Info Fub
Figure 2.4 Critical Region for Testing 2 against 4 for n 4 This gives us an idea of how, for a fixed a 0.05, the minimum decreases with larger sample size n. As n increases from 4 to 9 to 16, the var x a2 n decreases and the two distributions shown in Figure 2.4 shrink in dispersion still centered around o 2 and i 4, respectively. This allows better decision making based on larger sample size as reflected by the critical region shrinking from x gt 3.65 for n 4 to x gt 2.8225 for n 16, and the...
Introduction Xte
There has been an enormous amount of research in time-series econometrics, and many economics departments have required a time-series econometrics course in their graduate sequence. Obviously, one chapter on this topic will not do it justice. Therefore, this chapter will focus on some of the basic concepts needed for such a course. Section 14.2 defines what is meant by a stationary time-series, while sections 14.3 and 14.4 briefly review the Box-Jenkins and Vector Autoregression VAR methods for...
Problems Hom
Vt PVt-1 et t 1, 2, ,T with p lt 1 and et - IIN 0, a Show that if yo N 0, a2 1 p2 , then E yt 0 for all t and var yt a2 1 p2 so that the mean and variance are independent of t. Note that if p 1 then var yt is to. If p gt 1 then var yt is negative b Show that cov yt,yt-s psa2 which is only dependent on s, the distance between the two time periods. Conclude from parts a and b that this AR 1 model is weakly stationary. c Generate the above AR 1 series for T 250, a2 0.25 and various values of p...
Empirical Examples
To illustrate the logit and probit models, we consider the PSID data for 1982 used in Chapter 4. In this example, we are interested in modelling union participation. Out of the 595 individuals observed in 1982, 218 individuals had their wage set by a union and 377 did not. The explanatory variables used are years of education ED , weeks worked WKS , years of full-time work experience EXP , occupation OCC 1, if the individual is in a blue-collar occupation , residence SOUTH 1, SMSA 1, if the...
Example 4 For the Normal case
oMLE n 1 s2 n and e oMLE n 1 a2 n. But as n to, lim E crMle a2. Hence, o le is asymptotically unbiased for a2. Similarly, an estimator which attains the Cramer-Rao lower bound in the limit is asymptotically efficient. Note that var X a2 n, and this tends to zero as n to. Hence, we consider nX which has finite variance since var i nX n var X a2. We say that the asymptotic variance of X denoted by asymp.var X a2 n and that it attains the Cramer-Rao lower bound in the limit. X is therefore...
The NeymanPearson Theory
The classical theory of hypothesis testing, known as the Neyman-Pearson theory, fixes a Pr type I error lt a constant and minimizes p or maximizes 1 p . The latter is known as the Power of the test under the alternative. The Neyman-Pearson Lemma If C is a critical region of size a and k is a constant such that Lo Li lt k inside C then C is a most powerful critical region of size a for testing Ho 6 6o, against Hi 6 6i. Note that the likelihood has to be completely specified under the null and...
Preface
This book is intended for a first year graduate course in econometrics. Courses requiring matrix algebra as a pre-requisite to econometrics can start with Chapter 7. Chapter 2 has a quick refresher on some of the required background needed from statistics for the proper understanding of the material in this book. For an advanced undergraduate masters class not requiring matrix algebra, one can structure a course based on Chapter 1 Section 2.6 on descriptive statistics Chapters 3-6 Section 11.1...