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.
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.
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
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
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
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
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
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
Many mathematical models use a large number of poorly-known parameters as inputs. Quantifying the influence of each of these parameters is one of the aims of sensitivity analysis. Global Sensitivity Analysis is an important paradigm for understanding model behavior, characterizing uncertainty, improving model calibration, etc. Inputs’ uncertainty is modeled by a probability distribution. There exist various measures built in that paradigm. This tutorial focuses on the so-called Sobol’ indices, based on functional variance analysis. Estimation procedures will be presented, and the choice of the designs of experiments these procedures are based on will be discussed. As Sobol’ indices have no clear interpretation in the presence of statistical dependences between inputs, it also seems promising to measure sensitivity with Shapley effects, based on the notion of Shapley value, which is a solution concept in cooperative game theory.
In classical data analysis, data are single values. This is the case if you consider a dataset of n patients which age and size you know. But what if you record the blood pressure or the weight of each patient during a day ? Then, for each patient, you do not have a single-valued data but a set of values since the blood pressure or the weight are not constant during the day.
Suppose now that you do not want to record blood pressure a thousand times for each patient and to store it into a database because your memory space is limited. Therefore, you need to aggregate each set of values into symbols: intervals (lower and upper bounds only), box plots, histograms or even distributions (distribution law with mean and variance)...
Thus, the issue is to adapt classical statistical tools to symbolic data analysis. More precisely, this article is aimed at proposing a method to fit a regression on Gaussian distributions. This paper is divided as follows: first, it presents the computation of the maximum likelihood estimator and then it compares the new approach with the usual least squares regression.
01. Differentiation-Theory & solved example Module-3.pdfRajuSingh806014
Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
A Case Study of Teaching the Concept of Differential in Mathematics Teacher T...theijes
In high schools of Viet Nam, teaching calculus includes the knowledge of the real function with a real variable. A mathematics educator in France, Artigue (1996) has shown that the methods and approximate techniques are the centers of the major problems (including number approximation and function approximation...) in calculus. However, in teaching mathematics in Vietnam, the problems of approximation almost do not appear. With the task of training mathematics teachers in high schools under the new orientations, we present a part of our research with the goal of improving the contents and methods of teacher training
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A Memorandum of Association (MOA) is a legal document that outlines the fundamental principles and objectives upon which a company operates. It serves as the company's charter or constitution and defines the scope of its activities. Here's a detailed note on the MOA:
Contents of Memorandum of Association:
Name Clause: This clause states the name of the company, which should end with words like "Limited" or "Ltd." for a public limited company and "Private Limited" or "Pvt. Ltd." for a private limited company.
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Registered Office Clause: It specifies the location where the company's registered office is situated. This office is where all official communications and notices are sent.
Objective Clause: This clause delineates the main objectives for which the company is formed. It's important to define these objectives clearly, as the company cannot undertake activities beyond those mentioned in this clause.
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Liability Clause: It outlines the extent of liability of the company's members. In the case of companies limited by shares, the liability of members is limited to the amount unpaid on their shares. For companies limited by guarantee, members' liability is limited to the amount they undertake to contribute if the company is wound up.
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Capital Clause: This clause specifies the authorized capital of the company, i.e., the maximum amount of share capital the company is authorized to issue. It also mentions the division of this capital into shares and their respective nominal value.
Association Clause: It simply states that the subscribers wish to form a company and agree to become members of it, in accordance with the terms of the MOA.
Importance of Memorandum of Association:
Legal Requirement: The MOA is a legal requirement for the formation of a company. It must be filed with the Registrar of Companies during the incorporation process.
Constitutional Document: It serves as the company's constitutional document, defining its scope, powers, and limitations.
Protection of Members: It protects the interests of the company's members by clearly defining the objectives and limiting their liability.
External Communication: It provides clarity to external parties, such as investors, creditors, and regulatory authorities, regarding the company's objectives and powers.
https://seribangash.com/difference-public-and-private-company-law/
Binding Authority: The company and its members are bound by the provisions of the MOA. Any action taken beyond its scope may be considered ultra vires (beyond the powers) of the company and therefore void.
Amendment of MOA:
While the MOA lays down the company's fundamental principles, it is not entirely immutable. It can be amended, but only under specific circumstances and in compliance with legal procedures. Amendments typically require shareholder
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"𝐄𝐯𝐞𝐫𝐲 𝐞𝐯𝐞𝐧𝐭 𝐢𝐬 𝐚 𝐬𝐭𝐨𝐫𝐲, 𝐚 𝐬𝐩𝐞𝐜𝐢𝐚𝐥 𝐣𝐨𝐮𝐫𝐧𝐞𝐲. 𝐖𝐞 𝐚𝐥𝐰𝐚𝐲𝐬 𝐛𝐞𝐥𝐢𝐞𝐯𝐞 𝐭𝐡𝐚𝐭 𝐬𝐡𝐨𝐫𝐭𝐥𝐲 𝐲𝐨𝐮 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐚 𝐩𝐚𝐫𝐭 𝐨𝐟 𝐨𝐮𝐫 𝐬𝐭𝐨𝐫𝐢𝐞𝐬."
Unveiling the Secrets How Does Generative AI Work.pdfSam H
At its core, generative artificial intelligence relies on the concept of generative models, which serve as engines that churn out entirely new data resembling their training data. It is like a sculptor who has studied so many forms found in nature and then uses this knowledge to create sculptures from his imagination that have never been seen before anywhere else. If taken to cyberspace, gans work almost the same way.
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India Orthopedic Devices Market: Unlocking Growth Secrets, Trends and Develop...Kumar Satyam
According to TechSci Research report, “India Orthopedic Devices Market -Industry Size, Share, Trends, Competition Forecast & Opportunities, 2030”, the India Orthopedic Devices Market stood at USD 1,280.54 Million in 2024 and is anticipated to grow with a CAGR of 7.84% in the forecast period, 2026-2030F. The India Orthopedic Devices Market is being driven by several factors. The most prominent ones include an increase in the elderly population, who are more prone to orthopedic conditions such as osteoporosis and arthritis. Moreover, the rise in sports injuries and road accidents are also contributing to the demand for orthopedic devices. Advances in technology and the introduction of innovative implants and prosthetics have further propelled the market growth. Additionally, government initiatives aimed at improving healthcare infrastructure and the increasing prevalence of lifestyle diseases have led to an upward trend in orthopedic surgeries, thereby fueling the market demand for these devices.
Business Valuation Principles for EntrepreneursBen Wann
This insightful presentation is designed to equip entrepreneurs with the essential knowledge and tools needed to accurately value their businesses. Understanding business valuation is crucial for making informed decisions, whether you're seeking investment, planning to sell, or simply want to gauge your company's worth.
Cracking the Workplace Discipline Code Main.pptxWorkforce Group
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A personal brand exploration presentation summarizes an individual's unique qualities and goals, covering strengths, values, passions, and target audience. It helps individuals understand what makes them stand out, their desired image, and how they aim to achieve it.
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Dummy variable
1. What is dummy variable?
Qualitative variable usually indicates the presence and absence of quality or an
attribute such as male and female, black and white, democrat and republican. If the
qualitative variables takes only two values 0 and 1 (absence or presence) then the
variable is called dummy variable. Example: suppose a qualitative variable sex
indicates the presence or absence of attribute such as male or female.
“1” may indicate that a person is a male and “0” may indicate that a person is
female. Variables that assume that “0” and “1” values are called dummy variables.
Alternative name of dummy variable
-indicator variable
-binary variable
-qualitative variable
-categorical variable
-dichotomous variable
Explain dummy variables in term of model or ANOVA model.
Dummy variables can be used in regression model just as easily as qualitative
variables. As a matter of fact that a linear regression model may contain
explanatory variables that are exclusively dummy or qualitative in nature. Such
model are called analysis of variance model or ANOVA model.
Let us consider the following modelYi=α+βDi+µi …………. (i)
where Yi= annual salary of a college professor
1 𝑖𝑓 𝑚𝑎𝑙𝑒 𝑝𝑟𝑜𝑓𝑒𝑠𝑠𝑜𝑟
𝐷𝑖 = {
0 𝑖𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 𝑝𝑟𝑜𝑓𝑒𝑠𝑠𝑜𝑟
2. Di is called dummy variable
µi ~ NID (0, σ²)
we get from equation (i)
Mean salary of female professor,
E(Yi│Di=o)= α
Mean salary of male professor,
E(Yi│Di=1)= α+β
Interpretation:
Here the intercept term α gives the mean salary of female college
professor. The slope coefficient β tells by how much the mean salary of male
professor differs from the mean salary of his female counter part.
α+β reflecting the mean salary of college professor.
Write down the advantages of dummy variables.
1. Dummy variables are data classifying device that is they divide a sample into
various subgroups based on qualitative or attributes.
2. If a model has several qualitative variables with several classes introduction of
dummy variables can consume a large number of d.f.
3. Since the dummy variable are non-stochastic they create no special problems
in the application of OLS.
What is a dummy variable trap? How will you avoid dummy variable
trap?
Let us consider a modelYi= α1+ α2D2i+ α3D3i+βXi+µi ………………… (i)
Here Yi are the annual salary of a college professor.
Xi is the years of teaching experience of college professor
D2i = {
1 𝑖𝑓 𝑚𝑎𝑙𝑒 𝑝𝑟𝑜𝑓𝑒𝑠𝑠𝑜𝑟
0 𝑖𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 𝑝𝑟𝑜𝑓𝑒𝑠𝑠𝑜𝑟
D3i = {
1 𝑖𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 𝑝𝑟𝑜𝑓𝑒𝑠𝑠𝑜𝑟
0 𝑖𝑓 𝑚𝑎𝑙𝑒 𝑝𝑟𝑜𝑓𝑒𝑠𝑠𝑜𝑟
The model (i) cannot be estimated because of perfect collinearity between D2 and
D3. To see this we have a sample of 3 male professors and 2 female professors.
The design matrix is-
3. Male
Male
Female
Male
Female
Y1
Y2
Y3
Y4
Y5
α1
1
1
1
1
1
D2
1
1
0
1
0
D3
0
0
1
0
1
X
X1
X2
X3
X4
X5
The first column denote the common intercept term α1. We see that,
D2 =1-D3 and D3 =1-D2.
That means, D2 and D3 are perfectly collinear. Thus avoiding the perfect
collinearity the general rule is if a qualitative variable has m categories then it has
only (m-1) dummy variables. If this rule is not followed we shall fall into dummy
variable trap.
To avoid the dummy variable trap we can write the above model asYi= α2D2i+ α3D3i+βXi+µi
In this mode we have drop the intercept term αi. If we drop the intercept term αi we
will not fall into perfect multicollinearity/the dummy variable trap because we
have no longer the perfect collinearity.
Comparing two regression lines in terms dummy variable approach
Let us consider, pool all n1 and n2 observations together and estimating the
following regressionYi= α1+ α2Di+ β1Xi+ β2DiXi +µi …………………….(i)
Where, Yi and Xi are savings and income and
Di = {
1 𝑓𝑜𝑟 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 1𝑠𝑡
0 𝑓𝑜𝑟 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑛𝑒𝑥𝑡
To see the implication of model (i) and assuming that, E(µi)=0 we obtain
E (Yi│Di=0; Xi) = αi+β1Xi ………….. (ii)
E (Yi│Di=1; Xi) = (α1+ α2) + (β1+ β2)Xi……………… (iii)
4. Let,
α1 = γ1
β1 = γ 2
α1+ α2= λ1
β1+ β2= λ2
So the equation of (ii) and (iii) is,
E (Yi│Di=0; Xi) = γ1+ γ2Xi…………….. (iv)
E (Yi│Di=1; Xi) = λ1+ λ2Xi……………... (v)
Therefore estimating equation (i) is equivalent to estimating the two individuals,
Re-construction period and post-reconstruction period. Where in equation (i) α1 is
the differential intercept term and α2 is the differential slope coefficient.
Find out the aggregate saving income relationship has changed between
the two periods.
Let us consider two linear regression model are, Re-construction period
Yi= λ1+ λ2Xi+ µ1i………………….(i)
i=1,2,…,ni
Post-construction period,
Yi= γ1+ γ2Xi+ µ2i………………….(i)
i=1,2,…,n2i
where, Yi= savings
X= income
µ1i and µ2i are the disturbance term in the two regression model.
Now regression model (i) and (ii) present the following four possibility
1) If λ1= γ1 and λ2= γ2 that means, the two regression model are identical then
it is called coincident regression.
Y
λ2= γ2
λ1= γ1
X
Income
5. (a) Coincident
2) If λ1≠γ1 and λ2= γ2 that means the two regression differ only in their
locations that means intercept then it is called parallel regression.
Y
λ2= γ2
λ2= γ2
γ1
λ1
X
(b) Coincident
3) If λ1=γ1 and λ2≠ γ2 that means the two regression have same intercept
different slopes. Then it is called concurrent regression
Y
γ2
λ2
λ1= γ1
6. X
(c) concurrent
4) If λ1≠γ1 and λ2≠ γ2 then the two regression equation are completely
different that means the regression is called dissimilar regression.
Y
γ2
λ2
λ1
γ1
X
(d) dissimilar
Question: Suppose the college professor salary regression model defined asYi= α1+ α2D2i+ α3D3i + α4(D2iD3i)+BXi +µi
Where Yi=annual salary of a college professor
Xi= years of experience
D2=
1 if male professor
0 if female professor
D3= 1 if the professor is white or 0 otherwise
Explain the terms (i) α2 (ii) α4 (iii) D2iD3i
(v) What about the effect of female and non-white professor
(vi) Find,
7. E(Yi│D2=1, D3=1,Xi=10) and interpret it.
Solution:
1. α2 is the differential effect of being male professor
2. α4 is the differential effect of male-white professor
3. D2iD3i be the interaction between two qualitative variables D2 and D3. It
means non-white have lower mean salary i. e they are male or female. A
female non-white may earn lower salary than a male non-white. So
interaction may be expressed such kind of assumption which may be
untrainable
4. The effect of female and non-white professor are the followingE(Yi│D2i=0, D3i=0)= α1+βXi
5. So it can be concluded that the mean salary depends on only the slope
coefficient and the coefficient of years of experience.
6.
E(Yi│D2=1, D3=1,Xi=10)
So the mean salary of male and white professor is which is the mean salary of
male and white professor when years of experience are 10 years.