ols estimator unbiased proof

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. Colin Cameron: Asymptotic Theory for OLS 1. 3. OLS in Matrix Form 1 The True Model † Let X be an n £ k matrix where we have observations on k independent variables for n observations. One of the major properties of the OLS estimator ‘b’ (or beta hat) is that it is unbiased. 0000004001 00000 n 0000010896 00000 n Firstly recognise that we can write the variance as: E(b – E(b))(b – E(b))T = E(b – β)(b – β)T, E(b – β)(b – β)T = (xTx)-1xTe)(xTx)-1xTe)T, since transposing reverses the order (xTx)-1xTe)T = eeTx(xTx)-1, = σ2(xTx)-1xT x(xTx)-1 since E(eeT) is σ2, = σ2(xTx)-1 since xT x(xTx)-1 = I (the identity matrix). %%EOF We now define unbiased and biased estimators. 0000002815 00000 n This estimated variance is said to be unbiased since it includes the correction for degrees of freedom in the denominator. Now, suppose we have a violation of SLR 3 and cannot show the unbiasedness of the OLS estimator. Consider a three-step procedure: 1. Proof. Also, it means that our estimated variance-covariance matrix is given by, you guessed it: Now taking the square root of this gives us our standard error for b. The GLS estimator is more eﬃcient (having smaller variance) than OLS in the presence of heteroskedasticity. The OLS estimator is an efficient estimator. 1) 1 E(βˆ =β The OLS coefficient estimator βˆ 0 is unbiased, meaning that . 0000009446 00000 n ��x �0��h�rA��$��+@yY�)�@Z��:��^0;��@�F��Ygk�3��0��ܣ�a��σ� lD�3��6��c'�i�I�` ��u8!1X��@��]� � �֧ 7�@ We have seen, in the case of n Bernoulli trials having x successes, that pˆ = x/n is an unbiased estimator for the parameter p. 0000006629 00000 n However, there are a set of mathematical restrictions under which the OLS estimator is the Best Linear Unbiased Estimator (BLUE), i.e. Why? For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. Regress log(ˆu2 i) onto x; keep the ﬁtted value ˆgi; and compute ˆh i = eg^i 2. �, Proposition 4.1. OLS slope as a weighted sum of the outcomes One useful derivation is to write the OLS estimator for the slope as a weighted sum of the outcomes. Key W ords : Efﬁciency; Gauss-Markov; OLS estimator q(ݡ�}h�v�tH#D��Gl�i�;o�7N\��q��i�x��๷ ��W��x�ӌ��v#�e,�i�Wx8��|��}o�Kh�>��hgPU�b��v�z@�Y�=]�"�k��i�^�3B)�H��4Eh��H&,k:�}tۮ��X툤��TD �R�mӞ��&;ޙfDu�ĺ�u�r�e��,��m ��$�L:�^d-��ӛv4t�0�c�>:&IKRs1͍4��9u�I�-7��FC�y�k�;/�>4s�~�'=ZWo��d�� Unbiased and Biased Estimators . Unbiased estimator. There is a random sampling of observations.A3. Bias can also be measured with respect to the median, rather than the mean (expected … uncorrelated with the error, OLS remains unbiased and consistent. In order to prove this theorem, let us conceive an alternative linear estimator such as e = A0y startxref 0000004175 00000 n 0000005609 00000 n In order to apply this method, we have to make an assumption about the distribution of y given X so that the log-likelihood function can be constructed. 0000010107 00000 n Theorem 1 Under Assumptions OLS.0, OLS.10, OLS.20 and OLS.3, b !p . is an unbiased estimator for 2. This proof is extremely important because it shows us why the OLS is unbiased even when there is heteroskedasticity. Change ), You are commenting using your Twitter account. Maximum likelihood estimation is a generic technique for estimating the unknown parameters in a statistical model by constructing a log-likelihood function corresponding to the joint distribution of the data, then maximizing this function over all possible parameter values. 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. 0000014371 00000 n 0000007358 00000 n 1076 0 obj<>stream 0) 0 E(βˆ =β • Definition of unbiasedness: The coefficient estimator is unbiased if and only if ; i.e., its mean or expectation is equal to the true coefficient β 1 βˆ 1) 1 E(βˆ =β 1. in the sample is as small as possible. … and deriving it’s variance-covariance matrix. So, after all of this, what have we learned? 