Exploring Data Dynamics with Local Projections Òscar Jordà Department of Economics

Exploring Data Dynamics with Local
Projections
Òscar Jordà
Department of Economics
U.C. Davis
Web
Web, e-mail
e mail, address:
1
Exploring Data Dynamics with Local Projections
November 08
Wh t iis a LLocall P
What
Projection?
j ti ?
y It refers to regressions of the form:
yt+h = ¯0hyt + ::: + ¯phyt¡p + ut+h
for h = 1, …, H
y In forecasting, these regressions are sometimes
called direct forecasts.
y What
Wh are they
h useful
f l ffor?? E
Essentially,
i ll they
h
provide a semi-parametric method to estimate
the coefficients of the Wold decomposition
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Exploring Data Dynamics with Local Projections
November 08
S
Some
Applications
A li ti
off LLocall P
Projections
j ti
y As a flexible way to calculate the impulse response
function (looks like a “treatment effect”):
IR(yt+hh ; ±) =E(yt+hh j²t = ±; yt¡11; :::)¡
)
E(yt+h j²t = 0; yt¡1 ; :::)
y As a flexible way to estimate parameters from
dynamic moment conditions, e.g., the parameters of a
Phillips curve
Philli
¼t = ¯Et (¼t+1 ) + °yt + ²t
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Exploring Data Dynamics with Local Projections
November 08
… and
d
y As a flexible way to compute path forecasts
2
3
E(yt+1jyt ; :::)
..
4
5
.
E(yt+H jyt ; :::)
and hence derive the joint predictive density
and appropriate statistics
statistics.
4
Exploring Data Dynamics with Local Projections
November 08
What is the motivation for using local
projections?
j ti
?
y Models for vector time series are well-known
(e.g. VARs, etc), and their likelihoods and
their statistical p
properties
p
are well-understood.
y However, the model’s parameters are rarely of
interest themselves. Usually it is a nonlinear
function of these parameters that interest us
(such as an impulse response or a forecast).
y Therefore, restrictions that are sensible from
the perspective of a model, may not be useful
f th
for
the objects
bj t th
thatt we wantt tto estimate.
ti t
5
Exploring Data Dynamics with Local Projections
November 08
E
Example:
l Properties
P
ti off IR
IRs ffrom a VAR
y Symmetry: the response of a variable to a
positive treatment has the same shape if the
shock is negative
g
instead.
y Shape-invariance: the size/sign of the
treatment does not affect the shape of the
impulse response (it scales it).
y History independence: the impulse response is
independent of the value of recent
observations.
6
Exploring Data Dynamics with Local Projections
November 08
B i IIntuition
Basic
t iti
Projection
DGP
Projection
VAR
7
Exploring Data Dynamics with Local Projections
November 08
Th Pl
The
Plan
y I hope to review 3 useful applications of local
projections:
1. Estimation and Inference of Impulse Responses (AER,
2005 + ReStat forthcoming)
2. Projection
j
Minimum Distance (under
(
review))
3. Path Forecasting (JAE, forthcoming)
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Exploring Data Dynamics with Local Projections
November 08
1 Impulse
1.
I
l R
Responses
y Suppose the data have a Wold (impulse
response) representation
yt = ²t + B1 ²t¡1 + B2 ²t¡2 + :::
y and that it is invertible
yt = A1 ytt¡11 + A2 ytt¡22 + ::: + ²t
y Notice this includes all VARs, VARMAs, etc.
but excludes some others
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Exploring Data Dynamics with Local Projections
November 08
L
Local
l Projections
P j ti
y Then,
Then iterating the VAR(∞) representation
forward
yt+h =Ah1 yt + Ah2 yt¡1 + :::+
²t+h + B1 ²t+h¡1 + ::: + Bh¡1 ²t+1
y with the convenient result
Ah1 = Bh for h ¸ 1
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Exploring Data Dynamics with Local Projections
November 08
E ti ti
Estimation
y Truncate the iterated VAR(∞) representation at
some lag k (not terribly important how chosen).
0
0
; :::; yt+H
g
y Let Y collect T obs.
obs of fyt+1
0
0
y Let Z collect T obs. of f1; yt¡1 ; :::; yt¡k+1 g
y Let X collect T obs.
obs of yt
y Let M = I ¡ Z(Z 0 Z)¡1 Z 0
y Let B stack the impulse response matrices Bh
bT
b = M Y ¡ M XB
y Let V
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Exploring Data Dynamics with Local Projections
November 08
E ti ti ((cont.)
