derivation of ols estimators multiple regression

x6���_Q6�=`��������8.j/A�"ؓ*~ߟn���ѢH���$��Έ��g�)DnXb��,�@��G��lECr{��Y�u�gtS�4γ�`I��r��b1�6B��ƃM��Y|j&[��*�@�O��F-K&���� =#�`p�!KU�h�Eh#�E���L�|��{b&���L���M��4-˙,��6��G!�ς��?��K�8?yT����!����v�T��)�Eh[D�K=X�Ǔ�p�[�#R ޘ�3��GM�ʊ���������,VIz���b�%�Vb*X(���`�|�P8@�1�:�?���!\DE>�Ժ.���W�9^���1[���S/r���qw��+������β�v�+����.\ٶ���Jp��!����[��ack���A�)���=�p�|��n��ɑ�` The multiple regression model is the study if the relationship between a dependent variable and one or more independent variables. In multiple regression we are looking for a plane which can best fit our data. You can find the same material in Applied Linear Statistical Models, 5th Edition, page 207. The equations must be linear in the parameters. Notice, the matrix form is much cleaner than the simple linear regression form. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. The derivative of a squared term is found using the power rule. Other factors such as competitors� price and the general state of the economy will affect your sales. H o w e v e r , t h i s v a l u e f o r x t 1 b e c o mes very important later in the derivation. By example, we have shown that equation (4) is much simpler to write than equation (2). The discussion will return to these assumptions and additional assumptions as the OLS estimator is continually derived. It allows to estimate the relation between a dependent variable and a set of explanatory variables. � � � � �4 c+ �! Page 5 of 12 pages . We can now estimate more complicated equations. Recall, from the simple linear regression case, linear in parameters refers to linear in the unknown parameters and not linear in the x�s. Step 5. Linear regression models find several uses in real-life problems. �q�R2���������ͮ]�4��F}u���Ƕ�������4�ܶM�/�����_�� f�0gJ�/U^�4�'���|����bWV͝�ֱ��6��]_ k�D�]�,�r7���/k62��_2�b&�LIdp5�p�y������J@�`чz�nj�&�o����\G}�Gyݕ}_���F���X��ȁ�2Z��4lC endobj ! The Multiple Linear Regression Model 1 Introduction The multiple linear regression model and its estimation using ordinary least squares (OLS) is doubtless the most widely used tool in econometrics. b) if and only if the omitted variable has a positive coefficient. Division by zero is not defined. The assumption that the FOC can be solved requires the determinate of X�X to not equal zero. Step 2. This is simply for ease. Ordinary Least Squares is the most common estimation method for linear models—and that’s true for a good reason.As long as your model satisfies the OLS assumptions for linear regression, you can rest easy knowing that you’re getting the best possible estimates.. Regression is a powerful analysis that can analyze multiple variables simultaneously to answer complex research questions. Division by matrices is not defined, but multiplying by the inverse is a similar operation. Q� � $ �# �# �# �# �# �# n% o% �% �% �% �% �% �% �% �% �% �% �% �% &. {" |" �" �" �" �" �# ��� � ��� � � Ľ� � ��� � ��� ��~�re~� j� EH��UmH nH u jD�C You will not have to take derivatives of matrices in this class, but know the steps used in deriving the OLS estimator. We will continue the discussion on R2 later in this class, when model specification is discussed. With these matrices, the property is EMBED Equation.3 We have shown by example the property holds and results in a scalar. 56 0 obj Further, equation (4) will be simpler to manipulate. Algebraic Property 1. 77 0 obj Again, this variation leads to uncertainty of those estimators which we seek to describe using their sampling distribution(s). If the mean of each independent variable is used in the estimated equation, the resulting y will equal the mean of the y observations. The equation is called the regression equation.. �! 