Re-estimation of Keynesian model by considering critical events and multiple cointegrating vectors.
Hina, Hafsa ; Qayyum, Abdul
This study employs the Mundell (1963) and Fleming (1962)
traditional flow model of exchange rate to examine the long run
behaviour of rupee/US $ exchange rate for Pakistan economy over the
period 1982:Q1 to 2010:Q2. This study investigates the effect of output
levels, interest rates and prices and different shocks on exchange rate.
Hylleberg, Engle, Granger, and Yoo (HEGY) (1990) unit root test confirms
the presence of non-seasonal unit root and finds no evidence of biannual
and annual frequency unit root in the level of series. Johansen and
Juselious (1988, 1992) likelihood ratio test indicates three long-run
cointegrating vectors. Cointegrating vectors are uniquely identified by
imposing structural economic restrictions on purchasing power parity
(PPP), uncovered interest parity (U1P) and current account balance.
Finally, the short-run dynamic error correction model is estimated on
the basis of identified cointegrated vectors. The speed of adjustment
coefficient indicates that 17 percent of divergence from long-run
equilibrium exchange rate path is being corrected in each quarter. US
war with Afghanistan has significant impact on rupee in short run
because of high inflows of US aid to Pakistan after 9/11. Finally, the
parsimonious short run dynamic error correction model is able to beat
the naive random walk model at out of sample forecasting horizons.
JEL Classification: F31, F37, F47
Keywords: Exchange Rate Determination, Keynesian Model,
Cointegration, Out of Sample Forecasting, Random Walk Model
1. INTRODUCTION
Stability of exchange rate is crucial for economic development. It
provides the macroeconomic links among the countries via goods and
asserts market [Moosa and Bhatti (2009)]. In literature different
approaches have been developed to analyse the behaviour of exchange
rate. Among them, purchasing power parity (PPP) is the earliest approach
for exchange rate determination, introduced by Swedish economist Gustav
Cassel in 1920s. Empirical evidence of PPP theory has been rather mixed,
In case of Pakistan, for example, Chisti and Hasan (1993) do not support
PPP model to explain the exchange rate variations. Bhatti and Moosa
(1994) argued that the failure of PPP under flexible exchange rate is
due to the negligence of expectations in exchange rate determination.
Bhatti (1997) investigated and proved the ex-ante version of PPP, in
which exchange rate is explained not only by current relative prices but
also by the expected real exchange rate. Moreover, Bhatti (1996),
Qayyum, el al. (2004) and Khan and Qayyunt's (2008) results do
support the validity of relative form of PPP in Pakistan.
PPP theory is based on the concept of good arbitrage and ignores
the importance of capital movements in exchange rate determination. To
fill this gap Keynesian approach of exchange rate determination is
initiated by introducing the capital flows into current account balance
of payment approach [Mundell (1962) and Fleming (1962)]. The empirical
validity of this structural model is tested by Bhatti (2001) for
determining Pak rupee exchange rates against six industrial
countries' currencies. He suggested that nominal exchange rate of
Pakistan is determined by relative price level, relative income level
and interest rates differentials. The relative version of exchange rate
model assumes symmetry in the coefficients of domestic and foreign
coefficients. However, no former information is available to assume this
symmetry. Moreover, relative version of exchange rate models is unable
to find the multiple cointegrating vectors. Multiple cointegrating
vectors contain valuable information and should be carefully interpreted
[Dibooglu and Enders (1995)]. In international literature a lots of
studies are available that established and uniquely identified the
multiple cointegrating vectors [see for example, Juselius (1995);
Dibooglu and Enders (1995); Helg and Serati (1996); Diamandis, et al.
(1998); Cushman (2007); Tweneboah (2009) among others]. This study,
therefore, considesr the non-relative version of Keynesian exchange rate
and test the symmetry among the domestic and foreign price level, output
level and the interest rate. Keynesian model also incorporates the
uncovered interest parity (UIP) and purchasing power parity (PPP)
conditions. The identification of these parity conditions are also the
aim of this paper.
One of the objectives of structural exchange rate models, like
Keynesian flow model, is to explain the exchange rate variations and
provide better forecast. In this regard, literature on exchange rate
forecasting is divided into two categories. One which emphasises the
importance of economic theory for exchange rate prediction and
recommends a theory based on plausible channel to stabilise it [see,
Khalid (2007); Abbas, et al. (2011)]. Similarly, Cushman (2007)
empirically tested the out of sample forecasting performance of dynamic
portfolio balance model of exchange rate with benchmark random walk by
adopting Mark (1995) technique. On the basis of Root Mean Square Error
(RMSE) and Diebold-Mariano (DM) test he suggested that structural model
outperforms the random walk models at longer horizons. Likewise,
MacDonald (1997), Hwang (2001), Korap (2008), and Anaraki
(2007)'have used multilateral cointegration technique and presented
the superiority of fundamental models over random walk models. Cheung,
et al. (2002) documented that the better performance of structural
models are credited to the dynamic error correction model with
stochastically varying coefficients and recursively updating the long
run cointegrating vectors. On the other hand the promoters of random
walk model argued that exchange rate is a random walk phenomenon. It
efficiently analyses the exchange rate fluctuations and provides better
future forecast such as Rashid (2006) and Malik (2011). According to
these studies there is no need to worry about the macroeconomic
variables of exchange rate determination. Meese and Rogoff (1983) and
Najand and Bond (2000) suggested that the poor performance of structural
models is characterised by unstable parameters. The stability of
parameters is usually disturbed by the existence of outlier in the
series. Therefore, it is necessary to control the outliers in order to
get better forecast [Balke and Famby (1994) and Dijk, et al. (1999)].
Therefore, to judge the out of sample forecasting performance of the
dynamic error correction model of Keynesian model as compared to naive
random walk model is the other objective of this paper.
Brief overview of exchange rate systems confirms that currencies
under flexible exchange rate system generally tend to depreciate more
than currencies having fixed exchange rate system due to the occurrence
of critical events [Ltaifa, et al. (2009)]. Pakistan had adopted a
flexible exchange rate system since 2000 and its currency is freely
floating against US dollar. Therefore, any shocks in US economy directly
hit the Pakistan rupee. After 2001, nominal exchange rate of Pakistan is
highly volatile, though, the other economic fundamentals remain the
same. Its instability is attached to the happening of critical events
during this era. 9/11 event and US war against terror in Afghanistan had
appreciated the rupee against US dollar. This appreciation was driven by
high inflows of remittances and foreign capital inflows into Pakistan.
The trend of the appreciation of rupee was reversed into depreciation
when Global Financial Crisis (GFC) occurred in 2007. In the period of
GFC the foreign exchange reserves declined from $14.2 billion in 2007 to
$3.4 billion in 2008. Pakistan rupee against US dollar lost its value by
21 percent during 2008. So far no study is available to test the
significance of these critical events on the exchange rate in the
framework of Keynesian model. This paper fills this gap by examining the
effect of critical events on the exchange rate of Pakistan in terms of
intervention dummies.
The rest of the study is organised as follows: Section 2 presents
the theoretical framework of Keynesian model. Section 3 deals with the
econometric methodology. Data and construction of variables is subject
of Section 4. Section 5 describes the empirical results and Section 6
reports the out of sample forecasts. Section 7 concludes the study and
identifies some policy implications.
2. THEORETICAL FRAMEWORK
The traditional Keynesian approach is developed by Mundell (1962)
and Fleming (1962). They extended the Keynesian IS-LM framework to an
open economy by incorporating the capital flows via balance of payments.
The objective of this section is to derive the reduced form
equation of the equilibrium exchange rate under the Keynesian approach.
In the literature a number of studies, for example Gylfason and
Helliwell (1983), Pearce (1983), Bhatti (2001) and Moosa and Bhatti
(2009), have derived the Keynesian equilibrium exchange rate model by
utilising BOP Equation (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
Equation (1) defines the balance of payments. [DELTA]f denotes the
change in foreign reserves which equals zero under the flexible exchange
rates. Current account (CA) is positively related to real exchange rate
([SP.sup.f]/P), where S denotes nominal exchange rate measured by
domestic currency per unit of foreign currency, P represents domestic
prices and [P.sup.f] the foreign price level. An increase in foreign
output (K) and depreciation of domestic currency has favourable effect
on the balance of trade (BOT) by enhancing the demand for domestic
exports. However, it deteriorates due to an increase in domestic output
level (Y). The traditional flow model also assumes that foreign and
domestic assets are imperfect substitutes, which implies that interest
rate differentials may causes finite capital flows into or out of a
country. Thus, the net capital inflow (K) is a positive function of
domestic interest rate (i) and negative function of foreign interest
rate ([i.sup.f]). To derive the fundamental equation of exchange rate,
the BOP, Equation (1) can be written as:
BOP = a([SP.sup.f]/P) + [b.sup.f] [Y.sup.f] - bY + ci - [c.sup.f]
[i.sup.f]. (2)
All variables of Equation (2) except interest rate are in logarithm
form and denoted it by small letters. For simplicity a restriction
[b.sup.f] = b and c = [c.sup.f] is imposed. The equilibrium exchange
rate is determined when BOP is in equilibrium i.e. the net of current
and capital account is zero and solving for nominal exchange rate
's', we have
s = (p - [p.sup.f]) + b/a (y - [y.sup.f]) - c/a (i - [i.sup.f]).
(3)
which explains that the equilibrium exchange rate is positively
related to relative prices and relative incomes, but inversely related
to relative interest rates. In general form, the above Equation (3) is
written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
MacDonald (1995) defined the theory of long-run exchange rate
modeling by relating the concepts of uncovered interest rate parity,
absolute and efficient markets PPP to a standard balance of payments
equilibrium condition. In order to link the absolute PPP with the
current account balance he asserted that under a long-run net capital
flows were zero when savings were at their desired level. This
specification reduces the BOP account to current account balances. Thus
we can write the Equation (3) as:
s = (p - [p.sup.f]) + b/a (y - [y.sup.f]). (5)
The current account balance approaches to PPP only when the
difference between domestic and foreign income level i.e. (y -
[y.sup.f]) tends to be zero. This would be possible if the price
elasticity of domestic exports is infinitely large (a [right arrow]
[infinity]) [MacDonald (1995) and Moosa and Bhatti (2009)], in this case
the exchange rate is exclusively determined by the PPP that is:
s = (p -[p.sup.f]). (6)
On the other hand, the non-zero value of (y - [y.sup.f]) is likely
to be most important when comparing countries at different stages of
development, but less important for countries at a similar level of
development. Allowing a constant in Equation (6) would represent a
permanent deviation from absolute PPP due to productivity differentials
and other factors [MacDonald (1995) and Taylor and Taylor (2004)].
The efficient market view of PPP suggests that in a world of high
or perfect capital mobility it is not goods arbitrage that matters for
the relationship between an exchange rate and relative prices, but
interest rate arbitrage. Hence, a slow speed goods market arbitrage
causes a temporary deviation of the exchange rate from PPP. This
requires that the exchange rate drifts in such a manner as to restore
the relative PPP. Algebraically these deviations can be expressed as:
[DELTA]s = p - [p.sup.f] - s. (7)
A perfectly mobile capital immediately diverts the attention to
focus on the capital account of the balance of payments. The assumption
of perfect capital mobility may be represented as:
[DELTA][s.sup.e] = i - [i.sup.f]. (8)
Equation (8) represents the uncovered interest parity condition.
This condition defines that the difference between the domestic interest
rate (i) and foreign interest rate (if) produces an expected
depreciation of the exchange rate. Frenkel (1978) and Juselius (1995)
among others, argued that the fluctuations in exchange rate are
attributed by both goods and assets market development. Therefore, PPP
and UIP conditions may not be independent of each other in the long run.
This allows us to substitute Equation (8) into Equation (7) to combine
PPP with UIP and model the nominal exchange rate as:
s = p - [p.sup.f] - i + [i.sup.f]. (9)
The above discussion makes it clear that it is not worthwhile to
empirically analyse the short run relationship between exchange rate,
domestic and foreign price level, interest rate and output and ignore
their long run associations (defined in Equations (5) to (9)). Hence,
long run relationship(s) would be combined with the short run dynamics
of exchange rate by employing the vector error correction mechanism.
3. EMPIRICAL METHODOLOGY
3.1. Unit Root Test
Cointegration analysis is based on the assumption that variables
are integrated of same order. Pre-testing for unit root is necessary to
avoid the problem of spurious regression. Hylleberg, Engle, Granger, and
Yoo (HEGY) (1990) is used to test for non-seasonal zero frequency unit
root and biannual and annual frequency seasonal unit roots in quarterly
data. HEGY provide following auxiliary regression equation:
[[DELTA].sub.4][y.sub.i] = [[mu].sub.i] + [[pi].sub.1][y.sub.1,t-1]
+ [[pi].sub.2][y.sub.2,t-1] + [[pi].sub.3][y.sub.3,t-1] +
[[pi].sub.4][y.sub.3,t-2] + [l.summation over (i=1)] [[gamma].sub.i]
[[DELTA].sub.4][y.sub.t-1] + [[epsilon].sub.t]. (10)
Where [[mu].sub.i] is a deterministic term which can include any
combination of a drift term, trend term and a set of seasonal dummies.
[y.sub.1,t], [y.sub.2,t], [y.sub.3,t], and [y.sub.4,t] are linearly
transformed series as proposed by HEGY i.e, [y.sub.1,t], = (1 + B)(1 +
[B.sup.2]) [y.sub.i],[y.sub.2,t] = -(1 - B)(1 + [B.sub.2])[y.sub.i],
[y.sub.3,t] = -(1 - B)(1 + B) [y.sub.i], and [y.sub.4,t] = (l -
[B.sup.4]) [y.sub.i], where B is a lag operator such that [B.sup.k]
[y.sup.i] = [y.sub.t-k]. [[epsilon].sub.t] ~ (0, [[sigma].sup.2.sub.e])
is
Gaussian error term and white noise Cov = ([[epsilon].sub.t],
[[epsilon].sub.t-1]) = 0. The auxiliary regression (10), comes from the
fact that [[DELTA].sub.4] = (1 - [B.sup.4]) can be decomposed as (1 - B)
x (1 + B) x (1 - iB)(1 + iB) where each term in bracket corresponds to
non-seasonal zero frequency unit root 1, biannual frequency unit root -1
and annual frequency unit root [+ or -]i.
HEGY method tests the significance of [[pi].sub.j] (j = 1,2,3,4)
parameters. If [[pi].sub.i] = 0 is statistically significant then series
contain non-seasonal zero frequency unit root. If [[pi].sub.2] = 0 is
accepted this implies the presence of biannual frequency seasonal unit
root. If [[pi].sub.3] = [[pi].sub.4] = 0, then series has seasonal unit
root at annual frequency. The appropriate filter corresponding to the
acceptance of each null hypothesis are (1-B), (1+B) and (1+[B.sup.2])
required to make the series stationary. Critical values for one sided
t-test for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for
the joint F-test for [[pi].sub.3] and [[pi].sub.4] ([F.sub.34]) are
provided by HEGY.
3.2. Johansen and Juselius Cointegration Methodology
Johansen and Juselius (1990) cointegration technique is useful to
construct a multiple long-run equilibrium relationships over
multivariate system. Generally, this technique is applied to 7(1)
variables. Johansen's method in k dimensional error correction (EC)
form is presented as follows:
[DELTA][z.sub.i] = [l-1.summation over
(i=1)][[GAMMA].sub.i][DELTA][z.sub.t-1] + [PI][z.sub.t-1] +
[phi][D.sub.i] + [mu] + [[epsilon].sub.t]. (11)
Where [z.sub.t] is (k x 1) is dimensional vector of I(1) variables,
D, consists of centered seasonal dummies, intervention and policy
dummies such that all are I(0), [mu] is deterministic trend component,
which consist of different combinations of constant and trend terms in
the long-run cointegrating equation and short-run vector auto regressive
(VAR) model, [] f N (0,1) is (it x 1) vector of Gaussian random error
terms and [SIGMA] is (k x k) variance covariance matrix of error terms.
(i = 1,2, ........, l-1) is the lag length. [[LAMBDA].sub.l] = -{I -
[A.sub.1] - ......... - [A.sub.l]) is short-run dynamic coefficients.
[PI] = -(I - [A.sub.1] - ......... - [A.sub.l]) is (k x k) matrix
containing long-run information regarding equilibrium cointegration
vectors.
The number of cointegrating vectors (r) are determined by rank of
[PI] matrix. If 0 < rank([PI]) < k - 1 then it is further
decomposed into two matrices i.e. [PI] = [alpha][beta]': [alpha] is
a (k x r) matrix containing error correction coefficients, which measure
the speed of adjustment to disequilibrium. [beta]' is (r x k)
matrix of r(El) cointegrating vectors. The rank of [PI] matrix is
measured by likelihood ratio trace and maximum eigenvalue statistics. In
case of multiple cointegrating vectors Johansen and Juselius (1990)
allow the imposition of linear economic restrictions on [beta] matrix to
obtain long-run structural relationships.
3.3. Short-Run Dynamic Error Correction Model
According to Granger (1983) Representation Theorem, if there is
long-run stable relationship among the variables then there will be a
short-run error correction relationship related with it. Short-run
vector error correction representation is as follows:
[DELTA][z.sub.i] + [l-1.summation over
(i=1)][[GAMMA].sub.i][DELTA][z.sub.t-i] +
[alpha]([beta]'[z.sub.t-1]) + [phi][D.sub.i] + [mu] +
[[epsilon].sub.t]. (12)
[beta]'[z.sub.t-1] is the error correction term. The
traditional methodology uses the residuals from the identified
cointegrating vector(s) to form [beta]'[z.sub.t-1] x [alpha] in
dynamic error correction model measures the speed of adjustment toward
equilibrium state. Theoretically speed of adjustment coefficient must be
negative and significant to confirm that long-run relationship can be
attained.
4. DATA AND CONSTRUCTION OF VARIABLES
This study considers quarterly data from 1982:Q1 to 2010:Q2. A
start from 1982 is on account of implementation of flexible exchange
rate policy in Pakistan. All variables are measured in the currency
units of each country. The data are obtained from International
Financial Statistics (IFS) and State Bank of Pakistan (SBP) Monthly
Statistical Bulletin (Various Issues).
The nominal exchange rate is measured in terms of Pakistan rupee
(PKR) per unit of US dollar (US $). Real Gross domestic product (GDP) is
commonly used as a measure of real output level. Quarter wise real GDP
of US is accessible from IFS. In case of Pakistan only annual real GDP
is available. Quarterisation of annual real GDP is done by using the
methodology of Kemal and Arby (2004). Consumer price index (2000=100) is
used as a proxy of domestic and foreign price level. Call money rate for
Pakistan and federal fund rate for US are used as a measures of interest
rates. During the analysis period exchange rate of Pakistan is also
influenced by the critical events such as 1998 Pakistan's nuclear
test, 9/11 event, US war against terror in Afghanistan after 9/11, 2005
stock market crash and recent global financial crisis (2007). Dummy
variables [D.sub.98] (0 for t < 1998: Q2 and 1 for t 1998: Q2),
[D.sub.911] (1 for t = 2001: Q3 and 0 otherwise), [D.sub.afgwar] (0 for
t < 2001: Q4 and 1 otherwise), [D.sub.SMC](1 for t = 2005: Q1 and 0
otherwise) and [D.sub.fc] (0 for t < 2007: Q1 and 0 otherwise) are
used to capture the influence of these events on the exchange rate.
5. RESULTS AND DISCUSSION
This section implements the Johansen and Juselius (1988, 1992)
multivariate cointegration methodology to detect the stable long run
relationships between the exchange rate and fundamental variables. The
preliminary time series properties for cointegration analysis are as
follows:
5.1. Order of Integration (Unit Root Test)
The presence of seasonal and non-seasonal unit roots for each
quarterly series is determined via HEGY (1990) test. All variables are
transformed in logarithmic form except the interest rate. The results of
the HEGY test are presented in Table 1. It can be observed that the null
hypothesis of a non-seasonal unit root cannot be rejected whereas the
null hypothesis of seasonal unit root at both biannual and annual
frequency are rejected at 5 percent critical values for all of the
variables. (1-B) is an appropriate filter to make the series stationary.
The results of HEGY test after applying required filter are presented in
Table 2 and we found no evidence of seasonal and non-seasonal unit roots
at 5 percent level of significance. Therefore, all variables in our
cointegration analysis are integrated of order one and we may suspect
multiple long run cointegrating vectors.
5.2. Unrestricted VAR Model Specification
The next step after implementing the unit root test is to decide
the optimal lag length of the multivariate system of equations, which
ensures that residuals of VAR model are white noise. We have used
Johansen (1995) multivariate LM test and 3 quarters have been selected
as appropriate lag structure of the model. Three central seasonal
dummies and four intervention dummies [D.sub.98], [D.sub.911],
[D.sub.afgwar], [D.sub.fc] are also included. The residual of the VAR(3)
passed the diagnostic test of no serial correlation ([[chi
square].sub.(49)] = 52.31 with four lags), no heterosedasticity ([[chi
square].sub.(1372)] = 1355.36) at 5 percent level of significance, but
fail to pass the null hypothesis of normally distributed error terms
under Jarque-Bera (JB) test ([[chi square].sub.(14)] = 73.24). However,
lack of normality does not affect the results of Johansen (1988)
likelihood ratio tests [Gonzalo (1994); Paruolo (1997); Cheung and Lai
(1993); Eitrheim (1992) and Goldberg and Frydman (2001)].
5.3. Multivariate Cointegration Analysis
After selecting the lag length of VAR model, another fundamental
issue is the suitable treatment of deterministic components such as
drift and trend terms in the cointegrating and the VAR part of the VECM.
Most of the series in our analysis exhibit a linear trend in the level
of the series. Therefore, we introduce intercept term unrestrictedly
both in long run (cointegrating part) and short run (VAR) model while
performing cointegration analysis [Johansen (1995); Harris, et al.
(2003) and Qayyum (2005)]. Table 3 presents the trace and maximum
eigenvalue statistic after adjusting by factor (T-kl)/T to correct the
small sample bias.
The trace test shows that the null hypothesis of no cointegration
(r=0), one cointegration (r < 1) and two cointegrating vectors (r
< 2) can be rejected, but fails to reject the null of three
cointegrating vectors at 5 percent level of significance. Therefore,
variables of Keynesian exchange rate model are found to be cointegrated
with three cointegrating vectors. Whereas, the maximum eigenvalue
statistic with the null hypothesis r=l is rejected, but the null
hypothesis of r=2 is not rejected and refers to one long run
relationship among the variables. (1) This contradiction among the tests
for cointegrating vector is common. We continue our analysis on the
basis of trace test, as it is a more powerful test as compared to
maximum eigenvalue statistics in case of not normally distributed error
terms [Cheung and Lai (1993); Hubrich, et al. (2001)]. Kasa (1992) and
Serletis and King (1997) also preferred trace statistics as it considers
all k-r (k is no. of variables in the system and r is the cointegrating
vectors) values of smallest eigenvalues.
The first three cointegrating vectors with the maximum eigenvalue
have been normalised on log of nominal exchange rate to determine the
sign and magnitude of the long-run elasticities in Keynesian exchange
rate model Equation (4). The results of normalised vectors are presented
in Equation (13);
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
Result shows that the sign of all variables except the foreign
price level are consistent with Keynesian theory in first cointegrating
vector. In second cointegrating vector the signs of domestic and foreign
price level, while, in the third vector the signs of domestic price
level, foreign price level and domestic interest rate support the
theory. The contradiction of results among the vectors arises due to
arbitrary normalisation. It restricts to draw a meaningful conclusion.
As described earlier that multiple cointegrating vectors contain
valuable information and must be identified properly and carefully
interpreted. To obtain this information we start by imposing
proportionality and symmetry restrictions on all vectors in proceeding
section.
5.4. Proportionality and Symmetry Restrictions
Before the identification of cointegrating vectors, we proceed to
test the proportionality and symmetry restrictions of prices, interest
rates and output through likelihood ratio test on all cointegrating
vectors. The acceptance of these restrictions provides the validity of
strict form PPP and UIP. The likelihood ratio (LR) test statistics along
with their probability values for the proportionality and symmetry
restrictions are reported in Table 4.
First part of Table 4, reports the results of symmetry restrictions
on prices, output and interest rates on all three cointegrating vectors
in order to find whether they enter in the equilibrium relation or not.
The symmetry restriction implies that prices, output and interest rates
influence the exchange rate regardless of where they originate.
According to LR test statistics, symmetry restrictions hold for prices
and output. Under [H.sub.3], we found no evidence of interest rate
symmetry. The joint symmetry restrictions implied by [H.sub.4] through
[H.sub.7] are mostly rejected at 95 percent level of significance.
Further, the proportionality restriction (Hs) holds for prices but
not for output and interest rate in all three cointegrating vectors.
Symmetry and proportionality of prices is opposite to the finding of
Khan and Qayyum (2008). The basic reason for this contradiction is the
absence of other fundamental variables such as output levels and
interest rate in their analysis. In our analysis we can predict the long
run strong form PPP in the presence of other fundamental variables.
5.5. Identification of Cointegrating Vectors
In Table 5, we proceed by imposing the theoretical restrictions on
PPP, UIP and their combinations. First part of Table 5 reports
individual parity conditions. Under [H.sub.11], strict version of PPP is
tested in all cointegrating vectors. The LR test statistics for this
hypothesis yields to accept the strong form of PPP with other
fundamental variables at 10 percent level of significance. Similarly
strong PPP form with unrestricted output coefficients ([H.sub.24]) and
with unrestricted interest rate coefficients ([H.sub.22]) are also
accepted at 5 percent level of significance.
[H.sub.12] analysed the strict form of PPP in the first
cointegrating vector. This was done by executing unity restriction on
exchange rate and prices and zero restriction on output and exchange
rates coefficients in the first cointegrating vector. This hypothesis is
rejected by LR test. This result suggests that strong form of PPP does
not hold on its own.
Weak form of PPP is investigated under [H.sub.13] and [H.sub.14],
both of these hypothesis are rejected by LR test.
The rejection of both strict and weak forms of PPP on its own (in
the absence of other fundamental variables) is consistent with Khan and
Qayyum (2008), Helg and Serati (1996), Dibooglu and Enders (1995) and
Macdonald (1993). Last two authors argued that this is due to the
different ways of finding national indices, which result into the non
proportionality of price adjustments. According to Helg and Serati
(1996), standard PPP does not hold on its own during the period of
flexible exchange rate. Khan and Qayyum (2008) argue that rejection of
strong form of PPP is due to the significance of transportation and
transaction cost.
After investigating the different versions of PPP restrictions, we
now analyse the UIP condition. First we examine whether strong form of
UIP restriction enters in all three cointegrating vectors or not, by
formulating H15. This hypothesis is strongly rejected by LR test.
However, under [H.sub.16], we set out that UIP relationship is
stationary by itself by imposing unity restriction on interest rate
coefficients and zero restriction on prices and output coefficients in
first cointegration vector. The LR test result supports that one of the
cointegrating vectors contains a stationary relationship between the
interest rate variables. This result is consistent with Johanson and
Juselius (1992).
Further, the weak form of UIP is tested in all cointegrating
vectors by H17 and in first cointegrating vectors through His with zero
restriction on the coefficient of prices and output. [H.sub.17] is
rejected by the LR test, whereas, the later hypothesis is not rejected
by LR test. From the results of various forms of UIP conditions, we can
conclude that UIP holds without the fundamental variables in one
cointegrating vector only.
Following this, we combined PPP and UIP restrictions by [H.sub.19]
through [H.sub.23]. On the basis of LR statistic the strong form of PPP
along with strong form of UIP ([H.sub.19]), weak form of PPP with strong
form of UIP ([H.sub.21]) and strong form of PPP with weak UIP
([H.sub.22]) enter in the cointegrating vector.
Finally the joint hypothesis of PPP, UIP and output symmetry in one
cointegrating vector is not rejected under [H.sub.27].
The general hypothesis tested through Hi to [H.sub.27], are
informative to formulate unique vectors in the multiple cointegration
space. These results suggest that strong form of PPP with output
relationship ([H.sub.24]) is considerable in one vector while the weak
form of UIP relationship ([H.sub.18]) is in the second vector and the
strict form of PPP and unrestricted interest rate is in the third
vector. All cointegrating vectors are normalised on nominal exchange
rate. Thus, it would seem plausible to specify the long run
cointegrating vector P' matrix as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The LR test statistics for these restrictions are [[chi
square].sup.2.sub.df] = 6 = 11.88 which do not reject this hypothesis.
The results of long-run cointegrating vectors are presented as:
[s.sub.t] = [P.sub.t] ~ [p.sup.*.sub.t] + 6.37 [y.sup.t] - 8.667
[y.sup.*.sub.t] + 55.06. (14)
[s.sub.t] = 0.65 [i.sub.t] - 0.76 [i.sup.*.sub.t] + 2.06. (15)
[s.sub.t] = [p.sub.t] - [p.sup.*.sub.t] + 0.19 [i.sub.t] -
0.24[i.sup.*.sub.t] + 3.47 (16)
The results of restricted vectors suggest that exchange rate is
determined by both current account balance and net capital inflows. The
estimated signs of all variables except the domestic and foreign
interest rates are consistent with Keynesian theory. On the basis of
cointegrating vectors following results can be made:
Strong form of PPP does not hold on its own but holds with other
fundamental variables. This result supports the arguments by Helg and
Sarati (1996) and Khan and Qayyum (2008) i.e., the rejection of strong
form of PPP on its own is due to the significance of transportation and
transaction cost. However, increase in domestic (foreign) price level
will lead to depreciation (appreciation) of the domestic currency.
Positive (negative) coefficient of domestic (foreign) output
reveals that increase in domestic (foreign) output level results in
depreciation (appreciation) of domestic currency via higher demand of
imported (exported) commodities. Hence, stronger economic growth of
Pakistan tends to cause depreciation in the exchange rate. This is
because the growth is led by higher consumer spending, this will cause a
rise in imports which could lower the exchange rate.
Positive impact of domestic interest rate on exchange rate suggests
that increase in domestic interest rate leads to depreciation of the
domestic currency against US dollar. Whereas, increase in foreign
interest rate results in the appreciation of domestic currency. The
estimated coefficients of both interest rates are not according to the
theory, the opposite signs of interest rates were also observed in
Bhatti (2001).
5.6. The Short-Run Function for Keynesian Exchange Rate:
Dynamic Error Correction Model
This section presents the short-run dynamic error correction model
(ECM) of the Keynesian exchange rate model. The residuals of the long
run cointegration functions (from Equations 14 to 16) are used as an
important determinant of ECM. These residuals are also known as
disequilibrium estimates or error correction terms. They measure the
divergence from long run equilibrium in period t-1 and provide the speed
of adjustment information toward equilibrium.
The ECM is estimated by ordinary least squares (OLS) method. The
estimation process considers the Hendray 'general-to-specific'
strategy (1992). General model is started by having drift term, three
seasonal dummies, intervention dummies ([D.sub.98], [D.sub.911],
[D.sub.afgwar], [D.sub.fc]), lag of error correction terms and lag
length of eight for each first difference variables (exchange rate,
prices, outputs, interest rates). The specific model is achieved by
dropping the insignificant lags. The parsimonious ECM model with
t-ratios in parentheses is as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
Adj [R.sup.2] = 0.40 [F.sup.(1392)] = 9.93 prob (0.000)
The residual of parsimonious ECM satisfied the diagnostic tests of
Breusch and Godfrey (1981) LM test of no serial correlation ([[chi
square].sub.(4)] = 4.28), Engle's (1982) no autocorrelation
conditional heteroskedasticity (ARCH) LM test ([[chi square].sub.(1)] =
1.40 and [[chi square].sub.(4)] = 3.56 and Jarque-Bera normality test
([[chi square].sub.(2)] = 5.47) at 5 percent level of significance.
The estimated coefficients of ECM in Equation (17), show that in
short run exchange rate immediately responds to change in foreign price
level, domestic and foreign real output and domestic and foreign
interest rates. The presence of lag of dependent variable makes the
short run dynamic ECM as an autoregressive model. Its estimated
coefficient indicates that a one percent depreciation in preceding
seventh quarter (approximately two years back) results in the
appreciation of current exchange rate by 0.22 percent.
In short-run change in foreign price level has dominant effect on
the nominal exchange rate among the other variables, due to its higher
coefficient. The positive sign of change in foreign price level
indicates that increase in foreign price level immediately depreciates
the domestic currency in the short run rather than appreciating it as
suggested by the theory. It confirms the finding of Alam and Ahmed
(2010) that Pakistan is a growth driven economy and increase in relative
price of imports may not reduce the import demand. Pakistan's major
imports consist of petroleum products, essential capital goods and
machinery goods. These goods contributed more than 50 percent share of
total imports and among these goods Petroleum Group only constituted the
largest share in our import bill that is 32 percent in 2010 (State Bank
of Pakistan). An increase in oil prices disturbs balance of payment and
puts downward pressure on exchange rate which makes imports more
expensive [Malik (2008); Kiani (2010)].
A change in domestic output level in preceding quarter depreciates
the domestic currency by 0.22 percent, but a change in four quarter
previous output level results in the appreciation of currency by 0.42
percent. This result is consistent with Ahmed and Ali (1999) study, in
which they suggest that a shock in output initially depreciates the
domestic currency but after four periods it appreciates the domestic
currency.
The estimated coefficients of lagged change in domestic and foreign
interest rate are significant and negative. According to estimates,
nominal exchange rate immediately appreciates due to change in domestic
and foreign interest rates.
Among the intervention dummy variables only DafgWar is found to be
significant in short run dynamic model. Its negative coefficient
signifies the appreciation of rupee. During the period of US war against
terror in Afghanistan the total US foreign assistance received by
Pakistan since fiscal year 2002 is $ 20 billion. This is more than the
aid Pakistan received from the US between 1947 and 2000, which is $12
billion [Epstein and Kronstadt (2012)].
The absence of financial crisis dummy variable does not imply that
nominal exchange rate of Pakistan is independent of financial crisis.
But the reason is the ignorance of financial sector in the Keynesian
model. Therefore, the effect of financial crisis will be clearly
measured in those models that incorporate the financial sector such as
monetary and portfolio models of exchange rate.
Theoretically, sign of error correction term should be negative and
significant. The negative sign confirms adjustment towards equilibrium
state. In our analysis, coefficient of first error correction term is
positive and statistically significant, while the coefficients of second
and third error correction terms obey theoretical definition that is
negative and significant.
The result of ECIm and EC2t-i indicates that exchange rate
overshoots from long run equilibrium path by 10 percent. The third error
correction term demonstrates that long run deviation of nominal exchange
rate from its equilibrium path is being corrected by 27 percent every
quarter. Therefore, the net convergence of exchange rate towards its
equilibrium state is 17 percent per quarter. The time required to remove
50 percent of disequilibrium from its exchange rate equilibrium path is
three quarters (nine months). (2)
Finally, the stability of ECM's parameters are examined by
utilising Cumulative Sum of Recursive Residuals (CUSUM) and Cumulative
Sum of Squares of Residuals (CUSUMSQ) test. The plots, provided in
Figure 1 and Figure 2, show that CUSUM and CUSUMSQ remain within the 5
percent critical bound, suggesting that there is no significant
structural instability and residual variance is stable during the
analysis period.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
6. OUT OF SAMPLE FORECASTS
Mark (1995) and Cushman (2007) methodology (3) of recursive
regression has been adopted to generate multi-step-ahead forecast from
Keynesian and random walk models. The methodology starts by dividing the
data set, containing t = 1, 2, ......, T number of observations that is
from 1982:1 to 2010:2, into thirty seven subsamples [t.sub.1],
[t.sub.2], ..., [t.sub.37]. The first subsample contains T-31 (smallest
subsample) number of observations. We denote it by [t.sub.1] (ends at
period 2001:1). The next subsample [t.sub.2] is extended by one
observation; it contains T-36 number of observations (ends at period
2001:2), and so on the largest and last sample ends with T-1 number of
observations, we denote it by [t.sub.37] with ending period 2010:1.
The parsimonious error correction model Equation (17) is estimated
for each subsample. This recursive procedure updates the estimated
parameters in each subsample due to the inclusion of new data point.
Each subsample estimated error correction model has been used to
construct a one quarter ahead forecast to sixteen quarter ahead
forecast. This results in 37 one quarter ahead forecast, 36 two quarter
ahead forecast and so on till 22 sixteen quarter ahead forecast.
Forecasted values are also obtained from random walk models for each
subsample.
The forecasting performance of each forecast horizon under
Keynesian exchange rate and random walk models are evaluated by using
standard root mean squared error (RMSE) and Theil's U statistics.
Theil's U statistics computes the ratio of the RMSE of the
Keynesian model to the RMSE of random walk models. If this ratio is less
than one then structural model on average provides better forecast than
benchmark. Finally, statistical significance of each forecasting horizon
is judged with the Diebold and Mariano (DM) (1995) test statistics.
Table 6 gives the result for RMSE of different models at 1, 4, 8,
12 and 16 forecasting horizons. It can be noted that RMSE of Keynesian
model is smaller than the RMSE of benchmark random walk models, with and
without drift, at all out of sample forecast horizons. Therefore, it is
easy to conclude that Keynesian model yields better forecast for
exchange rate than random walk models. Theil's U coefficient at
each forecasting horizon is reported in Table 7. This coefficient again
supports the dominance of structural model over the random walk models
at every horizon.
RMSE and Theil's U factor do not provide any idea of the
significance of the difference in the forecasting performance.
Therefore, final conclusion will draw on DM test statistics. Table 8
lists the DM statistics and its associated probability values at various
horizons, to significantly test whether the mean square error of one
forecast is better than another.
First part of Table 8 takes random walk model without drift as
benchmark model in loss differential function. The DM test statistics
confirm that the predictive accuracy of Keynesian model is significantly
more accurate than the random walk model at long forecast horizon i.e.
k=\2, and 16. The success of structural models at long horizons is
consistent with Mark (1995) and Chinn and Meese (1992). Second part of
Table 8 compares the difference in the forecasting performance of the
structural models to the benchmark random walk with drift model. DM test
statistics clearly states that parsimonious cointegrated Keynesian model
easily beat the random walk model with drift at every horizon except the
first. This finding confirms the remarks of Faust, et al. (2003) that it
is easy to beat the random walk model with drift than the random walk
model without drift.
7. CONCLUSION
This paper has empirically analysed the Keynesian exchange rate
model by employing Johansen and Juselius (1988, 1992) cointegration
method. Trace test has found three long run relationships among exchange
rate, prices, interest rates and output levels. The symmetry
restrictions on price coefficients and output coefficients and
proportionality restriction on price coefficients are only satisfied by
maximum likelihood ratio test. This study has tested the various forms
of PPP, UIP and their combinations to identify the cointegrating
vectors. The results support the validity of PPP with the presence of
other fundamental variables such as unrestricted output level and
interest rates. However, UIP condition holds on its own. Based on these
restrictions, further, the first cointegration vector has defined the
current account, the second vector has explained the UIP and the last
vector has described the Keynesian approach to exchange rate
determination. The entire coefficients (except the interest rates)
estimated in the system are significant and according to theory. The
error correction terms suggest that the net convergence of exchange rate
towards its equilibrium state is 17 percent per quarter and three
quarters are required to remove 50 percent of exchange rate misalignment
from equilibrium path.
The out of sample forecasting results suggests that in case of
Pakistan Keynesian exchange rate model outperforms the random walk
model, with and without drift, to accurately predict the nominal
exchange rate. This finding is attributable to the parsimonious error
correction model, which includes lag of dependent variable and
fundamental variables to exchange rate determination, error correction
terms and financial crisis dummy variable. Therefore, it captures the
interruptions in the economy and explains the significant part of
instability and outliers in exchange rate series.
The main policy implications drawn from this study are:
* The maintenance of PPP ensures that obtaining unlimited benefits
from arbitrage in traded goods is not possible. Therefore, Pakistan is
unlikely to improve its external competitiveness against U.S.
* Validity of PPP and UIP allows the use of inflation differentials
and interest rate differentials to forecast long-run movements in
exchange rates.
* The exchange rate of rupee against US dollar is significantly
determined by output levels, prices and interest rates. Therefore,
interaction between good and capital assets market is required to study
exchange rate dynamics in Pakistan.
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(1) Before adjusting trace test reports four while maximum
eigenvalue test indicates three cointegrating equations at 5 percent
level of significance (results are not presented here).
(2) The time required to remove the x percent of disequilibrium
from its equilibrium path is determined as [(1 - EC).sup.41] = (1 - x),
where t is required number of periods to dissipate x percent of
disequilibrium.
(3) Only the difference is in the construction of subsamples, Mark
(1995) has considered forty subsamples and Cushman (2007) has followed
Hansen and Juselius (1995) methodology and constructed thirty seven
subsamples. This study has considered the later approach to elude the
problem of smaller sample size at long horizon forecast and make the
DM test statistics more reliable.
Flafsa Hina <hafsahina@pide.org.pk> is Assistant Professor at
Department of Econometrics, Pakistan Institute of Development Economics,
Islamabad. Abdul Qayyum<qayyumdr@gmail.com> is Joint Director at
Pakistan Institute of Development Economics, Islamabad, Pakistan.
Table 1
HEGY Test at Level of Series
HEGY Regression Model
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Null & Alternative
Hypothesis
[[pi].sub.1] = 0
[[pi].sub.1] < 0
Regressors
Test Statistic
Lags Drift Trend Seasonal
Variable Dummies [[pi].sub.1]
s 0 Yes No No -0.81
Y 3 Yes No No -2.10
[y.sup.f] 0 Yes No No -3.06
P 0 Yes Yes Yes -1.69
[p.sup.f] 0 Yes No No -2.46
i 0 No No No -0.23
[i.sup.f] 0 Yes Yes Yes -3.14
Null & Alternative Hypothesis
[[pi].sub.2] = 0 [[pi].sub.3]
= [[pi].sub.4]
= 0
[[pi].sub.2] < 0 [[pi].sub.3]
[not equal to]
[[pi].sub.4]
[not qual to] 0
Variable [[pi].sub.2] [[pi].sub.3], Roots
[[pi].sub.4] (Filter)
s -5.76 55.37 1(1-B)
Y -8.81 29.61 1(1-B)
[y.sup.f] -4.50 101.23 1(1-B)
P -8.66 27.92 1(1-B)
[p.sup.f] -9.89 20.52 1(1-B)
i -4.74 22.96 1(1-B)
[i.sup.f] -8.12 73.87 1(1-B)
Table 2
HEGY Test on Filtered Series
Regressors
Lags Drift Trend Seasonal
Variable Dummies
(1-B) s 0 Yes No No
(1-B) y 2 Yes No No
(1-B) [y.sup.f] 1 Yes No No
(l-B) p 0 Yes No No
(1-B) [p.sup.f] 0 Yes No No
(1-B) i 0 No No No
(1-B) [i.sup.f] 0 Yes No Yes
Null & Alternative Hypothesis
[[pi].sub.1] = 0 [[pi].sub.2] = 0
[[pi].sub.1] < 0 [[pi].sub.2] < 0
Test Statistic
Variable [[pi].sub.1] [[pi].sub.2]
(1-B) s -4.86 -4.79
(1-B) y -2.96 -8.45
(1-B) [y.sup.f] -3.69 -4.05
(l-B) p -3.07 -6.77
(1-B) [p.sup.f] -4.34 -6.64
(1-B) i -6.20 -3.72
(1-B) [i.sup.f] -4.94 -6.31
Null & Alternative
Hypothesis
[[pi].sub.3]
= [[pi].sub.4] = 0
[[pi].sub.3]
[not equal to]
[[pi].sub.4]
[not equal to] 0
Test Statistic
Variable [[pi].sub.3], Roots
[[pi].sub.4]
(1-B) s 26.77 --
(1-B) y 36.91 --
(1-B) [y.sup.f] 39.85 --
(l-B) p 15.54 --
(1-B) [p.sup.f] 19.13 --
(1-B) i 13.27 --
(1-B) [i.sup.f] 51.09 --
Table 3
Cointegration Test Results
0.05
Null Hypothesis Alternative Chi-Square Critical
Hypothesis Value
Trace Statistic
r = 0 r > 0 155.05 (a) 125.62
r [less than or equal to] 1 r > 1 104.24 (a) 95.75
r [less than or equal to] 2 r > 2 71,43 (a) 69.82
r [less than or equal to] 3 r > 3 40.78 47.86
r [less than or equal to] 4 r > 4 20.94 29.80
r [less than or equal to] 5 r > 5 5.77 15.49
r [less than or equal to] 6 r > 6 0.29 3.84
Maximum Eigenvalue Statistic
r = 0 r = 1 50.81 (a) 46.23
r = 1 r = 2 32.81 40.08
r = 2 r = 3 30.65 33.88
r = 3 r = 4 19.85 27.58
r = 4 r = 5 15.16 21.13
r = 5 r = 6 5.49 14.26
r = 6 r = 7 0.29 3.84
Note: Indicates the rejection of null hypothesis at the 5 percent
level.
Table 4
Restricted Cointegrating Vectors
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hypothesis Restrictions
Symmetry Restrictions
Price Symmetry [H.sub.1]: [[alpha].sub.1] =
-[[alpha].sub.2]
Output Symmetry [H.sub.2]: [[alpha].sub.3] =
-[[alpha].sub.4]
Interest Rate Symmetry [H.sub.3]: [[alpha].sub.5] =
-[[alpha].sub.6]
Price and Output Symmetry [H.sub.4]: [H.sub.1] [union]
[H.sub.2]
Price and Interest Rate [H.sub.1] [union]
Symmetry [H.sub.5]: [H.sub.3]
Output and Interest Rate [H.sub.2] [union]
Symmetry [H.sub.6]: [H.sub.3]
Joint Symmetry of Prices, [H.sub.7]: [H.sub.1] [union]
Interest Rate and Output [H.sub.2] [union]
[H.sub.3]
Proportionality Restrictions
[H.sub.8]: [[alpha].sub.1] =
-[[alpha].sub.2] = 1
[H.sub.9]: [[alpha].sub.3] =
-[[alpha].sub.4] = 1
[H.sub.10]: [[alpha].sub.5] =
-[[alpha].sub.6] = 1
Hypothesis [chi square](df) P-Value
Symmetry Restrictions
Price Symmetry [H.sub.1]: 9.33(3) (a) 0.03
Output Symmetry [H.sub.2]: 7.13(3) (aa) 0.08
Interest Rate Symmetry [H.sub.3]: 16.41(3) 0.00
Price and Output Symmetry [H.sub.4]: 15.73(6) (a) 0.02
Price and Interest Rate
Symmetry [H.sub.5]: 23.24(6) 0.00
Output and Interest Rate
Symmetry [H.sub.6]: 23.00(6) 0.00
Joint Symmetry of Prices, [H.sub.7]: 25.92(9) 0.00
Interest Rate and Output
Proportionality Restrictions
[H.sub.8]: 14.80(6) (a) 0.02
[H.sub.9]: 32.85(6) 0.00
[H.sub.10]: 32.85(6) 0.00
Note: (a), (aa), and (aaa) indicate the significance at 1 percent,
5 percent and 10 percent.