Estimation
t)
y Then,
Then the least
least-squares
squares estimate of the system
system’ss impulse
responses is
¡11
0
b
BT = (X M X) (X 0 M Y )
p
d
b
T ¡ k ¡ Hvec(B
Hvec(BT ¡ B0 ) ! N (0; Ðb )
£ 0
¤
¡1
Ðb = (X M X) Ð §v
^ V^ 0
V
bv =
§
T ¡k¡H
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Exploring Data Dynamics with Local Projections
November 08
P
Properties
ti
y When the model is correctly specified, slightly less
efficient than VAR-based IRs
y Consistent (and robust to misspecification)
y Can be estimated equation
equation-by-equation
by equation
(convenient for panels, non-linearities and
nonparametrics)
y Hence
H
th
they can b
be generalized
li d easily
il with
ith
univariate nonlinear models: stress testing
(Drehmann, Patton and Sorensen, 2006);
thresholds (Jordà
(Jordà, 2005); STAR (Jordà and Taylor,
Taylor
2008); spatial correlation in housing markets (Brady,
2007)
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Exploring Data Dynamics with Local Projections
November 08
Local Projections for Cointegrated
S t
Systems
y A state-space representation of a VECM
2
3
2
0
0
zt+1
(Ik ¡ A B) A ª1
6 ¢yyt+1 7 6
¡B
ª1
6
7 6
6 ¢yt 7 = 6
0k;k
In
6
7
6
..
..
..
4
5 4
.
.
.
¢yt¡p+1
0k;k
0n;n
t p+1
kk
nn
0
0
32
3
2
0
::: A ªp¡2 A ªp¡1
zt
A "t+1
6
7 6
7
::: ªpp¡22
ªpp¡11 7
7 6 ¢yyt 7 6 "t+1 7
6 ¢yt¡1 7 + 6 0 7
::: 0n;n
0n;n 7
7
7 6 . 7
..
.. 5 6
..
4
5 4 .. 5
:::
.
.
.
:::
In
0n;n
¢yt¡p+2
0
nn
t p+2
y …or compactly
Yt+1 = GYt + ²t+1
14
3
Exploring Data Dynamics with Local Projections
November 08
E ti t and
Estimator
d Properties
P
ti
0
0
¡H
y Let ZH collect fzt+1
; :::; zt+H
gTt=p+1
0
0
¡H
y Let YH collectf¢yt+1
; :::; ¢yt+H
gTt=p+1
¡H
y Let X collect fzt0 ; ¢yt0 gTt=p+1
y Let
0
0
¡H
; :::; ¢yt¡p+2
gTt=p+1
W collect f1;¢yt¡1
y Let MW = I ¡ W (W 0 W )¡1 W 0
y Then
c t+h ; ±j ) = G
bh A
b0 ±j + G
b h ±j
IR(
IR(z
1;1
1;2
h b0
h
c
b
b
;
±
)
=
G
A
±
+
G
±j
IR(¢y
t j
j
2;1
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Exploring Data Dynamics with Local Projections
2;2
November 08
… estimator
ti t continued
ti
d
2
bz;H
G
by;H
G
p
16
b11;1
G
6 ..
=4 .
bH
G
1;1
2
b12;1
G
;
6 ..
=4 .
bH
G
2;1
;
3
b11;2
G
.. 7 = Z 0 M X (X 0 M X)¡1
. 5
W
H W
bH
G
1;2
3
b12;2
G
;
.. 7 = Y 0 M X (X 0 M X)¡1
. 5
W
H W
bH
G
2;2
;
³
´
d
d
T ¡ p ¡ H vec(G
i;H ) ¡ vec(Gi;H ) ! N (0; Ði ) i = z; y
Exploring Data Dynamics with Local Projections
November 08
Ad
Advantages
t g
y Given the cointegrating vector (estimated or
imposed), recall that:
c t+h ; ±j ) = G
bh A
b0 ±j + G
b h ±j
IR(z
1;1
1;2
c ( yt ; ±j ) = G
bh A
b0 ±j + G
b h ±j
IR(¢y
21
2;1
Response to
Long-Run
E ilib i
Equilibrium
22
2;2
Response to
Short-run
D
Dynamics
i
y Example: effect of PPP on carry trade (with
Alan Taylor)
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Exploring Data Dynamics with Local Projections
November 08
.5
0
Percentage
1
1.5
PPP Adj
Adjustment
t
t Speeds
S
d
0
3
6
9
12
15
Months
Total response
Short run response
Short-run
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Exploring Data Dynamics with Local Projections
18
21
24
27
30
Long-run response
November 08
2 P
2.
