ASYMPTOTIC ESTIMATION THEORY

Chapter 10 ASYMPTOTIC ESTIMATION THEORY

  ∧

  Let θ n (p × 1) be an estimator, applied to a sample of size n, of a vector parameter

  θ 0 . Both and must be elements of the set Θ of all admissible values of the para- meters, called the parameter space, which can in principle be defined to be R p , or

  p-dimensional Euclidian space. For technical reasons, Θ must be a compact subset of R p

  i.e. bounded and closed, i.e. it contains its boundary points. Furthermore, θ 0

  must be an interior point of Θ. This is to say that θ 0 ∈ int (Θ) if there exist a real number δ > 0 such that θ ∈ Θ whenever kθ − θ 0 k < δ. This excludes θ 0 being at the

  boundary of the set.

  ∧

  ∧

  Definition 32 θ n is a consistent estimator of θ 0 if plim θ n =θ 0 .

  Consistency might be a minimum requirement for a useful estimator. Proofs of consistency play an important role in econometric theory. The forms that these µ

  ∧ ¶

  proofs is that if lim n→∞ E θ n =θ 0 , i.e. the estimator is asymptotically unbiased,

  µ ∧ ¶

  and lim n→∞

  V ar θ n =0 suffices for the consistency of the estimator.

  Now suppose θ n is consistent, and n θ n −θ 0 =O p (1) for some k > 0, and

  has a non-degenerate limit distribution as n → ∞. This distribution is called the

  ∧

  asymptotic distribution of θ n .

  ∧

  ∧

  Definition 33 θ n is said to be consistent and asymptotically normal (CAN) for θ n µ ¶ ∈

  int (Θ) if there exist k > 0 such that n d θ

  ∧

  k

  n −θ 0 → N (0, V ), where V is a finite

  variance-covariance matrix.

  Asymptotic Estimation Theory

  In most applications k = 12, although it can be larger than this for models containing determinist trend terms. There are also case where k > 12 but the limiting distribution is not normal, e.g. when there are stochastic trends. Asymptotic normality is an important property to establish for an estimator, as it is often the only basis for constructing interval estimates and tests of hypotheses.

  ∧

  Let denote by C the class of CAN estimators of θ 0 , and write θ n ∈ C to denote

  that the estimator belongs to this class.

  ∧

  Definition 34 θ n ∈ C is said to best asymptotically normal for θ 0 (BAN) in the class

  if AV ar θ n − AV ar θ n is positive semi-definite for every θ n ∈ C.

  This property is also called asymptotic efficiency. BAN can be seen as an asymptotic counterpart of the BLUE property.

  10.1 Asymptotics of the Stochastic Regressor Model Assume the regression model:

  y

  t =x t β+u t t = 1, 2, ..., n

  Let the following assumptions hold for all n > k

  E (u) = 0 a.s.

  rank (X) = k a.s.

  where u is the (n × 1) vector of the errors, X is the (n × k) matrix of the explana- tory variables and I n is the identity matrix of dimension n. These are the usual assumptions. Furthermore, assume that

  E x t x t =M xx < ∞(p.d.)

  ≤ B < ∞ δ > 0, ∀ fixed λ

  Asymptotics of the Stochastic Regressor Model

  The first additional condition can be written as plimn −1 X X=M xx has two com-

  ponents. The weak law of large numbers must apply to the squares and the cross- products of the elements of x t , and M xx must have full rank. The latter can fail even if the matrix X has rank k for every finite n. To see this take the fairly trivial

  P n 1 π 2 P n 1

  example x t = 1t . Then lim n→∞ t=1 t 2 = . Hence lim

  n 6 −1 n→∞ t=1 t 2 =0 .

  The least squared estimator can be written as

  2 Consider now the k × 1 vector x 2

  t u t . Since E(u t |x t )=0 and E(u t |x t )=σ the Law

  of Iterated Expectations gives

  Furthermore, the u t 0s are independent hence, the Weak Law of Large Numbers can

  be applied on x t u t , i.e.

  which is written as p lim 1 n X u =0 . Then by the Continuous Mapping Theorem we

  have that

  which is the consistency result.

  Let us now consider the sequence λ x

  t u t . We have that

  2 Since now M

  xx is positive definite 0 < σ λ M xx λ< ∞. Hence the denominator of the

  ¯ ¯ 2+δ

  condition of Theorem 20 is bounded and bigger than 0. Furthermore, E ¯λ x t u t

  ≤

  B ensures the condition of the same Theorem and consequently we have that

  1 X d ³ 2 ´

  √

  λ x t u t →N

  0, σ λ M xx λ

  n t=1

  Asymptotic Estimation Theory

  for each specific λ. But this is equivalent to

  1 d ¡ 2 ¢

  √ X u →N

  0, σ M xx .

  n

  Finally

  µ ∧

  ¶ µ

  ¶

  √

  1 −1 1 d ¡ 2 ¢

  n β

  X X −β −1 √ X u →N 0, σ M xx .

  n

  n

  Part IV

  Likelihood Function