Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: asymptotic properties of estimators: plims and consistency Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

1 ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY The asymptotic properties of estimators are their properties as the number of observations in a sample becomes very large and tends to infinity. We shall be concerned with the concepts of probability limits and consistency, and the central limit theorem. These topics are usually given little attention in standard statistics texts, generally without an explanation of why they are relevant and useful. However, asymptotic properties lie at the heart of much econometric analysis and so for students of econometrics they are important.

2 The asymptotic properties of estimators are their properties as the number of observations in a sample becomes very large and tends to infinity. We shall be concerned with the concepts of probability limits and consistency, and the central limit theorem. These topics are usually given little attention in standard statistics texts, generally without an explanation of why they are relevant and useful. However, asymptotic properties lie at the heart of much econometric analysis and so for students of econometrics they are important. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

3 W e will start with an abstract definition of a probability limit and then illustrate it with a simple example. Probability limits ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

4 A sequence of random variables X n is said to converge in probability to a constant a if, given any positive , however small, the probability of X n deviating from a by an amount greater than  tends to zero as n tends to infinity. Probability limits ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

5 The constant a is described as the probability limit of the sequence, usually abbreviated as plim. Probability limits ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

6 We will take as our example the mean of a sample of observations, X, generated from a random variable X with population mean  X and variance  2 X. We will investigate how X behaves as the sample size n becomes large. n 150 probability density function of X n = ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

7 For convenience we shall assume that X has a normal distribution, but this does not affect the analysis. If X has a normal distribution with mean  X and variance  2 X, X will have a normal distribution with mean  X and variance  2 X / n. n 150 probability density function of X n = ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

8 For the purposes of this example, we will suppose that X has population mean 100 and standard deviation 50, as in the diagram. n 150 probability density function of X n = ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n The sample mean will have the same population mean as X, but its standard deviation will be 50/, where n is the number of observations in the sample n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n The larger is the sample, the smaller will be the standard deviation of the sample mean n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n If n is equal to 1, the sample consists of a single observation. X is the same as X and its standard deviation is n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n We will see how the shape of the distribution changes as the sample size is increased n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n The distribution becomes more concentrated about the population mean n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n To see what happens for n greater than 100, we will have to change the vertical scale n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n We have increased the vertical scale by a factor of n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n The distribution continues to contract about the population mean n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

n In the limit, the variance of the distribution tends to zero. The distribution collapses to a spike at the true value. The plim of the sample mean is therefore the population mean n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

Formally, the probability of X differing from  X by any finite amount, however small, tends to zero as n becomes large. 18 ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

Hence we can say plim X =  X. 19 ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

Consistency An estimator of a population characteristic is said to be consistent if it satisfies two conditions: (1)It possesses a probability limit, and so its distribution collapses to a spike as the sample size becomes large, and (2)The spike is located at the true value of the population characteristic. Hence we can say plim X =  X. 20 ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

21 The sample mean in our example satisfies both conditions and so it is a consistent estimator of  X. Most standard estimators in simple applications satisfy the first condition because their variances tend to zero as the sample size becomes large n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

22 The only issue then is whether the distribution collapses to a spike at the true value of the population characteristic. A sufficient condition for consistency is that the estimator should be unbiased and that its variance should tend to zero as n becomes large n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

23 It is easy to see why this is a sufficient condition. If the estimator is unbiased for a finite sample, it must stay unbiased as the sample size becomes large n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

24 Meanwhile, if the variance of its distribution is decreasing, its distribution must collapse to a spike. Since the estimator remains unbiased, this spike must be located at the true value. The sample mean is an example of an estimator that satisfies this sufficient condition n = probability density function of X ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

25 However the condition is only sufficient, not necessary. It is possible that an estimator may be biased in a finite sample … n = 20 Z  probability density function of Z ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

26 … but the bias becomes smaller as the sample size increases n = 100 n = 20 probability density function of Z  Z ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

27 … to the point where the bias disappears altogether as the sample size tends to infinity. Such an estimator is biased for finite samples but nevertheless consistent because its distribution collapses to a spike at the true value. n = 100 n = 1000 n = 20 probability density function of Z  Z n = ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

28 A simple example of an estimator that is biased in finite samples but consistent is shown above. We are supposing that X is a random variable with unknown population mean  X and that we wish to estimate  X. Consistency ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

29 The estimator is biased for finite samples because its expected value is n  X /(n + 1). But as n tends to infinity, n /(n + 1) tends to 1 and the estimator becomes unbiased. Consistency ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

30 The variance of the estimator is given by the expression shown. This tends to zero as n tends to infinity. Thus Z is consistent because its distribution collapses to a spike at the true value. Consistency ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

31 Consistency In practice we deal with finite samples, not infinite ones. So why should we be interested in whether an estimator is consistent? One reason is that sometimes it is impossible to find an estimator that is unbiased for small samples. If you can find one that is at least consistent, that may be better than having no estimate at all. A second reason is that often we are unable to say anything at all about the expectation of an estimator. The expected value rules are weak analytical instruments that can be applied in relatively simple contexts. In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)} applies only when X and Y are independent, and in most situations of interest this will not be the case. By contrast, we have a much more powerful set of rules for plims. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

