sheet

... The result of lm() is an object that contains various quantities related to the linear model. The following functions return some of these: > residuals(ld) > fitted(ld) > coef(ld) Printing the model object ld itself returns the coefficients. Inference for the Linear Model ...

... The result of lm() is an object that contains various quantities related to the linear model. The following functions return some of these: > residuals(ld) > fitted(ld) > coef(ld) Printing the model object ld itself returns the coefficients. Inference for the Linear Model ...

Simultaneous Equation Models

... one equation at a time (this is called limited information estimation) ...

... one equation at a time (this is called limited information estimation) ...

Let v denote a column vector of the nilpotent matrix Pi(A)(A − λ iI)ni

... Pi (A)(A − λi I)ni −1 where ni is the so called nilpotency. Theorem 3 in [1] shows that APi (A)(A − λi I)ni −1 = λi Pi (A)(A − λi I)ni −1 . which means a column vector v of the matrix is an eigenvector corresponding to the eigenvalue λi . The symbols are explained in [1]. However it is worth noting ...

... Pi (A)(A − λi I)ni −1 where ni is the so called nilpotency. Theorem 3 in [1] shows that APi (A)(A − λi I)ni −1 = λi Pi (A)(A − λi I)ni −1 . which means a column vector v of the matrix is an eigenvector corresponding to the eigenvalue λi . The symbols are explained in [1]. However it is worth noting ...

Macro

... • Put formulas on the top row and the variable we wish to vary in the column • Highlight the table area • Data/sensitivity analysis/ Data table • Select the cell input in row or in column ...

... • Put formulas on the top row and the variable we wish to vary in the column • Highlight the table area • Data/sensitivity analysis/ Data table • Select the cell input in row or in column ...

(pdf).

... (b) Let a = 0 in C. Assume that such a matrix is an echelon form of some matrix A. What value of c and d so that rank(A) = 2? (c) Let d = 1 and c = 1 and a = 0 in the matrix C. Assume that the matrix you obtain is the reduced echelon form of some matrix A. Write the last column of A as linear combin ...

... (b) Let a = 0 in C. Assume that such a matrix is an echelon form of some matrix A. What value of c and d so that rank(A) = 2? (c) Let d = 1 and c = 1 and a = 0 in the matrix C. Assume that the matrix you obtain is the reduced echelon form of some matrix A. Write the last column of A as linear combin ...

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

... 16. Given the following Revenue (R) and Cost (C) functions for a firm = 20 + 2, find the equilibrium level of output, price, total revenue, total cost and profit. ...

... 16. Given the following Revenue (R) and Cost (C) functions for a firm = 20 + 2, find the equilibrium level of output, price, total revenue, total cost and profit. ...

3. Model Fitting 3.1 The bivariate normal distribution

... What precisely does the P-value mean? “If the galaxy flux density really is constant, and we repeatedly obtained sets of 15 measurements under the same conditions, then only 2% of the F 2 values derived from these sets would be expected to be greater than our one actual measured value of 26.76” From ...

... What precisely does the P-value mean? “If the galaxy flux density really is constant, and we repeatedly obtained sets of 15 measurements under the same conditions, then only 2% of the F 2 values derived from these sets would be expected to be greater than our one actual measured value of 26.76” From ...

PresentationAbstracts2012

... Sparse factor modelling for high dimensional time series Analyzing multiple time series via factor model is one of the frequently used methods to achieve dimension reduction. Modern time series analysis concentrates on the situation when the number of time series p is as large as or even larger than ...

... Sparse factor modelling for high dimensional time series Analyzing multiple time series via factor model is one of the frequently used methods to achieve dimension reduction. Modern time series analysis concentrates on the situation when the number of time series p is as large as or even larger than ...

OLS regression in the SAS system-

... 4. Look over these results, checking the t-statistic results to determine which regerssors need to be booted out. Cook up different sets of regressors and run the models, again looking over the results. Consider not only the "raw" statistics when choosing your regressors, but also the underlying the ...

... 4. Look over these results, checking the t-statistic results to determine which regerssors need to be booted out. Cook up different sets of regressors and run the models, again looking over the results. Consider not only the "raw" statistics when choosing your regressors, but also the underlying the ...

Parameter estimation in multivariate models Let X1,..., Xn be i.i.d.

