendor.
@Pi_vendor_247
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
If you are looking for a pi coin investor. Then look no further because I have the right one he is a pi vendor (he buy and resell to whales in China). I met him on a crypto conference and ever since I and my friends have sold more than 10k pi coins to him And he bought all and still want more. I will drop his telegram handle below just send him a message.
@Pi_vendor_247
how to sell pi coins in South Korea profitably.DOT TECH
Yes. You can sell your pi network coins in South Korea or any other country, by finding a verified pi merchant
What is a verified pi merchant?
Since pi network is not launched yet on any exchange, the only way you can sell pi coins is by selling to a verified pi merchant, and this is because pi network is not launched yet on any exchange and no pre-sale or ico offerings Is done on pi.
Since there is no pre-sale, the only way exchanges can get pi is by buying from miners. So a pi merchant facilitates these transactions by acting as a bridge for both transactions.
How can i find a pi vendor/merchant?
Well for those who haven't traded with a pi merchant or who don't already have one. I will leave the telegram id of my personal pi merchant who i trade pi with.
Tele gram: @Pi_vendor_247
#pi #sell #nigeria #pinetwork #picoins #sellpi #Nigerian #tradepi #pinetworkcoins #sellmypi
Introduction to Indian Financial System ()Avanish Goel
The financial system of a country is an important tool for economic development of the country, as it helps in creation of wealth by linking savings with investments.
It facilitates the flow of funds form the households (savers) to business firms (investors) to aid in wealth creation and development of both the parties
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1. Multiple Regression Analysis: Statistical Inference: I
Introductory Econometrics: A Modern Approach, 5e
Haoming Liu
National University of Singapore
August 21, 2022
1 . Sampling Distributions of the OLS Estimators
2 . Testing Hypotheses About a Single Population Parameter
3 . Confidence Intervals
4 . Testing Single Linear Restrictions
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 1 / 104
2. Recap
So far, what do we know how to do with the population model
y = β0 + β1x1 + ... + βkxk + u?
1 Mechanics of OLS for a given sample. We only need MLR.2 insofar as
it introduces the data, and MLR.3 (no perfect collinearity) so that the
OLS estimates exist. Interpretation of OLS regression line – ceteris
paribus effects – R2 goodness-of-fit measure. Some functional form
(natural logarithm).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 2 / 104
3. Recap: MLRs
1 : y = β0 + β1x1 + β2x2 + ... + βkxk + u
2 : random sampling from the population
3 : no perfect collinearity in the sample
4 : E(u|x1, ..., xk) = E(u) = 0 (exogenous explanatory variables)
5 : Var(u|x1, ..., xk) = Var(u) = σ2 (homoskedasticity)
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 3 / 104
4. Recap
Unbiasedness of OLS under MLR.1 to MRL.4. Obtain bias (or at
least the direction) when MLR.4 fails due to an omitted variable.
Obtain the variances, Var(β̂j ), under MLR.1 to MLR.5.
The Gauss-Markov Assumptions also imply OLS is the best linear
unbiased estimator (BLUE) (conditional on the values of the
explanatory variables).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 4 / 104
5. Sampling Distributions of the OLS Estimators
We now want to test hypotheses about the βj . This means we hypothesize
that a population parameter is a certain value, then use the data to
determine whether the hypothesis is likely to be false.
EXAMPLE: (Motivated by ATTEND.DTA)
final = β0 + β1missed + β2priGPA + β3ACT + u
where ACT is the achievement test score. The null hypothesis, that
missing lecture has no effect on final exam performance (after accounting
for prior MSU GPA and ACT score), is
H0 : β1 = 0
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 5 / 104
6. Sampling Distributions of the OLS Estimators
To test hypotheses about the βj using exact (or “finite sample”) testing
procedures, we need to know more than just the mean and variance of the
OLS estimators.
MLR.1 to MLR.4: We can compute the expected value as
E(β̂j ) = βj
MLR.1 to MLR.5: We know the variance is
Var(β̂j ) =
σ2
SSTj (1 − R2
j )
And, σ̂2 = SSR/(n − k − 1) is an unbiased estimator of σ2
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 6 / 104
7. Sampling Distributions of the OLS Estimators
But hypothesis testing relies on the entire sampling distributions of
the β̂j . Even under MLR.1 through MLR.5, the sample distributions
can be virtually anything.
Write
β̂j = βj +
n
X
i=1
wij ui ,
where the wij are functions of {(xi1, ..., xik) : i = 1, ..., n}.
Conditional on {(xi1, ..., xik) : i = 1, ..., n}, β̂j inherits its distribution
from that of {ui : i = 1, .., n}, which is a random sample from the
population distribution of u.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 7 / 104
8. Assumption MRL.6 (Normality)
Normality
The population error u is independent of (x1, ..., xk) and is normally
distribution with mean zero and variance σ2:
u ∼ Normal(0, σ2
)
MLR.4: E(u|x1, ..., xk) = E(u) = 0
MLR.5: Var(u|x1, ..., xk) = Var(u) = σ2
Now MLR.6 imposes full independence between u and (x1, x2, ..., xk)
(not just mean and variance independence), which is where the label
of the xj as “independent variables” originated.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 8 / 104
9. The important part of MLR.6 is that we have now made a very
specific distributional assumption for u: the familiar bell-shaped curve:
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 9 / 104
10. Assumption MRL.6 (Normality)
Normality is by far the most common assumption, but the usual
arguments about why normality is a good assumption are not always
operative.
Usually, the argument starts with the claim that u is the sum of many
independent factors, say u = a1 +a2 +...+am for “large” m, and then
we can apply the central limit theorem. But what if the factors have
very different distributions, or are multiplicative rather than additive?
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 10 / 104
11. Assumption MRL.1-6
Ultimately, like Assumption MLR.5, Assumption MLR.6 is maintained
for convenience. Fortunately, we will later see that, for approximate
inference in large samples, we can drop MLR.6. For now we keep it.
It is very difficult to perform exact statistical inference without
Assumption MLR.6.
