ECON 61001: Lecture 7
Alastair R. Hall
The University of Manchester
Alastair R. Hall ECON 61001: Lecture 7 1 / 29
Outline of this lecture
Time series regression models with non-spherical errors
OLS-based inference GLS-based inference Testing for serial correlation
Instrumental Variables estimation
Models with endogenous regressors
OLS as MoM → IV
Examples of instruments
Alastair R. Hall ECON 61001: Lecture 7 2 / 29
OLS
In Lecture 5 we show that conventional OLS inference framework goes through under the following assumptions.
Assumption TS1: yt = xt′β0 + ut, t = 1,2,…T Assumption TS2: (yt , ht′ ) is a weakly stationary, weakly
dependent time series.
Assumption TS3: E[xtxt′] = Q, a finite, positive definite matrix.
Assumption TS4: E[ut |xt] = 0 for all t = 1,2,…,T. A s s u m p t i o n T S 5 : V a r [ u t | x t ] = σ 02 .
Assumption TS6: For all t ̸= s, E[ut us | xt, xs] = 0.
Alastair R. Hall ECON 61001: Lecture 7 3 / 29
OLS – with conditionally heteroscedastic errors
In video on “OLS-based inference in time series regression models with conditionally heteroscedastic errors” we consider OLS based inference in models satisfying:
Assumption TS1: yt = xt′β0 + ut, t = 1,2,…T Assumption TS2-SS: (yt , ht′ ) is a strongly stationary, weakly
dependent time series.
Assumption TS3: E[xtxt′] = Q, a finite, positive definite matrix.
Assumption TS4: E[ut |xt] = 0 for all t = 1,2,…,T. Assumption TS5-H: Var[yt |It] = σt2.
Assumption TS6: For all t ̸= s, E[ut us | xt, xs] = 0
Alastair R. Hall ECON 61001: Lecture 7 4 / 29
Dynamic completeness
Alternatively, we can impose Assumptions TS1, TS2, TS3 and TS5 (or TS1,TS2-SS, TS3 and (TS5-H) and
Assumption TS7: E[yt |It] = xt′β0, where It is information set at time t.
– if Assumption TS7 holds then the model is said to be dynamically complete.
Alastair R. Hall ECON 61001: Lecture 7 5 / 29
Serial correlation in errors
The framework adopted affects how we view the consequences of serial correlation in the errors and so how we proceed.
If Assumptions TS1, TS2-SS, TS3, TS4, and TS5-H hold but TS6 does not (so E[ut us | xt, xs] ̸= 0) then:
OLS is still consistent as Assumption TS4 holds.
Scope for serial correlation robust inference based on OLS
If Assumptions TS1, TS2, TS3 & TS5 (or TS1, TS2-SS, TS3 & TS5-H) hold but Assumption TS7 does not (so E[yt |It] ̸= xt′β0) then:
OLS is likely inconsistent as TS4 may not hold.
Need to reconsider specification.
Alastair R. Hall ECON 61001: Lecture 7 6 / 29
OLS-based inference
If Assumptions TS1, TS2-SS, TS3, TS4, & TS5-H hold then: βˆT is consistent and
T1/2(βˆT−β0)→d N(0,Vsc) Vsc = Q−1ΩQ−1,
Ω=Γ0+∞(Γi+Γ′),forΓi=E[utut−ixtx′ ]. i=1 i t−i
Estimation of long run variance Ω?
where
Alastair R. Hall ECON 61001: Lecture 7 7 / 29
Covariance matrix estimation
Use heteroscedasticity and autocorrelation covariance (HAC) matrix estimator:
Ωˆ H A C = Γˆ 0 +
T−1 i=1
ω ( i , T ) ( Γˆ i + Γˆ ′i ) t=i+1 t−i
where
Γˆi =T−1T etet−ixtx′
ω(·, ·) known as kernel
ω( · , · ) chosen so that Ωˆ HAC is both consistent and psd.
Alastair R. Hall ECON 61001: Lecture 7 8 / 29
Covariance matrix estimation
Popular choice of kernel:
Bartlett (Newey & West, 1987): putting ai = i/(bT + 1)
ω(i,T) = 1−ai,forai<1 = 0, else
This is an example of a truncated kernel estimator:
bT ΩˆHAC=Γˆ0+ ω(i,T) Γˆi+Γˆ′i
i=1
Alastair R. Hall ECON 61001: Lecture 7 9 / 29
OLS-based inference
L e t Qˆ = T − 1 X ′ X .
S e t Vˆ s c = Qˆ − 1 Ωˆ H A C Qˆ − 1 . Then can show Vˆsc →p Vsc.
So can perform inference based on OLS provided substitute Vˆsc for σˆT2 Qˆ−1 in statistics discussed in Lecture 5. (See Tutorial 7)
Alastair R. Hall ECON 61001: Lecture 7 10 / 29
OLS if Assumption TS7 fails
If justify standard OLS-based inference framework via Assumptions TS1-TS3, TS5 & TS7 then serial correlation arises through failure of Assumption TS7 and so:
E [ y t | I t ] ̸ = x t′ β 0
This means model is misspecified and so either:
model is linear but have omitted variables model is nonlinear
So need to re-consider specification.
Alastair R. Hall ECON 61001: Lecture 7 11 / 29
GLS
Recall need model for Σ.
Here consider model with AR(1) errors:
yt =xt′β0+ut ut =ρ0ut−1+εt
where
εt ∼iid(0,σε2).
Regularity conditions hold (esp E [xt ut ] = 0) ⇒ βˆT →p β0.
Alastair R. Hall ECON 61001: Lecture 7 12 / 29
GLS
Then GLS is OLS applied to:
y ̈t =x ̈t′β0+u ̈t
where
y ̈t = yt − ρ0yt−1 etc. ∼ “quasi-difference”.
Consistency of GLS requires
E[x ̈tu ̈t] = E[(xt − ρ0xt−1)(ut − ρ0ut−1)] = 0,
→ (in general) need E[xs ut ] = 0 for s = t − 1, t, t + 1.
So conditions for consistency of GLS are stronger than for OLS. Thus, the conditions for the consistency of GLS are stronger than those for the consistency of OLS.
Alastair R. Hall ECON 61001: Lecture 7 13 / 29
GLS: alternative view
If sub ut = yt − xt′ β0 in regression model then: y=ρy +x′β−x′ (ρβ)+ε,
t 0t−1 t0 t−100 t which is same as (re-written) transformed model in GLS.
Also special case of the the linear regression model y=δy +x′δ+x′ δ+ε,
t 0t−1 t1 t−12 t in which the parameters satisfy:
δ2+δ0∗δ1 =0,
known as the common factor or COMFAC restrictions.
(c.f. response to violation of TS7.)
Alastair R. Hall ECON 61001: Lecture 7 14 / 29
Breusch-Godfrey test for serial correlation
Assume errors follow AR(p) process that is,
ut = ρ0,1ut−1 +ρ0,2ut−2 +...+ρ0,put−p +εt, where εt ∼ iid(0,σε2).
H0: ρ0,i =0,foralli=1,2,...,p
HA : ρ0,i ̸= 0, for at least one i = 1,2,...,p
Test stat is:
where R2 is from regression of et on xt,et−1,et−2,...,et−p.
U n d e r H 0 : L M p →d χ 2p .
Alastair R. Hall ECON 61001: Lecture 7 15 / 29
LMp = TR2
Breusch-Godfrey test for serial correlation
Choice of p? trade-off
may reflect sampling frequency of data Interpretation of significant test:
can be caused by AR(p) errors
can also be caused by misspecified model
Empirical example: See Section 4.4.5 of Lecture Notes
Alastair R. Hall ECON 61001: Lecture 7 16 / 29
Instrumental Variables
Consequences of E [xt ut ] ̸= 0 (violation of CS4 or TS4). Recall
T −1 T
βˆT −β0 = T−1xtxt′ T−1xtut,
t=1 t=1
and using the WLLN, we have
T
T−1xtxt′ →p E[xtxt′] = Q,
t=1
−1T p
T xtut → E[xtut]=μ̸=0. t=1
So using Slutsky’s Theorem: βˆT →p β0 + Q−1μ ̸= β0.
Alastair R. Hall ECON 61001: Lecture 7 17 / 29
Instrumental Variables
So OLS is an inconsistent estimator of β0.
Source of problem is endogenous regressor(s).
This can arise in three main ways in econometric models:
reverse causality omitted variables measurement error
Now consider an example of each.
Alastair R. Hall ECON 61001: Lecture 7 18 / 29
Example 1: Economic development and institutions
Acemoglu et al (2001):
ln[yi] = β0 + riβ1 + controls+ ui
where
yi is income per capita in developing country i
ri is quality of institutions in developing country i
r is likely correlated with ui due to reverse causality.
Alastair R. Hall ECON 61001: Lecture 7 19 / 29
Example 2: returns to education
Angrist & Krueger (1991):
ln[wi] = θ1 + θeedi + controls + ui
where
wi equals the weekly wage of individual i;
edi is the number of years of education of individual i;
edi likely correlated with ui due to omitted variables such as “ability” that affect wi and edi .