... Let X1 , . . . , Xn be i.i.d. sample from the Pθ distribution, where θ ∈ Θ and Θ ⊂ Rk is the parameter space. The unknown parameter θ is estimated by means of a T = T(X1 , . . . , Xn ) = T(X) ∈ Rk statistic, which depends on the sample condensed into the p × n matrix X column-wise. Here XT is the da ...

... Let X1 , . . . , Xn be i.i.d. sample from the Pθ distribution, where θ ∈ Θ and Θ ⊂ Rk is the parameter space. The unknown parameter θ is estimated by means of a T = T(X1 , . . . , Xn ) = T(X) ∈ Rk statistic, which depends on the sample condensed into the p × n matrix X column-wise. Here XT is the da ...

Simulation Methods Based on the SAS System

... are the non negative square roots of the eigenvalues of X' X j they are called the singular values. Using the singular value decomposition defined above we get the following representation of the OLS estimator b : b = VE-1U'y, ...

... are the non negative square roots of the eigenvalues of X' X j they are called the singular values. Using the singular value decomposition defined above we get the following representation of the OLS estimator b : b = VE-1U'y, ...

Least Squares Adjustment

... where b is the number of histogram classes, ai is the observed frequency in class i and ei is the expected frequency in class i. Failure of the chi2 goodness of fit test is an indication that the observations are not normally distributed and/or that there is a problem in the model. Another test for ...

... where b is the number of histogram classes, ai is the observed frequency in class i and ei is the expected frequency in class i. Failure of the chi2 goodness of fit test is an indication that the observations are not normally distributed and/or that there is a problem in the model. Another test for ...

Problem Set 2

... to overﬂow (e.g., in the case a, b ≈ 10180 ), underﬂow (e.g., in the case a, b ≈ 10−180 ), and severe cancellation (e.g., in the case |a| |b|). Write (on paper) a matlab program to evaluate F that should be more robust against overﬂow, underﬂow and cancellation than the direct implementation. It i ...

... to overﬂow (e.g., in the case a, b ≈ 10180 ), underﬂow (e.g., in the case a, b ≈ 10−180 ), and severe cancellation (e.g., in the case |a| |b|). Write (on paper) a matlab program to evaluate F that should be more robust against overﬂow, underﬂow and cancellation than the direct implementation. It i ...

Testing Time Reversibility of Markov Processes

... Using a central limit theorem for strongly mixing sequences of random variables and Slutsky's Theorem it can be shown that, under some appropriate conditions, L(Tn) ) 2p as M; n ! 1 provided M=n ! 0. Here L(X ) denotes the law of a random variable X and ` )0 weak convergence. Apart from this asymp ...

... Using a central limit theorem for strongly mixing sequences of random variables and Slutsky's Theorem it can be shown that, under some appropriate conditions, L(Tn) ) 2p as M; n ! 1 provided M=n ! 0. Here L(X ) denotes the law of a random variable X and ` )0 weak convergence. Apart from this asymp ...

1 The Chain Rule - McGill Math Department

... are two transformations such that (x1 , x2 , · · · , xn ) = G(F (x1 , x2 , · · · , xn )) then the Jacobian matrices DF and DG are inverse to one another. This is because, if I(x1 , x2 , · · · , xn ) = (x1 , x2 , · · · , xn ) then DI is the identity matrix n × n matrix In . Hence, In = D(I) = D(F ◦ G ...

... are two transformations such that (x1 , x2 , · · · , xn ) = G(F (x1 , x2 , · · · , xn )) then the Jacobian matrices DF and DG are inverse to one another. This is because, if I(x1 , x2 , · · · , xn ) = (x1 , x2 , · · · , xn ) then DI is the identity matrix n × n matrix In . Hence, In = D(I) = D(F ◦ G ...

Multivariable Linear Systems and Row Operations

... The algorithm used to transform a system of linear equations into an equivalent system in row-echelon form is called Gaussian elimination. The operations used to produce equivalent systems are given below. ...

