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The Least Squares Assumptions for Multiple Regression

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Assumptions of Multiple Regression These are the conditions under which the OLS estimator is valid and has the nice statistical properties we rely on (like unbiasedness and consistency). Assumption 1: Zero Conditional Mean E ( u | X 1 = x 1 ,…, X k = x k ) = 0 Meaning: On average, the omitted factors uuu are unrelated to the included regressors XXX. Put differently: Once you control for the regressors, there’s no leftover systematic relationship between uuu and XXX. Why it matters: If this fails, your regression suffers from omitted variable bias . Example: If PctEL (percent English learners) belongs in the model but you leave it out, and it’s correlated with STR, then the STR coefficient gets biased. Solution: Include the omitted variable (if you can measure it). Assumption 2: i.i.d. Sampling   ( X 1 i ,…, X ki ,Y i ), i =1,…, n , are i.i.d. Meaning: Each observation comes from the same population and ...
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What is error term (u) and Omitted Variable Bias in Regression?

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Omitted Variable Bias? What is "error u"? Imagine you want to predict students’ grades (Y) using hours studied (X) . But grades are not decided only by study hours. Some students are naturally smarter. Some had better teachers. Some were sick on exam day. All of these extra things (not included in your formula) are captured in the error term (u) . So u = "all the other factors we didn’t include in the equation." Could you please explain why there are always omitted variables?  In real life, it's impossible to include every single factor that affects Y. Example: For grades, you can’t measure “motivation”, “sleep quality”, or “stress” perfectly. So, some variables will always be omitted. Could you please let me know when this might become a concern If the omitted factors are unrelated to X (study hours), it’s not a big deal. OLS (our method) still works fine. BUT if the omitted factors are related ...
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Commerce Subjects PDF Books

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Hello Everyone I have uploaded Free PDF books related to commerce subjects. Here is the list of available books: 1. Auditing by A.H. Millichamp  2. International Standards on Auditing by IFAC 3. Auditing by Basu 4. Auditing by Depaula  5. Advance Auditing by Prof. Dr. Khawaja Amjad Saeed 6. Practical Auditing by Spicer & Pegler  7. Restall, G. Logic: An Introduction by Tylor & Francis  8. Hurlez, P.J. A consise Introduction to logic  9. Copi, I.M (2002) Introduction to Logic 10. Vaughn, L. The power of Critical Thinking  11. Companies Ordinance, 1984 12. Companies Act, 2017 13. Code of Corporate Governance (for listed companies) Issued by SECP 14. Corporate Governance by Christine A. Mallin                                                        Please Click here to download PDF Books No...
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CSS Past Papers Analysis Current Affairs

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Passing CSS exam is not hard work it's the game of smart work. Many intelligent and hard working students fail this examination in one paper or other because they fail to predict the anticipated questions, and do not prepare accordingly.  Here I have analyzed past papers of Current Affairs subject from year 2016-2024, and I have segregated the CSS IR paper two question topic or theme wise. By reading this post you can easily comprehend that what type of questions repeat each year in CSS current affairs paper, and from which theme or category.    World Affairs CSS Questions on Pakistan-China Relations in Current Affairs paper: Q. No. 5. Discuss the project idea China Pakistan Economic Corridor (CPEC) which is the great leap forward of economic regionalization in the globalized world. Explain its potential advantages, challenges, and future prospects. Q. No. 2. Examine China Strategic Vision behind ‘The Belt and Road Initiative (BRI)’, also known as the One Belt One Road ...
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CSS Past Paper Analysis IR-2

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CSS Past Paper Analysis IR-2 Passing CSS exam is not hard work it's the game of smart work. Many intelligent and hard working students fail this examination in one paper or other because they fail to predict the anticipated questions, and do not prepare accordingly.  Here I have analyzed past papers of International Relations (paper II) from year 2016-2024, and I have segregated the CSS IR paper two question topic or theme wise.  By reading this post you can easily comprehend that what type of questions repeat each year, and from which theme or category.   CSS Questions About Pakistan-India Conflict (Kashmir, Water, Nuclear rivalry) in IR paper-II: Q. No. 6. One of the most serious dilemmas of the South-Asian politics is the contentious relations between Pakistan and India. What measures would you consider for normalization between the two neighbours? (20) Q. No. 2. Evaluate the significance of conflict between India and Pakistan in global perspective of terrorism. (...
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How to Write Research Proposal? Easy Guide

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  Steps in Research proposal writing Research Proposal Contents Title It should be attractive and precise. Ideally the keywords used in title should give idea about theory used, dependent variable of research, geography, industry and sample on which study is conducted.  For-example: "Determinant of Pakistani young-users purchase intention towards green products: An application of Theory of Planned Behavior" Abstract It is usually written without citations. It contains: Background, Purpose, Design or methods which will be used to conduct the study. Keywords (3-5) are mentioned in the last. 1. Introduction 1.1. Research Objectives: Mention two to three clear and concise objectives 1.2. Research Questions: Provide two to three clear research question, this will decide what research Methodology will be used in your research 1.3. Problem Statement: Discuss whether problem is basic (i.e. existing literature is missing something which your trying to cover up) or problem is applie...
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Quotable Quotes For Competitive Exams CSS / PMS

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Quotes for International Relation Subject   “The matter of International Relations is very subtle and exquisite.” -Vladimir Zhirinovsky “Realism can break a writer’s heart.” -Salman Rushdie, Shame “An idealist is one who, on noticing that a rose smells better than a cabbage, concludes that it makes a better soup.” -H.L. Mencken “Power always has to be kept in check; power exercised in secret, especially under the cloak of national security, is doubly dangerous.” -William Proxmire “Domestic policy can only defeat us; foreign policy can kill us.” -John F. Kennedy “No foreign policy – no matter how ingenious – has any chance of success if it is born in the minds of a few and carried in the hearts of the none.” -Henry A. Kissinger “Sovereignty is not given, it is taken.” -Kemal Ataturk “Great is the guilt of unnecessary war.” -John Adams “If everyone fought for their own convictions there would be no war.” -Leo Tolstoy “Mankind must put an end to w...
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Location: Pakistan
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