3.2 Ordinary Least Squares (OLS) 3.2.1 Key assumptions in Regression Analysis; 3.2.2 Derivation of the Ordinary Least Squares Estimator. / ? ��m�� The sum of the squared errors or residuals is a scalar, a single number. /Filter /FlateDecode Using this rule puts equation (11) into a simpler form for derivation. This simplifies the derivation of the multiple linear regression case. This point will be covered in more detail later in the class. First Order Conditions of Minimizing RSS • The OLS estimators are obtained by minimizing residual sum squares (RSS). 2949605 . Multiplying any matrix, A, by I results in A, similar to multiplying by one in linear algebra. Three points one should recognize in equation (6) are: 1) each row corresponds to an individual observation, 2) the column of ones in the X matrix represent the intercept term, and 3) without subscripts the notation denotes a matrix, whereas with subscripts the notation denotes elements of a matrix. 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. This is important, because most equations that are estimated are not simple linear equations, but rather multiple regressions. The linear regression model is “linear in parameters.”A2. 5�CJ \�9 3 L M . Bl�)��zS���̠�g'w��`�C��O%FS @ͬG�e Ü= 4+ 5xi1 + 6xi2+ 6xi3 • Ceteris paribus interpretation of 5 = 5 x1 let x2=0 and x3=0 When x1 increases by one unit, y changes by 5units, controlling for the variable x2 and x3. Equation (8) in matrix notation is (9) EMBED Equation.3 where Y and X are as previously defined, EMBED Equation.3 is the vector of estimated parameters, EMBED Equation.3 is the vector of estimated dependent variables, and EMBED Equation.3 is the vector of estimated error terms. Abbott ¾ PROPERTY 3: Variance of βˆ 1. • Definition: The variance of the OLS slope coefficient estimator is defined as 1 βˆ {[]2} 1 1 1) Var βˆ ≡ E βˆ −E(βˆ . k -1 =2). In multiple regression we are looking for a plane which can best fit our data. The argument is made in the matrix form. @,%a���wu�x�x��ʇp��Yv�9^y�rh�r��1k��BȧS�_�#�G�����G!���c��1��R����k��NgZk�g���38/q��R�P�@�ÿMt�nj�2����͓D���^�NĚ vNjG3Ylt #�F0�,�~o������3 =t �����z�{��[{����_�D��'a�ė�MD��cG��I��t��*K7�� W����>O5���8A�F�Q)IYe�\���@�AC�Tu�T��R��P*��F���#"#����ڂx�B�&�� �4����xiI�����m������r 7���[�澉���g�+MWC�-�\3e*��� :Nr����b�':�L�%��7ɡ�.�`q�P�m2S��a�.���;%�͆�4E�S'���D���}����yq�2�(���[�f������`���� au����,3���cl�v�e�;��pie���PZ�P�e�����,D?����68��:n2� ��?��qHóD� �3��\�#Ĺ(^M�0 �m��fQ-��+뭋���QJ8�O��GDA�Ε�28�) 왢�τsƇs�ѝ��= ¹�g���,�0Z�%m��/��#B,�=|�­5ę.u�y�T-�^- 0��l%G�Y����s��$�fR��L�0{��۵#xi��_�FF�KF���DH�JSǘ�J�9ϣ8�Co(sn��8�Y�ۑ�"��>m�s���=�q��IZpa�� I�.bN=VQ�{����S��� �"�c}����2�\����'�_����WR��T�=g�ݭ��ݪ���&xn��Cu�U�M� The equation is called the regression equation.. Applying this rule one obtains EMBED Equation.3 . N e i t h e r a s s u m p t i o n i s p a r t i c u l a r l y r e s t r i c t i v e . SOC The math necessary to show the SOC in matrix form is beyond the matrix algebra presented in the prerequisites for this class. The Nature of the Estimation Problem ... Derivation of the OLS Coefficient Estimators. In this equation, the only unknowns are the EMBED Equation.3 , both the Y and X matrix are know. Finally, EMBED Equation.3 is found by premultiplying both sides by EMBED Equation.3 . Thus when k -1 =2 we obtain: and a parallel expression for b3 can be obtained by exchanging X2 and X3. ! Each step is described here. Linear Equation Again, similar to the simple linear regression case, OLS is used to estimate linear equations. ˆ.  �7�b�~:��B�ɘ]>G�I�;�TPzC��'��o4�Y�h���#w��ws�w�����ޣG#�C�����̽�lS���ʢm�-���*�{7��~dS�����)���]�[~޸w/;�OǽĮ,���OL��J�ޕ7@�'��hʗ�l(����/-Z�š�nf7��V ,��PB{}EOo�`w���Qd�t���˼Ku�� ��쇛. CJ UVaJ '3 � � . Linear regression models have several applications in real life. If X explains no variation in y, the SSE will equal zero. I n g e n e r a l , t h e m u l t i p l e r e g r e s s i o n c a s e c a n b e w r i t t e n a s ( 3 ) E M B E D E q u a t i o n . With the sum of squared residuals defined and using the definition for calculating the error term, the objective function of OLS can be written as follows: (11) EMBED Equation.3 . The OLS estimators of the coefficients in multiple regression will have omitted variable bias a) if and only if the omitted variable is not mean zero. • Reason: ... Variance of the OLS estimators Wenow obtain the variance of the OLS estimators, so th tthat we have a measureof the spread in th itheir sampling distributions. Problem Set-up. 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. �)M����#�t�um�0�:tz�¼@W�H�LJ�|����Gǣս䅓����W��ݼ�7�̻��i��78�6�A�髛e��^��@$c*y�IIydИ�Hߓ�A�4�U���2����i���x�N~�Za+rxV��R"�����r��2�����0�N4)��Nĉ.q����L#��d0�O�ʦ���;b���_Nm2�|ljeeO�k6>'&� 5�S����,�W����u�U��*����P���U/�^.��f��W�m���XN endstream Problem Set-up N-Paired Observations Similar to the simple linear regression problem, you have N-paired observations. 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. The sample covariance between each individual xi and the OLS residual EMBED Equation.3 is equal to zero. Recall, EMBED Equation.3 is a vector or coefficients or parameters. • Measure of fit 1 %���� Algebraic Property 4. Observations must still be paired. Now let us jump back to the example of test scores and class sizes. EMBED Equation.3 is zero. << 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. 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 done nothing new, except expand the methodology to more than two unknown parameters. 'B In many applications, there is more than one factor that influences the response. Using the procedure to derive the simple linear case, the derivation of the OLS estimator results in k equations (a FOC for each unknown). Why? This is given by the equation (15) EMBED Equation.3 As shown in this equation, the coefficient of determination, R2, is one minus the ratio of the amount of variation not explained relative to the total variation. CJ UVaJ j� EHJ�Uj�|�C This video screencast was created with Doceri on an iPad. • Assumption 4: Homoskedasticity. Summary of Lecture 11 • Multiple regression model • Interpretation of coefficients in multiple regression • Derivation of the OLS estimators • Omitted variable bias (again!) 3 . Asymptotic Efficiency of OLS . The individual equations for k parameters and n observations are (4) EMBED Equation.3 . Rarely, are you interested in only one independent variable�s potential affect on the dependent variable, y. With these matrices, the OLS estimates for EMBED Equation.3 are: EMBED Equation.3 This is the same result that was obtained for the simple linear regression case. Because so few assumptions have been made, OLS is a powerful estimation technique. Because the OLS can be obtained easily, this also results in OLS being misused. �n�ɇ6�v F�1��a�d�C��%����NC�����q�:�p�냅��=6��ۏ���f��. • The key to work with matrix is keeping track of the dimension. Quantitative Methods of Economic Analysis . Example Sum of Squared Errors Matrix Form. Consistency 2. CJ UVaJ H* j� EH��Uj���C Multiple Regression Analysis: Estimation . The estimated model object is mult.mod. Multicollinearity is often a dire threat to our model. Chairat Aemkulwat Interpretation • Total differentiate the equation. Key point: the paired observations are one y associated with a set of … Instead of including multiple independent variables, we start considering the simple linear regression, which includes only one independent variable. 