An Introduction To Econometric Theory 1st Edition James Davidson - Solutions

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2 In the simple linear regression model y 5 b0 1 b1x 1 u, suppose that E(u)  0. Letting a0 5 E(u), show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value.
1 Let kids denote the number of children ever born to a woman, and let educ denote years of education for the woman. A simple model relating fertility to years of education is kids 5 b0 1 b1educ 1 u, where u is the unobserved error. (i) What kinds of factors are contained in u? Are these likely to
C5 The data in FERTIL2.DTA were collected on women living in the Republic of Botswana in 1988. The variable children refers to the number of living children. The variable electric is a binary indicator equal to one if the woman’s home has electricity, and zero if not. (i) Find the smallest and
C4 The data in JTRAIN2.RAW come from a job training experiment conducted for lowincome men during 1976–1977; see Lalonde (1986) (i) Use the indicator variable train to determine the fraction of men receiving job training. (ii) The variable re78 is earnings from 1978, measured in thousands of
C3 The data in MEAP01.RAW are for the state of Michigan in the year 2001. Use these data to answer the following questions. (i) Find the largest and smallest values of math4. Does the range make sense? Explain. (ii) How many schools have a perfect pass rate on the math test? What percentage is this
C2 Use the data in BWGHT.RAW to answer this question (i) How many women are in the sample, and how many report smoking during pregnancy? (ii) What is the average number of cigarettes smoked per day? Is the average a good measure of the “typical” woman in this case? Explain. (iii) Among women
C1 Use the data in WAGE1.RAW for this exercis (i) Find the average education level in the sample. What are the lowest and highest years of education? (ii) Find the average hourly wage in the sample. Does it seem high or low? (iii) The wage data are reported in 1976 dollars. Using the Economic
3 Suppose at your university you are asked to find the relationship between weekly hours spent studying (study) and weekly hours spent working (work). Does it make sense to characterize the problem as inferring whether study “causes” work or work “causes” study? Explain.
2 A justification for job training programs is that they improve worker productivity. Suppose that you are asked to evaluate whethe r more job training makes workers more productive. However, rather than having data on individual workers, you have access to data on manufacturing firms in Ohio. In
1 Suppose that you are asked to conduct a study to determine whether smaller class sizes lead to improved student performance of fourth graders. (i) If you could conduct any experiment you want, what would you do? Be specific. (ii) More realistically, suppose you can collect observational data on
In a regression model for count data, with a dependent variable yi taking non-negative integer values, suppose that yi = ????xi + ui, where xi ≥ 0 is a non-stochastic explanatory variable and E(ui) = 0. Assume that yi is Poisson-distributed, so that p(yi|xi) =????yi i e −????i yi!where ????i =
In a random trial, the odds in favour of success are pi 1 − pi, where pi is the probability of success depending on an explanatory variable xi through the relation log pi 1 − pi= ???? + ????xi.(a) Show that the probability of success is pi = e????+????xi 1 + e????+????xi.(b) Suppose there are
Consider the regression model with heteroscedastic disturbances:yt = ????xt + ut ut ∼ N(0, ????z2 t ), t = 1, ..., T, where xt and zt are both conditionally fixed in repeated samples, and E(utus) = 0 for t ≠ s.(a) Write down the log-likelihood function for this model.(b) Obtain the first-order
Consider the linear model y = X???? + u (T × 1), where X is T × k and ???? is k × 1, plim T−1X′X = MXX < ∞ having full rank, but plim T−1X′u ≠ ????.(a) Show that the ordinary least squares estimator is inconsistent.(b) Suppose there exists a matrix W (T × k) such that plim
Consider the pair of equations yt = ????xt + ut xt = ????zt + ????t for t = 1,…, T, where ut and ????t are independent normally distributed disturbances withmeans of zero, variances ????uu, ????????????, and covariance ????u????. zt is an observed variable, also normally distributed, with mean
Are the following statements true or false?(a) Least squares is asymptotically efficient in the class of linear unbiased estimators.(b) Identification of relationships by instrumental variables depends on the existence of correlations between the instruments and the explanatory variables. At least
Consider a regression model yt = x′t ???? + ut t = 1,…, T where ut is identically and independently distributed with mean 0 and variance ????2 and xt (2 × 1) is the vector with elements 1 and t.(a) Does the matrix T−1Σxtx′t converge to a finite limit?(b) Is the least squares estimator
Consider a regression model yt = x′t ???? + ut t = 1,…, T where ut is identically and independently distributed with mean 0 and variance ????2, and xt = (1 rt) where rt = t modulo 5. Thismeans that rt is the remainder after dividing t by 5, so it takes the values 1, 2, 3, 4, and 0 in
Consider the model y = X???? + u, where y isT × 1, ???? is a k × 1 vector of parameters, andX (T × k) is a stochasticmatrix having full column rank with probability 1; E(u|X) = ???? w.p. 1 and E(uu′|X) = ????2IT w.p. 1; and plim 1T X′X = Q (k × k) is a finite positive definite matrix. Let
Consider a random sequence X1, X2,…, XN,…. generated by the rule XN = N with probability 1∕N XN = 0 with probability 1 − 1∕N.(a) What is the expected value of XN?(b) What is the variance of XN?(c) What is the probability limit of XN as N → ∞?(d) Comment.
