DEC
02
2020

# ols blue assumptions

These assumptions are presented in Key Concept 6.4. dependent on X’s), then the linear regression model has heteroscedastic errors and likely to give incorrect estimates. More the variability in X's, better are the OLS estimates in determining the impact of X's on Y. OLS Assumption 5: Spherical errors: There is homoscedasticity and no autocorrelation. Properties of the O.L.S. This assumption of OLS regression says that: OLS Assumption 3: The conditional mean should be zero. If this variance is not constant (i.e. For example, consider the following: A1. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. Thank you for your patience! If a number of parameters to be estimated (unknowns) equal the number of observations, then OLS is not required. a)quad Y={ beta }_{ 0 }+{ beta }_{ 1 }{ X }_{ 1 }+{ beta }_{ 2 }{ X }_{ 2 }+varepsilon, b)quad Y={ beta }_{ 0 }+{ beta }_{ 1 }{ X }_{ { 1 }^{ 2 } }+{ beta }_{ 2 }{ X }_{ 2 }+varepsilon, c)quad Y={ beta }_{ 0 }+{ beta }_{ { 1 }^{ 2 } }{ X }_{ 1 }+{ beta }_{ 2 }{ X }_{ 2 }+varepsilon. How to Find Authentic Texts Online when Preparing for the AP® French Exam, How to Calculate Medians: AP® Statistics Review. The sample taken for the linear regression model must be drawn randomly from the population. Linear regression models have several applications in real life. OLS assumptions 1, 2, and 4 are necessary for the setup of the OLS problem and its derivation. This assumption states that the errors are normally distributed, conditional upon the independent variables. Why BLUE : We have discussed Minimum Variance Unbiased Estimator (MVUE) in one of the previous articles. Having said that, many times these OLS assumptions will be violated. If the OLS assumptions 1 to 5 hold, then according to Gauss-Markov Theorem, OLS estimator is Best Linear Unbiased Estimator (BLUE). Estimator 3. These should be linear, so having Î² 2 {\displaystyle \beta ^{2}} or e Î² {\displaystyle e^{\beta }} would violate this assumption.The relationship between Y and X requires that the dependent variable (y) is a linear combination of explanatory variables and error terms. Now, if you run a regression with dependent variable as exam score/performance and independent variables as time spent sleeping, time spent studying, and time spent playing, then this assumption will not hold. Gauss Markov theorem. In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. between the two variables. 5. In other words, the distribution of error terms has zero mean and doesn’t depend on the independent variables X's. The OLS Assumptions. For example, if you run the regression with inflation as your dependent variable and unemployment as the independent variable, the. Therefore, it is an essential step to analyze various statistics revealed by OLS. IntroductionAssumptions of OLS regressionGauss-Markov TheoremInterpreting the coe cientsSome useful numbersA Monte-Carlo simulationModel Speci cation Assumptions of OLS regression Assumption 1: The regression model is linear in the parameters. Learn how your comment data is processed. Linear regression models are extremely useful and have a wide range of applications. This makes the dependent variable random. The Gauss-Markov Theorem is telling us that in a â¦ Share this: In such a situation, it is better to drop one of the three independent variables from the linear regression model. The OLS assumption of no multi-collinearity says that there should be no linear relationship between the independent variables. 1. Spherical errors: There is homoscedasticity and no autocorrelation. We will not go into the details of assumptions 1-3 since their ideas generalize easy to the case of multiple regressors. For c) OLS assumption 1 is not satisfied because it is not linear in parameter { beta }_{ 1 }. The dependent variable Y need not be normally distributed. OLS estimators minimize the sum of the squared errors (a difference between observed values and predicted values). Learn more about our school licenses here. The model must be linear in the parameters.The parameters are the coefficients on the independent variables, like Î± {\displaystyle \alpha } and Î² {\displaystyle \beta } . The conditional mean should be zero.A4. Consider the linear regression model where the outputs are denoted by , the associated vectors of inputs are denoted by , the vector of regression coefficients is denoted by and are unobservable error terms. Meaning, if the standard GM assumptions hold, of all linear unbiased estimators possible the OLS estimator is the one with minimum variance and is, therefore, most efficient. However, below the focus is on the importance of OLS assumptions by discussing what happens when they fail and how can you look out for potential errors when assumptions are not outlined. Check 2. runs.test ... (not OLS) is used to compute the estimates, this also implies the Y and the Xs are also normally distributed. by Marco Taboga, PhD. Ordinary Least Squares is a method where the solution finds all the Î²Ì coefficients which minimize the sum of squares of the residuals, i.e. yearly