Class Notes in Statistics and Econometrics Part 13 pps
Class Notes in Statistics and Econometrics Part 13 pps
... linear in P y are also linear in y and vice versa,ˆβ is also the BLUE in the original GLSmodel.Problem 319. 2 points Show that(26.0.3)ˆβ −β = (XΨ−1X)−1XΨ−1εεε and derive from this ... than those familiar from linear estimators. The problem at hand can bereduced to a minimum trace problem as defined in Rao [Rao73, pp. 65–66], and part of the following proof draws on a private ... is uninteresting, sincethe (biased) best bounded MSE quadratic estimator is better. The class of unbiasedestimators only then becomes interesting when the kurtosis is not known: for certainX-matrices,...
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Class Notes in Statistics and Econometrics Part 4 pps
... independent. Problem 136 . If the vector random variables x and y have the property thatxiis independent of every yjfor all i and j, does that make x and y independentrandom vectors? Interestingly, ... argumentinside.220 8. VECTOR RANDOM VARIABLES1), and if cov[x, y] = 0, then x and y are independent. (I.e., in this respect indicatorvariables have similar properties as jointly normal random ... 2 points Given two independent random variables x and y withdensity functions fx(x) and gy(y). Write down their joint, marginal, and conditionaldensities.Answer. Joint density: fx,y(x,...
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Class Notes in Statistics and Econometrics Part 5 pps
... the inner integral becomes −12πe−t∞0=12π; therefore the whole integral is 1. Therefore the product of the integrals of themarginal densities is 1, and since each such marginal integral ... Firstwe will introduce the multivariate standard normal, which one obtains by applyingonly operations (1) and (2), i.e., it is a vector composed of independent univariatestandard normals, and give ... dr, and the integral becomes12π∞0e−qdq. • d. 1 point Are r and φ independent?Answer. Yes, because joint density function is the product of the marginals. 10.4. Multivariate Standard...
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Class Notes in Statistics and Econometrics Part 12 ppsx
... lefthand side; it is amazing and surprising that it is exactly the populationequivalent of the F -test for testing α = 0 in the regression with intercept. It can be estimated byreplacing α2with ... thefollowing:Theorem 24.2.2.ˆβ is a linear minimax est imator of the parameter vector β in the following sense: for every nonrandom coefficient vector t, tˆβ is the linear640 24. SAMPLING PROPERTIES OF THE ... the unit vector having 1 in the firstplace and zeros elsewhere, and ι is the vector which has ones everywhere.Answer. Write both ¯y and ˆβ in terms of y, i.e., ¯y =1nιy and ˆβ = (XX)−1Xy....
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Class Notes in Statistics and Econometrics Part 14 ppsx
... best linear bounded MSE predictor of z based on y, µ, and ν.• a. Give special cases of this specification in which µ and ν are constant and y and z random, and one in which µ and ν and y are random ... are random and z is constant, and one in which µ and ν are random and y and z are constant.27.1. MINIMUM MEAN SQUARED ERROR, UNBIASEDNESS NOT REQUIRED 713 information itself, and the joint MSE ... Normallydistributed and independent of each other, and E[ε] = 0. Define y = α + β1x1+β2x2+ ε. Again express β1 and β2 in terms of variances and covariances of x1, x2, and y.Answer. Since x1and...
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Class Notes in Statistics and Econometrics Part 17 pps
... within-group or between-group phenomena or both.:• negative relationship of protein % and oil % (both within • and ◦);• p osit ive relationship of oil % and seed size (both within • and ◦ and ... of seed size and height;• p osit ive relationship of height and lodging (within ◦ and between groups);• negative relationship of oil % and lodging (between groups and possibly within •);• negative ... searches for an interesting and informative projec-tion of the data by maximizing a criterion function. A logical candidate would forinstance be the variance ratio as defined in (8.6.7), but...
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Class Notes in Statistics and Econometrics Part 18 pps
... do something which he do e s not quite understand is in the above eliciation ofprior information. It often happens that prior information gained in this way is self-contradictory, and the researcher ... and variance alone.Problem 395. 4 points Show that, if y is normally distributed, s2 and ˆβ areindependent.Answer. We showed in question 300 thatˆβ and ˆε are uncorrelated, therefore in ... BAYESIAN ESTIMATION IN THE LINEAR MODEL and that our prior knowledge about α is not affected by (is independent of ) ourprior knowledge concerning β and γ. Assume that σ2is known and that it has...
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Class Notes in Statistics and Econometrics Part 20 pps
... Scenario: Minimizing relative increase in Mahalanobisdistance i f distribution is knownWe start with a situation where the e xpec ted values of the random vectors y and z are known, and their joint ... minimize the relative increase in the Maha-lanobis distance (instead of selecting its minimax value) because all parameters ofthe underlying distribution were known. The simplest situation in ... a n + 1st observation in such a way as to minimize the worst possiblerelative increase in this Mahalanobis distance?Minimizing the maximum possible relative increase in the Mahalanobis distancedue...
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Class Notes in Statistics and Econometrics Part 29 ppsx
... utis independent of εs−1. Since s < t, therefore in particular s = t, utis also independent ofus. Therefore utis independent of any function of εs−1 and us, in particular, it is independent ... process independent of the pre-sample disturbanceε0, and α0> 0 and 0 < α1< 1 are two constants.• a. 2 points Show that utis independent of all εswith 0 ≤ s < t. (Hint: useinduction).Answer. ... 1000 families in acertain town, and denote the income of family k by zk. Let µ =110001000k=1zkbe the population average of all 1000 incomes in this fin ite population, and letσ2=110001000k=1(zk−...
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Class Notes in Statistics and Econometrics Part 34 pps
... back, and you regress y on X with a constant term. (The under-lining does not denote taking out of the mean, but the taking out of the seasonalmeans and adding back of the overall mean.) In the ... causality should be included in statistics textbooks.Innovation accounting or impulse response functions: make a moving averagerepresentation, and then you can pick the timepath of the innovations: ... 1 · δ3+ ε4autumnHere the winter intercept is α + δ1, the spring intercept α + δ2, summer α + δ3, and autmumnα −δ1−δ2−δ3. Summing this and dividing by 4 shows that the constant...
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