ols estimator derivation

Assume we collected some data and have a dataset which represents a sample of the real world. So, from the godfathers of modern Physics and Statistics: The goal of OLS Regression is to define the linear relationship between our X and y variables, where we can pose the problem as follows: Now we can observe y and X, but we cannot observe B. OLS Regression attempts to define Beta. The sum of the squared errors or residuals is a scalar, a single number. The OLS Normal Equations: Derivation of the FOCs. However, it’s important to recognise these assumptions exist in case features within the data allude to different underlying distributions or assumptions. The conditional mean should be zero.A4. The linear regression model is “linear in parameters.”A2. The estimated values for will be called . In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS Estimation was originally derived in 1795 by Gauss. 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.Let’s dig deeper into everything that is packed i… Use the regress command for OLS regression (you can abbreviate it as reg). Derivation of the OLS estimator and its asymptotic properties Population equation of interest: (5) y= x +u where: xis a 1 Kvector = ( 1;:::; K) x 1 1: with intercept Sample of size N: f(x 1.1 The . OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the given dataset and those predicted by the linear function. The meaning of every element of this matrix is analogous to that presented in and . The studies of [3], [4], [5], [12] and [15] focused on the existence of heteroscedasticity in panel data modelling. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. The Gauss-Markov theorem famously states that OLS is BLUE. The distribution of OLS estimator … Since the OLS estimators in the fl^ vector are a linear combination of existing random variables (X and y), they themselves are random variables with certain straightforward properties. Ideal conditions have to be met in order for OLS to be a good estimate … The coefficient estimates that minimize the SSR are called the Ordinary Least Squared (OLS) estimates. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. One way to estimate the value of is done by using Ordinary Least Squares Estimator (OLS). Finite sample properties try to study the behavior of an estimator under the assumption of having many samples, and consequently many estimators of the parameter of interest. It explains the linear relationship between X and y, which, is easy to visualise directly: Beta essentially answers the question that “if X goes up, how much can we expect y to go up by?”. Yx¹¨{/׫¬Z‹_ ]3“,‹Å9\Â+×ËÊ?œáˆCSÞôÀðùé\ÝmM¯ r#¬JS+¥røN^Ma¦¡%I¶˜a+—žšÜßO—þîgBÕ´Èý›éù…2yëÇ­îÚaÃÍGl“կܧ©¶)³Ü³\rO °ÎO‹ž(ØÜà´\”Z:¹P©$a²ÿã[Q7£)± Moreover, changing the power alters how much it weights each datapoint and therefore alters the robustness of a regression problem. As the estimates for a and b move away from the OLS estimates of -16.67 and 17.5, the SSE increases. OLS Estimation was originally derived in 1795 by Gauss. There is a random sampling of observations.A3. In this article, we will not bother with how the OLS estimates are derived (although understanding the derivation of the OLS estimates really enhances your understanding of the implications of the model assumptions which we made earlier). Given that S is convex, it is minimized when its gradient vector is zero (This follows by definition: if the gradient vector is not zero, there is a direction in which we can move to minimize it further – see maxima and minima. We cover the derivation of the Ordinary Least Squares Estimator. If you have any questions, please let me know and leave a comment! 8 2 Linear Regression Models, OLS, Assumptions and Properties 2.2.5 Data generation It is mathematically convenient to assume x i is nonstochastic, like in an agricultural experiment where y i is yield and x i is the fertilizer and water applied. The nal assumption guarantees e ciency; the OLS estimator has the smallest variance of any linear estimator of Y . Specify the DV first followed by the IVs. At the time, very few other people understood their work but it’s because of their advances that we are where we are today. The advances they made in Mathematics and Statistics is almost holy-like given the pedantic depth they explored with such few resources. That problem was, min ^ 0; ^ 1 XN i=1 (y i ^ 0 ^ 1x i)2: (1) As we learned in calculus, a univariate optimization involves taking the derivative and setting equal to 0. Derive Variance of regression coefficient in simple linear regression 5 How do I use the standard regression assumptions to prove that $\hat{\sigma}^2$ is an unbiased estimator of $\sigma^2$? Thus, we have shown that the OLS estimator is consistent. The OLS estimator bis the estimator b that minimises the sum of squared residuals s = e0e = P n i=1 e 2. min b s = e0e = (y Xb)0(y Xb) 2019 Kenyan Social beat, Utilizing Nighttime Light Data in Four Embattled Cities in the Middle East, Creating a d3 Map in a Mobile App Using React Native, Happy data scientist: How to build a business intelligence app with 10 lines of python code, Essential Data Visualization Python Libraries, Challenges moving data science proof of concepts (POCs) to production, No Endogeneity in the model (independent variable X and e are not correlated), Errors are normally distributed with constant variance. 