data of unemployment), then the regression is likely to suffer from autocorrelation because unemployment next year will certainly be dependent on unemployment this year. However, in the case of multiple linear regression models, there are more than one independent variable. The dependent variable is assumed to be a â¦ If the OLS assumptions 1 to 5 hold, then according to Gauss-Markov Theorem, OLS estimator is Best Linear Unbiased Estimator (BLUE). There is a random sampling of observations.A3. Model is linear in parameters 2. OLS assumptions are extremely important. The expected value of the mean of the error terms of OLS regression should be zero given the values of independent variables. Hence, error terms in different observations will surely be correlated with each other. This is because a lack of knowledge of OLS assumptions would result in its misuse and give incorrect results for the econometrics test completed. For example, a multi-national corporation wanting to identify factors that can affect the sales of its product can run a linear regression to find out which factors are important. Hence, this OLS assumption says that you should select independent variables that are not correlated with each other. In simple terms, this OLS assumption means that the error terms should be IID (Independent and Identically Distributed). This video details the first half of the Gauss-Markov assumptions, which are necessary for OLS estimators to be BLUE. Gauss-Markov Assumptions, Full Ideal Conditions of OLS The full ideal conditions consist of a collection of assumptions about the true regression model and the data generating process and can be thought of as a description of an ideal data set. In the multiple regression model we extend the three least squares assumptions of the simple regression model (see Chapter 4) and add a fourth assumption. Proof under standard GM assumptions the OLS estimator is the BLUE estimator. This makes sense mathematically too. The OLS estimator is the vector of regression coefficients that minimizes the sum of squared residuals: As proved in the lecture entitled Linear regresâ¦ The multiple regression model is the study if the relationship between a dependent variable and one or more independent variables. OLS assumptions are extremely important. Following points should be considered when applying MVUE to an estimation problem MVUE is the optimal estimator Finding a MVUE requires full knowledge of PDF (Probability Density Function) of the underlying process. This chapter is devoted to explaining these points. Do you believe you can reliably run an OLS regression? Key Concept 5.5 The Gauss-Markov Theorem for $$\hat{\beta}_1$$. The following post will give a short introduction about the underlying assumptions of the classical linear regression model (OLS assumptions), which we derived in the following post.Given the Gauss-Markov Theorem we know that the least squares estimator and are unbiased and have minimum variance among all unbiased linear estimators. The assumption of no perfect collinearity allows one to solve for first order conditions in the derivation of OLS estimates. Note that only the error terms need to be normally distributed. Y = 1 + 2X i + u i. Under certain conditions, the Gauss Markov Theorem assures us that through the Ordinary Least Squares (OLS) method of estimating parameters, our regression coefficients are the Best Linear Unbiased Estimates, or BLUE (Wooldridge 101). Components of this theorem need further explanation. That is, it proves that in case one fulfills the Gauss-Markov assumptions, OLS is BLUE. In a simple linear regression model, there is only one independent variable and hence, by default, this assumption will hold true. The data are a random sample of the population 1. The above diagram shows the difference between Homoscedasticity and Heteroscedasticity. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameter of a linear regression model. The next section describes the assumptions of OLS regression. This above model is a very simple example, so instead consider the more realistic multiple linear regression case where the goal is to find beta parameters as follows:yÌ = Î²Ì0 + Î²Ì1x1 + Î²Ì2x2 + ... + Î²ÌpxpHow does the model figure out what Î²Ì parameters to use as estimates? A4. Mathematically, Covleft( { { varepsilon }_{ i }{ varepsilon }_{ j } }|{ X } right) =0enspace forenspace ineq j. While OLS is computationally feasible and can be easily used while doing any econometrics test, it is important to know the underlying assumptions of OLS regression. The Seven Classical OLS Assumption. This is because there is perfect collinearity between the three independent variables. The First OLS Assumption Thank you for your patience! When the dependent variable (Y) is a linear function of independent variables (X's) and the error term, the regression is linear in parameters and not necessarily linear in X's. The Gauss-Markov theorem famously states that OLS is BLUE. OLS Assumption 6: Error terms should be normally distributed. Ordinary Least Squares is the most common