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If the regressors in x are nonrandom and the

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If the regressors in X are nonrandom and the disturbances are 2 (0, ) IID , we do not assume normality of the disturbance distributions, it can still be argued that the distributions of OLS estimators ( ˆ i ) is approximately normal provided that the sample size is reasonably large. 3. Sampling Distribution and Repeated Sampling Suppose we perform the experiment of taking what is called a repeated sample: keeping the values of the independent variables unchanged, we obtain new observations for the dependent variable by drawing a new set of disturbances. This could be repeated, ay, 5000 times, obtaining 5000 of these repeated samples. For each of these repeated samples we could use an estimator ˆ i to calculate an estimate of i . Because the samples differ, there 5000 estimated will not be the same. The histogram of these estimates or the manner in which these estimates are distributed is called the sampling distribution of ˆ i . This concept of a sampling distribution, the distribution of estimates produced by an estimator (for example, OLS estimator) in repeated sampling, is crucial to an understanding of econometrics (Kennedy, 2001, pp.13-14). 4. The Sampling Distribution of ˆ i Hence ˆ i has a sampling distribution. To carry out hypothesis tests about the ˆ i , we need to use its distributions. However, there is no any certainty that they will follow any type of distribution. In classical theory, in case of small samples , it is assumed that they have normal
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 13 distributions. If we do not assume that condition, one cannot carry out any hypothesis tests in case of small samples. However, this is a strong assumption. Consequently, as we have stated before, the modern approach to econometrics drops the normality assumption and simply assumes that t u are independently draws from an identical distribution ( IID ). We can simply summarize as follows: What is the distribution of 1 ˆ in small samples ? If we assume that t u has a normal distribution, then the distribution of OLS estimators ( ˆ i ) becomes normal, since the OLS estimators ( ˆ i ) are the functions of random variable Y t , which in turn is a function of the stochastic disturbance, t u . What is the distribution of 1 ˆ in large samples ? When the t u are independently and identically distributed [ 2 (0, ) t u IID ] and when T is large (at least 30 T ), the sampling distribution of 1 ˆ is well approximated by a standard normal distribution (CLT, Central Limit Theorem): 1 1 1 ˆ ˆ ( ) ˆ var( ) E ~ N(0,1) Central Limit Theorem : Suppose { Y t }, t =1,…, T is IID (independently and identically distributed 5 ) with ( ) E Y and 5 When Y 1 , Y 2 , …,Y T are drawn from the same distribution and are independently distributed, they are said to be independently and identically distributed , or iid (Stock and Watson, 2011, p.87).
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR
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