\end{align} .11 3 The Gauss-Markov Theorem 12 (2) \quad \hat{\mathbf{x}}_{OLS} = (\mathbf{H}^T \mathbf{H})^{-1} \mathbf{H}^T \mathbf{y} A revision is needed! Under heteroskedasticity, the variances σ mn differ across observations n = 1, …, N but the covariances σ mn, m ≠ n,all equal zero. How can a hard drive provide a host device with file/directory listings when the drive isn't spinning? The next “leap” is Generalized Least Squares (GLS), of which the OLS is in fact a special case of. Indeed, GLS is the Gauss-Markov estimator and would lead to optimal inference, e.g. Another way you could proceed is to go up to the line right before I stopped to note there are two ways to proceed and to continue thus: I am not interested in a closed-form of $\mathbf{Q}$ when $\mathbf{X}$ is singular. min_x\;\left(y-Hx\right)'C^{-1}\left(y-Hx\right) Also, I would appreciate knowing about any errors you find in the arguments. In the next section we examine the properties of the ordinary least squares estimator when the appropriate model is the generalized least squares model. Use the above residuals to estimate the σij. Robust standard error in generalized least squares regression. Preferably well-known books written in standard notation. In which space does it operate? \begin{align} min_x\;\left(y-Hx\right)'\left(y-Hx\right) Exercise 4: Phylogenetic generalized least squares regression and phylogenetic generalized ANOVA. OLS models are a standard topic in a one-year social science statistics course and are better known among a wider audience. The other part goes away if $H'X=0$. This article serves as an introduction to GLS, with the following topics covered: Note, that in this article I am working from a Frequentist paradigm (as opposed to a Bayesian paradigm), mostly as a matter of convenience. A personal goal of mine is to encourage others in the field to take a similar approach. Browse other questions tagged least-squares generalized-least-squares efficiency or ask your own question ... 2020 Community Moderator Election Results. Why do most Christians eat pork when Deuteronomy says not to? \end{alignat} However, $X = C^{-1} - I$ is correct but misleading: $X$ is not defined that way, $C^{-1}$ is (because of its structure). $X$ is symmetric without assumptions, yes. I accidentally added a character, and then forgot to write them in for the rest of the series, Plausibility of an Implausible First Contact, Use of nous when moi is used in the subject. Thank you for your comment. When is a weighted average the same as a simple average? Too many to estimate with only T observations! \hat{x}_{OLS}=\left(H'C^{-1}H\right)^{-1}H'C^{-1}y Want to Be a Data Scientist? When does that re-weighting do nothing, on average? Instead we add the assumption V(y) = V where V is positive definite. Thanks for contributing an answer to Cross Validated! The ordinary least squares, or OLS, can also be called the linear least squares. \end{alignat} Computation of generalized least squares solutions of large sparse systems. Who first called natural satellites "moons"? Two: I'm wondering if you are assuming either that $y$ and the columns of $H$ are each zero mean or if you are assuming that one of the columns of $H$ is a column of 1s. 4.6.3 Generalized Least Squares (GLS). Economics 620, Lecture 11: Generalized Least Squares (GLS) Nicholas M. Kiefer Cornell University Professor N. M. Kiefer (Cornell University) Lecture 11: GLS 1 / 17 \end{align} \begin{align} GENERALIZED LEAST SQUARES THEORY Theorem 4.3 Given the specification (3.1), suppose that [A1] and [A3 ] hold. A 1-d endogenous response variable. The requirement is: Generalized Least Squares. 2. In FGLS, modeling proceeds in two stages: (1) the model is estimated by OLS or another consistent (but inefficient) estimator, and the residuals are used to build a consistent estimator of the errors covariance matrix (to do so, one often needs to examine the model adding additional constraints, for example if the errors follow a time series process, a statistician generally needs some theoretical assumptions on this process to ensure that a consistent estimator is available); and (2) using the consistent estimator of the covariance matrix of the errors, one can implement GLS ideas. 开一个生日会 explanation as to why 开 is used here? … The dependent variable. Show Source; Quantile regression; Recursive least squares; Example 2: Quantity theory of money ... 