estimation method for linear modelsâand thatâs true for a good reason.As long as your model satisfies the OLS assumptions for linear regression, you can rest easy knowing that youâre getting the best possible estimates.. Regression is a powerful analysis that can analyze multiple variables simultaneously to answer complex research questions. If you want to get a visual sense of how OLS works, please check out this interactive site. The error terms are random. You should know all of them and consider them before you perform regression analysis.. Assumptions in the Linear Regression Model 2. However, that should not stop you from conducting your econometric test. For more information about the implications of this theorem on OLS estimates, read my post: The Gauss-Markov Theorem and BLUE OLS Coefficient Estimates. We are gradually updating these posts and will remove this disclaimer when this post is updated. These are desirable properties of OLS estimators and require separate discussion in detail. A6: Optional Assumption: Error terms should be normally distributed. Ideal conditions have to be met in order for OLS to be a good estimate (BLUE, unbiased and efficient) Are you a teacher or administrator interested in boosting AP® Biology student outcomes? The linear regression model is “linear in parameters.”. The independent variables are not too strongly collinear 5. Instead, the assumptions of the GaussâMarkov theorem are stated conditional on . However, below the focus is on the importance of OLS assumptions by discussing what happens when they fail and how can you look out for potential errors when assumptions are not outlined. Rather, when the assumption is violated, applying the correct fixes and then running the linear regression model should be the way out for a reliable econometric test. Save my name, email, and website in this browser for the next time I comment. However, if these underlying assumptions are violated, there are undesirable implications to the usage of OLS. Unlike the acf plot of lmMod, the correlation values drop below the dashed blue line from lag1 itself. There is no multi-collinearity (or perfect collinearity). This OLS assumption is not required for the validity of OLS method; however, it becomes important when one needs to define some additional finite-sample properties. The number of observations taken in the sample for making the linear regression model should be greater than the number of parameters to be estimated. Inference in the Linear Regression Model 4. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. OLS Assumption 2: There is a random sampling of observations. The expected value of the errors is always zero 4. Varleft( { varepsilon }|{ X } right) ={ sigma }^{ 2 }, Covleft( { { varepsilon }_{ i }{ varepsilon }_{ j } }|{ X } right) =0enspace forenspace ineq j. Albert.io lets you customize your learning experience to target practice where you need the most help. Under the GM assumptions, the OLS estimator is the BLUE (Best Linear Unbiased Estimator). This OLS assumption of no autocorrelation says that the error terms of different observations should not be correlated with each other. The theorem now states that the OLS estimator is a BLUE. In order to use OLS correctly, you need to meet the six OLS assumptions regarding the data and the errors of your resulting model. Mathematically, Varleft( { varepsilon }|{ X } right) ={ sigma }^{ 2 }. The Gauss Markov theorem says that, under certain conditions, the ordinary least squares (OLS) estimator of the coefficients of a linear regression model is the best linear unbiased estimator (BLUE), that is, the estimator that has the smallest variance among those that are unbiased and linear in the observed output variables. Assumptions of Linear Regression. Even if the PDF is known, [â¦] We are gradually updating these posts and will remove this disclaimer when this post is updated. Albert.io lets you customize your learning experience to target practice where you need the most help. We’ll give you challenging practice questions to help you achieve mastery of Econometrics. The importance of OLS assumptions cannot be overemphasized. ... (BLUE). Random sampling, observations being greater than the number of parameters, and regression being linear in parameters are all part of the setup of OLS regression. OLS is the basis for most linear and multiple linear regression models. If a number of parameters to be estimated (unknowns) are more than the number of observations, then estimation is not possible. are likely to be incorrect because with inflation and unemployment, we expect correlation rather than a causal relationship. The errors are statistically independent from one another 3. This is sometimes just written as Eleft( { varepsilon } right) =0. Linear Regression Models, OLS, Assumptions and Properties 2.1 The Linear Regression Model The linear regression model is the single most useful tool in the econometricianâs kit. Time spent sleeping = 24 – Time spent studying – Time spent playing. In order for OLS to be BLUE one needs to fulfill assumptions 1 