Show that a F(m, n)-distributed random variable converges in distribution to????2(m)∕m as n → ∞.
If x1, x2,…, xT is a random sample from a distribution with mean ???? and variance ????2 and ̄xT = T−1ΣTt=1 xt, show that(a) E(̄xT) = ????(b) Var(̄xT) = ????2∕T(c) plim ̄xT = ????
Are the following statements true or false?(a) The probability limit of a sequence of ratios is equal to the ratio of the probability limits of the respective sequences, when these exist and the limit of the denominator is nonzero.(b) A sequence of ratios is distributed in the limit like the ratio
Let a non-linear regression model have the form yt = ???? + ????xt + ????x2t+ ut, t = 1,…, T where E(ut |x1,…, xT) = 0 w.p.1 and E(utus |x1,…, xT) = {????2 t = s 0 t ≠ s w.p.1.Assume that E(xt) = 0 for t = 1,…, T.(a) What is E(yt |x1,…, xT )?(b) Find the coefficients a and b that
In the regression model yt = ????xt + ????zt + ut, t = 1,…, T, yt, xt, and zt are random variables with zero means. You may assume that E(yt|xt, zt) = ????xt + ????zt w.p.1 and also that the sample is independently drawn.(a) What is E(ut|xt, zt)?(b) What is E(ut|x1,…xT , z1,…zT )?(c) Write
Let xt and yt be random variables, having mean zero, and related by E(yt|xt) = ????xt w.p.1.Given a random sample of the pairs (yt, xt), for t = 1,…, T, consider the ordinary least squares estimator of ????????̂ =ΣΣtxtyt tx2t.(a) Is ????̂ unbiased for ?????Why? (or,Why not?)(b) If E(u2t|xt) =
Consider the estimator ????̂ = (X′X)−1X′y for ???? (k × 1) in the model y = X???? + u(T × 1) where E(u|X) = ???? w.p.1 and E(uu′|X) = ????2IT w.p.1.(a) What assumptions on X are implicit in this setup?(b) Derive the covariance matrix of ????̂ , being careful to justify each step in the
Are the following statements true or false?(a) Conditional probability is defined as a “degree of belief”. It has no meaning in classical statistics.(b) The law of iterated expectations allows us to calculate covariances using information about predictability.(c) The law of iterated
(a) Show that if A (n × n) is a nonsingular matrix and b any n-vector, then A + bb′is nonsingular and(A + bb′)−1 = A−1 − A−1bb′A−1 1 + b′A−1b(the Sherman-Morrison formula).(b) Let????̂ T = (X′X)−1X′y be the least squares regression from a sample of T observations. Show
In model (11.27) where ????1 = ????2, define the estimators.???? = (X′1X1 + X′2X2)−1 (X′1y1 + X′2y2)(from fitting (11.26)) and????̂ 2 = (X′2X2)−1X′2y2.(a) Show that both estimators are unbiased and that Var(????̂ 2 −.????) = ????2((X′2X2)−1 − ((X′1X1 + X′2X2)−1)).(b)
Linear restrictions on the coefficients of a regression model can be represented as zero restrictions for implementation of a significance test with the statistic (11.21), by linear transformations of the variables. If the equation is yt = ????1x1t + ????2x2t + ????3x3t + ????4 + ut, show how to
Consider the following two regressions:y = X???? + Z???? + ????, (1)where y is T × 1, X is T × k and includes the column of ones representing the intercept, Z is T × l, and̂u = X???? + Z???? +e, (2)where û = y − X????̂ is the vector of least squares residuals fromthe regression of y on
Consider the regression y = ???????? + X???? + u where ???? is the column of ones and X contains other regressors.(a) If the columns of X are in mean deviation form, what is the least squares estimator of ?????(b) If the columns of X are not in mean deviation form, how does this modify the answer
