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University of York
Department of Economics
PhD Course 2006
VAR ANALYSIS IN MACROECONOMICS
Lecturer: Professor Mike Wickens
Lecture 5
Macroeconomic Fluctuations:
Keynesian, RBC and DSGE Models
Contents
1. Blanchard SVAR model
2. Blanchard and Quah VAR model
3. King, Plosser, Stock and Watson VMA model
4. RBC models and the VAR model
(i) General theoretical considerations
(ii) Cooley and Dwyer model
(iii) Using a VAR to calibrate a DSGE model
5. Fiscal policy using a VAR
(i) Fiscal shocks
(ii) Fiscal sustainablility
6. Open economy VAR models
(i) General theoretical considerations
0
(ii) A sticky-price monetary model with a �oating exchange rate
(iii) Garrett, Lee, Pesaran and Shin model
(iv) DSGE models
(v) The sustainability of the current account using a VAR
1
VARs have been widely used to study business cycle �uctuations in macroeconomics. We
consider some of the principal investigations.
The aim in most of these papers is either to uncover key features of the macroeconomy or, in
the case of RBC and DSGE models, to use the data to learn how best to formulate the model. The
latter often address the issue: if the predictions of these models are inconsistent with the data, or
require the model parameters to take implausible values, how can the models be re-formulated so
that its predictions match the data.?
1 Blanchard SVAR model (AER 1989)
Blanchard proposed a Keynesian model of the macroeconomy and used a structural VAR.
Model
AD : y = c12es + ed
OL : u = a21y + es
PS : p = a34w + a31y + c32es + ep
WS : w = a43p+ a42u+ c42es + ew
MR : m = a51y + a52u+ a53p+ a54w + em
where y =log GDP, u =unemployment rate, p =log price level, w =log wage rate,
m =log M1.
The AD equation is the aggregate demand function, OL is Okun�s law, the remaining equations
are the price, the wage and the money rule
2
(i) Structural VAR
Azt = A(L)zt�1 +Bxt + Cet
E(ete0t) = D; diagonal matrix
where z0t = (�yt; ut;�pt;�wt;�mt) and e0t = (ed; es; ep; ew; em):
Thus, where necessary, non-stationary variables are �rst di¤erenced to make them stationarity
(i.e. all except unemployment)
A =
2666666666666664
1 0 0 0 0
�a21 1 0 0 0
�a31 0 1 �a34 0
0 �a42 �a43 1 0
�a51 �a52 �a53 �a54 1
3777777777777775
C =
2666666666666664
1 c12 0 0 0
0 1 0 0 0
0 c32 1 0 0
0 c42 0 1 0
0 0 0 0 1
3777777777777775(ii) Reduced form VAR
zt = A�1A(L)zt�1 +A�1Bxt +A
�1Cet
= F (L)zt�1 +Gxt + vt
E(vtv0t) = �; unrestricted matrix
We note that in the SVAR there are two elements that break the recursive structure of the
model: a34 and c12:
3
Given c12 and either a34 or a43 the SVAR is just identi�ed. In the absence of this knowledge
it is necessary to use structural estimation and not OLS.
Main empirical conclusions
(i) relations between the real variables and relations between the nominal variables are stronger
than across real and nominal variables
(ii) Estimates are broadly consistent with the Keynesian model. The main exception is the
strong unemployment e¤ect (supply shock) in the aggregate demand function.
(iii) Demand shocks explain most of the short-run �uctuations in output, prices and wages
(iv) Supply shocks dominate in the medium to long term.
(v) The main problem is that unemployment seems to strongly Granger-cause output, instead
of the other way round.
4
2 Blanchard and Quah VAR model (AER 1989)
The aim here is to use long-run restrictions to identify the di¤erence between demand and supply
shocks. Blanchard and Quah postulate the following Keynesian model:
yt = mt � pt + �at
yt = nt + at
pt = wt � at
wt = wjfEt�1nt =_ng
ut =_n� nt
�mt = edt
�at = est
Notes:
(i) at = productivity shock
(ii) Wages are chosen one period in advance so as to achieve full employment.
(iii) The disturbances edt and est are iid with zero mean and constant variances.
The system can be re-written as
�yt = �edt + ��est + est
ut = �edt � �est
Thus, yt is I(1) and ut is I(0).
The model implies that demand shocks have no long run e¤ect on output, and the impulse
response function of output to demand shocks is zero for all lags- i.e. there is only a contempora-
neous e¤ect.
5
The model can be written as the stationary VMA in z0t = (�yt; ut) �I(0):
zt = C(L)et
= C(1)et + C1�et
e0t = (est; edt) � i:i:d(0;�)
C(1) =
2664 1 0
�1 ��
3775 ; C1 =
2664 1 �
0 0
3775As (�mt;�at)
0 = et,
yt = y0 +
�1 0
�C(1)� t +
�1 0
�C1et
= y0 + at + ��at +�mt
Thus, in the long run, yt is determined by a stochastic trend derived from the productivity
(supply) shock, but in the short run it is also a¤ected by the money (demand) shock.
Identi�cation
The issue is how to identify the shocks from knowledge of the VMA
zt = B(L)ut
Eutu0t = �
where � is unrestricted?
Proceeding as in Lecture 4 we can de�ne
1. K such that
et = K�1ut
Eete0t = K�10�K = I
6
2. C(L) such that
C(L) = B(L)K
Hence
�zt = B(L)KK�1ut
= C(L)et
We now have orthogonal shocks
K has n2 = 4 elements and the restriction K�10�K = I provides n(n+1)2 = 3 of these.
3. Now impose the long-run restriction that the demand shock has no long-run e¤ect on yt:
Thus
C(1)12 = 0
This implies that C(1) and K are de�ned to satisfy the restriction
C(1)12 = B(1)11K21 +B(1)12K22 = 0
This is the 4th restriction required to identify K:
Estimation
How do we estimate the VMA?
Answer we estimate the VAR
A(L)zt = ut
In principle we need to invert the VAR to get the VMA
zt = A(L)�1ut = B(L)ut
In practice we do NOT need to take the last step as we only need estimates of C(1) and � in
order to construct K, and we can use
C(1) = A(1)�1
7
Note: Blanchard and Quah do something slightly di¤erent.
Main empirical conclusions: (Figs 1 and 2)
(i) Output:
Supply shock: Output is permanently (and positively) a¤ected by a positive supply shock.
The full e¤ect takes about
2 years and is slow to take e¤ect.
Demand shock: output responds quickly but the e¤ect disappears after about 5 years.
(ii) Unemployment:
Supply shock: Unemployment increases in the short run, before falling after about 2 years.
The e¤ect disappears after about 4 years. B & Q interpret the behaviour in the short run to price
and wage rigidities: a productivity shock raises real wages and hence increases unemployment.
Could also argue that there is capital-labour substitution.
Demand shock: Causes unemployment to fall in the short run due to increased demand.
Comment
The evidence on the impulse response function of output to the demand shock is not strictly
consistent with the theoretical model, nor are the two IRFs for unemployment. One might expect
the supply shock to reduce unemployment even in the short run. The theoretical modelalso says
that the e¤ects should be instantaneous and not distributed over time. But this was only a stylised
model.
8
3 King, Plosser, Stock andWatson VMAmodel (AER 1991)
The aim here is to use a VMA and to identify the common stochastic trends using information
about long-run relations among the variables obtained from cointegrating regressions.
Model
Long-run relations
m� p = �yy � �ii
c� y = �1(i��p)
v � y = �2(i��p)
v =investment.
These equations were estimated by �dynamic�OLS (using 5 leads and lags of di¤erences in
each of the variables). This is a way of exploiting the super-consistency property of cointegrating
regressions to eliminate the e¤ects of dynamics. This is Stock and Watson�s recommendation but
is hardly ever used.
De�ne z0 = (y; c; v;m�p; i;�p) � I(1): Thus it is assumed that p;m � I(2) and m�p � I(1):
Identi�cation
Consider the VMA
�zt = �+ C(L)et
= �+ C(1)et + C�(L)�et
= �+ �0et + C�(L)�et
Eete0t = �
9
As there are 3 long-run relations and 6 non-stationary variables, there are 3 stochastic trends.
The problem now is to construct and �:
If, for example, we start with the CVAR
�zt = Azt�1 + et
= ��0zt�1 + et
and the associated VMA is
�zt = C(1)et + C�(L)�et
= �0et + C�(L)�et
Then using �� t = et we can write
zt = z0 + �0� t + C
�(L)et
Since �0zt � I(0);
�0zt = �0z0 + (�0 )�0� t + �
0C�(L)et � I(0)
But this can only be true if the term in � t is eliminated as it is I(1): Thus should be
constructed so that
�0 = 0
i.e. must be orthogonal to �0:
Although the matrix � must then be chosen so that 0� = C(1); this does not uniquely de�ne
�:
The matrix de�ned by KPSW satis�es the above requirement and can be obtained from the
(assumed) known (or pre-estimated) cointegrating relations. KPSW choose � to give a recursive
10
structure to the stochastic trends. Thus
=
26666666666666666664
1 0 0
1 0 �1
1 0 �2
�y ��i ��i
0 1 1
0 1 0
37777777777777777775
; � =
26666666666666666664
1 �12 �13
0 1 �23
0 0 1
0 0 0
0 0 0
0 0 0
37777777777777777775 is known, but � require estimation.
