ECON 61001: Lecture 3
Alastair R. Hall
The University of Manchester
Alastair R. Hall ECON 61001: Lecture 3 1 / 25
Outline of today’s lecture
Testing hypothesis about β0 in the CLR model based on OLS estimators.
Testing hypotheses about individual coefficients
Testing hypotheses about linear combinations of the coefficients
Restricted Least Squares Variable selection
Alastair R. Hall ECON 61001: Lecture 3 2 / 25
Examples of parameter restrictions in econometric models
assetreturns:R−Rf =β0(Rm−Rf)+error
β0 < 0 ⇒ stock is inversely related to market index;
β0 = 1 ⇒ stock moves in line with market index. returns to education (dropping exp to simplify and
reparameterizing):
ln(w) = β0,1 + β0,2 ∗ ed + β0,3 ∗ D + β0,4 ∗ (D ∗ ed) + error
D = 1 is female and zero else
β0,3 = 0, β0,4 = 0 ⇒ no difference between men and women.
Alastair R. Hall ECON 61001: Lecture 3 3 / 25
Examples of parameter restrictions in econometric models
aggregate production function:
lnQ = β0,1 + β0,2 ∗ln(L) + β0,3 ∗ln(K) + error
⎧⎨ < ⎫⎬ β0,2 + β0,3 =
⎩ > ⎭ Alastair R. Hall
⎧⎨ diminishing ⎫⎬
constant returns to scale
⎩ increasing ⎭ ECON 61001: Lecture 3
1 ⇒
4 / 25
Testing hypotheses about β0,i
Consider inference about β0,i based on βˆT,i.
Recall that we showed last lecture that βˆ T , i − β 0 , i
σ0√mi,i ∼ N(0,1),
where mi,i is the ith main diagonal element of (X′X)−1,
and noted that if we replace σ0 by σˆT then,
ˆ
βT,i √
− β0,i ∼ Student’s t distribution with T-k df
σˆ T m i , i
Alastair R. Hall ECON 61001: Lecture 3 5 / 25
Inference about β0,i
Consider the two-sided test: H0 : β0,i = β∗,i vs. H1 : β0,i ̸= β∗,i .
Natural to base test statistic on: τˆT,i(β∗,i)= σˆT√
because under H0:
τˆT,i(β∗,i) ∼ Student’s t distribution with T-k df
βˆ T , i − β ∗ , i mi ,i
Alastair R. Hall ECON 61001: Lecture 3 6 / 25
Inference about β0,i
Decision rule: reject H0 at 100α% significance level if |τˆT,i(β∗,i)| > τT−k(1−α/2) Note: significance level is 100 × P(Type I error).
Alastair R. Hall ECON 61001: Lecture 3 7 / 25
For you to print out and annotate if you wish
Alastair R. Hall ECON 61001: Lecture 3 8 / 25
Inference about β0,i
Suppose H0 is false. How does our test statistic behave?
βˆT,i − β∗,i βˆT,i − β0,i β0,i − β∗,i τˆT,i(β∗,i)=σˆ√m =σˆ√
m +σˆ√m T i,i T i,i T i,i
So under Assumptions CA1-CA6:
τˆT,i(β∗,i)∼ Student’stdistributionwithT−kdfandncpν
where the non-centrality parameter (ncp) is:
ν = β0,i√
− β∗,i σ0 mi,i
Alastair R. Hall ECON 61001: Lecture 3 9 / 25
Inference about β0,i
As a result:
P(reject H0 |H1 true) > α ⇒ unbiased test. power ↑ as |ν| ↑.
power depends on |ν| and not |β0,i − β∗,i| per se.
Alastair R. Hall ECON 61001: Lecture 3 10 / 25
Example: traffic fatalities
From Lecture 1:
yˆt = controls − 0.030 ∗ beltt + 0.0671 ∗ mpht
Did passage of seat belt law affect % of accidents with fatalities?
H0 : βbelt,0 = 0 (no effect) vs H1 : βbelt,0 ̸= 0 (has effect) test statistic:
significance levels.
βˆbelt −0 −0.030
|τˆ |= = =1.304
belt
s.e.(βˆbelt) 0.023
p-value is 0.195 and so fail to reject at all conventional
Alastair R. Hall ECON 61001: Lecture 3 11 / 25
Example: traffic fatalities
Did passage of seat belt law reduce % of accidents with fatalities? H0 : βbelt,0 ≥0(no)vsH1 : βbelt,0 <0(yes)
Example of one-sided test: H0 : β0,i ≥ β∗,i vs H1 : β0,i < β∗,i Test statistic is now:
βˆ T , i − β ∗ , i τˆT,i(β∗,i) = σˆT√mi,i
Decision rule is to reject H0 in favour of H1 at the 100α% significance level if
τˆT,i(β∗,i) < τT−k(α)
In our example, the critical value is −1.291 (−1.662) for the 10% (5%) significance level test and so marginal evidence against H0.
Alastair R. Hall ECON 61001: Lecture 3 12 / 25
For you to print out and annotate if you wish
Alastair R. Hall ECON 61001: Lecture 3 13 / 25
Inference about Rβ0 = r
Consider testing: H0 : Rβ0 = r vs H1 : Rβ0 ̸= r where R, r are
nr × k and nr × 1 are specified constants.