to 4 of the assumptions of the classical linear regression model. Attention: This post was written a few years ago and may not reflect the latest changes in the AP® program. For example, when we have time series data (e.g. The following website provides the mathematical proof of the Gauss-Markov Theorem. In the above three examples, for a) and b) OLS assumption 1 is satisfied. In addition, the OLS estimator is no longer BLUE. These are desirable properties of OLS estimators and require separate discussion in detail. ols-assumptions Assumptions Required for OLS to be Unbiased Assumption M1: The model is linear in the parameters Assumption M2: The data are collected through independent, random sampling Assumption M3: The data are not perfectly multicollinear. We’ll give you challenging practice questions to help you achieve mastery of Econometrics. The first component is the linear component. Linear regression models find several uses in real-life problems. So, the time has come to introduce the OLS assumptions.In this tutorial, we divide them into 5 assumptions. However, the ordinary least squares method is simple, yet powerful enough for many, if not most linear problems.. Inference on Prediction CHAPTER 2: Assumptions and Properties of Ordinary Least Squares, and Inference in the Linear Regression Model Prof. Alan Wan 1/57 Let us know in the comment section below! A2. There is a random sampling of observations. With Assumptions (B), the BLUE is given conditionally on Let us use Assumptions (A). 1. Thus, there must be no relationship between the X's and the error term. You can simply use algebra. 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If the relationship (correlation) between independent variables is strong (but not exactly perfect), it still causes problems in OLS estimators. We assume to observe a sample of realizations, so that the vector of all outputs is an vector, the design matrixis an matrix, and the vector of error termsis an vector. The necessary OLS assumptions, which are used to derive the OLS estimators in linear regression models, are discussed below. For example, suppose you spend your 24 hours in a day on three things – sleeping, studying, or playing. Mathematically, Eleft( { varepsilon }|{ X } right) =0. For example, if you have to run a regression model to study the factors that impact the scores of students in the final exam, then you must select students randomly from the university during your data collection process, rather than adopting a convenient sampling procedure. The fact that OLS estimator is still BLUE even if assumption 5 is violated derives from the central limit theorem, ... Assumptions of Classical Linear Regressionmodels (CLRM) Overview of all CLRM Assumptions Assumption 1 Assumption 2 Assumption 3 Assumption 4 Assumption 5. The linear regression model is âlinear in parameters.âA2. LEAST squares linear regression (also known as âleast squared errors regressionâ, âordinary least squaresâ, âOLSâ, or often just âleast squaresâ), is one of the most basic and most commonly used prediction techniques known to humankind, with applications in fields as diverse as statistics, finance, medicine, economics, and psychology. When you use them, be careful that all the assumptions of OLS regression are satisfied while doing an econometrics test so that your efforts don’t go wasted. Given the assumptions A â E, the OLS estimator is the Best Linear Unbiased Estimator (BLUE). Linearity. Assumptions of OLS regression 1. The independent variables are measured precisely 6. You can find thousands of practice questions on Albert.io. These assumptions are extremely important, and one cannot just neglect them. OLS Assumption 4: There is no multi-collinearity (or perfect collinearity). Like many statistical analyses, ordinary least squares (OLS) regression has underlying assumptions. This does not mean that Y and X are linear, but rather that 1 and 2 are linear. Assumptions (B) E(If we use Assumptions (B), we need to use the law of iterated expectations in proving the BLUE. OLS Assumption 1: The linear regression model is “linear in parameters.”. A5. According to this OLS assumption, the error terms in the regression should all have the same variance. The variance of errors is constant in case of homoscedasticity while it’s not the case if errors are heteroscedastic. Analysis of Variance, Goodness of Fit and the F test 5. But, often people tend to ignore the assumptions of OLS before interpreting the results of it. Suppose that the assumptions made in Key Concept 4.3 hold and that the errors are homoskedastic.The OLS estimator is the best (in the sense of smallest variance) linear conditionally unbiased estimator (BLUE) in this setting. So autocorrelation canât be confirmed. An important implication of this assumption of OLS regression is that there should be sufficient variation in the X's. If the form of the heteroskedasticity is known, it can be corrected (via appropriate transformation of the data) and the resulting estimator, generalized least squares (GLS), can be shown to be BLUE. BLUE is an acronym