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
Econometrics notes (Introduction, Simple Linear regression, Multiple linear r...Muhammad Ali
Econometrics notes for BS economics students
Muhammad Ali
Assistant Professor of Statistics
Higher Education Department, KPK, Pakistan.
Email:Mohammadale1979@gmail.com
Cell#+923459990370
Skyp: mohammadali_1979
Brief notes on heteroscedasticity, very helpful for those who are bigners to econometrics. i thought this course to the students of BS economics, these notes include all the necessary proofs.
Heteroscedasticity is the condition which refers to the violation of the Homoscedasticity condition of the linear regression model used in econometrics study. In simple words, it can be described as the situation which leads to increase in the variance of the residual terms with the increase in the fitted value of the variable. Copy the link given below and paste it in new browser window to get more information on Heteroscedasticity:- http://www.transtutors.com/homework-help/economics/heteroscedasticity.aspx
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
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
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
Econometrics notes (Introduction, Simple Linear regression, Multiple linear r...Muhammad Ali
Econometrics notes for BS economics students
Muhammad Ali
Assistant Professor of Statistics
Higher Education Department, KPK, Pakistan.
Email:Mohammadale1979@gmail.com
Cell#+923459990370
Skyp: mohammadali_1979
Brief notes on heteroscedasticity, very helpful for those who are bigners to econometrics. i thought this course to the students of BS economics, these notes include all the necessary proofs.
Heteroscedasticity is the condition which refers to the violation of the Homoscedasticity condition of the linear regression model used in econometrics study. In simple words, it can be described as the situation which leads to increase in the variance of the residual terms with the increase in the fitted value of the variable. Copy the link given below and paste it in new browser window to get more information on Heteroscedasticity:- http://www.transtutors.com/homework-help/economics/heteroscedasticity.aspx
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
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxjeremylockett77
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w.
Chapter 4 part3- Means and Variances of Random Variablesnszakir
Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,
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.
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
how to sell pi coins effectively (from 50 - 100k pi)DOT TECH
Anywhere in the world, including Africa, America, and Europe, you can sell Pi Network Coins online and receive cash through online payment options.
Pi has not yet been launched on any exchange because we are currently using the confined Mainnet. The planned launch date for Pi is June 28, 2026.
Reselling to investors who want to hold until the mainnet launch in 2026 is currently the sole way to sell.
Consequently, right now. All you need to do is select the right pi network provider.
Who is a pi merchant?
An individual who buys coins from miners on the pi network and resells them to investors hoping to hang onto them until the mainnet is launched is known as a pi merchant.
debuts.
I'll provide you the Telegram username
@Pi_vendor_247
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
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
when will pi network coin be available on crypto exchange.DOT TECH
There is no set date for when Pi coins will enter the market.
However, the developers are working hard to get them released as soon as possible.
Once they are available, users will be able to exchange other cryptocurrencies for Pi coins on designated exchanges.
But for now the only way to sell your pi coins is through verified pi vendor.
Here is the telegram contact of my personal pi vendor
@Pi_vendor_247
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
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
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
what is the best method to sell pi coins in 2024DOT TECH
The best way to sell your pi coins safely is trading with an exchange..but since pi is not launched in any exchange, and second option is through a VERIFIED pi merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and pioneers and resell them to Investors looking forward to hold massive amounts before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade pi coins 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.
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
how can I sell my pi coins for cash in a pi APPDOT TECH
You can't sell your pi coins in the pi network app. because it is not listed yet on any exchange.
The only way you can sell is by trading your pi coins with an investor (a person looking forward to hold massive amounts of pi coins before mainnet launch) .
You don't need to meet the investor directly all the trades are done with a pi vendor/merchant (a person that buys the pi coins from miners and resell it to investors)
I Will leave The telegram contact of my personal pi vendor, if you are finding a legitimate one.
@Pi_vendor_247
#pi network
#pi coins
#money
2. 2
What is Heteroskedasticity
Recall the assumption of homoskedasticity
implied that conditional on the explanatory
variables, the variance of the unobserved error, u,
was constant
If this is not true, that is if the variance of u is
different for different values of the x’s, then the
errors are heteroskedastic
Example: estimating returns to education and
ability is unobservable, and think the variance in
ability differs by educational attainment
4. 4
Why Worry About
Heteroskedasticity?
