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Time series tests of the small open
economy model of the current
account
Birmingham MSc International Macro
Autumn 2015
Tony Yates
Intro
• Note: not examined directly in the MCQ test.
• Material for the main end of year exam.
• Exploits some of the multivariate time series
we covered when we did Clarida-Gali’s work
on the DMF model.
• Reinforces importance of taking macro to the
data, and connection between time series
econometrics and macro.
Testing we already did: 1
• We experimented with stationary and nonstationary endowment processes…
• … And saw that this flips the sign of the
relationship between the ca and output.
• In the data, current account worsens in a
boom.
• Need non-stationary endowment to get this.
Testing we already did: 2
• We derived analytical expressions for the
variance of consumption and output growth in
terms of the variance of the shock to the
endowment.
• And we got conditions for when var(C)>var(Y)
• And we noted Uribe’s comment that
var(C)>var(Y) in the data for small open
economies.
Test 1: ca vs VAR-based infinite stream
forecasts
• First test based on comparison with the SOE’s
prediction that the ca=f(stream of future
forecasts of endowment growth), and the
infinite stream of forecasts using a VAR.
SOE TIME SERIES TEST 1: COMPARING
MODEL AND DATA BASED INFINITE
STREAM FORECASTS
ca as a f(forecast endowments)
Recall the definition of the current
account.
ca t y t c t rd t1

r
c t rd t1 1
E t
r t
y tj
And a relationship we found for
consumption in terms of forecast
endowments.

1
rj
j0

ca t 
1 ry t rE t 
j1
y tj

1
rj
Now substitute in definition of the
current account.
Recall definition of the (eg quarterly)
change in a generic time series x_t
x t1 x t1 x t

y t rE t 
j1
y tj

1 rj


1 r
j1
1
E t y tj

1 rj

 ca t 
j1
1
E t y tj [1]

1 rj
Derive a relationship between the current account and expected future growth
rates of output.
We are going to exploit fact that we can also use a VAR to do this.
xt 
y t
ca t
Dx t1 e t
E t x tj |Ht Dj x t
A VAR for output and the current account.
The VAR’s infinite sequence forecast.

ca t 
j1
1

1
rj
E t y tj
Model: ca=saving=f(infinite sequence
forecast of future income)
Here, we sub in the VAR into RHS of equation
for current account…


j1
1

1
rj
1
E t y tj 
1 0

I D/
1 r

D/
1 r
y t
ca t
Where does that formula come from?!
Picks out output , selects out ca.


1

1
rj
1
E t y tj 
1 0

I D/
1 r
 D/
1 r
j1
y t
ca t
Matrix formula for the infinite sequence sum, seen many
times so far…. And a reminder of what that looks like,
expanded….

1 0 1 D
1 r
y t
ca t
...

1 0 1 3 D3

1 r

1 0 1 2 D2

1 r
y t
ca t
y t
ca t

...

...

1 0 1 j Dj

1 r
y t
ca t

...
Where does that come from…ctd!

1 0 1 D
1 r
y t
ca t
...

1 0 1 3 D3

1 r
y t
ca t
1
D
1
r
A 
I K

1 0 1 2 D2

1 r
y t
ca t
y t
ca t

...

...

1 0 1 j Dj

1 r
y t
ca t

...
This is a common factor in this infinite sequence, so we can take
it out….
Left with a sequence whose first term is this, and where each
successive term is also multiplied by this….
This is a reminder of the matrix formula for an infinite geometric
sequence, provided abs(max(eig(A)))<1.


1

1
rj
1
E t y tj 
1 0

I D/
1 r
 D/
1 r
j1
y t
ca t
1
F 
1 0

I D/
1 r

D/
1 r Define F to be the coefficient on [dy,ca]’
ca t 
0 1
F 
0 1
y t
ca t
This is true, trivially. Ie the current account =
itself.
For both the top equation on this slide, and
the trivial eq to be true, this must also be true,
when we use the estimated VAR based
forecasts using D_hat.
Result: fails, of course!
• Like all the models in the course….
• This one also strongly rejected by the data in
Nason and Rogers’ paper.
• [Recall my promise only to present false
models in the course, still kept here].
SOE TIME SERIES TEST 2:
ORTHOGONALITY RESTRICTIONS
Informal account of the orthogonality
approach
• Find combinations of the current account that = 0
in expectation…
• And are therefore predicted to be orthogonal to
data available at time expectations formed.
• Use estimated VAR-based equivalents for the
forecasts in the model-based orthogonality
restrictions.
• See if they are correlated with available data or
not.
• [They will be, of course….!]

ca t 
1

1
rj
E t y tj
j1
Recall this expression from our small open
economy microfounded model.
E t ca t1 
1 r
ca t E t y t1 0 We can show this to be true. [It’s an exercise
to do it. Use above eq to sub out for ca and
related terms…]
xt 
y t
ca t
Dx t1 e t
E t ca t1 
0 1
Dx t

1 r
ca t 
0 1

1 r
Ix t
E t y t1 
1 0
Dx t
We can also construct data-based versions of
these variables from a bivariate VAR
representation of the data.
Here is each element in the 2nd eq above in
terms of the VAR for [dy, ca]’
E t ca t1 
1 r
ca t E t y t1 0
G t 
0 1
Dx t 
0 1