3 w h e r e t h e � s a r e k u n k n o w n p a r a m e t e r s , t h e u s a r e t h e e r r o r o r r e s i d u a l t e r m s , t r e f e r s t o t h e o b s e r v a t i o n n u m b e r , a n d x t i r e f e r s t o t h e i t h i n d e p e n d e n t v a r i a b l e f o r o b s e r v a t i o n t . T h i s v a l u e o f o n e f o r x t 1 i s u s u a l l y o m i t t e d w h e n w r i t i n g t h e e q u a t i o n . This makes it difficult to impossible to show the error or deviations graphically. Similar to the simple linear regression problem, you have N-paired observations. c) always d) if and only if the omitted variable is a determinant … o! /Length 3727 Using the example from the simple linear case, we can show using the matrix form will result in the same OLS estimates. deriving the OLS coefficient estimators 1 for any given sample of N observations (Yi, Xi), ... Derivation of the OLS Coefficient Estimators. Key point: the paired observations are one y associated with a set of x�s. 5. The above discussion provides the background for this formulation. ), and K is the number of independent variables included. �1�V� As in the simple linear case, for any observation, the estimated error term is defined as the actual y value minus the estimated y value EMBED Equation.3 . A derivation of the OLS estimators of beta0 and beta1 for simple linear regression. f�x�:�;g���i�j�p����9�(Q,��ݘQ eMT��,�X��A�l��b��ɔCu�g�es�&v��� ��?? Because it holds for any sample size . I m p l i c i t l y , �1 , t h e i n t e r c e p t p a r a m e t e r , i s m u l t i p l i e d b y o n e . I t i s c l e a r t h a t equation (6) is much simpler to write than writing the equations in equation (4). Deriving the Inconsistency in OLS Suppose the true model is: =𝛽0+𝛽1 1+𝛽2 2+ If we omit 2 from the regression and do the simple regression of on 1, =𝛽0+𝛽1 1+ , then =𝛽2 2+ . That is, the matrix forms adds nothing to the derivation. CJ UVaJ H* j EH��UjE��C The last term, EMBED Equation.3 , is simply a squared term in EMBED Equation.3 with X�X as constants. Algebraic Properties of the OLS Estimator Several algebraic properties of the OLS estimator were shown for the simple linear case. This clip establishes under which assumptions the OLS estimators for a linear multiple regression model are unbiased. 𝛽 1 =𝛽1+𝛽2𝛿 ��0V�\i=jB�p`�8J�_�P��E�xW.⌍k 9$��`���7��#6�a��ض�a�Ь$����l���a��2D �R���bgX�$�8r˪�H0��E��T�F>�A��SBv`!��֌o}y��C�� �G�XM�qT���)�/�;_ �;w�ʹn� �����v$�����5��B��,��s��(�'��b�ڪ-L7���"g��vI�;_�'d�]���M;�� dN�`lQO�9B"���%���I��_:g%ZoLz� �f�ln�L�]�3_x�4I� ϼ�ό��%�)Ҵ��)V2}`��8�ƕ-�pz����������Ҕv1��&H����G�Km�����hA���wMa�VN���sp���}�3�C�#� �+�p���� {37��x�hk��{É������:�����ʹ��p8�E_H0��*j� |�1�K������6Fm�}f�A�3�3�@m�y��ݍ��O�� S.=��=��"����R�������T���/@�0���ߟ� �+��00�|���И�ƛ�Ɵ�{�؉Ш��� қT�����l&�T��jC����2�:�c4��T(Ou�B&��� �uFi�oijp���X�6`�,��~ 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. Nevertheless some insights into the general nature of the slope estimators in multiple regression can be obtained by examining the estimators for the particular case where there are two explanatory variables (i.e. Expected Value and Variance of Estimation of Slope Parameter $\beta_1$ in Simple Linear Regression 5 How do I use the standard regression assumptions to prove that $\hat{\sigma}^2$ is an unbiased estimator of $\sigma^2$? Multicollinearity is often a dire threat to our model. The derivation of these properties is not as simple as in the simple linear case. Table 1. Applying transpose rule in equation (12) to equation (11) and then expanding, the following equation is obtained; (13) EMBED Equation.3 . Y�Y does not include EMBED Equation.3 , therefore, the partial of Y�Y w.r.t. One will not usually see either of these values when running a regression. This is where matrix algebra enters the mix. Let 𝛽 1 denote the simple regression slope estimator. Algebraic Property 3. Important Terms / Concepts n-paired observations Error Term vs. estimated error term Residual Hat symbol Sum of squares SSR SST SSE Why OLS is powerful? The values of [0 - 1] are just the theoretical range for the coefficient of determination. The sum / difference rule is applied to each set of matrices in the equation. {! OLS Estimation of the Multiple (Three-Variable) Linear Regression Model. 3 Y a n d U a r e c o l u m n v e c t o r s o f d i m e n s i o n n x 1 , � i s c o l u m n v e c t o r o f d i m e n s i o n k x 1 , a n d X i s a m a t r i x o f d i m e n s i o n n x k . >> Expected Values and Variances of the Estimators. Step 3. �f��՛?��˘gtg"c�(Zw �Ǩ��1���\��ij�H�"���1�3���AO3:��m}�$�������#*�c+��&��鐮Ԫ+k���3%��.�� IB��O��A�i^�����G�w�D�i��>�PWܨN�llL8���Wk���l˾����4��Ot�5��4���K�ep�pbB��5�$b͜4%�5�u̓�����U+�L�縩!~�X��.�IN� ��L�8�jܔ�<2G�nhB ɤ�I3�����ի�Ȭ��氥f1�7������dcAW�8�qإl:�� On the left hand side, the two terms EMBED Equation.3 and EMBED Equation.3 cancel each other out leaving the null matrix. The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, ... Derivation of simple linear regression estimators 2Pߺ��L�m�ކ�UCjf(hl�`�7�e�*��_�1&R�6��n��Ut(��%>-T.5�F^I�R/��ॣ�N��` CJ UVaJ j U j EH:�Uj�|�C • Multiple regression analysis is more suitable for causal (ceteris paribus) analysis. Simple linear regression. In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. � � ! " We cover the derivation of the Ordinary Least Squares Estimator. In matrix form, the estimated sum of squared errors is: (10) EMBED Equation.3 where the dimensions of each matrix are shown below each matrix and the symbol EMBED Equation.3 represents the matrix transpose operation. Although, the coefficient of determination is the most common measure, it is not the only measure of the fit of an equation. x��[m�������&�g��� ���&i�[$�൵�jmiO��n��;CR�h���%h��.EQ$g8/��pل�l23|���F�3��m�(�F19�!Z2-`�䂒�,��c�qLf�Χ�1�0a��b�`:�� ��`ac�����_//����I p=���(��ˉ���'���O��Θ֊�n�)7E��΄��]��6��m�,��4aT�h�v]�XVtSV�lm��\~}����C�N���Ė���/t��w_}���On�z"�!x1YM�]|�rT�TE�'�߼cә���o6�5����-�����w]�Ŷn���?w�j�ϛ����\y�ȌOSn��j��eu��5K�F�|�,�p��I"����>߭��|�v��>+��S��yWϯV���U�߫������WB�eB8y���Y�V�,�T�'C�m��CFi��� /Length 3718 Key point: the two assumptions made are the same as in the simple linear case. ECON 351* -- Note 12: OLS Estimation in the Multiple CLRM … Page 1 of 17 pages ECON 351* -- NOTE 12 . The simple maths of OLS regression coefficients for the simple (one-regressor) case. N-Paired Observations. The multiple linear regression model is used to study the relationship between a dependent variable and one or more independent variables. A short hand notation is necessary. $\endgroup$ – … The Y and X matrix are made up of elements associated with your n-paired observations. With multiple independent variables, there is a chance that some of them might be correlated. Step 1. The necessary OLS assumptions, which are used to derive the OLS estimators in linear regression models, are discussed below. Here, this step is writing the equation that the partial derivatives will be taken in matrix form. 2. Derivation and Interpretation 3. The conditional mean should be zero.A4. Further, this example shows how the equations are used. �! Steps Involved in Obtaining the OLS Estimator in Matrix Form StepMathematical DeviationStep involves1 EMBED Equation.3 Original problem min. Nothing new is added, except addressing the complicating factor of additional independent variables. Note, the only difference from the simple linear regression case is the addition of independent variables. �4 c+ � c+ �3 � @ � �3 z ��s3��� � r ) D �3 �3 � �5 0 �5 �3 D9 c+ D9 �3 c+ � � � � � � � Derivation of the Ordinary Least Squares Estimator Multiple Regression Case In the previous reading assignment the ordinary least squares (OLS) estimator for the simple linear regression case, only one independent variable (only one x), was derived. They will hold using the set up presented here. Derivation in Matrix Form The steps necessary to derive the OLS estimator in matrix form are given in mathematical form in table 1. The rules of differentiation are applied to the matrix as follows. Four Algebraic properties Goodness-of-fit R2 - Coefficient of Determination Range of R2 EMBED Equation.3 Meaning of n, k, i PAGE PAGE 1 L - / E � � p q � � � � � � � � n The OLS estimator is derived for the multiple regression case. Algebraic Property 2. A g a i n i t i s i m p o r t a n t t o n o t e , t h e a s s u m p t i o n s s a y n o t h i n g a b o u t t h e s t a t i s t i c a l d i s t r i b u t i o n o f t h e e s t i m a t e s , j u s t t h a t we can get the estimates. Three matrix algebra operations are necessary, multiplication, transpose, and inverse. The objective of the OLS estimator is to minimize the sum of the squared errors. Recall, the coefficient of determination, R2, measures the amount of the sample variation in y that is explained by x. The following example illustrates why this definition is the sum of squares. L’expression à minimiser sur 2Rp+1 s’écrit : Xn i=1 (y i 0 1x 1 i 2x 2 px p i) 2 = ky X k2 = (y X )0(y X ) = y 0y 2 X 0y + X0X : Par dérivation matricielle de la dernière équation on obtient les “équations To derive the estimator, it is useful to use the following rule of transposing matrices. The difference between the simple linear and multiple linear case is the complicating issue of additional independent variables, x�s. Why OLS is misused? For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. Simple matrix algebra is used to rearrange the equation. Note, by matrix multiplication, both sides of this equation results in a scalar. SSR2 EMBED Equation.3 FOC for minimization3 EMBED Equation.3 Use the sum and power rules to take first partial derivative and set equal to zero4 EMBED Equation.3 Divide both sides by 2 and rearrange by adding X�Y to both sides5 EMBED Equation.3 OLS estimator obtained by premultiplying both sides by the inverse of X�X OLS Estimator Matrix Form The OLS estimator in matrix form is given by the equation, EMBED Equation.3 . Multiple regression expands the regression model using more than 1 regressor / explanatory ... First we turn our attention back to the technical aspects of estimating the OLS parameters with multiple regressors. Equations in Matrix Form To write equation (4) in matrix form, four matrices must be defined, one for the dependent variables, one for the independent variables, one for the unknown parameters, and finally one for the error terms. Let’s take a step back for now. ECONOMICS 351* -- NOTE 4 M.G. One reason OLS is so powerful is that estimates can be obtained under these fairly unrestrictive assumptions. The OLS estimator is derived for the multiple regression case. N o t i c e t h e n u m b e r i n g o f t h e x v a r i a b l e s b e g i n s w i t h t w o . Ordinary Least Squares is the most common estimation method for linear models—and that’s true for a good reason.As long as your model satisfies the OLS assumptions for linear regression, you can rest easy knowing that you’re getting the best possible estimates.. Regression is a powerful analysis that can analyze multiple variables simultaneously to answer complex research questions. With these elements, the sum of squares equals 22 + 42 + 62 = 56. 