Assumptions MLR.1 to MLR.6 are called the classical linear model
(CLM) assumptions (for cross-sectional regression).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 11 / 104
12. Normality
For practical purposes, think of
CLM = Gauss-Markov + normality
An important fact about independent normal random variables: any
linear combination is also normally distributed. Because the ui are
independent and identically distributed (iid) as Normal(0, σ2),
β̂j = βj +
n
X
i=1
wij ui ∼ Normal[βj , Var(β̂j )]
where we already know the formula for Var(β̂j ):
Var(β̂j ) =
σ2
SSTj (1 − R2
j )
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 12 / 104
13. THEOREM (Normal Sampling Distributions)
Under the CLM Assumptions (and conditional on the sample outcomes of
the explanatory variables),
β̂j ∼ Normal[βj , Var(β̂j )]
and so
β̂j − βj
sd(β̂j )
∼ Normal(0, 1)
The second result follows from a feature of the normal distribution: if
W ∼ Normal then a + bW ∼ Normal for constants a and b.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 13 / 104
14. Normality
The standardized random variable
β̂j − βj
sd(β̂j )
always has zero mean and variance one. Under MLR.6, it is also
normally distributed.
Notice that the standard normal distribution holds even when we do
not condition on {(xi1, xi2, ..., xik) : i = 1, ..., n}.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 14 / 104
15. Testing Hypotheses About a Single Population Parameter
We cannot directly use the result
β̂j − βj
sd(β̂j )
∼ Normal(0, 1)
to test hypotheses about βj : sd(β̂j ) depends on σ = sd(u), which is
unknown.
But we have σ̂ as an estimator of σ. Using this in place of σ gives us
the standard error, se(β̂j ).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 15 / 104
16. THEOREM (t Distribution for Standardized Estimators)
Under the CLM Assumptions,
β̂j − βj
se(β̂j )
∼ tn−k−1 = tdf
We will not prove this as the argument is somewhat involved.
It is replacing σ (an unknown constant) with σ̂ (an estimator that
varies across samples), that takes us from the standard normal to the
t distribution.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 16 / 104
17. Distribution for Standardized Estimators
The t distribution also has a bell shape, but is more spread out than
the Normal(0, 1).
E(tdf ) = 0 if df > 1
Var(tdf ) =
df
df − 2
> 1 if df > 2
We will never have very small df in this class.
When df = 10, Var(tdf ) = 1.25, which is 25% larger than the
Normal(0, 1) variance.
When df = 120, Var(tdf ) ≈ 1.017 – only 1.7% larger than the
standard normal.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 17 / 104
18. Distribution for Standardized Estimators
As df → ∞,
tdf → Normal(0, 1)
The difference is practically small for df > 120.
The next graph plots a standard normal pdf against a t6 pdf.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 18 / 104
19. Testing
We use the result on the t distribution to test the null hypothesis that
xj has no partial effect on y:
H0 : βj = 0
lwage = β0 + β1educ + β2exper + β3tenure + u
H0 : β2 = 0
In words: Once we control for education and time on the current job
(tenure), total workforce experience has no affect on
lwage = log(wage).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 19 / 104
20. Testing
To test H0 : βj = 0, we use the t statistic (or t ratio),
tβ̂j
=
β̂j
se(β̂j )
This is the estimated coefficient divided by our estimate of β̂j ’s
sampling standard deviation. In virtually all cases β̂j is not exactly
equal to zero. When we use tβ̂j
, we are measuring how far β̂j is from
zero relative to its standard error.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 20 / 104
21. Testing
Because se(β̂j ) > 0, tβ̂j
always has the same sign as β̂j . To use tβ̂j
to
test H0 : βj = 0, we need to have an alternative.
Some like to define tβ̂j
as the absolute value, so it is always positive.
This makes it cumbersome to test against one-sided alternatives.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 21 / 104
22. Testing Against One-Sided Alternatives
First consider the alternative
H1 : βj > 0
which means the null is effectively
H0 : βj ≤ 0
Using a positive one-sided alternative, if we reject βj = 0 than we
reject any βj < 0, too. We often just state H0 : βj = 0 and act like
we do not care about negative values.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 22 / 104
23. Testing Against One-Sided Alternatives
If the estimated coefficient β̂j is negative, it provides no evidence
against H0 in favor of H1 : βj > 0.
If β̂j is positive, the question is: How big does tβ̂j
= β̂j /se(β̂j ) have
to be before we conclude H0 is “unlikely”?
Traditional approach to hypothesis testing:
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 23 / 104
24. Testing Against One-Sided Alternatives
1 . Choose a null hypothesis: H0 : βj = 0 (or H0 : βj ≤ 0)
2 . Choose an alternative hypothesis: H1 : βj > 0
3 . Choose a significance level (or simply level, or size) for the test.
That is, the probability of rejecting the null hypothesis when it is in
fact true. (Type I Error). Suppose we use 5%, so the probability of
committing a Type I error is .05.
4 . Choose a critical value, c > 0, so that the rejection rule
tβ̂j
> c
leads to a 5% level test.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 24 / 104
25. Testing Against One-Sided Alternatives
The key is that, under the null hypothesis,
tβ̂j
∼ tn−k−1 = tdf
and this is what we use to obtain the critical value, c.
Suppose df = 28 and we use a 5% test. The critical value is
c = 1.701, as can be gotten from Table G.2 (page 833 in 5e).
The following picture shows that we are conducting a one-tailed test
(and it is these entries that should be used in the table).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 25 / 104
26. Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 26 / 104
27. Testing Against One-Sided Alternatives
So, with df = 28, the rejection rule for H0 : βj = 0 against
H1 : βj > 0, at the 5% level, is
tβ̂j
> 1.701
We need a t statistic greater than 1.701 to conclude there is enough
evidence against H0.
If tβ̂j
≤ 1.701, we fail to reject H0 against H1 at the 5% significance
level.
Suppose df = 28, but we want to carry out the test at a different
significance level (often 10% level or the 1% level).
c.10 = 1.313
c.05 = 1.701
c.01 = 2.467
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 27 / 104
28. Testing Against One-Sided Alternatives
If we want to reduce the probability of Type I error, we must increase
the critical value (so we reject the null less often).
