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Date Created: 10/03/15
Wooldridge Introductory Econometrics 2d ed Appendix C Fundamentals of mathematical statistics A short review of the principles of mathematical statistics Econometrics is concerned with statistical inference learning about the characteristics of a population from a sample of the population The population is a well de ned group of subjects and it is important to de ne the population of interest Are we trying to study the unemployment rate of all labor force participants or only teenaged workers or only AHANA workers Given a population we may de ne an economic model that contains parameters of interest coef cients or elasticities which express the effects of changes in one variable upon another Let Y be a random variable rv representing a population with probability density function pdf f y 6 with 9 a scalar parameter We assume that we know f but do not know the value of 9 Let a random sample from the population be Y1 YN with Yi being an independent random variable drawn from f y 6 We speak of Yi being it39d independently and identically distributed We often assume that random samples are drawn from the Bernoulli distribution for instance that if I pick a student randomly from my class list what is the probability that she is female That probability is y where 7 of the students are female so PYZ 1 y and PYZ 0 1 7 For many other applications we will assume that samples are drawn from the Normal distribution In that case the pdf is characterized by two parameters LL and 02 expressing the mean and spread of the distribution respectively Finite sample properties of estimators The nite sample properties as opposed to asymptotic properties apply to all sample sizes large or small These are of great relevance when we are dealing with samples of limited size and unable to conduct a survey to generate a larger sample How well will estimators perform in this context First we must distinguish between estimators and estimates An estimator is a rule or algorithm that speci es how the sample information should be manipulated in order to generate a numerical estimate Estimators have properties they may be reliable in some sense to be de ned they may be easy or dif cult to calculate that dif culty may itself be a function of sample size For instance a test which involves measuring the distances between every observation of a variable involves an order of calculations which grows more than linearly with sample size An estimator with which we are all familiar is the sample average or arithmetic mean of N numbers add them up and divide by N That estimator has certain properties and its application to a sample produces an estimate We will often call this a point estimate since it yields a single number as opposed to an interval estimate which produces a range of values associated with a particular level of con dence For instance an election poll may state that 55 are expected to vote for candidate A with a margin of error of i4 If we trust those results it is likely that candidate A will win with between 51 and 59 of the vote We are concerned with the sampling distributions of estimators that is how the estimates they generate will vary when the estimator is applied to 2 repeated samples What are the nite sample properties which we might be able to establish for a given estimator and its sampling distribution First of all we are concerned with unbiasedness An estimator W of 9 is said to be unbiased if 9 for all possible values of 9 If an estimator is unbiased then its probability distribution has an expected value equal to the population parameter it is estimating Unbiasedness does not mean that a given estimate is equal to 9 or even very close to 9 it means that if we drew an in nite number of samples from the population and averaged the W estimates we would obtain 9 An estimator that is biased exhibits BiasW 9 The magnitude of the bias will depend on the distribution of the Y and the function that transforms Y into W that is the estimator In some cases we can demonstrate unbiasedness or show that bias0 irregardless of the distribution of Y for instance consider the sample average Y which is an unbiased estimate of the population mean LL E07 Egim Y gimp ii 3 1 Z 5W6 Z M Any hypothesis tests on the mean will require an estimate of the variance 02 from a population with mean LL Since we do not know LL but must estimate it with Y the estimate of sample variance is de ned as TL 32 1 gag 02 n 1 21 with one degree of freedom lost by the replacement of the population statistic LL with its sample estimate 7 This is an unbiased estimate of the population variance whereas the counterpart with a divisor of n will be biased unless we know LL Of course the degree of this bias will depend on the difference between and unity which disappears as n gt 00 Two dif culties with unbiasedness as a criterion for an estimator some quite reasonable estimators are unavoidably biased but useful and more seriously many unbiased estimators are quite poor For instance picking the rst value in a sample as an estimate of the population mean and discarding the remaining n 1 values yields an unbiased estimator of LL since 2 LL but this is a very imprecise estimator What additional information do we need to evaluate estimators We are concerned with the precision of the estimator as well as its bias An unbiased estimator with a smaller sampling variance will dominate its counterpart with a larger sampling variance eg we can demonstrate that the estimator that uses only the rst observation to estimate LL has a much 4 larger sampling variance than the sample average for nontrivial n What is the sampling variance of the sample average 1 quot Var Var n A Y 1 TL Eva 1 TL E Z VarltKgt 1 TL 1 2 2 2quot TL 239l 1 2 02 Z 710 Z so that the precision of the sample average depends on the sample size as well as the unknown variance of the underlying distribution of Y Using the same logic we can derive the sampling variance of