The multiple regression model in partitioned matrix notation is y = x1????1 + X2????2 + u (T × 1), where x1 is T × 1, and X2 is T × k − 1, and the multiple least squares estimator of ????1 is????̂1 =x′1M2y x′1M2x1.(a) Show that for the case k = 3, where themodel contains an intercept,
In the regression model yt = ????1x1t + ????2x2t + ????3 + ut, (1)the multiple least squares estimator of ????1 can be written as????̂1 =ΣTt=1 x̃1tyt ΣTt=1 ̃x2 1t. (2)(a) Give a definition of ̃x1t under which this formula is valid.(b) In formula (2), suppose that yt is replaced by ̃yt where
Are the following statements true or false?(a) Multiple regressions can be computed by a recursive sequence of simple regressions.(b) If regressors are in mean deviation form, including an intercept in the regression leaves the regression coefficients unchanged.(c) Omitted variable bias vanishes if
The regression model yt = ????1x1t + ????2x2t + ut, t = 1,…, T is estimated by constrained least squares, subject to the restriction ????2 = 0.(a) Find the constrained least squares estimators .????1 and .????2.(b) Find the formula for the Lagrangian multiplier .????.(c) Show that .????1 −
Consider the following two-variable model with T = 21 observations, where the data are expressed in sample mean-deviation form.y = x1????1 + x2????2 + u(a) If x′1x1 = 3, x′2x2 = 2, x′1x2 = x′1y = x′2y = 1, and y′y = 4∕5, calculate i. The least squares coefficients ????̂1 and
In the model y = X???? + u (T × 1) where X is T × k, consider the test statistic W = (R????̂ − c)′(R(X′X)−1R′)−1(R????̂ − c)rs2 where R (r × k) and c (r × 1) are fixed and s2 = ̂u′ ̂u∕(T − k).(a) For W to have the F(r, T − k) distribution requires that the numerator
Consider the linear regression model yt = ????′xt + ut, t = 1,…, T, where yt and xt (k × 1) are observed and ut is a random disturbance.The following assumptions are made about this model:(i) E(ut) = 0 for each t.(ii) E(u2t) = ????2 (constant) for each t.(iii) E(utus) = 0 for each t and s ≠
In the regression model yt = ????1x1t + ????2x2t + ut, t = 1,…, 12, x1t and x2t are fixed in repeated samples and are expressed as deviations from their sample means, and ut ∼ N(0, ????2). The following is the moment matrix of the data:⎡⎢⎢⎢⎣Σy2tΣx1tytΣΣ x2tyt x1tytΣx2 1tΣΣ
Are the following statements true or false?(a) The size of a test is the probability of rejecting the null hypothesis when it is true. If there are different cases of the null hypothesis on which the probability depends, the size is the largest such probability.(b) The least squares estimator is
Consider the matrix B =⎡⎢⎢⎢⎢⎣23−13−13−13 23 −13−13−13 23⎤⎥⎥⎥⎥⎦(a) Show that B has eigenvalues ????1 = ????2 = 1 and ????3 = 0.(b) Find eigenvectors c1 for ????1 and c3 for ????3. Verify that these vectors are mutually orthogonal, and normalize them to have unit
Consider the matrix A =⎡⎢⎢⎢⎢⎣13 13 13 13 13 13 13 13 13⎤⎥⎥⎥⎥⎦(a) Show that the eigenvalues of A are ????1 = 1 and ????2 = ????3 = 0.(b) Find eigenvectors c1 for ????1 and c2 for ????2. Verify that these vectors are mutually orthogonal. Normalize them to have unit length.(c)
Obtain the eigenvalues and eigenvectors of the following 3 × 3matrices. Ineachcase, determine whether the matrix is: (i) positive definite, (ii) positive semidefinite, (iii)idempotent.(a) ⎡⎢⎢⎣1 0 1 0 1 0 1 0 1⎤⎥⎥⎦(b) ⎡⎢⎢⎣2 1 1 1 2 0 1 0 2⎤⎥⎥⎦