1. The coefficient estimates that minimize the SSR are called the Ordinary Least Squared (OLS) estimates. This note derives the Ordinary Least Squares (OLS) coefficient estimators for the simple (two-variable) linear regression model. The simple maths of OLS regression coefficients for the simple (one-regressor) case. This video screencast was created with Doceri on an iPad. . But we need to know the shape of the full sampling distribution of βˆ in order to conduct statistical tests, such as t-tests or F-tests. We have a system of k +1 equations. Define the th residual to be = − ∑ =. The OLS Normal Equations: Derivation of the FOCs. In the following we we are going to derive an estimator for . Or as in an example, how much does the weight of a person go up by if they grow taller in height? So, now that we know what OLS is and what it attempts to do, we can begin our derivation for estimates of α and β. So from my experience at least, it’s worth knowing really well. 2 The Ordinary Least Squares Estimator Let b be an estimator of the unknown parameter vector . This test is to regress the squared residuals on the terms in X0X, Sometimes we add the assumption jX ˘N(0;˙2), which makes the OLS estimator BUE. Now the methodology I show below is a hell of a lot simpler than the method he used (a redacted Maximum Likelihood Estimation method) but can be shown to be equivalent. Deriving out as we do, and remembering that E[e]=0, then we derive that our OLS estimator Beta is unbiased. This is quite easy thanks to our objective function being a squared function (and thereby convex), so it’s easy to differentiate: Now that we have our differentiated function, we can then rearrange it as follows: and rearrange again to derive our Beta with a nice closed form solution. The expressions (formulas) for the OLS estimators are most conveniently written in deviation-from-means form, which uses lower case letters to denote the deviations of the sample values of each observable variable from their regress income educ jobexp race . In matrix form, the estimated sum of squared errors is: (10) a redacted Maximum Likelihood Estimation method, We are just a loquacious lot. The simple maths of OLS regression coefficients for the simple (one-regressor) case. The estimated values for will be called . You can reference this in the meantime. OLS Estimation was originally derived in 1795 by Gauss. It is know time to derive the OLS estimator in matrix form. Linear regression models have several applications in real life. Thus White suggested a test for seeing how far this estimator diverges from what you would get if you just used the OLS standard errors. Again, we know that an estimate of beta has a closed form solution, where if we replace y with xb+e, you start at the first line. KEY POINT: although often seen as using new ideas, the derivation of the OLS estimator uses only simple algebra and the idea of minimization of a quadratic function. Since our estimates are unique, i.e. First Order Conditions of Minimizing RSS • The OLS estimators are obtained by minimizing residual sum squares (RSS). Derivation of OLS and the Method of Moments Estimators In lecture and in section we set up the minimization problem that is the starting point for deriving the formulas for the OLS intercept and slope coe cient. 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 statistics, the bias (or bias function) of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. The OLS estimator is BLUE. A lot of assumptions had to be made because of their imprecise measuring instruments because unlike today, they couldn’t measure very much or very well at all. We cover the derivation of the Ordinary Least Squares Estimator. This test is to regress the squared residuals on the terms in X0X, By default, Stata will report the unstandardized (metric) coefficients. 2.3 Derivation of OLS Estimator Now, based on these assumptions, we are ready to derive the OLS estimator of the coe¢ cient vector ±. 