'd $-D@ M� STAGE 1. consists of . In this case for each y observation, there is an associated set of x�s. CJ UVaJ j UmH nH u mH nH uCJ mH nH u j� EH��Uj��C Using the three observations in equation (2), the appropriate matrices are EMBED Equation.3 Equation (6) is EMBED Equation.3 Equation (2) is obtained from equation (6) by multiplying out the matrices and using the definition of matrix addition: EMBED Equation.3 . ��x4co pg�&�|�+��.�������֮��x �ve(�&�w��(��x�v��EzE7Ê6�d��?�T�yA}�r+�"\�P��/� �����q����7����V���,/�ą�0��E4��[Vk��[�����kւ��K;��q�?�T���q�U���KuL.>��rT4KhS5~�������P�������&0�ұ�;�O����5����Aa This property is illustrated in the following example. The linear multiple regression model in matrix form is Y = X + U • Read Appendix D of the textbook. This step moves EMBED Equation.3 to the right hand side. ~ O� � $ P� � ! ����;��3�p:!��t���fk��Gԡ�U��Ă���e�G���*)�"v�7����;���d :wc�"��|�M�! 2 Outline 1. Second, EMBED Equation.3 is added to both side of the equation. Using our knowledge of calculus, we know that if we want to minimize an equation, we can take the first derivative, set the resulting equations equal to zero, and solve for the unknown EMBED Equation.3 . This article has multiple issues. Multiple Regression Analysis: OLS Asymptotics . Assumptions Made to this Point As in the simple linear case, very few assumptions have been made to derive the OLS estimator. To show the formulation, matrix multiplication must be used. In the multiple linear regression case, the estimated error term is defined in the same manner; the only difference is in the number of independent variables. Derivation of the OLS formulas for the regression coefficient estimators is performed in two stages: 0 1 ˆ and β β. Multiple Linear Regression So far, we have seen the concept of simple linear regression where a single predictor variable X was used to model the response variable Y. To show EMBED Equation.3 holds, define the following matrices: EMBED Equation.3 . Asymptotic Normality and Large Sample Inference 3. >> / E F j k � m We will assume the SOC hold. << T h e s e k e y p o i n t s a r e t h e s a m e a s i n t h e s i m p l e l i n e a r c a s e . the first-order conditions (or FOCs) for minimizing the residual sum of squares function . Further, in economic data it is not uncommon to have low R2 values. n Example of equation (6). CJ UVaJ 5�\� j� EH��UjVάC U s i n g t h e s e m a t r i c e s , e q u a t i o n ( 4 ) c a n b e w r i t t e n a s ( 6 ) E M B E D E q u a t i o n . ~ The two assumptions are 1) the equation to b e e s t i m a t e d i s l i n e a r i n t h e u n k n o w n p a r a m e t e r s , �, a n d 2 ) t h e F O C c o u l d b e s o l v e d . The properties are simply expanded to include more than one independent variable. More important, the matrix form allows for k unknowns, whereas the simple linear form allowed for only two unknowns an intercept and a slope. OLS estimators minimize the sum of the squared errors (a difference between observed values and predicted values). Assumptions of OLS Regression. CJ UVaJ j+ EH��Uj�C The generic form of the linear regression model is y = x 1β 1 +x 2β 2 +..+x K β K +ε where y is the dependent or explained variable and x 1,..,x K are the independent or explanatory variables. Key point: the derivation of the OLS estimator in the multiple linear regression case is the same as in the simple linear case, except matrix algebra instead of linear algebra is used. 0 2 4 � � � � � � 3 K * + , - B C V W X Y � � � � � � ~!

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