basis of theory and methods of research

... of samples of fU (u) and (independently) a set v1 , v2 , of samples of fV (v) , and then building the sequence w1 = {u1 , v1 } , w2 = {u2 , v2 } , , does not provide a set of samples of ... study is a random realization of a Gaussian random field, with given mean mprior and given covariance CM (examples of realizations of such random fields are displayed in chapter (Figure 2.4) and chapter ... termed a random variable The mathematical theory of statistics is the natural tool for analyzing the outputs of a random process The second interpretation is in terms of a subjective degree of knowledge...

Ngày tải lên: 31/03/2014, 15:24

... square root of the variance, σ, is called the standard deviation of the distribution Estimates of standard deviations are often referred to as standard errors, especially when the random variable ... on X2 , the expectation of a product of another random variable X1 and a deterministic function of X2 is the product of that deterministic function and the expectation of X1 conditional on X2 ... X11 of dimensions n1 × k1 , X12 of dimensions n1 × k2 , X21 of dimensions n2 × k1 , and X22 of dimensions n2 × k2 , with n1 + n2 = n and k1 + k2 = k Thus X11 and X12 have the same number of rows,...

Ngày tải lên: 08/04/2014, 12:28

... function estimation for any family of nonstationary random processes” by J Sandberg and M Hansson-Sandsten, the covariance function estimate of a zero-mean nonstationary random process in discrete time ... design and analysis, and a discussion of the main open problems in this area In the paper “Tree image growth analysis using instantaneous phase modulation” by J Ramachandran et al., the use of Amplitude-Modulation ... robust in the sense that it is often close to optimal for many other random processes as well Three papers are concerned with applications of wavelet and wideband signal processing In the paper...

Ngày tải lên: 21/06/2014, 05:20

... square root of the variance, σ, is called the standard deviation of the distribution Estimates of standard deviations are often referred to as standard errors, especially when the random variable ... on X2 , the expectation of a product of another random variable X1 and a deterministic function of X2 is the product of that deterministic function and the expectation of X1 conditional on X2 ... X11 of dimensions n1 × k1 , X12 of dimensions n1 × k2 , X21 of dimensions n2 × k1 , and X22 of dimensions n2 × k2 , with n1 + n2 = n and k1 + k2 = k Thus X11 and X12 have the same number of rows,...

Ngày tải lên: 04/07/2014, 15:20

... ABC, with hypotenuse AC, and two other sides, AB and BC, of lengths x1 and x2 respectively The squares on each of the three sides of the triangle are drawn, and the area of the square on the hypotenuse ... combination of x1 and x2 , Xβ ˆ as shown The numerical value of β1 can be computed as the ratio of the ˆ1 x1 to that of x1 , and similarly for β2 ˆ length of the vector β In panel c) of Figure ... the right-hand side of (2.38) as a linear combination of the columns of X1 and of X2 In the present general context, there is an operation analogous to that of centering The result of centering...

Ngày tải lên: 04/07/2014, 15:20

... times, ˆ1 and the β2 Use these averages to estimate ˆ and find the averages of the β the bias of the OLS estimators of β1 and β2 Repeat this exercise for sample sizes of 50, 100, and 200 What ... matrix of its central second moments This matrix is the analog, for vector random variables, of the variance of a scalar random variable If b is any random vector, we will denote its matrix of central ... covariance of bi and bj The concept of covariance was introduced in Exercise 1.10 In terms of the random variables bi and bj , the definition is Cov(bi , bj ) ≡ E bi − E(bi ) bj − E(bj ) (3.21) Many of...

Ngày tải lên: 04/07/2014, 15:20

... (unrestricted) estimate of σ and can be thought of as a measure of the scale of the random noise Since u ∼ N (0, σ I), every element of u has the same variance, and so every component of (4.37), if centered ... some of the residuals exactly once, some of them more than once, and some of them not at all Therefore, the value of each drawing must be the value of one of Copyright c 1999, Russell Davidson and ... combinations of standard normal ones plus constants We will tackle the proof in several steps, each of which is important in its own right To begin with, let z1 and z2 be standard normal and mutually...

Ngày tải lên: 04/07/2014, 15:20

... intervals are symmetric, not all of them share this property The symmetry of (5.06) is a consequence of the symmetry of the standard normal distribution and of the form of the test statistic (5.04) ... distribution, and find the 95 and 99 “quantiles” of the EDF, that is the 950 th and 990 th entries in the sorted list of the realizations Compare these with the 95 and 99 quantiles of the χ2 (2) ... Clearly, c∗ 1−(α/2) and cα/2 , which are approximately the − (α/2) and α/2 quantiles of the EDF of the bootstrap tests, play the same roles as the − (α/2) and α/2 quantiles of the exact Student’s...

Ngày tải lên: 04/07/2014, 15:20

... Similar asymptotic results, and similar methods of proof, apply to all of them 6.10 Exercises 6.1 Let the expectation of a random variable Y conditional on a set of other random variables X1 , ... the n×k matrix of partial derivatives of the vector of regression functions x(β) = x(β1 , β2 ) of (6.69) Similarly, let X1 (β) and X2 (β) denote the n × k1 and n × k2 submatrices of partial derivatives ... (1990), Goffe, Ferrier, and Rogers (1994), Dorsey and Mayer (1995), and Andrews (1997) In difficult cases, one or more of these methods should work better than simply using a number of starting values...

Ngày tải lên: 04/07/2014, 15:20

... correlation of ut and ut−1 , we divide Cov(ut , ut−1 ) by the square root of the product of the variances of ut and ut−1 , that is, by σu We then find that the correlation of ut and ut−1 is just ... matrices UX and UZ by multiplying the t th row of X and th the t row of Z by ut for all t ˜ ˜ ˜ Create the matrices U −1X and U −1Z by dividing the t th row of X and the t th row of Z by ut for ... Regress each of the columns of U −1X and U −1Z on UX and UZ jointly ¯ ¯ Save the resulting matrices of fitted values and call them X and Z, respectively Copyright c 1999, Russell Davidson and James...