the estimator that uses only the rst observation of a sample as 02 Even for a sample of size 2 the sample mean will be twice as precise This leads us to the concept of ef ciency given two unbiased estimators of 9 an estimator W1 is ef cient relative to W2 when VarltW1 S VarltWg V6 with strict inequality for at least one 9 A relatively ef cient unbiased estimator dominates its less ef cient counterpart We can compare two estimators even if one or both is biased by comparing mean squared error MSE MSEW E 62 This expression can be shown to equal the variance of the estimator plus the square of the bias thus it equals the variance for an unbiased estimator Large sample asymptotic properties of estimators We can compare estimators and evaluate their relative usefulness by appealing to their large sample properties or asymptotic properties That is how do they behave as sample size goes to in nity We see that the sample average has a sampling variance with limiting value of zero as n gt 00 The rst asymptotic property is that of consistency If W is an estimate of 6 based on a sample Y17 Yn of size n W is said to be a consistent estimator of 6 if for every 6 gt 0 Pan 6l gt 6 gt0asn gtoo Intuitively a consistent estimator becomes more accurate as the sample size increases without bound If an estimator does not possess this property it is said to be inconsistent In that case it does not matter how much data we have the recipe that tells us how to use the data to estimate 6 is awed If an estimator is unbiased and its variance shrinks as n gt 00 then the estimator is consistent A consistent estimator has probability limit or plim equal to the population parameter plim Y 2 LL Some mechanics of plims let 6 be a parameter and g a continuous function so that y 96 Suppose plimWn 6 and we devise an estimator of y Gn Then plimGn y or plim gltWn g plimWn 39 6 This allows us to establish the consistency of estimators which can be shown to be transformations of other consistent estimators For instance we can demonstrate that the estimator given above of the population variance is not only unbiased but consistent The standard deviation is the square root of the variance a nonlinear function continuous for positive arguments thus the standard deviation 8 is a consistent estimator of the population standard deviation Some additional properties of plims if plimTn oz and plimUn plimTn U 2 oz B plim TnUn oz plimTnUn oz 7 0 Consistency is a property of point estimators the distribution of the estimator collapses around the population parameter in the limit but that says nothing about the shape of the distribution for a given sample size To work with interval estimators and hypothesis tests we need a way to approximate the distribution of the estimators Most estimators used in econometrics have distributions that are reasonably approximated by the Normal distribution for large samples leading to the concept of asymptotic normality PZn 2 gtltIgtz asn gtoo where lt1 is the standard normal cumulative distribution function cdf We will often say ZnN0 1 or Z is asy N This relates to one form of the central limit theorem CLT If Y17 is a random sample with mean LL and variance 02 Yn u 0N5 has an asymptotic standard normal distribution Regardless of the population distribution of Y this standardized version of Y will be asy N and the entire distribution of Z will become arbitrarily close to the standard normal as n gt 00 Since many of the estimators we will derive in econometrics can be viewed as sample averages the law of large numbers and the central limit theorem can be combined to show that these estimators will be asy N Indeed the above estimator will be asy N even if we replace 0 with a consistent estimator of that parameter 8 General approaches to parameter estimation What general strategies will provide us with estimators with desirable properties such as unbiasedness consistency and ef ciency One of the most fundamental strategies for estimation is the method of moments in which we replace population moments with their sample counterparts We have seen this above where a consistent estimator of sample variance is de ned by replacing the unknown population LL with a consistent estimate thereof 37 A second widely employed strategy is the principle of maximum likelihood where we choose an estimator of the population parameter 9 by nding the value that maximizes the likelihood of observing the sample data We will not focus on maximum likelihood estimators in this course but note their importance in econometrics Most of our work here is based on the least squares principle that to nd an estimate of the n 8 population parameter we should solve a minimization problem We can readily show that the sample average is a method of moments estimator and is in fact a maximum likelihood estimator as well We demonstrate now that the sample average is a least squares estimator TL 2 mnin m will yield an estimator m which is identical to that de ned as 7 We may show that the value m minimizes the sum of squared deviations about the sample mean and that any other value 777 would have a larger sum or would not be least squares Standard regression techniques to which we will devote much of the course are often called OLS ordinary least squares Interval estimation and con dence intervals Since an estimator will yield a value or point estimate as well as a sampling variance we may generally form a con dence interval around the point estimate in order to make probability statements about a population parameter For instance the fraction of Firestone tires involved in fatal accidents is surely not 00005 of those sold Any number of samples would yield estimates of that mean differing from that number and for a continuous random variable the probability of a point is zero But we can test the hypothesis that 00005 of the tires are involved with fatal accidents if we can generate both a point and interval estimate for that parameter and if the interval estimate cannot reject 00005 as a plausible