程序代写代做代考 ECON 61001: Lecture 4

ECON 61001: Lecture 4
Alastair R. Hall
The University of Manchester
Alastair R. Hall ECON 61001: Lecture 4 1 / 22

Outline of this lecture
Stochastic regressors
Large sample theory: concepts & Limit Theorems Large sample behaviour of OLS
Large sample inference in models estimated from cross-section data
Alastair R. Hall ECON 61001: Lecture 4 2 / 22

Stochastic regressors
Recall that so far our model has been:
y = Xβ0 + u
where
CA1: true model is: y = Xβ0 + u.
CA2: X is fixed in repeated samples. CA3: X is rank k.
CA4: E[u] = 0.
C A 5 : V a r [ u ] = σ 02 I T .
CA6: u ∼ Normal.
Alastair R. Hall ECON 61001: Lecture 4 3 / 22

Stochastic regressor model
Now consider model with stochastic regressors. Assumptions:
SR1: true model is: y = Xβ0 + u. SR2: X is stochastic.
SR3: X is rank k with probability 1. SR4: E[u|X] = 0.
S R 5 : V a r [ u | X ] = σ 02 I T . SR6: u|X ∼ Normal.
⇒ y | X ∼ N ( X β 0 , σ 02 I T ) .
Alastair R. Hall ECON 61001: Lecture 4 4 / 22

Stochastic regressor model
Results discussed in Lecture 2 before still go through although with some additional conditions: e.g.
E[βˆT] = β0 + EX 􏰜Eu|X 􏰜(X′X)−1X′u􏰝􏰝 and provided the expectation on the rhs exists then,
E[βˆT] = β0 + EX 􏰜(X′X)−1X′Eu|X [u]􏰝 = β0, using SR4.
Distributions of test statistics also goes through but depend crucially on SR6. If distribution is not normal then inference methods discussed in previous lectures are invalid. → alternative approach using large sample theory.
Alastair R. Hall ECON 61001: Lecture 4 5 / 22

Large sample theory
Recall that
βˆT = (X′X)−1X′y = β0 + (X′X)−1X′u 􏰂T 􏰃−1T
= β0 + 􏰈xtxt′ 􏰈xtut t=1 t=1
S o { βˆ T ; T = k , k + 1 , . . . } i s a s t o c h a s t i c s e q u e n c e i n d e x e d b y T . Large sample theory rests on considering how statistics of interest
(βˆT , t-stats etc) behave as T → ∞.
Our large sample analysis rests on two key concepts: “convergence
in probability” and “convergence in distribution”.
Alastair R. Hall ECON 61001: Lecture 4 6 / 22

Convergence in probability
Consider stochastic sequence {VT ; T = 1, 2, . . .} and random variable V.
VT is said to converge in probability to V if
P(|VT −V| < ε)→1foranyε>0asT→∞,andisdenoted b y V T →p V .
This definition holds for rv V . If V is a degenerate rv (that is, a constant) then we have two important items of terminology.
Definition: If VT →p c where c is a constant then c is referred to as the probability limit of VT and this is written as plimVT = c.
Definition: If θˆT is an estimator of the unknown parameter θ0 and θˆT →p θ0 then θˆT is said to be a consistent estimator of θ0.
Alastair R. Hall ECON 61001: Lecture 4 7 / 22

Convergence in probability
Let {MT; T = 1,2,…} be a sequence of random matrices and M be random matrix (MT , M are p × q).
Then
MT →p M iff MT,i,j →p Mi,j for i = 1,2,…p, j = 1,2,…q,
where i , j subscript denotes i − j th element of the matrix in question.
Slutsky’s theorem: Let {VT } be a sequence of r × 1 random vectors (or matrices) which converge in probability to the random vector (or matrix) V and let f (.) be a real- valued vector of continuous functions then f (VT ) →p f (V ).
Alastair R. Hall ECON 61001: Lecture 4 8 / 22