Projection
j ti Mi
Minimum
i
Di
Distance
t
The idea
y The first order Euler conditions of many
dynamic macroeconomic models provide
moment conditions, often estimated by GMM
y However, models often simplify reality
considerably
id bl
y Hence we want a method to cast these
moment conditions against a statistical model
of the data that imposes the least constraints
possible (certainly avoid the restrictive
economic
i model)
d l)
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Exploring Data Dynamics with Local Projections
November 08
Ill giti t IInstruments
Illegitimate
t
t
y Suppose we want to estimate
y = Y¯ +u
with a set of instruments Z for Y
y Suppose the DGP is instead
y = Y ¯ + x° + "
where Z are valid instruments for Y
y
20
Are the
A
th Z valid
lid iinstruments
t
t ffor th
the equation
ti
we want to estimate? Usually, no.
Exploring Data Dynamics with Local Projections
November 08
H
Here
is
i Wh
Why
y If E(Z 0 x) 6= 0 and ° 6= 0 then the moment
condition required to ensure that instruments
are valid is
E(Z 0 u) = E(Z 0 x)° + E(Z 0 ") = E(Z 0x)° 6= 0
which
h h is violated…
l d
y What can one do? Even though the x where
not included in the economic model
model, they can
be used to restore the legitimacy of the
instruments.
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Exploring Data Dynamics with Local Projections
November 08
T ways tto restore
Two
t
legitimacy
l giti
1 Orthogonalize the explanatory variable and
1.
the regressors with respect to the omitted x
2 Orthogonalize the instruments with respect
2.
to the omitted x, e.g. regress
Z = x± + v
and do the usual IV with the v^. This is
different than the usual TSLS – here it is the
residuals (not the predicted values) that are
the valid instruments.
22
Exploring Data Dynamics with Local Projections
November 08
What do Local Projections Have to Do
with
ith All This?
Thi ?
y Plenty: local projections provide a semisemi
parametric method to consistently estimate
the first H coefficients of the Wold
representation.
i
y Moment conditions expressed in terms of their
Wold representation are simply a collection of
restrictions between the parameters of interest
and the impulse response coefficient matrices
y Hence, consistent and asymptotically normal
estimates can be obtained by minimum
distance methods
methods.
23
Exploring Data Dynamics with Local Projections
November 08
Th M
The
Mechanics
h i
y Suppose the Euler conditions from a model
can be summarized in the system:
yt = ©F Et yt+1 + ©B yt¡1 + ut
y Stability of the system means that it has a
reduced-form Wold representation
yt = ²t + B1 ²t¡1 + B2 ²t¡2 + :::
y Plugging back to the Euler equations
equations…
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Exploring Data Dynamics with Local Projections
November 08
Th M
The
Mechanics
h i ((cont.)
t)
y … we get the set of conditions
Bh = ©F Bh+1 + ©B Bh¡1 for h ¸ 1
y These conditions are linear in the parameters
unique.
©F ; ©B and hence unique
y Do not require structural identification since
they are based on serial correlation properties
of the data.
y Can be estimated by GLS-type
GLS type step
25
Exploring Data Dynamics with Local Projections
November 08
Th E
The
Estimator
ti t iin a nutshell
t h ll
b T ; Á) = vec(B
^ ¡ ©F B
^ F ¡ ©B B
^ B)
y Let f (b
(b
vec(B
y Then a consistent and asymptotically normal
estimate of the Á = vec(©F ; ©B ) is found from
0
^
^ f (b
^ T ; Á)
min f (bT ; Á) W
Á
^ T ; Á)
^ T ; Á)
@f (b
@f (b
y Let Fb =
and
; FÁ =
@b
@Á
^ = (F 0 С1 Fb )¡1
W
b b
26
Exploring Data Dynamics with Local Projections
November 08
Then…
Th
^ = (F
^ F^Á )¡11 (F
^ b^T )
Á
(F^Á0 W
(F^Á0 W
³
´
p
d
b
T ¡ H ¡ k ÁT ¡ Á0 ! N (0;Ð
(0 ÐÁ )
^ Á = ((F 0 W
^ FÁ )¡1
Ð
Á
y with overidentifying restrictions test
d
b
b
Q(bT ; ÁT ) ! Â2dim(f(bb
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Exploring Data Dynamics with Local Projections
T ;Á))¡dim(Á)
November 08
So What is the Connection between
PMD and
d Ill
Illegitimate
iti t IInstruments?
t
t ?
y The constraints implied by the Euler equations
are not used to restrict the data generating
process used as in, e.g., MLE, Bayesian
estimation or GMM
estimation,
y The impulse responses are estimated semiparametrically
no restrictions on the dynamics
of the data but also, one is free to include other
variables not originally in the analysis.
y Instruments are sequentially orthogonalized for
omitted dynamics and/or omitted variables – this
becomes useful for model checking.
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Exploring Data Dynamics with Local Projections
November 08
E
Example
l
y You consider estimating the forward-looking
Phillips curve: ¼t = ¯Et ¼t+1 + ut using ¼t¡2 as
an instrument,, hoping
p g to avoid biases with ¼tt¡11
y The true model is: ¼t = °f Et ¼t+1 + °b ¼t¡1 + "t
y GMM:
PT
Á1
p
b̄GM M = P 1 ¼t¡2¼t !