... The algorithm used to transform a system of linear equations into an equivalent system in row-echelon form is called Gaussian elimination. The operations used to produce equivalent systems are given below. ...

july 22

... (d) Do the columns of A form a linearly independent set? (e) Is the set {~a3 , ~a4 , ~a5 } a linearly independent set? If linear transformation T (~x) = A~x, (f) What is the domain of T ? the codomain of T ? (g) Is the linear transformation T (~x) = A~x onto its codomain? one-to-one ? (h) What is th ...

... (d) Do the columns of A form a linearly independent set? (e) Is the set {~a3 , ~a4 , ~a5 } a linearly independent set? If linear transformation T (~x) = A~x, (f) What is the domain of T ? the codomain of T ? (g) Is the linear transformation T (~x) = A~x onto its codomain? one-to-one ? (h) What is th ...

Linear Algebra Exam 1 Spring 2007

... 6. [15 − 3each] True or False: Justify each answer by citing an appropriate definition or theorem. If the statement is false and you can provide a counterexample to demonstrate this then do so. If the statement is false and be can slightly modified so as to make it true then indicate how this may b ...

... 6. [15 − 3each] True or False: Justify each answer by citing an appropriate definition or theorem. If the statement is false and you can provide a counterexample to demonstrate this then do so. If the statement is false and be can slightly modified so as to make it true then indicate how this may b ...

USE OF LINEAR ALGEBRA I Math 21b, O. Knill

... projections along lines reduces to the solution of the Radon transform. Studied first in 1917, it is today a basic tool in applications like medical diagnosis, tokamak monitoring, in plasma physics or for astrophysical applications. The reconstruction is also called tomography. Mathematical tools de ...

... projections along lines reduces to the solution of the Radon transform. Studied first in 1917, it is today a basic tool in applications like medical diagnosis, tokamak monitoring, in plasma physics or for astrophysical applications. The reconstruction is also called tomography. Mathematical tools de ...

Analytic Models and Empirical Search: A Hybrid Approach to Code

... Why is Speed Important? • Adaptation may have to be applied at runtime, where running time is critical. • Adaptation may have to be applied at compile time (e.g., with feedback from a fast simulator) • Library routines can be used as a benchmark to evaluate alternative machine designs. ...

... Why is Speed Important? • Adaptation may have to be applied at runtime, where running time is critical. • Adaptation may have to be applied at compile time (e.g., with feedback from a fast simulator) • Library routines can be used as a benchmark to evaluate alternative machine designs. ...

Sol 2 - D-MATH

... multiple of the other. But if ~v2 were a scalar multiple of ~v1 , it would have to lie along the line going through ~v1 . In the picture, this is clearly not the case, thus the two vectors are linearly independent. However, ~v1 , ~v2 and ~v3 are linearly dependent, as with a correct scaling of ~v1 a ...

... multiple of the other. But if ~v2 were a scalar multiple of ~v1 , it would have to lie along the line going through ~v1 . In the picture, this is clearly not the case, thus the two vectors are linearly independent. However, ~v1 , ~v2 and ~v3 are linearly dependent, as with a correct scaling of ~v1 a ...

leastsquares

... Quick and Dirty Approach Multiply by AT to get the normal equations: AT A x = AT b For the mountain example the matrix AT A is 3 x 3. The matrix AT A is symmetric . However, sometimes AT A can be nearly singular or singular. Consider the matrix A = 1 1 e 0 0 e The matrix AT A = 1+ e2 ...

... Quick and Dirty Approach Multiply by AT to get the normal equations: AT A x = AT b For the mountain example the matrix AT A is 3 x 3. The matrix AT A is symmetric . However, sometimes AT A can be nearly singular or singular. Consider the matrix A = 1 1 e 0 0 e The matrix AT A = 1+ e2 ...

In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the unknown parameters in a linear regression model, with the goal of minimizing the differences between the observed responses in some arbitrary dataset and the responses predicted by the linear approximation of the data (visually this is seen as the sum of the vertical distances between each data point in the set and the corresponding point on the regression line - the smaller the differences, the better the model fits the data). The resulting estimator can be expressed by a simple formula, especially in the case of a single regressor on the right-hand side.The OLS estimator is consistent when the regressors are exogenous and there is no perfect multicollinearity, and optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors be normally distributed, OLS is the maximum likelihood estimator. OLS is used in economics (econometrics), political science and electrical engineering (control theory and signal processing), among many areas of application. The Multi-fractional order estimator is an expanded version of OLS.