1 r
Ix t 
1 0
Dx t
G uses VAR-based terms to replicate the LHS of the top equation.
Since top equation=0 [it is a sum of 2 expectations remember]… we should find
G=0, or at least is unrelated to current and past values of x_t.
Test fails miserably on actual data.
Attempting a rescue of the SOE model
• SOE model assumed only 1 shock, a shock to
the exogenous endowment of our SOE
consumers.
• We’ll modify it to include a demand shock
[will also give you a clue as to how generally
RBC/DSGE models are modified in this way]
• Leads to a new ‘orthogonality’ or zero
condition.
SOE with demand shock
u
c0. 5
c c
u
c0. 5
c 
c t 

E t ca t1 
1 r
ca t E t y t1 t
We include a serially uncorrelated ‘bliss point shock’ in the SOE model.
This wrecks our orthogonality condition.
LHS no longer equal to zero.
Explaining why our orthogonality
condition is wrecked by that RHS u_t
term
c t t E t c t1


1 r
d t1 E t 
j0
c t t E t c tj
This is the new Euler equation with the
demand shock. [Confirm in an exercise]
y tj c tj

1
r
j
Recall that this relation holds from the
infinite period budget constraint.
It turns out by use of the EE, and the law
of iterated expectations, that this modified
substitution can be made to eliminate the
sequence of forecast c’s on the RHS of the
infinite period budget constraint.
Deriving our wrecked orthoganality
condition


1 r
d t1 E t 

y tj

1
r
j

c t t 
j0
1

1
rj
t
j0
We get this after eliminating the sum of infinite future forecast c’s.

ca t  1

1
r
t E t 
y tj

1
rj
j1
And then this after a bit of re-arranging, and substituting in defn of the
current account.

ca t 
1

1
r
t E t 
y tj

1
rj
Using this expression for the current
account…
j1
E t ca t1 
1 r
ca t E t y t1 t
Substitute in here to confirm that this
equation holds, and therefore
expresses the new, wrecked
orthogonality condition.
We will rescue this shortly to form a
new condition and a closely related
regression test, that will fail anyway.
Rescuing our orthogonality condition
E t ca t1 
1 r
ca t E t y t1 t
Our old condition, now wrecked by
the demand shock.
E t1 
E t ca t1 
1 r
ca t E t y t1 E t1 
t 
E t1 ca t1 
1 r
E t1 ca t E t1 y t1 0
But take expectations
conditional on time t-1
of both sides…
..and because mu_t is
serially uncorrelated,
RHS now=0.
So we have a new
orthogonality condition
that leads to similar
regression test of our
SOE model.
New ‘lagged’ version of our regression
test
E t1 ca t1 
1 r
E t1 ca t E t1 y t1 0
E t1 ca t1 
0 1
D2 x t1

1 r
E t1 ca t 
0 1

1 r
. I. Dx t1
E t1 y t1 
1 0
D2 x t1
New orth. Condition.
VAR-based forecasts
conditional on time t1 substituted in.
D^2 because 2 periodahead forecasts.
G t 
0 1
D2 x t1 
0 1

1 r
. I. Dx t1 
1 0
D2 x t1
Analogous to previous test: we now run regression of G on variables [eg x]
dated t-1 and earlier.
If model is true, expect zero coefficients. But fails spectacularly again.
Recap
• We wanted to test implications of the SOE
model.
• Up to this point, we had tests…
– … based on the sign of the correlation between
the current account and output growth
– …and on relative variance of consumption and
output growth
In particular
• We saw that model matches data, which has
that current account worsens in good times,
only if we assume non-stationary endowment
process.
• And we derived certain plausible restrictions
on model parameters that could produce fact
from data that var(C)>var(Y) in the SOE.
New tests
• 2 tests.
• First test based on equating
• Second test derives an ‘orthogonality
condition’ or really an expression involving
actual and expected current accounts that
should be zero, therefore uncorrelated with
things dated t.
2nd test modified
• We saw how enriching the model with a
demand shock wrecks the orthogonality
conditions.
• But how we could rescue it by taking
expectations of the ‘wrecked’ conditoin at
time t-1.
• Nevertheless, all tests failed.
Remarks
• Demand shock seems like an ok complication,
but many people question what this means
[though neuroeconomists believe in them].
• Is the demand shock a shock or just a residual
indicating model failure?
• Tests assume rational expectations… if this
doesn’t hold in reality, then we might reject
the model altogether when most of it might
be true.
Model tests and learning
• The infinite stream forecasts are based on
taking RE of recursively substituted period by
period budget constraints.
• What would happen if these constraints are
not wholly appreciated?
• Or if agents make mistakes using or
comprehending the matrix formula [like you
lot when you first saw them]
Model tests and learning
• Tests assume that the forecasts can use a
constant, whole-sample VAR forecast.
• This rules out some models of least squares
learning.
• Those models imagine agents continually
updating forecasting models as new data
confounds the old.
• Generates a time-varying-parameter-VAR.
Conclusion
• Used the micro theory to construct macro
model of the SOE.
• Then used that theory to derive time-series,
VAR-based tests.
• Model failed badly, but Nason and Rogers
explored ways to account for this [wrong test,
also wrong model].
• ‘Scientific’ progress that echoes the good bits
of applied modern macro.