Note that if we let �� t = et then the stochastic trends �t = �0� t are formed from cumulating
the �rst three elements of et: Hence from ��t = �0�� t = �0et
��1t = e1t
��2t = �12e1t + e2t
��3t = �13e1t + �23e2t + e3t
and the stochastic trends have a recursive structure.
It is also assumed that � can be partitioned conformably with e0t = (e1t; e2t; e3tje4t; e5t; e6t)
� =
2664 �11 0
0 �22
3775 �11 is diagonal
Thus, as part of the recursive structure, the stochastic trends are constrained to be uncorre-
lated.
11
It follows that the long-run structure of the model is
yt = �1t
�pt = �12�1t + �2t
it = �13�1t + �23�2t + �3t
ct = yt + �1(it ��pt)
vt = yt + �2(it ��pt)
mt � pt = �yyt � �iit
Hence
�1 is the �real balanced growth�shock - i.e. output (supply?) shock
�2 is the �neutral in�ation�shock - i.e. price (nominal?) shock
�3 is the �real interest rate�shock - demand shock?
Main empirical conclusions (Fig 4)
(i) The output shock has a large permanent e¤ect on output, consumption and investment. In
the case of investment this is initially negative.
(ii) The in�ation shock has little e¤ect on output and consumption. Investment is a¤ected in
the short run but not the long run.
(iii) The real interest rate shock has no long run e¤ects, but in the short run all three variables
increase.
(iv) KPSW conclude that US data are not consistent with the view that a single permanent
shock is the dominant source of business-cycle �uctuations.
(v) and that contrary to Monetarist thought in�ation shocks have little e¤ect on real variables
even in the short run.
Comment on the KPSW model
12
It is not clear that the restrictions imposed are consistent with the data. In particular, is it
really correct to restrict � in this way?
Levtchenkova, Pagan and Robertson (J. of Econ Surveys, 1998) argue that the shocks do not
have the structure assumed by KPSW. They suggest that a better way to proceed would be
construct and � by directly by exploiting the long-run structural information.
13
4 RBC and VAR models
4.1 Ellen McGrattan (Fed Res. Bank of Minn QR 1994)
She summarizes the typical RBC methodology and its �ndings to that date very clearly.
The original Kydland-Prescott analysis attempts to explain business cycles as being due to
technology shocks. The KP model can account for some features of the data but not others.
Features captured:
(i) cyclical variability of output
(ii) the lower variability of consumption than income
(iii) higher variability of investment
Features not captured:
(i) the variability of consumption is too low
(ii) the variability of hours worked is too low
(iii) the variability of investment is too low
(iv) the correlation between hours worked and productivity is far too high.
See Charts 1-3 and Table 1
Hansen (1985) tries to improve the performance of the model by altering the labour supply
structure. He argues that employment is indivisible: individuals work either a set number of hours
or none. This enables the model to better capture the variability of hours worked, but still has a
too high correlation between hours and productivity. (see Table 1).
McGrattan argues that the standard RBC model omits �scal shocks. Her results suggest that in
response to changes in tax rates to �nance government expenditures households substitute between
taxable and non-taxable activities thereby altering the variability of consumption, investment,
14
hours worked and productivity. The result is a lower correlation of hours worked and productivity.
(see Tables 2 and 3).
(i) General theoretical issues
There is a huge literature on RBCmodels. Notable papers are Prescott (FRBMQR 1986), King,
Plosser & Rebelo (JME 1988), Kydland and Prescott (FRBMQR 1990), McGrattan (FRBMQR
1994), Campbell (JME 1994), Cooley and Dwyer (J.Ectcs 1998).
The classical problem is that the economy is seeking to choose consumption C; labour L and
capital K to maximise
Et
1Xs=0
�sU(Ct+s)
where � = 11+� subject to a budget constraint (the national income identity), a production function
and a capital accumulation equation:
Yt = Ct + It
Yt = F (At;Kt; Lt)
�Kt+1 = It � �Kt
or, just to the combined constraint,
F (At;Kt; Lt) = Kt+1 + Ct � (1� �)Kt
The main agenda of the RBC literature became to show how it is that productivity shocks
(i.e. shocks to At) are the main cause of business cycle �uctuations.
Rather than work in such generality it is convenient to consider speci�c functional forms. We
assume a Cobb-Douglas production
Yt = AtK�t L
1��t
15
with labour growing at the constant rate n, implying that
Lt = (1 + n)tL0
and where log productivity is a random walk about a constant rate of long-run growth �
At = (1 + �)tZt
lnZt = zt; �zt = et � i:i:d(0; !2)
The only shock to this economy is et and this gets propagated throughout the economy.
In order to obtain the solution to the model, �rst we re-de�ne all variables in terms of deviations
about their long-run growth paths. In equilibrium, the balanced growth rate for the economy is
� = n+ �1�� .
We can treat technical progress in this model as labour augmenting, hence we re-write the
production function as
yt = Ztk�t
where
yt =Yt
L#t=
Yt
[(1 + �)1
1�� ]tLt=
Yt(1 + �)tL0
kt =Kt
L#t=
Kt
[(1 + �)1
1�� ]tLt
=Kt
(1 + �)tL0
L#t = (1 + �)t
1��Lt = [(1 + �)1
1��]t(1 + n)tL0 = (1 + �)
tL0
where we have used the approximation
[(1 + �)1
1��]t(1 + n)t ' (1 + �)t
� ' n+�
1� �
The national income identity is now
yt = ct + it
16
where
ct =Ct
L#t=
Ct(1 + �)tL0
it =It
L#t=
It(1 + �)tL0
and the capital accumulation equation can be written as
�Kt+1 = It � �Kt
Kt+1
L#t+1
L#t+1
L#t=
It
L#t+ (1� �)Kt
L#t
(1 + �)kt+1 = it + (1� �)kt
sinceL#t+1
L#t= 1 + �:
Finally, we assume that the utility function is
U(Ct) =Ct
1��
1� �
=[(1 + �)tct]
1��
1� �
The problem facing the economy can now be formulated in terms of maximising the value function
Vt = U(Ct) + �Et(Vt+1)
=[(1 + �)tct]
1��
1� � + �Et(Vt+1)
subject to
Ztk�t = ct + (1 + �)kt+1 � (1� �)kt
The �rst-order conditions for this stochastic dynamic programming problem are
@Vt@ct
= (1 + �)(1��)tc��t + �Et[@Vt+1@ct+1
� @ct+1@ct
] = 0
17
We note that
Vt+1 = U(Ct+1) + �Et+1(Vt+2)
hence,
@Vt+1@ct+1
= (1 + �)(1��)(t+1)c��t+1
To evaluate @ct+1@ct
we re-express it as
@ct+1@ct
=
@ct+1@kt+1@ct@kt+1
and then use the budget constraints for periods t and t+1 to evaluate numerator and denominator.
This gives
@ct+1@ct
=�Zt+1k
��1t+1 + 1� �
�(1 + �)
Hence
@Vt@ct
= (1 + �)(1��)tc��t � �Et[(1 + �)(1��)(t+1)c��t+1 ��Zt+1k
��1t+1 + 1� �1 + �
]
= (1 + �)(1��)tfc��t � �Et[(1 + �)��c��t+1(�Zt+1k��1t+1 + 1� �)]g = 0
implying the Euler equation
Et
��[(1 + �)
ct+1ct]��(�Zt+1k
��1t+1 + 1� �)
�= 1
In evaluating the Euler equation we should take account of the conditional covariance terms
involving ct+1; kt+1 and Zt+1 - particularly covt(ct+1; kt+1) because we can ignore those involving
Zt+1 as it is predicted to be Zt in every future period - but instead, for convenience, we shall omit
them. This is equivalent to assuming certainty equivalence or, in a decentralised economy, that
there is a real risk-free interest rate.
Steady-state solution
In equilibrium �ct+1 = �kt+1 = 0 and Zt = 1 for each time period. Hence we can drop the
time subscript in the steady state to obtain
�(1 + �)��[�k��1 + 1� �] = 1
18
implying that in equilibrium
k '�� + � + �(n+ �
1�� )
�
� �11��
Although kt is constant in equilibrium, the capital stock per capita of the economy, Kt
Lt, is growing
through time. As kt = Kt
[(1+�)1
1�� ]tLt; the optimal path for capital per capita is
(� + � + �(n+ �
1�a )
�)�11�� [(1 + �)
11�� ]t
and hence grows at approximately the rate �1�� :
Optimal per capita growth rates
The optimal growth rate of per capita output YtLt is determined from that of yt =Yt
[(1+�)1
1�� ]tLt
.
As yt = k�t and �kt+1 = 0; it follows that �yt+1 = 0 too. Thus, the growth rate of YtLtis also
approximately �1�� : The optimal growth rate of consumption per capita
CtLt
can be obtained
from the condition that �ct+1 = 0 and ct = Ct
[(1+�)1
1�� ]tLt
: Thus the growth rate of CtLt is also
approximately �1�� :
Optimal total growth rates
The optimal growth rates of total output, total capital and total consumption are obtained by
taking into account population growth. Adding on the growth rate of the population, we obtain
their common rate of growth � = n+ �1�� . As the growth rates of output, capital and consumption
are the same, the optimal solution is a balanced growth path.