If we reject at, say, the 1% level, then we must also reject at any
larger level.
If we fail to reject at, say, the 10% level – so that tβ̂j
≤ 1.313 – then
we will fail to reject at any smaller level.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 28 / 104
29. Testing Against One-Sided Alternatives
With large sample sizes – certain when df > 120 – we can use critical
values from the standard normal distribution. These are the df = ∞
entry in Table G.2.
c.10 = 1.282
c.05 = 1.645
c.01 = 2.362
which we can round to 1.28, 1.65, and 2.36, respectively. The value
1.65 is especially common for a one-tailed test.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 29 / 104
30. EXAMPLE: Factors Affecting lwage (WAGE2.DTA)
In applications, it is helpful to label parameters with variable names
to state hypotheses. So βeduc, βIQ, and βexper , for example. Then
H0 : βexper = 0
is that workforce experience has no effect on a wage once education,
and IQ have been accounted for.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 30 / 104
31. EXAMPLE: Factors Affecting lwage (WAGE2.DTA)
lwage = −.229
(.230)
+ .107
(.012)
educ + .0080
(.0016)
IQ + .0435
(.0084)
exper
n = 759, R2
= .217
The quantities in parentheses are still standard errors, not t statistics!
Easiest to read the t statistic off the Stata output, when available:
texper = 5.17,
which is well above the one-sided critical value at the 1% level, 2.36.
In fact, the .5% critical value is about 2.58.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 31 / 104
32. EXAMPLE: Factors Affecting lwage (WAGE2.DTA)
The bottom line is that H0 : βexper = 0 can be rejected against
H1 : βexper > 0 at very small significance levels. A t of 5.17 is very
large.
The estimated effect of exper – that is, its economic importance – is
apparent. Another year of experience, holding educ and IQ fixed, is
estimated to be worth about 4.4%.
The t statistics for educ and IQ are also very large; there is no need
to even look up critical values.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 32 / 104
33. . reg lwage educ IQ exper
Source | SS df MS Number of obs = 759
-------------+------------------------------ F( 3, 755) = 69.78
Model | 57.0352742 3 19.0117581 Prob > F = 0.0000
Residual | 205.71337 755 .27246804 R-squared = 0.2171
-------------+------------------------------ Adj R-squared = 0.2140
Total | 262.748644 758 .346634095 Root MSE = .52198
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ | .1069849 .0116513 9.18 0.000 .084112 .1298578
IQ | .0080269 .0015893 5.05 0.000 .0049068 .0111469
exper | .0435405 .0084242 5.17 0.000 .0270028 .0600783
_cons | -.228922 .2299876 -1.00 0.320 -.6804132 .2225692
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 33 / 104
34. EXAMPLE: Does ACT score help predict college GPA?
In the GPA1.DTA n = 141 MSU students from mid-1990s. All variables
are self reported.
Consider controlling for high school GPA:
colGPA = β0 + β1hsGPA + β2ACT + u
H0 : β2 = 0
From the Stata ouput, β̂2 = β̂ACT = .0094 and tACT = .87. Even at
the 10% level (c = 1.28), we cannot reject H0 against H1 : βACT > 0.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 34 / 104
35. Does ACT score help predict college GPA?
Because we fail to reject H0 : βACT = 0, we say that “β̂ACT is
statistically insignificant at the 10% level against at one-sided
alternative.”
It is also very important to see that the estimated effect of ACT is
small, too. Three more points (slightly more than one standard
deviation) only predicts colGPA that is .0094(3) ≈ .028 – not even
three one-hundreths of a grade point.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 35 / 104
36. Does ACT score help predict college GPA?
By contrast, β̂hsGPA = .453 is large in a practical sense – each point
on hsGPA is associated with about .45 points on colGPA – and
thsGPA = 4.73 is very large.
No critical values in Table G.2 with df = 141 − 3 = 138 are even
close to 4. So “β̂hsGPA is statistically significant” at very small
significance levels.
Notice what happens if we do not control for hsGPA. The simple
regression estimate is .0271 with tACT = 2.49. The magnitude is still
pretty modest, but we would conclude it is statistically different from
zero at the 1% significance level using the standard normal critical
value, 2.36.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 36 / 104
37. Does ACT score help predict college GPA?
Not clear why ACT has such a small, statistically insignificant effect.
The sample size is small and the scores were self-reported. The survey
was done in a couple of economics courses, so it is not a random
sample of all MSU students.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 37 / 104
38. . des colGPA hsGPA ACT
storage display value
variable name type format label variable label
-----------------------------------------------------------------------------
colGPA float %9.0g MSU GPA
hsGPA float %9.0g high school GPA
ACT byte %9.0g ’achievement’ score
. sum colGPA hsGPA ACT
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
colGPA | 141 3.056738 .3723103 2.2 4
hsGPA | 141 3.402128 .3199259 2.4 4
ACT | 141 24.15603 2.844252 16 33
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 38 / 104
39. . reg colGPA hsGPA ACT
Source | SS df MS Number of obs = 141
-------------+------------------------------ F( 2, 138) = 14.78
Model | 3.42365506 2 1.71182753 Prob > F = 0.0000
Residual | 15.9824444 138 .115814814 R-squared = 0.1764
-------------+------------------------------ Adj R-squared = 0.1645
Total | 19.4060994 140 .138614996 Root MSE = .34032
------------------------------------------------------------------------------
colGPA | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hsGPA | .4534559 .0958129 4.73 0.000 .2640047 .6429071
ACT | .009426 .0107772 0.87 0.383 -.0118838 .0307358
_cons | 1.286328 .3408221 3.77 0.000 .612419 1.960237
------------------------------------------------------------------------------
. reg colGPA ACT
Source | SS df MS Number of obs = 141
-------------+------------------------------ F( 1, 139) = 6.21
Model | .829558811 1 .829558811 Prob > F = 0.0139
Residual | 18.5765406 139 .133644177 R-squared = 0.0427
-------------+------------------------------ Adj R-squared = 0.0359
Total | 19.4060994 140 .138614996 Root MSE = .36557
------------------------------------------------------------------------------
colGPA | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ACT | .027064 .0108628 2.49 0.014 .0055862 .0485417
_cons | 2.402979 .2642027 9.10 0.000 1.880604 2.925355
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 39 / 104
40. For the negative one-sided alternative,
H0 : βj < 0,
we use a symmetric rule. But the rejection rule is
tβ̂j
< −c
where c is chosen in the same way as in the positive case.