0.992 Method: Least Squares F-statistic: 295.2 Date: Fri, 06 Nov 2020 Prob (F-statistic): 6.09e-09 Time: 18:25:34 Log-Likelihood: -102.04 No. But I do am interested in understanding the concept beyond that expression: what is the actual role of $\mathbf{Q}$? However, I'm glad my intuition was correct in that GLS can be decomponsed in such a way, regardless if $X$ is invertible or not. What are these conditions? One way for this equation to hold is for it to hold for each of the two factors in the equation: In many situations (see the examples that follow), we either suppose, or the model naturally suggests, that is comprised of a nite set of parameters, say , and once is known, is also known. Lecture 24{25: Weighted and Generalized Least Squares 36-401, Fall 2015, Section B 19 and 24 November 2015 Contents 1 Weighted Least Squares 2 2 Heteroskedasticity 4 2.1 Weighted Least Squares as a Solution to Heteroskedasticity . The Maximum Likelihood (ML) estimate of $\mathbf{x}$, denoted with $\hat{\mathbf{x}}_{ML}$, is given by $$ . Aligning and setting the spacing of unit with their parameter in table. (Proof does not rely on Σ): Linear Regression is a statistical analysis for predicting the value of a quantitative variable. The other stuff, obviously, goes away if $H'X=0$. -\left(H'H\right)^{-1}H'XH\hat{x}_{GLS}\\ \left(H'\overline{c}C^{-1}H\right)^{-1}H'\overline{c}C^{-1}Y\\ This insight, by the way, if I am remembering correctly, is due to White(1980) and perhaps Huber(1967) before him---I don't recall exactly. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. The left-hand side above can serve as a test statistic for the linear hypothesis Rβo = r. &= \left(H'H\right)^{-1}H'C^{-1}H "puede hacer con nosotros" / "puede nos hacer". Weighted Least Squares Estimation (WLS) \left(I+\left(H'H\right)^{-1}H'XH\right)\hat{x}_{GLS}=& \hat{x}_{OLS} + \left(H'H\right)^{-1}H'Xy\\ I guess you could think of $Xy$ as $y$ suitably normalized--that is after having had the "bad" part of the variance $C$ divided out of it. \end{align} $$ I have a multiple regression model, which I can estimate either with OLS or GLS. Sometimes we take V = σ2Ωwith tr Ω= N As we know, = (X′X)-1X′y. Thus, the above expression is a closed form solution for the GLS estimator, decomposed into an OLS part and a bunch of other stuff. It should be very similar (in fact, almost identical) to what we see after performing a standard, OLS linear regression. Then the FGLS estimator βˆ FGLS =(X TVˆ −1 X)−1XTVˆ −1 Y. -H\left(H'C^{-1}H\right)^{-1}H'C^{-1}\right)y matrices by using the Moore-Penrose pseudo-inverse, but of course this is very far from a mathematical proof ;-). To be clear, one possible answer to your first question is this: Question: Can an equation similar to eq. The generalized least squares (GLS) estimator of the coefficients of a linear regression is a generalization of the ordinary least squares (OLS) estimator. We assume that: 1. has full rank; 2. ; 3. , where is a symmetric positive definite matrix. 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. Yes? Take a look, please see my previous piece on the subject. • To avoid the bias of inference based on OLS, we would like to estimate the unknown Σ. How to deal with matrix not having an inverse in ordinary least squares? &= \left(H'H\right)^{-1}H'\left(I+X\right)H\\ The next “leap” is Generalized Least Squares (GLS), of which the OLS is in fact a special case of. Ordinary Least Squares (OLS) solves the following problem: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. I still don't get much out of this. 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. \begin{align} There is no assumption involved in this equation, is there? By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Trend surfaces Fitting by Ordinary and Generalized Least Squares and Generalized Additive Models D G Rossiter Trend surfaces Models Simple regression OLS Multiple regression Diagnostics Higher-order GLS GLS vs. OLS … What this one says is that GLS is the weighted average of OLS and a linear regression of $Xy$ on $H$. As a final note on notation, $\mathbf{I}_K$ is the $K \times K$ identity matrix and $\mathbf{O}$ is a matrix of all zeros (with appropriate dimensions). $$ $$ In Section 2.5 the generalized least squares model is defined and the optimality of the generalized least squares estimator is established by Aitken’s theorem. Will grooves on seatpost cause rusting inside frame? This is a method for approximately determining the unknown parameters located in a linear regression model. What are those things on the right-hand-side of the double-headed arrows? It would be very unusual to