Tests of Intertemporal Models of Current Account
(Lecture partly based on Ghosh and OR)
CAt  Bt 1  Yt  rBt  Ct  Gt  I t  rBt  Ct  Z t
St  rBt  Yt  Ct  Gt
Optimal current account assuming equality between subjective discount rate and market
rate:
~
~
~
CAt  (Yt  EtY )  (Gt  Et G)  ( It  Et I )
In order to estimate the optimal CA, we need to get series on Y, G, I and estimate
(1)
r
~
Et X 
Et
1 r
 1 


t 1 1  r




s t
X s , where X=Y, G, I.
How do we model expectations of future NOs?
First generation models
They used one equation regression models. Measuring problems:
(i)
Which real interest rate to use to discount future flows? Most studies use a
constant low value (3-5%). Is this value correct to evaluate a risky future?
Bernanke used 14%.
~
(ii)
Measures of X are sensitive to differences in time series generating X. Thus
results are not robust.
~
From (1) we need to calculate X t 

r
X t  (1  r )1 Et X t 1  (1  r ) 2 Et X t  2  ...
1 r

Ex: assume X follows an AR(1) process:
X t  X t 1  ut where 0    1 and u ~ i.i.d.
Solving recursively,
Et X t 1  X t
Et X t  2  X t 1   2 X t …. Thus Et X t  k   k X t
Replacing these values into (1) we get:



r
rX t
~
Xt 
X t  (1  r )1 X t  (1  r ) 2  2 X t  ... =
1  (1  r )1   (1  r ) 2  2  ...
1 r
1 r



rX t  1 
r
~
Xt 
X t  X t


1  r 1    1  r  
 1 r 
~
If  is large (small) X is (not) sensitive to X.
1
~
~
With   1, X =X. If   0.97, and r=3%,  =0.5, permanent X =0.5X. In practice,
  1 because the time series have usually unit root. With unit root,  will always be large
~
and thus estimates of X will be sensitive to the choice of X.
Ex: assume X is determined with a distributed lag function: we can use past average into
the future:
~
Et X  0 


X
t 1 i t i
This approach is inadequate since agents use a richer information set, which includes
contemporaneous variables and announcements/leading indicators predicting future
values.
Ahmed (1986, JME) uses UK data for 1908-1980, 1732-1913.
The null hypothesis: CA deficit happens during wars since households perceive an
increase in G as a temporary event.
Accordingly, permanent increases in G should have no effect on CA; but temporary
deviations of G from its permanent levels should cause CA deficit during wars. This is
one of the earlier attempts at modeling deviation from permanent G. But there is not I.
He finds support for the argument using the intertemporal model. For this, he runs the
trade balance on transitory changes of G and permanent changes of G:
~
~
TBt   0.21(Gt  G )  0.44Gt   t , R2=0.28, D.W.=2.32, SMPL=1908-80
( 0.05)
( 0.85)
The hypothesis cannot be rejected, so temporary component G has a significant negative
effect on TB (-0.21) whereas the permanent does not.
Problem with the early approach: the results are observationally equivalent to the result
one would get with a static Keynesian model, where the only significant term would be
G. It is not clear whether we are testing the intertemporal model or the assumptions of
the model.
Glick and Rogoff (1995) separate shocks into local and global and run country and pooled
regressions for 1961-90:
CAt  b0  b1 Atc  b2 AtW  b3 I t 1  b4T  b5 (Gtc  Et 1Gtc  Gtc )
where independent variables are: % change in country-specific and world productivities
(A), lagged real gross investment, time trend, and the term in G is the temporary
component of the changes in government spending follows an ARIMA process (same as
in Ahmed).
AW is a mean-GDP weighted average for the group of countries in the sample.
AC  A  AW , demeaned. The productivity terms are constructed as Solow residuals
(accounting for changes in labor and for capital, separately).
They also have a similar equation for I. So it is a system of equations determining CA
and I simultaneously.
Their model improves the ability of the intertemporal approach to explain actual CA.
2
They find that the CA responds negatively and significantly to country-specific shocks
but not to global shocks: b1, b5 significant, b2 not significant.
.
Second generation models: the intertemporal or PV models of CA
Sheffrin and Woo (1990, JIE) are the first to adapt the methodology developed by
Campbell and Shiller to predict stock prices (1987, JPE) to testing the intertemporal
current account (ICA). The idea is to use the information in past CA to predict the
permanent values of NO components. You take the prediction of the ICA and
superimpose over a VAR. If the intertemporal model is true, and we have perfect capital
mobility (more on this later), including the CA in the information set of the agents can
capture agents’ expectations of shocks to Y, I, G. The main difference between the PV
models and earlier ones is how one proxies private agents’ expectations of future net
output. This methodology assumes that past values of CA contain information useful to
construct estimates of agents’ expectations of future net outputs.
~
Recall CAt  (Zt  Et Zt ) where Z= output net of I, G. But Z is likely to be I(1) i.e., have
unit root, while CA is likely to be I(0) (stationary) in the long run (this is also shown to
be a condition for sustainability, see Trehan and Walsh, 1990, Wickens and Uctum,
1993). So following Sheffrin and Woo, we need to transform the RHS into I(0) variables,
such as first-differences:
s
 1 

