The Canadian Dollar Determination Puzzle
Ali Emre Konukoglu
Jordi Mondria
Rotman School of Management
University of Toronto
(Very Preliminary. Comments are welcome)
May 2008
Abstract
The exchange rate determination puzzle refers to the empirical finding that nominal
exchange rate models based on macroeconomic fundamentals are outperformed by a
driftless random walk. Several recent papers have developed models in international
macroeconomics with good performance for several currencies of developed countries,
and claim the puzzle no longer exists. Others claim the puzzle is still there since there is
no single standard model that is able to explain exchange rate movements of all
developed currencies, even though they assert that most currencies of developed
countries have a structural model with high explanatory power. In this paper, we claim
there is no single traditional structural model that beats the random walk for the Canadian
dollar. We asses the performance of several recent models based on macroeconomic
fundamentals of nominal exchange rate as well as models based on commodity prices.
The models are estimated in first-difference, dynamic OLS and error correction
specifications. Their performance is evaluated at forecast horizons of 1, 2, 4, 12,16 and
20 quarters using the mean squared error ratio. The statistical significance of the results
are assessed by the Diebold and Mariano (1995) t-statistics, Clark and West (2006)
statistics and change of direction statistics.
1. INTRODUCTION
What are the determinants of the Canadian dollar? There has been a great deal of research
on identifying the relationship between macroeconomic fundamentals and exchange rates.
Recently, several papers have developed structural models with macroeconomic
fundamentals that are able to explain how the exchange rate values of most currencies in
developed countries are determined. However, none of these fundamental variables seem
to be able to predict the behavior of the Canadian exchange rate. This paper re-assess if
standard macro models could significantly outperform a driftless random walk in out-ofsample forecasts of the Canadian dollar.
The empirical finding that exchange rates are poorly predicted by macroeconomic
fundamentals is usually called the exchange rate determination puzzle. In a seminal
paper Richard Messe and Kenneth Rogoff (1983) conducted a monthly out-of -sample
analysis for several exchange rates. Time series and structural models based on
macroeconomic fundamentals were outperformed by a driftless random walk.
Mark (1995) was the first to shed light on this puzzle. He presented evidence that long
horizon changes in the exchange rate depend on the current deviation from its
fundamentals. His out-of-smaple forecasts beat the random walk at the 12- and 16-quarter
horizons for the Deutsche mark, Swiss franc and the yen and concluded that exchange
rates are predictable at a three year horizon. However, there was no evidence that there is
an economically and statistically significant component in long horizon changes of the
Canadian dollar.
More recently, Gourinchas and Rey (2007) have developed fundamental approach to
exchange rates that beats the random walk over less than a year horizon. They generated
2
a new variable of external imbalances, which is based on an intertemporal budget
constraint, in order to predict exchange rates. This new variable combines linearly
components of exports, imports, assets and liabilities. It is a new measure of external
imbalances that unlike the current account, it incorporates information both the trade
balance and the foreign asset position. They found that this new variable beats the
random walk at all horizons starting one quarter ahead. They found significant predictive
power for currencies such as the US dollar against the Sterling, the Japanese yen, the
German D-Mark (Euro after 1999) and the Swiss Franc. However, there was no evidence
that this new measure of external imbalances had any significant predictive power for the
Canadian dollar.
Among others, Mark (1995) and Gourinchas and Rey (2007) have built models based on
macroeconomic fundamentals that are able to predict exchange rate movements and
outperform a driftless random walk for several currencies of developed countries. These
empirical findings have lead some economists to say that the exchange rate determination
puzzle is no longer a puzzle. Others such as Cheung, Chinn and Pascual (2005) claim the
puzzle is still there since there is no single standard model that consistently explains
exchange rate movements and outperforms a driftless random walk for most currencies.
This paper provides empirical evidence that there is no single standard structural model
that significally beats the random walk for the Canadian dollar. The traditional
fundamental variables used as potential determinants of the Canadian dollar against the
US dollar are the interest rate differential, output differential, inflation differential, money
supply differential and productivity differential. The standard macroeconomic models are
augmented with a commodity price index as in Amano and Van Norden (1995) and Chen
3
(2004) and an energy price index as in Amano and Van Norden (1995). These
fundamental variables are estimated using first-difference, dynamic OLS and error
correction specifications. Their performance is evaluated at forecast horizons of 1, 2, 4,
12, 16 and 20 quarters, using the mean squared error ratio. Diebold and Mariano (1995) tstatistics, Clark and West (2006) statistics and change of direction statistics are reported
to assess the statistical significance of the results. The evidence in this paper supports the
theory of Engel and West (2005), at least for the Canadian dollar, which suggests that
fundamental variables do not help predict exchange rate values since exchange rates
manifests near–random walk behavior.
2. LITERATURE REVIEW
A more detailed and extensive literature review to be added.
3. EMPIRICAL RESULTS
Throughout the paper three specifications of various theoretical models are estimated: (1)
an error correction specification, (2) Dynamic OLS (DOLS) specification and (3) first
differences specification. Estimation with cointegrated variables raises concerns about
spurious regression bias and Engle and Granger (1987) representation theorem shows the
use of error correction specification for cointegrated variables, which has been widely
used in the exchange rate forecasting literature (Amano and Norden (1993) and Cheung,
Chinn and Pascual (2002)). Although our commodity and energy price indices don’t
show unit roots (Appendix A) and unit roots for other macro variables are not considered
4
by the previous literature we are still using the error correction specification with the
primary goal of being consistent with the previous literature.
Similarly, DOLS procedure, proposed by Stock and Watson (1993) produces efficient
estimates of the cointegrating vector and we are employing it as the alternative estimation
method in this study as in Chen (2004). The first differences specification basically
estimates the differenced series of the nominal exchange rates with the differenced series
of the explanatory variables. It is used by the previous literature as an alternative to error
correction specification (Cheung, Chinn and Pascal (2002) and Chinn and Meese (1995) )
and therefore we include it in our analysis.