for the following:Best Linear Unbiased EstimatorIn this context, the definition of âbestâ refers to the minimum variance or the narrowest sampling distribution. More specifically, when your model satisfies the assumptions, OLS coefficient estimates follow the tightest possible sampling distribution of unbiased estimates compared to other linear estimation methods.Letâs dig deeper into everything that is packed iâ¦ Most linear and multiple linear regression model 6: error terms need to be normally distributed this when., ols blue assumptions is BLUE order for OLS estimators and require separate discussion detail! Regression model you believe you can reliably run an OLS regression says that the errors are.... Are extremely important, and one can not be correlated with each other assumed to be because! Then estimation is not linear in parameters. ” drop below the dashed BLUE line from lag1 itself is no (. Wide range of applications squares ( OLS ) method is widely used to estimate the of! Post is updated better to drop one of the classical linear regression models, there are undesirable to. Be incorrect because with inflation as your dependent variable Y need not be overemphasized three things sleeping! Given conditionally on Let us use assumptions ( a ) 2X i u! C ) OLS assumption 4: there is homoscedasticity and Heteroscedasticity this is because there is homoscedasticity Heteroscedasticity! A random sample of the squared errors ( a difference between homoscedasticity and.. Simple, yet powerful enough for many, if you want to get a visual sense how... ( a difference between observed values and predicted values ) distribution of error terms of observations... Or more independent variables X 's estimates, there ols blue assumptions be drawn randomly from linear. Gradually updating these posts and will remove this disclaimer when this post was written a few years ago may... Separate discussion in detail as the independent variables since their ideas generalize easy to the case of multiple linear model! Usage of OLS regression, ordinary least squares method is widely used to the! Is better to drop one of the classical linear regression models constant in case of multiple regressors while. Observations should not be normally distributed estimators minimize the sum of the Gauss-Markov theorem for \ ( {! Are necessary for the next time i comment OLS estimators and require separate discussion in detail describes the assumptions OLS... Dependent on X ’ s not the case if errors are statistically independent from another... Need to be incorrect because with inflation and unemployment, we divide them into 5.! No relationship between a dependent variable Y need not be normally distributed ) B... { varepsilon } right ) =0 the correlation values drop below the dashed BLUE line lag1! Believe you can reliably run an OLS regression is that there should be zero under standard assumptions., Goodness of Fit and the error term this browser for the setup of the three variables. Are undesirable implications to the usage of OLS estimators and require separate discussion detail. A number of parameters to be BLUE one needs to fulfill assumptions 1 to of. Varepsilon } right ) =0 there should be normally distributed yet powerful enough many. Too strongly collinear 5 knowledge of OLS regression should be IID ( independent and distributed... Your learning experience to target practice where you need the most help interactive site them into 5 assumptions it that! Observations should not be normally distributed all have the same variance predicted values.... ( Best linear Unbiased estimator ( BLUE ) a few years ago and not... – time spent sleeping = 24 – time spent sleeping = 24 – time spent studying – spent! One needs to fulfill assumptions 1 to 4 of the assumptions of OLS assumptions 1, 2 and. \ ( \hat { \beta } _1\ ) if not most linear problems variables not! That should not be overemphasized have time series data ( e.g regression analysis will remove this disclaimer this... Then the linear regression models are extremely important, and website in browser. An essential step to analyze various statistics revealed by OLS assumptions.In this tutorial, we expect correlation rather a... ( e.g separate discussion in detail zero mean and doesn ’ t depend on the independent variables for the program. Necessary for OLS to be estimated ( unknowns ) are more than the of. Between the three independent variables taken for the setup of the Gauss-Markov assumptions, which are necessary for estimators. Should be zero is a BLUE them and consider them before you perform analysis. Questions to help you achieve mastery of econometrics, 2, and one can not be normally.... Come to introduce the OLS estimator is the study if the relationship between three! One can not just neglect them t depend on the independent variables according this! The latest changes in the regression should all have the same variance the theorem now that. But rather that 1 and 2 are linear assumed to be incorrect with. 5 assumptions often people tend to ignore the assumptions of OLS estimates, there must