°f + °b
T
Á3
h ¼t¡22 ¼t+1
y PMD:
29
PT
0
p
1 ¼t¡2 Mt¡1 ¼t
b̄
¯P M D = PT
! °f + °b = °f
±3
h ¼t¡2 Mt¡1 ¼t+1
Exploring Data Dynamics with Local Projections
November 08
Other Advantages: An ARMA(1,1)
M t Carlo
Monte
C l
yt = ½yt¡1 + "t + μ"
μ t¡1
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Exploring Data Dynamics with Local Projections
November 08
31
Exploring Data Dynamics with Local Projections
November 08
3 P
3.
Path
th F
Forecasting
ti g
y What is a path forecast? A collection of 1 to
H step-ahead forecasts
2
3
2
3
yb¿ (1)
y¿ +1
Yb¿ (H) = 4 ... 5 ; Y¿;H = 4 ... 5
yb¿ (H)
y¿ +H
y Suppose for convenience
´
p ³
d
b
T Y¿ (H) ¡ Y¿;H jy¿ ; y¿ ¡1 ; ::: ! N (0; ¥H )
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Exploring Data Dynamics with Local Projections
November 08
What is the Uncertainty Associated
with
ith the
th Path
P th Forecast?
F
t?
y Formally,
Formally invert the statistic for the null
³
´
H0 : E Yb¿ (H) ¡ Y¿;H jy¿ ; y¿ ¡1; ::: = 0
y …but this is a multidimensional ellipse since
³
WH = T Yb¿ (H) ¡ Y¿;H
´0
³
´
d
¡1 b
¥H Y¿ (H) ¡ Y¿;H ! Â2H
y and
d th
the region
i we wantt tto plot
l t iis th
thatt which
hi h
satisfies
£
Pr WH ·
33
Exploring Data Dynamics with Local Projections
c2®(H)
¤
=1¡®
November 08
S h ff Bands
Scheffe
B d
y Scheffe
Scheffe’ss (1953) S
S-Method:
Method: provides a method
to obtain statistics for nulls that involve linear
combinations of the original
g
jjoint null
y A particularly appealing linear combination
allows one to maximize the joint variation of
the path
y Let P be the Cholesky factor for ¥H = P P 0
then bands can be constructed as…
34
Exploring Data Dynamics with Local Projections
November 08
S h ff Bands
Scheffe
B d ((cont.)
t)
"r
Yb¿ (H) § P
c2® (h)
h
#H
h=1
y instead of the usual
Yb¿ (H) § z®=2diag(¥H )1=2
y or the
th Bonferroni
B f
i version
i
1=2
Yb¿ ((H)) § z®=2H
diag(¥
(
)
H
=
35
Exploring Data Dynamics with Local Projections
November 08
I t iti
Intuition
4
95% Scheffe Upper
Bound (1.73, 3.03)
3
Traditional 2 S.E. Box
2
Estimated Values
1
0
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
95% Scheffe Lower
Bound ((-1.73,
1 73 -3.03)
3 03)
95% Confidence Ellipse
-4
36
Exploring Data Dynamics with Local Projections
November 08
S
Some
Pi
Pictures…
t
37
Exploring Data Dynamics with Local Projections
November 08
… and
d some M
Monte
t C
Carlos
l
38
Exploring Data Dynamics with Local Projections
November 08
Summary
y Some General Principles:
1. Tailor the statistics for the question you
want to answer
2. When you are unsure about the “truth,”
choose methods that are robust to
misspecification
3. Simple
p is often times better: if y
you hear
hooves, think horses, not zebras…
4. Models can sometimes be too restrictive
39
Exploring Data Dynamics with Local Projections
November 08
Wh
Where
next?
t?
y IRs with Local Projections:
y In a panel setting with cointegration and
nonlinearities, investigate the returns to carry trade
and long-run
g
PPP adjustment
j
y Harrod-Balassa-Samuelson
y PMD
y Estimation of GARCH models,
models specifically
multivariate GARCH
y Path Forecasting:
y Mahalanobis vs.
vs RMSE and other alternatives
y Value at Risk and Path-dependent options
y Path Predictive Ability Testing
40
Exploring Data Dynamics with Local Projections
November 08
Pi iin the
Pie
th Sk
Sky
y Using IV to estimate structural IRs with local
projections
y Semiparametric Dynamic Treatment Effects:
use the Mahalanobis distance instead of
propensity scores for dynamic treatments, e.g.
the effects of monetary policy
41
Exploring Data Dynamics with Local Projections
November 08