For all three specifications we report in-sample results as well as out-of-sample forecasts.
In-sample tests are conducted for the whole sample (1972Q1-2002Q4) and for the
subsample of 1983Q1 to 2002Q4 to account for the end of monetary targeting in US as
also done in the previous literature (Cheung, Chinn and Pascual (2002)). We should
clearly state that we don’t hold a stance against the existence for a structural break in the
CDN/USD exchange rate series and we keep it out of our main goal of evaluating the outof-sample forecasting power of the empirical exchange rate models. To our knowledge
Issa, Lafrance and Murray (2006) is the only paper that specifically deals with the issue
of a structural break in the CDN/USD exchange rate series and our results don’t explicitly
deal with it.
The out-of-sample forecasting is done for two alternative out-of-sample intervals and
with constant moving window of in-sample regressions. For the out-of-sample window
that starts in 1981Q1 we start our first observation at 1972Q1 and use the next 39
observations to forecast the nth observation in the out-of-sample. For the next forecast in
5
the out-of-sample we move the in-sample estimation forward by one observation. We
use six different values of n starting from one quarter up to twenty quarters out-of-sample
to evaluate the forecasting power of the model at different horizons. We also use an
alternative out-of-sample window that starts in 1991Q1 with an in-sample estimation
period that starts in 1981Q1 to test the out-of-sample forecasting power of the empirical
models in the later subsample.
The forecasting power at out-of-sample are evaluated with two methods: Mean Squared
Error (MSE) ratios are reported for the out-of-sample forecasting period as the ratio of
the MSE of the random walk over the MSE of the empirical model that is being tested. A
MSE ratio that is significantly less than one shows the superior forecasting power of the
empirical model with respect to the benchmark of random walk forecasts. The statistical
significance of the MSE ratios is evaluated with two alternative t-statistics as done in the
previous literature. We report the Diebold and Mariano (1995) t-statistics as well as Clark
and West (2006) statistic for that purpose. Clark-West (2006) statistics basically adjusts
the Diebold and Mariano (1995) statistics for the finite sample bias by the squared sum of
the predicted values of the model and is also used by Gourinchas and Rey (2007).
In addition to MSE ratios we also use the change of direction statistics that is used by
Cheung, Chinn and Pascual (2002). Change of direction statistics measures the number of
times the empirical model forecasts the direction of change in the nominal exchange rate
correctly in the out-of-sample. It is a useful measure in terms of giving an indication of
whether the empirical model can be used to time the future exchange rate changes. A
change of direction measure that is greater than 0.5 means that the model performs better
than the random walk in the out-of-sample.
6
3.1. Error Correction Specification
3.1.1. Amano and Norden (1993) Specification
This section follows the Amano and Norden (1993) specification that uses to explain real
CDN/USD exchange rate with the terms of trade and real exchange rates. The basic idea
behind that study is to use stationary and non-stationary variables together to explain the
differences in the real exchange rate series.
/ us
Δε tcdn / us = α + β1ε tcdn
+ β 2 ft −1 + ε t
−1
(1)
As stated by Equation 1, the forward difference in log CDN/USD exchange rate series is
estimated using the lagged value of the log exchange rate as well as the lagged values of
the log commodity and price indices and the difference in real interest rates between US
and Canada.
This study modifies the Amano and Norden (1993) framework in two dimensions: Firstly,
we use the nominal CDN/USD exchange rate series since the paper is about the nominal
CDN/USD exchange rate. And secondly, we extend the Amano and Norden (1993) study
for the out-of-sample forecast results.
In-sample results, reported in Table3, clearly show the poor performance of the
explanatory variables for the whole sample period. Low R-squared values and
insignificant coefficient estimates suggest the poor fit of the Amano and Norden (1993)
specification for the nominal CDN/USD exchange rate series. One issue of having very
poor in-sample explanatory power of the model is the difficulty of interpreting the out-ofsample results relative to random walk specification. Poor in-sample results such as in
Table 3 signals the possibility of comparing noise with random walk at out-of-sample and
we fully consider it below while discussing the out-of-sample results.
7
The subsample results show significantly stronger explanatory power of the Amano and
Norden (1993) specification for the later part of the sample (Table 4). One important
result is the positive coefficient estimate for energy price index, which has been reported
previously as a counterintuitive Canada being a net exporter of energy products like crude
oil and natural gas.
The out-of-sample results are documented in Tables 5 and 6. Both tables show that terms
of trade specification doesn’t have superior forecasting power compared to the random
walk. The out-of-sample forecasts starting from 1981Q1 seem to be better than the
shorter sample that starts in 1991Q1 but considering the poor in-sample results we can’t
make much out of the MSE ratios that are significantly less than one. Change of direction
statistics are also significantly less than 0.5 showing inferior predictability compared to
random walk. The results collectively show that the terms of trade specification suggested
by Amano and Norden (1993) performs poorly in the out-of-sample for our sample period.
3. 1.2. Cheung, Chinn and Pascual (2002) Specification
This section uses the two-step error-correction specification for predicting the differences
in CDN/USD exchange rates. The 1st step estimates the coefficients using the levels of
exchange rates and fundamentals. The estimated coefficients are then used to predict the
change in the exchange rates for k periods ahead:
8
1st step:
ε tcdn / us = α + β ft + ε t
(2)
2nd step:
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
(3)
In error-correction model the choice of k is crucial and since we don’t have a specific
model for the choice of k we are conducting the tests for four different levels. Also two
sets of macroeconomic fundamentals are used in the estimation:
The first model uses the sticky price monetary model as the basis and uses the log
differences in real output levels and money stocks as well as differences in real interest
and inflation rates as explanatory variables in the first and second steps. The in-sample
results of the second step for the whole sample are reported in Table 7. We can clearly
see that the in-sample explanatory power increases in k. The subsample starting 1983Q1
shows better in-sample predictive power like in previous section (Table 8).