Ngày tải lên: 04/07/2014, 15:20

... of parameters, γd and γs are scalar parameters, and ud and us are the error terms in the demand and t t supply functions Economic theory predicts that, in most cases, γd < and γs > 0, which is ... premultiplying the sum of a g vector of linear combinations of the exogenous and predetermined variables and a g vector of error terms If we want to estimate the full system of equations, there ... (8.53) and (8.54), respectively ´ Because both H0 and H1 are linear models, the value of β used to evaluate the regressands of (8.53) and (8.54) has no effect on the difference between the SSRs of...

Ngày tải lên: 04/07/2014, 15:20

... heteroskedastic and/ or serially correlated In this case, W typically consists of X augmented by a number of functions of the ˆ columns of X, such as squares and cross-products, and Ω has squared ... treatment of the asymptotic theory of GMM estimation, see Newey and McFadden (1994) Copyright c 1999, Russell Davidson and James G MacKinnon 9.6 The Method of Simulated Moments 381 9.6 The Method of ... values of the incomes of a random sample of households from a given population Describe in detail how to use this sample in order to estimate the parameters a, b, and c of (9.127) by the method of...

Ngày tải lên: 04/07/2014, 15:20

... 394 The Method of Maximum Likelihood probability density function, or PDF, of a scalar random variable and of the joint density function, or joint PDF, of a set of random variables If we ... loglikelihood function, and thus also the matrix of first derivatives of the gradient Let θ(j) denote the value of the vector of estimates at step j of the algorithm, and let g(j) and H(j) denote, respectively, ... of the support of the random variable X, that is, the set of real numbers for which the density of X is nonzero, and the support contains more than one point See Exercise 10.4 for the proof of...

Ngày tải lên: 04/07/2014, 15:20

... simulation-based methods, including the method of simulated moments, which was discussed in Section 9.6 See Hajivassiliou and Ruud (1994) and Gouri´roux and Monfort (1996) for discussions e of some of the methods ... Xt and Wt are vectors of observations on exogenous or predetermined variables, β and γ are unknown parameter vectors, σ is the standard deviation of ut , and ρ is the correlation between ut and ... assuming that β1 = and β2 = 1, 100 assuming that β1 = −1 and β2 = 1, 100 assuming that β1 = and β1 = 2, 100 assuming that β1 = and β1 = −2, and 100 assuming that β1 = and β2 = For each of the 500 samples,...

Ngày tải lên: 04/07/2014, 15:20

... second line above makes use of the expression of Ω in terms of Γ and Σ, and of the result (12.108) It is straightforward to see that (12.113) is one row of the left-hand side of (12.83), which therefore ... regression, the inner product of the regressand with the matrix of regressors yields the left-hand side of the moment conditions (12.45), and the inverse of the inner product of the regressor matrix ... equation i of (12.54) in the form [ yi Yi ] −β2i = Zi β1i + ui (12.69) All of the columns of Yi are also columns of Y, as is yi itself, and so column i of the matrix Γ has for element i, and the...

Ngày tải lên: 04/07/2014, 15:20

... autocovariances, and autocorrelations of an AR(p) process by using equations (13.11), (13.12), and (13.13), after we have solved for α(L) However, these methods make use of the theory of functions of a complex ... values of µ and α1 , then we may define the pseudo-true values of the parameters γ and ρ of the auxiliary model (13.43) as those values for which the expectations of the left-hand sides of equations ... AR, MA, and ARMA Models 563 limit and the factor of n−1 in expression (13.47) and replace Ω by a suitable estimate, we obtain an estimate of the covariance matrix of γ and ρ Instead ˆ ˆ of estimating...

Ngày tải lên: 04/07/2014, 15:20

... Davidson and James G MacKinnon 14.2 Random Walks and Unit Roots 597 This model is often called a random walk with drift, and the constant term is called a drift parameter To understand this terminology, ... element in the first row and column of the covariance matrix Σ of the innovations et1 and et2 , we see that the random walk vt1 can 1/2 be expressed as σ11 wt , for a standardized random walk wt We ... estimation of a system of nonlinear equations was treated in Section 12.3 We saw there that one approach is to minimize the determinant of the matrix of sums of squares and cross-products of the...

Ngày tải lên: 04/07/2014, 15:20

... 1, , N, where N is the number of elements of r, and Zt and rt are, respectively, the t th row and t th element of Z and r Copyright c 1999, Russell Davidson and James G MacKinnon 15.2 Specification ... size n, and R(θ) is a matrix with as many rows as r(θ) and k columns For example, in the case of the GNR, r(θ) is a vector of residuals, written as a function of the data and parameters, and R(θ) ... the Specification of Econometric Models ˆ where β is the vector of NLS estimates of the parameters, and Xt (β) is the k vector of derivatives of xt (β) with respect to the elements of β ˆ Since the...

Ngày tải lên: 04/07/2014, 15:20

... development of high-performance computers and progress of computational methods, more and more large-scale optimization problems have been studied and solved As pointed out by Professor Yuqi He of Harvard ... found from lots of schemes by means of scientific methods and tools It involves the study of optimality conditions of the problems, the construction of model problems, the determination of algorithmic ... numbers p and q are each larger than 1, and p + = If x and y are also real numbers, then q xy ≤ xp y q + , p q (1.2.37) and equality holds if and only if xp = y q Proof Set s = xp and t = y...

Ngày tải lên: 20/08/2015, 15:18

Ngày tải lên: 17/11/2016, 11:32