With df = 18 and a 5% test, the critical value is c = −1.734, so the
rejection rule is
tβ̂j
< −1.734
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 40 / 104
41. Now we must see a significantly negative value for the t statistic to
reject H0 : βj = 0 in favor of H1 : βj < 0.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 41 / 104
42. EXAMPLE: Does missing lectures affect final exam
performance?
final = β0 + β1missed + β2priGPA + β3ACT + u
H0 : β1 = 0, H1 : β1 < 0
We get β̂1 = −.079, tβ̂1
= −2.25. The 5% cv is −1.65 and the 1% cv
is −2.36. So we reject H0 in favor of H1 at the 5% level but not at
the 1% level.
The effect is not huge: 10 missed lectures, out of 32, lowers final
exam score by about .8 points – so not even one point.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 42 / 104
43. . reg final missed priGPA ACT
Source | SS df MS Number of obs = 680
-------------+------------------------------ F( 3, 676) = 56.79
Model | 3032.09408 3 1010.69803 Prob > F = 0.0000
Residual | 12029.853 676 17.7956405 R-squared = 0.2013
-------------+------------------------------ Adj R-squared = 0.1978
Total | 15061.9471 679 22.1825435 Root MSE = 4.2185
------------------------------------------------------------------------------
final | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
missed | -.0793386 .0352349 -2.25 0.025 -.1485216 -.0101556
priGPA | 1.915294 .372614 5.14 0.000 1.183674 2.646914
ACT | .4010639 .0532268 7.54 0.000 .2965542 .5055736
_cons | 12.37304 1.171961 10.56 0.000 10.07192 14.67416
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 43 / 104
44. If we do not control for ACT score, the effect of missed goes away. It turns out that missed and ACT are
positively correlated: those with higher ACT scores miss more classes, on average.
. reg final missed priGPA
Source | SS df MS Number of obs = 680
-------------+------------------------------ F( 2, 677) = 52.48
Model | 2021.72415 2 1010.86207 Prob > F = 0.0000
Residual | 13040.2229 677 19.2617768 R-squared = 0.1342
-------------+------------------------------ Adj R-squared = 0.1317
Total | 15061.9471 679 22.1825435 Root MSE = 4.3888
------------------------------------------------------------------------------
final | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
missed | .0172012 .0341483 0.50 0.615 -.0498481 .0842504
priGPA | 3.237554 .3419779 9.47 0.000 2.56609 3.909019
_cons | 17.41567 1.000942 17.40 0.000 15.45035 19.381
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 44 / 104
45. Reminder about Testing
Our hypthoses involve the unknown population values, βj . If in a our
set of data we obtain, say, β̂j = 2.75, we do not write the null
hypothesis as
H0 : 2.75 = 0
(which is obviously false).
Nor do we write
H0 : β̂j = 0
(which is also false except in the very rare case that our estimate is
exactly zero).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 45 / 104
46. Testing Against Two-Sided Alternatives
We do not test hypotheses about the estimate! We know what it is
once we collect the sample. We hypothesize about the unknown
population value, βj .
Sometimes we do not know ahead of time whether a variable definitely
has a positive effect or a negative effect. Even in the example
final = β0 + β1missed + β2priGPA + β3ACT + u
it is conceivable that missing class helps final exam performance.
(The extra time is used for studying, say.)
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 46 / 104
47. Generally, the null and alternative are
H0 : βj = 0
H1 : βj ̸= 0
Testing against the two-sided alternative is usually the default. It
prevents us from looking at the regression results and then deciding
on the alternative. Also, it is harder to reject H0 against the two-sided
alternative, so it requires more evidence that xj actually affects y.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 47 / 104
48. Two-Sided Alternatives
Now we reject if β̂j is sufficiently large in magnitude, either positive or
negative. We again use the t statistic tβ̂j
= β̂j /se(β̂j ), but now the
rejection rule is
tβ̂j
> c
This results in a two-tailed test, and those are the critical values we
pull from Table G.2.
For example, if we use a 5% level test and df = 25, the two-tailed cv
is 2.06. The two-tailed cv is, in this case, the 97.5 percentile in the
t25 distribution. (Compare the one-tailed cv, about 1.71, the 95th
percentile in the t25 distribution).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 48 / 104
50. EXAMPLE: Factors affecting math pass rates.
(MEAP98.DTA)
Run a multiple regression of math4 on lunch, str, avgsal, enrol.
A priori, we might expect lunch to have a negative effect (it is
essentially a school-level poverty rate), str to have a negative effect,
and avgsal to have a positive effect. But we can still test against a
two-sided alternative to avoid specifying the alternative ahead of time.
enrol is clearly ambiguous.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 50 / 104
51. With 923 observations, we can use the standard normal critical values. For a 10% test it is 1.65, for a 5%,
1.96, and for 1%, cv = 2.58.