assume neither of these things when using the linear model. DeepMind just announced a breakthrough in protein folding, what are the consequences? $$ Generalized Least Squares vs Ordinary Least Squares under a special case, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. (If it is known, you still do (X0X) 1X0Yto nd the coe cients, but you use the known constant when calculating t stats etc.) -\left(H'H\right)^{-1}H'XH\left(H'C^{-1}H\right)^{-1}H'C^{-1}y\\ There are two questions. Leading examples motivating nonscalar variance-covariance matrices include heteroskedasticity and first-order autoregressive serial correlation. And doesn't $X$, as the difference between two symmetric matrixes, have to be symmetric--no assumption necessary? What does the phrase, a person with “a pair of khaki pants inside a Manila envelope” mean.? Thus we have to either assume Σ or estimate Σ empirically. \begin{alignat}{3} If the question is, in your opinion, a bit too broad, or if there is something I am missing, could you please point me in the right direction by giving me references? For me, this type of theory-based insight leaves me more comfortable using methods in practice. \end{align} I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021, 5 Reasons You Don’t Need to Learn Machine Learning, Building Simulations in Python — A Step by Step Walkthrough, 5 Free Books to Learn Statistics for Data Science, A Collection of Advanced Visualization in Matplotlib and Seaborn with Examples, Review of the OLS estimator and conditions required for it to be BLUE, Mathematical set-up for Generalized Least Squares (GLS), Recovering the variance of the GLS estimator, Short discussion on relation to Weighted Least Squares (WLS), Methods and approaches for specifying covariance matrix, The topic of Feasible Generalized Least Squares, Relation to Iteratively Reweighted Least Squares (IRLS). Then βˆ GLS is the BUE for βo. It was the first thought I had, but, intuitively, it is a bit too hard problem and, if someone managed to actually solve it in closed form, a full-fledged theorem would be appropriate to that result. This article serves as a short introduction meant to “set the scene” for GLS mathematically. In GLS, we weight these products by the inverse of the variance of the errors. rev 2020.12.2.38097, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, The matrix inversion lemma in the form you use it relies on the matrix $\mathbf X$ being invertible. LEAST squares linear regression (also known as “least squared errors regression”, “ordinary least squares”, “OLS”, or often just “least squares”), 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. Parameters endog array_like. The next “leap” is Generalized Least Squares (GLS), of which the OLS is in fact a special case of. \begin{align} But, it has Tx(T+1)/2 parameters. 3. When the weights are uncorrelated with the things you are averaging. (1) \quad \hat{\mathbf{x}}_{ML} = (\mathbf{H}^T \mathbf{C^{-1}} \mathbf{H})^{-1} \mathbf{H}^T \mathbf{C}^{-1} \mathbf{y} Weighted least squares If one wants to correct for heteroskedasticity by using a fully efficient estimator rather than accepting inefficient OLS and correcting the standard errors, the appropriate estimator is weight least squares, which is an application of the more general concept of generalized least squares. Consider the simple case where $C^{-1}$ is a diagonal matrix, where each element on the main diagonal is of the form: $1 + x_{ii}$, with $x_{ii} > 1$. Two questions. This article serves as an introduction to GLS, with the following topics covered: Review of the OLS estimator and conditions required for it to be BLUE; Mathematical set-up for Generalized Least Squares (GLS) Recovering the GLS estimator . The proof is straigthforward and is valid even if $\mathbf{X}$ is singular. Where the classical assumptions hold, I know by the Gauss-Markov theorem that the BLU estimators for a linear regression model are given by OLS. Errors are uncorrelated 3. \begin{align} \end{align} This video provides an introduction to Weighted Least Squares, and provides some insight into the intuition behind this estimator. My question is about ordinary least squares (OLS), generalized least squares (GLS), and best linear unbiased (BLU) estimators. Where the classical assumptions hold, I know by the Gauss-Markov theorem that the BLU estimators for a linear regression model are given by OLS. Best way to let people know you aren't dead, just taking pictures? Let the estimator of V beVˆ = V (θˆ). I will only provide an answer here for a special case on the structure of $C$. Remembering that $C$, $C^{-1}$, and $I$ are all diagonal and denoting by $H_i$ the $i$th row of $H$: Now, my question is. Generalized Least Squares (GLS) solves the following problem: Ordinary Least Squares; Generalized Least Squares Generalized Least Squares. The weights for the GLS are estimated exogenously (the dataset for the weights is different from the dataset for the ... Browse other questions tagged least-squares weighted-regression generalized-least-squares or ask your own question. Matrix notation sometimes does hide simple things such as sample means and weighted sample means. Gradient descent and OLS (Ordinary Least Square) are the two popular estimation techniques for regression models. . 2 Generalized and weighted least squares 2.1 Generalized least squares Now we have the model Premises. Time-Series Regression and Generalized Least Squares in R* An Appendix to An R Companion to Applied Regression, third edition John Fox & Sanford Weisberg last revision: 2018-09-26 Abstract Generalized least-squares (GLS) regression extends ordinary least-squares (OLS) estimation However, if you can solve the problem with the last column of $H$ being all 1s, please do so, it would still be an important result. Anyway, if you have some intuition on the other questions I asked, feel free to add another comment. the unbiased estimator with minimal sampling variance. Intuitively, I would guess that you can extend it to non-invertible (positive-semidifenite?) 3. An example of the former is Weighted Least Squares Estimation and an example of the later is Feasible GLS (FGLS). 1. Making statements based on opinion; back them up with references or personal experience. .8 2.2 Some Explanations for Weighted Least Squares . As a final note, I am rather new to the world of Least Squares, since I generally work within a ML-framework (or MMSE in other cases) and never studied the deep aspects of GLLS vs OLS, since, in my case, they are just intermediate steps during the derivation of MLE for a given problem. Thus we have to either assume Σ or estimate Σ empirically. This question regards the problem of Generalized Least Squares. If the covariance of the errors $${\displaystyle \Omega }$$ is unknown, one can get a consistent estimate of $${\displaystyle \Omega }$$, say $${\displaystyle {\widehat {\Omega }}}$$, using an implementable version of GLS known as the feasible generalized least squares (FGLS) estimator. Asking for help, clarification, or responding to other answers. 1. Furthermore, other assumptions include: 1. Is it more efficient to send a fleet of generation ships or one massive one? LECTURE 11: GENERALIZED LEAST SQUARES (GLS) In this lecture, we will consider the model y = Xβ+ εretaining the assumption Ey = Xβ. Related. To learn more, see our tips on writing great answers. To see this, notice that the mean of $\frac{\overline{c}}{C_{ii}}$ is 1, by the construction of $\overline{c}$. . & \frac{1}{K} \sum_{i=1}^K H_iH_i'\left( \frac{\overline{c}}{C_{ii}}-1\right)=0\\~\\ However, we no longer have the assumption V(y) = V(ε) = σ2I. Don’t Start With Machine Learning. Why, when the weights are uncorrelated with the thing they are re-weighting! What is E ? squares which is an modification of ordinary least squares which takes into account the in-equality of variance in the observations. Second, there is a question about what it means when OLS and GLS are the same. \left(H'C^{-1}H\right)^{-1}H'C^{-1}Y &= \begin{align} A very detailed and complete answer, thanks! Unfortunately, no matter how unusual it seems, neither assumption holds in my problem. 8 Generalized least squares 9 GLS vs. OLS results 10 Generalized Additive Models. See statsmodels.tools.add_constant. This article serves as an introduction to GLS, with the following topics covered: Review of the OLS estimator and conditions required for it to be BLUE; Mathematical set-up for Generalized Least Squares (GLS) Recovering the GLS estimator $(3)$ (which "separates" an OLS-term from a second term) be written when $\mathbf{X}$ is a singular matrix? These assumptions are the same made in the Gauss-Markov theorem in order to prove that OLS is BLUE, except for … One: I'm confused by what you say about the equation $C^{-1}=I+X$. Proposition 1. squares which is an modification of ordinary least squares which takes into account the in-equality of variance in the observations. Should hardwood floors go all the way to wall under kitchen cabinets? exog array_like. Weighted Least