Questions and Answers of Econometric Analysis

Suppose the true regression model is given by (4-8). The result in (4-10) shows that if either P1.2 is nonzero or β2 is nonzero, then regression of y on X1 alone produces a biased and inconsistent
Show that in the multiple regression of y on a constant, x1 and x2 while imposing the restriction β1 + β2 = 1 leads to the regression of y − x1 on a constant and x2 − x1.
Prove that under the hypothesis that Rβ = q, the estimator where J is the number of restrictions, is unbiased for σ2. (у — Хь,У (у — Хb,) п — К+J
Use the test statistic defined in Exercise 7 to test the hypothesis in Exercise 1. e'e, x² = X {Est. Var[2.]}¯'a, = (n – K) e'e
An alternative way to test the hypothesis Rβ ?? q = 0 is to use a Wald test of the hypothesis that λ?? = 0, where λ?? is defined in (5-23). Prove that the fraction in brackets is the ratio of two
Prove the result that the R2 associated with a restricted least squares estimator is never larger than that associated with the unrestricted least squares estimator. Conclude that imposing
Prove the result that the restricted least squares estimator never has a larger covariance matrix than the unrestricted least squares estimator.
The expression for the restricted coefficient vector in (5-23) may be written in the form b∗ =[I − CR]b + w, where w does not involve b. What is C? Show that the covariance matrix of the
The regression model to be analyzed is y = X1β1 + X2β2 + ε, where X1 and X2 have K1 and K2 columns, respectively. The restriction is β2 = 0.a. Using (5-23), prove that the restricted estimator is
Using the results in Exercise 1, test the hypothesis that the slope on x1is 0 by running the restricted regression and comparing the two sums of squared deviations. Test the hypothesis that the two
A multiple regression of y on a constant x1and x2produces the following results: ŷ = 4 + 0.4x1+ 0.9x2, R2= 8/60, e'e = 520, n = 29, Test the hypothesis that the two slopes sum to 1. [29 0 0 50 10
Example 4.10 presents a regression model that is used to predict the auction prices of Monet paintings. The most expensive painting in the sample sold for $33.0135M (log = 17.3124). The height and
In Section 4.7.3, we consider regressing y on a set of principal components, rather than the original data. For simplicity, assume that X does not contain a constant term, and that the K variables
In (4-13), we find that when superfluous variables X2 are added to the regression of y on X1 the least squares coefficient estimator is an unbiased estimator of the true parameter vector, β = (β'1,
Consider a data set consisting of n observations, nc complete and nm incomplete, for which the dependent variable, yi, is missing. Data on the independent variables, xi, are complete for all n
For the simple regression model yi = μ + εi, εi ∼ N[0, σ2], prove that the sample mean is consistent and asymptotically normally distributed. Now consider the alternative estimator μ̂ = Σi
Let ei be the ith residual in the ordinary least squares regression of y on X in the classical regression model, and let εi be the corresponding true disturbance. Prove that plim(ei − εi) = 0.
For the classical normal regression model y = Xβ + ε with no constant term and K regressors, what is plim F[K, n − K] = plim R2/K / (1−R2)/(n−K), assuming that the true value of β is zero?
Prove that E[b'b] = β'β + σ2ΣKk=1(1/λk) where b is the ordinary least squares estimator and λk is a characteristic root of X'X.
For the classical normal regression model y = Xβ + ε with no constant term and K regressors, assuming that the true value of β is zero, what is the exact expected value of F[K, n − K] =
Consider the multiple regression of y on K variables X and an additional variable z. Prove that under the assumptions A1 through A6 of the classical regression model, the true variance of the least
The following sample moments for x = [1, x1, x2, x3] were computed from 100 observations produced using a random number generator: The true model underlying these data is y = x1 + x2 + x3 + ε. a.
Prove that the least squares intercept estimator in the classical regression model is the minimum variance linear unbiased estimator.
As a profit-maximizing monopolist, you face the demand curve Q = α+βP+ε. In the past, you have set the following prices and sold the accompanying quantities: Suppose that your marginal cost is
Suppose that the regression model is yi = α + βxi + εi, where the disturbances εi have f (εi) = (1/λ) exp(−εi /λ), εi ≥ 0. This model is rather peculiar in that all the disturbances are
Suppose that the classical regression model applies but that the true value of the constant is zero. Compare the variance of the least squares slope estimator computed without a constant term with
Consider the simple regression yi= βxi+ εiwhere E[ε | x] = 0 and E[ε2| x] = σ2 a. What is the minimum mean squared error linear estimator of β? b. For the estimator in part a, show that ratio
Suppose that you have two independent unbiased estimators of the same parameter θ̂, say θ̂1 and θ̂2, with different variances v1 and v2. What linear combination θ̂ = c1 θ̂1 + c2 θ̂2 is
In the December 1969, American Economic Review (pp. 886–896), Nathaniel Leff reports the following least squares regression results for a cross section study of the effect of age composition on
Using the matrices of sums of squares and cross products immediately preceding Section 3.2.3, compute the coefficients in the multiple regression of real investment on a constant, real GNP and the
Three variables, N, D, and Y, all have zero means and unit variances. A fourth variable is C = N+ D. In the regression of C on Y, the slope is 0.8. In the regression of C on N, the slope is 0.5. In
Suppose that you estimate a multiple regression first with, then without, a constant. Whether the R2 is higher in the second case than the first will depend in part on how it is computed. Using the
Adata set consists of n observations on Xnand yn. The least squares estimator based on these n observations is bn= (X'nXn)??1X'nyn. Another observation, xsand ys, becomes available. Prove that the
Prove that the adjusted R2 in (3-30) rises (falls) when variable xk is deleted from the regression if the square of the t ratio on xk in the multiple regression is less (greater) than 1.
Let Y denote total expenditure on consumer durables, nondurables, and services and Ed, En, and Es are the expenditures on the three categories. As defined, Y = Ed + En + Es. Now, consider the
A common strategy for handling a case in which an observation is missing data for one or more variables is to fill those missing variables with 0s and add a variable to the model that takes the value
What is the result of the matrix product M1M where M1 is deﬁned in (3-19) and M is deﬁned in (3-14)?
In the least squares regression of y on a constant and X, to compute the regression coefficients on X, we can first transform y to deviations from the mean y̅ and, likewise, transform each column of
Suppose that b is the least squares coefficient vector in the regression of y on X and that c is any other K × 1 vector. Prove that the difference in the two sums of squared residuals is(y − Xc)'
For the regression model y = α + βx + ε,a. Show that the least squares normal equations imply Σi ei = 0 and Σi xi ei = 0.b. Show that the solution for the constant term is a = y̅ − bx̅.c.

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