. des math4 lunch str avgsal enrol
storage display value
variable name type format label variable label
------------------------------------------------------------------------------
math4 byte %9.0g pass rate, 4th grade math test
lunch float %9.0g % students eligible free lunch
str float %9.0g student-teacher ratio
avgsal float %9.0g average teacher salary
enrol int %9.0g school enrollment
. sum math4 lunch str avgsal enrol
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
math4 | 923 60.54713 19.71111 3 100
lunch | 923 37.34231 26.21696 0 98.78
str | 923 23.50704 3.755936 7.6 41.1
avgsal | 923 47557.53 8577.373 13976 81045
enrol | 923 403.5655 162.6491 18 1176
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 51 / 104
52. . reg math4 lunch str avgsal enrol
Source | SS df MS Number of obs = 923
-------------+------------------------------ F( 4, 918) = 68.82
Model | 82641.3258 4 20660.3315 Prob > F = 0.0000
Residual | 275581.374 918 300.197575 R-squared = 0.2307
-------------+------------------------------ Adj R-squared = 0.2273
Total | 358222.7 922 388.527874 Root MSE = 17.326
------------------------------------------------------------------------------
math4 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lunch | -.2911477 .0237168 -12.28 0.000 -.3376931 -.2446023
str | -.8354922 .1776196 -4.70 0.000 -1.18408 -.4869046
avgsal | .0003744 .000079 4.74 0.000 .0002194 .0005294
enrol | .0050858 .0036523 1.39 0.164 -.002082 .0122537
_cons | 71.20066 4.302933 16.55 0.000 62.75593 79.64539
------------------------------------------------------------------------------
The variables lunch, str, and avgsal all of coefficients with the anticipated signs, and the absolute
values of the t statistics are above 4. So we easily reject H0 : βj = 0 against H1 : βj ̸= 0.
enrol is a different situation. tenroll = 1.39 < 1.65, so we fail to reject H0 at even the 10% signficance
level.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 52 / 104
53. Functional form can make a difference. The math pass rates are capped at 100, so a diminishing effect in
avgsal and enrol seem appropriate; these variables have lots of variation. So use the logarithm instead.
. reg math4 lunch str lavgsal lenrol
Source | SS df MS Number of obs = 923
-------------+------------------------------ F( 4, 918) = 71.09
Model | 84715.9491 4 21178.9873 Prob > F = 0.0000
Residual | 273506.751 918 297.937637 R-squared = 0.2365
-------------+------------------------------ Adj R-squared = 0.2332
Total | 358222.7 922 388.527874 Root MSE = 17.261
------------------------------------------------------------------------------
math4 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lunch | -.2886697 .0235046 -12.28 0.000 -.3347986 -.2425408
str | -.9549563 .1824296 -5.23 0.000 -1.312984 -.5969288
lavgsal | 18.13305 3.605116 5.03 0.000 11.05782 25.20827
lenrol | 2.622179 1.256434 2.09 0.037 .1563616 5.087996
_cons | -116.6793 37.28153 -3.13 0.002 -189.8462 -43.51239
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 53 / 104
54. Of course, all estimates change, but it is those on the lavgsal and
lenrol that are now much different. Before, we were measure a dollar
effect. But now, holding the other variables fixed,
∆
math4 = (18.13/100)%∆avgsal = .1813(%∆avgsal)
So if, say, %∆avgsal = 10 – teacher salaries are 10 percent higher –
math4 is estimated to increase by about 1.8 points.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 54 / 104
55. Also,
∆
math4 = (2.62/100)%∆enroll = .0262(%∆enroll)
so a 10% increase in enrollment is associated with a .26 point increase in
math4.
Notice how lenrol = log(enrol) is statistically significant at the 5% level:
tlenrol = 2.09 > 1.96.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 55 / 104
56. Reminder: When we report the results of, say, the second regression,
it looks like
math4 = −116.68
(37.28)
− .289
(.024)
lunch − .955
(.182)
str + 18.13
(3.61)
lavgsal + 2.62
(1.26)
lenro
n = 903, R2
= .237
so that standard errors are below coefficients.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 56 / 104
57. When we reject H0 : βj = 0 against H1 : βj ̸= 0, we often say that β̂j is
statistically different from zero and usually mention a significance level. For
example, if we can reject at the 1% level, we say that. If we can reject a the
10% level but not the 5%, we say that.
As in the one-sided case, we also say β̂j is “statistically significant” when we
can reject H0 : βj = 0.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 57 / 104
58. Testing Other Hypotheses about the βj
Testing the null H0 : βj = 0 is by far the most common. That is why
Stata and other regression packages automatically report the t
statistic for this hypothesis.
It is critical to remember that
tβ̂j
=
β̂j
se(β̂j )
is only for H0 : βj = 0.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 58 / 104
59. What if we want to test a different null value? For example, in a
constant-elasticity consumption function,
log(cons) = β0 + β1 log(inc) + β2famsize + β3pareduc + u
we might want to test
H0 : β1 = 1
which means an income elasticity equal to one. (We can be pretty
sure that β1 > 0.)
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 59 / 104
60. More generally, suppose the null is
H0 : βj = aj
where we specify the value aj (usually zero, but, in the consumption
example, aj = 1).
It is easy to extend the t statistic:
t =
(β̂j − aj )
se(β̂j )
This t statistic just measures how far our estimate, β̂j , is from the
hypothesized value, aj , relative to se(β̂j ).
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 60 / 104
61. A useful general expression for general t testing:
t =
(estimate − hypothesized value)
standard error
The alternative can be one-sided or two-sided.
We choose critical values in exactly the same way as before.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 61 / 104
62. The language needs to be suitably modified. If, for example,
H0 : βj = 1
H1 : βj ̸= 1
is rejected at the 5% level, we say “β̂j is statistically different from
one at the 5% level.” Otherwise, β̂j is “not statistically different from
one.” If the alternative is H1 : βj > 1, then “β̂j is statistically greater
than one at the 5% level.”
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 62 / 104
63. EXAMPLE: Crime and enrollment on college campuses
(CAMPUS.DTA)
A simple regression model:
log(crime) = β0 + β1 log(enroll) + u
H0 : β1 = 1
H1 : β1 > 1
We get β̂1 = 1.27, and so a 1% increase in enrollment is estimated to
increase crime by 1.27% (so more than 1%). Is this estimate statistically
greater than one?
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 63 / 104
64. Crime and enrollment on college campuses
(CAMPUS.DTA)
We cannot pull the t statistic off of the usual Stata output. We can
compute it by hand (rounding the estimate and standard error):
t =
(1.270 − 1)
.110
≈ 2.45
(Note how this is much smaller than the t for H0 : β1 = 0, reported
by Stata.)
We have df = 97 − 2 = 95, so we use the df = 120 entry in Table
G.2. The 1% cv for a one-sided alternative is about 2.36, so we reject
at the 1% significance level.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 64 / 104
65. . reg lcrime lenroll
Source | SS df MS Number of obs = 97
-------------+------------------------------ F( 1, 95) = 133.79
Model | 107.083654 1 107.083654 Prob > F = 0.0000
Residual | 76.0358244 95 .800377098 R-squared = 0.5848
-------------+------------------------------ Adj R-squared = 0.5804
Total | 183.119479 96 1.90749457 Root MSE = .89464
------------------------------------------------------------------------------
lcrime | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lenroll | 1.26976 .109776 11.57 0.000 1.051827 1.487693
_cons | -6.63137 1.03354 -6.42 0.000 -8.683206 -4.579533
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 65 / 104
66. Alternatively, we can let Stata do the work using the lincom (“linear combination” command). Here the
null is stated equivalent as
H0 : β1 − 1 = 0
. lincom lenroll - 1
( 1) lenroll = 1
------------------------------------------------------------------------------
lcrime | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .2697603 .109776 2.46 0.016 .0518273 .4876932
------------------------------------------------------------------------------
The t = 2.46 is the more accurate calculation of the t statistic.
The lincom lenroll - 1 command is Stata’s way of saying “test whether βlenroll − 1 equals zero.”
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 66 / 104
67. Computing p-Values for t Tests
The traditional approach to testing, where we choose a significance
level ahead of time, can be cumbersome.
Plus, it can conceal information. For example, suppose that, for
testing against a two-sided alternative, a t statistical is just below the
5% cv. I could simply say that “I fail to reject H0 : βj = 0 against the
two-sided alternative at the 5% level.” But there is nothing sacred
about 5%. Might I reject at, say, 6%?
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 67 / 104
68. Computing p-Values for t Tests
Rather than have to specify a level ahead of time, or discuss different
traditional significance levels (10%, 5%, 1%), it is better to answer
the following question: Given the observed value of the t statistic,
what is the smallest significance level at which I can reject H0?
The smallest level at which the null can be rejected is known as the
p-value of a test. It is a single number that automatically allows us to
carry out the test at any level.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 68 / 104
69. One way to think about the p-values is that it uses the observed
statistic as the critical value, and then finds the significance level of
the test using that critical value.
It is most common to report p-values for two-sided alternatives. This
is what Stata does. The t tables are not detailed enough.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 69 / 104
70. For t testing against a two-sided alternative,
p-value = P(|T| > |t|)
where t is the value of the t statistic and T is a random variable with
the tdf distribution.
The p-value is a probability, so it is between zero and one.
Perhaps the best way to think about p-values: it is the probability of
observing a statistic as extreme as we did if the null hypothesis is true.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 70 / 104
71. So smaller p-values provide more are evidence against the null. For
example, if p-value = .50, then there is a 50% chance of observing a
t as large as we did (in absolute value). This is not enough evidence
against H0.
If p-value = .001, then the chance of seeing a t statistic as extreme
as we did is .1%. We can conclude that we got a very rare sample –
which is not helpful – or that the null hypothesis is very likely false.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 71 / 104
72. From
p-value = P(|T| > |t|)
we see that as |t| increases the p-value decreases. Large absolute t
statistics are associated with small p-values.
Suppose df = 40 and, from our data, we obtain t = 1.85 or
t = −1.85. Then
p-value = P(|T| > 1.85) = 2P(T > 1.85) = 2(.0359) = .0718
where T˜t40. Finding the actual numbers required using Stata.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 72 / 104
73. Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 73 / 104
74. Given p-value, we can carry out a test at any significance level. If α is
the chosen level, then
Reject H0 if p-value < α
For example, in the previous example we obtained p-value = .0718.
This means that we reject H0 at the 10% level but not the 5% level.
We reject at 8% but (not quite) at 7%.
Knowing p-value = .0718 is clearly much better than just saying “I
fail to reject at the 5% level.”
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 74 / 104
75. Computing p-Values for One-Sided Alternatives
Stata and other packages report the two-sided p-value. How can we
get a one-sided p-value?
With a caveat, the answer is simple:
one-sided p-value =
two-sided p-value
2
We only want the area in one tail, not two tails. The two-sided
p-value gives us the area in both tails.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 75 / 104
76. This is the correct calculation when it is interesting to do the
calculation. The caveat is simple: if the estimated coefficient is not in
the direction of the alternative, the one-sided p-value is above .50,
and so it is not an interesting calculation.
In Stata, the two-sided p-values for H0 : βj = 0 are given in the
column labeled P |t|.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 76 / 104
77. EXAMPLE: Factors Affecting NBA Salaries
(NBASAL.DTA)
des wage games mingame points rebounds assists
storage display value
variable name type format label variable label
------------------------------------------------------------------------------------------------
wage float %9.0g annual salary, thousands $
games byte %9.0g average games per year
mingame float %9.0g minutes per game
points float %9.0g points per game
rebounds float %9.0g rebounds per game
assists float %9.0g assists per game
. sum wage games mingame points rebounds assists
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
wage | 269 1423.828 999.7741 150 5740
games | 269 65.72491 18.85111 3 82
mingame | 269 23.97925 9.731177 2.888889 43.08537
points | 269 10.21041 5.900667 1.2 29.8
rebounds | 269 4.401115 2.892573 .5 17.3
-------------+--------------------------------------------------------
assists | 269 2.408922 2.092986 0 12.6
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 77 / 104
78. Factors Affecting NBA Salaries (NBASAL.DTA)
Use lwage = log(wage) to get constant percentage effects.
. reg lwage games mingame points rebounds assists
Source | SS df MS Number of obs = 269
-------------+------------------------------ F( 5, 263) = 40.27
Model | 90.2698185 5 18.0539637 Prob > F = 0.0000
Residual | 117.918945 263 .448361006 R-squared = 0.4336
-------------+------------------------------ Adj R-squared = 0.4228
Total | 208.188763 268 .776823743 Root MSE = .6696
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
games | .0004132 .002682 0.15 0.878 -.0048679 .0056942
mingame | .0302278 .0130868 2.31 0.022 .0044597 .055996
points | .0363734 .0150945 2.41 0.017 .0066519 .0660949
rebounds | .0406795 .0229455 1.77 0.077 -.0045007 .0858597
assists | .0003665 .0314393 0.01 0.991 -.0615382 .0622712
_cons | 5.648996 .1559075 36.23 0.000 5.34201 5.955982
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 78 / 104
79. Forgetting the intercept (or “constant”), none of the variables is
statistically significant at the 1% level against a two-sided alternative.
