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118 PART ONE: SINGLE-EQUATION REGRESSION MODELS

where fi1 = mean of Y\ 1^2 = mean of Y2 o1 = standard deviation of Yi a2 = standard deviation of Y2 p = coefficient of correlation between Yi and Y2

4.2. By applying the second-order conditions for optimization (i.e., second-derivative test), show that the ML estimators of p1, f}2, and o2 obtained by solving Eqs. (9), (10), and (ii) do in fact maximize the likelihood function (4).

4.3. A random variable X follows the exponential distribution if it has the following probability density function (PDF):

f (X) = (1/0 )e-X/e for X > 0 = 0 elsewhere where 9 > 0 is the parameter of the distribution. Using the ML method, show that the ML estimator of 9 is 9 = YI Xi /n, where n is the sample size. That is, show that the ML estimator of 9 is the sample mean X.

Gujarati: Basic I I. Single-Equation I 5. Two-Variable I I © The McGraw-Hill

Econometrics, Fourth Regression Models Regression: Interval Companies, 2004 Edition Estimation and Hypothesis

Testing

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