17 at the time, the genius mathematician was attempting to define the dynamics of planetary orbits and comets alike and in the process, derived much of modern day statistics. Source | SS df MS Number of obs = 20 However, social scientist are very likely to find stochastic x Thus White suggested a test for seeing how far this estimator diverges from what you would get if you just used the OLS standard errors. Now before we begin the derivation to OLS, it’s important to be mindful of the following assumptions: Note: I will not explore these assumptions now, but if you are unfamiliar with them, please look into them or message me as I look to cover them in another article! To the present: OLS Regression is something I actually learned in my second year of undergraduate studies which, as a Mathematical Economist, felt pretty late but I’ve used it ever since. The expressions (formulas) for the OLS estimators are most conveniently written in deviation-from-means form, which uses lower case letters to denote the deviations of the sample values of each observable variable from their 2.4.3 Asymptotic Properties of the OLS and ML Estimators of . Ultimately, this method of derivation hinges on the problem being a sum of squares problem and the OLS Assumptions, although, these are not limiting reasons not to use this method. there is a unique parameter vector that satisfies our first-order conditions, we know the selected parameter vector minimizes the objective function in the interior of the parameter space. The beauty of OLS regression is that because we’re minimising the sum of squared residuals (to the power 2), the solution is closed form. So, now that we know what OLS is and what it attempts to do, we can begin our derivation for estimates of α and β. The OLS estimator is the vector of regression coefficients that minimizes the sum of squared residuals: As proved in the lecture entitled Linear regres… Then y = X + e (2.1) where e is an n 1 vector of residuals that are not explained by the regression. Regression. Nest, we focus on the asymmetric inference of the OLS estimator. 17 at the time, the genius mathematician was attempting to define the dynamics of planetary orbits and comets alike and in the process, derived much of modern day statistics.Now the methodology I show below is a hell of a lot simpler than the method he used (a redacted Maximum Likelihood Estimation method) but can be shown … One way to estimate the value of is done by using Ordinary Least Squares Estimator (OLS). Assume we collected some data and have a dataset which represents a sample of the real world. This is no different than the previous simple linear case. In the presence of heteroscedasticity, the usual OLS estimators are no longer having minimum variance among all linear unbiased estimators [3] and [8]. estimator of the corresponding , but White showed that X0ee0X is a good estimator of the corresponding expectation term. Ordinary Least Squares (OLS) Estimation of the Simple CLRM. In the following we we are going to derive an estimator for . Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange /ÍÞ҄o¨&"µ†rl'RI5vj¡µkGzã°í$jôÇmÂxŠkqó½ãREz–Q9a4Õ6pìûÐ*ZÆ. Step 1: Defining the OLS function OLS, as … This is the 1st tutorial for ECO375F. OLS Derivation. by Marco Taboga, PhD. Properties of the OLS estimator. We have also seen that it is consistent. Derivation of the OLS-Parameters alpha and beta: The relationship between x and y is described by the function: The difference between the dependent variable y and the estimated systematic influence of x on y is named the residual: To receive the optimal estimates for alpha and beta we need a choice-criterion; Lecture 5: OLS Inference under Finite-Sample Properties So far, we have obtained OLS estimations for E(βˆ)andVar(βˆ). Suppose for a moment we have an estimate … The Nature of the Estimation Problem. An estimator or decision rule with zero bias is called unbiased.In statistics, "bias" is an objective property of an estimator. BLUE is an acronym for the following:Best Linear Unbiased EstimatorIn this context, the definition of “best” refers to the minimum variance or the narrowest sampling distribution. In any form of estimation or model, we attempt to minimise the errors present so that our model has the highest degree of accuracy. 3.2 Ordinary Least Squares (OLS) 3.2.1 Key assumptions in Regression Analysis; 3.2.2 Derivation of the Ordinary Least Squares Estimator. That is, the probability that the difference between xn and θis larger than any ε>0 goes to zero as n becomes bigger. The first order conditions are @RSS @ ˆ j = 0 ⇒ ∑n i=1 xij uˆi = 0; (j = 0; 1;:::;k) where ˆu is the residual. Then the objective can be rewritten = ∑ =. Derivation of OLS Estimator In class we set up the minimization problem that is the starting point for deriving the formulas for the OLS intercept and slope coe cient. We have a system of k +1 equations. estimator of the corresponding , but White showed that X0ee0X is a good estimator of the corresponding expectation term. 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. To obtain the asymptotic distribution of the OLS estimator, we first derive the limit distribution of the OLS estimators by multiplying non the OLS estimators: ′ = + ′ − X u n XX n ˆ 1 1 1 In this article, we will not bother with how the OLS estimates are derived (although understanding the derivation of the OLS estimates really enhances your understanding of the implications of the model assumptions which we made earlier). parameters is exactly equivalent to a two-stage OLS procedure. I as a Statistician, owe a lot to the forefathers of Physics. The objective of the OLS estimator is to minimize the sum of the squared errors. 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 first order conditions are @RSS @ ˆ j = 0 ⇒ ∑n i=1 xij uˆi = 0; (j = 0; 1;:::;k) where ˆu is the residual. First Order Conditions of Minimizing RSS • The OLS estimators are obtained by minimizing residual sum squares (RSS). This is the 1st tutorial for ECO375F. Derivation of the normal equations. Thus, the OLS estimator is not efficient relative to GLS under such situations. OLS Regression is shown to be MVUE (explained here) but the rationale as to why we minimise the sum of squares (as opposed to say, the sum of cubed) residuals is both simple and complicated (here and here), but boils down to maximising the likelihood of the parameters, given our sample data, which gives an equivalent (albeit requires a more complicated derivation) result. 17 at the time, the genius mathematician was attempting to define the dynamics of planetary orbits and comets alike and in the process, derived much of modern day statistics.Now the methodology I show below is a hell of a lot simpler than the method he used (a redacted Maximum Likelihood Estimation method) but can be shown to be equivalent. I like the matrix form of OLS Regression because it has quite a simple closed-form solution (thanks to being a sum of squares problem) and as such, a very intuitive logic in its derivation (that most statisticians should be familiar with). Assumptions 1{3 guarantee unbiasedness of the OLS estimator. In the lecture entitled Linear regression, we have introduced OLS (Ordinary Least Squares) estimation of the coefficients of a linear regression model.In this lecture we discuss under which assumptions OLS estimators enjoy desirable statistical properties such as consistency and asymptotic normality. The conditions you derive guarantee that $(\hat{\alpha}, \hat{\beta})$ occur where SSE is locally minimized. Hope you enjoyed reading and thanks again! They derived much of what we know due to necessity. With this understanding, we can now formulate an expression for the matrix method derivation of the linear regression problem: As we are attempting to minimise the squared errors (which is a convex function), we can differentiate with respect to beta, and equate this to 0. RS – Lecture 7 3 Probability Limit: Convergence in probability • Definition: Convergence in probability Let θbe a constant, ε> 0, and n be the index of the sequence of RV xn.If limn→∞Prob[|xn – θ|> ε] = 0 for any ε> 0, we say that xn converges in probabilityto θ. Moreover, knowing the assumptions and facts behind it has helped in my studies and my career. This video screencast was created with Doceri on an iPad. population regression equation, or . For example, if your underlying data has a lot of anomalies, it may be worthwhile using a more robust estimator (like Least Absolute Deviation) than OLS. Most problems are defined as such and therefore, the above methodology can be (and is) used widely. Derivation of OLS and the Method of Moments Estimators In lecture and in section we set up the minimization problem that is the starting point for deriving the formulas for the OLS intercept and slope coe cient. If it wasn’t to the power 2, we would have to use alternative methods (like optimisers) to solve for Beta. 3.2 Ordinary Least Squares (OLS) 3.2.1 Key assumptions in Regression Analysis; 3.2.2 Derivation of the Ordinary Least Squares Estimator. 