The out-of-sample results (Tables 9 and 10) indicate poor out-of-sample forecasting
power of the error correction specification for sticky price monetary model. 1 out of 24
MSE ratios are significantly less than one for the longer out-of-sample period of 1981Q12002Q4 and only 4 out of 24 MSE ratios are significantly less than one for the later outof-sample period of 1991Q1-2002Q4. Change of direction also confirm the poor out-ofsample forecasting power, none of them is greater than 0.5. Another important
observation from this section is that the MSE ratios are decreasing in k, meaning that the
model performs better in differences in CDN/USD exchange rate in longer periods.
9
The second model is the generic productivity differential exchange rate equation from
Chinn, Cheung and Pascual (2002), which uses differences in log real output levels,
money stocks and productivity levels and differences in real interest rates between
Canada and US. The in-sample and out-of-sample performances of the productivity
differential model are also similar to the ones of sticky price monetary model (Tables 1114). Only 1 out of 24 MSE ratios is significantly less than one for the out-of-sample
period of 1981Q1-2002Q4 and 3 out of 24 for the shorter out-of-sample period of
1991Q1-2002Q4.
3.2. Chen (2004) Specification
In DOLS specification the levels of CDN/USD nominal exchange rates are estimated
using the levels of macroeconomic and terms of trade variables and their lagged and
forward differences to control for cointegration. Given the single unit root in CDN/USD
nominal exchange rates and no unit root in energy and commodity price indices we have
decided to include the current, one forward and one lagged differences in fundamentals
(DOLS(1,-1)). Equation 4 summarizes the idea behind the DOLS specification:
ε
p
cdn / us
t
= α + β ft + ∑ Δft −i +ε t
(4)
i =− p
In this section we are testing three different empirical models: First model is a modified
version of the asset approach which uses log commodity and energy price indices in
addition to log differences in money stocks and real output levels as the explanatory
variables. The modified flex-price monetary model adds the log differences in money
stocks as the additional explanatory variable. And thirdly, sticky price monetary model
10
uses the log differences in money stocks in addition to the variables used in modified
flex-price model. The additional use of energy and commodity price indices in this
section should be seen as an attempt to improve the in-sample and out-of-sample
performances of the conventional models that have been documented to perform poorly
for nominal CDN/USD exchange rate by the previous literature.
The in-sample results are summarized in Table 15 for the whole sample (1972Q12002Q4) and in Table 16 for the later subsample (1983Q1-2002Q4). The in-sample
performances of all three models are pretty strong and even stronger in the latter half of
the sample. Another important observation from Tables 15 and 16 is that the additional
explanatory variables improve the in-sample explanatory powers of the models. Also the
sign of the log differences in output levels reverses sign in the latter subsample, which
may signal a structural break in the data.
Surprisingly, the out-of-sample predictive power of the models is very poor. For none of
the models the MSE ratios are less than one at any out-of-sample forecast horizon. The
change of direction values are also very low (Table 17 for the larger out-of-sample and
Table 18 for the latter out-of-sample).
11
3.3. First Differences Specification
First-differences specification predicts the first differences in nominal CDN/USD
exchange rate with first differences of the macroeconomic fundamentals and terms of
trade variables. Equation 5 shows the first-differences empirical equation.
Δε tcdn / us = α + βΔft + ε t
(5)
In this section we are testing two empirical models: (1) Classical sticky-price model and
(2) modified sticky-price model. Given the in-sample success of the modified sticky-price
model with the DOLS specification we decided to concentrate on it for the firstdifferences specification as well.
For the classical sticky-price model Table 19 reports the in-sample result for the whole
sample and Table 20 for the latter subsample. In-sample fit seems to be better than errorcorrection specification in Section 1 and like in all the other results in the paper the latter
subsample shows a better fit of the model.
Tables 20 and 21 summarize the out-of-sample predictive power of the classical stickyprice model. For neither out-of-sample the first-differences specification seems to
produce MSE ratios that are significantly less than 1. The change of direction criteria also
confirms the poor out-of-sample power of the model.
For the modified sticky-price model the in-sample results show that the addition of the
energy and commodity price indices don’t add on the explanatory power of the model
(Tables 22 and 23). Out-of-sample forecasts also confirm the poor performance of the
modified sticky-price model with first-differences specification.
12
4. CONCLUSION
This paper shows that there is no nominal exchange rate determination model based on
macroeconomic fundamental variables that is able to explain changes in the Canadian
dollar and outperform a driftless random walk in out-of sample forecasts.
This empirical evidence either supports the theory of Engel and West (2005), which
suggests that exchange rates behave as a near–random walk or raises the question about
the Canadian dollar having a unique characteristic not yet analyzed.
5. REFERENCES
Amano, R. and S. Norden, 1993, A forecasting equation for the Canada-U.S dollar exchange rate,
in The Exchange Rate and the Economy, 201-65 (Bank of Canada, Ottawa).
Chen, Y., 2004, Exchange Rate and Fundamentals: Evidence from Commodity Economies,
University of Washington Working Paper.
Cheung, Y., Chinn, M. and A. Pascual, 2002, Empirical exchange rate models of the nineties: Are
any fit to survive?, NBER Working Paper 9393.
Chinn, M. and R. Meese, 1995, Banking on Currency Forecasts: How Predictable Is Change in
Money? , Journal of International Economics 38, 161-178.
Clark, T. and K. West, 2006, Using out-of-sample mean squared prediction errors to test the
martingale difference hypothesis, Journal of Financial Econometrics, 135, 155-186.
