# ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

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ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Properties as • probability limits • consistency • central limit theorem 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. 1

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Properties as • probability limits • consistency • 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. For students of econometrics they are important. 2

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Properties as • probability limits • consistency • central limit theorem This sequence is addressed to probability limits and consistency. A subsequent one will treat the central limit theorem. 3

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits We will start with an abstract definition of a probability limit and then illustrate it with a simple example. 4

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits A sequence of random variables Zn is said to converge in probability to a constant a if, given any positive e, however small, the probability of Zn deviating from a by an amount greater than e tends to zero as n tends to infinity. 5

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits The constant a is described as the probability limit of the sequence, usually abbreviated as plim. 6

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 n = 1 We will take as our example the mean of a sample of observations, X, generated from a random variable X with population mean mX and variance s2X. We will investigate how X behaves as the sample size n becomes large. 7

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 n = 1 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 mX and variance s2X, X will have a normal distribution with mean mX and variance s2X / n. 8

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 n = 1 For the purposes of this example, we will suppose that X has population mean 100 and standard deviation 50, as in the diagram. 9

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 n = 1 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. 10

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 n = 1 The larger is the sample, the smaller will be the standard deviation of the sample mean. 11

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 n = 1 If n is equal to 1, the sample consists of a single observation. X is the same as X and its standard deviation is 50. 12

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 4 25 n = 4 n = 1 We will see how the shape of the distribution changes as the sample size is increased. We have added the distribution of X when n = 4. 13

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 4 25 25 10 n = 25 n = 4 n = 1 We add the distribution for n = 25. The distribution becomes more concentrated about the population mean. 14

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 4 25 25 10 100 5 n = 100 n = 25 n = 4 n = 1 We add the distribution for n = To see what happens for n greater than 100, we will have to change the vertical scale. 15

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 4 25 25 10 100 5 n = 100 We have increased the vertical scale. 16

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 4 25 25 10 100 5 n = 400 n = 100 We increase the sample size to The distribution continues to contract about the population mean. 17

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
1 50 4 25 25 10 100 5 n = 1600 n = 400 n = 100 We increase the sample size again. 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. 18

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits Sample mean as estimator of population mean Formally, the probability of X differing from mX by any finite amount, however small, tends to zero as n becomes large. 19

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits Sample mean as estimator of population mean Hence we can say plim X = mX. 20

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. An estimator of a population characteristic is said to be consistent if it satisfies two conditions. The first is that the estimator possesses a probability limit, and so its distribution collapses to a spike as the sample size becomes large. 21

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. The second is that the spike is located at the true value of the population characteristic. 22

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
The sample mean in our example satisfies both conditions and so it is a consistent estimator of mX. 23

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Most standard estimators in simple applications satisfy the first condition because their variances tend to zero as the sample size becomes large. The only issue then is whether the distribution collapses to a spike at the true value of the population characteristic. 24

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
A sufficient condition for consistency is that the estimator should be unbiased and that its variance should tend to zero as n becomes large. 25

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
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. 26

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
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. 27

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
However the condition is only sufficient, not necessary. It is possible for a biased estimator to be consistent, if the bias vanishes as the sample size becomes large. In this example, the true value is 100, and the estimator is biased for sample size 25. 28

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Here the sample size is greatr and the bias is smaller. 29

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
A further reduction in the bias with a further increase in the sample size. 30

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
This is an example 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. 31

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
We will encounter estimators of this type when we come to Model B, and they will be important to us. 32

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Example The foregoing example was just a general graphical illustration of what might happen as the sample size increases. Here is a simple mathematical example. 33

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Example We are supposing that X is a random variable with unknown population mean mX and that we wish to estimate mX. 34

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Example As defined, the estimator Z is biased for finite samples because its expected value is nmX/(n + 1). But as n tends to infinity, n /(n + 1) tends to 1 and the bias disappears. 35

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Example 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. 36

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Why should we be interested in consistency? In practice we deal with finite samples, not infinite ones. So why should we be interested in whether an estimator is consistent? 37

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Why should we be interested in consistency? • If no unbiased estimator exists, a consistent estimator may be preferred to an inconsistent one. One reason is that often, in practice, 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. 38

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Why should we be interested in consistency? • If no unbiased estimator exists, a consistent estimator may be preferred to an inconsistent one. • It is often not possible to determine the expectation of an estimator, but still possible to evaluate probability limits. Reason: plim rules are stronger than the rules for handling expectations. 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. 39

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Two conditions: (1) The estimator possesses a probability limit. (2) The limit is the true value of the population characteristic. Why should we be interested in consistency? • If no unbiased estimator exists, a consistent estimator may be preferred to an inconsistent one. • It is often not possible to determine the expectation of an estimator, but still possible to evaluate probability limits. Reason: plim rules are stronger than the rules for handling expectations. 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. 40

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. 4. 5. 6. Here are six rules for decomposing plims. Note that, in each case, the validity of the rule depends on plims existing for each of the components on the right side of the equation. 41

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. The first three rules are straightforward counterparts for corresponding rules for decomposing expectations. The first rule, the additive rule, depends on X, Y, and Z each having their individual plims. 42

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. The second rule, a multiplicative rule, b being a constant, depends on X having a plim. The third rule states the obvious fact that a constant is its own limit. 43

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. The fourth rule, a multiplicative rule for two (or more) variables, depends on each variable having a plim. It does not require X and Y to be independent. 44

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. 4. By contrast, the corresponding rule for expectations E{g(X)h(Y)} = E{g(X)} E{h(Y)} applies only when X and Y are independent. This is often not the case. 45

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. 4. The quotient rule for plims, which we shall use very frequently in Model B, requires that X and Y both have plims and that plim Y is not zero. 46

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. 4. 5. There is no counterpart for expectations, even if X and Y are independent. If X and Y are independent, E(X/Y) = E(X) E(1/Y), provided that both expectations exist, and that is as far as one can go. 47

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
plim rules 1. 2. 3. 4. 5. 6. The 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. 48

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: 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 49

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Z is generated randomly from a fixed distribution with population mean mZ and variance s2Z. a is unknown and we wish to estimate it. We have a sample of n observations. 50

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Y is measured accurately but Z is measured with random error w with population mean zero and constant variance s2w. Thus in the sample we have observations on X, where X = Z + w, rather than Z. 51

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : One estimator of a (not necessarily the best) is Y/X. 52

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : We can decompose the estimator into the true value and an error term, as shown. 53

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : We can decompose the estimator into the true value and an error term, as shown. 54

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : To investigate whether the estimator is biased or unbiased, we need to take the expectation of the error term. But we cannot do this. 55

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : The random variable w appears in both the numerator and the denominator and the expected value rules are too weak to allow us to determine the expectation of a ratio when both the numerator and the denominator are functions of the same random variable. 56

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : However, we can show that the error term tends to zero as the sample becomes large. 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 = mZ. 57

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : Hence plims exist for both the numerator and denominator of the estimator. 58

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : Since the plims of the numerator and the denominator of the error term both exist, we are able to take the plim of the estimator. 59

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis Model: Estimator of a : Thus, provided that mZ ≠ 0, we are able to show that the estimator is consistent, despite the fact that we cannot say anything analytically about its finite sample properties. 60