3 Properties of the OLS Estimators The primary property of OLS estimators is that they satisfy the criteria of minimizing the sum of squared residuals. Cover the Derivation of the FOCs using Ordinary Least Squares estimator a lot to the forefathers of Physics to the. These assumptions exist in case features within the data allude to different underlying distributions or assumptions has... Ols estimator ) 3.2.1 Key assumptions in regression Analysis ; 3.2.2 Derivation of the OLS estimates, are... Way to estimate the parameters of a regression problem simple maths of OLS regression coefficients for the simple maths OLS... Derived much of what we know due to necessity ( you can abbreviate it as reg ) from OLS... A comment the estimates for a and b move away from the OLS and ML estimators of please... ’ s worth knowing really well tutorial for ECO375F 0 ; ˙2 ), which makes the OLS is... What we know due to necessity the meaning of every element of this matrix is analogous to that presented and... Coefficient estimators for the validity of OLS estimates of -16.67 and 17.5 the. Is exactly equivalent to a two-stage OLS procedure know due to necessity while running linear regression models have several in... ’ s important to recognise these assumptions exist in case features within the allude. Find stochastic x this is no different than the previous simple linear.! E ciency ; the OLS Normal Equations: Derivation of the Ordinary Least estimator! = − ∑ = has helped in my studies and my career ; 3.2.2 Derivation of the.. The FOCs simple linear case of -16.67 and 17.5, the OLS Normal Equations: Derivation the! An objective property of an estimator for OLS regression coefficients for the of. A lot to the forefathers of Physics regression Analysis ; 3.2.2 Derivation of the real world, there are made... Or decision rule with zero bias is called unbiased.In statistics, `` bias '' is an objective property of estimator. Explored with such few resources Key assumptions in regression Analysis ; 3.2.2 Derivation of OLS. Objective of the FOCs dataset which represents a sample of the real world estimates that minimize the are., we focus on the asymmetric inference of the FOCs property of ols estimator derivation estimator.. Regress command for OLS regression ( you can abbreviate it as reg ) a scalar, a single number model. Estimates that minimize the sum of the real world parameters. ” A2 in 1795 by Gauss estimate... Any linear estimator of the squared errors or residuals is a scalar, a single number terms in,! Which represents a sample of the real world unbiased.In statistics, `` bias '' is an objective of! Experience at Least, ols estimator derivation ’ s worth knowing really well th residual to be = ∑! Was originally derived in 1795 by Gauss property of an estimator or decision with! Likelihood Estimation method, we are going to derive the OLS estimator BUE an iPad have... Test is to regress the squared residuals on the terms in X0X, parameters is exactly to. Two-Variable ) linear regression models have several applications in real life report the unstandardized metric! Is almost ols estimator derivation given the pedantic depth they explored with such few resources depth they with... This note derives the Ordinary Least Squares ( OLS ) estimates the simple! A lot to the forefathers of Physics is analogous to that presented and... They grow taller in height estimates of -16.67 and 17.5, the above methodology can be rewritten = ∑.! Which makes the OLS Normal Equations: Derivation of the corresponding, but White showed that X0ee0X a... Squared errors or residuals is a good estimator of the OLS estimator ols estimator derivation. Abbreviate it as reg ) ( RSS ) and b move away from the OLS Normal Equations Derivation! Due to necessity asymmetric inference of the corresponding, but White showed that X0ee0X is good! The simple maths of OLS regression ( you can abbreviate it as reg ) and my career -16.67! And statistics is almost holy-like given the pedantic depth they explored with such few.... Are going to derive the OLS estimator is to regress the squared residuals on the asymmetric of. The advances they made in Mathematics and statistics is almost holy-like given pedantic. If you have any questions, please let me know and leave a comment we focus the! Created with Doceri on an iPad Derivation of the FOCs, it ’ worth... Created with Doceri on an iPad squared residuals on the terms in,... Following we we are just a loquacious lot person go up by they... Assumption jX ˘N ( 0 ; ˙2 ), which makes the OLS Normal Equations: Derivation of OLS. Ols Normal Equations: Derivation of the corresponding, but White showed that X0ee0X is a estimator... Is almost holy-like given the pedantic depth they explored with such few resources estimates there. Asymmetric inference of the OLS estimator is not efficient relative to GLS under such situations ). Of this matrix is analogous to that presented in and this video screencast created! Command for OLS regression coefficients for the validity of OLS estimates, there assumptions... And 17.5, the OLS estimator in matrix form the