Diebold, F. and Mariano R., 1995, Comparing Predictive Accuracy, Journal of Business and
Economic Statistics, 13, 253-265
Djoudad R., J. Murray, T. Chan and J. Daw, 2000, The role of chartists and fundamentalists in
currency markets: the experience of Australia, Canada and New Zealand, in Revisiting the Case
for Flexible Exchange Rates, proceedings of a conference held at the Bank of Canada,
Engel, C., K. West, 2005, Exchange Rates and Fundamentals, Journal of Political Economy, 113,
485-517.
Engle, R and C. Granger,1987, Cointegration and Error Correction: Representation, Estimation
and Testing, Econometrica, 55. 251-276
Issa, R., R. Lafrance and J. Murray, 2006, The turning black tide: Energy prices and Canadian
dollar, Bank of Canada Working paper.
13
Gourinchas, P. and H. Rey, 2007, International Financial Adjustment, NBER working paper
Mark, N., 1995, Exchange rates and fundamentals: evidence on long-horizon prediction,
American Economic Review, 85, 201-218.
Messe, R., K. Rogoff, 1983, Empirical Exchange rate models of the Seventies: Do they fit out of
sample, Journal of International Economics, 14, 3-24.
6. APPENDICES
Data Appendix
End of period quarterly nominal exchange rates for CDN/USD are taken from IMF’s
International Financial Statistics (IFS) for the period 1972Q1 to 2002Q4.
Short term Interest Rates: 3-month US and Canadian Treasury bill rates are taken from
IFS.
Consumer Price Index (CPI): Quarterly consumer prices are taken from IFS.
Money Stock (M1): M1 series are taken from IFS.
Real Output: GDP volume (1995=100) is taken from IFS.
Commodity and Energy Prices: The country specific commodity and energy export price
indices are constructed by using the weights in Djoudad, Murray, Chan and Daw (2001)
and world market prices in US dollar from IFS.
Productivity: The productivity series are labor productivity indices, measured as real
GDP per employee (converted to indices (1995=100)). The data are drawn from
CANSIM and IFS.
Description of variable symbols:
ε tcdn / us : Quarterly CDN/USD exchange rate
pteng : Quarterly Canadian energy index
ptcom : Quarterly Canadian commodity index
rt cdn : Quarterly Canadian short term interest rates
rtus : Quarterly US short term interest rates
m1tcdn : Quarterly Canadian M1 base
m1us
t : Quarterly US M1 base
ytus
: Quarterly US GDP
cdn
t
y
: Quarterly Canadian GDP
us
t
cpi
: Quarterly US CPI
cpitcdn
: Quarterly Canadian CPI
π tus : Quarterly US inflation
π tcdn : Quarterly Canadian inflation
14
prod tus : Quarterly US labor productivity index (1995=100)
prod tcdn : Quarterly Canadian labor productivity index (1995=100)
VARIABLE
ε
cdn / us
t
MEAN
MEDIAN
ST-DEV
SKEWNESS
KURTOSIS
MIN
MAX
1.2528
1.2324
0.1685
0.0463
-0.8392
0.9664
1.5870
pteng
0.0143
0.0134
0.0038
0.5869
-0.5217
0.0072
0.0241
com
t
0.4428
0.4294
0.0442
0.9606
0.4597
0.3593
0.5732
8.0584
8.1400
3.6200
0.7375
0.4610
2.0000
19.3500
6.5131
5.7400
2.7841
1.1463
1.7781
1.6300
15.6600
711.2845
743.8000
336.3540
0.0355
-1.5734
231.7000
1203.5000
m1
88.9434
82.3600
66.4691
0.9132
-0.0956
17.7600
263.4900
ytus
5080.7504
4695.8000
2789.2710
0.3459
-1.0691
1192.5000
10486.1000
551.5889
536.9350
297.1366
0.2009
-1.0717
103.52000
1138.2100
cpi
74.1433
74.2300
27.9146
-0.1527
-1.2192
27.0900
118.5200
cpitcdn
p
rt
cdn
rtus
us
t
m1
cdn
t
cdn
t
y
us
t
73.8113
77.9000
28.3215
-0.3267
-1.2925
24.5500
115.0900
π
us
t
0.0122
0.0097
0.0084
1.0854
0.7855
-0.00283
0.0391
π
cdn
t
0.1279
0.0109
0.0094
0.4018
-0.5762
-0.0088
0.0341
prod tus
73.8909
72.9763
30.6037
0.0439
-1.1942
24.5473
128.5064
prod tcdn
71.5074
71.5240
31.2967
-0.0648
-1.1715
18.8108
125.3481
Table 1: Summary statistics for the variables used in the study. All the variables are measured
quarterly for 1972Q1-2002Q4 period. CDN/US exchange rate gives the value of CDN with
respect to one USD.
15
Appendix A
Time Series Persistence of CDN/USD exchange rate and commodity and energy price
indices:
To answer whether CDN/USD exchange rate and commodity and energy price indices
contain unit roots we are using Adjusted Dickey Fuller (ADF). Equation 1 describes the
basic idea behind the ADF.
n
Δyt = α + β1 yt −1 + β 2t + ∑ γ k Δyt −k + ε t
(1a)
k =1
According to ADF if yt has unit root or a stochastic trend the joint hypothesis of β1 =0
and β2 =0 should not be rejected. The lags of differenced yt series are used to control for
the time series persistence of the differenced series and the number of lags for the
differenced series is determined depending on the time series persistence of yt .
Table 2 reports the ADF results for CDN/USD exchange rate and commodity and energy
price indices. The results suggest the existence of a single unit root for the exchange rate
series (Table 2-Panel A). For commodity and energy price indices ADF tests couldn’t
detect unit roots for neither series (Table 2-Panels B and C). Since commodity and energy
price indices show no unit roots further tests of cointegration are omitted in this section.