value This is the concept of a 9 con dence interval which is de ned with regard to a given level of con dence or level of probability For a standard normal N0 1 variable P 1 96 Y 39 lt 1M which de nes the interval estimate Y 17 We do not conclude from this that the probability that ILL lies in the interval is 095 the population parameter either lies in the interval or it does not The proper way to consider the con dence interval is that if we construct a large number of random samples from the population 95 of them will contain ILL Thus if a hypothesized value for LL lies outside the con dence interval for a single sample that would occur by chance only 5 of the time But what if we do not have a standard normal variate for which we know the variance equals unity If we have a variable X which we conclude is distributed as N u 02 we arrive at the dif culty that we do not know 02 and thus cannot specify the con dence interval Via the method of moments we replace the unknown 02 with a consistent estimate 82 to form the transformed statistic lt 196 095 Y u S W denoting that its distribution is no longer standard normal but student s t with n degrees of freedom The t distribution has fatter tails than does the normal above 20 or 25 degrees of freedom it is approximated quite well by the normal Thus con dence intervals constructed with the t distribution will be 10 in wider for small n since the value will be larger than 196 A 95 con dence interval given the symmetry of the t distribution will leave 25 of probability in each tail a two tailed t test If ca is the 100l a percentile in the t distribution a 100l a con dence interval for the mesan will be de ned as y Cox2 y CaZ n where s is the estimated standard deviation of Y We often refer to in as the standard error of the parameter in this case the standard error of our estimate of Lt Note well the difference between the concepts of the standard deviation of the underlying distribution an estimate of a and the standard error or precision of our estimate of the mean LL We will return to this distinction when we consider regression parameters A simple rule of thumb for large samples is that a 95 con dence interval is roughly two standard errors on either side of the point estimate the counterpart of a t of 2 denoting signi cance of a parameter If an estimated parameter is more than two standard errors from zero a test of the hypothesis that it equals zero in the population will likely be rejected Hypothesis testing We want to test a speci c hypothesis about the value of a population parameter 9 We may believe that the parameter equals 042 so that we state the null and alternative hypotheses H 0 c9 042 H A c9 7 042 In this case we have a two sided alternative we will reject ll the null if our point estimate is signi cantly below 042 or if it is signi cantly above 042 In other cases we may specify the alternative as one sided For instance in a quality control study our null might be that the proportion of rejects from the assembly line is no more than 003 versus the alternative that it is greater than 003 A rejection of the null would lead to a shutdown of the production process whereas a smaller proportion of rejects would not be cause for concern Using the principles of the scienti c method we set up the hypothesis and consider whether there is suf cient evidence against the null to reject it Like the principle that a nding of guilt must be associated with evidence beyond a reasonable doubt the null will stand unless suf cient evidence is found to reject it as unlikely Just as in the courts there are two potential errors of judgment we may nd an innocent person guilty and reject a null even when it is true this is Type I error We may also fail to convict a guilty person or reject a false null this is Type II error Just as the judicial system tries to balance those two types of error especially considering the consequences of punishing the innocent or even putting them to death we must be concerned with the magnitude of these two sources of error in statistical inference We construct hypothesis tests so as to make the probability of a Type I error fairly small this is the level of the test and is usually denoted as oz For instance if we operate at a 95 level of con dence then the level of the test is oz 2 005 When we set 04 we are expressing our tolerance for committing a Type I error and rejecting a true null Given oz we would like to minimize the probability of a Type II error or 12 equivalently maximize the power of the test which is just one minus the probability of committing a Type 11 error and failing to reject a false null We must balance the level of the test and the risk of falsely rejecting the truth with the power of the test and failing to reject a false null When we use a computer program to calculate point and interval estimates we are given the information that will allow us to reject or fail to reject a particular null This is usually phrased in terms of p values which are the tail probabilities associated with a test statistic If the p value is less than the level of the test then it leads to a rejection a p value of 0035 allows us to reject the null at the level of 005 One must be careful to avoid the misinterpretation of a p value of say 094 which is indicative of the massive failure to reject that null One should also note the duality between con dence intervals and hypothesis tests They utilize the same information the point estimate the precision as expressed in the standard error and a value taken from the underlying distribution of the test statistic such as 196 If the boundary of the 95 con dence interval contains a value 5 then a hypothesis test that the population parameter equals 5 will be on the borderline of acceptance and rejection at the 5 level We can consider these quantities as either de ning an interval estimate for the parameter or alternatively supporting an hypothesis test for the parameter l3
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