Alastair R. Hall ECON 61001: Lecture 7 20 / 29
Example 3: monetary policy reaction function
Clarida, Gali & Gertler (2000):
rt =c+βπE[πt+1|Ωt]+βyE[yt+1|Ωt]+β1rt−1+β2rt−2+ut,
rt = Federal Funds rate in year.quarter t πt+1 =inflationint+1
Ωt information set at time t
yt+1 = output gap in t +1
ut is error satisfying E[ut | Ωt] = 0
Alastair R. Hall ECON 61001: Lecture 7 21 / 29
Example 3: monetary policy reaction function
Expectations not observed if replace by actual values then introduce measurement error because:
rt = c+βππt+1+βy yt+1+β1rt−1+β2rt−2+(ut − βπvπ,t+1 − βy vy,t+1)
where
=et
v(·),t+1 =(·)t+1−E[(·)t+1|Ωt], E[(·)t+1et]=β(·)Var[v(·),t+1] ̸= 0.
Alastair R. Hall ECON 61001: Lecture 7 22 / 29
OLS as MoM
To motivate IV estimation, we reinterpret OLS as a Method of Moments (MoM) estimator.
From normal equations: OLS is MoM based on the population moment condition:
E[xtut(β0)] = 0 (⋆) w h e r e u t ( β ) = y t − x t′ β ( a n d s o u t ( β 0 ) = u t ) .
From this perspective, estimation is based on information in (⋆) about β0. So if
E[xtut(β0)] = 0 → consistent estimator. E[xtut(β0)] ̸= 0 → inconsistent estimator.
Alastair R. Hall ECON 61001: Lecture 7 23 / 29
IV estimation
Find a q × 1 vector zt such that:
E[ztut(β0)] = 0 (♯)
need q ≥ k and zt “sufficiently related to” xt + certain other conditions discussed in the next lecture
zt known as an instrument
Use the population moment equation in (♯) as basis for estimation
→ Instrumental Variables (IV) estimator of β0.
Discuss next time how to calculate IV estimator and what its properties are. Conclude by looking at choices of instruments in our three examples.
Alastair R. Hall ECON 61001: Lecture 7 24 / 29
Example 1: Economic development and institutions
Acemoglu et al (2001):
ln[yi] = β0 + riβ1 + controls+ ui
where
yi is income per capita in developing country i
ri is quality of institutions in developing country i
r is likely correlated with ui due to reverse causality.
Base estimation on the population moment condition: E[ziui] = 0
where zi = [1, controls, Mi ]′ and Mi is settler mortality in country i . Alastair R. Hall ECON 61001: Lecture 7 25 / 29
Example 2: returns to education
Angrist & Krueger (1991):
ln[wi] = θ1 + θeedi + controls + ui
where
wi equals the weekly wage of individual i;
edi is the number of years of education of individual i;
edi likely correlated with ui due to omitted variables such as “ability” that affect wi and edi .
Base estimation on
where zi = [1, controls, Qi ]′ and Qi is quarter of birth of i .
E[ziui] = 0
Alastair R. Hall ECON 61001: Lecture 7 26 / 29
Example 3: monetary policy reaction function
Clarida, Gali & Gertler (2000):
rt =c+βπE[πt+1|Ωt]+βyE[yt+1|Ωt]+β1rt−1+β2rt−2+ut,
rt = Federal Funds rate in year.quarter t πt+1 =inflationint+1
Ωt information set at time t
yt+1 = output gap in t +1
ut is error satisfying E[ut | Ωt] = 0
Alastair R. Hall ECON 61001: Lecture 7 27 / 29
Example 3: monetary policy reaction function
Expectations not observed if replace by actual values then introduce measurement error because:
rt = c+βππt+1+βy yt+1+β1rt−1+β2rt−2+(ut − βπvπ,t+1 − βy vy,t+1)
where
=et
v(·),t+1 =(·)t+1−E[(·)t+1|Ωt], E[(·)t+1et]=β(·)Var[v(·),t+1] ̸= 0.
However: E [zt et ] = 0 for any zt ∈ Ωt (e.g. macro variables).
Alastair R. Hall ECON 61001: Lecture 7 28 / 29
Further reading
Notes: Sections 4.4, 5.1, 5.2 Greene:
Time series regression models with serial correlation,
OLS 20.4 (but 20.4.1 and 20.4.2. go into more detail on “other conditions” in limit theorems than needed for the course), 20.5
GLS/FGLS 20.8, 20.9
Testing for serial correlation 20.7
IV 8.1, 8.2 and 8.3
Alastair R. Hall ECON 61001: Lecture 7 29 / 29