Obtain the eigenvalues and eigenvectors of the following 2 ×2 matrices. In each case, determine whether the matrix is: (i) positive definite, (ii) positive semidefinite,(iii) idempotent.(a) [2 1 1 2](b) [1 2 2 4](c) [12 12 12 12](d) [12−12−12 12]
Are the following statements true or false?(a) The eigenvalues of a symmetric matrix are always real.(b) Odd numbers of complex-valued eigenvalues of real-valued matrices cannot occur.(c) The eigenvalues of an idempotent matrix include at least one zero in every case.(d) Eigenvectors of idempotent
Suppose that yt = ???? + ut, t = 1,…, T where E(ut) = 0 but E(u2t) = ????2 t , and it is known that the observations fall into two groups, having more and less dispersion. In the ‘noisy’ group labelled t = 1,…, T1,????2 t = 4, while in the group labelled t = T1 + 1,…, T, ????2 t = 1.(a)
In themodel y = X???? + u where X (T × k) is fixed with rank k, consider transforming the data as y∗ = Py and X∗ = PX, where P is a T × T lower triangular nonsingular matrix.(a) Show that the generalized least squares estimator can be computed by running the OLS regression of y∗ onto X∗
In the regressionmodel y = X???? + u (T × 1), X (T × k) is fixed with rank k, E(u) = ????, and E(uu′) = ????2????.(a) Consider the function S(b) = (y − Xb)′????−1(y − Xb).Given a vector ̃ ????, define ̃u = y − X ̃ ????. Show that if X′????−1 ̃u = ????, (8.21)then S is
A ‘random coefficient’ model of the relationship between a random series y1,…, yT and a1,…, aT , a collection of observed constants, is yt = at(???? + ut), where ut ∼ i.i.d.(0, ????2). Compare and contrast the following proposed estimators of ????.(a) The sample mean of the series
In the model of Question 2, suppose that E(ut) = 0, E(u2t) = ????2????2t, and E(utus) = 0 for t ≠ s, where ????t is an observed data series, fixed in repeated samples.(a) Show that the weighted least squares estimator of ???? has a lower variance than the OLS estimator.(b) Show that the weighted
of ???? than in the case of Estimator 1, this means that the difference of the two covariance matrices is positive definite.(d) The Gauss-Markov theorem compares the variances of linear functions of the regression coefficients. It has nothing to tell us about non-linear functions.(e) If the
Are the following statements true or false?(a) ‘Efficiency’ is the property of an estimator of having a smaller variance than any other estimator.(b) If the difference of the covariance matrices of two alternative estimators of coefficient vector ???? is positive definite, this means that the
Consider the quadratic formu′Mu, whereMis defined by (7.13) and X is T × k with rank k.(a) Show that this is a sum of squares.(b) In the case when k = 1 and X (T × 1) is the unit column ???? = (1, 1,…, 1)′, write out u′Mu in ‘sigma’ notation.(c) Show that if T − k of the elements of
In the regressionmodel y = X???? + u whereX = (x1 x2 · · · xT )′, assume E(u) = ???? and E(uu′) = ????2IT .(a) What is the variance of ût, the tth element of the vector ̂u of least squares residuals?(b) What is the covariance of ût with ûs when t is different from s?(c) What do your
Let Q = X(X′X)−1X′ where X (T × k) has rank k.(a) Show that Q has rank k.(b) Show thatM = I − Q has rank T − k.
Let ????̂ be defined in (7.2) and let assumptions CRM hold.(a) Let a be a k-vector of constants.What are the mean and variance of a′????̂ ?(b) Let A be a r × k matrix of constants. What are the mean vector and variance matrix of A????̂ ?(c) In part (b), suppose r > k and A has rank k. Show
Consider the multiple regression equation y = X???? + u Suppose that the independent variables are subjected to a linear transformation Z = XA where the matrix A is square and nonsingular. In the regression y = Z???? + ????(a) What is the relationship between the OLS estimate ????̂ and the
For all choices of A and B such that the following matrices exist, are the matrices(i) symmetric? (ii) idempotent? (iii) positive definite or semidefinite?(a) A(A′A)−1A′(b) AB−1A′(c) A(B′A)−1B′(d) A(A′BA)−1A′B(e) BA(A′A)−1A′B−1(f)
Are these matrices idempotent?(a) ⎡⎢⎢⎣−12 12 12 −12⎤⎥⎥⎦(b) [0 0 0 1](c) ⎡⎢⎢⎢⎢⎢⎣23−13−13−13 23−13−13−13 23⎤⎥⎥⎥⎥⎥⎦(d) ⎡⎢⎢⎢⎢⎢⎣13 13 13 13 13 13 13 13 13⎤⎥⎥⎥⎥⎥⎦(e) ⎡⎢⎢⎢⎢⎣12 0 12 0 0 0 12 0
Are the following statements true or false?(a) If the square matrix A is idempotent, then the matrix I − A is also idempotent.(b) If the square matrix A is idempotent, then the matrix A − I is also idempotent.(c) If two square matrices are orthogonal, they must be singular.(d) If A and B are n
Consider a pair of random variables generated in the following way. X is standard normal (mean 0, variance 1), and Y = WX whereW is an independently drawn discrete random variable, taking the values 1 and −1 with equal probabilities of 0.5.(a) Is the marginal distribution of Y standard normal?(b)
Suppose that the n1 + n2-vector x =[x1 x2]n1 n2 is multivariate normal with mean zero and covariance matrix ???? =[????11 ????12????21 ????22]where ????21 = ????′12.(a) Write out the marginal density function of x2.(b) Write out the joint density function of x in partitioned form, using the
Let x (n × 1) be a standard normal random vector.(a) What is the joint density function of the m-vector y = Ax + b where A (m × n)is a fixed matrix with rank m and b is a fixed vector?(b) Consider the vectors u = Ax and ???? = Bx where A (m × n) and B (p × n) are fixed matrices. Show that if
Let X and Y be a pair of independent standard normal random variables.(a) What is the mean of XY?(b) What is the variance of XY?(c) Explain why Y X has the same distribution as Y|X| .(d) What is this distribution?(e) What can you say about the mean of Y X ?
If y is a normal n-vector and y′Ay has the ????2????2(n) distribution for positive constant????2, what can you say about(a) the rank of A?(b) The definiteness of A?(c) The mean of y?(d) The variance matrix of y?(e) The mean of y′Ay?(f) The variance of y′Ay?
Let X and Y be independent standard normal random variables, and let Z = aX + bY where a and b are constants.What is the probability density function of the random vector[Z X]?
Are the following statements true or false?(a) If u ∼ N(????, ????) (n × 1) where ???? is nonsingular then u′????−1u ∼ ????2(n).(b) If a sample x1…, xn is identically and independently distributed asN(0, ????2), then if ̄x denotes the sample mean and s2 the sample variance, the
The mean of the Poisson distribution with parameter ???? has the formula E(X) =Σ∞k=0 k e−????????k k!Use the series expansion of e???? (see (B.2)) to show that E(X) = ????.
Let X be distributed as binomial(6, 0.2).(a) What is the probability that X = 3?(b) What is the probability that X < 2?(c) What is E(X)?(d) What is Var(X)?
What is the probability of having x heads appear in n successive tosses of a fair coin, in the following cases?(a) x = n = 1(b) x = n = 5(c) x = 2, n = 5(d) x = 3, n = 6
(a) If x1,…, xT is a random sample with mean ???? and variance ????2, and s2 is defined in (1.3), show that E(s2) = ????2.(b) If (xt, yt) for t = 1,…, T is a random sample of pairs with means ????x and ????y and covariance ????xy, and sxy is defined in (1.9), show that E(sxy) = ????xy.
Let a pair (X, Y) of independent standard normal random variables represent a point in the plane. Using the tabulation of the normal distribution, or otherwise, calculate the following:(a) The probability that X falls in the interval [1, 2].(b) The probability that (X, Y) falls in the rectangle [1,
Let X and Y be independent standard normal random variables, and let a and b be constants.(a) Find the probability density function of the random variable Z = aX + bY.(b) Find the joint probability density function of the pair (U,V) when U = aX and V = bY.