Short-run dynamics
We consider short-run deviations about the logarithm of the growth path. The model reduces
to two equations: the Euler equation and the budget constraint. We need to take logarithmic
approximations of each based on the Taylor series approximation
f(xt) ' f(x�t ) + f0(x�t )
�@xt@ lnxt
�x�t
[lnxt � lnx�t ]
' f(x�t ) + f0(x�t )x
�t [lnxt � lnx�t ]
19
Leaving out the intercept, and using certainty equivalence in which Et lnZt+1 = lnZt = zt;
the Euler equation can be shown to approximately satisfy
Et� ln ct+1 = �(� +� + �
�)(1� �)Et ln kt+1 + (� +
� + �
�)zt
The log-linearized budget constraint is
ln kt+1 = �[� + �(� � 2)] ln ct + (1 + � + ��) ln kt
+� + � + (� � �)�
�zt
These two equations can be written as the system2664 1 + � + �� �[� + �(� � 2)]
0 1
37752664 ln kt
ln ct
3775 =
2664 1 0
(� + �+�� )(1� �) 1
3775Et2664 ln kt+1
ln ct+1
3775
�
2664 �+�+(���)��
� + �+��
3775 ztWe note that each equation involves zt which is I(1): This implies that neither the linearized
Euler equation nor the budget constraint give a cointegrating relation. This is despite the fact
that along the steady-state growth path the ratio of consumption to capital is constant. Although
the ratio of consumption to capital is expected to be constant, in practice, future shocks cause it
to be non-stationary.
The general form of the solution is
Bxt = CEtxt+1 +Dzt
xt = B�1CEtxt+1 +B�1zt
The roots of the matrix A = B�1C determine the type of solution: roots < 1 in absolute value
are stable, and those greater than 1 are unstable. These can be obtained from the determinental
equation
jAj � (trA)L+ L2 = 0
20
In this example there are two roots. If we set L = 1 in the equation then there are three cases
(i) jAj � (trA) + 1 > 0 implies both roots are either stable or unstable.
(ii) jAj � (trA) + 1 < 0 implies a saddlepath solution (one root is stable and the other is
unstable).
(iii) jAj � (trA) + 1 = 0 then at least one root is 1:
It can be shown that
jAj � (trA) + 1 = �[� + �(� � 2)](� + �+�
� )(1� �)1 + � + ��
< 0
if � � 2; a su¢ cient but not necessary condition. The short-run dynamics about the steady-state
growth path therefore follow a saddlepath.
Solving the unstable root forwards and noting that Etzt+s = zt implies that the general solution
is the VAR
xt = Qxt�1 +Rzt
where the error term is I(1). Since the vector xt is not cointegrated, we must take �rst di¤erences.
This gives the stationary VAR
�xt = Q�xt�1 +Ret
or, in terms of the original variables,2664 � ln Kt
Lt
� ln CtLt
3775 = constant + Q
2664 � ln Kt�1Lt�1
� ln Ct�1Lt�1
3775+RetIt is important to note that there is only one disturbance term in this VAR and this drives
both ln Kt
Ltand ln CtLt . This is because there is only one shock in the orginal model, a productivity
shock. This is characteristic of RBC models. As the joint distribution of ln Kt
Ltand ln CtLt is not
degenerate, there is obviously a need to identify and introduce an additional shock. Sometimes
this is assumed to be a shock to U(Ct); i.e. a preference shock.
21
In the literature it is common to assume that the productivity shock is a stationary autore-
gression, eg Campbell (1994). This raises the issue of what makes real variables non-stationary.
It seems likely that productivity and money supply shocks are two key sources of real and
nominal non-stationarities.
22
4.2 Cooley and Dwyer (J. of Ectcs 1998)
C&D compare the Blanchard and Quah (1989) results, which are widely regarded as issuing a
strong challenge to RBC theory despite the inconsistencies between the theory and the evidence,
with three RBC/DSGE models.
Their argument is that the B&Q methodology is based on atheoretical assumptions and the
B&Q results are not capable of being produced by RBC/DSGE models based on �a lot of economic
theory�.
They consider three di¤erent versions of the RBC/DSGE model.
(a) Their second model has one shock: a technology shock.
It is an RBC model in which the aim is to maximise
E��t(ln ct �Ah t )
subject to
ct + it � yt
kt+1 = it + (1� �)kt
yt = k�t (eatht)
1��
�at = g + �(L)"t
�(L) = �0 + �1L+ �2L2 + �1L
4 + �0L4
�0 < �1 < �2; �(1) = 1
This model is calibrated.
Data is generated from the model for di¤erent samples of random errors.
Impulse response functions are calculated and the average IRF is plotted in Fig 4.
23
This is compared with the IRF based on using the B&Q procedure for actual data.
The calibrated IRFs are remarkably similar to the B&Q IRFs even to the extent of producing
an even larger sized demand shock when in theory none exists.
This raises doubts about what the B&Q method is really measuring.
(b) The third model has two shocks: a technology and a preference shock. The model is
E��t(ln ct �Ah t ez1t)
z1t = �1(L)"1t
subject to
ct + it � yt
kt+1 = it + (1� �)kt
yt = ez2tk1��t h�t
�z2t = �2(L)"2t
where the roots of �1(L) and �2(L) are assumed to lie outside the unit circle.
The solution takes the form
lnht = �11z1t + �12z2t + �13 ln kt
ln yt = �21z1t + �22z2t + �23 ln kt
ln kt+1 = �31z1t + �32z2t + �33 ln kt
It is possible to eliminate capital from the model using the last equation which can be written
ln kt =�31z1t�1 + �32z2t�1
(1� �33L)
24
to obtain a VAR in � lnht and � ln yt :
� lnht = �33� lnht + �11�z1t + (�13�31 � �11�33)�z1t�1
+�12�z2t + (�13�32 � �12�33)�z2t�1
� ln yt = �33� ln yt + �21�z1t + (�23�31 � �21�33)�z1t�1
+�22�z2t + (�23�32 � �22�33)�z2t�1
It is in �rst di¤erences because the technology shock z2t is I(1).
The model can also be written
(1� �33L)
2664 � lnht
� ln yt
3775 = B(L)
2664 �"1t
"2t
3775This may be compared with the equivalent B&Q model (using hours instead of unemployment)
A(L)
2664 lnht
� ln yt
3775 =2664 "dt
"st
3775This shows that
(i) the equation in lnht involves an I(1) variable and hence lnht must be I(1) and not I(0) as
in B&Q:
(ii) the model has a VARMA structure and not a VAR structure
This will a¤ect the estimated IRFs from the B&Q model which will be biased.
(iii) the shocks "dt and "st do not match the distinction found in the theoretical model which
implies that the shocks in the VAR are functions of both technology and preference shocks.
(iv) the theoretical model implies that both "dt and "st have permanent e¤ects on both vari-
ables.
(v) the true technology shock has a permanent e¤ect on lnht:
(vi) the preference shock does not have a permanent e¤ect on ln yt - as in B&Q - but this does
not require any restrictions of the sort imposed by B&Q to produce it.
25
4.3 Using a VAR to calibrate a DSGE model
Rotemberg and Woodford (NBER Macroeconomics Annual 1997), Christiano, Eichenbaum and
Evans (NBER 2001) and Smets and Wouter (ECB WP 2001)
1. formulate a DSGE model
2. Construct and estimate a VAR based on the variables in the DSGE
3. Calibrate the DSGE to the impulse response functions obtained from the VAR
They can then carry out policy analysis using the calibrated DSGE model.
The calibration chooses the DSGE parameters � to minimise the distance function
[vec(IRFV AR)� vec(IRFDSGEM (�))]0W�1[vec(IRFV AR)� vec(IRFDSGEM (�))]
where W is a diagonal weighting matrix of the di¤erent IRFs and time periods that has the
variances of the VAR IRFs on the leading diagonal.
26
5 Fiscal policy using a VAR
We consider two issues
(a) The e¤ects of �scal shocks (government spending and taxation shocks) on GDP
(b) The sustainability of the current �scal stance.
5.1 Fiscal shocks
We consider the paper by Blanchard and Perotti (1999)
Their basic model is an SVAR in three variables taxes, government spending and GDP.
A(L)zt = et
where
z0t = (lnTt; lnGt; lnYt).
- all variables in the VAR are logarithms, real and in per capita terms.
- all are I(1) variables.
- the data are either de-trended
- or cointegration is assumed between lnTt and lnGt
- the cointegrating relation is assumed to be lnTt � lnGt
- the data are quarterly
It is assumed that "0t = ("Tt ; "
Gt ; "
Yt ) are uncorrelated tax, spending and output shocks.
The identi�cation scheme relating "t to et is de�ned as266666641 0 �a1
0 1 b1
�c1 �c2 1
37777775
26666664eTt
eGt
eYt
37777775 =266666641 a2 0
b2 1 0
0 0 1
37777775
26666664"Tt
"Gt
"Yt
3777777527
All of the parameters are assumed known and replaced by prior estimates. It is assumed that
b1 = 0 and either a2 = 0 or b2 = 0.