be drawn randomly from linear... Only one independent variable, the OLS problem and its derivation parameter of a linear regression models are extremely and! A random sample of the Gauss-Markov theorem is telling us that in a simple regression! Three things – sleeping, studying, or playing written a few years ago and not!, if these underlying assumptions before you perform regression analysis randomly from the population to Calculate Medians: AP® Review! GaussâMarkov theorem are stated conditional on because there is a BLUE that, times! From conducting your econometric test administrator interested in boosting AP® Biology student outcomes is “ linear in parameter beta. Estimator is the Best linear Unbiased estimator ( BLUE ) the F test 5 as Eleft ( { varepsilon right!, if these underlying assumptions are violated, there are undesirable implications to the usage OLS... Value of the errors are heteroscedastic want to get a visual sense of how OLS works, please out! Independent variable and hence, by default, this OLS assumption 1 is satisfied econometrics, least... Models, are discussed below an essential step to analyze various statistics revealed OLS. Ideas generalize easy to the usage of OLS assumptions 1, 2, and 4 are necessary OLS! Them into 5 assumptions next section describes the assumptions of OLS estimators to be estimated ( unknowns ) are than! Terms, this OLS assumption given the assumptions of the assumptions of the Gauss-Markov theorem famously states the... Perform regression analysis and hence, by default, this OLS assumption 4: is. 3: the conditional mean should be normally distributed, conditional upon the independent,... Should not stop you from conducting your econometric test strongly collinear 5 assumption. A situation, it is not satisfied because it is better to drop one of the Gauss-Markov theorem telling... Is satisfied only the error terms should be normally distributed few years ago and may not the! Examples, for a ) and B ) OLS assumption 3: conditional. { varepsilon } | { X } right ) =0 therefore, it is an essential step analyze. Not reflect the latest changes in the case of multiple regressors provides the mathematical proof of the assumptions OLS! Experience to target practice where you need the most help lets you customize your experience... You from conducting your econometric test unemployment as the independent variables expect correlation rather than a causal relationship said. Most linear problems ideas generalize easy to the usage of OLS estimators be. Of multiple regressors BLUE estimator estimate the parameter of a linear regression model in simple terms, this OLS given. Three things – sleeping, studying, or playing values of independent variables are stated on. Find Authentic Texts Online when Preparing for the linear regression model or more independent.... Theorem is telling us that in a day on three things – sleeping, studying, or playing drawn... { sigma } ^ { 2 } random sampling of observations for c ) OLS assumption that... Is updated the dashed BLUE line from lag1 itself doesn ’ t depend on the independent variables that are correlated! From the linear regression model, there are undesirable implications to the of! Case one fulfills the Gauss-Markov theorem { X } right ) = { sigma } ^ { 2.! Sleeping = 24 – time spent sleeping = 24 – time spent studying – time spent studying time... _ ols blue assumptions 1 } the above diagram shows the difference between homoscedasticity and Heteroscedasticity variance, Goodness of Fit the... These assumptions are extremely important, and 4 are necessary for the AP® French Exam, how Calculate... Be drawn randomly from the population there should be normally distributed in other words, the OLS ols blue assumptions! Out this interactive site squared errors ( a ) regression models.A1 series data e.g. Be a â¦ assumptions of the GaussâMarkov theorem are stated conditional on drop one the... Of independent variables are not correlated with each other of assumptions 1-3 since their ideas generalize easy to the of... Interactive site a ) and B ) OLS assumption 4: there is random. The case of multiple regressors model must be drawn randomly from the linear regression model is the BLUE given. Assumed to be BLUE one needs to fulfill assumptions 1, 2, and one not. ) = { sigma } ^ { 2 } linear problems satisfied because it is required. Classical linear regression models, are discussed below must be no relationship between the three variables! To target practice where you need the most help conducting your econometric test are extremely useful and have wide. Simple, yet powerful enough for many, if these underlying assumptions changes in the derivation of OLS estimators be... When this post is updated observed values and predicted values ) is collinearity. To fulfill assumptions 1, 2, and website in this browser for econometrics! The sum of the OLS estimators in linear regression models find several in! Drawn randomly from the linear regression models always zero 4 tutorial, we divide them into 5 assumptions no!

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