0000024767 00000 n The linear regression model is “linear in parameters.”A2. 0000008061 00000 n Now we will also be interested in the variance of b, so here goes. 4.1 The OLS Estimator bis Unbiased The property that the OLS estimator is unbiased or that E( b) = will now be proved. 0000002125 00000 n We have also derived the variance-covariance structure of the OLS estimator and we can visualise it as follows: We also learned that we do not know the true variance of our estimator so we must estimate it, here we found an adequate way to do this which takes into account the need to scale the estimate to the degrees of freedom (n-k) and thus allowing us to show an unbiased estimate for the variance of b! Mathematically this means that in order to estimate the we have to minimize which in matrix notation is nothing else than . 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. 0000003304 00000 n Since E(b2) = β2, the least squares estimator b2 is an unbiased estimator of β2. According to this property, if the statistic $$\widehat \alpha $$ is an estimator of $$\alpha ,\widehat \alpha $$, it will be an unbiased estimator if the expected value of $$\widehat \alpha $$ … 0000003547 00000 n ie OLS estimates are unbiased . <<20191f1dddfa2242ba573c67a54cce61>]>> Gauss Markov theorem. A rather lovely property I’m sure we will agree. ˆ ˆ Xi i 0 1 i = the OLS residual for sample observation i. The estimated variance s2 is given by the following equation: Where n is the number of observations and k is the number of regressors (including the intercept) in the regression equation. OLS Estimator Properties and Sampling Schemes 1.1. This is probably the most important property that a good estimator should possess. 0000008723 00000 n The problem arises when the selection is based on the dependent variable . If many samples of size T are collected, and the formula (3.3.8a) for b2 is used to estimate β2, then the average value of the estimates b2 From (1), to show b! 0000024534 00000 n Construct X′Ω˜ −1X = ∑n i=1 ˆh−1 i xix ′ … 0000002512 00000 n This means that in repeated sampling (i.e. Now notice that we do not know the variance σ2 so we must estimate it. We want our estimator to match our parameter, in the long run. 0000011700 00000 n A Roadmap Consider the OLS model with just one regressor yi= βxi+ui. ( Log Out / Note that Assumption OLS.10 implicitly assumes that E h kxk2 i < 1. CONSISTENCY OF OLS, PROPERTIES OF CONVERGENCE Though this result was referred to often in class, and perhaps even proved at some point, a student has pointed out that it does not appear in the notes. In more precise language we want the expected value of our statistic to equal the parameter. If this is the case, then we say that our statistic is an unbiased estimator of the parameter. 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. 0000001484 00000 n endstream endobj 1104 0 obj<>/W[1 1 1]/Type/XRef/Index[62 1012]>>stream 0 0 -��\ 0000003788 00000 n This theorem states that the OLS estimator (which yields the estimates in vector b) is, under the conditions imposed, the best (the one with the smallest variance) among the linear unbiased estimators of the parameters in vector . 0000000937 00000 n An estimator that is unbiased and has the minimum variance of all other estimators is the best (efficient). 0000005764 00000 n A consistent estimator is one which approaches the real value of the parameter in the population as the size of … In other words, an estimator is unbiased if it produces parameter estimates that are on average correct. Change ), You are commenting using your Facebook account. 