32 Consistency In practice we deal with finite samples, not infinite ones. So why should we be interested in whether an estimator is consistent? One reason is that sometimes it is impossible to find an estimator that is unbiased for small samples. If you can find one that is at least consistent, that may be better than having no estimate at all. A second reason is that often we are unable to say anything at all about the expectation of an estimator. The expected value rules are weak analytical instruments that can be applied in relatively simple contexts. In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)} applies only when X and Y are independent, and in most situations of interest this will not be the case. By contrast, we have a much more powerful set of rules for plims. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

33 Consistency In practice we deal with finite samples, not infinite ones. So why should we be interested in whether an estimator is consistent? One reason is that sometimes it is impossible to find an estimator that is unbiased for small samples. If you can find one that is at least consistent, that may be better than having no estimate at all. A second reason is that often we are unable to say anything at all about the expectation of an estimator. The expected value rules are weak analytical instruments that can be applied in relatively simple contexts. In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)} applies only when X and Y are independent, and in most situations of interest this will not be the case. By contrast, we have a much more powerful set of rules for plims. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

34 Consistency In practice we deal with finite samples, not infinite ones. So why should we be interested in whether an estimator is consistent? One reason is that sometimes it is impossible to find an estimator that is unbiased for small samples. If you can find one that is at least consistent, that may be better than having no estimate at all. A second reason is that often we are unable to say anything at all about the expectation of an estimator. The expected value rules are weak analytical instruments that can be applied in relatively simple contexts. In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)} applies only when X and Y are independent, and in most situations of interest this will not be the case. By contrast, we have a much more powerful set of rules for plims. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

35 Plim rules Plim rule 1The plim of the sum of several variables is equal to the sum of their plims. For example, if you have three random variables X, Y, and Z, each possessing a plim, plim (X + Y + Z) = plim X + plim Y + plim Z Plim rule 2If you multiply a random variable possessing a plim by a constant, you multiply its plim by the same constant. If X is a random variable and b is a constant, plim bX = b plim X Plim rule 3The plim of a constant is that constant. For example, if b is a constant, plim b = b ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

36 Plim rules Plim rule 1The plim of the sum of several variables is equal to the sum of their plims. For example, if you have three random variables X, Y, and Z, each possessing a plim, plim (X + Y + Z) = plim X + plim Y + plim Z Plim rule 2If you multiply a random variable possessing a plim by a constant, you multiply its plim by the same constant. If X is a random variable and b is a constant, plim bX = b plim X Plim rule 3The plim of a constant is that constant. For example, if b is a constant, plim b = b ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

37 Plim rules Plim rule 1The plim of the sum of several variables is equal to the sum of their plims. For example, if you have three random variables X, Y, and Z, each possessing a plim, plim (X + Y + Z) = plim X + plim Y + plim Z Plim rule 2If you multiply a random variable possessing a plim by a constant, you multiply its plim by the same constant. If X is a random variable and b is a constant, plim bX = b plim X Plim rule 3The plim of a constant is that constant. For example, if b is a constant, plim b = b ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

38 Plim rules Plim rule 4The plim of a product is the product of the plims, if they exist. For example, if Z = XY, and if X and Y both possess plims, plim Z = (plim X)(plim Y) Plim rule 5The plim of a quotient is the quotient of the plims, if they exist. For example, if Z = X/Y, and if X and Y both possess plims, and plim Y is not equal to zero, plim Z = plim X plim Y ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

39 Plim rules Plim rule 4The plim of a product is the product of the plims, if they exist. For example, if Z = XY, and if X and Y both possess plims, plim Z = (plim X)(plim Y) Plim rule 5The plim of a quotient is the quotient of the plims, if they exist. For example, if Z = X/Y, and if X and Y both possess plims, and plim Y is not equal to zero, plim Z = plim X plim Y ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

40 Plim rules Plim rule 6The plim of a function of a variable is equal to the function of the plim of the variable, provided that the variable possesses a plim and provided that the function is continuous at that point. plim f(X) = f(plim X) ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

41 Example use of asymptotic analysis To illustrate how the plim rules can lead us to conclusions when the expected value rules do not, consider this example. Suppose that you know that a variable Y is a constant multiple of another variable Z ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

42 Example use of asymptotic analysis Z is generated randomly from a fixed distribution with population mean  Z and variance  2 Z. is unknown and we wish to estimate it. We have a sample of n observations. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

43 Example use of asymptotic analysis Y is measured accurately but Z is measured with random error w with population mean zero and constant variance  2 w. Thus in the sample we have observations on X, where X = Z + w, rather than Z. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

44 Example use of asymptotic analysis One estimator of l (not necessarily the best) is  Y i /  X i ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

45 Example use of asymptotic analysis Substituting from the first two equations, the estimator can be rewritten as shown. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

46 Example use of asymptotic analysis The expression can be simplified as shown. Hence we have decomposed the estimator into the true value,, and an error term. To investigate whether the estimator is biased or unbiased, we need to take the expectation of the error term. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

47 Example use of asymptotic analysis But we cannot do this. The random quantity appears in both the numerator and the denominator and the expected value rules are too weak to allow us to investigate the expectation analytically. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

48 Example use of asymptotic analysis However, we know that a sample mean tends to a population mean as the sample size tends to infinity, and so plim w = 0 and plim Z =  Z. ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

49 Since the plims of the numerator and the denominator of the error term both exist, we are able to take the plim of the error term. Thus we are able to show that the estimator is consistent, despite the fact that we cannot say anything about its finite sample properties. Example use of asymptotic analysis ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.14 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics