Using a real exchange rate model to make real and
nominal forecasts
Peter Sellin
Sveriges Riksbank
November 2006
Abstract
In this paper we evaluate the forecasting performance of a real
exchange rate model applied to the Swedish Krona’s real and
nominal effective exchange rate, as well as the bilateral
exchange rates against the US and the Euro area and the
implicit USD/EUR exchange rate. The theoretical model implies
a cointegrating relation between the real exchange rate, relative
output, terms of trade and net foreign assets. The vector error
correction model’s forecast performance is quite satisfactory for
most of these exchange rates once the dynamics of the model
have been augmented with an interest rate differential.
JEL classification: C52, C53, F31.
Keywords: real exchange rate, nominal exchange rate,
forecasting.

I am grateful for helpful comments by Paolo Giordani, Jesper Hansson, Tor Jacobson, Anders Vredin
and Mattias Villani on a previous version of this paper.
1 Introduction
A standard inter-temporal open economy model typically finds that the steady-state
real exchange rate is related to the relative productivity in the tradables sector, the
terms of trade and the net foreign asset position (see for example Lane and MilesiFerretti (2004) and Obstfeld and Rogoff (2004)). In this paper we will test if forecasts
from a vector error correction model based on the Lane and Milesi-Ferretti (2004)
model can beat a random walk at horizons of 1 to 12 quarters. A simple extension of
the empirical model allows us to consider forecasts of the nominal as well as the real
effective exchange rate. We also forecast the effective exchange rate indirectly as a
weighted average of forecasts of the SEK/EUR and SEK/USD exchange rates.
Finally, we evaluate the implicit forecast of the USD/EUR exchange rate. We find that
the model forecasts are quite satisfactory once the dynamics of the model have been
properly specified.
The out-of-sample forecasting performance of exchange rate models poses a
special challenge to open economy macroeconomists. The standard finding in the
empirical literature, starting with the seminal article by Meese and Rogoff (1983)
evaluating the models of the 1970’s, is that no model can consistently outperform a
random walk (for an evaluation of the models of the 1990’s see Cheung, Chinn and
Pascual (2005)). However, the tests used have not been appropriate for testing
nested models, as has been pointed out by for example Clark and McCracken
(2001). Now research in this area has started using inference procedures that have
recently been developed by Clark and West (2006a, 2006b). Their test procedure
corrects for an upward bias in the alternative model’s sample mean squared
prediction error. The test is easy to use and can also be applied to multi-step
forecasts, using standard normal inference. Gourinchas and Rey (2005), Alquist and
Chinn (2006) and Melodtsova and Papell (2006) use the new test. The results are
somewhat more supportive regarding the ability of forecasts from exchange rate
models to beat a random walk.
Gourinchas and Rey (2005) estimate a model of the effective US dollar
exchange rate using the ratio of net exports to net foreign assets as explanatory
variable. They present forecasts that are consistently better than a random walk at
horizons of 1 to 16 quarters over the period 1978:1 to 2004:1. Alquist and Chinn
(2006) consider three different models: the sticky price monetary model, the
2
uncovered interest rate parity relation and Gourinchas and Rey’s (2005) model.
Alquist and Chinn investigate the dollars bilateral exchange rates relative to the
Canadian dollar, the UK Pound, the Japanese yen and the euro. They could not
identify a model that was able to outperform a random walk forecast for all horizons
and all bilateral exchange rates considered. However, the Gourinchas and Rey
model did quite well for short forecast horizons of 1 and 4 quarters, except for the
JPY/USD exchange rate. Molodtsova and Papell (2006) consider several models: the
flexible price monetary model, uncovered interest parity, purchasing power parity and
two types of Taylor rule models. They also consider a greater number of US bilateral
exchange rates compared to Alquist and Chinn. They find that the models perform
quite well at short forecast horizons. This is especially true for the Taylor rule models.
The Swedish real effective exchange rate has been modelled recently by
Bergvall (2002), Lindblad and Sellin (2003), Nilsson (2004), and Lane (2006). A
common finding in these studies is that an improvement in the terms of trade lead to
an appreciation of the real exchange rate. The first three studies also find that an
increase in relative productivity (or relative output) leads to an appreciation, while
Lane (2006) finds such a variable to be insignificant and excludes it from the
analysis. However, none of the papers mentioned above focused on forecasting,
which is the main focus of the present paper.
The outline of the paper is the following. In Section 2 we present the steadystate exchange rate equation derived by Lane and Milesi-Ferretti (2004). In Section 3
we estimate a vector autoregressive model and analyse a possible cointegrating
relation among the variables that is consistent with the theoretical model. The model
is then extended to a model for the nominal exchange rate. In Section 4 we evaluate
forecasts of the real and nominal effective exchange rate. In Section 5 we model the
bilateral SEK/EUR and SEK/USD exchange rates. We then weight these forecasts
together to form a forecast of the effective exchange rate and we also compute the
implicit USD/EUR exchange rate. These forecasts are then evaluated using the
inference procedures developed by Clark and West (2006a, 2006b). Section 6
concludes.
2 A model of the real exchange rate
We will base our analysis on the theoretical model derived in Lane and Milesi-Ferretti
(2004). The main result in their paper is that higher steady state tradable output
3
endowment (YT ) , terms of trade ( PTx ) , and net foreign assets (B) lead to a higher
relative price of nontradables. The latter follows since an increase in any of the
factors mentioned leads to greater wealth and hence increased demand for goods
and leisure, the latter inducing a reduction in the supply of labour used (exclusively)
in the production of nontradables.
The derived steady-state variation in the real exchange rate is (log) linear in the
price of nontradables:
- q  (1   ) log( PN )
 (1   )  (1   ) log( YT )  (1   ) log( PTx ) 
(1   )r B

Y0
(1)
where q is the log of the real exchange rate defined as the foreign price level relative
to the domestic price level, (1   ) is the weight placed on consumption of nontraded
goods in the utility function,  is a constant,   (1   )[(1   )  (   ) ]1 with 
the intertemporal elasticity of substitution and  the elasticity of substitution between
traded and nontraded goods, r is the exogenously given real interest rate, Y0 is the
steady-state output. Thus, higher net foreign assets and tradable output as well as an
improvement in the terms of trade lead to an appreciation of the real exchange rate.
Equation (1) rewritten in obvious notation as
0  q    1 log( YT )   2 log( PTx )   3
B
Y0
(2)
forms the basis for our empirical analysis. To get some rough idea of the size of the
coefficients we should expect to see we shall make some assumptions about the size
of the elasticities above. We will assume an elasticity of substitution between traded
and nontraded goods of   0.74 , as estimated by Mendoza (1995) for a sample of
industrialised countries. Adolfsson et al (2005) set the share of imports in aggregate
consumption to 0.31 for the Euro area. In addition to imports, traded goods should
also include exportable goods. In view of this it does not seem unreasonable to
choose the weight placed on consumption of traded goods in the utility function to
  0.5 . The elasticity of intertemporal substitution is set to   0.67 as in Mendoza
(1995). With the final assumption that the real interest rate r = 0.01 per quarter we
get the parameter values ( 1 ,  2 ,  3 )  ( 0.7, 0.7, 0.014).
4
Following Lane and Milesi-Ferretti (2002), we can use an alternative empirical
formulation by noting that it is possible for a country to run a trade deficit in steadystate equal to the net investment income on its net foreign asset position:
TB
B
 r ,
Y0
Y0
(3)
where TB is the country’s trade balance. Equation (2) can then be rewritten in the
alternative form,
0  q    1 log( YT )   2 log( PTx )   3*
TB
,
Y0
(4)
where we now expect ( 1 ,  2 ,  3* )  ( 0.7, 0.7, -1.4). We will estimate both
specification (2) and (4) as a check on the robustness of the results and to see which
model yields the better forecasts.
3 Estimating a VAR model for the effective exchange rate
3.1Preliminaries
The benchmark model consists of four variables: real exchange rate, qt , relative
output, yt , terms of trade,  t , and net foreign assets relative to GDP , bt . The first
three variables are measured in natural logarithms. The real exchange rate is
measured as the foreign price level relative to the Swedish price level expressed in
the Swedish currency. Net foreign assets are measured as a fraction of gross
domestic product. The alternative model also consists of four variables: real
exchange rate, qt , relative output, yt , terms of trade,  t , and the trade balance
relative to GDP , tbt . The quarterly value of the trade balance is expressed as a
fraction of annual gross domestic product to conform to the measure of net foreign
assets. More details about the data are given in an appendix.
The levels and first differences of all the abovementioned variables are depicted
in Figure 1a-b. In addition some descriptive statistics for the time series are provided
in Table 1. The sample covers the period 1985Q1 – 2005Q3. The main reason for
starting in 1985 is that we will consider an augmented version of the model that
includes a short-term interest rate differential and there was no functioning money
market in Sweden before the early 1980s. Allowing for up to four lags would then
take us back to the first quarter of 1984, by which time there was a working money
5
market. By using 1985Q1 as the starting date we also avoid the devaluations of the
early 1980s and more than half of the observations are from the floating period
1993Q1-2005Q3.
A look at the time series in Figure 1 shows that the real exchange rate has been
more volatile since the Krona was allowed to float on 19 November 1992. The
Swedish economy has experienced a period of relatively high growth since the crisis
years of the early 1990s and the net foreign asset position has improved dramatically
since then. This is in large part due to continuous trade balance surpluses but
valuation effects have also played a role. Another thing worth noting is the
deterioration in the Swedish terms of trade during the past ten years. At the end of
the sample period the terms of trade were at the same level as before the OPEC
collapse of 1986.
Because of the small sample size some of the critical values of interest have
been bootstrapped (using 4,999 replications in each case). The “Structural VAR
Program” developed by Anders Warne has been used for this purpose. This program
has also been used in order to investigate parameter stability.1 Otherwise, the results
presented in most of the tables have been produced by the Eviews 5.0 program.
3.2 The statistical model
We will be using a 4-dimensional vector autoregressive (VAR) model of order k. This
model can be expressed in vector error correction form (VEC) as:
k 1
X t   ' X t 1   i X t i   ´ t ,
(5)
i 1
where X t  (qt , yt , t , bt )' or alternatively X t  (qt , yt , t , tbt )' . The errors are assumed
to be independently and normally distributed with mean zero and covariance matrix
 . Under certain conditions the X t -process is non-stationary while both the first-
differenced process X t and the linear combinations  ' X t 1 are stationary (see
Johansen (1996)). Based on the theoretical considerations discussed above we
expect to find one such stationary linear combination, with   (1, 0.7, 0.7, 0.014) or
alternatively (1, 0.7, 0.7, -1.4), when the vector is normalised with respect to the real
exchange rate.
1
See the Structural VAR homepage at http://texlips.hypermart.net/svar/index.html.
6
3.3 The estimated benchmark model (BM)
In order to get a good estimate of the cointegration vector it is important to specify the
correct number of lags for the VAR. In order to determine the lag order we
considered a number of lag order selection criteria. The LR and AIC criteria indicated
that a lag order of k=3 is optimal, the FPE criterion favoured 2 lags and the SC and
HQ criteria preferred 1 lag. Lag exclusion tests indicated that we should keep 3 lags
in the model. Multivariate tests of serially uncorrelated residuals for the k=3 model
showed that the null hypothesis of no serial correlation in the residuals cannot be
rejected at the 5 percent significance level against the alternative hypotheses of 1st –
4th order serial correlation. Thus 3 lags were used. The null hypothesis that the
residuals are normally distributed was rejected, using the test statistics derived by
Doornik and Hansen (1994). The reason for this is that the kurtosis values are lower
than those from a normal distribution. There was no indication of skewness in the
residuals, which according to Juselius (2005) would be a more serious problem.
Tests for the cointegration rank (r) are reported in Table 1 for the VAR(3) model.
We conclude from Johansen’s likelihood ratio tests that there is one cointegrating
vector, which is what we would expect from the theoretical model. Bootstrapping was
then used for these test statistics. First, we find that bootstrapped critical values for
the trace statistics are higher than the asymptotic values and we cannot reject the
null hypothesis of no cointegration. On the other hand a likelihood ratio test does not
reject the hypothesis of r=1 against the alternative r=0 at conventional significance
levels. There are three near unit roots in the unrestricted system. The choice of r=1
leaves a root of 0.937 in the system. Nevertheless, we judge that the overall
evidence against cointegration is not strong enough for us to give up our prior belief
in one cointegration vector.
We computed the sample means and standard deviations of the firstdifferenced series. None of the means in the first-differenced series are significantly
different from zero, see Table 1. We therefore restrict the constant term to the
cointegration space to exclude deterministic drifts in the individual series. This implies
that in the long run, or in the steady state, output in Sweden will grow at the same
rate as in its main trading partners, export and import prices will grow at the same
rate, and net foreign assets will converge to some constant fraction of GDP.
7
We have included two dummy variables in the estimated vector error correction
model. The first is a devaluation dummy that takes the value one in the 4th quarter of
1992 and zero otherwise and captures the real depreciation after letting the Krona
float on 19 November 1992. This dummy variable is included contemporaneously and
lagged one period. The second is a shift dummy and captures the change to an
inflation targeting regime in the 1st quarter of 1995. It takes the value one up to and
including the 4th quarter of 1994 and zero thereafter.
Estimating the model by maximum likelihood is equivalent to solving an
eigenvalue problem. We use the estimated non-zero eigenvalue to check for nonconstant parameters as suggested in Hansen and Johansen (1999). If the estimated
eigenvalue fluctuates over time this can be due to fluctuations in  or  or both and
additional testing is required to determine which is the case. The time path of the
estimated non-zero eigenvalue is shown in Figure 2. It looks quite stable indicating
no problem with non-constant parameters in the cointegration relation. The
cointegrating relation is shown in Figure 3, Panel A.
The cointegration vector normalized with respect to the real SEK/TCW
exchange rate is given by2
̂ ´ 1.0, 0.72 , 0.50 ,  0.17  ,

( 0.54, 2.31) ( 0.02,1.05) ( 0.39, 0.03)

with 80 percent bootstrapped confidence intervals in parentheses under the
estimated coefficients (using bootstrapped t-values). The coefficients are imprecisely
estimated. However, we note that the point estimates of ˆ1 and ˆ 2 are quite close to
the values we expected them to take. On the other hand, ̂ 3 does not even take the
expected sign. The estimated adjustment coefficients are
̂´  0.11 , 0.03 ,  0.04 , 0.13  ,

( 0.20, 0.02) ( 0.01, 0.05) ( 0.10, 0.03) ( 0.00, 0.27)

with 80 percent bootstrapped confidence intervals. The adjustment of the real
exchange rate is quite slow with only 11 percent of the gap being closed each
quarter. We also note that the confidence interval includes the zero value. Thus, the
adjustment is not even significantly different from zero at the chosen level. The
evidence of statistically significant adjustment in the second equation indicates that a
2
The complete estimated model is presented in Table 7.
8
weak exchange rate has a positive effect on output, which makes sense for an
economy with a large export sector.
We next restrict the model by imposing the theoretically derived cointegration
vector of the previous section. The restriction is not rejected (the probability value is
0.34). The estimated adjustment coefficients for this model are
̂´  0.09 , 0.02 ,  0.05 , 0.05  .

( 0.14, 0.00) ( 0.01, 0.04) ( 0.09, 0.00) ( 0.02, 0.14)

The cointegration relation for the restricted model is shown in Figure 3, Panel B. For
this specification the confidence interval for the adjustment in the real exchange rate
does not include zero (by a small margin). From this perspective the restrictions can
be deemed successful.
Next we consider a model with richer dynamics. Since the theory does not have
anything to say about the dynamics of the model we can freely choose to include
variables that we feel could be important in the short run. A natural candidate is the
short-term real interest rate differential, which we add to the restricted model.3 . Thus
our new vector of endogenous variables is X t*  (qt , yt , t , bt , cit )' , where cit is the
cumulated real interest rate differential. The cumulated variable is restricted to zero in
the cointegration vector so that we only end up with the interest rate differential, it , in
the dynamics. The interest rate differential lagged one period enters with a negative
sign and the coefficient is statistically significant at the 10 percent level (using
bootstrapped t-values); the two period lag is not significant. The estimated
adjustment coefficients for this model are
̂´  0.12 , 0.03 ,  0.04 , 0.03 , 0.52  .

( 0.17, 0.02) ( 0.01, 0.04) ( 0.08, 0.02) ( 0.06, 0.13)
( 3.89, 3.00)

The cointegration relation for the restricted model is shown in Figure 3, Panel C. The
adjustment in the real exchange rate is now clearly significant at the 80 percent
confidence level.
3.4 The estimated alternative model (AM)
In the alternative model we have replaced the net foreign asset variable with the
trade balance relative to GDP. Another difference is that we have excluded the
inflation regime shift dummy from this model. Using the inflation regime shift variable
3
The differential consists of the Swedish 3-month interest rate compared to a weighted 3-month rate
consisting of interest rates for Sweden’s main trading partners (using the TCW weights for Sweden).
9
in this model leads to very strange parameter estimates. The shift dummy is highly
significant in the trade balance equation but the estimated coefficients in the
cointegration vector become extremely large. We have therefore excluded the shift
dummy from the model.
All of the information criteria considered suggest that one lag should be
adequate. But the lag exclusion test favours keeping 2 lags in the model. Estimating
a VAR with one lag there is some evidence of serial correlation in the terms of trade
residuals. With two lags this serial correlation goes away. Since it is not clear which
model is to be preferred we choose to proceed with both k=1 and k=2.
Tests for the cointegration rank for the VAR(1) and VAR(2) models respectively
give no reason to give up our prior belief in one cointegration vector. As in the
benchmark model, bootstrapping the trace test critical values we cannot reject the
null hypothesis of no cointegration. But a likelihood ratio test of r=1 against r=0 does
not reject r=1 (with a probability value of 0.55). We will restrict the constant term to
the cointegration space to exclude deterministic drifts in the individual series. This
implies that in the long run the trade balance will converge to some constant fraction
of GDP.
We check for non-constant parameters using the single non-zero eigenvalue, as
explained in the previous section. The time path of the estimated non-zero
eigenvalue is depicted in Figure 4. It looks reasonably stable. The estimated
cointegrating relation is shown in Figure 5, Panel A.
The cointegration vector normalized with respect to the real exchange rate and
the vector of adjustment coefficients are given by
̂ ´ 1.0,  0.67 , 0.34 ,  0.88  ,

( 1.87, 0.69) ( 0.19, 0.82) ( 2.37, 0.54)

̂´  0.01 , 0.03 ,  0.05 , 0.01  ,
( 0.03,0.06)
( 0.02, 0.05) ( 0.09, 0.01) ( 0.01, 0.03)

for the VEC with k=1 and for the VEC with k=2 we have4
̂ ´ 1.0,  0.56 , 0.45 ,  1.18  ,

( 1.66, 0.63) ( 0.01, 0.85) ( 2.47, 0.14)

̂´   0.04 , 0.04 ,  0.06 , 0.02  ,
( 0.10,0.03)
( 0.02, 0.05) ( 0.09, 0.01) ( 0.01, 0.04)

with 80 percent bootstrapped confidence intervals in parentheses below the
estimated coefficients (using bootstrapped t-values). The estimated coefficients of the
VAR(2) model comes closest to what we expected to find. We note that the point
estimates of ˆ 2 and ̂ 3 are quite close to the values we expected them to take.
4
The complete estimated models are presented in Tables 12 and 13 respectively.
10
Disappointingly, ˆ1 does not take the expected sign. However, the 80 percent
confidence interval also contains positive values. The adjustment coefficient in the
exchange rate equation is halved compared with the benchmark model and it is not
statistically significantly different from zero at the 20 percent level.
We proceed by imposing the theoretically derived coefficient values in the
cointegration vector of the VEC with k=2. The restriction is not rejected at the 1
percent significance level (but is rejected at the 5 percent level). The estimated
adjustment coefficients for this model are
̂´  0.08 , 0.03 ,  0.06 , 0.01  .

( 0.14, 0.01) ( 0.02, 0.05) ( 0.10, 0.01) ( 0.02, 0.03)

The adjustment coefficient in the exchange rate equation is significantly different from
zero at the 20 percent level and is equal in size to that estimated for the restricted
benchmark model: -0.08 compared to -0.09. The cointegration relation for the
restricted model is shown in Figure 5, Panel B.
The estimated adjustment coefficients for the restricted model with an interest
rate differential are
̂´  0.14 , 0.03 ,  0.03 , 0.02 ,  1.48  ,

( 0.19, 0.06) ( 0.02, 0.04) ( 0.07, 0.02) ( 0.01, 0.05) ( 4.03,1.82)

with 80 percent bootstrapped confidence intervals. The interest rate differential
lagged one period enters with a negative sign, which is significantly different from
zero at the 5 percent level (using bootstrapped t-values).
3.5 Estimating the nominal effective exchange rate
When forecasting exchange rates the interest is usually in the nominal rather than in
the real exchange rate. We will consider a very simple way of extending the analysis
to get a forecasting model for the nominal exchange rate. This simply involves
splitting up the real exchange rate into two components: the nominal exchange rate
and the relative price level between the foreign and domestic country, qt  et  pt
(the variables are shown in Figure 1c). We estimate the nominal versions of the
restricted models augmented with real interest rate differentials. Thus our new vector
of endogenous variables is X t*  (et , pt , yt , t , bt , cit )' for the benchmark model and
X t*  (et , pt , yt , t , tbt , cit )' for the alternative model.
11
Estimating the nominal version of the benchmark model we obtain the following
adjustment parameters:5

( 0.17, 0.02)


̂ '    0.12 ,  0.05 , 0.03 ,  0.03, 0.03 ,  1.53  ,
( 0.08, 0.02) ( 0.01, 0.05) ( 0.07, 0.03) ( 0.06, 0.14) ( 4.97, 2 , 46)
with 80 percent bootstrapped confidence intervals within parenthesis below the
estimated coefficients. We can see that the adjustment of the real exchange rate
takes place both through the adjustment of the nominal exchange rate and through
the adjustment of the relative price level. However, the point estimate for the
adjustment coefficient in the nominal exchange rate equation is twice as big as the
one in the relative price level equation. This is not surprising given that most of the
sample covers the floating exchange rate period.
Estimating the nominal version of the alternative model we get the following
adjustment parameters:6

( 0.16, 0.03)


̂ '    0.11 ,  0.04 , 0.03 ,  0.02, 0.02 ,  1.74  ,
( 0.06, 0.02) ( 0.01, 0.04) ( 0.06, 0.02) ( 0.01, 0.05) ( 4.21,1.36)
with 80 percent bootstrapped confidence intervals. These results are very similar to
those of the benchmark model.
3.6 Discussion
Generally speaking, most of the estimated coefficients in the cointegration vector are
imprecisely estimated and are not different from zero at the 20 percent significance
level, using bootstrapped t-values. However, most of the point estimates are close to
the values we would expect from the calibrated model. When we impose the
theoretically derived cointegration vectors they are not rejected by the data, i.e. we
cannot reject cointegration. The estimated adjustment coefficients in the exchange
rate equation and the relative output equation are significant with the expected signs.
The real test of the models will be their out-of-sample forecast performance, to which
we turn next.
5
The complete estimated model is presented in Table 4.
The complete estimated model is presented in Table 5. We note that the output from the Eviews
program in Table 5 gives an adjustment coefficient of -5.6 in the interest rate differential equation,
which is not even inside the 80 percent confidence interval presented in the text, which is generated
by Warne’s Structural VAR program.
6
12
4 Forecast evaluation
4.1 The evaluation procedure
We will evaluate forecasts of the percentage change in the exchange rate with
horizons h=1,2,…,12 quarters. The first forecast will be based on the model
estimated for the period 1985Q1 - 1997Q4. The estimation window is then moved
forward by one quarter at a time using a rolling scheme.7 Thus, we are always using
the latest 13 years of data to estimate the model, which gives us 52 observations.
The final 1-period forecast is for 2005Q3, which is the last observation in the sample.
We thus get a total of P-h+1 h-period forecasts with P=31. Forecast accuracy is
measured by mean squared prediction error (MSPE). With at  s  h being the actual
value and f t  s  h the forecast value, the MSPE for an h-period forecast is defined as:
ˆ 2 (h) 
Ph
1
(at  s  h  f t  s  h ) 2 ,

P  h  1 s 0
(6)
where t=1997Q4. We will compare our forecasts with those of the random walk
model, which says that the exchange rate will remain unchanged at the present level
during the forecast period. We will also compare our forecasts to those of a VAR in
first differences, i.e. leaving out the error correction term. This should tell us
something about the value for forecasting purposes of using the theoretically derived
relationship between the variables in levels.
Meese and Rogoff (1983) found that none of the exchange rate models
developed during the 1970s could beat a random walk and this has been the
benchmark for exchange rate forecasters ever since. In the spirit of Meese and
Rogoff, Cheung, Chinn and Pascual (2005) evaluate the forecasting performance of
the models developed during the 1990s, with equally disappointing results. However,
as shown by Clark and McCracken (2001) among others, the tests employed in these
studies are not appropriate when the models under comparison are nested. The tests
are shown to be under-sized. Critical values are derived for the correct asymptotic
distributions, but only for 1-step ahead forecasts.8
Clark and West (2006a, 2006b) show that there is an upward bias in the nesting
model’s sample mean squared prediction error and how one can correct for this. The
7
We also conducted the analysis using a recursive scheme. The results were very similar. We do not
use a recursive scheme since the inference of our tests are not strictly valid for such a scheme (see
Clark and West (2006a)).
8 West (2006) gives a useful overview of forecast evaluation techniques.
13
test is applicable also to multi-step forecasts and is easy to implement. We will use
the bias correction suggested by Clark and West. Their MSPE-adjusted test statistic
is defined as
MSPEadj  ˆ12  (ˆ 22  adj.) ,
(7)
where subscript 1 indicates the parsimonious (null) model and subscript 2 indicates
the larger (alternative) model that nests model 1 (we have suppressed the horizon
index h). The adjustment term is defined as
adj.(h) 
N h
1
 ( f1,t  s h  f 2,t  s h ) 2 .
N  h  1 s 0
(8)
This term adjusts for the upward bias in MSPE produced by the estimation of
parameters that are zero under the null.
We implement the test by first defining


2
2
2
~
f t  s h  at  s h  f1,t  s h   at  s h  f 2,t  s h    f1,t  s h  f 2,t  s h  ,
(9)
~
The null hypothesis is equal MSPE. We test for equal MSPE by regressing f t  s  h on
a constant and using the resulting t-statistic to test for a zero coefficient. Clark and
West (2006a, 2006b) argue that standard normal critical values are approximately
correct. We thus reject the null hypothesis of equal MSPE if this statistic is greater
than +1.282 (for a one-tailed 0.10 test) or +1.645 (for a one-tailed 0.05 test). A onetailed test is used since for nested models it does not make sense for the population
MSPE of the parsimonious model to be smaller than that of the larger model. We use
Newey-West autocorrelation consistent standard errors to construct the t-statistic
because multi-period forecasts will in general be autocorrelated.
Finally, we would like to draw attention to the fact that the seminal paper by
Meese and Rogoff (1983) and many subsequent studies use the actual values for
their exogenous regressors and still could not beat a random walk. Our exercise is
more challenging in that the forecasts only involve endogenous variables. We make
fully dynamic forecasts, i.e. a multi-period forecast uses predicted rather than actual
values for all variables during the forecast period.
4.2 Evaluating the forecasts
In Table 6 we present the MSPEs for forecasts of the benchmark real exchange rate
model (VEC), a random walk (RW), and a VAR in first differences, as well as the
adjusted MSPEs. In the last two columns of the table we present the usual ratio of
14
the (unadjusted) MSPEs and the t-values of the Clark-West statistic (for the adjusted
MSPEs). We first compare the change in the exchange rate forecasted by the model
to a random walk forecast, which says that the exchange rate will remain unchanged.
We see that the forecasts of the unrestricted model (reported in Panel A) as well as
those of the restricted model (in Panel B) cannot beat a random walk. However, the
forecasts of the restricted model augmented with an interest rate differential (reported
in Panel C) are significantly better than those of a random walk at horizons of 5-10
quarters at the 10 percent significance level, using a one-tailed test. Thus, it is not
enough to impose the theoretical cointegration vector. We also need to model the
dynamics better by introducing some financial variable like the real interest rate
differential. Comparing the forecasts of the benchmark model to those from a simple
VAR in first differences, in Panel C, we see that the restricted cointegration vector of
the VEC model helps us to make better forecasts, which lends some support to the
theoretical model. Figure 6a well illustrates the curse put on anyone attempting to
forecast the SEK/TCW exchange rate using fundamentals. It seems that no matter
what, the Krona is always expected to appreciate in value. Figure 6b shows how the
curse is lifted by including the interest rate differential in the dynamics of the model.
Especially noteworthy are the forecasts in the quarters preceeding the substantial
depreciation of the Krona in 2000.
In Table 7 we analyse the forecast performance of the alternative real exchange
rate model. The results are similar to those reported for the benchmark model in that
we need to augment the dynamics with an interest rate differential in order to beat a
random walk forecast. The alternative model yields better forecasts in the short run
but worse forecasts in the long run compared to the benchmark model (Panel C).
The forecasts from the nominal versions of the benchmark and alternative
models are presented in Table 8, Panels A and B respectively. The predictive ability
is not quite as good as for the real models, even though the forecasts of the
alternative model are still fairly good. The AM forecasts outperform a random walk at
all horizons. They do not outperform the forecasts from a simple VAR in first
differences at conventional significance levels, but most of the t-values are close to
one for all forecast horizons.
The overall assessment is that we cannot reject the theoretical model as a
description of how the real effective exchange rate is related to fundamental
macroeconomic variables. But this conclusion is dependent on our ability to
15
successfully model the dynamics about which the theoretical model has very little to
say. We are successful in beating the random walk forecasts for both the real and
nominal SEK/TCW exchange rates.
5 Forecasting the effective exchange rate indirectly
Given that one is interested in forecasting not only the effective exchange rate but
also a few of its largest components one would like these forecasts to be consistent
with each other. One way of assuring that this is indeed the case is to model the
component bilateral rates and then weight these together to form an effective
exchange rate that is approximately equal to the broader effective exchange rate
index. In this section we will estimate models of the SEK/EUR and SEK/USD
exchange rates. The forecasts from these models will then be weighted together to
form an approximate SEK/TCW forecast. The use of the theoretical model in
modelling the bilateral exchange rates gives rise to a few issues that we first need to
address.
When we analyse a country’s real effective exchange rate the implementation of
the theory is quite straightforward. There is one issue concerning the output variable.
Lane and Milesi-Ferretti state that the level of output should be understood to be
measured relative to output overseas, since a global increase in tradable output
would increase the relative price on non-tradables in all countries (leaving the real
effective exchange rate unchanged). With regard to the real effective exchange rate
the net foreign asset position as well as the terms of trade will have symmetric
effects. The home country’s increased wealth, whether as a result of an increase in
net foreign assets or improved terms of trade, reflects the decrease in wealth in the
rest of the world (due to the same factor). However, this is not necessarily the case if
we are considering a bilateral exchange rate. Both countries’ wealth could increase
(or decrease) reflecting a decrease (or increase) in the wealth of the rest of the world.
Thus, when considering a bilateral exchange rate the implementation of the theory is
less straightforward. However, it seems that a similar way of reasoning as above
concerning the output variable should apply to the other two variables as well; we
should be considering the relative net foreign asset position and the relative terms of
trade of the two countries whose bilateral exchange rate we wish to model.
16
5.1 The estimated SEK/EUR model
To determine the number of lags to use in the VAR we start with a look at some lag
order selection criteria. The LR, FPE and AIC criteria indicated that a lag order of k=3
is optimal, while the SC and HQ criteria deemed 1 lag to be sufficient. Lag exclusion
tests indicated that we should keep 3 lags in the model. Multivariate tests of serially
uncorrelated residuals for the k=3 model showed that the null hypothesis of no serial
correlation in the residuals cannot be rejected at the 5 percent significance level
against the alternative hypotheses of 1st – 4th order serial correlation. Based on the
above we decided to use 3 lags. Next we performed tests of the normality of the
residuals. The null hypothesis that the residuals are normally distributed was
rejected, using the test statistics derived by Doornik and Hansen (1994). As in the
SEK/TCW model the reason for this is that the kurtosis values are lower than those
from a normal distribution, while there was no indication of skewness in the residuals.
Tests for the cointegration rank (r) are reported in Table 9 for the VAR(3) model.
We conclude from this test that there is one cointegrating vector, which is what we
would expect from the theoretical model. Bootstrapping was then employed. First, we
find that bootstrapped critical values for the trace statistics are quite a bit larger than
the asymptotic critical values so that we cannot reject the null of no cointegration. On
the other hand a likelihood ratio test does not reject the hypothesis of r=1 against the
alternative of r=0 at conventional significance levels. In our judgement the evidence
against cointegration is not strong enough for us to give up our prior belief in one
cointegration vector.
We computed the sample means and standard deviations of the firstdifferenced series. None of the means of these series are significantly different from
zero, see Table 9. We therefore restrict the constant term to the cointegration space
to exclude deterministic drifts in the individual series. This implies that in the long run,
or in the steady state, output in Sweden will grow at the same rate as in the Euro
area, export and import prices will grow at the same rate, and net foreign assets will
converge to some constant fraction of GDP.
We have included one dummy variable in the estimated vector error correction
model. This is a devaluation dummy that takes the value one in the 4 th quarter of
1992 and zero otherwise and captures the real depreciation after letting the Krona
float on 19 November 1992. This dummy variable was found to be significant both
contemporaneously and lagged one period. We also include the shift dummy used in
17
the SEK/TCW model, which captures the change to an inflation targeting regime in
the 1st quarter of 1995.
We used the estimated non-zero eigenvalue to check for non-constant
parameters. There were no signs of instability, indicating no problem with nonconstant parameters in the cointegration relation.
The cointegration vector normalized with respect to the real SEK/EUR
exchange rate is given by9
̂ ´ 1.0, 0.50 , 0.40 ,  0.17  ,

( 0.15,1.29) ( 0.22, 0.91) ( 0.38, 0.00)

with 80 percent bootstrapped confidence intervals in parentheses under the
estimated coefficients (using bootstrapped t-values). The coefficients are imprecisely
estimated. However, we note that the point estimates of ˆ1 and ˆ 2 are fairly close to
the values we expected them to take. On the other hand, ̂ 3 does not take the
expected sign. The estimated adjustment coefficients are
̂´  0.10 , 0.04 ,  0.02 , 0.16  ,

( 0.18, 0.02) ( 0.02, 0.07) ( 0.06, 0.03) ( 0.04, 0.26)

with 80 percent bootstrapped confidence intervals. The adjustment of the real
exchange rate is taking place with about 10 percent of the gap being closed each
quarter.
We next restrict the model by imposing the theoretically derived cointegration
vector of the previous section. The restriction is rejected at the 5 percent level but not
at the 1 percent level (the probability value is 0.04). The estimated adjustment
coefficients for this model are
̂´  0.08 , 0.03 ,  0.04 , 0.10  .

( 0.13, 0.03) ( 0.01, 0.05) ( 0.07, 0.00) ( 0.02, 0.18)

The results are very similar to those of the unrestricted model.
The estimated adjustment coefficients for the restricted model with an interest
rate differential are
̂´  0.08 , 0.03 ,  0.04 , 0.11 ,  2.65  ,

( 0.13, 0.03) ( 0.02, 0.05) ( 0.07, 0.00) ( 0.02, 0.20) ( 5.72, 0.64)

with 80 percent bootstrapped confidence intervals. Again, the results are basically the
same as in the previous specifications.
9
The complete estimated model is presented in Table 11.
18
For the nominal SEK/EUR model we obtain the following adjustment
parameters:

( 0.14, 0.05)


̂ '    0.07 ,  0.06 , 0.04 ,  0.02, 0.11 ,  1.97  .
( 0.08, 0.03) ( 0.02, 0.06) ( 0.06, 0.03) ( 0.01, 0.22) ( 5.05,1.38)
According to the point estimates the adjustment of the real exchange rate takes place
in equal measure through the adjustment of the nominal exchange rate and the
relative price levels (the first two coefficients). However, the adjustment coefficient in
the nominal exchange rate equation is not significantly different from zero at the 20
percent significance level, since the 80 per cent confidence interval contains the zero
value.
Overall, the evidence supporting the theoretical model is rather weak, perhaps
reflecting the fact that the model is more appropriately regarded as a model of the
real effective exchange rate.
5.2 The estimated SEK/USD model
We start with a look at some lag order selection criteria in order to determine the
number of lags to use in the VAR. The LR, FPE and AIC criteria indicated that a lag
order of k=4 is optimal, while the SC and HQ criteria found 1 lag to be sufficient. Lag
exclusion tests indicated that we should keep 4 lags in the model. Multivariate tests
of serially uncorrelated residuals for the k=4 model showed that the null hypothesis of
no serial correlation in the residuals cannot be rejected at the 5 percent significance
level against the alternative hypotheses of 1st – 5th order serial correlation. Based on
the evidence presented above we decided to use 4 lags. Next we performed tests of
the normality of the residuals. The null hypothesis that the residuals are normally
distributed was rejected, using the test statistics derived by Doornik and Hansen
(1994). As in the previously estimated models the reason for this is that the kurtosis
values are lower than those from a normal distribution, while there was no indication
of skewness in the residuals.
Tests for the cointegration rank (r) are reported in Table 13 for the VAR(4)
model. We conclude from this test that there is one cointegrating vector, which is
what we would expect from the theoretical model. Bootstrapping was then used in
deriving more accurate critical values for the test statistics. First, we find that
bootstrapped critical values for the trace statistics are so large that we cannot reject
the null of no cointagration. The choice of r=1 leaves a root of 0.84 in the system.
19
Based on these tests we find that the evidence against cointegration is not so strong
that we are willing to give up our theoretically motivated prior belief in one
cointegration vector.
We computed the sample means and standard deviations of the firstdifferenced series. None of the means in the first-differenced series are significantly
different from zero, see Table 12. We therefore restrict the constant term to the
cointegration space to exclude deterministic drifts in the individual series. This implies
that in the long run output in Sweden will grow at the same rate as in the U.S., export
and import prices will grow at the same rate, and net foreign assets will converge to
some constant fraction of GDP.
We have used the same set of dummy variables that were used in the
SEK/EUR model, described in the previous subsection.
The cointegration vector normalized with respect to the real SEK/USD
exchange rate is given by10
̂ ´ 1.0,  1.25 , 1.22 ,  0.14  ,

( 4.53,1.18) ( 1.03, 2.57) ( 0.62, 0.38)

with 80 percent bootstrapped confidence intervals in parentheses under the
estimated coefficients (using bootstrapped t-values). The coefficients are imprecisely
estimated. Contrary to the previous models, the point estimates of ˆ1 and ˆ 2 are not
close to the values we expected them to take. As before, ̂ 3 does not take the
expected sign. Somewhat disturbingly, none of the coefficients are significantly
different from zero at the 20 percent significance level. The estimated adjustment
coefficients are
̂´  0.12 , 0.02 , 0.01 , 0.01  ,

( 0.34, 0.03) ( 0.00, 0.06) ( 0.02, 0.06) ( 0.05, 0.09)

with 80 percent bootstrapped confidence intervals. The adjustment coefficients in the
exchange rate equation and the relative output equation take the expected sign and
are significantly different from zero at the 20 percent level. The adjustment of the real
exchange rate indicates that 12 percent of the gap is being closed each quarter.
However, since the estimated cointegration vector is far from the one we expected it
is difficult to see what one should make of this.
10
The complete estimated model is presented in Table 14.
20
We next restrict the model by imposing the theoretically derived cointegration
vector of the previous section. The restriction is not rejected (the probability value is
0.17). The estimated adjustment coefficients for this model are
̂´  0.08 , 0.01 , 0.01 , 0.00  .

( 0.12, 0.03) ( 0.00, 0.02) ( 0.01, 0.02) ( 0.03, 0.03)

The adjustment coefficients in the exchange rate equation and the relative output
equation have the expected signs and are significantly different from zero at the 20
percent level. Hence, we get some support for the theoretical model here.
The estimated adjustment coefficients for the restricted model with an interest
rate differential are
̂´  0.10 , 0.01 , 0.01 ,  0.02 , 0.35  ,

( 0.13, 0.04) ( 0.00, 0.02) ( 0.00, 0.03) ( 0.05, 0.01) ( 1.13,1.68)

with 80 percent bootstrapped confidence intervals. The results are basically the same
as in the previous specification, with an additional variable (the real interest rate
differential) that is long-run weakly exogenous with regard to the cointegrating
relationship.
For the nominal SEK/USD restricted model with interest rate differential we
obtain the following adjustment parameters:


̂ '    0.08 ,  0.01 , 0.01 , 0.02 ,  0.02, 0.05  .
( 0.14, 0.06) ( 0.02, 0.00) ( 0.00, 0.02) ( 0.00, 0.03) ( 0.05, 0.01) ( 0.95,1.59) 
We can see that most of the adjustment of the real exchange rate takes place
through the adjustment of the nominal exchange rate and very little through the
adjustment of the relative price levels.
The theoretically derived restrictions seem to fit better for the SEK/USD model
than for the SEK/EUR model. In the next subsection we will see if this tentative
conclusion has any implications for the forecast evaluation exercises.
5.3 Evaluating the forecasts
In Table 15 we present the MSPEs for the benchmark real SEK/EUR exchange rate
model; the unrestricted model (reported in Panel A), the restricted model (in Panel B)
and the restricted model augmented with an interest rate differential (reported in
Panel C). Almost all of the t-values are negative and thus our VEC model is unable to
beat a random walk. With the help of Figure 10 it is easy to see why the random walk
has been so difficult to beat during the evaluation period. The SEK/EUR exchange
21
rate has been quite stable during this period. The results for the nominal model with
an interest rate differential look somewhat better (reported in Panel D). Most of the tvalues are positive and the ones at the 10 – 12 quarter horizons are even statistically
significant at the 10 percent level, even though the unadjusted ratios of the MSPEs
actually exceed one. However, at these longest horizons we have few forecasts and
we should be a bit careful interpreting these results.
The initially estimated SEK/USD model yielded forecast MSPEs that rapidly
grew very large with increasing horizon. We therefore tried to trim the number of lags
and found that the model with one lag (in differenced form) yielded the smallest
MSPEs. This then became our preferred model.11 The results for the SEK/USD
model overall look more promising than those for the SEK/EUR model (see Figure
16). The restricted model is able to beat a random walk at short horizons (1 – 6
quarters), and the forecasts are better than those of a VAR in first differences (Panel
B). When we look at the forecasts of the model with an interest rate differential (in
Panel C) we see that it is able to beat a random walk and a VAR at all horizons. The
unadjusted ratios of MSPEs are quite small; most of them below 0.5.12 It is clear from
Figure 11 why the random walk is a very poor model for the SEK/USD during the
evaluation period, with large swings in the exchange rate. The figure also shows that
the VEC model yields surprisingly good forecasts. The results are very similar for the
nominal version of the model (presented in Panel D).
We next consider an approximate SEK/TCW forecast, which we will compare to
the actual SEK/TCW effective exchange rate. We use renormalized TCW weights to
weigh together the nominal SEK/EUR forecasts and the nominal SEK/USD forecasts
both based on the restricted model with an interest rate differential. In order to start
off the forecast at the correct level a factor of proportionality between the actual
SEK/TCW index and the weighted SEK/EUR – SEK/USD index is computed for the
quarter preceding each forecast. This factor is then applied to the forecast path. In
Table 17 the MSPEs for the approximate SEK/TCW forecasts are evaluated. The
model forecasts are not able to beat a random walk at any horizon; only a few of the
t-values are positive. It is small comfort that the forecasts are better than those from a
VAR in first differences at horizons above four quarters.
11
We had previously also tried to trim the SEK/TCW and SEK/EUR models, but this resulted in poorer
forecast performance.
12 We note that the some of the adjusted MSPEs in the third column of Panel C are negative, making
the interpretation of an adjusted MSPE a bit problematic.
22
In Table 18 we evaluate forecasts of the implicit nominal USD/EUR exchange
rate obtained as the cross of the nominal SEK/EUR forecasts and the nominal
SEK/USD forecasts, again based on the restricted model with an interest rate
differential. In view of the poor forecasting performance of the SEK/EUR model,
these results look surprisingly good. The VEC model is able to beat both a random
walk and a VAR at horizons of 2 – 9 quarters at the 10 percent significance level or
better.
6 Conclusions
In this paper we have estimated models of the real and nominal SEK/TCW,
SEK/EUR and SEK/USD exchange rates using the Johansen approach. Imposing
values derived from a calibrated theoretical model on the cointegrating vectors is
shown to improve the forecasting performance in general.
Augmenting the dynamics of the model by including an interest rate differential
further improves the forecast performance, generating forecasts for the real as well
as nominal SEK/TCW exchange rates that are superior to those of a random walk at
horizons of around one to two years.
For the real and nominal SEK/EUR exchange rate the results are not as
impressive. The model is not able to beat a random walk at any horizon. Model
forecasts of the real and nominal SEK/USD exchange rate are superior to those of a
random walk at all horizons.
Weighting the nominal SEK/EUR and SEK/USD forecasts together into an
approximate SEK/TCW index did not yield good forecasts. On the other hand, and
somewhat surprisingly, the model forecasts of the implicit nominal USD/EUR
exchange rate are superior to those of a random walk at most horizons.
Generally speaking, it seems like a good theoretical model and careful
modelling of the dynamics would enable you to construct a model that yields
forecasts that are superior to those of a random walk. One caveat is that the sample
considered in this paper is quite short, so it would be interesting to follow up on these
results at a later date.
23
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25
Appendix: Data
The data set spans the period 1984Q1 to 2005Q3. The data set thus allows for up to
4 lags in the models. Current data series are imported from EcoWin to the largest
extent possible (the series name is given within square brackets below). Some of the
series have been linked back in time to earlier series as indicated. Some earlier data
for the Euro zone are taken from the Area Wide Model (See Fagan, Henry, and
Mestre (2001)).
Real exchange rates
The real SEK/EUR is obtained from crossing the SEK/USD and EUR/USD, which are
both available in EcoWin. Consumer price indexes are used to convert nominal
exchange rates into real exchange rates.







SEK/USD Kronor per one US dollar [oecd:swe_ccusma02_stq].
EUR/USD Euros per one US dollar [oecd:emu_ccusma02_stq].
SEK/TCW The nominal effective exchange rate using the IMF:s
competitiveness weights (1991 vintage). The fixed exchange rate of the Krona
was abandoned on 19 November 1992. Therefore, the index is set equal to
100 on 18 November 1992, which can be translated into a value of 106.17 for
the 4th quarter of 1992. The real index has as its base November 1992=100.
This translates into a value of 105.75 for the 4th quarter of 1992. (Source:
Sveriges Riksbank).
Sweden, consumer price index [oecd:swe_cpaltt01_ixobq].
Euro zone, HICP consumer price index [emu_cphptt01_ixnbq]. This series in
available starting in 1990Q1. Data for 1984Q2-1989Q4 are taken from the
Area Wide Model.
U.S.A., consumer price index [oecd:usa_cpaltt01_ixobq].
TCW-weighted consumer price index (Source: Sveriges Riksbank).
Terms of trade
Terms of trade is computed as the ratio between export prices and import prices.






Sweden, export price index [ifs:s1447400dzfq].
Sweden, import price index [ifs:s1447500dzfq].
Euro zone, export price index. Data for 1984Q1-2003Q4 are taken from the
Area Wide Model. Data for 2004Q1-2005Q3 are from ECB Monthly Bulletin.
Euro zone, import price index. Data for 1984Q1-2003Q4 are taken from the
Area Wide Model. Data for 2004Q1-2005Q3 are from ECB Monthly Bulletin.
U.S.A., export price index [ifs:s1117400dzfq].
U.S.A., import price index [ifs:s1117500dzfq].
Output
26




Sweden, gross domestic product, seasonally adjusted [ew:swe_01950]. This
series is available starting 1993Q1. Data for 1984Q2-1992Q4 are from
Sveriges Riksbank. An index is created which is set equal to 100 in 2000.
Euro zone, gross domestic product, seasonally adjusted [ifs:s16399b0rywq].
The series is available starting 1999Q1. Data for 1984Q2-1998Q4 is from
Sveriges Riksbank (this series is updated on a more timely basis compared to
Ecowin[ifs:s16399b0rywq], which could serve as an alternative source
together with the Area Wide Model for extensions back in time). An index is
created which is set equal to 100 in 2000.
U.S.A., gross domestic product [ifs:s11199b0rzfq]. An index is created which
is set equal to 100 in 2000.
TCW-weighted gross domestic product (Source: Sveriges Riksbank).
Net foreign assets
The net foreign asset position relative to nominal GDP is the measure used. Our
preferred choice for net foreign assets is the net international investment position with
market valued foreign direct investment. Unfortunately such a series is available only
for the last few quarters for the Euro zone, which is why cumulated current accounts
have been used in this case.






Sweden, net international investment position, market valued FDI (Source:
Sveriges Riksbank) This is an annual series. A quarterly series has been
obtained by linearly interpolating so that the annual figure is reached in the 4 th
quarter.
Sweden, gross domestic product, current prices [oe:swe_gdpq].
Euro zone, net foreign assets, data 1984Q2-2003Q4 from the Area Wide
Model. This series is updated for 2004Q1-2005Q3 using the computed growth
factors from the cumulated current account balances based on [oe:euro_cbq].
Euro zone, gross domestic product, current prices [oe:euro_gdpq].
U.S.A., net international investment position, market valued FDI
[ew:usa16101]. This is an annual series. A quarterly series has been obtained
by linearly interpolating so that the annual figure is reached in the 4th quarter.
U.S.A., gross domestic product, current prices [oe:usa_gdpq].
27
Table 1. Descriptive statistics of endogenous variables in levels and first differences:
real effective exchange rate, qt , relative output, yt , terms of trade,  t , net foreign
assets, bt , trade balance, tbt , nominal effective exchange rate, et , and relative price,
pt .
Variable
qt
yt
Standard
JarqueMean Maximum Minimum deviation Skewness Kurtosis Bera
4.793
4.977
4.597
0.098
-0.15
2.14
2.86
0.012
0.059
-0.037
0.026
0.04
1.77
5.27
t
bt
4.630
4.711
4.485
0.062
-0.79
2.23
10.59
-0.160
0.215
-0.410
0.154
0.34
2.49
2.52
tbt
0.045
0.089
0.002
0.022
-0.23
1.74
6.25
et
pt
q t
yt
 t
bt
tbt
et
p t
4.749
4.937
4.563
0.115
-0.24
1.46
9.01
-0.004
0.112
-0.073
0.053
0.64
2.46
6.66
0.002
0.110
-0.076
0.025
0.88
6.82
61.09
0.000
0.010
-0.012
0.005
-0.14
2.76
0.47
0.000
0.054
-0.031
0.014
1.29
6.06
55.18
0.001
0.068
-0.070
0.030
0.15
2.67
0.69
0.0004
0.035
-0.031
0.009
-0.01
6.46
41.34
0.004
0.130
-0.073
0.025
1.40
10.24
208.48
-0.001
0.013
-0.034
0.008
-1.72
7.99
126.99
28
Table 2. Johansen cointegration test
Sample (adjusted): 1985Q1 2005Q3
Included observations: 83 after adjustments
Trend assumption: No deterministic trend (restricted constant)
Series: qt , yt ,  t , bt
Lags interval (in first differences): 1 to 2
Unrestricted Cointegration Rank Test (Trace)
Hypothesized
No. of CE(s)
Eigenvalue
Trace
Statistic
0.05
Critical Value
Prob.**
None *
At most 1
At most 2
At most 3
0.315642
0.159804
0.112177
0.035330
58.79283
27.31303
12.86102
2.985445
54.07904
35.19275
20.26184
9.164546
0.0179
0.2734
0.3750
0.5834
Trace test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized
No. of CE(s)
Eigenvalue
Max-Eigen
Statistic
0.05
Critical Value
Prob.**
None *
At most 1
At most 2
At most 3
0.315642
0.159804
0.112177
0.035330
31.47980
14.45201
9.875576
2.985445
28.58808
22.29962
15.89210
9.164546
0.0207
0.4215
0.3458
0.5834
Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):
qt
yt
t
bt
-20.16761
5.246184
-2.055866
3.086618
-18.86939
51.49897
-10.96593
-15.08201
-10.68840
11.27004
-22.01646
6.920338
3.581216
-6.387902
-5.696369
-3.734962
Constant
147.1160
-79.08079
110.9626
-46.92688
Unrestricted Adjustment Coefficients (alpha):
q t
yt
 t
bt
0.005788
-0.006232
0.002556
0.002256
-0.001579
-0.000933
0.000654
-0.000241
0.001731
0.002892
0.002887
-0.001149
-0.005841
0.003652
0.002762
0.003342
29
Table 3. The estimated real SEK/TCW unrestricted model
Vector Error Correction Estimates
Sample (adjusted): 1985Q1 2005Q3
Included observations: 83 after adjustments
t-statistics in [ ]
Cointegrating Eq:
CointEq1
qt 1
1.000000
y t 1
0.721723
[ 1.27477]
 t 1
0.497981
[ 2.37698]
bt 1
-0.167318
[-2.03297]
Constant
-7.145720
[-7.38777]
Error Correction:
Adjustment coeff.
qt 1
qt 2
y t 1
yt 2
 t 1
 t  2
q t
yt
 t
bt
-0.105660
[-1.88693]
0.031653
[ 2.88481]
-0.036372
[-0.99109]
0.131003
[ 2.00194]
0.158565
[ 1.50949]
-0.010586
[-0.51428]
0.087042
[ 1.26430]
0.274976
[ 2.23998]
-0.009676
[-0.09124]
0.035030
[ 1.68568]
-0.020194
[-0.29054]
-0.020162
[-0.16268]
1.379069
[ 2.34125]
-0.061501
[-0.53285]
0.158269
[ 0.40997]
-0.884125
[-1.28440]
0.496111
[ 0.85026]
0.021049
[ 0.18411]
0.067015
[ 0.17525]
-1.704901
[-2.50033]
-0.010619
[-0.05877]
-0.017518
[-0.49485]
0.353144
[ 2.98248]
0.067466
[ 0.31955]
-0.032999
[-0.18329]
0.033940
[ 0.96202]
-0.061293
[-0.51944]
0.009220
[ 0.04382]
30
bt 1
0.151809
[ 1.69914]
0.016944
[ 0.96787]
0.069842
[ 1.19274]
0.307913
[ 2.94907]
-0.150554
[-1.67905]
0.017709
[ 1.00790]
-0.071242
[-1.21229]
0.337025
[ 3.21632]
DUMMY92_4
0.073540
[ 3.29461]
-0.003204
[-0.73245]
-0.024501
[-1.67481]
0.043336
[ 1.66131]
DUMMY93_1
0.102368
[ 4.33172]
0.014474
[ 3.12564]
-0.025079
[-1.61919]
-0.049254
[-1.78345]
DUMMY95
-0.005726
[-1.26178]
-2.68E-05
[-0.03013]
0.001742
[ 0.58556]
0.000994
[ 0.18746]
0.408592
0.316966
0.030410
0.020696
4.459325
210.5692
-4.784800
-4.435088
0.002486
0.025041
0.391096
0.296758
0.001168
0.004055
4.145717
345.8510
-8.044602
-7.694890
-0.000207
0.004836
0.217618
0.096404
0.013062
0.013564
1.795317
245.6386
-5.629846
-5.280134
-0.000324
0.014269
0.428525
0.339987
0.041530
0.024185
4.840000
197.6354
-4.473143
-4.123431
0.001050
0.029770
Determinant resid covariance (dof adj.)
Determinant resid covariance
Log likelihood
Akaike information criterion
Schwarz criterion
6.83E-16
3.66E-16
1003.993
-22.91550
-21.37094
bt 2
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
31
Table 4. The nominal SEK/TCW restricted BM model with interest rate differential
Vector Error Correction Estimates
Sample: 1985Q1 2005Q3
Included observations: 83
t-statistics in [ ]
Cointegration Restrictions:
B(1,1)=1,B(1,2)=1,B(1,3)=0.7,B(1,4)=0.7,B(1,5)=0.014,B(1,6)=0
Convergence achieved after 14 iterations.
Restrictions identify all cointegrating vectors
LR test for binding restrictions (rank = 1):
Chi-square(5)
12.39716
Probability
0.029733
Cointegrating Eq:
CointEq1
et 1
1.000000
pt 1
1.000000
y t 1
0.700000
 t 1
0.700000
bt 1
0.014000
cit 1
0.000000
Constant
-8.045304
[-668.579]
Error Correction:
Adjustment coeff.
et 1
et 2
pt 1
pt 2
et
p t
yt
 t
bt
it
-0.115907
[-2.09705]
-0.053863
[-2.83798]
0.030434
[ 2.55059]
-0.027537
[-0.73143]
0.029047
[ 0.40093]
-6.918386
[-2.11493]
0.104140
[ 1.03373]
-0.006360
[-0.18385]
-0.015603
[-0.71745]
0.148649
[ 2.16624]
0.285725
[ 2.16373]
2.151027
[ 0.36077]
-0.031211
[-0.31189]
0.032419
[ 0.94344]
0.047160
[ 2.18299]
-0.041647
[-0.61099]
0.009048
[ 0.06898]
0.733746
[ 0.12389]
0.519122
[ 1.15123]
0.163929
[ 1.05869]
0.202988
[ 2.08521]
-0.683171
[-2.22420]
0.841427
[ 1.42355]
-82.96637
[-3.10877]
-0.404652
[-0.83907]
0.134393
[ 0.81154]
-0.172185
[-1.65385]
0.369047
[ 1.12343]
0.548080
[ 0.86700]
32.45194
[ 1.13697]
32
y t 1
1.000602
[ 1.86566]
0.493932
[ 2.68200]
-0.029123
[-0.25153]
0.053979
[ 0.14776]
-0.591317
[-0.84111]
62.25682
[ 1.96133]
0.552272
[ 0.99935]
0.211955
[ 1.11694]
0.026893
[ 0.22542]
0.162829
[ 0.43256]
-1.357045
[-1.87336]
8.239300
[ 0.25191]
-0.027523
[-0.15843]
0.066514
[ 1.11500]
-0.037143
[-0.99039]
0.350609
[ 2.96289]
0.149993
[ 0.65868]
7.013416
[ 0.68212]
0.036406
[ 0.21206]
0.013802
[ 0.23413]
0.016823
[ 0.45391]
-0.035936
[-0.30730]
-0.010483
[-0.04658]
3.780164
[ 0.37204]
0.159991
[ 1.90818]
-0.024464
[-0.84970]
0.025644
[ 1.41676]
0.056033
[ 0.98112]
0.297877
[ 2.71034]
-3.086917
[-0.62207]
-0.141594
[-1.69909]
0.009289
[ 0.32460]
0.015065
[ 0.83741]
-0.067500
[-1.18912]
0.346751
[ 3.17432]
2.802869
[ 0.56828]
it 1
-0.005457
[-1.89125]
-0.000522
[-0.52704]
-0.000907
[-1.45529]
0.003119
[ 1.58674]
-0.001299
[-0.34341]
0.843587
[ 4.93947]
it  2
0.000805
[ 0.27529]
-0.000267
[-0.26614]
0.001341
[ 2.12303]
-0.000690
[-0.34627]
0.000531
[ 0.13842]
-0.161764
[-0.93423]
DUMMY92_4
0.091619
[ 3.81003]
-0.002579
[-0.31237]
0.001017
[ 0.19589]
-0.036385
[-2.22135]
0.029258
[ 0.92823]
-2.648199
[-1.86075]
DUMMY93_1
0.121065
[ 5.21569]
-0.010765
[-1.35060]
0.011535
[ 2.30198]
-0.031271
[-1.97784]
-0.053330
[-1.75278]
-2.810359
[-2.04573]
DUMMY95
0.001617
[ 0.21476]
-0.006547
[-2.53198]
0.000572
[ 0.35157]
-0.006353
[-1.23867]
0.004544
[ 0.46036]
-0.131004
[-0.29395]
0.531189
0.426232
0.024227
0.019016
5.060991
220.0027
-4.915727
-4.449445
0.003723
0.025104
0.392669
0.256699
0.002857
0.006530
2.887916
308.7220
-7.053542
-6.587260
-0.001195
0.007574
0.411196
0.279374
0.001129
0.004105
3.119327
347.2440
-7.981783
-7.515501
-0.000207
0.004836
0.326723
0.175989
0.011241
0.012953
2.167552
251.8714
-5.683647
-5.217365
-0.000324
0.014269
0.427204
0.298966
0.041626
0.024926
3.331338
197.5396
-4.374447
-3.908165
0.001050
0.029770
0.632793
0.550582
84.86066
1.125423
7.697220
-118.6920
3.245589
3.711872
1.339998
1.678768
yt 2
 t 1
 t 2
bt 1
bt 2
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
Determinant resid covariance (dof adj.)
Determinant resid covariance
Log likelihood
1.81E-20
5.00E-21
1233.313
33
Table 5. The nominal SEK/TCW restricted AM model with interest rate differential.
Vector Error Correction Estimates
Sample: 1985Q1 2005Q3
Included observations: 83
t-statistics in [ ]
Cointegration Restrictions:
B(1,1)=1,B(1,2)=1,B(1,3)=0.7,B(1,4)=0.7,B(1,5)=-1.4,B(1,6)=0
Convergence achieved after 11 iterations.
Restrictions identify all cointegrating vectors
LR test for binding restrictions (rank = 1):
Chi-square(5)
16.13065
Probability
0.006481
Cointegrating Eq:
CointEq1
et 1
1.000000
pt 1
1.000000
y t 1
0.700000
 t 1
0.700000
tbt 1
-1.400000
cit 1
0.000000
Constant
-7.956576
[-700.331]
Error Correction:
Adjustment coeff.
et 1
pt 1
yt 1
et
p t
yt
 t
tbt
it
-0.109513
[-2.37372]
-0.038682
[-2.51555]
0.027775
[ 2.66481]
-0.021765
[-0.70152]
0.024292
[ 1.17594]
-5.609177
[-2.25646]
0.114614
[ 1.24112]
-0.021746
[-0.70651]
0.002813
[ 0.13482]
0.094372
[ 1.51964]
0.059470
[ 1.43820]
2.961879
[ 0.59526]
0.108387
[ 0.36715]
0.334972
[ 3.40431]
0.089771
[ 1.34597]
-0.345692
[-1.74130]
-0.026555
[-0.20089]
-70.64450
[-4.44122]
0.737089
[ 1.44386]
0.505509
[ 2.97094]
0.061541
[ 0.53359]
0.086956
[ 0.25329]
-0.134147
[-0.58686]
59.07517
[ 2.14770]
34
 t 1
-0.023725
[-0.14793]
0.061247
[ 1.14578]
0.004183
[ 0.11545]
0.286975
[ 2.66087]
0.019762
[ 0.27519]
6.644800
[ 0.76896]
0.197186
[ 0.82438]
-0.044205
[-0.55447]
0.011651
[ 0.21560]
-0.013073
[-0.08128]
-0.373864
[-3.49073]
-37.80452
[-2.93332]
it 1
-0.002556
[-1.96337]
-0.001400
[-3.22529]
1.21E-05
[ 0.04100]
0.001011
[ 1.15455]
0.001028
[ 1.76284]
0.787893
[ 11.2312]
DUMMY92_4
0.075059
[ 3.55958]
0.002775
[ 0.39489]
-0.003736
[-0.78413]
-0.027771
[-1.95846]
0.001948
[ 0.20635]
-1.851271
[-1.62941]
DUMMY93_1
0.121437
[ 5.68051]
-0.012642
[-1.77418]
0.013971
[ 2.89267]
-0.026281
[-1.82809]
-0.006357
[-0.66408]
-3.374280
[-2.92940]
0.467910
0.410387
0.027497
0.019276
8.134283
214.7482
-4.957789
-4.695505
0.003723
0.025104
0.350565
0.280356
0.003055
0.006425
4.993145
305.9403
-7.155188
-6.892904
-0.001195
0.007574
0.268105
0.188981
0.001403
0.004355
3.388423
338.2160
-7.932915
-7.670631
-0.000207
0.004836
0.255203
0.174684
0.012435
0.012963
3.169492
247.6817
-5.751367
-5.489083
-0.000324
0.014269
0.187712
0.099897
0.005513
0.008631
2.137581
281.4374
-6.564757
-6.302473
0.000375
0.009098
0.654571
0.617228
79.82767
1.038630
17.52833
-116.1546
3.015774
3.278058
1.339998
1.678768
tbt 1
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
Determinant resid covariance (dof adj.)
Determinant resid covariance
Log likelihood
2.21E-21
1.11E-21
1295.749
35
Table 6. Forecast evaluation: Real SEK/TCW benchmark model
Panel A. Unrestricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
2
-adj.
ˆ VEC
0.589
1.403
2.172
2.940
3.590
4.539
5.086
5.287
6.016
6.563
7.229
8.000
0.515
1.317
2.072
2.795
3.402
4.319
4.822
4.985
5.666
6.163
6.807
7.527
2
ˆ RW
2
-adj.
ˆ VEC
2
ˆ VAR
0.553
1.333
2.084
2.806
3.406
4.319
4.808
4.956
5.611
6.092
6.683
7.379
0.526
1.296
2.168
3.001
3.582
4.555
5.098
5.476
6.097
6.513
7.382
8.569
2
VEC
ˆ
0.474
1.067
1.719
2.407
2.970
3.692
4.114
4.465
4.963
5.240
5.772
6.410
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.24 (-0.67)
1.31 (-2.30)
1.26 (-2.41)
1.22 (-2.03)
1.21 (-1.89)
1.23 (-2.35)
1.24 (-2.37)
1.18 (-1.34)
1.21 (-1.38)
1.25 (-1.42)
1.25 (-1.28)
1.25 (-1.10)
1.12 (-0.79)
1.08 (-0.37)
1.00 ( 0.38)
0.98 ( 0.53)
1.00 ( 0.38)
1.00 ( 0.40)
1.00 ( 0.43)
0.97 ( 0.64)
0.99 ( 0.53)
1.01 ( 0.41)
0.98 ( 0.50)
0.93 ( 0.65)
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.15 ( 0.43)
1.30 (-0.44)
1.30 (-0.85)
1.26 (-0.81)
1.27 (-0.85)
1.31 (-1.06)
1.38 (-1.21)
1.45 (-1.28)
1.51 (-1.33)
1.17 (-1.31)
1.69 (-1.39)
1.79 (-1.55)
1.03 ( 0.35)
1.07 ( 0.11)
1.03 ( 0.28)
1.01 ( 0.39)
1.06 ( 0.28)
1.07 ( 0.28)
1.11 ( 0.15)
1.18 (-0.07)
1.23 (-0.22)
1.29 (-0.35)
1.18 (-0.49)
1.34 (-0.61)
Panel B. Restricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
ˆ
0.544
1.387
2.239
3.035
3.786
4.854
5.667
6.464
7.499
8.376
9.782
11.456
2
-adj.
VEC
ˆ
0.431
1.185
2.012
2.743
3.386
4.329
4.999
5.627
6.478
7.155
8.320
9.786
0.474
1.067
1.719
2.407
2.970
3.692
4.114
4.465
4.963
5.240
5.772
6.410
2
RW
ˆ
-adj.
0.493
1.261
2.035
2.734
3.355
4.287
4.940
5.553
6.384
7.032
8.195
9.629
2
VAR
0.526
1.296
2.168
3.001
3.582
4.555
5.098
5.476
6.097
6.513
7.382
8.569
Panel C. Restricted model with real interest rate differential
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
0.553
1.191
1.471
1.812
2.012
2.550
2.915
3.267
3.673
4.187
5.333
6.654
ˆ
2
-adj.
VEC
ˆ
0.291
0.635
0.889
0.999
1.006
1.373
1.585
1.727
2.044
2.490
3.462
4.649
0.474
1.067
1.719
2.407
2.970
3.692
4.114
4.465
4.963
5.240
5.772
6.410
2
RW
ˆ
2
VEC
-adj.
0.404
0.900
1.155
1.386
1.500
2.002
2.321
2.607
2.952
3.392
4.456
5.678
ˆ
2
VAR
0.489
1.199
1.951
2.706
3.243
4.263
4.946
5.551
6.218
6.881
8.149
9.610
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.17 (1.19)
1.05 (0.98)
0.86 (1.11)
0.75 (1.18)
0.68 (1.33)
0.69 (1.41)
0.71 (1.49)
0.73 (1.58)
0.74 (1.73)
0.80 (1.56)
0.92 (1.25)
1.04 (0.97)
1.13 (0.98)
0.99 (1.41)
0.86 (2.14)
0.67 (2.15)
0.62 (2.20)
0.60 (2.45)
0.59 (2.64)
0.73 (2.94)
0.59 (3.21)
0.61 (3.62)
0.65 (4.07)
0.69 (3.88)
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded.
36
Table 7. Forecast evaluation: Real SEK/TCW alternative model
Panel A. Unrestricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
2
-adj.
ˆ VEC
0.532
1.172
1.899
2.522
3.075
3.851
4.346
4.620
5.093
5.327
5.846
6.610
0.468
1.113
1.826
2.440
2.979
3.741
4.223
4.479
4.933
5.144
5.639
6.376
2
ˆ RW
2
-adj.
ˆ VEC
2
ˆ VAR
0.521
1.166
1.894
2.517
3.068
3.841
4.330
4.599
5.065
5.290
5.798
6.551
0.485
1.109
1.847
2.489
3.006
3.742
4.213
4.482
4.897
5.105
5.597
6.288
2
VEC
ˆ
0.474
1.067
1.719
2.407
2.970
3.692
4.114
4.465
4.963
5.240
5.772
6.410
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.12 ( 0.12)
1.10 (-0.36)
1.10 (-0.69)
1.05 (-0.17)
1.04 (-0.03)
1.04 (-0.13)
1.06 (-0.24)
1.03 (-0.03)
1.03 ( 0.05)
1.02 ( 0.16)
1.01 ( 0.20)
1.03 ( 0.04)
1.10 (-0.88)
1.06 (-1.35)
1.03 (-1.72)
1.01 (-0.95)
1.02 (-0.98)
1.03 (-0.84)
1.03 (-0.70)
1.03 (-0.50)
1.04 (-0.62)
1.04 (-0.64)
1.04 (-0.62)
1.05 (-0.64)
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.05 ( 0.62)
1.09 (-0.08)
1.12 (-0.43)
1.09 (-0.36)
1.10 (-0.56)
1.12 (-0.86)
1.17 (-1.34)
1.19 (-1.43)
1.20 (-1.22)
1.20 (-1.11)
1.18 (-0.99)
1.19 (-1.07)
1.03 ( 0.27)
1.05 (-0.07)
1.04 (-0.08)
1.05 (-0.17)
1.08 (-0.48)
1.10 (-0.70)
1.14 (-1.03)
1.19 (-1.31)
1.22 (-1.46)
1.23 (-1.40)
1.22 (-1.34)
1.21 (-1.46)
Panel B. Restricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
ˆ
0.498
1.163
1.925
2.617
3.259
4.120
4.797
5.319
5.962
6.272
6.832
7.638
2
-adj.
VEC
ˆ
0.425
1.082
1.825
2.500
3.116
3.948
4.589
5.069
5.663
5.924
6.422
7.166
0.474
1.067
1.719
2.407
2.970
3.692
4.114
4.465
4.963
5.240
5.772
6.410
2
RW
ˆ
-adj.
0.466
1.119
1.866
2.539
3.159
3.994
4.640
5.124
5.727
5.996
6.509
7.268
2
VAR
0.485
1.109
1.847
2.489
3.006
3.742
4.213
4.482
4.897
5.105
5.597
6.288
Panel C. Restricted model with real interest rate differential
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
0.417
0.945
1.481
2.045
2.516
3.298
3.854
4.359
4.962
5.346
5.984
7.011
ˆ
2
-adj.
VEC
ˆ
0.307
0.819
1.323
1.887
2.341
3.108
3.644
4.125
4.707
5.094
5.691
6.667
0.474
1.067
1.719
2.407
2.970
3.692
4.114
4.465
4.963
5.240
5.772
6.410
2
RW
ˆ
2
VEC
-adj.
0.372
0.897
1.432
2.005
2.484
3.273
3.836
4.343
4.945
5.323
5.952
6.962
ˆ
2
VAR
0.451
1.013
1.659
2.282
2.778
3.573
4.046
4.482
4.998
5.336
5.892
6.839
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
0.88 ( 2.25)
0.89 ( 1.75)
0.86 ( 1.82)
0.85 ( 2.35)
0.85 ( 2.23)
0.89 ( 2.00)
0.94 ( 1.42)
0.98 ( 0.84)
1.00 ( 0.49)
1.02 ( 0.26)
1.04 ( 0.15)
1.09 (-0.40)
0.92 ( 1.22)
0.93 ( 1.19)
0.89 ( 1.59)
0.90 ( 1.57)
0.91 ( 1.69)
0.92 ( 1.78)
0.95 ( 1.45)
0.97 ( 1.10)
0.99 ( 0.45)
1.00 ( 0.08)
1.02 (-0.25)
1.03 (-0.33)
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded.
37
Table 8. Forecast evaluation: Nominal SEK/TCW, restricted model
with real interest rate differential
Panel A. Benchmark model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
0.522
1.194
1.659
2.290
3.133
4.429
5.696
6.958
9.024
11.516
15.052
20.274
ˆ
2
-adj.
VEC
ˆ
0.323
0.834
1.248
1.667
2.237
3.093
3.732
4.112
5.019
5.912
7.225
9.716
0.425
0.984
1.568
2.220
2.755
3.394
3.750
4.069
4.450
4.704
5.052
5.568
2
RW
ˆ
2
VEC
-adj.
ˆ
2
VAR
0.466
1.111
1.601
2.238
3.085
4.348
5.523
6.607
8.381
10.404
13.230
17.376
0.454
1.164
1.827
2.420
2.989
3.804
4.389
4.742
5.452
6.281
7.603
9.388
2
-adj.
ˆ VEC
2
ˆ VAR
0.370
0.865
1.294
1.699
2.041
2.544
2.911
3.188
3.655
3.931
4.269
4.853
0.409
0.936
1.454
1.942
2.334
2.870
3.147
3.335
3.717
3.955
4.352
5.105
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.23 ( 0.95)
1.21 ( 0.52)
1.06 ( 0.78)
1.37 ( 0.88)
1.14 ( 0.64)
1.30 ( 0.27)
1.52 ( 0.01)
1.71 (-0.02)
2.03 (-0.18)
2.45 (-0.31)
2.98 (-0.48)
3.64 (-0.75)
1.15 (-0.19)
1.03 ( 0.44)
0.91 ( 1.77)
0.95 ( 1.26)
1.05 (-0.53)
1.16 (-1.41)
1.30 (-1.43)
1.47 (-1.37)
1.66 (-1.35)
1.83 (-1.34)
1.98 (-1.32)
2.16 (-1.34)
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
0.96 (1.46)
0.95 (1.17)
0.89 (1.52)
0.82 (2.14)
0.79 (2.27)
0.79 (2.23)
0.81 (2.04)
0.82 (1.85)
0.86 (1.52)
0.87 (1.36)
0.88 (1.41)
0.91 (1.32)
0.99 (0.70)
1.00 (0.60)
0.96 (0.81)
0.93 (0.89)
0.93 (0.93)
0.93 (0.88)
0.81 (0.56)
1.00 (0.30)
1.03 (0.12)
1.04 (0.04)
1.02 (0.13)
0.99 (0.40)
Panel B. Alternative model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
0.406
0.932
1.397
1.815
2.172
2.679
3.055
3.342
3.819
4.105
4.457
5.057
2
-adj.
ˆ VEC
0.328
0.787
1.171
1.559
1.874
2.354
2.666
2.892
3.319
3.608
3.970
4.604
2
ˆ RW
0.425
0.984
1.568
2.220
2.755
3.394
3.750
4.069
4.450
4.704
5.052
5.568
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded
38
Table 9. Descriptive statistics of endogenous variables in levels and first differences:
real SEK/EUR exchange rate, qt , relative output, yt , relative terms of trade,  t ,
relative net foreign assets, bt , nominal SEK/EUR exchange rate, et , and relative
price, p t .
Variable
qt
yt
t
bt
et
pt
q t
yt
 t
bt
et
p t
Mean
Standard
JarqueMaximum Minimum deviation Skewness Kurtosis Bera
2.111
2.274
1.926
0.093
-0.27
1.93
4.94
0.006
0.063
-0.062
0.033
-0.13
1.97
3.92
0.017
0.096
-0.137
0.061
-1.03
2.95
14.79
-0.145
0.216
-0.377
0.141
0.36
2.73
2.10
2.110
2.273
1.846
0.115
-0.51
1.86
8.15
0.002
0.131
-0.074
0.057
0.74
2.63
8.06
0.003
0.106
-0.069
0.026
0.58
5.45
25.42
<0.001
0.016
-0.015
0.006
-0.13
3.24
0.44
-0.002
0.029
-0.025
0.010
0.57
3.45
5.23
0.001
0.067
-0.070
0.030
0.10
2.74
0.38
0.005
0.125
-0.067
0.026
0.95
7.70
88.99
-0.001
0.013
-0.033
0.008
-1.49
6.98
85.48
39
Table 10. Johansen cointegration test
Sample: 1985Q1 2005Q3
Included observations: 83
Trend assumption: No deterministic trend (restricted constant)
Series: LNRSEKEUR LNY_SEEU LNTOT_SEEU NFA_SEEU
Lags interval (in first differences): 1 to 2
Unrestricted Cointegration Rank Test (Trace)
Hypothesized
No. of CE(s)
None *
At most 1
At most 2
At most 3
Eigenvalue
Trace
Statistic
0.05
Critical Value
Prob.**
0.363604
0.160632
0.070195
0.024015
60.10271
22.59218
8.058406
2.017603
54.07904
35.19275
20.26184
9.164546
0.0132
0.5556
0.8195
0.7742
Trace test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized
No. of CE(s)
None *
At most 1
At most 2
At most 3
Eigenvalue
Max-Eigen
Statistic
0.05
Critical Value
Prob.**
0.363604
0.160632
0.070195
0.024015
37.51053
14.53377
6.040803
2.017603
28.58808
22.29962
15.89210
9.164546
0.0028
0.4144
0.7834
0.7742
Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
40
Table 11. The estimated real SEK/EUR unrestricted model
Vector Error Correction Estimates
Sample: 1985Q1 2005Q3
Included observations: 83
t-statistics in [ ]
Cointegrating Eq:
CointEq1
qt 1
1.000000
yt 1
0.503960
[ 1.58173]
 t 1
0.402381
[ 1.66822]
bt 1
-0.170795
[-2.19522]
Constant
-2.152460
[-137.587]
Error Correction:
Adjustment coeff.
q t  1
q t  2
y t  1
y t  2
 t 1
 t 2
bt 1
q t
yt
 t
bt
-0.097566
[-1.71016]
0.041227
[ 3.20999]
-0.016361
[-0.61275]
0.160939
[ 2.68749]
0.115841
[ 1.02775]
-0.017513
[-0.69020]
0.019580
[ 0.37117]
0.288783
[ 2.44085]
-0.024933
[-0.22247]
0.029075
[ 1.15233]
-0.012079
[-0.23027]
-0.061913
[-0.52627]
0.911675
[ 1.77925]
-0.045989
[-0.39868]
0.035760
[ 0.14912]
-1.095665
[-2.03714]
0.567378
[ 1.08505]
-0.007916
[-0.06724]
0.140779
[ 0.57524]
-1.491781
[-2.71786]
-0.012187
[-0.04698]
-0.019382
[-0.33189]
0.261356
[ 2.15269]
0.420652
[ 1.54485]
0.106949
[ 0.40775]
-0.018412
[-0.31182]
-0.116826
[-0.95166]
-0.178961
[-0.65000]
0.121486
0.013586
0.080486
0.320806
41
[ 1.23532]
[ 0.61366]
[ 1.74867]
[ 3.10772]
-0.182753
[-1.88521]
0.010288
[ 0.47140]
-0.097852
[-2.15671]
0.322369
[ 3.16805]
DUMMY92_4
0.056403
[ 2.37508]
0.001929
[ 0.36076]
-0.013549
[-1.21901]
0.043974
[ 1.76407]
DUMMY93_1
0.093192
[ 3.84800]
0.019027
[ 3.48978]
-0.023468
[-2.07041]
-0.038744
[-1.52408]
DUMMY95
-0.003121
[-0.69208]
-4.31E-05
[-0.04245]
-0.000448
[-0.21206]
0.001930
[ 0.40779]
0.342649
0.240805
0.035332
0.022308
3.364473
204.3436
-4.634787
-4.285075
0.003391
0.025602
0.361226
0.262261
0.001791
0.005022
3.650037
328.1057
-7.617005
-7.267293
5.31E-05
0.005847
0.126251
-0.009118
0.007739
0.010441
0.932643
267.3600
-6.153252
-5.803540
-0.002326
0.010393
0.459449
0.375702
0.038929
0.023416
5.486142
200.3197
-4.537825
-4.188113
0.000561
0.029635
Determinant resid covariance (dof adj.)
Determinant resid covariance
Log likelihood
5.62E-16
3.01E-16
1012.142
bt 2
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
42
Table 12. Descriptive statistics of endogenous variables in levels and first
differences: real SEK/USD exchange rate, qt , relative output, yt , relative terms of
trade,  t , relative net foreign assets, bt , nominal SEK/USD exchange rate, et , and
relative price, p t .
Variable
qt
yt
t
bt
et
pt
q t
yt
 t
bt
et
p t
Standard
JarqueMean Maximum Minimum deviation Skewness Kurtosis Bera
1.976
2.361
1.590
0.185
0.24
2.48
1.75
0.059
0.162
-0.010
0.058
0.54
1.65
10.29
-0.005
0.076
-0.139
0.059
-0.59
2.02
8.09
-0.080
0.443
-0.391
0.220
0.59
2.15
7.37
2.002
2.360
1.677
0.164
0.29
2.47
2.15
-0.026
0.061
-0.111
0.053
-0.17
1.70
6.23
-0.002
0.164
-0.081
0.050
0.83
3.94
12.65
-0.002
0.010
-0.021
0.006
-0.68
3.80
8.61
0.000
0.034
-0.043
0.014
-0.45
3.24
2.97
0.004
0.087
-0.069
0.032
0.33
3.00
1.49
-0.002
0.186
-0.086
0.050
1.08
4.97
29.45
0.000
0.013
-0.034
0.009
-1.30
5.68
48.43
43
Table 13. Johansen cointegration test
Sample: 1985Q1 2005Q3
Included observations: 83
Trend assumption: No deterministic trend (restricted constant)
Series: LNNSEKUSD LNCPI_USSE LNY_SEUS LNTOT_SEUS NFA_SEUS
Lags interval (in first differences): 1 to 3
Unrestricted Cointegration Rank Test (Trace)
Hypothesized
No. of CE(s)
Eigenvalue
Trace
Statistic
0.05
Critical Value
Prob.**
None *
At most 1 *
At most 2
At most 3
At most 4
0.427051
0.252907
0.164417
0.138233
0.055842
102.4536
56.22615
32.02618
17.11723
4.769332
76.97277
54.07904
35.19275
20.26184
9.164546
0.0002
0.0318
0.1056
0.1282
0.3097
Trace test indicates 2 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized
No. of CE(s)
None *
At most 1
At most 2
At most 3
At most 4
Eigenvalue
Max-Eigen
Statistic
0.05
Critical Value
Prob.**
0.427051
0.252907
0.164417
0.138233
0.055842
46.22749
24.19997
14.90895
12.34790
4.769332
34.80587
28.58808
22.29962
15.89210
9.164546
0.0015
0.1646
0.3826
0.1667
0.3097
Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
44
Table 14. The estimated real SEK/USD unrestricted model
Vector Error Correction Estimates
Sample: 1985Q1 2005Q3
Included observations: 83
t-statistics in [ ]
Cointegrating Eq:
CointEq1
qt 1
1.000000
yt 1
-1.253421
[-1.37556]
 t 1
1.222797
[ 2.43515]
bt 1
-0.142756
[-0.91365]
Constant
-2.056432
[-57.1279]
Error Correction:
Adjustment coeff.
qt 1
qt 2
qt 3
yt 1
yt 2
yt 3
q t
yt
 t
bt
-0.124461
[-2.82573]
0.018874
[ 2.67165]
0.008044
[ 0.52320]
0.014644
[ 0.51270]
0.270349
[ 2.31960]
-0.014144
[-0.75664]
0.015654
[ 0.38478]
0.182530
[ 2.41499]
-0.016937
[-0.14393]
0.004447
[ 0.23561]
-0.005554
[-0.13520]
0.030153
[ 0.39513]
0.392245
[ 3.53430]
-0.018475
[-1.03786]
-0.118550
[-3.06013]
0.013855
[ 0.19250]
1.684907
[ 2.11584]
0.025671
[ 0.20099]
0.074889
[ 0.26941]
-0.697160
[-1.35000]
0.946670
[ 1.18578]
0.172823
[ 1.34965]
-0.307933
[-1.10498]
-0.653517
[-1.26228]
-1.889067
[-2.62575]
-0.034710
[-0.30080]
0.262028
[ 1.04339]
0.423850
[ 0.90847]
45
 t 1
-0.358333
[-0.97282]
0.031580
[ 0.53454]
0.228351
[ 1.77600]
0.530422
[ 2.22055]
0.030354
[ 0.08149]
0.005104
[ 0.08543]
0.125631
[ 0.96620]
0.233551
[ 0.96683]
0.574684
[ 1.62930]
-0.007307
[-0.12916]
-0.280459
[-2.27789]
-0.249025
[-1.08870]
0.401559
[ 2.52597]
0.011859
[ 0.46508]
-0.109977
[-1.98185]
0.558570
[ 5.41812]
-0.457066
[-2.65153]
0.018615
[ 0.67327]
0.091866
[ 1.52674]
0.436390
[ 3.90378]
0.129938
[ 0.78507]
-0.010912
[-0.41106]
-0.008449
[-0.14625]
-0.459735
[-4.28325]
DUMMY92_4
0.218089
[ 5.27036]
-0.011917
[-1.79547]
-0.025851
[-1.78967]
0.022862
[ 0.85194]
DUMMY93_1
0.136390
[ 2.77252]
0.014325
[ 1.81549]
-0.038009
[-2.21343]
-0.085596
[-2.68311]
DUMMY95
-0.045599
[-3.84033]
0.002025
[ 1.06342]
0.006198
[ 1.49537]
0.001185
[ 0.15395]
0.587697
0.495390
0.084447
0.035502
6.366783
168.1829
-3.667057
-3.200774
-0.001763
0.049978
0.284021
0.123727
0.002172
0.005694
1.771879
320.0837
-7.327318
-6.861035
-0.002167
0.006083
0.393588
0.257824
0.010290
0.012393
2.899063
255.5397
-5.772040
-5.305758
0.000300
0.014385
0.586723
0.494198
0.035514
0.023023
6.341258
204.1301
-4.533255
-4.066972
0.004210
0.032372
Determinant resid covariance (dof adj.)
Determinant resid covariance
Log likelihood
2.68E-15
1.14E-15
956.9135
 t 2
 t 3
bt 1
bt 2
bt 3
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
46
Table 15. Forecast evaluation: SEK/EUR benchmark model
Panel A. Real exchange rate: unrestricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
2
-adj.
ˆ VEC
0.764
1.847
2.661
3.406
3.996
4.849
5.219
4.820
4.289
3.932
3.545
3.405
0.636
1.517
2.213
2.792
3.178
3.861
4.066
3.653
3.024
2.462
1.857
1.505
2
ˆ RW
2
-adj.
ˆ VEC
2
ˆ VAR
0.663
1.620
2.266
2.816
3.143
3.764
3.902
3.436
2.735
2.081
1.379
0.924
0.623
1.556
2.310
2.945
3.085
3.573
3.629
3.503
3.298
3.407
4.019
4.995
0.519
1.135
1.811
2.421
2.642
2.921
2.878
2.671
2.545
2.460
2.705
3.109
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.47 (-0.92)
1.62 (-1.20)
1.47 (-1.32)
1.41 (-1.23)
1.51 (-1.09)
1.66 (-1.35)
1.81 (-1.55)
1.80 (-1.54)
1.69 (-1.09)
1.60 (-0.00)
1.31 ( 1.00)
1.10 ( 1.11)
1.23 (-0.33)
1.19 (-0.15)
1.20 ( 0.07)
1.16 ( 0.16)
1.30 (-0.06)
1.36 (-0.19)
1.44 (-0.26)
1.38 ( 0.07)
1.30 ( 0.68)
1.15 ( 1.08)
0.88 ( 1.22)
0.68 ( 1.25)
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.44 (-0.68)
1.75 (-1.15)
1.54 (-1.11)
1.40 (-0.99)
1.39 (-0.83)
1.20 (-1.13)
1.26 (-1.37)
1.37 (-1.67)
1.55 (-1.95)
1.70 (-3.06)
1.71 (-4.49)
1.64 (-4.34)
1.20 (-0.48)
1.28 (-0.66)
1.21 (-0.48)
1.15 (-0.21)
1.19 (-0.18)
1.20 (-0.11)
1.26 (-0.22)
1.37 (-0.42)
1.55 (-1.00)
1.70 (-1.64)
1.71 (-2.13)
1.64 (-2.27)
Panel B. Real exchange rate: restricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
ˆ
0.749
1.988
2.794
3.397
3.681
4.287
4.590
4.789
5.125
5.794
6.856
8.201
2
-adj.
VEC
ˆ
0.609
1.572
2.324
2.865
2.991
3.423
3.577
3.614
3.786
4.280
5.155
6.322
0.519
1.135
1.811
2.421
2.642
2.921
2.878
2.671
2.545
2.460
2.705
3.109
2
RW
ˆ
2
VEC
-adj.
0.679
1.814
2.541
3.061
3.199
3.660
3.811
3.848
4.018
4.529
5.450
6.676
ˆ
2
VAR
0.623
1.556
2.310
2.945
3.085
3.573
3.629
3.503
3.298
3.407
4.019
4.995
Panel C. Real exchange rate: restricted model with real interest rate differential
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
0.656
1.663
2.476
3.164
3.549
4.141
4.206
3.967
3.846
3.958
4.391
5.446
ˆ
2
-adj.
VEC
ˆ
0.558
1.437
2.258
2.919
3.219
3.717
3.713
3.383
3.175
3.232
3.587
4.560
0.519
1.135
1.811
2.421
2.642
2.921
2.878
2.671
2.545
2.460
2.705
3.109
2
RW
ˆ
2
VEC
-adj.
0.619
1.590
2.391
3.074
3.423
3.989
4.023
3.747
3.574
3.631
3.996
4.983
ˆ
2
VAR
0.561
1.411
2.149
2.857
3.117
3.638
3.608
3.225
2.870
2.632
2.816
3.576
47
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.26 (-0.41)
1.47 (-1.28)
1.37 (-1.42)
1.31 (-1.58)
1.34 (-1.86)
1.42 (-2.40)
1.46 (-3.23)
1.49 (-2.68)
1.51 (-1.71)
1.61 (-2.09)
1.62 (-1.70)
1.75 (-2.12)
1.17 (-0.65)
1.18 (-0.70)
1.15 (-0.75)
1.11 (-0.57)
1.14 (-0.71)
1.14 (-0.70)
1.17 (-0.78)
1.23 (-1.01)
1.34 (-1.42)
1.50 (-2.11)
1.56 (-2.30)
1.52 (-2.44)
Panel D. Nominal exchange rate: restricted model with real interest rate differential
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
0.630
1.476
1.968
2.553
2.977
3.523
3.540
3.116
2.742
2.384
2.232
2.456
2
-adj.
ˆ VEC
0.536
1.308
1.805
2.354
2.707
3.180
3.127
2.622
2.185
1.794
1.552
1.675
2
ˆ RW
0.490
1.107
1.745
2.342
2.594
2.845
2.762
2.548
2.319
2.140
2.118
2.264
2
-adj.
ˆ VEC
2
ˆ VAR
0.606
1.439
1.943
2.528
2.949
3.500
3.523
3.103
2.731
2.373
2.220
2.441
0.576
1.391
2.056
2.742
3.199
3.733
3.710
3.230
2.764
2.343
2.173
2.366
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.29 (-0.65)
1.33 (-0.69)
1.13 ( 0.23)
1.09 ( 0.43)
1.15 ( 0.11)
1.24 (-0.25)
1.28 (-0.32)
1.22 ( 0.25)
1.18 ( 0.89)
1.11 ( 1.59)
1.05 ( 1.87)
1.08 ( 2.48)
1.09 (-0.67)
1.06 (-0.43)
0.96 ( 1.37)
0.93 ( 2.46)
0.93 ( 1.90)
0.94 ( 1.81)
0.95 ( 1.81)
0.96 ( 1.52)
0.99 ( 0.44)
1.02 (-0.42)
1.03 (-0.67)
1.04 (-1.10)
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded.
48
Table 16. Forecast evaluation: SEK/USD benchmark model
Panel A. Real exchange rate: unrestricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
1.547
3.923
7.289
12.227
19.323
27.951
35.670
42.826
50.717
57.611
64.375
70.646
2
-adj.
ˆ VEC
1.255
3.111
5.799
10.023
16.407
24.322
31.417
37.823
45.117
51.235
57.146
63.346
2
ˆ RW
1.862
4.760
8.258
13.246
19.916
27.660
35.443
43.204
51.402
58.889
64.803
68.936
2
-adj.
ˆ VEC
2
ˆ VAR
1.478
3.810
7.062
11.859
18.733
27.106
34.476
41.333
49.061
56.116
62.582
68.528
1.513
3.944
7.106
11.405
17.752
25.992
34.149
41.761
50.792
58.400
65.245
71.479
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
0.83 (2.09)
0.82 (1.60)
0.88 (1.18)
0.92 (0.95)
0.97 (0.70)
1.01 (0.50)
1.01 (0.49)
0.99 (0.55)
0.99 (0.53)
0.98 (0.55)
0.99 (0.53)
1.02 (0.39)
1.02 ( 0.39)
0.99 ( 0.43)
1.03 ( 0.07)
0.92 (-0.55)
0.97 (-0.83)
1.01 (-0.69)
1.04 (-0.15)
1.03 ( 0.16)
1.00 ( 0.52)
0.99 ( 0.52)
0.99 ( 0.52)
0.99 ( 0.54)
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
0.73 (2.76)
0.75 (1.94)
0.78 (1.49)
0.79 (1.46)
0.79 (1.46)
0.82 (1.34)
0.84 (1.27)
0.85 (1.29)
0.89 (1.04)
0.93 (0.88)
0.98 (0.72)
1.06 (0.36)
0.90 (2.62)
0.90 (2.18)
0.91 (1.95)
0.91 (1.76)
0.89 (1.95)
0.87 (2.07)
0.87 (2.00)
0.88 (1.71)
0.90 (1.41)
0.94 (1.01)
0.97 (0.74)
1.02 (0.40)
Panel B. Real exchange rate: restricted model
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
1.364
3.561
6.458
10.423
15.809
22.695
29.778
36.731
45.955
54.754
63.226
72.983
2
-adj.
ˆ VEC
1.126
2.911
5.299
8.735
13.546
19.826
26.212
32.364
40.675
48.409
55.643
64.240
2
ˆ RW
1.862
4.760
8.258
13.246
19.916
27.660
35.443
43.204
51.402
58.889
64.803
68.936
2
-adj.
ˆ VEC
2
ˆ VAR
1.315
3.415
6.151
9.880
14.932
21.397
28.073
34.608
43.143
51.346
58.851
67.436
1.513
3.944
7.106
11.405
17.752
25.992
34.149
41.761
50.792
58.400
65.245
71.479
Panel C. Real exchange rate: restricted model with real interest rate differential
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
1.145
2.090
2.855
4.036
4.989
5.824
7.120
9.234
11.863
15.970
19.594
25.966
ˆ
2
-adj.
VEC
0.742
0.761
0.021
-0.963
-3.075
-6.119
-9.540
-12.979
-15.994
-18.652
-20.268
-20.477
ˆ
2
RW
1.862
4.760
8.258
13.246
19.916
27.660
35.443
43.204
51.402
58.889
64.803
68.936
ˆ
2
VEC
-adj.
0.912
1.367
1.295
1.355
0.706
-0.472
-1.155
-0.986
-1.503
-0.124
-0.551
0.893
ˆ
2
VAR
1.292
2.592
4.017
6.583
10.592
16.125
22.155
29.509
39.080
48.222
57.438
70.118
49
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
0.61 (2.74)
0.44 (2.72)
0.35 (2.58)
0.30 (2.56)
0.25 (3.14)
0.21 (3.41)
0.20 (3.58)
0.21 (3.62)
0.23 (3.63)
0.27 (3.61)
0.30 (3.52)
0.38 (3.18)
0.89 (2.84)
0.81 (2.22)
0.71 (2.10)
0.61 (2.06)
0.47 (2.44)
0.36 (2.62)
0.32 (2.63)
0.31 (2.58)
0.30 (2.65)
0.33 (2.52)
0.34 (2.49)
0.37 (2.42)
Panel D. Nominal exchange rate: restricted model with real interest rate differential
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
1.297
2.441
3.382
4.796
5.528
6.284
7.999
10.030
12.139
16.684
22.311
31.766
2
-adj.
ˆ VEC
0.940
1.347
1.122
0.810
-0.883
-3.290
-5.634
-8.685
-11.666
-13.977
-15.042
-15.053
2
ˆ RW
1.696
4.395
7.764
12.666
18.994
26.294
33.997
41.719
49.463
57.038
63.054
67.513
2
-adj.
ˆ VEC
2
ˆ VAR
1.110
1.927
2.366
3.111
2.951
2.563
3.325
4.424
4.628
7.596
10.269
15.525
1.172
2.224
3.290
5.246
8.018
12.139
17.353
24.374
32.915
42.863
54.619
70.822
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
0.76 (1.66)
0.56 (2.21)
0.44 (2.52)
0.38 (2.74)
0.29 (2.81)
0.24 (2.91)
0.24 (3.34)
0.24 (3.37)
0.25 (3.24)
0.29 (3.18)
0.35 (3.06)
0.47 (2.87)
1.11 (0.40)
1.10 (0.63)
1.03 (0.91)
0.91 (1.10)
0.69 (1.74)
0.52 (2.11)
0.46 (2.21)
0.41 (2.38)
0.37 (2.58)
0.39 (2.55)
0.41 (2.57)
0.45 (2.50)
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded
Table 17. Forecast evaluation: approximate nominal SEK/TCW exchange rate
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
ˆ
2
VEC
0.739
3.791
6.272
8.769
11.061
12.863
13.420
11.687
9.347
6.970
5.526
5.635
ˆ
2
-adj.
VEC
ˆ
0.286
1.994
3.287
4.715
5.716
6.532
6.239
4.731
3.602
2.421
1.322
1.400
0.425
0.984
1.568
2.220
2.755
3.394
3.750
4.069
4.450
4.704
5.052
5.568
2
RW
ˆ
2
VEC
-adj.
0.590
3.526
6.013
8.449
10.577
12.236
12.671
10.814
8.206
5.596
3.745
3.300
ˆ
2
VAR
0.563
3.230
6.154
9.160
11.975
14.293
15.283
13.898
12.031
9.412
8.273
9.181
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.74 ( 0.89)
3.85 (-1.37)
4.00 (-1.50)
3.95 (-1.58)
4.01 (-1.49)
3.79 (-1.28)
3.58 (-0.98)
2.87 (-0.28)
2.10 ( 0.37)
1.48 ( 1.00)
1.09 ( 1.66)
1.01 ( 1.79)
1.31 (-0.30)
1.17 (-0.55)
1.02 ( 0.24)
0.96 ( 1.07)
0.92 ( 1.56)
0.90 ( 1.89)
0.88 ( 2.04)
0.84 ( 2.13)
0.78 ( 2.24)
0.74( 2.00)
0.67 ( 2.22)
0.61 ( 2.72)
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded
Table 18. Forecast evaluation: implicit nominal USD/EUR exchange rate
Horizon
1
2
3
4
5
6
7
8
9
10
11
12
2
ˆ VEC
1.767
2.964
3.883
5.698
7.104
8.933
12.047
15.035
18.310
24.063
31.361
42.240
2
-adj.
ˆ VEC
1.437
2.172
2.126
2.346
1.614
0.514
-0.174
-1.923
-3.641
-4.640
-3.703
-1.730
2
ˆ RW
1.756
4.320
7.206
11.621
17.240
23.826
31.460
39.105
46.532
54.432
60.373
64.292
2
-adj.
ˆ VEC
2
ˆ VAR
1.620
2.523
2.914
4.029
4.598
5.315
7.518
9.607
11.030
15.262
19.650
26.351
1.697
3.055
4.155
6.355
9.514
14.363
20.845
28.972
38.704
51.028
64.728
82.164
2
2
/ ˆ RW
ˆ VEC
2
2
/ ˆ VAR
ˆ VEC
(t-value)
(t-value)
1.01 (0.75)
0.69 (1.62)
0.54 (1.74)
0.49 (1.83)
0.41 (2.27)
0.37 (2.47)
0.38 (2.61)
0.38 (2.74)
0.39 (2.73)
0.44 (2.64)
0.52 (2.43)
0.66 (2.13)
1.04 (0.46)
0.97 (1.06)
0.93 (1.34)
0.90 (1.35)
0.75 (1.86)
0.62 (2.10)
0.58 (2.12)
0.52 (2.26)
0.47 (2.45)
0.47 (2.46)
0.48 (2.46)
0.51 (2.35)
NOTE. Cases when the Clark-West test statistic is significant at the 10 percent level or better have been shaded
50
Figure 1a. Graphs of endogenous variables in levels and first differences: the real
effective exchange rate, relative output, and the terms of trade.
LNTCWR
D_LNTCWR
5.0
.12
4.9
.08
4.8
.04
4.7
.00
4.6
-.04
4.5
-.08
86
88
90
92
94
96
98
00
02
04
86
88
90
92
LNY
94
96
98
00
02
04
98
00
02
04
00
02
04
D_LNY
.06
.012
.008
.04
.004
.02
.000
-.004
.00
-.008
-.02
-.012
-.04
-.016
86
88
90
92
94
96
98
00
02
04
86
88
90
92
LNTOT
94
96
D_LNTOT
4.72
.06
4.68
.04
4.64
.02
4.60
.00
4.56
-.02
4.52
4.48
-.04
86
88
90
92
94
96
98
00
02
04
86
51
88
90
92
94
96
98
Figure 1b. Graphs of endogenous variables in levels and first differences: net foreign
assets, trade balance and differential between 3-month interest rates in Sweden and
its trading partners.
NFA
D_NFA
.3
.08
.2
.06
.1
.04
.0
.02
-.1
.00
-.2
-.02
-.3
-.04
-.4
-.06
-.5
-.08
86
88
90
92
94
96
98
00
02
04
86
88
90
92
EXIM
94
96
98
00
02
04
98
00
02
04
00
02
04
D_EXIM
.09
.04
.08
.03
.07
.02
.06
.01
.05
.00
.04
-.01
.03
-.02
.02
-.03
.01
-.04
.00
86
88
90
92
94
96
98
00
02
04
86
88
90
92
94
96
I_SE_TCW
8
6
4
2
0
-2
86
52
88
90
92
94
96
98
Figure 1c. Graphs of endogenous variables in levels and first differences: the nominal
effective exchange rate and the relative consumer price level.
LNTCWN
D_LNTCWN
5.0
.16
.12
4.9
.08
4.8
.04
4.7
.00
4.6
-.04
4.5
-.08
86
88
90
92
94
96
98
00
02
04
86
88
LNCPI_TCWSE
90
92
94
96
98
00
02
04
02
04
D_LNCPI_TCWSE
.12
.02
.01
.08
.00
.04
-.01
.00
-.02
-.04
-.03
-.08
-.04
86
88
90
92
94
96
98
00
02
04
86
88
90
92
94
96
98
00
Figure 2. Recursive estimates of the eigenvalue with 95% confidence bands –
benchmark model.
53
Figure 3. Cointegration relations for the real SEK/TCW benchmark model.
Panel A. Unrestricted model
.12
.08
.04
.00
-.04
-.08
-.12
-.16
-.20
86
88
90
92
94
96
98
00
02
04
92
94
96
98
00
02
04
Panel B. Restricted model.
.15
.10
.05
.00
-.05
-.10
-.15
-.20
86
88
90
Panel C. Restricted model with interest rate differential.
.12
.08
.04
.00
-.04
-.08
-.12
-.16
-.20
-.24
86
88
90
92
94
96
98
54
00
02
04
Figure 4. Recursive estimates of the eigenvalue with 95% confidence bands –
alternative model.
Figure 5. Cointegration relations for the real SEK/TCW alternative model.
Panel A. Unrestricted model
.15
.10
.05
.00
-.05
-.10
-.15
86
88
90
92
94
96
98
00
02
04
92
94
96
98
00
02
04
Panel B. Restricted model.
.15
.10
.05
.00
-.05
-.10
-.15
-.20
86
88
90
55
Panel C. Restricted model with interest rate differential.
.10
.05
.00
-.05
-.10
-.15
-.20
86
88
90
92
94
96
98
56
00
02
04
Figure 6a. Real SEK/TCW exchange rate and 1-8 quarter forecasts: BM model with
restrictions (corresponding to Table 6, Panel B).
150
145
140
135
130
125
120
jun
-9
8
9
0
1
2
3
4
5
98
99
00
01
02
03
04
-9
-0
-0
-0
-0
-0
-0
cccccccn
n
n
n
n
n
e
e
e
e
e
e
e
u
u
u
u
u
u
j
j
j
j
j
j
jun
d
d
d
d
d
d
d
Figure 6b. Real SEK/TCW exchange rate and 1-8 quarter forecasts: BM model with
restrictions and real interest rate differential (corresponding to Table 6, Panel C).
150
145
140
135
130
125
120
jun
-9
8
9
0
1
2
3
4
5
98
99
00
01
02
03
04
-9
-0
-0
-0
-0
-0
-0
cccccccjun
jun
jun
jun
jun
jun
jun
de
de
de
de
de
de
de
57
Figure7a. Graphs of endogenous variables in levels and first differences: the real
SEK/EUR exchange rate, relative output, and relative terms of trade.
LNRSEKEUR
D_LNRSEKEUR
2.28
.12
2.24
.08
2.20
2.16
.04
2.12
2.08
.00
2.04
2.00
-.04
1.96
-.08
1.92
86
88
90
92
94
96
98
00
02
04
86
88
90
LNY_SEEU
92
94
96
98
00
02
04
00
02
04
00
02
04
D_LNY_SEEU
.08
.020
.06
.015
.04
.010
.02
.005
.00
.000
-.02
-.005
-.04
-.010
-.06
-.08
-.015
86
88
90
92
94
96
98
00
02
04
86
88
90
LNTOT_SEEU
92
94
96
98
D_LNTOT_SEEU
.10
.03
.02
.05
.01
.00
.00
-.05
-.01
-.10
-.02
-.15
-.03
86
88
90
92
94
96
98
00
02
04
86
58
88
90
92
94
96
98
Figure 7b. Graphs of endogenous variables in levels and first differences: relative net
foreign assets, nominal SEK/EUR exchange rate and relative consumer price level.
NFA_SEEU
D_NFA_SEEU
.3
.08
.2
.06
.04
.1
.02
.0
.00
-.1
-.02
-.2
-.04
-.3
-.06
-.4
-.08
86
88
90
92
94
96
98
00
02
04
86
88
90
LNNSEKEUR
92
94
96
98
00
02
04
00
02
04
00
02
04
D_LNNSEKEUR
2.3
.16
.12
2.2
.08
2.1
.04
2.0
.00
1.9
-.04
1.8
-.08
86
88
90
92
94
96
98
00
02
04
86
88
90
LNCPI_EUSE
92
94
96
98
D_LNCPI_EUSE
.16
.02
.12
.01
.08
.00
.04
-.01
.00
-.02
-.04
-.03
-.08
-.04
86
88
90
92
94
96
98
00
02
04
86
59
88
90
92
94
96
98
Figure 8a. Graphs of endogenous variables in levels and first differences: the real
SEK/USD exchange rate, relative output, and relative terms of trade.
LNRSEKUSD
D_LNRSEKUSD
2.4
.20
2.3
.15
2.2
2.1
.10
2.0
.05
1.9
1.8
.00
1.7
-.05
1.6
-.10
1.5
86
88
90
92
94
96
98
00
02
04
86
88
90
LNY_SEUS
92
94
96
98
00
02
04
00
02
04
00
02
04
D_LNY_SEUS
.20
.010
.16
.005
.000
.12
-.005
.08
-.010
.04
-.015
.00
-.020
-.04
-.025
86
88
90
92
94
96
98
00
02
04
86
88
90
LNTOT_SEUS
92
94
96
98
D_LNTOT_SEUS
.08
.04
.03
.04
.02
.00
.01
.00
-.04
-.01
-.08
-.02
-.03
-.12
-.04
-.16
-.05
86
88
90
92
94
96
98
00
02
04
86
60
88
90
92
94
96
98
Figure 8b. Graphs of endogenous variables in levels and first differences: relative net
foreign assets, nominal SEK/EUR exchange rate and relative consumer price level.
NFA_SEUS
D_NFA_SEUS
.5
.12
.4
.08
.3
.2
.04
.1
.0
.00
-.1
-.2
-.04
-.3
-.08
-.4
86
88
90
92
94
96
98
00
02
04
86
88
90
LNNSEKUSD
92
94
96
98
00
02
04
00
02
04
00
02
04
D_LNNSEKUSD
2.4
.20
2.3
.15
2.2
.10
2.1
2.0
.05
1.9
.00
1.8
-.05
1.7
1.6
-.10
86
88
90
92
94
96
98
00
02
04
86
88
90
LNCPI_USSE
92
94
96
98
D_LNCPI_USSE
.08
.02
.01
.04
.00
.00
-.01
-.04
-.02
-.08
-.03
-.12
-.04
86
88
90
92
94
96
98
00
02
04
86
61
88
90
92
94
96
98
Figure 9. Graphs of endogenous variables in levels and first differences: Differentials
between 3-month interest rates in Sweden and Euroland and between Sweden and
the US.
I_SE_EU
I_SE_US
6
16
5
12
4
3
8
2
4
1
0
0
-1
-2
-4
86
88
90
92
94
96
98
00
02
04
86
62
88
90
92
94
96
98
00
02
04
Figure 10. Real SEK/EUR exchange rate and 1-8 quarter forecasts: BM model with
restrictions and real interest rate differential (corresponding to Table 15, Panel C).
11
10
9
8
7
jun
-9
8
c
de
8
-9
jun
-9
9
99
cde
jun
-0
0
1
2
3
4
5
00
01
02
03
04
-0
-0
-0
-0
-0
cccccn
n
n
n
e
e
e
e
e
u
u
u
u
j
j
j
j
jun
d
d
d
d
d
Figure 11. Real SEK/USD exchange rate and 1-8 quarter forecasts: BM model with
restrictions and real interest rate differential (corresponding to Table 16, Panel C).
11
10
9
8
7
jun
-9
8
c
de
8
-9
jun
-9
9
99
cde
jun
-0
0
1
2
3
4
5
00
01
02
03
04
-0
-0
-0
-0
-0
cccccjun
jun
jun
jun
jun
de
de
de
de
de
63