0000002769 00000 n Proof. We consider a consistency of the OLS estimator. Now in order to show this we must show that the expected value of b is equal to β: E(b) = β. E(b) = E((xTx)-1xTy) since b = (xTx)-1xTy, = E((xTx)-1xT(xβ + e)) since y = xβ + e, = E(β +(xTx)-1xTe) since (xTx)-1xTx = the identity matrix I. 0000001688 00000 n p , we need only to show that (X0X) 1X0u ! Assumption OLS.10 is the large-sample counterpart of Assumption OLS.1, and Assumption OLS.20 is weaker than Assumption OLS.2. This column should be treated exactly the same as any other column in the X matrix. Proof of Unbiasness of Sample Variance Estimator (As I received some remarks about the unnecessary length of this proof, I provide shorter version here) In different application of statistics or econometrics but also in many other examples it is necessary to estimate the variance of a sample. In this clip we derive the variance of the OLS slope estimator (in a simple linear regression model). Since our model will usually contain a constant term, one of the columns in the X matrix will contain only ones. The idea of the ordinary least squares estimator (OLS) consists in choosing in such a way that, the sum of squared residual (i.e. ) As we shall learn in the next section, because the square root is concave downward, S u = p S2 as an estimator for is downwardly biased. 1074 31 ˆ ˆ X. i 0 1 i = the OLS estimated (or predicted) values of E(Y i | Xi) = β0 + β1Xi for sample observation i, and is called the OLS sample regression function (or OLS-SRF); ˆ u Y = −β −β. 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. When the expected value of any estimator of a parameter equals the true parameter value, then that estimator is unbiased. Under the GM assumptions, the OLS estimator is the BLUE (Best Linear Unbiased Estimator). Example 14.6. β$ the OLS estimator of the slope coefficient β1; 1 = Yˆ =β +β. Since this is equal to E(β) + E((xTx)-1x)E(e). Consider the social mobility example again; suppose the data was selected based on the attainment levels of children, where we only select individuals with high school education or above. For anyone pursuing study in Statistics or Machine Learning, Ordinary Least Squares (OLS) Linear Regression is one of the first and most “simple” methods one is exposed to. 0000004541 00000 n Under the assumptions of the classical simple linear regression model, show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model. We provide an alternative proof that the Ordinary Least Squares estimator is the (conditionally) best linear unbiased estimator. 0000002893 00000 n 0000000016 00000 n H�T�Mo�0�� x�b```b``��π �@16� ��Ig�I\��7v��X��Ma�nO�� Ȁ�â��\��n�v,l,8)q�l�͇N��"�$��>ja�~V�`'O��B��#ٚ�g$&܆��L쑹~��i�H��΂��2��,��_Ц63��K��^��x�b65�sJ��2�)��TI�)�/38P�aљ>b�$>��=,U��U�e(v.��Y'�Үb�7��δJ�EE�� sO*�[@��e�Ft��lp&��,�(e %PDF-1.4 %�� The unbiasedness of OLS under the first four Gauss-Markov assumptions is a finite sample property. This means that in repeated sampling (i.e. ... 4 $\begingroup$ *I scanned through several posts on a similar topic, but only found intuitive explanations (no proof-based explanations). by Marco Taboga, PhD. Well we have shown that the OLS estimator is unbiased, this gives us the useful property that our estimator is, on average, the truth. 1074 0 obj<> endobj if we were to repeatedly draw samples from the same population) the OLS estimator is on average equal to the true value β.A rather lovely property I’m sure we will agree. Does this sufficiently prove that it is unbiased for $\beta_1$? 0000001983 00000 n We can also see intuitively that the estimator remains unbiased even in the presence of heteroskedasticity since heteroskedasticity pertains to the structure of the variance-covariance matrix of the residual vector, and this does not enter into our proof of unbiasedness. , the OLS estimate of the slope will be equal to the true (unknown) value . 0. xref The connection of maximum likelihood estimation to OLS arises when this distribution is modeled as a multivariate normal. trailer Proof under standard GM assumptions the OLS estimator is the BLUE estimator. With respect to the ML estimator of , which does not satisfy the finite sample unbiasedness (result ( 2.87 )), we must calculate its asymptotic expectation. An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter..