Chapter 3: The Multiple Linear Regression Model


 Emory Fleming
 1 years ago
 Views:
Transcription
1 Chapter 3: The Multiple Linear Regression Model Advanced Econometrics  HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
2 Section 1 Introduction Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
3 1. Introduction The objectives of this chapter are the following: 1 De ne the multiple linear regression model. 2 Introduce the ordinary least squares (OLS) estimator. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
4 1. Introduction The outline of this chapter is the following: Section 2: The multiple linear regression model Section 3: The ordinary least squares estimator Section 4: Statistical properties of the OLS estimator Subsection 4.1: Finite sample properties Subsection 4:2: Asymptotic properties Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
5 1. Introduction References Amemiya T. (1985), Advanced Econometrics. Harvard University Press. Greene W. (2007), Econometric Analysis, sixth edition, Pearson  Prentice Hil (recommended) Pelgrin, F. (2010), Lecture notes Advanced Econometrics, HEC Lausanne (a special thank) Ruud P., (2000) An introduction to Classical Econometric Theory, Oxford University Press. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
6 1. Introduction Notations: In this chapter, I will (try to...) follow some conventions of notation. f Y (y) F Y (y) Pr () y Y probability density or mass function cumulative distribution function probability vector matrix Be careful: in this chapter, I don t distinguish between a random vector (matrix) and a vector (matrix) of deterministic elements. For more appropriate notations, see: Abadir and Magnus (2002), Notation in econometrics: a proposal for a standard, Econometrics Journal. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
7 Section 2 The Multiple Linear Regression Model Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
8 2. The Multiple Linear Regression Model Objectives 1 De ne the concept of Multiple linear regression model. 2 Semiparametric and Parametric multiple linear regression model. 3 The multiple linear Gaussian model. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
9 2. The Multiple Linear Regression Model De nition (Multiple linear regression model) The multiple linear regression model is used to study the relationship between a dependent variable and one or more independent variables. The generic form of the linear regression model is y = x 1 β 1 + x 2 β x K β K + ε where y is the dependent or explained variable and x 1,.., x K are the independent or explanatory variables. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
10 2. The Multiple Linear Regression Model Notations 1 y is the dependent variable, the regressand or the explained variable. 2 x j is an explanatory variable, a regressor or a covariate. 3 ε is the error term or disturbance. IMPORTANT: do not use the term "residual".. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
11 2. The Multiple Linear Regression Model Notations (cont d) The term ε is a random disturbance, so named because it disturbs an otherwise stable relationship. The disturbance arises for several reasons: 1 Primarily because we cannot hope to capture every in uence on an economic variable in a model, no matter how elaborate. The net e ect, which can be positive or negative, of these omitted factors is captured in the disturbance. 2 There are many other contributors to the disturbance in an empirical model. Probably the most signi cant is errors of measurement. It is easy to theorize about the relationships among precisely de ned variables; it is quite another to obtain accurate measures of these variables. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
12 2. The Multiple Linear Regression Model Notations (cont d) We assume that each observation in a sample fy i, x i1, x i2..x ik g for i = 1,.., N is generated by an underlying process described by Remark: y i = x i1 β 1 + x i2 β x ik β K + ε i x ik = value of the k th explanatory variable for the i th unit of the sample x unit,variable Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
13 2. The Multiple Linear Regression Model Notations (cont d) Let the N 1 column vector x k be the N observations on variable x k, for k = 1,.., K. Let assemble these data in an N K data matrix, X. Let y be the N 1 column vector of the N observations, y 1, y 2,.., y N. Let ε be the N 1 column vector containing the N disturbances. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
14 2. The Multiple Linear Regression Model Notations (cont d) y = N 1 0 y 1 y 2.. y i.. y N 1 C A x k = N 1 0 x 1k x 2k.. x ik.. x Nk 1 C A ε = N 1 0 ε 1 ε 2.. ε i.. ε N 1 C A β = K 1 0 β 1 β 2.. β K 1 C A Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
15 2. The Multiple Linear Regression Model Notations (cont d) or equivalently X = N K X = (x 1 : x 2 :.. : x K ) N K 0 x 11 x 12.. x 1k.. x 1K x 21 x 22.. x 2k.. x 2K x i1 x i2.. x ik.. x ik x N1 x N2.. x Nk.. x NK 1 C A Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
16 2. The Multiple Linear Regression Model Fact In most of cases, the rst column of X is assumed to be a column of 1s so that β 1 is the constant term in the model. X = N K 0 x 1 = 1 N 1 N 1 1 x 12.. x 1k.. x 1K 1 x 22.. x 2k.. x 2K x i2.. x ik.. x ik x N2.. x Nk.. x NK 1 C A Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
17 2. The Multiple Linear Regression Model Remark More generally, the matrix X may as well contain stochastic and non stochastic elements such as: Constant; Time trend; Dummy variables (for speci c episodes in time); Etc. Therefore, X is generally a mixture of xed and random variables. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
18 2. The Multiple Linear Regression Model De nition (Simple linear regression model) The simple linear regression model is a model with only one stochastic regressor: K = 1 if there is no constant or K = 2 if there is a constant: y i = β 1 x i + ε i y i = β 1 + β 2 x i2 + ε i for i = 1,.., N, or y = β 1 + β 2 x 2 + ε Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
19 2. The Multiple Linear Regression Model De nition (Multiple linear regression model) The multiple linear regression model can be written y = X N 1 β N K K 1 + ε N 1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
20 2. The Multiple Linear Regression Model One key di erence for the speci cation of the MLRM: Parametric/semiparametric speci cation Parametric model: the distribution of the error terms is fully characterized, e.g. ε N (0, Ω) SemiParametric speci cation: only a few moments of the error terms are speci ed, e.g. E (ε) = 0 and V (ε) = E εε > = Ω. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
21 2. The Multiple Linear Regression Model This di erence does not matter for the derivation of the ordinary least square estimator But this di erence matters for (among others): 1 The characterization of the statistical properties of the OLS estimator (e.g., e ciency); 2 The choice of alternative estimators (e.g., the maximum likelihood estimator); 3 Etc. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
22 2. The Multiple Linear Regression Model De nition (Semiparametric multiple linear regression model) The semiparametric multiple linear regression model is de ned by where the error term ε satis es y = Xβ + ε E ( εj X) = 0 N 1 V ( εj X) = σ 2 I N N N and I N is the identity matrix of order N. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
23 2. The Multiple Linear Regression Model Remarks 1 If the matrix X is non stochastic ( xed), i.e. there are only xed regressors, then the conditions on the error term u read: E (ε) = 0 V (ε) = σ 2 I N 2 If the (conditional) variance covariance matrix of ε is not diagonal, i.e. if V ( εj X) = Ω the model is called the Multiple Generalized Linear Regression Model Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
24 2. The Multiple Linear Regression Model Remarks (cont d) The two conditions on the error term ε E ( εj X) = 0 N 1 V ( εj X) = σ 2 I N are equivalent to E (yj X) = Xβ V (yj X) = σ 2 I N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
25 2. The Multiple Linear Regression Model De nition (The multiple linear Gaussian model) The (parametric) multiple linear Gaussian model is de ned by y = Xβ + ε where the error term ε is normally distributed ε N 0, σ 2 I N As a consequence, the vector y has a conditional normal distribution with yj X N Xβ, σ 2 I N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
26 2. The Multiple Linear Regression Model Remarks 1 The multiple linear Gaussian model is (by de nition) a parametric model. 2 If the matrix X is non stochastic ( xed), i.e. there are only xed regressors, then the vector y has marginal normal distribution: y N Xβ, σ 2 I N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
27 2. The Multiple Linear Regression Model The classical linear regression model consists of a set of assumptions that describes how the data set is produced by a data generating process (DGP) Assumption 1: Linearity Assumption 2: Full rank condition or identi cation Assumption 3: Exogeneity Assumption 4: Spherical error terms Assumption 5: Data generation Assumption 6: Normal distribution Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
28 2. The Multiple Linear Regression Model De nition (Assumption 1: Linearity) The model is linear with respect to the parameters β 1,.., β K. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
29 Remarks The model speci es a linear relationship between the dependent variable and the regressors. For instance, the models are all linear (with respect to β). y = β 0 + β 1 x + u y = β 0 + β 1 cos (x) + v y = β 0 + β 1 1 x + w In contrast, the model y = β 0 + β 1 x β 2 + ε is non linear Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
30 2. The Multiple Linear Regression Model Remark The model can be linear after some transformations. Starting from y = Ax β exp (ε), one has a loglinear speci cation: ln (y) = ln (A) + β ln (x) + ε Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
31 2. The Multiple Linear Regression Model De nition (Loglinear model) The loglinear model is ln (y i ) = β 1 ln (x i1 ) + β 2 ln (x i2 ) β K ln (x ik ) + ε i This equation is also known as the constant elasticity form as in this equation, the elasticity of y with respect to changes in x does not vary with x ik : β k = ln (y i ) ln (x ik ) = y i x ik x ik y i Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
32 2. The Multiple Linear Regression Model The classical linear regression model consists of a set of assumptions that describes how the data set is produced by a data generating process (DGP) Assumption 1: Linearity Assumption 2: Full rank condition or identi cation Assumption 3: Exogeneity Assumption 4: Spherical error terms Assumption 5: Data generation Assumption 6: Normal distribution Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
33 2. The Multiple Linear Regression Model De nition (Assumption 2: Full column rank) X is an N K matrix with rank K. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
34 2. The Multiple Linear Regression Model Interpretation 1 There is no exact relationship among any of the independent variables in the model. 2 The columns of X are linearly independent. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
35 2. The Multiple Linear Regression Model Example Suppose that a crosssection model satis es: y i = β 0 + β 1 non labor income i + β 2 salary i +β 3 total income i + ε i The identi cation condition does not hold since total income is exactly equal to salary plus non labor income (exact linear dependency in the model). Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
36 2. The Multiple Linear Regression Model Remarks 1 Perfect multicollinearity is generally not di cult to spot and is signalled by most statistical software. 2 Imperfect multicollinearity is a more serious issue (see further). Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
37 2. The Multiple Linear Regression Model De nition (Identi cation) The multiple linear regression model is identi able if and only if one the following equivalent assertions holds: (i) rank (X) = K (ii) The matrix X > X is invertible (iii) The columns of X form a basis of L (X) (iv) Xβ 1 = Xβ 2 =) β 1 = β 2 (v) Xβ = 0 =) β = 0 8β 2 R K (vi) ker (X) = f0g 8 (β 1, β 2 ) 2 R K R K Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
38 2. The Multiple Linear Regression Model The classical linear regression model consists of a set of assumptions that describes how the data set is produced by a data generating process (DGP) Assumption 1: Linearity Assumption 2: Full rank condition or identi cation Assumption 3: Exogeneity Assumption 4: Spherical error terms Assumption 5: Data generation Assumption 6: Normal distribution Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
39 2. The Multiple Linear Regression Model De nition (Assumption 3: Strict exogeneity of the regressors) The regressors are exogenous in the sense that: E ( εj X) = 0 N 1 or equivalently for all the units i 2 f1,..ng E ( ε i j X) = 0 or equivalently E ( ε i j x jk ) = 0 for any explanatory variable k 2 f1,..k g and any unit j 2 f1,..ng. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
40 2. The Multiple Linear Regression Model CommentsComments 1 The expected value of the error term at observation i (in the sample) is not a function of the independent variables observed at any observation (including the i th observation). The independent variables are not predictors of the error terms. 2 The strict exogeneity condition can be rewritten as: E (y j X) = Xβ 3 If the regressors are xed, this condition can be rewritten as: E (ε) = 0 N 1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
41 2. The Multiple Linear Regression Model Implications The (strict) exogeneity condition E ( εj X) = 0 N 1 has two implications: 1 The zero conditional mean of ε implies that the unconditional mean of u is also zero (the reverse is not true): E (ε) = E X (E ( εj X)) = E X (0) = 0 2 The zero conditional mean of ε implies that (the reverse is not true): E (ε i x jk ) = 0 8i, j, k or Cov (ε i, X) = 0 8i Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
42 2. The Multiple Linear Regression Model The classical linear regression model consists of a set of assumptions that describes how the data set is produced by a data generating process (DGP) Assumption 1: Linearity Assumption 2: Full rank condition or identi cation Assumption 3: Exogeneity Assumption 4: Spherical error terms Assumption 5: Data generation Assumption 6: Normal distribution Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
43 2. The Multiple Linear Regression Model De nition (Assumption 4: Spherical disturbances) The error terms are such that: and V ( ε i j X) = E ε 2 i X = σ 2 for all i 2 f1,..ng Cov ( ε i, ε j j X) = E ( ε i ε j j X) = 0 for all i 6= j The condition of constant variances is called homoscedasticity. The uncorrelatedness across observations is called nonautocorrelation. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
44 2. The Multiple Linear Regression Model Comments 1 Spherical disturbances = homoscedasticity + nonautocorrelation 2 If the errors are not spherical, we call them nonspherical disturbances. 3 The assumption of homoscedasticity is a strong one: this is the exception rather than the rule! Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
45 2. The Multiple Linear Regression Model Comments Let us consider the (conditional) variance covariance matrix of the error terms: V ( εj X) = E εε > X {z } {z } N N N N 0 E ε X E ( ε 1 ε 2 j X).. E ( ε 1 ε j j X).. E ( ε 1 ε N j X) E ( ε 2 ε 1 j X) E ε 2 2 X.. E ( ε 2 ε j j X).. E ( ε 2 ε N j X) = B E ( ε i ε 1 j X).. E ( ε i ε j j X).. E ( ε i ε N j X) A E ( ε N ε 1 j X).. E ( ε N ε j j X).. E ε 2 N X Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
46 2. The Multiple Linear Regression Model Comments The two assumptions (homoscedasticity and nonautocorrelation) imply that: V ( εj X) = E εε > X = σ 2 I N {z } {z } N N N N 0 = σ σ σ 2 1 C A Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
47 2. The Multiple Linear Regression Model The classical linear regression model consists of a set of assumptions that describes how the data set is produced by a data generating process (DGP) Assumption 1: Linearity Assumption 2: Full rank condition or identi cation Assumption 3: Exogeneity Assumption 4: Spherical error terms Assumption 5: Data generation Assumption 6: Normal distribution Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
48 2. The Multiple Linear Regression Model De nition (Assumption 5: Data generation) The data in (x i1 x i2...x ik ) may be any mixture of constants and random variables. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
49 2. The Multiple Linear Regression Model Comments 1 Analysis will be done conditionally on the observed X, so whether the elements in X are xed constants or random draws from a stochastic process will not in uence the results. 2 In the case of stochastic regressors, the unconditional statistical properties of are obtained in two steps: (1) using the result conditioned on X and (2) nding the unconditional result by averaging (i.e., integrating over) the conditional distributions. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
50 2. The Multiple Linear Regression Model Comments Assumptions regarding (x i1 x i2...x ik y i ) for i = 1,.., N is also required. This is a statement about how the sample is drawn. In the sequel, we assume that (x i1 x i2...x ik y i ) for i = 1,.., N are independently and identically distributed (i.i.d). The observations are drawn by a simple random sampling from a large population. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
51 2. The Multiple Linear Regression Model The classical linear regression model consists of a set of assumptions that describes how the data set is produced by a data generating process (DGP) Assumption 1: Linearity Assumption 2: Full rank condition or identi cation Assumption 3: Exogeneity Assumption 4: Spherical error terms Assumption 5: Data generation Assumption 6: Normal distribution Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
52 2. The Multiple Linear Regression Model De nition (Assumption 6: Normal distribution) The disturbances are normally distributed. ε i j X N 0, σ 2 or equivalently εj X N 0 N 1, σ 2 I N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
53 2. The Multiple Linear Regression Model Comments 1 Once again, this is a convenience that we will dispense with after some analysis of its implications. 2 Normality is not necessary to obtain many of the results presented below. 3 Assumption 6 implies assumptions 3 (exogeneity) and 4 (spherical disturbances). Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
54 2. The Multiple Linear Regression Model Summary The main assumptions of the multiple linear regression model A1: linearity The model is linear with β A2: identi cation X is an N K matrix with rank K A3: exogeneity E ( εj X) = 0 N 1 A4: spherical error terms V ( εj X) = σ 2 I N A5: data generation X may be xed or random A6: normal distribution εj X N 0 N 1, σ 2 I N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
55 2. The Multiple Linear Regression Model Key Concepts 1 Simple linear regression model 2 Multiple linear regression model 3 Semiparametric multiple linear regression model 4 Multiple linear Gaussian model 5 Assumptions of the multiple linear regression model 6 Linearity (A1), Identi cation (A2), Exogeneity (A3), Spherical error terms (A4), Data generation (A5) and Normal distribution (A6) Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
56 Section 3 The ordinary least squares estimator Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
57 3. The ordinary least squares estimator Introduction 1 The simple linear regression model assumes that the following speci cation is true in the population: y = Xβ + ε where other unobserved factors determining y are captured by the error term ε. 2 Consider a sample fx i1, x i2,.., x ik, y i g N i=1 of i.i.d. random variables (be careful to the change of notations here) and only one realization of this sample (your data set). 3 How to estimate the vector of parameters β? Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
58 3. The ordinary least squares estimator Introduction (cont d) 1 If we assume that assumptions A1A6 hold, we have a multiple linear Gaussian model (parametric model), and a solution is to use the MLE. The MLE estimator for β coincides to the ordinary least squares (OLS) estimator (cf. chapter 2). 2 If we assume that only assumptions A1A5 hold, we have a semiparametric multiple linear regression model, the MLE is unfeasible. 3 In this case, the only solution is to use the ordinary least squares estimator (OLS). Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
59 3. The ordinary least squares estimator Intuition Let us consider the simple linear regression model and for simplicity denote x i = x i2 : y i = β 1 + β 2 x i + ε i The general idea of the OLS consists in minimizing the distance between the points (x i, y i ) and the regression line by i = bβ 1 + bβ 2 x i or the points (x i, by i ) for all i = 1,.., N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
60 3. The ordinary least squares estimator Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
61 3. The ordinary least squares estimator Intuition (cont d) Estimates of β 1 and β 2 are chosen by minimizing the sum of the squared residuals (SSR): N This SSR can be written as: N i=1 i=1 bε 2 i 2 bε 2 i = y i bβ 1 bβ 2 x i Therefore, bβ 1 and bβ 2 are the solutions of the minimization problem b β 1, bβ 2 = arg min (β 1,β 2 ) N i=1 (y i β 1 β 2 x i ) 2 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
62 3. The ordinary least squares estimator De nition (OLS  simple linear regression model) In the simple linear regression model y i = β 1 + β 2 x i + ε i, the OLS estimators bβ 1 and bβ 2 are the solutions of the minimization problem b β 1, bβ 2 = arg min (β 1,β 2 ) N i=1 (y i β 1 β 2 x i ) 2 The solutions are: bβ 1 = y N bβ 2 x N bβ 2 = N i=1 (x i x N ) (y i y N ) N i=1 (x i x N ) 2 where y N = N 1 N i=1 y i and x N = N 1 N i=1 x i respectively denote the sample mean of the dependent variable y and the regressor x. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
63 3. The ordinary least squares estimator Remark The OLS estimator is a linear estimator (cf. chapter 1) since it can be expressed as a linear function of the observations y i : with in the case where y N = 0. ω i = bβ 2 = N ω i y i i=1 (x i x N ) N i=1 (x i x N ) 2 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
64 3. The ordinary least squares estimator De nition (Fitted value) The predicted or tted value for observation i is: by i = bβ 1 + bβ 2 x i with a sample mean equal to the sample average of the observations by N = 1 N N by i = y N = 1 N i=1 N y i i=1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
65 3. The ordinary least squares estimator De nition (Fitted residual) The residual for observation i is: bε i = y i bβ 1 bβ 2 x i with a sample mean equal to zero by de nition. bε N = 1 N N i=1 bε i = 0 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
66 3. The ordinary least squares estimator Remarks 1 The t of the regression is good if the sum N i=1 bε2 i (or SSR) is small, i.e., the unexplained part of the variance of y is small. 2 The coe cient of determination or R 2 is given by: R 2 = N i=1 (by i y N ) 2 N N i=1 (y i y N ) 2 = 1 i=1 bε2 i N i=1 (y i y N ) 2 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
67 3. The ordinary least squares estimator Orthogonality conditions Under assumption A3 (strict exogeneity), we have E ( ε i j x i ) = 0. This condition implies that: E (ε i ) = 0 E (ε i x i ) = 0 Using the sample analog of this moment conditions (cf. chapter 6, GMM), one has: N 1 N y i i=1 bβ 1 bβ 2 x i = 0 N 1 N y i i=1 bβ 1 bβ 2 x i x i = 0 This is a system of two equations and two unknowns bβ 1 and bβ 2. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
68 3. The ordinary least squares estimator De nition (Orthogonality conditions) The ordinary least squares estimator can be de ned from the two sample analogs of the following moment conditions: E (ε i ) = 0 E (ε i x i ) = 0 The corresponding system of equations is justidenti ed. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
69 3. The ordinary least squares estimator OLS and multiple linear regression model Now consider the multiple linear regression model or y i = y = Xβ + ε K β k x ik + ε i k=1 Objective: nd an estimator (estimate) of β 1, β 2,.., β K and σ 2 under the assumptions A1A5. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
70 3. The ordinary least squares estimator OLS and multiple linear regression model Di erent methods: 1 Minimize the sum of squared residuals (SSR) 2 Solve the same minimization problem with matrix notation. 3 Use moment conditions. 4 Geometrical interpretation Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
71 3. The ordinary least squares estimator 1. Minimize the sum of squared residuals (SSR): As in the simple linear regression, bβ = arg min β N i=1 ε 2 i = arg min β N i=1 y i! 2 K β k x ik k=1 One can derive the rst order conditions with respect to β k for k = 1,.., K and solve a system of K equations with K unknowns. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
72 3. The ordinary least squares estimator 2. Using matrix notations: De nition (OLS and multiple linear regression model) In the multiple linear regression model y i = x > i β + ε i, with x i = (x i1,.., x ik ) >, the OLS estimator b β is the solution of bβ = arg min β The OLS estimators of β is: bβ = N x i x i > i=1 N i=1 y i! 1 2 x i > β! N x i y i i=1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
73 3. The ordinary least squares estimator 2. Using matrix notations: De nition (Normal equations) Under suitable regularity conditions, in the multiple linear regression model y i = x > i β + ε i, with x i = (x i1 :.. : x ik ) >, the normal equations are N x i y i x i > i=1 bβ = 0 K 1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
74 3. The ordinary least squares estimator 2. Using matrix notations: De nition (OLS and multiple linear regression model) In the multiple linear regression model y = Xβ + ε, the OLS estimator b β is the solution of the minimization problem bβ = arg min β ε > ε = arg min β (y Xβ) > (y Xβ) The OLS estimators of β is: bβ = 1 X > X X > y Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
75 3. The ordinary least squares estimator 2. Using matrix notations: De nition The ordinary least squares estimator β b of β minimizes the following criteria s (β) = k(y Xβ)k 2 I N = (y Xβ) > (y Xβ) Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
76 3. The ordinary least squares estimator 2. Using matrix notations: The FOC (normal equations) are de ned by: s (β) β = 2 X > bβ {z} y X b β K N {z } N 1 = 0 K 1 The secondorder conditions hold: s (β) β β > = 2 X {z > X} is de nite positive bβ K K since by de nition X > X is a positive de nite matrix. We have a minimum. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
77 3. The ordinary least squares estimator 2. Using matrix notations: De nition (Normal equations) Under suitable regularity conditions, in the multiple linear regression model y = Xβ + ε, the normal equations are given by: X > {z} (y Xβ) {z } K N N 1 = 0 K 1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
78 3. The ordinary least squares estimator De nition (Unbiased variance estimator) In the multiple linear regression model y = Xβ + ε, the unbiased estimator of σ 2 is given by: bσ 2 = 1 N K N i=1 bε 2 i SSR N K Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
79 3. The ordinary least squares estimator 2. Using matrix notations: The estimator bσ 2 can also be written as: bσ 2 = 1 N K N y i i=1 x > i 2 bβ bσ 2 = (y Xβ)> (y Xβ) N K bσ 2 = k(y Xβ)k2 I N N K Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
80 3. The ordinary least squares estimator 3. Using moment conditions: Under assumption A3 (strict exogeneity), we have E ( εj X) = 0. This condition implies: E (ε i x i ) = 0 K 1 with x i = (x i1 :.. : x ik ) >. Using the sample analogs, one has: N 1 N x i y i x i > i=1 bβ = 0 K 1 We have K (normal) equations with K unknown parameters bβ 1,.., bβ K. The system is just identi ed. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
81 3. The ordinary least squares estimator 4. Geometric interpretation: 1 The ordinary least squares estimation methods consists in determining the adjusted vector, by, which is the closest to y (in a certain space...) such that the squared norm between y and by is minimized. 2 Finding by is equivalent to nd an estimator of β. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
82 3. The ordinary least squares estimator 4. Geometric interpretation: De nition (Geometric interpretation) The adjusted vector, by, is the (orthogonal) projection of y onto the column space of X. The tted error terms, bε, is the projection of y onto the orthogonal space engendered by the column space of X. The vectors by and bε are orthogonal. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
83 3. The ordinary least squares estimator 4. Geometric interpretation: Source: F. Pelgrin (2010), Lecture notes, Advanced Econometrics Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
84 3. The ordinary least squares estimator 4. Geometric interpretation: De nition (Projection matrices) The vectors by and bε are de ned to be: by = P y bε = M y where P and M denote the two following projection matrices: 1 P = X X > X X > 1 M = I N P = I N X X > X X > Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
85 3. The ordinary least squares estimator Other geometric interpretations: Suppose that there is a constant term in the model. 1 The least squares residuals sum to zero: N i=1 bε i = 0 2 The regression hyperplane passes through the point of means of the data (x N, y N ). 3 The mean of the tted (adjusted) values of y equals the mean of the actual values of y: by N = y N Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
86 3. The ordinary least squares estimator De nition (Coe cient of determination) The coe cient of determination of the multiple linear regression model (with a constant term) is the ratio of the total (empirical) variance explained by model to the total (empirical) variance of y: R 2 = N i=1 (by i y N ) 2 N N i=1 (y i y N ) 2 = 1 i=1 bε2 i N i=1 (y i y N ) 2 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
87 3. The ordinary least squares estimator Remark 1 The coe cient of determination measures the proportion of the total variance (or variability) in y that is accounted for by variation in the regressors (or the model). 2 Problem: the R 2 automatically and spuriously increases when extra explanatory variables are added to the model. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
88 3. The ordinary least squares estimator De nition (Adjusted Rsquared) The adjusted Rsquared coe cient is de ned to be: R 2 = 1 N 1 N p 1 1 R2 where p denotes the number of regressors (not counting the constant term, i.e., p = K 1 if there is a constant or p = K otherwise). Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
89 3. The ordinary least squares estimator Remark One can show that 1 R 2 <R 2 2 if N is large R 2 'R 2 3 The adjusted Rsquared R 2 can be negative. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
90 3. The ordinary least squares estimator Key Concepts 1 OLS estimator and estimate 2 Fitted or predicted value 3 Residual or tted residual 4 Orthogonality conditions 5 Normal equations 6 Geometric interpretations of the OLS 7 Coe cient of determination and adjusted Rsquared Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
91 Section 4 Statistical properties of the OLS estimator Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
92 4. Statistical properties of the OLS estimator In order to study the statistical properties of the OLS estimator, we have to distinguish (cf. chapter 1): 1 The nite sample properties 2 The large sample or asymptotic properties Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
93 4. Statistical properties of the OLS estimator But, we have also to distinguish the properties given the assumptions made on the linear regression model 1 Semiparametric linear regression model (the exact distribution of ε is unknown) versus parametric linear regression model (and especially Gaussian linear regression model, assumption A6). 2 X is a matrix of random regressors versus X is a matrix of xed regressors. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
94 4. Statistical properties of the OLS estimator Fact (Assumptions) In the rest of this section, we assume that assumptions A1A5 hold. A1: linearity The model is linear with β A2: identi cation X is an N K matrix with rank K A3: exogeneity E ( εj X) = 0 N 1 A4: spherical error terms V ( εj X) = σ 2 I N A5: data generation X may be xed or random Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
95 Subsection 4.1. Finite sample properties of the OLS estimator Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
96 4.1. Finite sample properties Objectives The objectives of this subsection are the following: 1 Compute the two rst moments of the (unknown) nite sample distribution of the OLS estimators b β and bσ 2 2 Determine the nite sample distribution of the OLS estimators b β and bσ under particular assumptions (A6). 3 Determine if the OLS estimators are "good": e cient estimator versus BLUE. 4 Introduce the GaussMarkov theorem. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
97 4.1. Finite sample properties First moments of the OLS estimators Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
98 4.1. Finite sample properties Moments In a rst step, we will derive the rst moments of the OLS estimators 1 Step 1: compute E bβ and V bβ 2 Step 2: compute E bσ 2 and V bσ 2 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
99 4.1. Finite sample properties De nition (Unbiased estimator) In the multiple linear regression model y = Xβ 0 + ε, under the assumption A3 (strict exogeneity), the OLS estimator β b is unbiased: E bβ = β 0 where β 0 denotes the true value of the vector of parameters. This result holds whether or not the matrix X is considered as random. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
100 4.1. Finite sample properties Proof Case 1: xed regressors (cf. chapter 1) bβ = 1 1 X > X X > y = β 0 + X > X X > ε So, if X is a matrix of xed regressors: 1 E bβ = β 0 + X > X X > E (ε) Under assumption A3 (exogeneity), E ( εj X) = E (ε) = 0. Then, we get: E bβ = β 0 The OLS estimator is unbiased. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
101 4.1. Finite sample properties Proof (cont d) Case 2: random regressors bβ = 1 1 X > X X > y = β 0 + X > X X > ε If X is includes some random elements: E bβ 1 X = β 0 + X > X X > E ( εj X) Under assumption A3 (exogeneity), E ( εj X) = 0. Then, we get: E bβ X = β 0 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
102 4.1. Finite sample properties Proof (cont d) Case 2: random regressors The OLS estimator β b is conditionally unbiased. E bβ X = β 0 Besides, we have: E bβ = E X E bβ X = E X (β 0 ) = β 0 where E X denotes the expectation with respect to the distribution of X. So, the OLS estimator β b is unbiased. E bβ = β 0 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
103 4.1. Finite sample properties De nition (Variance of the OLS estimator, nonstochastic regressors) In the multiple linear regression model y = Xβ + ε, if the matrix X is nonstochastic, the unconditional variance covariance matrix of the OLS estimator β b is: 1 V bβ = σ 2 X > X Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
104 4.1. Finite sample properties Proof bβ = 1 1 X > X X > y = β 0 + X > X X > ε So, if X is a matrix of xed regressors: bβ > V bβ = E β0 bβ β0 1 1 = E X X > X > εε > X X > X = 1 X X > X > E εε > 1 X X > X Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
105 4.1. Finite sample properties Proof (cont d) Under assumption A4 (spherical disturbances), we have: V (ε) = E εε > = σ 2 I N The variance covariance matrix of the OLS estimator is de ned by: V bβ 1 = X X > X > E εε > X X > X 1 1 = X X > X > σ 2 I N X X > X 1 1 = σ 2 X > X X > X X > X = σ 2 X > X 1 1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
106 4.1. Finite sample properties De nition (Variance of the OLS estimator, stochastic regressors) In the multiple linear regression model y = Xβ 0 + ε, if the matrix X is stochastic, the conditional variance covariance matrix of the OLS estimator β b is: V bβ 1 X = σ 2 X > X The unconditional variance covariance matrix is equal to: 1 V bβ = σ 2 E X X > X where E X denotes the expectation with respect to the distribution of X. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
107 4.1. Finite sample properties Proof bβ = 1 1 X > X X > y = β 0 + X > X X > ε So, if X is a stochastic matrix: bβ V bβ > X = E β0 bβ β0 X 1 1 = E X X > X > εε > X X X > X = 1 X X > X > E εε > 1 X X X > X Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
108 4.1. Finite sample properties Proof (cont d) Under assumption A4 (spherical disturbances), we have: V ( εj X) = E εε > X = σ 2 I N The conditional variance covariance matrix of the OLS estimator is de ned by: V bβ X = = 1 X X > X > E εε > X X X > X 1 1 X X > X > σ 2 I N X X > X = σ 2 X > X 1 1 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
109 4.1. Finite sample properties Proof (cont d) We have: V bβ 1 X = σ 2 X > X The (unconditional) variance covariance matrix of the OLS estimator is de ned by: V bβ = E X V bβ 1 X = σ 2 E X X > X where E X denotes the expectation with respect to the distribution of X. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
110 4.1. Finite sample properties Summary Mean Variance Cond. mean Cond. var Case 1: X stochastic Case 2: X nonstochastic E bβ = β 0 E bβ = β V bβ = σ 2 E X X > X V bβ = σ 2 X > X E bβ X = β 0 V bβ 1 X = σ 2 X X > Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
111 4.1. Finite sample properties Question How to estimate the variance covariance matrix of the OLS estimator? 1 V bβols = σ 2 X X > if X is nonstochastic 1 V bβols = σ 2 E X X > X if X is stochastic Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
112 4.1. Finite sample properties Question (cont d) De nition (Variance estimator) An unbiased estimator of the variance covariance matrix of the OLS estimator is given: bv 1 bβols = bσ 2 X > X where bσ 2 = (N K ) 1 bε > bε is an unbiased estimator of σ 2. This result holds whether X is stochastic or non stochastic. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
113 4.1. Finite sample properties Summary Variance Estimator Case 1: X stochastic 1 V bβ = σ 2 E X X > X Case 2: X nonstochastic V bβ = σ 2 X > X 1 bv 1 bβols = bσ 2 X X > bv 1 bβols = bσ 2 X > X Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
114 4.1. Finite sample properties De nition (Estimator of the variance of disturbances) Under the assumption A1A5, in the multiple linear regression model y = Xβ + ε, the estimator bσ 2 is unbiased: E bσ 2 = σ 2 where bσ 2 = 1 N K N i=1 bε 2 i = bε> bε N K This result holds whether or not the matrix X is considered as random. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
115 4.1. Finite sample properties Proof We assume that X is stochastic. Let M denotes the projection matrix ( residual maker ) de ned by: with M = I N bε = (N,1) 1 X X > X X > M y (N,N )(N,1) The N N matrix M satis es the following properties: 1 if X is regressed on X, a perfect t will result and the residuals will be zero, so M X = 0 2 The matrix M is symmetric M > = M and idempotent M M = M Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
116 4.1. Finite sample properties Proof (cont d) The residuals are de ned as to be: bε = M y Since y = Xβ + ε, we have bε = M (Xβ + ε) = MXβ + Mε Since MX = 0, we have bε = Mε Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
117 4.1. Finite sample properties Proof (cont d) The estimator bσ 2 is based on the sum of squared residuals (SSR) bσ 2 = The expected value of the SSR is E bε > bε X bε> bε N K = ε> Mε N K = E ε > Mε X The scalar ε > Mε is a 1 1 scalar, so it is equal to its trace. tr E ε > Mε X = tr E εε > M X = tr E Mεε > X since tr (AB) = tr (AB). Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
118 4.1. Finite sample properties Proof (cont d) Since M = I N E bε > bε X 1 X X X > X > depends on X, we have: = tr E Mεε > X = tr M E εε > X Under assumptions A3 and A4, we have E εε > X = σ 2 I N As a consequence E bε > bε X = tr σ 2 M I N = σ 2 tr (M) Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
119 4.1. Finite sample properties Proof (cont d) E bε > bε X = σ 2 tr (M) = σ 2 tr I N X X > X = σ 2 tr (I N ) σ 2 tr = σ 2 tr (I N ) σ 2 tr 1 X > 1 X > X X > X 1 X > X X > X = σ 2 tr (I N ) σ 2 tr (I K ) = σ 2 (N K ) Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
120 4.1. Finite sample properties Proof (cont d) By de nition of bσ 2, we have: E bσ 2 X = E bε > bε X N K = σ2 (N K ) N K = σ 2 So, the estimator bσ 2 is conditionally unbiased. E bσ 2 = E X E bσ 2 X = E X σ 2 = σ 2 The estimator bσ 2 is unbiased: E bσ 2 = σ 2 Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
121 4.1. Finite sample properties Remark Given the same principle, we can compute the variance of the estimator bσ 2. As a consequence, we have: bσ 2 = V bσ 2 X = V But, it takes... at least ten slides... bε> bε N K = ε> Mε N K 1 (N K ) V ε > Mε X bσ 2 = E X V bσ 2 X Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
122 4.1. Finite sample properties De nition (Variance of the estimator bσ 2 ) In the multiple linear regression model y = Xβ 0 + ε, the variance of the estimator bσ 2 is V bσ 2 = 2σ4 N K where σ 2 denotes the true value of variance of the error terms. This result holds whether or not the matrix X is considered as random. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
123 4.1. Finite sample properties Summary Mean Variance Cond. mean Cond. var Case 1: X stochastic Case 2: X nonstochastic E bσ 2 = σ 2 E bσ 2 = σ 2 V bσ 2 = 2σ4 N K V bσ 2 = 2σ4 N K E bσ 2 X = σ 2 V bσ 2 X = 2σ4 N K Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
124 4.1. Finite sample properties Finite sample distributions Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
125 4.1. Finite sample properties Summary 1 Under assumptions A1A5, we can derive the two rst moments of the (unknown) nite sample distribution of the OLS estimator β, b i.e. E bβ and V bβ. 2 Are we able to characterize the nite sample distribution (or exact sampling distribution) of b β? 3 For that, we need to put an assumption on the distribution of ε and use a parametric speci cation. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
126 4.1. Finite sample properties Finite sample distribution Fact (Finite sample distribution, I) (1) In a parametric multiple linear regression model with stochastic regressors, the conditional nite sample distribution of β b is known. The unconditional nite sample distribution is generally unknown: bβ X D β b?? where D is a multivariate distribution. Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
127 4.1. Finite sample properties Finite sample distribution Fact (Finite sample distribution, II) (2) In a parametric multiple linear regression model with nonstochastic regressors, the marginal (unconditional) nite sample distribution of b β is known: bβ D Christophe Hurlin (University of Orléans) Advanced Econometrics  HEC Lausanne November 23, / 174
Chapter 4: Statistical Hypothesis Testing
Chapter 4: Statistical Hypothesis Testing Christophe Hurlin November 20, 2015 Christophe Hurlin () Advanced Econometrics  Master ESA November 20, 2015 1 / 225 Section 1 Introduction Christophe Hurlin
More information1. The Classical Linear Regression Model: The Bivariate Case
Business School, Brunel University MSc. EC5501/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 018956584) Lecture Notes 3 1.
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationChapter 2. Dynamic panel data models
Chapter 2. Dynamic panel data models Master of Science in Economics  University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider
More informationOLS in Matrix Form. Let y be an n 1 vector of observations on the dependent variable.
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 Since our model will usually contain a constant term, one of the columns
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
More informationRegression III: Advanced Methods
Lecture 5: Linear leastsquares Regression III: Advanced Methods William G. Jacoby Department of Political Science Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Simple Linear Regression
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models  part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby
More information1 Another method of estimation: least squares
1 Another method of estimation: least squares erm: estim.tex, Dec8, 009: 6 p.m. (draft  typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA
More informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationPanel Data Econometrics
Panel Data Econometrics Master of Science in Economics  University of Geneva Christophe Hurlin, Université d Orléans University of Orléans January 2010 De nition A longitudinal, or panel, data set is
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationIDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011
IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION
More information2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measuresoffit in multiple regression Assumptions
More informationEC327: Advanced Econometrics, Spring 2007
EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Appendix D: Summary of matrix algebra Basic definitions A matrix is a rectangular array of numbers, with m
More information3.1 Least squares in matrix form
118 3 Multiple Regression 3.1 Least squares in matrix form E Uses Appendix A.2 A.4, A.6, A.7. 3.1.1 Introduction More than one explanatory variable In the foregoing chapter we considered the simple regression
More informationEC 6310: Advanced Econometric Theory
EC 6310: Advanced Econometric Theory July 2008 Slides for Lecture on Bayesian Computation in the Nonlinear Regression Model Gary Koop, University of Strathclyde 1 Summary Readings: Chapter 5 of textbook.
More informationNormalization and Mixed Degrees of Integration in Cointegrated Time Series Systems
Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 EMail: r.j.rossana@wayne.edu
More informationL10: Probability, statistics, and estimation theory
L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More informationChapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem
Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become
More informationHeteroskedasticity and Weighted Least Squares
Econ 507. Econometric Analysis. Spring 2009 April 14, 2009 The Classical Linear Model: 1 Linearity: Y = Xβ + u. 2 Strict exogeneity: E(u) = 0 3 No Multicollinearity: ρ(x) = K. 4 No heteroskedasticity/
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationThe Loss in Efficiency from Using Grouped Data to Estimate Coefficients of Group Level Variables. Kathleen M. Lang* Boston College.
The Loss in Efficiency from Using Grouped Data to Estimate Coefficients of Group Level Variables Kathleen M. Lang* Boston College and Peter Gottschalk Boston College Abstract We derive the efficiency loss
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationLinear and Piecewise Linear Regressions
Tarigan Statistical Consulting & Coaching statisticalcoaching.ch Doctoral Program in Computer Science of the Universities of Fribourg, Geneva, Lausanne, Neuchâtel, Bern and the EPFL Handson Data Analysis
More informationVariance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212.
Variance of OLS Estimators and Hypothesis Testing Charlie Gibbons ARE 212 Spring 2011 Randomness in the model Considering the model what is random? Y = X β + ɛ, β is a parameter and not random, X may be
More informationMultivariate Normal Distribution
Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #47/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues
More informationQuadratic forms Cochran s theorem, degrees of freedom, and all that
Quadratic forms Cochran s theorem, degrees of freedom, and all that Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 1, Slide 1 Why We Care Cochran s theorem tells us
More informationELECE8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems
Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed
More informationTopic 5: Stochastic Growth and Real Business Cycles
Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar
More informationLimitations of regression analysis
Limitations of regression analysis Ragnar Nymoen Department of Economics, UiO 8 February 2009 Overview What are the limitations to regression? Simultaneous equations bias Measurement errors in explanatory
More informationExact Nonparametric Tests for Comparing Means  A Personal Summary
Exact Nonparametric Tests for Comparing Means  A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More informationPortfolio selection based on upper and lower exponential possibility distributions
European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department
More informationMultivariate normal distribution and testing for means (see MKB Ch 3)
Multivariate normal distribution and testing for means (see MKB Ch 3) Where are we going? 2 Onesample ttest (univariate).................................................. 3 Twosample ttest (univariate).................................................
More informationMaximum Likelihood Estimation of an ARMA(p,q) Model
Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates
More informationHypothesis Testing in Linear Regression Models
Chapter 4 Hypothesis Testing in Linear Regression Models 41 Introduction As we saw in Chapter 3, the vector of OLS parameter estimates ˆβ is a random vector Since it would be an astonishing coincidence
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationReview of Bivariate Regression
Review of Bivariate Regression A.Colin Cameron Department of Economics University of California  Davis accameron@ucdavis.edu October 27, 2006 Abstract This provides a review of material covered in an
More information1 Vector Spaces and Matrix Notation
1 Vector Spaces and Matrix Notation De nition 1 A matrix: is rectangular array of numbers with n rows and m columns. 1 1 1 a11 a Example 1 a. b. c. 1 0 0 a 1 a The rst is square with n = and m = ; the
More information2. What are the theoretical and practical consequences of autocorrelation?