If, for example, b2 = 0 then
"t =
266666641 a2 0
0 1 0
0 0 1
37777775
�1 266666641 0 �a1
0 1 0
�c1 �c2 1
37777775 et
=
266666641 �a2 �a1
0 1 0
�c1 �c2 1
37777775 et= Qet
As a result
A(L)zt = et = Q�1"t
QA(L)zt = "t
A�(L)zt = "t
In other words, we can re-write the original VAR as a new VAR in zt which has uncorrelated
errors and a di¤erent dynamic structure. It is now straightforward to obtain the impulse response
functions.
Main �ndings
(i) The spending shock "Gt has a strong positive e¤ect on both GDP and taxes
(ii) The tax shock "Tt has a strong negative e¤ect on GDP but does not a¤ect spending.
5.2 Fiscal sustainability
Fiscal sustainablity concerns the evolution of btyt and whether it remains �nite or explodes.
28
The �scal stance is said to be sustainable if btyt is �nite and if �nancial markets are willing to
hold the level of debt that emerges. The current �scal de�cit can then be sustained inde�nitely
without the government incurring unsupportable debt obligations.
If the current �scal stance is not sustainable, then a change of policy will be required at some
point to make it sustainable.
Univariate and multivariate models, as well as cointegration analysis, have been used to try to
determine �scal sustainability.
The key condition that lies at the centre of this analysis is the present-value (or inter-temporal)
government budget constraint (PVGBC). Fiscal policy is said to be sustainable if the PVGBC is
satis�ed. This has been tested in a number of ways.
5.2.1 The government budget constraint (GBC)
Three ways of writing the GBC
1. The nominal GBC
Ptgt + (1 +Rt)Bt�1 = Bt +�Mt + PtTt
gt is real government expenditure including real transfers to households
Tt is total real taxes
Mt is the stock of outside nominal, non-interest bearing money at the start of period t
Bt is the nominal value of government bonds issued by the end of period t
Rt is the average interest rate on bonds issued at the end of period t� 1
RtBt�1 is total interest payments made in period t
29
2. The real GBC
gt + (1 + rt)bt�1 = Tt + bt +mt �1
1 + �tmt�1
�t =�PtPt�1
is the rate of in�ation
bt is the real stock of government debt
mt is the real stock of money
rt is the real rate of interest de�ned by
1 + rt =1 +Rt1 + �t
rt ' Rt � �t
3. The real GBC as a proportion of GDP
gtyt+
1 +Rt(1 + �t)(1 + t)
bt�1yt�1
=Ttyt+btyt+mt
yt� 1
(1 + �t)(1 + t)
mt�1yt�1
yt is real GDP
Ptyt is nominal GDP
t is the rate of growth of GDP
Ttytis the average tax rate
5.2.2 Four de�cits
1. Total nominal government de�cit (or public sector borrowing requirement, PSBR)
PtDt = Ptgt +RtBt�1 � PtTt ��Mt
2. The real government de�cit as a proportion of GDP Dt
yt
Dt
yt=
gtyt+
Rt(1 + �t)(1 + t)
bt�1yt�1
� Ttyt� mt
yt+
1
(1 + �t)(1 + t)
mt�1yt�1
=btyt� 1
(1 + �t)(1 + t)
bt�1yt�1
30
The right-hand side shows the net borrowing required to fund the de�cit expressed as a pro-
portion of GDP.
3. Nominal primary de�cit Ptdt (the total de�cit less debt interest payments)
Ptdt = PtDt �RtBt�1
4. The ratio of the primary de�cit to GDP dtyt
dtyt
=Dt
yt� Rt(1 + �t)(1 + t)
bt�1yt�1
=gtyt� Ttyt� mt
yt+
1
(1 + �t)(1 + t)
mt�1yt�1
=btyt� 1 +Rt(1 + �t)(1 + t)
bt�1yt�1
5.2.3 The evolution of the debt-GDP ratio
(i) In terms of the primary de�cit
This is described by a non-linear di¤erence equation that may be stable or unstable.
btyt= (1 + �t)
bt�1yt�1
+dtyt
where �t is real interest rate adjusted for economic growth
1 + �t =1 +Rt
(1 + �t)(1 + t)
�t ' Rt � �t � t = rt � t
(ii) In terms of the total de�cit
Concerned with the evolution of btyt over timecan also be written in terms of the total de�cit
btyt=
1
(1 + �t)(1 + t)
bt�1yt�1
+Dt
yt
For positive in�ation and growth this is a stable di¤erence equation
31
5.2.4 Existing tests
The literature on �scal sustainability distinguishes between two cases concerning the discount rate
�t
(i) �t (and hence Rt;�t and t) constant
(ii) �t time varying.
Constant discount rate If
1 + � =1 +R
(1 + �)(1 + )
� ' R� � �
btytevolves according to the di¤erence equation
btyt= (1 + �)
bt�1yt�1
+dtyt
Case1: � < 0 (stable case)
Have
1 +R
(1 + �)(1 + )< 1
As btyt evolves according to a stable di¤erence equation it can be solved backwards by successive
substitution
Et(bt+nyt+n
) = (1 + �)n btyt+n�1Xs=0
(1 + �)n�s
Et(dt+syt+s
)
Conditions for �scal sustainability
limn!1
(1 + �)n btyt
= 0
limn!1
Et(bt+nyt+n
) = limn!1
nXs=1
(1 + �)n�s
Et(dt+syt+s
)
32
The evolution of the debt-GDP ratio depends on that of dtyt
Suppose that dtyt may be stochastic but is expected to grow at the rate �, then
Et(dt+syt+s
) = (1 + �)s dtyt
and
limn!1
Et(bt+nyt+n
) = limn!1
nXs=1
(1 + �)n�s
(1 + �)s dtyt
= limn!1
(1 + �)
�(1 + �)n � (1 + �)n
�� �
�dtyt
Hence
(i) If �; � < 0
limn!1
Et(bt+nyt+n
) = 0
(ii) If � = 0
limn!1
Et(bt+nyt+n
) = �1�
dtyt
(iii) If � > 0
limn!1
Et(bt+nyt+n
)!1
Conclusions
1. The debt-GDP ratio will remain �nite and positive if the ratio of the primary surplus to
GDP (�dtyt) does not explode
2. If � < 0 then dtytis a stationary I(0) process and the expected, or long-run value of the
debt-GDP ratio is zero thereby satisfying �scal sustainability
3. If � = 0 then
(i) dtyt is a non-stationary I(1) process, and hencebtytwill also be I(1).
(ii) btytand dt
ytwill be cointegrated with cointegrating vector (1; 1� )
33
(iii) Fiscal policy is therefore sustainable provided btytdoes not grow over time
i.e. there is no drift.
Case 2: � > 0 (unstable case)
This is the usual case that is considered in the literature.
We now have
0 <(1 + �)(1 + )
1 +R< 1
Hence btytevolves as an unstable di¤erence equation.
This must be solved forwards, not backwards as follows:
btyt
=1
1 + �Et(
bt+1yt+1
� dt+1yt+1
)
= (1 + �)�n
Et(bt+nyt+n
)�nXs=1
(1 + �)�sEt(
dt+syt+s
)
Conditions for �scal sustainability
limn!1
(1 + �)�n
Et(bt+nyt+n
) = 0
btyt
=1Xs=1
(1 + �)�sEt(
�dt+syt+s
)
The RHS is the expected present value of current and future primary surpluses expressed as a
proportion of GDP
This condition implies that current and future surpluses will be su¢ cient to pay-o¤ current
debt.
If dtyt is expected to evolve as before
btyt
=1Xs=1
(1 + �)�s(1 + �)s(
�dtyt)
=1 + �
�� � (�dtyt) if � 1 < � < �; � > 0
34
Provided that the current level of the debt-GDP ratio does not exceed the right-hand side,
�scal policy is sustainable and the debt-GDP ratio will grow at the rate �, the same rate as �dtyt.
If �dtyt is stationary then �1 < � < 0 and btytwill also be stationary, thereby satifsying �scal
sustainability.
If � = 0, so that �dtyt is I(1) then we obtain the same condition as in the stable case, namely
btyt=1
�(�dtyt)
Implies that btytwill be I(1) and cointegrated with �dt
yt.
Comments on the rationale for existing empirical tests for �scal
sustainability
1. Hamilton and Flavin (1986)
Test the transversality condition by testing H0 : A0 = 0 in
btyt= A0 (1 + �)
�t �1Xs=1
(1 + �)�sEt(
dt+syt+s
)
Note H&F use real debt and the real primary de�cit
2. Trehan and Walsh (1988) and Hakkio and Rush (1991)
Use a cointegration test for �scal sustainability
If btyt anddtytor (bt and dt) have unit roots and are cointegrated with cointegrating vector (�; 1)
then �scal policy is sustainable
Or, if government expenditures and revenues are I(1), then the cointegrating vector with debt
must be (�; 1;�1).
Note: if the cointegrating relation between debt and the primary de�cit is
dtyt+ �
btyt= ut
35
where ut is I(0) then
(1 + �)btyt= (1 + �)
bt�1yt�1
+ ut
and btythas a unit root i¤ � = �.