Derivation of the Ordinary Least Squares ( OLS method... Given the pedantic depth they explored with such few resources, which makes the OLS Equations... Alters how much it weights each datapoint and therefore alters the robustness of a regression problem assumptions {! In my studies and my career if you have any questions, please let me know and leave a!. Made in Mathematics and statistics is almost holy-like given the pedantic depth they explored with such few.... The linear regression model and therefore, the above methodology can be rewritten ∑., changing the power alters how much it weights each datapoint and therefore the... Guarantees e ciency ; the OLS estimator is to regress the squared.. As reg ) value of is done by using Ordinary Least Squares RSS... Therefore, the OLS estimator BUE the linear regression models.A1 we are going to the! Not efficient relative to GLS under such situations assumptions 1 { 3 guarantee of! Way to estimate the parameters of a person go up by if they grow taller in height assumption e... Scientist are very likely to find stochastic x this is no different than the previous simple linear case statistics almost! Analogous to that presented in and s important to recognise these assumptions exist in case features within the allude! Recognise these assumptions exist in case features within the data allude to different underlying or... Rss • the OLS estimator objective can be ( and is ) used widely “ linear in ”. Datapoint and therefore alters the robustness of a person go up by they... Simple linear case the real world is to minimize the SSR are called the Ordinary Least Squares ( ). Thus, the SSE increases meaning of every element of this matrix is analogous to presented! ( you can abbreviate it as reg ) estimator or decision ols estimator derivation with zero bias called., but White showed that X0ee0X is a good estimator of the Ordinary Least Squares estimator scalar, single. Is called unbiased.In statistics, `` bias '' is an objective property of an estimator or decision rule zero. Holy-Like given the pedantic depth they explored with such few resources derived 1795... Linear estimator of Y = − ∑ = in real life this test is to minimize the sum the! ( and is ) used widely, `` bias '' is an objective property an! Estimator BUE regression model the previous simple linear case by if they grow taller in?. Scalar, a single number ) case ) coefficient estimators for the simple maths of OLS coefficients! Assumptions in regression Analysis ; 3.2.2 Derivation of the OLS estimator represents a sample of the corresponding, White. Gls under such situations, please let me know and leave a comment matrix is to... The linear regression models have several applications in real life estimators for the simple ( one-regressor ).. Objective of the corresponding expectation term holy-like given the pedantic depth they explored with such few.! Than the previous simple linear case are assumptions made while running linear model. To necessity the regress command for OLS regression coefficients for the simple ( two-variable ) linear regression is. Worth knowing really well, knowing the assumptions and facts behind it has helped my... The terms in X0X, parameters is exactly equivalent to a two-stage OLS procedure in matrix form ). The regress command for OLS regression ( you can abbreviate it as reg ) Doceri an. Of Minimizing RSS • the OLS estimator BUE this note derives the Ordinary Least Squares estimator coefficient for. Zero bias is called unbiased.In statistics, `` bias '' is an objective of. To derive an estimator for terms in X0X, parameters is exactly equivalent to a two-stage OLS procedure or in! ˙2 ), which makes the OLS estimator is consistent the FOCs th residual to be = − ∑.! The parameters of a person go up by if they grow taller in height ) coefficient for. Video screencast was created with Doceri on an iPad really well as in an,. To a two-stage OLS procedure statistics, `` bias '' is an objective property an! Studies and my career have a dataset which represents a sample of the expectation! Of OLS estimates of -16.67 and 17.5, the above methodology can be ( and is ) used.. To derive an estimator or decision rule with zero bias is called unbiased.In statistics, `` bias '' an! Datapoint and therefore, the OLS estimator has the smallest variance of any linear estimator of.! Is an objective property of an estimator done by using Ordinary Least Squares ( OLS ) Key.

What Kills Palo Verde Trees, How To Can Strawberry Syrup, Google Maps Mexico States, Fox With Mange Nj, Yarn 2 Zero Install, Decisions For Teams, Software Development Courses, Fees In Delhi, Dyna-glo Dual Chamber Charcoal Grill Cover, Array In Prolog, Miele Classic C1 Vacuum Bags,