 Et Z t  s where Z=Y-I-G=NO
s 1  1  r 

CAt  

Proof:
Define a lead operator L1 such that L1Et Z t  Et Z t 1 . Then,
~
r
Et Z t 
1 r
 1 


s t  1  r 


s t
Et Z s =
r
1 r
s
 1 
s

 Et L Z t
s 0  1  r 






r
1  s
r 
1
 
 Et Z t
L
E
Z
=
=


s 0
t t
1 r
1  r   1  1 
1 r 
1   1  r  L 
 
 
s
Thus, we can rewrite the CA as:



~
r 
1
 Et Z t .
CAt  (Z t  Et Z t ) = Z t 
1  r   1  1 
1   1  r  L 
 
 
Z (1  r )  rZ t
1 1  
1 1 
r
1

CAt 1 
L   1 
L Z t 
Et Z t  t

Et Z t 1
1

n
1

r
1

r
1

r
1

r

 

3
=

1
Z t  Et Z t 1  =  1 Et Z t 1  =  1 L1Et Z t
1 r
1 r
1 r
1
CAt  
s
 1  s

 L Et Z t  
s 1 1  r 

CAt  
1 1 
1 1 
1 1
L 1 
L  Et Z t = 
L
1 r
1

r
1

r




s
 1  s

 L Et Z t
s 0  1  r 


s
 1 

 Et Z t s
s 1  1  r 


(2)
This equation says that the current account is in deficit when the present discounted value
of future net output changes is positive. So, the contemporaneous CA is a predictor of
future increases in net output. Since the information set used by the econometrician
doesn’t have all the information available to private agents, we can use past values of CA
which contain information to construct agents’ expectations of future values of Z.
Assume that consumers’ forecast of Z s for s>t are based on a 1st order VAR model:
Z s  11 12  Z s 1  e1s 


    . From this, we can get the consumers’ forecasts, for

CAs  12  22  CAs 1  e2 s 
which they use current and lagged CA, current and lagged net output changes:
Z s  11 12 
Et 


CAs  12  22 
s t
Z t 
Z t 
s t 

  

CAt1 
CAt1 
To see why, consider the 1-variable case: X s  X s1  et . Solve it
forward: at s=t+1 X t 1  X t  et 1 .
At s=t+2, X t 2   2 X t  error terms . Thus, at s=t+n,
X t n   n X t  error terms  Et X s   st X t
In equations:
Et Z s  11Z t  12CAt
Et CAs  21Z t  22CAt
We can now get the forecast of CA by premultiplying the RHS with the 1x2 vector [0 1]
or the forecast of Z by premultiplying with the vector [1 0]:
s t
11 12  Z t 
ˆ
Et Zt  Zt  [1 0] 

  CA 

12
22 

(3)
t1 
We substitute this expression to calculate for the RHS of equation (2) and get the model’s
prediction of CA.
s t
  Z 

(similarly Et CAt  CAˆt  [0 1]  11 12   t  )


CA

12
22 

t1 
Defining   {ij } matrix and plugging the expression in (3) into equation (2)
4
CAˆ t  

 1 


s t 1  1  r 

s t
Z t 
 1 
[1 0] s t 

 =  [1 0] 
CA
 1 r 
 t

 1 


s t  1  r 

s t
Z t 
 s t 

CAt 
Z 
Z 
Z 
 [1 0] (1  r ) 1   t   [1 0] (1  r ) 2  2  t   [1 0] (1  r ) 3  3  t   ...
CAt 
CAt 
CAt 
 [1 0]
2
3
 Z 
 
    
  
  ...  t 
1 
 
1 r  1 r 1 r  1 r 
 CAt 

1
1
 1  
 Z t 
 [1 0] 
 
  I 

1  r  CAt 
1 r  
We thus get the intertemporal model’s prediction of the optimal CA:
CAˆ t  [Z
Z t 
Z 
CA ] 
  K  t
CA
 t
CAt 
(4)
The test consists of seeing whether the CAˆ at t can replicate the actual data at t, i.e., if the
coefficient vector K= [Z CA ]  [0 1] . This is not a regression but is calculated from
the VAR parameters. Thus, if the intertemporal approach is valid, then the theoretically
predicted value of K=[0 1] then CAˆ (obtained from model) = CA CAt (where CA is
observed). In other words, if agents have more information about future cash flows (ex., a
rise in G at t+k) then this information should be reflected in current CA (a CA surplus at
t, now). CA at t captures the consumer’s best estimates of the PV of future cash flow
changes.
Thus the methodology consists of:
(i)
Estimate a VAR (1st or higher-order) and get the K̂ .
(ii)
Calculate the ˆ Z , ˆ CA using the K̂ .
(iii)
Test if Z  0, CA  1 and compare CAˆ to CA.
Illustration: Ghosh article
He tests if Var (CAt )  Var (CAt* ) . If Var (CAt )  Var (CAt* ) then CA predicted by the model
didn’t vary enough to allow capital flows to smooth consumption as much as the actual
consumption.
Estimation results for unstructural VAR (1 lag) with US data :
NOt 1
CAt 1
NOt
-0.09
(1.0)
-0.17
CAt
-0.18
(2.3)
0.87
5
R2
DW
(3.7)
-0.12
2.01
(21.3)
0.81
2.14
In matrix notation:
NOt   0.09  0.18  NOt 1 
 CA   


t 
 0.17 0.87   CAt 1 

1
Z 
Calculate the optimal CA* given by the theory: CA*t  [1 0]  1  I  1    t 
1 r  
1  r  CAt 
Take 2%  r  6% , ex: r=6%