OLS is still unbiased and consistent, even if
we do not assume homoskedasticity
The standard errors of the estimates are
biased if we have heteroskedasticity
If the standard errors are biased, we can not
use the usual t statistics or F statistics or LM
statistics for drawing inferences
5. 5
Variance with Heteroskedasticity
residualsOLStheareareˆwhere,
ˆ
iswhenfor thisestimatorA valid
where,ˆ
so,ˆcase,simpleFor the
2
22
22
i
2
2
22
1
211
i
x
ii
ix
x
ii
i
ii
u
SST
uxx
xxSST
SST
xx
Var
xx
uxx
ss
s
6. 6
Variance with Heteroskedasticity
regressionthisfromresidualssquaredofsumtheis
ands,t variableindependenotherallonregressing
fromresidualtheisˆwhere,
ˆˆ
ˆˆ
isasticityheteroskedwithˆofestimator
validamodel,regressionmultiplegeneralFor the
th
2
2
j
j
ij
j
iij
j
j
SST
x
ir
SST
ur
rVa
Var
7. 7
Robust Standard Errors
Now that we have a consistent estimate of
the variance, the square root can be used as
a standard error for inference
Typically call these robust standard errors
Sometimes the estimated variance is
corrected for degrees of freedom by
multiplying by n/(n – k – 1)
As n → ∞ it’s all the same, though
8. 8
Robust Standard Errors (cont)
Important to remember that these robust
standard errors only have asymptotic
justification – with small sample sizes t
statistics formed with robust standard errors
will not have a distribution close to the t,
and inferences will not be correct
In Stata, robust standard errors are easily
obtained using the robust option of reg
9. 9
A Robust LM Statistic
Run OLS on the restricted model and save the
residuals ŭ
Regress each of the excluded variables on all of
the included variables (q different regressions) and
save each set of residuals ř1, ř2, …, řq
Regress a variable defined to be = 1 on ř1 ŭ, ř2 ŭ,
…, řq ŭ, with no intercept
The LM statistic is n – SSR1, where SSR1 is the
sum of squared residuals from this final regression
10. 10
Testing for Heteroskedasticity
Essentially want to test H0: Var(u|x1, x2,…,
xk) = s2, which is equivalent to H0: E(u2|x1,
x2,…, xk) = E(u2) = s2
If assume the relationship between u2 and xj
will be linear, can test as a linear restriction
So, for u2 = d0 + d1x1 +…+ dk xk + v) this
means testing H0: d1 = d2 = … = dk = 0
11. 11
The Breusch-Pagan Test
Don’t observe the error, but can estimate it with
the residuals from the OLS regression
After regressing the residuals squared on all of the
x’s, can use the R2 to form an F or LM test
The F statistic is just the reported F statistic for
overall significance of the regression, F =
[R2/k]/[(1 – R2)/(n – k – 1)], which is distributed
Fk, n – k - 1
The LM statistic is LM = nR2, which is distributed
c2
k
12. 12
The White Test
The Breusch-Pagan test will detect any
linear forms of heteroskedasticity
The White test allows for nonlinearities by
using squares and crossproducts of all the x’s
Still just using an F or LM to test whether
all the xj, xj
2, and xjxh are jointly significant
This can get to be unwieldy pretty quickly
13. 13
Alternate form of the White test
Consider that the fitted values from OLS, ŷ,
are a function of all the x’s
Thus, ŷ2 will be a function of the squares
and crossproducts and ŷ and ŷ2 can proxy
for all of the xj, xj
2, and xjxh, so
Regress the residuals squared on ŷ and ŷ2
and use the R2 to form an F or LM statistic
Note only testing for 2 restrictions now
14. 14
Weighted Least Squares
While it’s always possible to estimate
robust standard errors for OLS estimates, if
we know something about the specific form
of the heteroskedasticity, we can obtain
more efficient estimates than OLS
The basic idea is going to be to transform
the model into one that has homoskedastic
errors – called weighted least squares
15. 15
Case of form being known up to
a multiplicative constant
Suppose the heteroskedasticity can be
modeled as Var(u|x) = s2h(x), where the
trick is to figure out what h(x) ≡ hi looks like
E(ui/√hi|x) = 0, because hi is only a function
of x, and Var(ui/√hi|x) = s2, because we
know Var(u|x) = s2hi
So, if we divided our whole equation by √hi
we would have a model where the error is
homoskedastic
16. 16
Generalized Least Squares
Estimating the transformed equation by
OLS is an example of generalized least
squares (GLS)
GLS will be BLUE in this case
GLS is a weighted least squares (WLS)
procedure where each squared residual is
weighted by the inverse of Var(ui|xi)
17. 17
Weighted Least Squares
While it is intuitive to see why performing
OLS on a transformed equation is
appropriate, it can be tedious to do the
transformation
Weighted least squares is a way of getting
the same thing, without the transformation
Idea is to minimize the weighted sum of
squares (weighted by 1/hi)
18. 18
More on WLS
WLS is great if we know what Var(ui|xi)
looks like
In most cases, won’t know form of
heteroskedasticity
Example where do is if data is aggregated,
but model is individual level
Want to weight each aggregate observation
by the inverse of the number of individuals
19. 19
Feasible GLS
More typical is the case where you don’t
know the form of the heteroskedasticity
In this case, you need to estimate h(xi)
Typically, we start with the assumption of a
fairly flexible model, such as
Var(u|x) = s2exp(d0 + d1x1 + …+ dkxk)
Since we don’t know the d, must estimate
20. 20
Feasible GLS (continued)
Our assumption implies that u2 = s2exp(d0
+ d1x1 + …+ dkxk)v
Where E(v|x) = 1, then if E(v) = 1
ln(u2) = a0 + d1x1 + …+ dkxk + e
Where E(e) = 1 and e is independent of x
Now, we know that û is an estimate of u, so
we can estimate this by OLS
21. 21
Feasible GLS (continued)
Now, an estimate of h is obtained as ĥ =
exp(ĝ), and the inverse of this is our weight
So, what did we do?
Run the original OLS model, save the
residuals, û, square them and take the log
Regress ln(û2) on all of the independent
variables and get the fitted values, ĝ
Do WLS using 1/exp(ĝ) as the weight
22. 22
WLS Wrapup
When doing F tests with WLS, form the weights
from the unrestricted model and use those weights to
do WLS on the restricted model as well as the
unrestricted model
Remember we are using WLS just for efficiency –
OLS is still unbiased & consistent
Estimates will still be different due to sampling
error, but if they are very different then it’s likely that
some other Gauss-Markov assumption is false