Table 5
Identification of Cointegrating Vectors
Some Theoretical Restrictions
Hypothesis Restricted Cl
vectors
s p [p.sup.f]
y [y.sup.f]
I [i.sup.f]
Individual Parity Conditions
PPP in all Three Vectors [H.sub.11]: 1 -1 1 * * * *
(Strict PPP with other 1 -1 1 * * * *
Fundamental Variables) 1 -1 1 * * * *
PPP in One Vector [H.sub.12]: 1 -1 1 0 0 0 0
(Strict PPP on its Own) * * * * * * *
* * * * * * *
Weak PPP in all Three Vectors [H.sub.13]: 1 * * 0 0 0 0
1 * * 0 0 0 0
1 * * 0 0 0 0
Weak PPP in One Vector [H.sub.14]: 1 * * 0 0 0 0
(PPP on its Own) * * * * * * *
* * * * * * *
UIP in all Three Vectors [H.sub.15]: 1 * * * * 1 -1
(Strict UIP with other 1 * * * * 1 -1
Fundamental Variables) 1 * * * * 1 -1
UIP in One Vector [H.sub.16]: 1 0 0 0 0 1 -1
(Strict UIP on its Own) * * * * * * *
* * * * * * *
Weak UIP in all Three Vectors [H.sub.17]: 1 0 0 0 0 * *
1 0 0 0 0 * *
1 0 0 0 0 * *
Weak UIP in One Vector [H.sub.18]: 1 0 0 0 0 * *
(UIP on its Own) * * * * * * *
* * * * * * *
Combined Parity Conditions
PPP and UIP [H.sub.19]: 1 -1 1 0 0 1 -1
(Strict PPP and Strict UIP) * * * * * * *
* * * * * * *
PPP and UIP [H.sub.20]: 1 * * 0 0 1 -1
(Weak PPP and Strict UIP) 1 * * 0 0 1 -1
1 * * 0 0 1 -1
PPP and UIP [H.sub.21]: 1 * * 0 0 1 -1
(Weak PPP and Strict UIP) * * * * * * *
* * * * * * *
PPP, i, I * [H.sub.22]: 1 -1 1 0 0 * *
(Strict PPP and Weak UIP) * * * * * * *
* * * * * * *
Weak PPP and Weak UIP [H.sub.23]: 1 * * 0 0 * *
1 * * 0 0 * *
1 * * 0 0 * *
Other Restrictions
PPP, y, y * [H.sub.24]: 1 -1 1 * * 0 0
* * * * * * *
* * * * * * *
Relationship between [H.sub.26]: 1 0 0 * * 0 0
s,y,y * * * * * * * *
* * * * * * *
PPP, UIP and Output 1 -1 1 -1 1 1 -1
Symmetry * * * * * * *
* * * * * * *
Hypothesis
[chi sqaure]
(df)
Individual Parity Conditions
PPP in all Three Vectors [H.sub.11]: 14.80(6) (a)
(Strict PPP with other
Fundamental Variables)
PPP in One Vector [H.sub.12]: 15.98(4)
(Strict PPP on its Own)
Weak PPP in all Three Vectors [H.sub.13]: 52.83(12)
Weak PPP in One Vector [H.sub.14]: 16.44(2)
(PPP on its Own)
UIP in all Three Vectors [H.sub.15]: 32.85(6)
(Strict UIP with other
Fundamental Variables)
UIP in One Vector [H.sub.16]: 2.06(4) (aaa)
(Strict UIP on its Own)
Weak UIP in all Three Vectors [H.sub.17]: 70.84(12)
Weak UIP in One Vector [H.sub.18]: 0.58(2) (aaa)
(UIP on its Own)
Combined Parity Conditions
PPP and UIP [H.sub.19]: 1.48(4) (aaa)
(Strict PPP and Strict UIP)
PPP and UIP [H.sub.20]: 60.95(12)
(Weak PPP and Strict UIP)
PPP and UIP [H.sub.21]: 0.73(2) (aaa)
(Weak PPP and Strict UIP)
PPP, i, i * [H.sub.22]: 0.42(2) (aaa)
(Strict PPP and Weak UIP)
Weak PPP and Weak UIP [H.sub.23]: 26.35(6)
Other Restrictions
PPP, y, y * [H.sub.24]: 1.72(2) (aaa)
Relationship between [H.sub.26]: 4.30(2) (aaa)
s,y,y *
PPP, UIP and Output 0.38(4) (aaa)
Symmetry
Hypothesis P- Value
Individual Parity Conditions
PPP in all Three Vectors [H.sub.11]: 0.02
(Strict PPP with other
Fundamental Variables)
PPP in One Vector [H.sub.12]: 0.003
(Strict PPP on its Own)
Weak PPP in all Three Vectors [H.sub.13]: 0.00
Weak PPP in One Vector [H.sub.14]: 0.00
(PPP on its Own)
UIP in all Three Vectors [H.sub.15]: 0.00
(Strict UIP with other
Fundamental Variables)
UIP in One Vector [H.sub.16]: 0.73
(Strict UIP on its Own)
Weak UIP in all Three Vectors [H.sub.17]: 0.00
Weak UIP in One Vector [H.sub.18]: 0.75
(UIP on its Own)
Combined Parity Conditions
PPP and UIP [H.sub.19]: 0.83
(Strict PPP and Strict UIP)
PPP and UIP [H.sub.20]: 0.00
(Weak PPP and Strict UIP)
PPP and UIP [H.sub.21]: 0.69
(Weak PPP and Strict UIP)
PPP, i, I * [H.sub.22]: 0.82
(Strict PPP and Weak UIP)
Weak PPP and Weak UIP [H.sub.23]: 0.00
Other Restrictions
PPP, y, y * [H.sub.24]: 0.22
Relationship between [H.sub.26]: 0.12
s,y,y *
PPP, UIP and Output 0.85
Symmetry
Note: * In column three represents no restriction.
(a), (aa) and (aaa) in column four indicate the significance
at 1 percent, 5 percent and 10 percent.
Table 6
Out-of- Sample Forecast Evaluation: RMSE
RMSE Forecast Horizon
1 4 8 12 16
RW Model 0.048 0.103 0.162 0.201 0.247
RW with Drift Model 0.030 0.089 0.152 0.177 0.199
Keynesian Model 0.024 0.019 0.021 0.018 0.021
Table 7
Out-of-Sample Forecast Evaluation: Theil's U
Forecast Horizon
Model 1 4 8 12 16
Benchmark: RW Model
Keynesian 0.793 0.216 0.135 0.104 0.107
Benchmark: RW with Drift Model
Keynesian 0.507 0.187 0.127 0.091 0.086
Table 8
Out-of- Sample Forecast Evaluation: DM Test Statistic
Forecast Horizon
1 4 8 12 16
Benchmark Loss Function: RW Model
Keynesian 0.5733 1.458 (a) 1.869 (a) 2.201 2.268
(0.570) (0.154) (0.072) (0.037) (0.034)
Benchmark Loss Function: RW with Drift Model
Keynesian 1.133 (a) 3.011 4.146 3.583 2.902
(0.265) (0.005) (0.000) (0.001) (0.009)
* Note: Represents the acceptance of null hypothesis of equal forecast.
A probability value of DM statistics is in brackets.