Time-varying discount rate Assuming that the compound discount rate over an s-period
horizon satis�es
�t;s = �si=1
1
1 + �t+i� 1 for all s � 1
we solve the GBC forwards to obtain
btyt= Et[(�
ns=1
1
1 + �t+s)bt+nyt+n
]� Et[nXs=1
(�si=11
1 + �t+i)dt+syt+s
]
Conditions for �scal sustainability
limn!1
Et[(�ns=1
1
1 + �t+s)bt+nyt+n
] = 0
btyt
= Et[1Xs=1
(�si=11
1 + �t+i)(�dt+syt+s
)]
Again implies the present value of current and future primary surpluses must be su¢ cient
to o¤set current debt liabilities with the di¤erence that the discount rate is compounded from
time-varying rates.
Tests for �scal sustainability
Use an alternative representation of GBC
De�ne
xt = �t;nbtyt
zt = �t;ndtyt
36
The GBC is now
�xt = zt
Conditions for �scal sustainability
limn!1
Et(xt+n) = 0
xt = � limn!1
Et[nXs=1
zt+s]
1. Wilcox (1989)
Shows that �scal sustainability is satis�ed if xt is a zero-mean stationary process
2. Wickens and Uctum (2000)
Prove a more general and useful result that does not require xt to be stationary. They show
that �scal sustainability is satis�ed if zt is a zero-mean stationary process. It then follows that xt
will be an I(1) process.
3. Trehan and Walsh (1991)
Argue that �scal policy is sustainable with a variable discount rate if the total de�cit is sta-
tionary.
It can be shown that this result follows directly from the representation
btyt=
1
(1 + �t)(1 + t)
bt�1yt�1
+Dt
yt
as for �t; t > 0 this is a stable di¤erence equation implying thatbtytis �nite (and stationary)
if Dt+s
yt+sis stationary.
5.3 Stability and Growth Pact (SGP)
Maastricht conditions
37
btyt< 0:6 and
Dt
yt< 0:03
For constant �t and t and constantbtytand Dt
ytat these maximum values
b
y=
(1 + �)(1 + )
(1 + �)(1 + )� 1D
y
' 1
� +
D
y
Hence
� + 'Dy
by
=0:03
0:6� 5%
Some implications
1. The nominal rate of growth must not be less than 5%
2. If nominal growth were lower than 5% then debt would rise above 0:6 even if the de�cit
limit were satis�ed. But this may still be sustainable.
3. If the de�cit were above 3% then the sustainability of debt at 60% simply requires nominal
growth to be the appropriate number greater than 5%.
5.4 Assessment
1. These measures of �scal sustainability are of limited practicality
- it is necessary to forecast future de�cits, in�ation, growth and interest rates in order to
compute the present value of expected future de�cits
- may sometimes be possible to use o¢ cial forecasts as in Wickens and Uctum (2000)
- or may need to construct forecasts
38
- could use a structural economic model but this has the disadvantage of embodying prior
information that may prove contentious and di¢ cult for outsiders to replicate
- could use a VAR. This has the merit of being easily understood and replicable.
2. The time horizon in these tests is so distant that the tests provide an ine¤ective
constraint on �scal policy in the short run
- therefore follow Uctum and Wickens (2000) and examine �scal sustainability over a �nite
time horizon.
3. A government running persistent, and even large de�cits, may simply claim
that they
expect, or will generate, o¤setting surpluses at some point in the future
- it is therefore desirable to be able to evaluate �scal sustainability under alternative policies
- problem is how to do this.
4. IGBC is a non-linear function of the policy instruments interest rates and taxes
- therefore recast the analysis of �scal sustainability using a log-linear approximation
5. Need a measure of the departure from �scal sustainability instead of a test
statistic
- construct an index based on the ratio of the present value of the forecast �scal surplus to
the current level of debt.
Taken together, we believe these changes to standard practice constitute a considerable ad-
vance. Moreover, as the whole procedure can easily by automated, it has the potential to become
a standard desciptive statistic for the �scal stance.
39
5.5 A log-linear approach to �scal sustainability
5.5.1 The log-linearized GBC
As the primary de�cit can take negative values, it is necessary to write the GBC in terms of total
expenditures gt and total revenues vt both of which are strictly positive
btyt=gtyt� vtyt+ (1 + �t)bt�1
where
vtyt=Ttyt+mt
yt� 1
(1 + �t)(1 + t)
mt�1yt�1
The steady-state solution to the GBC
�b
y= �g
y+v
y
If h(xt) = exp[lnxt] then a �rst-order Taylor series approximation about lnx is
h(xt) = x[1 + (lnxt � lnx)]
De�ning
f(xt) = exp [lnbtyt]� exp [ln gt
yt]+ exp [ln
vtyt]� exp [ln (1 + �t) + ln
bt�1yt�1
] = 0
a log-linear approximation to the GBC is given by
lnbtyt
' c+g
blngtyt� v
blnvtyt+ (1 + �) ln(1 + �t) + (1 + �)ln
bt�1yt�1
c = �� ln by� g
blng
y+v
blnv
y� (1 + �) ln(1 + �)
Assuming that � > 0, we solve the equation forwards to obtain
lnbtyt
= (1 + �)�n
Et(lnbt+nyt+n
) +
nXs=1
(1 + �)�sEt(kt+s)
kt = �c� g
blngtyt+v
blnvtyt� (1 + �) ln(1 + �t)
40
kt is the logarithmic equivalent of the primary surplus
The conditions for �scal sustainability
limn!1
(1 + �)�n
Et(lnbt+nyt+n
) = 0
lnbtyt
=
1Xs=1
(1 + �)�sEt(kt+s)
Hence
- ln btyt(and hence bt
yt) remains �nite and stationary if kt is stationary.
- This could happen if each term of kt is stationary, or I(1) but cointegrated with the
approporiate contegrating vector
- Or, if kt and each component of kt are I(1) then, if they also cointegrated with cointegrating
vector given by the coe¢ cients in the de�nition of c, then �scal sustainability is still satis�ed
5.5.2 An index of sustainability for a �nite time horizon
For a �nite time horizon we need a di¤erent approach to �scal sustainability from one that depends
on considerations of stationarity and non-stationarity
We propose an index of sustainability that is a generalization of that proposed by Buiter (1985)
and Blanchard (1990)
Their indices are based on a comparison of the current debt-GDP ratio and that n periods
ahead with given �xed values of the de�cit and discount rate.
Our generalization allows the de�cit and discount rate to be time-varying and endogenous, and
the target level of the debt-GDP ratio to be a choice variable.
The IGBC for a �nite time horizon of n periods shows what level of bt+nyt+nis expected to occur
given btytand future values of kt
41
If we replace Et[lnbt+nyt+n
] by a target level ln( bt+nyt+n)� we obtain
lnbtyt
Q (1 + �)�nln(
bt+nyt+n
)� +nXs=1
(1 + �)�sEt(kt+s)
Actual Q Target + Fiscal stance
Consider the following measure of �scal sustainability which compares the LHS and RHS
FS(t; n) = (1 + �)�nln(
bt+nyt+n
)� +nXs=1
(1 + �)�sEt(kt+s)� ln
btyt
FS(t; n) = 0 Debt-GDP ratio is forecast to be on target over the horizon
FS(t; n) > 0 Debt-GDP ratio is forecast to be below target (fall)
FS(t; n) < 0 Debt-GDP ratio is forecast to be above target (increase) - current �scal stance is not sustainable.
Our proposed index of �scal sustainability is
FSI(t; n) = expFS(t; n)
=Kt;n
bt=yt
where
lnKt;n = (1 + �)�nln(
bt+nyt+n
)� +nXs=1
(1 + �)�sEt(kt+s)
FSI(t; n) = 1 Debt-GDP ratio is forecast to be constant over the horizon
FSI(t; n) > 1 Debt-GDP ratio is forecast to be below target
FS(t; n) < 1 Debt-GDP ratio is forecast to be above target - current �scal stance is not sustainable.
How to choose ln( bt+nyt+n)�?
- like Buiter and Blanchard assume that the aim is a constant debt-GDP ratio over the
42
planning horizon.
- this eliminates ( bt+nyt+n)� from the index
Now get the index to be calculated in the paper:
lnbtyt
Q 1
1� (1 + �)�nnXs=1
(1 + �)�sEt(kt+s)
FS(t; n) =1
1� (1 + �)�nnXs=1
(1 + �)�sEt(kt+s)� ln
btyt
FSI(t; n) =Kt;n
bt=yt
lnKt;n =1
1� (1 + �)�nnXs=1
(1 + �)�sEt(kt+s)
5.5.3 Forecasting the �scal variables
We obtain forecasts of the vector
zt =
�lnbtyt; ln
gtyt; ln
vtyt; ln (1 + �t) ; ln(1 + t); ln (1 + �t)
�using the VAR
zt = A0 +
pXi=1
Aizt�i + et
et � i:i:d:[0;�]
zt may be I(0) or I(1)
n�period ahead forecasts may be obtained using the companion form
Zt = B0 +BZt�1 + ut:
43
where Z0t=[z0t; z
0t�1; :::; zt�p+1], u
0t=[e
0t;0; :::;0], B
00 = [A
00; 0; :::; 0] and
B = A1 A2 : : Ap�1
0 I 0 : :
0 : I 0 :
: : : : :
: : 0 I 0
The forecast of Zt+n is
Et[Zt+s] =s�1Xi=0
BiB0 +BsZt
Expressing kt as the following linear function of zt
kt = �c+ �0zt
and de�ning the selection matrix S =[I;0;0; ::;0] such that
zt= SZt
we obtain
FS(t; n) =1
1� (1 + �)�nnXs=1
f(1 + �)�s [�c+ �0S(s�1Xi=0
BiB0 +BsZt)]g � ln
btyt
� an + b0nZt
44
5.6 Evaluating �scal sustainability under alternative policy rules
We are interested in assessing whether a change in policy is consistent with �scal sustainability.
