properties of estimators consistency

DESIRABLE PROPERTIES OF ESTIMATORS 6.1.1 Consider data x that comes from a data generation process (DGP) that has a density f( x). Least Squares Estimation - Large-Sample Properties In Chapter 3, we assume ujx ˘ N(0;˙2) and study the conditional distribution of bgiven X. (1) Small-sample, or finite-sample, properties of estimators The most fundamental desirable small-sample properties of an estimator are: S1. 11 Why are statistical properties of estimators important? The properties of consistency and asymptotic normality (CAN) of GMM estimates hold under regularity conditions much like those under which maximum likelihood estimates are CAN, and these properties are established in essentially the same way. Consistent estimators: De nition: The estimator ^ of a parameter is said to be consistent estimator if for any positive lim n!1 P(j ^ j ) = 1 or lim n!1 P(j ^ j> ) = 0 We say that ^converges in probability to (also known as the weak law of large numbers). The last property that we discuss for point estimators is consistency. More generally we say Tis an unbiased estimator of h( ) if and only if E (T) = h( ) … If an estimator is consistent, then more data will be informative; but if an estimator is inconsistent, then in general even an arbitrarily large amount of data will offer no guarantee of obtaining an estimate “close” to the unknown θ. Proof: omitted. Efficiency (2) Large-sample, or asymptotic, properties of estimators The most important desirable large-sample property of an estimator is: L1. (van der Vaart, 1998, Theorem 5.7, p. 45) Let Mn be random functions and M be The two main types of estimators in statistics are point estimators and interval estimators. We say that an estimate ϕˆ is consistent if ϕˆ ϕ0 in probability as n →, where ϕ0 is the ’true’ unknown parameter of the distribution of the sample. Most statistics you will see in this text are unbiased estimates of the parameter they estimate. A consistent estimator is one which approaches the real value of the parameter in the population as the size of the sample, n, increases. Asymptotic Normality. Lacking consistency, there is little reason to consider what other properties the estimator might have, nor is there typically any reason to use such an estimator. Consistency While not all useful estimators are unbiased, virtually all economists agree that consistency is a minimal requirement for an estimator. Definition: An estimator ̂ is a consistent estimator of θ, if ̂ → , i.e., if ̂ converges in probability to θ. Theorem: An unbiased estimator ̂ for is consistent, if → ( ̂ ) . Question: Although We Derive The Properties Of Estimators (e.g., Unbiasedness, Consistency, Efficiency) On The Basis Of An Assumed Population Model, These Models Are Thoughts About The Real World, Unlikely To Be True, So It Is Vital To Understand The Implications Of Using An Incorrectly Specified Model And To Appreciate Signs Of Such Specification Issues. The following are desirable properties for statistics that estimate population parameters: Unbiased: on average the estimate should be equal to the population parameter, i.e. A distinction is made between an estimate and an estimator. 2 Consistency One desirable property of estimators is consistency. The hope is that as the sample size increases the estimator should get ‘closer’ to the parameter of interest. When we say closer we mean to converge. Point estimation is the opposite of interval estimation. We call an estimator consistent if lim n MSE(θ) = 0 Consistency of θˆ can be shown in several ways which we describe below. Previously we have discussed various properties of estimator|unbiasedness, consistency, etc|but with very little mention of where such an estimator comes from. Estimation has many important properties for the ideal estimator. An estimator is consistent if ˆθn →P θ 0 (alternatively, θˆn a.s.→ θ 0) for any θ0 ∈ Θ, where θ0 is the true parameter being estimated. 2 Consistency of M-estimators (van der Vaart, 1998, Section 5.2, p. 44–51) Definition 3 (Consistency). Not even predeterminedness is required. If an estimator is consistent, then the distribution of becomes more and more tightly distributed around as … We will prove that MLE satisfies (usually) the following two properties called consistency and asymptotic normality. n)−θ| ≤ ) = 1 ∀ > 0. In class, we’ve described the potential properties of estimators. Three important attributes of statistics as estimators are covered in this text: unbiasedness, consistency, and relative efficiency. Consistency. 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