Suppose that X ∼ N(5, 2), Y ∼ N(1, 3), and X and Y are independent of each other.Calculate the mean, the variance, and the covariance with X of(a) X + Y(b) X − Y(c) XY.
Are the following statements true or false?(a) If and only if random variables are independent, they are uncorrelated.(b) If and only if continuously distributed random variables are independent, their joint density function factorizes into marginal densities.(c) A normal random variable can take
Suppose that P =[A B C D](n + m × n + m) has full rank n + m, andD(m × m) is nonsingular.(a) Find partitioned matrices Q and R such that PQ = R and the partition of R has A − BD−1C (n × n) as top-left block and |R| = |A −BD−1C|.(b) Hence (or otherwise) show that A − BD−1C (n × n)
In the equation y = X???? + u (T × 1):(a) Show that if X (T × k) has full rank, setting b = (X′X)−1X′y minimizes S(b) = (y − Xb)′(y − Xb).(b) Explain the role of the rank condition in the result of part (a). What does a failure of the condition imply, and how might you proceed in such
Let X =⎡⎢⎢⎢⎢⎣3 0 1 0 2 1 0 3 1 1 0 1⎤⎥⎥⎥⎥⎦.(a) Show that X′X is positive definite.(b) Calculate (X′X)−1. Is it also positive definite?(c) If y =⎡⎢⎢⎢⎢⎣3231⎤⎥⎥⎥⎥⎦, what is ????̂ = (X′X)−1X′y?What is û = y− X????̂ ?
For all choices of A and B such that both matrices have full rank, are the following matrices (assumed to exist) (i) positive semidefinite? (ii) positive definite?(a) (A′A)−1(b) B′AB(c) I − A(A′A)−1A′(d) A′B(B′AB)−1B′A(e) A(A′B′BA)−1A′
Are these matrices positive definite? positive semidefinite?(a) ⎡⎢⎢⎣2 1 0 1 2 1 0 1 2⎤⎥⎥⎦ (b) ⎡⎢⎢⎣2 −1 −1−1 2 1−1 1 2⎤⎥⎥⎦(c) ⎡⎢⎢⎣ 2 −2 −1−2 2 1−1 1 −1⎤⎥⎥⎦
Solve the following equation systems as far as possible.(a) ⎡⎢⎢⎣0 2 2 1 1 0 0 0 1⎤⎥⎥⎦⎡⎢⎢⎣xyz⎤⎥⎥⎦=⎡⎢⎢⎣000⎤⎥⎥⎦(b) ⎡⎢⎢⎣0 2 2 1 1 0−1 1 2⎤⎥⎥⎦⎡⎢⎢⎣xyz⎤⎥⎥⎦=⎡⎢⎢⎣000⎤⎥⎥⎦(c) ⎡⎢⎢⎣0 2 2 1 1 −1 1 3 1
What is the rank of the following matrices?(a) ⎡⎢⎢⎣1 2 3 1 2 3 1 2 3⎤⎥⎥⎦(b) ⎡⎢⎢⎢⎢⎣5 5 5 3 0 1 6 0 2 9 0 3 ⎤⎥⎥⎥⎥⎦(c) ⎡⎢⎢⎣1 0 1 1 1 0 1 0 1 ⎤⎥⎥⎦(d) ⎡⎢⎢⎢⎢⎣0 2 0 0 1 0 1 0 0 0 0 1 0 0 2 0 ⎤⎥⎥⎥⎥⎦(e) ⎡⎢⎢⎢⎢⎣3 6 9 1
Are the following statements true or false?(a) The row rank and column rank of a matrix are always equal.(b) The null space of a matrix is the collection of all the possible vectors of weights such that the weighted sum of the columns is the zero vector.(c) A homogeneous equation system has a
Obtain the partitioned inverse of the transposed matrix[A B C D]′, assumed to be nonsingular, and show that this is the transpose of the partitioned inverse of[A B C D].
Let A (n × n) and D (m × m) be nonsingular, and B (n × m) and C (m × n) be any matrices such that[A B C D]is nonsingular.(a) Show by direct calculation that[A B C D]−1=[E −EBD−1−D−1CE D−1 + D−1CEBD−1]where E = (A − BD−1C)−1.(b) Show by direct calculation that EBD−1 =
For the matrices A =[1 2 3 4], B =[2 1 1 2 calculate (a) (AB)−1 (b) B−1A−1 (c) (BA)−1 (d) A−1B−1
Use Cramer’s rule to solve the following systems of equations.(a) ⎡⎢⎢⎣1 0 1 1 1 0 0 1 1⎤⎥⎥⎦⎡⎢⎢⎣x1 x2 x3⎤⎥⎥⎦=⎡⎢⎢⎣100⎤⎥⎥⎦(b) ⎡⎢⎢⎣1 0 1 1 1 1 0 1 1⎤⎥⎥⎦⎡⎢⎢⎣x1 x2 x3⎤⎥⎥⎦=⎡⎢⎢⎣101⎤⎥⎥⎦
What is the determinant of this matrix?⎡⎢⎢⎢⎢⎢⎢⎢⎣3 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 2 0 0 0 5 2 33 29 2 0 0 5 2 0 5 3 1 0 5 0 4 4 0 0 2⎤⎥⎥⎥⎥⎥⎥⎥⎦
Calculate the inverses of these matrices.(a) ⎡⎢⎢⎣1 0 0 0 2 4 0 0 1⎤⎥⎥⎦(b) ⎡⎢⎢⎢⎣3 0 0 0 0 2 0 0 0 0 1 1 1 0 1 2⎤⎥⎥⎥⎦(c) ⎡⎢⎢⎢⎢⎣1 0 0 0 1 0 2 0 0 0 0 0 3 0 0 0 0 0 3 1 0 0 1 0 1⎤⎥⎥⎥⎥⎦
Do the following systems have (i) a unique solution, (ii) no solution, or (iii) many solutions? Draw graphs to illustrate your answers.(a) 2x1 + 3x2 = 1 x1 + 3x2 = 2 (b) 2x1 + 3x2 = 1 2x1 + 3x2 = 2 (c) 2x1 + 3x2 = 1 4x1 + 6x2 = 2
Are the following statements true or false?(a) The adjoint of a square matrix is the matrix having the principal minors as its elements.(b) If A and D are square matrices of the same dimension, D is nonsingular, and B = DAD−1, the determinant of B is equal to the determinant of A.(c) The
If A, B,c, and d are defined as in Questions 2 and 3, write down ten matrix expressions involving one ormore of these components that do not exist (violate conformability).
Let A and B be as in Question 2 and c =⎡⎢⎢⎣101⎤⎥⎥⎦, d =[11].Compute(a) Bc(b) B′d(c) A + cd′(d) c′c(e) cc′(f) ‖c‖(g) d′Bc(h) c′AA′c
Let A =⎡⎢⎢⎣1 4 2 5 3 6⎤⎥⎥⎦, B =[1 2 3 1 2 3].Compute (a) A + B′(b) 5B (c) AB (d) BA (e) A′A (f) BB′
Are the following statements true or false?(a) If for matrices A and B both of the products AB and BA exist, then A and B must be square.(b) Symmetric matrices must be square.(c) If A and B are square matrices conformable for multiplication, then AB = BA(d) If A, B, and C are matrices having the
Show that the ???? that minimizesΣTt=1 (yt − ????)2 is the sample mean of y1,…, yT .
(a) Solve the following equation system for ????̂1.????̂1ΣT t=1 x2 1t + ̂ ????2ΣT t=1 x1tx2t =ΣT t=1 x1tyt (1)????̂1ΣT t=1 x2tx1t + ????̂2ΣT t=1 x2 2t =ΣT t=1 x2tyt (2)(b) Define ????̂ = ȳ − ????̂1 x̄ 1 − ????̂1 ̄x2. Show that if T ̂????̄x1 is subtracted from the right-hand
Compute the regression of y on x (original sample).(a) Report the fitted slope and intercept coefficients, and the residuals ̂u.(b) Verify that ̂u has zero correlation with x. Draw the scatter plot of the two variables.(c) Consider the prediction equation̆y = 5 + 0.6x.Show that the prediction

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An Introduction To Econometric Theory 1st Edition James Davidson - Solutions

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