Convergence in distribution
The sequence of random variables {VT } with corresponding distribution functions {FT ( · )} converges in distribution to the random variable V with distribution function F ( · ) if and only if FT (c) → F(c) as T → ∞ at all points of continuity {c} of F(.).
Convergence in distribution is denoted by VT →d V .
The distribution of V is known as the limiting distribution of
VT.
Note: If VT →p V then VT →d V but reverse implication does not
hold unless V = c, a constant.
Alastair R. Hall ECON 61001: Lecture 4 9 / 22

Limit Theorems
Recall that
ˆ 􏰂T 􏰃−1T
βT−β0= 􏰈xtxt′ 􏰈xtut t=1 t=1
So large sample behaviour depends on: a random matrix􏰔􏰀􏰔Tt=1 xtxt′􏰁−1,
a random vector Tt=1 xtut.
Their large sample behaviour is specified by two fundamental Limit
Theorems:
the Weak Law of Large Numbers (WLLN) the Central Limit Theorem (CLT)
Alastair R. Hall ECON 61001: Lecture 4 10 / 22

Limit Theorems
Let {vt, t = 1,2,…T} be random vectors with E[vt] = μt. Weak Law of Large Numbers: Subject to certain conditions,
T
T − 1 􏰈 ( v t − μ t ) →p 0 .
t=1
Central Limit Theorem: Subject to certain conditions,
where
T
T − 1 / 2 􏰈 ( v t − μ t ) →d N ( 0 , Ω ) ,
t=1
􏰄 − 1 / 2 􏰈T Ω=limT→∞Var T (vt−μt)
t=1
􏰅
is a finite positive definite matrix of constants.
Alastair R. Hall ECON 61001: Lecture 4
11 / 22

Limit Theorems
In addition, we have the following two useful results:
Large sample behaviour of matrix-vector products:
IfbT =MTmT whereMT isaq×r randommatrixandmT isa r × 1 random vector and:
MT →p M, a matrix of finite constants,
mT →d N ( 0, Ω), Ω is finite p.d. matrix of constants,
rank(MΩM′) = q, Then
bT →d N(0,MΩM′)
Alastair R. Hall ECON 61001: Lecture 4 12 / 22

Limit Theorems
Large sample behaviour of quadratic forms:
If aT = m′ Ωˆ−1mT where mT is a r ×1 random vector and: TT
mT →d N ( 0, Ω), where Ω is a finite p.d. matrix of constants, Ωˆ T →p Ω ,
Then
aT →d χ2r
Alastair R. Hall ECON 61001: Lecture 4 13 / 22

Large sample analysis of OLS
To develop a large sample analysis for OLS with cross-section data we impose the following assumptions.
Assumption CS1:
yi = xi′β0 +ui = β0,1 + x2′,iβ0,2 + ui, i = 1,2,…,N
Assumption CS2: { (ui,xi′), i = 1,2,…N} forms an independent and identically distributed sequence.
Assumption CS3: E[xixi′] = Q, finite, p.d.
Assumption CS4: E [ui |xi ] = 0.
Assumption CS5: Var [ui |xi ] = σ02 , positive, finite constant.
Alastair R. Hall ECON 61001: Lecture 4 14 / 22

Large sample analysis of OLS
In our notation here, the OLS estimator is ˆ 􏰂N 􏰃−1N
βN = 􏰈xixi′ 􏰈xiyi i=1 i=1
and so using CS1, we have
ˆ 􏰂N􏰃−1N
βN = β0 + 􏰈xixi′ 􏰈xiui i=1 i=1
􏰂 N 􏰃−1 N
= β0 + N−1􏰈xixi′ N−1􏰈xiui
i=1 i=1
Under Assumptions CS1-CS5, we can use the WLLN to deduce:
N − 1 􏰔 Ni = 1 x i x i ′ →p Q
N−1 􏰔Ni=1 xiui →p E[xiui] = E [xi E[ui |xi]] = 0 (via LIE) Alastair R. Hall ECON 61001: Lecture 4
15 / 22