We need rank(R) = nr to rule out redundancies. Natural to base inference on: R βˆT − r .
Given sampling distribution of βˆT , we have:
R βˆ T − R β 0 ∼ N ( 0 , σ 02 R ( X ′ X ) − 1 R ′ )
and so under H0
R βˆ T − r ∼ N ( 0 , σ 02 R ( X ′ X ) − 1 R ′ )
Alastair R. Hall ECON 61001: Lecture 3 14 / 25
Inference about Rβ0 = r Test statistic:
F = (RβˆT − r)′[R(X′X)−1R′]−1(RβˆT − r) n r σˆ T2
Under H0, F ∼ Fnr,T−k, the F distribution with (nr,T −k) df. Decision rule: reject H0 : Rβ0 = r at the 100α% significance level
if:
F >Fnr,T−k(1−α)
where Fnr ,T −k (1 − α) is the 100(1 − α)th percentile of the F
distribution with (nr , T − k ) df.
Alastair R. Hall ECON 61001: Lecture 3 15 / 25
For you to print out and annotate if you wish
Alastair R. Hall ECON 61001: Lecture 3 16 / 25
Inference about Rβ0 = r
Alternative representation:
F = RSSR −RSSUT−k
RSSU nr
RSSU is RSS from regression without imposing Rβ = r
RSSR is RSS from regression imposing Rβ = r
where
Alastair R. Hall ECON 61001: Lecture 3 17 / 25
Example: traffic fatalities
Did two traffic laws offset each other?
H0:βbelt,0 +βmph,0=0vsH1:βbelt,0 +βmph,0̸=0.
Reject H0 at the 100α% significance level if F > F1,91(1 − α). F = 3.126, F1,91(.95) = 3.946 ⇒ Fail to reject at 5% level. p-value = 0.080 so reject at 10% significance level.
Alastair R. Hall ECON 61001: Lecture 3 18 / 25
Do regressors collectively help to explain y?
H0 : Rβ0 =0k−1 (no)vsH1 : Rβ0 ̸=0k−1 (yes)for R = [0k−1, Ik−1]
where 0k−1 is (k − 1) × 1 null vector.
Decision rule: reject at 100α% significance level if
F >Fk−1,T−k(1−α).
In this case, F test statistic given by:
F= R2 T−k 1 − R2 k − 1
Alastair R. Hall ECON 61001: Lecture 3 19 / 25
Example: traffic fatalities
Do regressors (monthly dummies, time trend, unem, wkends, belt, mph) collectively help to explain y (% of traffic accidents involving fatalities)?
R2 = 0.72 F = 14.625
F =F16,91(1−δ)⇒p-value≈0andsorejectH0 atall conventional sig. levels.
Alastair R. Hall ECON 61001: Lecture 3 20 / 25
Restricted Least Squares
Suppose we wish to impose linear restrictions on estimated coefficients. Can do this via method of Restricted Least Squares.
Recall that OLS is: βˆT = argminβ∈B QT (β).
Define: BR ={β;Rβ=r,β∈B}.
Restricted Least Squares (RLS) estimator of β0 is: βˆR,T = argminβ∈BR QT (β).
Note: RβˆR,T = r by construction.
Alastair R. Hall ECON 61001: Lecture 3 21 / 25
Restricted Least Squares
Obtain estimator via Lagrange’s method with Lagrangean: L(β,λ) = QT(β) + 2λ′(Rβ − r)
It can be shown that (see Lecture Notes)
βˆR,T = βˆT − (X′X)−1R′{R(X′X)−1R′}−1(RβˆT − r).
Alastair R. Hall ECON 61001: Lecture 3 22 / 25
Sampling distribution of RLS
If Assumptions CA1- CA6, rank(R) = nr and Rβ0 = r then
βˆR,T ∼Nβ0,σ02D
D = (X′X)−1 − (X′X)−1R′{R(X′X)−1R′}−1R(X′X)−1.
Under these assumptions, RLS is at least as efficient as OLS because:
Var[βˆT]−Var[βˆR,T] = σ02(X′X)−1R′{R(X′X)−1R′}−1R(X′X)−1 = psd
However, if Rβ0 ̸= r then: E[βˆR,T ] ̸= β0.
Alastair R. Hall ECON 61001: Lecture 3 23 / 25
where
Variable selection
So far have taken X as given but in practice need to choose regressors.
Choice may come from economic theory. Data-based methods:
Maximize R2? Not a good idea.
Maximize R ̄2 = 1 − RSS/T−k ? Ok, but equivalent to including
TSS /T −1 all variables with |tstat| > 1.
For further discussion please read Notes Section 2.9
Alastair R. Hall ECON 61001: Lecture 3 24 / 25
Further reading
Notes: Sections 2.8 – 2.10 and Section 2.13 (Appendix on Statistical Distributions)
Greene:
Classical hypothesis testing framework, Section C.7 Inference based on OLS estimators – Sections 5.1-5.5
Alastair R. Hall ECON 61001: Lecture 3 25 / 25