Lecture Notes on Confidence Intervals | AEB 6933, Study notes of Introduction to Macroeconomics

Download Lecture Notes on Confidence Intervals | AEB 6933 and more Study notes Introduction to Macroeconomics in PDF only on Docsity! Confidence Intervals Lecture XXI I. Interval Estimation A. As we discussed when we talked about continuous distribution functions, the probability of a specific number under a continuous distribution is zero. B. Thus, if we conceptualize any estimator, either a nonparametric estimate of the mean or a parametric estimate of a function, the probability of the true value equal to the estimated value is obviously zero. C. Thus, usually talk about estimated values in terms of confidence intervals. Specifically, as in the case when we discussed the probability of a continuous variable, we define some range of outcomes. However, this time we usually work the other way around defining a certain confidence level and then stating the values that contain this confidence interval. II. Confidence Intervals A. Amemiya notes a difference between confidence and probability. Most troubling is our classic definition of probability as “a probabilistic statement involving parameters.” This is troublesome due to our inability without some additional Bayesian structure to state anything concrete about probabilities. A. Example 8.2.1. Let i X be distributed as a Bernoulli distribution, 1,2,i n . Then, ~ , (1 ) / A T X N p p p n Therefore, we have ~ 0,1 1 AT p Z N p p n 1. Why? By the Central Limit Theory 2. Given this distribution, we can ask questions about the probability. Specifically, we know that if Z is distributed 0,1N , then we can define k P Z k Building on the normal probability, the one tailed probabilities for the normal distribution are: AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XXI Professor Charles Moss Fall 2007 2 Table 1. Confidence Levels k 2 1.0000 0.1587 0.3173 1.5000 0.0668 0.1336 1.6449 0.0500 0.1000 1.7500 0.0401 0.0801 1.9600 0.0250 0.0500 2.0000 0.0228 0.0455 2.3263 0.0100 0.0200 1. The values of k can be derived from the standard normal table as 1 k T p P k p p n 2. Assuming that the sample value of T is t , the confidence is defined by 1 k t p C k p p n 3. Building on the first term 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 0 1 2 0 k T p p p P k P T p k np p n p p P T p k n k k P T Tp p p p n n k k P p p T T n n AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XXI Professor Charles Moss Fall 2007 5 be the estimator of 2 . Then the probability distribution 1 1 1 1nt S T n or the Student’s t distribution with n-1 degrees of freedom is the probability distribution. 1. Theorem 5.4.1: Let 1 2 , , n X X X be a random sample from a 2,N distribution, and let 1 1 n ii X X n and 22 1 1 1 n ii S X X n Then a) X and 2S are independent random variables a) 2 ~ ,X N n b) 2 2 1N S has a chi-squared distribution with 1n degrees of freedom. 2. The proof of independence is based on the fact that 2S is a function of the deviations from the mean which, by definition, must be independent of the mean. 3. More interesting, is the discussion of the chi-squared statistic. a) The chi-square is defined as: 1 2 2 2 1 2 2 p x p f x x e p for all positive x , where p is called the degrees of freedom. b) In general, the gamma distribution function is defined through the gamma function: 1 0 t t e dt Dividing both sides of this expression by yields 1 1 0 1 1 t t t e t e dt f t AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XXI Professor Charles Moss Fall 2007 6 Substituting X t gives the traditional two parameter form of the distribution function 11 , x f x x e The expected value of the gamma distribution is and the variance is 2 . c) Lemma 5.4.1. (Facts about chi-squared random variables) We use the notation 2 p to denote a chi-squared random variable with p degrees of freedom. (1) If Z is a 0,1N random variable, then 2 2 1~Z , that is, the square of a standard normal random variable is a chi- squared random variable. (2) If 1 2 , , n X X X are independent, and 2 ~ ii p X , then 1 2 2 1 2 ~ nn p p p X X X , that is, independent chi- squared variables add to a chi-square variable, and the degrees of freedom also add. d) The first part of the Lemma follows from the transformation of random variables for 2 Y X which yields: 1 2 Y X Xf y f y f y y e) Returning to the proof at hand, we want to show that 2 2 1N S has a chi-squared distribution with 1n degrees of freedom. To demonstrate this, note that 22 2 1 1 2 2 1 1 2 2 1 1 12 1 1 1 2 2 12 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 n n n n n n n i ji j n n n n n j n i j nj i j n n n n n i j ni j n n S n S X X n S X X n n n S X X X X X X n n n n n n n S X X X X X n n n n 1 AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XXI Professor Charles Moss Fall 2007 7 If 2n , we get 22 2 2 1 1 2 S X X Given that 2 1 2 X X is distributed 0,1N , 2 2 2 1~S and by extension for 2 2 1 , 1 ~ k k n k k S . 4. Given these results for the chi-square, the distribution of the Student’s t then follows. 2 2 X nX S S n Note that this creates a standard normal random variable in the numerator and 2 1 1 n n in the denominator. The complete distribution found by multiplying the standard normal time the chi-squared distribution times the Jacobian of the transformation yields: 1 1 2 22 2 1 12 12 T p p f t p p t p