Large sample analysis of OLS
And so using Slutsky’s Theorem,
ˆ 􏰂 N 􏰃−1 N
βN − β0 = N−1 􏰈xixi′ N−1 􏰈xiui i=1 i=1
→p Q−1×0=0 ⇒ βˆ N →p β 0 .
So βˆN is consistent for β0.
Alastair R. Hall ECON 61001: Lecture 4 16 / 22

Large sample analysis of OLS
Under these conditions, we can also apply the CLT to deduce:
N
N−1/2􏰈xiui →d N(0,Ω)
i=1
To deduce form of Ω, define ΩN = Var[N−1/2 􏰔Ni=1 xiui].
U s i n g ( u i , x i′ ) ∼ i . i . d . a n d E [ x i u i ] = 0 , w e h a v e
NN
ΩN = N−1􏰈Var[xiui] = N−1􏰈E[ui2xixi′]
i=1 i=1
N
= N−1􏰈E􏰜E[ui2|xi]xixi′􏰝 = σ02Q
i=1
Alastair R. Hall ECON 61001: Lecture 4 17 / 22

Large sample analysis of OLS
So Ω = limN→∞ΩN = σ02Q. As we have
ˆ 􏰂 N 􏰃−1 N N1/2(βN − β0) = N−1 􏰈xixi′ N−1/2 􏰈xiui
i=1 i=1
and:
N−1 􏰔Ni=1 xi xi′ →p Q, nonsingular
N−1/2 􏰔Ni=1 xi ui →d N( 0, σ02Q). it follows that
N1/2(βˆN−β0)→d N􏰚0,σ02Q−1􏰛
Alastair R. Hall ECON 61001: Lecture 4 18 / 22

Inference based on large sample analysis
To perform inference, we need to estimate variance. Use
Ωˆ N = σˆ N2 N − 1 X ′ X
So, for example, an approximate 100(1 − α)% confidence interval
for β0,l based on limiting distribution is:
βˆN,l ± z1−α/2σˆN√ml,l
where
ml,l is the lth main diagonal element of (X′X)−1.
z1−α/2 is 100(1 − α/2)th percentile of the standard normal distribution
Alastair R. Hall ECON 61001: Lecture 4 19 / 22

Inference based on large sample analysis
Suppose we wish to test H0 : Rβ0 = r vs H1 : Rβ0 ̸= r. Test statistic
WN = N(RβˆN − r)′ 􏰜R(N−1X′X)−1R′ 􏰝−1 (RβˆN − r)/σˆN2 ThenunderH0:WN →d χ2nr.
(Note: WN = nr F from Lecture 3.)
Alastair R. Hall ECON 61001: Lecture 4 20 / 22

Inference about nonlinear restrictions
Suppose we wish to test: H0 : g(β0) = 0 vs H1 : g(β0) ̸= 0. Assume:
g ( · ) is a ng × 1 vector of continuous differentiable functions G(β ̄) = ∂g(β)/∂β′|β=β ̄ with rank{G(β0)} = ng
Can test H0 using:
W(g) = Ng(βˆN)′ 􏰉G(βˆN)(N−1X′X)−1G(βˆN)′ 􏰊−1 g(βˆN)/σˆ2
and under H0: W(g) →d χ2n . Ng
Proof based on so-called “Delta method”.
NN
Alastair R. Hall ECON 61001: Lecture 4 21 / 22

Further reading
Notes: Sections 2.11, 3.1-3.2
Greene:
convergence in probability, Section D.2.1 p.1107-1110;
Slutsky’s Theorem D.2.3, p.1113-1114
convergence in distribution, Section D.2.5, p.1116-1118. WLLN, Section D.2.2, p.1110-1113.
CLT, Section D.2.6, p.1118-1123.
Large sample behaviour of OLS, Section 4.4.1-4.4.3, p.103-108.
Alastair R. Hall ECON 61001: Lecture 4 22 / 22