And we wish to carry out the analysis using the same VAR.
We now write the VAR as
zt = A(L)zt�1 + et
A0 = 0; A(L) =
pXi=1
AiLi
zt�s = Lszt
z0t = (z01t; z02t)
e0t = (e01t; e02t)
z2t is a set of policy instruments to be determined by new rules that can be expressed as linear
functions of zt, and its lagged values. We wish to form a new VAR based on replacing the old
equations for z2t.
We cannot simply substitute the new equations for the old as this would also alter the cor-
relation structure of the disturbances of the VAR model. If the VAR equations for the policy
instruments are changed then the correlation structure of the VAR errors will change too. Only
if the original errors were uncorrelated would there be no problem.
We therefore seek a way of replacing the equation for the policy instruments so that the
correlation structure of the VAR errors is una¤ected.
This can be accomplished by transforming the VAR equations for z1t into a VAR that is
conditional on the current value of the policy instrument.
45
We de�ne "t, the component of e1t that is uncorrelated with e2t, by
e1t = "t +Ge2t
E("te2t) = 0
In e¤ect we are applying the block Choleski decomposition
et =
2664 e1t
e2t
3775 =2664 I G
0 I
37752664 "t
e2t
3775where G is derived from �, the covariance matrix of the VAR errors:
� = E[ete0t]
=
2664 I G
0 I
37752664 �"" 0
0 �22
37752664 I 0
G0 I
3775
=
2664 �"" +G�22G0 G�22
�22G0 �22
3775where E["t"0t] = �"". Hence,
G = �12��122
G can easily be estimated from the covariance matrix of VAR residuals.
Denoting
H =
2664 I G
0 I
3775we pre-multiply the VAR by
H�1 =
2664 I �G
0 I
3775with the result that the disturbances associated with z1t are uncorrelated with those of z2t
H�1zt = H�1A(L)zt�1 +H�1et
46
Partitioning A(L) conformably,
z1t = [A11(L)�GA21(L)]z1;t�1 +Gz2t + [A12(L)�GA22(L)]z2;t�1 + "t
z2t = A21(L)z1;t�1 +A22(L)z2;t�1 + e2t
We now replace the equation for z2t by the new policy rule. Suppose this takes the general
form
Fz1t + z2t = A�21(L)z1;t�1 +A�22(L)z2t + e
�2t
A Taylor rule for in�ation, for example, is non-stochastic and has no lagged dynamics, and so
A�21(L), A�22(L) and e
�2t
would all be zero.
The complete model is now2664 I �G
F I
37752664 z1t
z2t
3775 =2664 I �G
0 I
37752664 A11(L) A12(L)
A�21(L) A�22(L)
37752664 z1;t�1
z2;t�1
3775+2664 "t
e�2t
3775
We can re-write this as a new VAR that can be used for policy analysis, called a PVAR.
2664 z1t
z2t
3775 =
2664 I �G
F I
3775�1 2664 I �G
0 I
37752664 A11(L) A12(L)
A�21(L) A�22(L)
37752664 z1;t�1
z2;t�1
3775
+
2664 I �G
F I
3775�1 2664 "t
e�2t
3775
We note that the response of z1t to "t in the PVAR must take account of the fact that we have
carried out a transformation of the disturbances. Thus in the original VAR
@z1t@"t
= I
47
but in the new VAR
@z1t@"t
= I � F (I + FG)�1G
We also note that now z2t will in general respond to "t.
Application to �scal sustainability
Simply construct the PVAR and then calculate FSI(t; n) as before.
Consider imposing a Taylor rule. The policy instrument for this is the short rate rst, not Rt
the e¤ective rate on total government debt.
Total debt consists of bonds of di¤erent maturities and so the e¤ective rate of return is an
implicit weighted average of rates on each maturity, weighted by the number of bonds issued
at each maturity.
We therefore include rst in the VAR
We also include the nominal long rate rlt to help forecast Rt and to better capture the monetary
transmission mechanism.
As a result, we modify the de�nition of zt to be
zt =
�lnbtyt; ln
gtyt; ln
vtyt; ln (1 + �t) ; ln(1 + t); ln (1 + �t) ; ln (1 + rlt) ; ln (1 + rst)
�
48
6 Open economy VAR models
6.1 General theoretical considerations
A central issue for monetary policy is how to determine the nominal anchor for an economy.
Historically, this has been achieved in several di¤erent ways, depending in part on whether the
economy is essentially a closed or open economy.
1. Gold standard (prior to WWII)
Link money supply to gold holdings
Depends on how closely �at money is backed by gold
In�ation depends on national gold holding
2. Bretton Woods
An exchange rate targeting scheme
Link currency to the US dollar
US dollar on the gold standard
In�ation equal to US in�ation
What happens if US in�ation is unacceptable?
3. Floating exchange rates
(i) use money supply targeting
- direct control of money base as an instrument
- money as an intermediate target, interest rate is the instrument
(ii) use in�ation targeting
- with the interest rate as the instrument
- discretion v. rules
49
(ii) Exchange rate target
- ECU
- dollar standard
- currency board
4. Monetary Union
EMU + �oating exchange rate
The key implications of this are:
1. What is the instrument: MS or i?
2. Is the instrument exogenous due to pure discretion, or determined by a rule?
3. How should the instrument be treated in the VAR?
- as a VAR variable?
- as an exogenous variable
- as a separate equation?
4. When the exchange rate is �exible
- how important is the exchange rate channel in the transmission mechanism from the monetary
instrument to in�ation?
- ought the exchange rate be included as a variable in the VAR?
6.2 A sticky-price monetary model with a �oating exchange rate
The monetary model of the exchange rate is
1. PPP
p#t = st + p�t
2. Money demand
mt = pt + yt � �it
50
3. UIP
it = i�t + Et[�st+1]
where money (mt), output (yt), the foreign price level (p�t ) - all logs -and the foreign interest
rate (i�t ) are treated as exogenous.
The monetary model treats the domestic price level (pt) as perfectly �exible, but modern
monetary analysis assumes that prices are sticky. The Calvo model and the Taylor model predict
that prices are determined by a forward-looking model of the form
�pt = �(p#t � pt�1) + �Et[�pt+1]
where p#t = target price level.
If we assume that the target price level in the long run satis�es PPP then
p#t = st + p�t
and hence the PPP equation is replaced by
�pt = � (st + p�t � pt�1) + �Et[�pt+1]
The solution of the model and the speci�cation of a VAR depends on how monetary policy is
conducted:
- whether mt or it is the policy instrument
- and whether a rule is used.
(i) If mt is the policy instrument then the solution is obtained from
51
�Et[pt+1] + (1 + �)pt � (1� �)pt�1 � �st = �p�t
pt � �Et[st+1] + �st = mt � yt + �i�t
Assuming that it and i�t are I(1), this would suggest a CVAR in (pt; st; yt;mt; p�t ; i
�t ) with three
CVs
pt � p�t � st
it � i�t
mt � pt � yt � �it
If it and i�t are I(0) there would be only two CVs
pt � p�t � st
mt � pt � yt
(ii) If it is the policy instrument then the solution is obtained from
�Et[pt+1] + (1 + �)pt � (1� �)pt�1 � �st = �p�t
it � Et[st+1] + st = i�t
This would suggest a CVAR in (pt; st; p�t ; i�t ) with two CVs
pt � p�t � st
it � i�t
or,if it and i�t are I(0), just the one CV, the PPP relation.
52
6.3 Garratt, Lee, Pesaran and Shin (EJ 2002)
They utilise restrictions on the cointegrating vectors to construct a quarterly model for the UK.
GLPS formulate an economic model consisting of the following equations:
PPP, UIP, Production function, Money demand,and stock-�ow relations.
They express the log-linearized long-run structure as:
PPP : pt � p�t � et = a10 + a11t+ �18(p�t � pot ) + "1;t+1
UIP : rt � r�t = a20 + "2;t+1
PF : yt � y�t = a30 + "3;t+1
BOP : pt � p�t � et = a40 + a41t+ �43rt + �45yt + �48(p�t � pot ) + "4;t+1
MM : ht � yt = a50 + a51t+ �53rt + �55yt + "5;t+1
An asterisk denotes the foreign equivalent, po is the oil price, r is the nominal interest rate. The
PF relation takes the domestic and foreign log production functions and assumes that domestic
productivity di¤ers from foreign productivity by a stationary random variable.
This set up implies that the non-stationarity is being transmitted by foreign variables.
The BOP equation appears not to be identi�ed in the long run as it can be formed by a linear
combination of the PPP and BOP equations. In the estimation, KLPS alter this speci�cation
slightly. It transpires that this removes this identi�cation problem.
The model is a CVAR in zt = (pt � pot ; et; rt; r�t ; yt; y�t ; ht � yt; p�t � pot )0
with two exogenous variables t and �(p�t � pot )
�zt = ��[�0zt�1 � a0 � a1t] + �(L)�zt�1 + �(p�t � pot ) + ut
53
�0 =
2666666666666664
1 �1 0 0 0 0 0 �(1 + �18)
0 0 1 �1 0 0 0 0
0 0 0 0 1 �1 0 0
1 �1 ��43 0 ��45 0 0 �(1 + �48)
0 0 ��53 0 ��55 0 1 0
3777777777777775The model is estimated by FIML and takes account of the restrictions to �:
Thus the model contains 8 I(1) variables and 5 long-run relations.
It is natural to think of there being 5 endogenous variables (pt � pot ; et; rt; yt; ht � yt) and 3
exogenous variables (r�t ; y�t ; p
�t � pot ); but GLPS do not make this distinction.
GLPS then focus on an impulse response function analysis of the e¤ects of the shocks on �0zt
. This gives the speed of the return to long-run equilibrium of each variable
This is a¤ected by how fast all of the variables in the long-relation react, and not just one
variable in the long-run relation such as the variable normalised.
6.4 DSGE models
Modern open economy macro is based increasingly upon the use of DSGE models with monop-
olistically competitive goods markets which deliver real exchange rate e¤ects in forward-looking
aggregate demand functions, prices dependent on marginal cost (and through this output) and
price stickiness. Examples are Gali and Monacelli (2000), Devereux and Engel (2001) and Smets
and Wouters (2002).
Smets and Wouters examine the implications of imperfect exchange rate pass-through to do-
mestic prices for optimal monetary policy in the euro zone. The analysis is based on an open
economy DSGEM. This is then log-linearised, calibrated and simulated.
54
The simulation requires the calibration of the processes generating the shocks. They use an
unrestricted VAR with exogenous variables.
- there are 6 endogenous variables: real GDP, net trade, in�ation, short interest rate, real
e¤ective exchange rate and import price in�ation
- and 3 exogenous variables: US short rate, US real GDP and US in�ation.
They identify the shocks using a Choleski decomposition with this ordering.
They claim to be able to examine the e¤ects of 5 shocks: to money, the exchange rate, pro-
ductivity, foreign demand, exchange rate risk premium.
The conclusions reached from the model simulations are:
1. Import prices exhibit the same degree of price stickiness as domestic prices as a result
exchange rate pass-through is imperfect.
2. The imperfect pass-through reduces the e¤ectiveness of the exchange rate channel in sta-
bilising domestic in�ation.
3. This implies that monetary policy would need to be very active to work viw the exchnage
rate
4. It is therefore better to aim monetary policy in the EU directly at domestic in�ation.
6.5 The sustainability of the current account using a VAR
Another perennial open economy question is whether or not current policy leads to a sustainable
current account position. If not this could ultimately undermine monetary policy. We therefore
examine the sustainability of a country�s economic position by examining its balance of payments
(BOP).
The problem is analagous to the sustainability of �scal policy which was analysed by examining
the government�s budget constraint. Thus the BOP replaces the GBC in the analysis.
55
6.5.1 The balance of payments
The balance of payments written in nominal terms is
CAt = Ptxt � StP �t xmt +R�tStB�t �RtBFt = St�B�t+1 ��BFt+1
where
B�t = domestic nominal holding of foreign assets expressed in foreign currency
BFt = foreign holding of domestic assets expressed in domestic currency
Ft = StB�t �BFt = net asset position expressed in domestic currency
CAt = current account balance expressed in nominal domestic currency terms
The BOP in real terms is obtained by de�ating by the domestic price level, Pt. This can be
written in the following equivalent ways
xt �StP
�t
Ptxmt + (1 +R
�t )St
B�tPt� (1 +Rt)
BFtPt
=StPtB�t+1 �
BFt+1Pt
xt �StP
�t
Ptxmt + (1 +R
�t )StP
�t
Pt
B�tP �t
� (1 +Rt)BFtPt
=Pt+1Pt
StSt+1
P �t+1St+1
Pt+1
B�t+1P �t+1
xt �Qtxmt + (1 +R�t )Qtb�t � (1 +Rt)bFt = (1 + �t+1)
�Qt+1b
�t+1
1 + �st+1� bFt+1
�xt �Qtxmt + (1 +R�t )ft � (Rt �R�t )bFt =
�1 + �t+11 + �st+1
��ft+1 ��st+1bFt+1
�where Qt =
StP�t
Pt= real exchange rate, �t+1 =
Pt+1Pt
� 1 = in�ation rate and ft = FtPt= Qtb
�t � bFt ,
with lower-case letters denotes the real equivalent.
The balance of payments can also be expressed as a proportion of GDP by dividing by real
domestic GDP, yt, to obtain
xt �Qtxmtyt
+ (1 +R�t )ftyt� (Rt �R�t )
bFtyt=
�(1 + �t+1)(1 + t+1)
1 + �st+1
��ft+1yt+1
��st+1bFt+1yt+1
�(1)
where t is the rate of growth of GDP. This can be simpli�ed and written as a di¤erence equation
in ftyt:
56
� tyt+ (1 +R�t )
ftyt=
�(1 + �t+1)(1 + t+1)
1 + �st+1
�ft+1yt+1
(2)
where
� tyt=xt �Qtxmt
yt� (Rt �R�t )
bFtyt+
�(1 + �t+1)(1 + t+1)
1 + �st+1
��st+1
bFt+1yt+1
; (3)
can be interpreted as the �primary�current account surplus expressed as a proportion of GDP.
This is analagous to the primary government de�cit.
We note that if uncovered parity holds ex-post and the exchange rate is constant, or if there
is foreign holding of domestic assets, then � t is just the trade balance. The two additional terms
on the right-hand side of equation (3) occur if these conditions do not hold. The �rst is excess
interest payments to foreign holders of domestic debt due to domestic exceeding foreign interest
rates. The second captures the cost of a revaluation of real foreign holdings of domestic assets
due to a depreciation of the exchange rate; the higher is the domestic nominal rate of growth, the
greater is this cost.
6.5.2 Current account sustainability
Current account sustainability involves the existence of a solution for ftytfrom equation (2). First
we consider the case where �t, t, R�t and �st are constant, taking the values �, , R
� and �s = 0.
We also assume that R = R�, i.e. that UIP holds in the long run. Hence equation (2) becomes
ftyt=(1 + �)(1 + )
1 +R�ft+1yt+1
� 1
1 +R�xt �Qtxmt
yt(4)
The solution for ftyt now depends on whether (1+�)(1+ ) is greater or less than 1+R�: Under
long-run UIP this is approximately equivalent to whether � + is greater or less than R. This is
the same condition that arises in the analysis of the sustainability of �scal policy.
If R < �+ then (4) is a stable di¤erence equation. In this case the ratio of net assets to GDP
remain �nite whether the trade balance is positive or negative. A trade de�cit would therefore
57
be sustainable. We therefore focus on the unstable case where R > � + and solve equation (4)
forwards to obtain
ftyt=
�(1 + �)(1 + )
1 +R
�nft+nyt+n
� 1
1 +R
nXi=0
�(1 + �)(1 + )
1 +R
�iEt
�xt+i �Qt+ixmt+i
yt+i
�Taking limits as n!1 gives the transversality condition
limn!1
�(1 + �)(1 + )
1 +R
�nEt(
ft+nyt+n
) = 0 (5)
If this holds then we obtain the inter-temporal or present-value balance of payments condition
expressed as a proportion of GDP:
�ftyt� 1
1 +R
1Xi=0
�(1 + �)(1 + )
1 +R
�iEt
�xt+i �Qt+ixmt+i
yt+i
�This can be interpreted as follows. If the economy has a negative net asset position (i.e.
if ftyt< 0), then it needs the present value of current and expected future real trade surpluses
xt+i�Qt+ixmt+i
yt+ias a proportion of GDP to be positive and large enough to pay-o¤ net debt.
In the special case where the real trade surplus is constant and positive (i.e. x�Qxmy > 0) a
simple connection emerges between net indebtedness and the long-run real trade surplus that is
required to sustain a given level of net indebtedness:
�fy� 1
R� (� + )x�Qxm
y(6)
To �nd the implication of this analysis for the current account, we note that in the long run
the real current account is the real trade balance plus real net interest earnings, hence
CA
Py=x�Qxm
y+R
f
y
Substituting for x�Qxmy from equation (6) gives
(�CAPy
) � (� + )(�fy)
58
It follows that it is possible to have a permanent and sustainable current account de�cit (CA < 0)
- even if fy < 0 - provided that the current account de�cit does not exceed the right-hand side.
The higher the rate of growth of nominal GDP, the more likely that a permanent current account
de�cit can be sustained without increasing a country�s net indebtedness.
6.5.3 Dynamic e¢ ciency
Dynamic e¢ ciency requires that in the long run economic agents (households, government or the
economy) aim to exactly consume their assets. If their spending plans are not su¢ cient to use up
their assets then they are said to be dynamically ine¢ cient. Unless they have a bequest motive,
they should raise consumption. If the economy has a positive net foreign asset position then it
follows that it should run a long-run trade de�cit su¢ cient to run down its assets. Thus the
economy should ensure that for f > 0 it has a trade de�cit but that this does not exceed the
left-hand side of
[R� (� + )]fy� Qxm � x
y> 0
It would clearly be a mistake to have a positive net asset position and a trade balance or surplus.
6.5.4 The long-run equilibrium real exchange rate
The long-run equilibrium real exchange rate must satisfy equation (6). Hence, it is
Q � x+ [R� (� + )]fxm
This equilibrium real exchange rate requires neither a long-run real trade balance nor a long-run
current account balance. It may therefore be contrasted with the popular FEER, the fundamental
e¤ective exchange rate. This is not based on the notion of current sustainability, but on long-run
current account balance; the FEER is analagous to requiring a balanced government budget in the
long run. Our analysis has shown that in the long run it is possible to have a both a permanent
59
trade and current account de�cit. Put another way, long-run current account balance is su¢ cient,
but not necessary, for current account sustainability.
6.5.5 Econometric tests of current account sustainability
We now suggest some tests of current account sustainability. These are closely related to well-
known tests of �scal sustainablility. Suppose that the trade balance as a proportion of GDP is
stochastic and is expected to grow at the rate �, then
Et
�xt+i �Qt+ixmt+i
yt+i
�= (1 + �)
i
�xt �Qtxmt
yt
�, � 2 < � � 0 (7)
It follows that
�ftyt
� 1
1 +R
1Xi=0
�(1 + �)(1 + ) (1 + �)
1 +R
�i�xt �Qtxmt
yt
�(8)
� 1
R� (� + + �)xt �Qtxmt
ytif R > � + + �
We consider two cases: where xt�Qtxmt
ytis stationary and where it is non-stationary.
(i) xt�Qtxmt
ytstationary
In this case �2 < � < 0 and so ftytwill also be stationary and the current account position is
sustainable.
(ii) xt�Qtxmt
ytis non-stationary
If � = 0 then xt�Qtxmt
ytis I(1). It then follows that in the long run
�ftyt=
1
R� (� + )xt �Qtxmt
yt(9)
Hence, ftytwill be I(1) and will be cointegrated with xt�Qtx
mt
yt, the above equation de�ning the
cointegrating relation. It follows that if ftyt andxt�Qtx
mt
ytare I(1) and cointegrated with cointe-
grating vector (R� �� ; 1) then the current account position is sustainable. To see this we note
that if the cointegrating relation between ftytand xt�Qtx
mt
ytis
xt �Qtxmtyt
+ �ftyt= ut
60
where ut is I(0) then, from equation (4),
(1 + �)ftyt= (1 +R� � � )ft�1
yt�1+ ut
It follows that ftythas a unit root if � = R� � � .
6.5.6 Current account sustainability with a time-varying discount rate
The above analysis of current account sustainability has assumed for convenience that UIP holds
at each period of time and that in�ation, growth, interest rates and the nominal exchange rate
are constant. In practice, of course, none of these assumptions will hold. The correct no-arbitrage
equation for foreign exchange will usually involve a time-varying risk premium and in�ation,
growth, interest rates and the exchange rate will be time-varying. To analyse the general case,
we therefore revert to the original balance of payments equations (2) and 3). We also de�ne the
time-varying discount rate
1 + �t+1 =(1 + �t+1)(1 + t+1)
(1 +R�t )(1 + �st+1)(10)
We may now write the balance of payments as
1
1 +R�t
� tyt+ftyt=
1
1 + �t+1
ft+1yt+1
(11)
This may be solved forwards to obtain
ftyt= Et[(�
ni=1
1
1 + �t+i)ft+nyt+n
]� Et[nXi=1
(�ij=11
1 + �t+j)
1
1 +R�t+i
� t+iyt+i
]
if
�t;i = �ij=1
1
1 + �t+j� 1 for all i � 1
Hence current account sustainability requires that the transversality condition
limn!1
Et[(�ni=1
1
1 + �t+i)ft+nyt+n
] = 0 (12)
61
holds. This implies that
�ftyt
= Et[1Xi=1
(�ij=11
1 + �t+j)
1
1 +R�t+i
� t+iyt+i
] (13)
In other words, the present value of the trade surplus (adjusted for foreign holdings of domestic
debt) current must be su¢ cient to o¤set current net debt liabilities.
In order to analyse sustainability further we de�ne the variables
xt = �t;n(�ft)yt
zt = �t;n1
1 +R�t
� tyt
We may now re-write equation (11) as
�xt = zt
Current account sustainability now requires the transversality condition
limn!1
Et(xt+n) = 0
and implies that
xt = � limn!1
Et[nXi=1
zt+i]
It follows that current account sustainability is satis�ed if xt is a zero-mean stationary process.1
However, current account sustainability is also satis�ed if zt is a zero-mean stationary process. In
this case the adjusted trade de�cit is stationary.2 It then follows that xt will be an I(1) process.3
6.5.7 A log-linear approach to current account sustainability
The log-linearized balance of payments
1 This result is analagous to that of Wilcox (1989) for �scal sustainability.
2 This result is analagous to that of Trehan and Walsh (1991) for �scal sustainability.
3 This result is analagous to that of Wickens and Uctum (2000) for sustainability.
62
The �rst step is to log-linearize the balance of payments. This must be carried out term by
term in order to ensure that all terms are positive. We therefore start by writing the balance of
payments as a proportion of GDP as
xtyt�Qt
xmtyt+ (1 +R�t )St
b�tyt� (1 +Rt)
bFtyt= (1 + �t+1)(1 + t+1)[St
b�t+1yt+1
�bFt+1yt+1
]
We now approximate this about the steady-state solution which we assume exists, and in which
we assume that all variables are constant. We also assume that in the long run trade is in balance
so that xy = Qxm
y and UIP holds so that R = R�. The steady-state solution of the balance of
payments is then approximately
x
y�Qx
m
y+ (R� � � )[S b
�
y� bF
y] = 0
Given long-run trade balance this implies that either R = �+ or that S b�
y =bF
y ; we assume the
latter.
Writing each term zt in the balance of payments as exp[ln zt], the balance of payments may be
re-written as
0 = exp [lnxtyt]� exp [lnQt
xmtyt]+ exp [ln(1 +R�t )St
b�tyt]� exp [ln(1 +Rt)
bFtyt]
� exp[ln(1 + �t+1)(1 + t+1)Stb�t+1yt+1
] + exp[ln(1 + �t+1)(1 + t+1)bFt+1yt+1
]
Noting that a �rst-order Taylor series approximation of exp[ln zt] about ln z is
exp[ln zt] = z[1 + (ln zt � ln z)]
a log-linear approximation to the balance of payments is given by
0 =x
y[ ln
xtyt� lnQt
xmtyt]+(1 +R)S
b�
y[ln(1 +R�t )St
b�tyt� ln(1 +Rt)
bFtyt]
�(1 + �)(1 + )S b�
y[lnSt
b�t+1yt+1
� lnbFt+1yt+1
]
63
This can be re-written as a di¤erence equation in the logarithmic equivalent of the net asset
position ln at = lnStb�tyt� ln b
Ft
yt= ln
Stb�t
bFt:
ln at =(1 + �)(1 + )
1 +Rln at+1 �
x
(1 +R)Sb�[ ln
xtyt� lnQt
xmtyt] + (Rt �R�t )
� (1 + �)(1 + )1 +R
� lnSt+1 (14)
Although the change in the exchange rate appears as a separate term in this equation, in�ation
and growth do not. This is due to taking the approximation about a zero long-term net asset
position. A corollary is that time-varying in�ation and growth rates only a¤ect the balance of
payments when there is a non-zero long-run net asset position.
Assuming once more that R > � + , equation (14) is an unstable di¤erence equation and so
must be solved forwards. The solution is
ln at =
�(1 + �)(1 + )
1 +R
�nEt(ln at+n) +
n�1Xi=0
�(1 + �)(1 + )
1 +R
�iEt(kt+s) (15)
kt = � x
(1 +R)Sb�[ ln
xtyt� lnQt
xmtyt]+(Rt �R�t )�
(1 + �)(1 + )
1 +R� lnSt+1 (16)
where kt is, in e¤ect, proportional to the logarithmic equivalent of the trade balance adjusted for
an interest di¤erential and changes in the exchange rate. We note that if R = �+ then the last
two terms are approximately Rt �R�t�Et� lnSt+1, which is zero if UIP holds and is otherwise the
FOREX risk premium. Thus the adjustment is required if the FOREX risk premium is non-zero.
And if there is a non-zero net asset position in the long run, then further adjustment would be
needed to take account of in�ation and growth.4
The transversality condition is
limn!1
�(1 + �)(1 + )
1 +R
�nEt(ln at+n) = 0 (17)
4 In this case kt becomes
kt = � x
(1 +R)Sb�[ ln
xt
yt� lnQ
t
xmtyt]+(Rt �R�t )�
(1 + �)(1 + )
1 +R� lnSt+1
�(S b�
y� bF
y)(�t+1 + t+1)
This ignores the e¤ect of a non-zero net asset position on the interest di¤erential and on ln at.
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which implies that
ln at =1Xi=0
�(1 + �)(1 + )
1 +R
�iEt(kt+s) (18)
If kt is stationary then ln at, and hence at, remains �nite and stationary. This may occur due
to the individual terms of kt being stationary, or due to some being I(1) but being cointegrated
with the appropriate contegrating vector.
Testing current account sustainability
This can be done in a similar way to testing for �scal sustainability. First construct a VAR to
forecast future values of kt. This requires a VAR in zt = (ln at, lnxt, lnxmt , ln yt, lnQt, Rt, R�t ).
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