 NO
1
t
Then CA *  1 0 0.09 / 1.06  0.18 / 1.06 

t
 0.17 / 1.06 0.87 / 1.06   1 0  0.09 / 1.06  0.18 / 1.06   CA 


t 


   0.17 / 1.06 0.87 / 1.06  
0
1
 


 0.09  NOt 
 1
CAt *   0.08  0.16 


 0.31 0.56   CAt 
NOt 
CAt *  0.03  0.097 
  0.03NOt  0.097 CAt .
 CAt 
Compare CA* with the actual CA:
The traditional way of analyzing is to look at if 0.03 and 0.097 are significantly different
from zero. For the theory to be valid, we want not to reject the second hypothesis and
reject the first one. Ghosh compares the variances. He finds that Cov(CAt *, CAt )  0.99
Other Empirical Results
Overall the evidence is ambiguous. This is a joint test of consumption smoothing CA and
perfect capital mobility. It is not clear which of the assumptions is rejected.
Sheffrin and Woo (1990) conducted the analysis to test the null hypothesis using annual
data for four countries, annual data for 1955-85 with a second-order VAR. They find that
the theory works well for Belgium and Denmark but fails for Canada and UK.
Otto (1992, JIMF) used quarterly data for Canada and US. Bad results were repeated for
Canada and also for the US.
Ghosh (1995) does it for 5 countries with quarterly data, 1960-88. The theory is not
rejected in the case of US and Japan but rejected for UK, and Canada, He emphasizes the
volatility of the CA and concludes that the CA forecast based on theory is too smooth
6
compared to the actual CA that is more volatile. This is the same problem that Sheffrin
and Woo had with Belgium. If the actual CA is too volatile, this means that C is too
smooth compared with the model’s prediction.
The results are surprising because the evidence supports the theory for larger countries
but the theory is for small open economies that are assumed to be able to borrow as much
as they want in constant world rates. Perhaps relaxing the constant r assumption will
help.
Bergin and Sheffrin (2000, EJ) introduce time varying interest rates and relative price of
traded goods. Model: the consumer solves an intertemporal max problem and chooses
consumption and debt, subject to constant elasticity utility with NT and T goods, and the
BC. They derive a consumption-based real interest rate ( rC ) consisting of the change in
relative price of NT goods and world interest rate defined i.t.o traded goods (rc=i-inflT).
They use world real interest rate data and real exchange rate as proxy for relative price of
traded goods. They conduct the tests for Australia, UK and Canada (small open
economies).
s
1 
 
CAt*  s1 
 NOt s  rC ,t i , where  =intertemporal elasticity of substitution (the
 1 r 
consumption function is constant elasticity). Now the RHS also includes r. It says that if
the consumption-based r rises, CA falls because the consumer will smooth C below its
smoothed level.
The VAR becomes 3-variables:
NOt 


*
ˆ
CAt  K rC ,t 
CA 
 t 
where
 1 
0  
 

K   0   0   AI  A1
 
1  
 0 
. They choose the  that maximizes
the fit of the model to data, compute  from 1/(1+r) where r is the real interest rate using
G7 nominal rate net of CPI inflation rate.
They don’t reject the theory for Australia and Canada but do so for the UK. Every time,
including rC improves the results. But including the real exchange rate is more important
than the real interest rate alone to improve the fit of the model.
Nason and Rogers (2003, FRB IF DP) look at the question from the business cycle model
and explore why the optimal CA theory might be rejected. Potential reasons: the model
(i) ignores shocks to world interest rate; (ii) treats asymmetrically government spending
(temporary) and output shocks (permanent); (iii) ignores trading frictions in asset
markets.
It does not directly test the ICA theory but runs simulations to generate artificial CA
series.
7
Conclusion:
The ICA models are useful and give the correct concept of long term. Although the
earlier simple models are inconclusive, more recent work shows that extending the model
to incorporate other shocks improves the results.
Possible extensions:
 Add additional shocks: productivity shocks a la Glick and Rogoff, variable real
exchange rate (Dornbusch, JPE 1983: price of nontraded goods relative to traded
goods); these external shocks may affect CA more than the domestic shocks.
 Relax the assumption of Ricardian equivalence in the intertemporal model (e.g.
O-G model, or stochastic life times for individuals);
 Introduce liquidity constraints for countries; results seem to be working better for
larger countries than small, who may be having more difficulties in raising capital
from international markets.
 Use the IIP measures rather than the historic measures of the CA.
 Use a unified framework (country and date) to compare all the previous work
done on different countries.
This is an area of research that is still developing.
8