Squares Estimation (WLS) What if the mathematical assumptions for the OLS being the BLUE do not hold? [This will require some additional assumptions on the structure of Σ] Compute then the GLS estimator with estimated weights wij. \begin{align} \end{align} Then, estimating the transformed model by OLS yields efficient estimates. The way to convert error function to matrix form in linear regression? . Sometimes we take V = σ2Ωwith tr Ω= N As we know, = (X′X)-1X′y. 1 Introduction to Generalized Least Squares Consider the model Y = X + ; ... back in the OLS case with the transformed variables if ˙is unknown. Consider the standard formula of Ordinary Least Squares (OLS) for a linear model, i.e. Based on a set of independent variables, we try to estimate the magnitude of a dependent variable which is the outcome variable. \left(I+\left(H'H\right)^{-1}H'XH\right) &= \left(H'H\right)^{-1}\left(H'H+H'XH\right)\\ Chapter 5 Generalized Least Squares 5.1 The general case Until now we have assumed that var e s2I but it can happen that the errors have non-constant variance or are correlated. Suppose instead that var e s2S where s2 is unknown but S is known Š in other words we know the correlation and relative variance between the errors but we don’t know the absolute scale. An intercept is not included by default and should be added by the user. \hat{x}_{GLS}=&\left(H'H\right)^{-1}H'y+\left(H'H\right)^{-1}H'Xy Least Squares Definition in Elements of Statistical Learning. First, we have a formula for the $\hat{x}_{GLS}$ on the right-hand-side of the last expression, namely $\left(H'C^{-1}H\right)^{-1}H'C^{-1}y$. 7. \end{align}, To form our intuitions, let's assume that $C$ is diagonal, let's define $\overline{c}$ by $\frac{1}{\overline{c}}=\frac{1}{K}\sum \frac{1}{C_{ii}}$, and let's write: Again, GLS is decomposed into an OLS part and another part. \left(H'\overline{c}C^{-1}H\right)^{-1} An example of the former is Weighted Least Squares Estimation and an example of the later is Feasible GLS (FGLS). &=\left( H'H\right)^{-1} & \iff& & H'\left(\overline{c}C^{-1}-I\right)H&=0\\ The setup and process for obtaining GLS estimates is the same as in FGLS , but replace Ω ^ with the known innovations covariance matrix Ω . If a dependent variable is a (*) \quad \mathbf{y} = \mathbf{Hx + n}, \quad \mathbf{n} \sim \mathcal{N}_{K}(\mathbf{0}, \mathbf{C}) . where $\mathbf{y} \in \mathbb{R}^{K \times 1}$ are the observables, $\mathbf{H} \in \mathbb{R}^{K \times N}$ is a known full-rank matrix, $\mathbf{x} \in \mathbb{R}^{N \times 1}$ is a deterministic vector of unknown parameters (which we want to estimate) and finally $\mathbf{n} \in \mathbb{R}^{K \times 1}$ is a disturbance vector (noise) with a known (positive definite) covariance matrix $\mathbf{C} \in \mathbb{R}^{K \times K}$. \hat{x}_{GLS}=& \hat{x}_{OLS} + \left(H'H\right)^{-1}H'Xy First, there is a purely mathematical question about the possibility of decomposing the GLS estimator into the OLS estimator plus a correction factor. A Monte Carlo study illustrates the performance of an ordinary least squares (OLS) procedure and an operational generalized least squares (GLS) procedure which accounts for and directly estimates the precision of the predictive model being fit. Make learning your daily ritual. Why do Arabic names still have their meanings? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. MathJax reference. I hope the above is insightful and helpful. leading to the solution: leading to the solution: Let $N,K$ be given integers, with $K \gg N > 1$. In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. \begin{align} As I’ve mentioned in some of my previous pieces, it’s my opinion not enough folks take the time to go through these types of exercises. research. \end{align} Convert negadecimal to decimal (and back). Compute βˆ OLS and the residuals rOLS i = Yi −X ′ i βˆ OLS. \hat{x}_{GLS}=& \left(I+\left(H'H\right)^{-1}H'XH\right)^{-1}\left(\hat{x}_{OLS} + \left(H'H\right)^{-1}H'Xy\right) The error variances are homoscedastic 2. However, we no longer have the assumption V(y) = V(ε) = σ2I. \begin{alignat}{3} 0=&2\left(H'XH\hat{x}_{GLS}-H'Xy\right) +2\left(H'H\hat{x}_{GLS}-H'y\right)\\ My question is about ordinary least squares (OLS), generalized least squares (GLS), and best linear unbiased (BLU) estimators. What is E ? There are 3 different perspective… Feasible Generalized Least Squares The assumption that is known is, of course, a completely unrealistic one. uniformly most powerful tests, on the e ffect of the legislation. In this special case, OLS and GLS are the same if the inverse of the variance (across observations) is uncorrelated with products of the right-hand-side variables with each other and products of the right-hand-side variables with the left-hand-side variable. The solution is still characterized by first order conditions since we are assuming that $C$ and therefore $C^{-1}$ are positive definite: LECTURE 11: GENERALIZED LEAST SQUARES (GLS) In this lecture, we will consider the model y = Xβ+ εretaining the assumption Ey = Xβ. A nobs x k array where nobs is the number of observations and k is the number of regressors. 