The closest is points, with p-value = .017. (The one-sided p-value is
.017/2 = .0085 < .01, so it is significant at the 1% level against the
positive one-sided alternative.)
mingame is statistically significant a the 5% level because p-value
= .022 < .05.
rebounds is statistically significant a the 10% level (against a
two-sided alternative) because p-value = .077 < .10, but not at the
5% level. But the one-sided p-value is .077/2 = .0385
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 79 / 104
80. Both games and assists have very small t statistics, which lead to
p-values close to one (for example, for assists, p-value = .991).
These variables are statistically insignificant.
In some applications, p-values equal to zero up to three decimal
places are not uncommon. We do not have to worry about statistical
significance in such cases.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 80 / 104
81. Using WAGE2.DTA:
. reg lwage educ IQ exper motheduc
Source | SS df MS Number of obs = 759
-------------+------------------------------ F( 4, 754) = 54.26
Model | 58.7293322 4 14.682333 Prob > F = 0.0000
Residual | 204.019312 754 .270582642 R-squared = 0.2235
-------------+------------------------------ Adj R-squared = 0.2194
Total | 262.748644 758 .346634095 Root MSE = .52018
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ | .1006798 .0118813 8.47 0.000 .0773555 .124004
IQ | .00735 .0016068 4.57 0.000 .0041957 .0105043
exper | .0449386 .0084136 5.34 0.000 .0284217 .0614555
motheduc | .0239265 .0095623 2.50 0.013 .0051545 .0426985
_cons | -.3837064 .2373921 -1.62 0.106 -.8497344 .0823215
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 81 / 104
82. Language of Hypothesis Testing
If we do not reject H0 (against any alternative), it is better to say “we
fail to reject H0” as opposed to “we accept H0,” which is somewhat
common.
The reason is that many null hypotheses cannot be rejected in any
application. For example, if I have β̂j = .75 and se(β̂j ) = .25, I do
not say that I “accept H0 : βj = 1.”
I fail to reject because the t statistic is (.75 − 1)/.25 = −1.
But the t statistic for H0 : βj = .5 is (.75 − .5)/.25 = 1, so I cannot
reject H0 : βj = .5, either.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 82 / 104
83. Clearly βj = .5 and βj = 1 cannot both be true. There is a single,
unknown value in the population. So I should not “accept” either.
The outcomes of the t tests tell us the data cannot reject either
hypothesis. Nor can the data reject H0 : βj = .6, and so on. The data
does reject H0 : βj = 0 (t = 3) at a pretty small significance level (if
we have a reasonable df .)
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 83 / 104
84. Practical versus Statistical Significance
t testing is purely about statistical significance. It does not directly
speak to the issue of whether a variable has a practically, or
economically, large effect.
Practical (Economic) Significance depends on the size (and sign)
of β̂j .
Statistical Significance depends on tβ̂j
.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 84 / 104
85. It is possible estimate practically large effects but have the estimates
so imprecise that they are statistically insignificant. This is especially
an issue with small data sets (but not only small data sets).
Even more importantly, it is possible to get estimates that are
statistically significant – often with very small p-values – but are not
practically large. This can happen with very large data sets.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 85 / 104
86. EXAMPLE
Suppose that, using a large cross section data set for teenagers across
the U.S.,
we estimate the elasticity of alcohol demand with respect to price to
be −.013 with se = .002.
Then the t statistic is −6.5, and we need look no further to conclude
the elasticity is statistically different from zero. But the estimate
means that, say, a 10% increase in the price of alcohol reduces
demand by an estimated .13%. This is a small effect.
The bottom line: do not just fixate on t statistics! Interpreting the β̂j
is just as important.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 86 / 104
87. Confidence Intervals
Rather than just testing hypotheses about parameters it is also useful
to construct confidence intervals (also know as interval estimates)
Loosely, the CI is supposed to give a “likely” range of values for the
corresponding population parameter.
We will only consider CIs of the form
β̂j ± c · se(β̂j )
where c > 0 is chosen based on the confidence level.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 87 / 104
88. We will use a 95% confidence level, in which case c comes from the
97.5 percentile in the tdf distribution. In other words, c is the 5%
critical value against a two-sided alternative.
Stata automatically reports at 95% CI for each parameter, based on
the t distribution using the appropriate df .
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 88 / 104
90. Notice how the three estimates that are not statistically different from
zero at the 5% level – games, rebounds, and assists – all have 95%
CIs that include zero. For example, the 95% CI for βrebounds is
[−.0045, .0859]
By contrast, the 95% CI for βpoints is
[.0067, .0661]
which excludes zero.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 90 / 104
91. A simple rule-of-thumb is useful for constructing a CI given the
estimate and its standard error. For, say, df ≥ 60, an approximate
95% CI is
β̂j ± 2se(β̂j ) or [β̂j − 2se(β̂j ), β̂j + 2se(β̂j )]
That is, subtract and add twice the standard error to the estimate.
(In the case of the standard normal, the 2 becomes 1.96.)
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 91 / 104
92. Properly interpeting a CI is a bit tricky. One often sees statements
such as “there is a 95% chance that βpoints is in the interval
[.0067, .0661].” This is incorrect. βpoints is some fixed value, and it
either is or is not in the interval.
The correct way to interpret a CI is to remember that the endpoints,
β̂j − c · se(β̂j ) and β̂j + c · se(β̂j ), change with each sample (or at
least can change). That is, the endpoints are random outcomes that
depend on the data we draw.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 92 / 104
93. What a 95% CI means is that for 95% of the random samples that we
draw from the population, the interval we compute using the rule
β̂j ± c · se(β̂j ) will include the value βj . But for a particular sample
we do not know whether βj is in the interval.
This is similar to the idea that unbiasedness of β̂j does not means
that β̂j = βj . Most of the time β̂j is not βj . Unbiasedness means
E(β̂j ) = βj .
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 93 / 104
94. CIs and Hypothesis Testing
If we have constructed a 95% CI for, say, βj , we can test any null
value against a two-sided alternative, at the 5% level. So
H0 : βj = aj
H1 : βj ̸= aj
1. If aj is in the 95% CI, then we fail to reject H0 at the 5% level.
2. If aj is not in the 95% CI then we reject H0 in favor of H1 at the
5% level.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 94 / 104
95. Note that, measured as percents,
significance level = 100 − confidence level
. reg lwage educ IQ exper motheduc
Source | SS df MS Number of obs = 759
-------------+------------------------------ F( 4, 754) = 54.26
Model | 58.7293322 4 14.682333 Prob > F = 0.0000
Residual | 204.019312 754 .270582642 R-squared = 0.2235
-------------+------------------------------ Adj R-squared = 0.2194
Total | 262.748644 758 .346634095 Root MSE = .52018
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ | .1006798 .0118813 8.47 0.000 .0773555 .124004
IQ | .00735 .0016068 4.57 0.000 .0041957 .0105043
exper | .0449386 .0084136 5.34 0.000 .0284217 .0614555
motheduc | .0239265 .0095623 2.50 0.013 .0051545 .0426985
_cons | -.3837064 .2373921 -1.62 0.106 -.8497344 .0823215
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 95 / 104
96. The 95% CI for βIQ is about [.0042, .0105]. So we can reject
H0 : βIQ = 0 against the two-sided alternative at the 5% level. We
cannot reject H0 : βIQ = .01 (altough it is close).
We can reject a return to schooling of 7.5% as being too low, but
also 12.5% is too high.
Just as with hypothesis testing, these CIs are only as good as the
underlying assumptions. If we have omitted key variables, the β̂j are
biased. If the error variance is not constant, the standard errors are
improperly computed.
With df = 754, we will see later that normality is not very important.
But normality is needed for these CIs to be eact 95% CIs.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 96 / 104
97. Testing Single Linear Restrictions
So far, we have discussed testing hypotheses that involve only on
parameter, βj . But some hypotheses involve many parameters.
EXAMPLE: Are the Returns to a Year of Junior College the Same as
for a Four-Year University? (COLLEGE.DTA). Sample of high school
graduates.
lwage = β0 + β1jc + β2univ + β3exper + u
H0 : β1 = β2
H1 : β1 < β2
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 97 / 104
98. We could use a two-sided alternative, too.
We can also write
H0 : β1 − β2 = 0
Remember the general way to construct a t statistic:
t =
(estimate − hypothesized value)
standard error
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 98 / 104
99. Given the OLS estimates β̂1 and β̂2,
t =
β̂1 − β̂2
se(β̂1 − β̂2)
Problem: The OLS output gives us β̂1 and β̂2 and their standard
errors, but that is not enough to obtain se(β̂1 − β̂2).
Recall a fact about variances:
Var(β̂1 − β̂2) = Var(β̂1) + Var(β̂2) − 2Cov(β̂1, β̂2)
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 99 / 104
100. The standard error is an estimate of the square root:
se(β̂1 − β̂2) = {[se(β̂1)]2
+ [se(β̂1)]2
− 2s12}1/2
where s12 is an estimate of Cov(β̂1, β̂2). This is the piece we are
missing.
Stata will report s12 if we ask, but calcuting se(β̂1 − β̂2) is
cumbersome. There is also a trick of rewriting the model (see text,
Section 4.4).
These days, it is easiest to use a command for testing linear functions
of the coefficients. In Stata, it is lincom.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 100 / 104
101. . des lwage jc univ exper
storage display value
variable name type format label variable label
-----------------------------------------------------------------------------
lwage float %9.0g log(wage)
jc float %9.0g total 2-year credits
univ float %9.0g total 4-year credits
exper float %8.0g work experience, years
. sum lwage jc univ exper
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
lwage | 750 2.233674 .4906276 .6931472 3.901973
jc | 750 .3449006 .7731012 0 3.833333
univ | 750 1.817076 2.276202 0 7.5
exper | 750 10.26722 2.713302 .25 13.83333
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 101 / 104
102. . reg lwage jc univ exper
Source | SS df MS Number of obs = 750
-------------+------------------------------ F( 3, 746) = 86.25
Model | 46.4300797 3 15.4766932 Prob > F = 0.0000
Residual | 133.86575 746 .179444705 R-squared = 0.2575
-------------+------------------------------ Adj R-squared = 0.2545
Total | 180.295829 749 .240715393 Root MSE = .42361
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
jc | .0661471 .0202773 3.26 0.001 .0263398 .1059544
univ | .0836956 .0068935 12.14 0.000 .0701626 .0972287
exper | .0653706 .0057692 11.33 0.000 .0540448 .0766964
_cons | 1.387603 .0636388 21.80 0.000 1.262671 1.512536
------------------------------------------------------------------------------
Note that β̂jc − β̂univ = .0661 − .0837 = −.0176, so the estimated return to univ is about 1.8% higher.
But is the difference statistically significant?
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 102 / 104
104. . lincom jc - univ
( 1) jc - univ = 0
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -.0175485 .0206407 -0.85 0.395 -.0580694 .0229723
------------------------------------------------------------------------------
The two-sided p-value is .395, which means the one-sided p-value is .1975. Even against a one-sided
alternative, we cannot reject H0 : βjc = βuniv at even the 20% level.
Note how much more variation there is in univ compared with jc.
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 104 / 104
105. Of course, nothing changes (except the sign of the estimate) if we use βuniv − βjc:
. lincom univ - jc
( 1) - jc + univ = 0
------------------------------------------------------------------------------
lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .0175485 .0206407 0.85 0.395 -.0229723 .0580694
------------------------------------------------------------------------------
Liu, H (NUS) Multiple Regression Analysis: Statistical Inference: I August 21, 2022 105 / 104