Lecture 10 Serial Correlation In this lecture, you will learn the following: 1. What is the nature of autocorrelation? 2. What are the theoretical and practical consequences of autocorrelation? 3. Since
More informationC: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)}
C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)} 1. EES 800: Econometrics I Simple linear regression and correlation analysis. Specification and estimation of a regression model. Interpretation of regression
More informationRegression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture  2 Simple Linear Regression
Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture  2 Simple Linear Regression Hi, this is my second lecture in module one and on simple
More informationChapter 4: Constrained estimators and tests in the multiple linear regression model (Part II)
Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part II) Florian Pelgrin HEC SeptemberDecember 2010 Florian Pelgrin (HEC) Constrained estimators SeptemberDecember
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulationbased method for estimating the parameters of economic models. Its
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationRegression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur
Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture  7 Multiple Linear Regression (Contd.) This is my second lecture on Multiple Linear Regression
More informationEmpirical Methods in Applied Economics
Empirical Methods in Applied Economics JörnSte en Pischke LSE October 2005 1 Observational Studies and Regression 1.1 Conditional Randomization Again When we discussed experiments, we discussed already
More informationIntroduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by
Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes
More informationClustering in the Linear Model
Short Guides to Microeconometrics Fall 2014 Kurt Schmidheiny Universität Basel Clustering in the Linear Model 2 1 Introduction Clustering in the Linear Model This handout extends the handout on The Multiple
More informationEconometrics II. Lecture 9: Sample Selection Bias
Econometrics II Lecture 9: Sample Selection Bias Måns Söderbom 5 May 2011 Department of Economics, University of Gothenburg. Email: mans.soderbom@economics.gu.se. Web: www.economics.gu.se/soderbom, www.soderbom.net.
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationHypothesis Testing in the Classical Regression Model
LECTURE 5 Hypothesis Testing in the Classical Regression Model The Normal Distribution and the Sampling Distributions It is often appropriate to assume that the elements of the disturbance vector ε within
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,
More informationChapter 2 Principles of Linear Regression
Chapter 2 Principles of Linear Regression E tric Methods Winter Term 2012/13 Nikolaus Hautsch HumboldtUniversität zu Berlin 2.1 Basic Principles of Inference 2.1.1 Econometric Modelling 2 86 Outline I
More informationMathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions
Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions
More informationOnline Appendix to Impatient Trading, Liquidity. Provision, and Stock Selection by Mutual Funds
Online Appendix to Impatient Trading, Liquidity Provision, and Stock Selection by Mutual Funds Zhi Da, Pengjie Gao, and Ravi Jagannathan This Draft: April 10, 2010 Correspondence: Zhi Da, Finance Department,
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationc 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationproblem arises when only a nonrandom sample is available differs from censored regression model in that x i is also unobserved
4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a nonrandom
More informationAdvanced Microeconomics
Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions
More informationEconometric Methods for Panel Data
Based on the books by Baltagi: Econometric Analysis of Panel Data and by Hsiao: Analysis of Panel Data Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies
More information, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (
Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we
More informationPoisson Models for Count Data
Chapter 4 Poisson Models for Count Data In this chapter we study loglinear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationThe Real Business Cycle Model
The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the comovements in the uctuations
More information1 Orthogonal projections and the approximation
Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationCHAPTER 4 ASYMPTOTIC THEORY FOR LINEAR REGRESSION MODELS WITH I.I.D. OBSERVATIONS
CHAPTER 4 ASYMPTOTIC THEORY FOR LINEAR REGRESSION MODELS WITH I.I.D. OBSERVATIONS Key words: Asymptotics, Almost sure convergence, Central limit theorem, Convergence in distribution, Convergence in quadratic
More informationElasticity Theory Basics
G22.3033002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationMean squared error matrix comparison of least aquares and Steinrule estimators for regression coefficients under nonnormal disturbances
METRON  International Journal of Statistics 2008, vol. LXVI, n. 3, pp. 285298 SHALABH HELGE TOUTENBURG CHRISTIAN HEUMANN Mean squared error matrix comparison of least aquares and Steinrule estimators
More informationCredit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
More informationEconometrics Regression Analysis with Time Series Data
Econometrics Regression Analysis with Time Series Data João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationOn Marginal Effects in Semiparametric Censored Regression Models
On Marginal Effects in Semiparametric Censored Regression Models Bo E. Honoré September 3, 2008 Introduction It is often argued that estimation of semiparametric censored regression models such as the
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationSolución del Examen Tipo: 1
Solución del Examen Tipo: 1 Universidad Carlos III de Madrid ECONOMETRICS Academic year 2009/10 FINAL EXAM May 17, 2010 DURATION: 2 HOURS 1. Assume that model (III) verifies the assumptions of the classical
More informationData Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression
Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction
More informationA SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS
A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZVILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationDEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests
DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationConditional InvestmentCash Flow Sensitivities and Financing Constraints
WORING PAPERS IN ECONOMICS No 448 Conditional InvestmentCash Flow Sensitivities and Financing Constraints Stephen R. Bond and Måns Söderbom May 2010 ISSN 14032473 (print) ISSN 14032465 (online) Department
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationMC3: Econometric Theory and Methods
University College London Department of Economics M.Sc. in Economics MC3: Econometric Theory and Methods Course notes: Econometric models, random variables, probability distributions and regression Andrew
More informationOligopoly. Chapter 10. 10.1 Overview
Chapter 10 Oligopoly 10.1 Overview Oligopoly is the study of interactions between multiple rms. Because the actions of any one rm may depend on the actions of others, oligopoly is the rst topic which requires
More informationWhat s New in Econometrics? Lecture 8 Cluster and Stratified Sampling
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and
More informationSome probability and statistics
Appendix A Some probability and statistics A Probabilities, random variables and their distribution We summarize a few of the basic concepts of random variables, usually denoted by capital letters, X,Y,
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationTHE LEAST SQUARES ESTIMATOR Q
4 THE LEAST SQUARES ESTIMATOR Q 4.1 INTRODUCTION Chapter 3 treated fitting the linear regression to the data by least squares as a purely algebraic exercise. In this chapter, we will examine in detail
More information