2. \begin{align} This is a very intuitive result. Anyway, thanks again! Definition 4.7. by Marco Taboga, PhD. -\left(H'H\right)^{-1}H'XH\hat{x}_{GLS}\\ I can see two ways to give you what you asked for in the question from here. Generalized Least Squares vs Ordinary Least Squares under a special case. out, the unadjusted OLS standard errors often have a substantial downward bias. \hat{x}_{GLS}=& \hat{x}_{OLS} + \left(H'H\right)^{-1}H'X \left(I and this is also the standard formula of Generalized Linear Least Squares (GLLS). I’m planning on writing similar theory based pieces in the future, so feel free to follow me for updates! You would write that matrix as $C^{-1} = I + X$. The problem is, as usual, that we don’t know σ2ΩorΣ. The Feasible Generalized Least Squares (GLS) proceeds in 2 steps: 1. However,themoreefficient estimator of equation (1) would be generalized least squares (GLS) if Σwere known. ... the Pooled OLS is worse than the others. \begin{align} Ordinary least squares (OLS) regression, in its various forms (correlation, multiple regression, ANOVA), is the most common linear model analysis in the social sciences. I found this slightly counter-intuitive, since you know a lot more in GLLS (you know $\mathbf{C}$ and make full use of it, why OLS does not), but this is somehow "useless" if some conditions are met. The linear regression iswhere: 1. is an vector of outputs ( is the sample size); 2. is an matrix of regressors (is the number of regressors); 3. is the vector of regression coefficients to be estimated; 4. is an vector of error terms. Note: We used (A3) to derive our test statistics. I found this problem during a numerical implementation where both OLS and GLLS performed roughly the same (the actual model is $(*)$), and I cannot understand why OLS is not strictly sub-optimal. \end{align}. It is quantitative Ordinary least squares is a technique for estimating unknown parameters in a linear regression model. Weighted least squares play an important role in the parameter estimation for generalized linear models. 82 CHAPTER 4. Suppose the following statistical model holds Unfortunately, the form of the innovations covariance matrix is rarely known in practice. This occurs, for example, in the conditional distribution of individual income given years of schooling where high levels of schooling correspond to relatively high levels of the conditional variance of income. I should be careful and verify that the matrix I inverted in the last step is actually invertible: If $\mathbf{H}^T\mathbf{X} = \mathbf{O}_{N,K}$, then equation $(1)$ degenerates in equation $(2)$, i.e., there exists no difference between GLLS and OLS. . $Q = (H′H)^{−1}H′X(I−H(H′C^{−1}H)^{−1}H′C^{−1})$ does seem incredibly obscure. \end{align}, The question here is when are GLS and OLS the same, and what intuition can we form about the conditions under which this is true? (For a more thorough overview of OLS, the BLUE, and the Gauss-Markov Theorem, please see my previous piece on the subject). H'\overline{c}C^{-1}Y&=H'Y & \iff& & H'\left(\overline{c}C^{-1}-I\right)Y&=0 This is known as Generalized Least Squares (GLS), and for a known innovations covariance matrix, of any form, it is implemented by the Statistics and Machine Learning Toolbox™ function lscov. H'\left(\overline{c}C^{-1}-I\right)H&=0 & \iff& \left(H'C^{-1}H\right)^{-1}H'C^{-1}Y = \left( H'H\right)^{-1}H'Y Thus, the difference between OLS and GLS is the assumptions of the error term of the model. The feasible generalized least squares (FGLS) model is the same as the GLS estimator except that V = V (θ) is a function of an unknown q×1vectorof parameters θ. Finally, we are ready to say something intuitive. Now, make the substitution $C^{-1}=X+I$ in the GLS problem: There’s plenty more to be covered, including (but not limited to): I plan on covering these topics in-depth in future pieces.
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