Table 2-Panel A: ADF test for ε tcdn / us
α
parameter estimate
0.0039
t-value
1.08
p-value
0.28
-1.77
0.08
1.17
1.55
0.24
0.12
ε
-0.0479
cdn / us
Δε t −1
0.1081
t
0.0002
Joint significance of β1 β 2
F-stats = 1.57
Pr>F=0.21
cdn / us
t −1
Adj- R 2
0.01
F-stats
1.34
Pr>F
0.27
Table 2-Panel B: ADF test for ptcom
α
parameter estimate
-0.1415
t-value
-2.86
p-value
0.01
ptcom
−1
-0.1930
-2.93
0.00
-0.3000
-0.0003
-3.45
-1.74
0.00
0.09
Δptcom
−1
t
Joint significance of β1 β 2
F-stats = 4.33
Pr>F=0.02
16
Adj- R 2
0.20
F-stats
10.89
Pr>F
0.00
Table 2-Panel C: ADF test for pteng
α
parameter estimate
-0.7280
t-value
-3.43
p-value
0.00
pteng
−1
-0.1800
-3.50
0.00
0.0548
t
-0.0006
Joint significance of β1 β 2
0.60
-1.58
0.60
0.12
Δp
eng
t −1
Adj- R 2
0.07
F-stats
0.01
Pr>F
0.00
F-stats = 6.14
Pr>F=0.00
Table 2: ADF test results for log nominal CDN/USD series, log total commodity and log energy price indices for
1972Q1-2002Q4.
/ us
Δε tcdn / us = α + β1ε tcdn
+ β 2 ft −1 + ε t
−1
ft = ⎡⎣
coefficient estimate
0.0302
α
ε
rt cdn − rtus
ptcom
pteng ⎤⎦
t-value
1.03
p-value
0.30
cdn / us
t −1
-0.0145
-1.03
0.31
com
t −1
-0.0212
-1
0.31
0.0089
1.21
0.18
-0.1576
-1.63
0.11
p
eng
t −1
p
−r
cdn
t −1
us
t −1
r
Adj- R 2
0.01
F-stats
1.20
Pr>F
0.3163
Table 3: In-sample results for Amano and Norden (1993) specification are reported for the time period 1972Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the lagged values of
the log nominal CDN/USD exchange rate, log commodity price index, log energy price index and the difference
between US and Canadian real short term interest rates. The coefficient estimates that are statistically significant at 5%
are highlighted.
/ us
Δε tcdn / us = α + β1ε tcdn
+ β 2 ft −1 + ε t
−1
ft = ⎡⎣
coefficient estimate
0.0274
α
ε
ptcom
pteng ⎤⎦
t-value
0.79
p-value
0.43
cdn / us
t −1
-0.0826
-2.52
0.01
ptcom
−1
-0.0829
-2.66
0.01
0.0147
1.69
0.09
-0.4806
-3.01
-3.01
eng
t −1
p
cdn
t −1
r
rt cdn − rtus
−r
us
t −1
Adj- R 2
0.11
F-stats
3.48
Table 4: In-sample results for Amano and Norden (1993) specification are reported for the time period 1983Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the lagged values of
the log nominal CDN/USD exchange rate, log commodity price index, log energy price index and the difference
between US and Canadian real short term interest rates. The coefficient estimates that are statistically significant at 5%
are highlighted.
17
Pr>F
/ us
Δε tcdn / us = α + β1ε tcdn
+ β 2 ft −1 + ε t
−1
Forecast horizon
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of Direction
t-stats
p-value
1
0.00041
0.00041
1.003
-0.036
0.97
1.935
0.06
0.239
4.550
0.00
ft = ⎡⎣ rtcdn − rtus ptcom pteng ⎤⎦
2
4
12
16
0.00038 0.00038 0.00040 0.00039
0.00036 0.00038 0.00048 0.00049
1.201
1.239
0.956
0.985
0.499
1.699
-2.372
-2.340
0.62
0.09
0.02
0.02
2.218
2.887
-0.718
-0.993
0.03
0.01
0.48
0.32
0.259
0.254
0.194
0.194
4.924
4.737
3.986
3.986
0.00
0.00
0.00
0.00
20
0.00042
0.00050
1.214
-2.419
0.02
-1.026
0.31
0.206
4.015
0.00
Table 5: Out-of-sample results for Amano and Norden (1993) specification are reported for the time period 1982Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the lagged values of
the log nominal CDN/USD exchange rate, log commodity price index, log energy price index and the difference
between US and Canadian real short term interest rates. The MSE ratio estimates that are statistically significant
according to Clark and West (2006) t-statistics at 5% are highlighted as well as Change or Direction estimates that are
significant at 5%.
/ us
Δε tcdn / us = α + β1ε tcdn
+ β 2 ft −1 + ε t
−1
Forecast horizon
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
1
0.00042
0.00041
0.982
0.159
0.88
1.321
0.20
0.111
1.803
0.08
ft = ⎡⎣ rtcdn − rtus ptcom pteng ⎤⎦
2
4
12
16
0.00043 0.00029 0.00033 0.00037
0.00045 0.00040 0.00041 0.00040
1.045
1.020
1.224
1.099
-0.316
0.889
-1.500
-0.564
0.75
0.38
0.15
0.58
0.720
1.699
-0.286
0.342
0.48
0.10
0.78
0.74
0.185
0.185
0.148
0.148
2.431
2.431
2.126
2.126
0.02
0.02
0.04
0.04
20
0.00040
0.00048
1.2
-1.256
0.11
-0.320
0.76
0.148
2.126
0.04
Table 6: Out-of-sample results for Amano and Norden (1993) specification are reported for the time period 1991Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the lagged values of
the log nominal CDN/USD exchange rate, log commodity price index, log energy price index and the difference
between US and Canadian real short term interest rates. The MSE ratio estimates that are statistically significant
according to Clark and West (2006) t-statistics at 5% are highlighted as well as Change or Direction estimates that are
significant at 5%.
18
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
K
δ0
δ1
Adj- R 2
F-stats
Pr>F
1
0.004
(2.28)
-0.028
(-1.14)
0.00
1.31
0.26
rt cdn − rtus
2
0.008
(3.02)
-0.062
(-1.71)
0.02
2.93
0.09
m1tcdn − m1tus π tcdn − π tus ⎤⎦
4
0.016
(4.05)
-0.129
(-2.35)
0.04
5.50
0.02
12
0.048
(6.48)
-0.464
(-4.50)
0.15
20.25
0.00
Table 7: In-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1972Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
are log differences in US and Canadian real GDP levels and M1 money stocks as well as the differences in US and
Canadian inflation and real interest rates. T-statistics are reported in parenthesis below the coefficient estimates. The
coefficient estimates that are statistically significant at 5% are highlighted..
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
K
δ0
δ1
Adj- R 2
F-stats
Pr>F
1
0.003
(1.44)
-0.097
(-2.27)
0.05
5.17
0.03
rt cdn − rtus
2
0.005
(1.83)
-0.193
(-3.34)
0.11
11.15
0.00
m1tcdn − m1tus π tcdn − π tus ⎤⎦
4
0.011
(2.50)
-0.417
(-4.78)
0.22
22.81
0.00
12
0.027
(3.60)
-1.350
(-9.30)
0.55
86.41
0.00
Table 8: In-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1983Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
are log differences in US and Canadian real GDP levels and M1 money stocks as well as the differences in US and
Canadian inflation and real interest rates. T-statistics are reported in parenthesis below the coefficient estimates. The
coefficient estimates that are statistically significant at 5% are highlighted..
19
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
rt cdn − rtus
cdn
− π tus ⎤⎦
m1tcdn − m1us
t πt
K
n=1
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
1
2
4
12
0.00041
0.00043
1.049
-0.806
0.42
0.187
0.85
0.134
3.542
0.00
0.00086
0.00089
1.038
-0.514
0.61
0.955
0.34
0.207
4.603
0.00
0.00209
0.00206
0.986
0.167
0.87
1.972
0.05
0.207
4.603
0.00
0.00931
0.00874
0.939
0.645
0.52
3.121
0.00
0.049
2.038
0.04
n=2
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00039
0.00041
1.063
-1.011
0.32
-0.002
0.99
0.136
3.546
0.00
0.00080
0.00089
1.105
-1.169
0.25
0.103
0.92
0.173
4.089
0.00
0.00207
0.00215
1.040
-0.482
0.63
1.347
0.18
0.173
4.089
0.00
0.00933
0.00939
1.006
-0.064
0.95
2.426
0.02
0.049
2.039
0.04
n=4
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00038
0.00042
1.083
-1.316
0.19
-0.363
0.72
0.114
3.167
0.00
0.00081
0.00091
1.129
-1.903
0.06
-0.422
0.67
0.152
3.738
0.00
0.00209
0.00235
1.127
-1.896
0.06
0.023
0.98
0.139
3.552
0.00
0.00949
0.01049
1.106
-1.109
0.27
1.455
0.15
0.051
2.040
0.04
n=12
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00041
0.00045
1.102
-2.313
0.02
-1.525
0.13
0.099
2.767
0.01
0.00084
0.00103
1.226
-3.925
0.00
-3.121
0.00
0.113
2.981
0.00
0.00214
0.00280
1.308
-4.441
0.00
-3.636
0.00
0.113
2.981
0.00
0.01000
0.01400
1.402
-2.955
0.00
-1.593
0.12
0.028
1.424
0.16
n=16
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00039
0.00042
1.059
-1.257
0.21
-0.482
0.63
0.134
3.200
0.02
0.00082
0.00093
1.127
-1.947
0.06
-1.111
0.27
0.179
3.795
0.00
0.00205
0.00242
1.181
-2.271
0.03
-1.351
0.18
0.179
3.795
0.00
0.01000
0.01328
1.344
-2.564
0.01
-1.019
0.31
0
0
1
20
n=20
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00042
0.00043
1.033
-0.722
0.47
0.042
0.97
0.095
2.555
0.01
0.00087
0.00092
1.059
-1.180
0.24
-0.052
0.96
0.159
3.420
0.00
0.00217
0.00246
1.134
-2.445
0.02
-1.030
0.31
0.127
3.003
0.00
0.00960
0.01301
1.353
-3.046
0.00
-1.237
0.22
0
0
1
Table 9: Out-of-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1972Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
and second-step are log differences in US and Canadian real GDP levels and M1 money stocks as well as the
differences in US and Canadian inflation and real interest rates. The MSE ratio estimates that are statistically
significant according to Clark and West (2006) t-statistics at 5% are highlighted as well as Change or Direction
estimates that are significant at 5%.
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
K n=1
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
n=2
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
n=4
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
rt cdn − rtus
cdn
− π tus ⎤⎦
m1tcdn − m1us
t πt
1 0.00041 0.00040 0.968 0.411 0.68 1.221 0.23 0.071 1.776 0.08 2 0.00088 0.00083 0.946 0.584 0.56 1.832 0.07 0.071 1.776 0.08 4 0.00222 0.00205 0.922 1.015 0.32 2.798 0.01 0.071 1.776 0.08 12 0.00865 0.00733 0.848 1.173 0.25 3.111 0.00 0 0 1 0.00042 0.00042 0.995 0.064 0.95 0.828 0.41 0.073 1.777 0.08 0.00086 0.00087 1.020 ‐0.188 0.85 1.067 0.29 0.049 1.432 0.16 0.00223 0.00222 0.998 0.028 0.98 1.823 0.07 0.049 1.432 0.16 0.00886 0.00812 0.917 0.607 0.55 2.433 0.02 0 0 1 0.00039 0.00040 1.020 ‐0.230 0.82 0.507 0.62 0.051 1.433 0.16 0.00079 0.00085 1.082 ‐0.858 0.40 0.431 0.67 0.026 1 0.33 0.00193 0.00210 1.088 ‐1.131 0.27 0.908 0.37 0.026 1 0.33 0.00907 0.00934 1.030 ‐0.224 0.82 1.691 0.10 0 0 1 21
n=12
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00036 0.00034 0.947 1.012 0.32 1.381 0.18 0.161 2.402 0.02 0.00073 0.00069 0.944 1.057 0.30 1.617 0.12 0.161 2.402 0.02 0.00142 0.00130 0.922 1.223 0.23 2.103 0.04 0.161 2.402 0.02 0.00947 0.00568 1.665 ‐2.945 0.01 ‐2.011 0.05 0 0 1 n=16
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00037 0.00035 0.945 0.928 0.36 1.136 0.26 0.148 2.126 0.04 0.00072 0.00066 0.925 1.177 0.25 1.578 0.13 0.111 1.803 0.08 0.00158 0.00134 0.846 3.082 0.00 3.504 0.00 0.148 2.126 0.04 0.00420 0.00660 1.570 ‐1.730 0.10 ‐0.411 0.69 0 0 1 n=20
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00043 0.00044 1.014 ‐0.370 0.72 ‐0.220 0.82 0.043 1 0.33 0.00083 0.00087 1.040 ‐0.723 0.48 ‐0.494 0.63 0 0 1 0.00180 0.00193 1.067 ‐1.298 0.21 ‐0.978 0.34 0 0 1 0.00431 0.00556 1.289 ‐1.008 0.32 0.445 0.66 0 0 1 Table 10: Out-of-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1982Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
and second-step are log differences in US and Canadian real GDP levels and M1 money stocks as well as the
differences in US and Canadian inflation and real interest rates. The MSE ratio estimates that are statistically
significant according to Clark and West (2006) t-statistics at 5% are highlighted as well as Change or Direction
estimates that are significant at 5%.
22
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
K
δ0
rt cdn − rtus
1
0.004
(2.37)
-0.023
(-0.94)
0.00
0.8
0.89
δ1
Adj- R 2
F-stats
Pr>F
cdn
m1tcdn − m1us
− prod tus ⎤⎦ t prod t
2
0.008
(3.03)
-0.047
(-1.31)
0.01
1.71
0.20
4
0.016
(4.03)
-0.105
(-1.95)
0.02
3.80
0.05
12
0.048
(6.51)
-0.460
(-4.59)
0.16
21.08
0.00
Table 11: In-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1972Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
are log differences in US and Canadian real GDP levels, productivity levels and M1 money stocks as well as the
differences in US and Canadian real interest rates. T-statistics are reported in parenthesis below the coefficient
estimates. The coefficient estimates that are statistically significant at 5% are highlighted.
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
K
δ0
δ1
Adj- R 2
F-stats
Pr>F
rt cdn − rtus
1
0.003
(1.54)
-0.103
(-2.45)
0.06
5.98
0.02
cdn
m1tcdn − m1us
− prod tus ⎤⎦ t prod t
2
0.006
(1.91)
-0.213
(-3.70)
0.14
13.66
0.00
4
0.011
(2.65)
-0.458
(-5.37)
0.26
28.80
0.00
12
0.028
(3.80)
-1.396
(-9.79)
0.58
95.76
0.00
Table 12: In-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1972Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
are log differences in US and Canadian real GDP levels, productivity levels and M1 money stocks as well as the
differences in US and Canadian real interest rates. T-statistics are reported in parenthesis below the coefficient
estimates. The coefficient estimates that are statistically significant at 5% are highlighted.
23
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
rt cdn − rtus
cdn
m1tcdn − m1us
− prod tus ⎤⎦ t prod t
K N=1
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
1 0.00040 0.00043 1.080 ‐1.249 0.22 ‐0.195 0.85 0.136 3.546 0.00 2 0.00086 0.00091 1.057 ‐0.743 0.46 0.889 0.38 0.146 3.726 0.00 4 0.00209 0.00210 1.006 ‐0.066 0.95 1.995 0.05 0.170 4.084 0.00 12 0.00931 0.00878 0.943 0.616 0.54 2.963 0.00 0.024 1.423 0.16 N=2
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00039 0.00042 1.088 ‐1.298 0.20 ‐0.279 0.78 0.136 3.546 0.00 0.00080 0.00092 1.144 ‐1.517 0.13 ‐0.145 0.89 0.124 3.357 0.00 0.00207 0.00220 1.064 ‐0.778 0.44 1.186 0.24 0.148 3.730 0.00 0.00933 0.00934 1.000 ‐0.004 0.99 2.362 0.02 0.025 1.423 0.16 0.00038 0.00042 1.100 ‐1.536 0.13 ‐0.473 0.64 0.114 3.167 0.00 0.00081 0.00094 1.158 ‐2.206 0.03 ‐0.650 0.52 0.101 2.965 0.00 0.00209 0.00244 1.169 ‐2.370 0.02 ‐0.359 0.72 0.114 3.167 0.00 0.00949 0.01033 1.089 ‐0.969 0.34 1.495 0.14 0.025 1.423 0.16 N=12
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00041 0.00045 1.109 ‐2.342 0.02 ‐1.427 0.16 0.099 2.767 0.01 0.00084 0.00106 1.264 ‐4.043 0.00 ‐3.005 0.00 0.099 2.767 0.01 0.00214 0.00294 1.376 ‐4.617 0.00 ‐3.693 0.00 0.099 2.767 0.01 0.01000 0.01400 1.405 ‐3.201 0.00 ‐1.884 0.06 0 0 1 N=16
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00039 0.00041 1.050 ‐1.000 0.33 ‐0.130 0.90 0.149 3.403 0.00 0.00082 0.00097 1.174 ‐2.351 0.02 ‐1.351 0.18 0.119 2.992 0.00 0.00205 0.00026 1.273 ‐3.144 0.00 ‐2.128 0.04 0.119 2.992 0.00 0.00989 0.01339 1.355 ‐2.734 0.01 ‐1.288 0.20 0 0 1 N=4
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
24
N=20
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00042 0.00043 1.033 ‐0.652 0.52 0.149 0.88 0.127 3.003 0.00 0.00087 0.00096 1.112 ‐1.870 0.07 ‐0.613 0.54 0.111 2.784 0.01 0.00217 0.00259 1.193 ‐3.050 0.00 ‐1.539 0.13 0.095 2.555 0.01 0.00961 0.01262 1.313 ‐2.694 0.01 ‐1.012 0.32 0 0 1 Table 13: Out-of-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1982Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
and second-step are log differences in US and Canadian real GDP levels, productivity levels and M1 money stocks as
well as the differences in US and Canadian real interest rates. The MSE ratio estimates that are statistically significant
according to Clark and West (2006) t-statistics at 5% are highlighted as well as Change or Direction estimates that are
significant at 5%.
/ us
/ us
ε tcdn / us − ε tcdn
= δ 0 + δ1 (ε tcdn
− βˆ ft ) +ν t
−k
−k
ft = ⎡⎣ ytcdn − ytus
rt cdn − rtus
cdn
m1tcdn − m1us
− prod tus ⎤⎦ t prod t
K N=1
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
1 0.00040 0.00041 1.004 ‐0.041 0.97 0.866 0.39 0.049 1.432 0.16 2 0.00088 0.00085 0.971 0.286 0.78 1.678 0.11 0.049 1.432 0.16 4 0.00222 0.00212 0.953 0.540 0.59 2.453 0.02 0.071 1.776 0.08 12 0.00719 0.00865 0.831 1.266 0.21 3.188 0.00 0 0 1 N=2
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00042 0.00042 1.002 ‐0.022 0.98 0.808 0.42 0.049 1.432 0.16 0.00086 0.00090 1.047 ‐0.400 0.69 0.888 0.38 0.049 1.432 0.16 0.00223 0.00227 1.020 ‐0.210 0.83 1.512 0.14 0.049 1.432 0.16 0.00886 0.00793 0.895 0.752 0.46 2.572 0.01 0 0 1 0.00039 0.00040 1.017 ‐0.188 0.85 0.609 0.55 0.051 1.433 0.16 0.00079 0.00085 1.086 ‐0.870 0.39 0.393 0.70 0.026 1 0.33 0.00193 0.00213 1.108 ‐1.250 0.22 0.645 0.53 0.026 1 0.33 0.00907 0.00916 1.009 ‐0.068 0.95 1.813 0.08 0 0 1 N=4
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
25
N=12
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00036 0.00033 0.931 1.138 0.26 1.669 0.11 0.161 2.402 0.02 0.00073 0.00068 0.929 1.164 0.25 1.941 0.06 0.194 2.683 0.01 0.00142 0.00126 0.893 1.507 0.14 2.691 0.01 0.226 2.958 0.01 0.00568 0.00963 1.694 ‐3.151 0.00 ‐2.184 0.04 0 0 1 N=16
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00037 0.00034 0.924 1.000 0.33 1.269 0.22 0.111 1.803 0.08 0.00072 0.00064 0.892 1.171 0.25 1.539 0.14 0.111 1.803 0.08 0.00158 0.00127 0.806 2.855 0.01 3.219 0.00 0.185 2.431 0.02 0.00420 0.00690 1.642 ‐1.923 0.07 ‐0.577 0.57 0 0 1 N=20
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
0.00043 0.00044 1.018 ‐0.294 0.77 ‐0.036 0.97 0.043 1 0.33 0.00083 0.00087 1.039 ‐0.502 0.62 ‐0.157 0.88 0 0 1 0.00180 0.00193 1.072 ‐1.117 0.28 ‐0.620 0.54 0 0 1 0.00431 0.00561 1.302 ‐0.976 0.34 0.520 0.61 0 0 1 Table 14: Out-of-sample results for Cheung, Chinn and Pascual (2002) specification are reported for the time period
1982Q1-2002Q4. Differences in log nominal CDN/USD exchange rates k periods ahead are estimated using estimated
using the long-run cointegrating relation estimated in the first-step (Equation (2)). The variables used in the first-step
and second-step are log differences in US and Canadian real GDP levels, productivity levels and M1 money stocks as
well as the differences in US and Canadian real interest rates. The MSE ratio estimates that are statistically significant
according to Clark and West (2006) t-statistics at 5% are highlighted as well as Change or Direction estimates that are
significant at 5%.
26
p
ε tcdn / us = α + β ft + ∑ Δft −i +ε t
i =− p
Asset approachprice flex monetary
0.8893
(5.41)
Flex-price
monetary
2.6111
(7.92)
sticky price monetary
3.7850
(8.69)
ptcom
0.0644
(0.53)
0.3964
(3.76)
0.1839
(1.53)
pteng
0.0171
(0.53)
-0.0451
(-1.21)
-0.0621
(-1.75)
0.2333
(8.06)
0.3542
(11.05)
0.2647
(6.90)
α
m1tcdn − m1us
t
rt cdn − rtus
-3.5788
(-3.84)
π tcdn − π tus
ytcdn − ytus
Adj- R 2
F-stats
Pr>F
0.5575
(3.98)
0.60
11.01
0.00
-3.3782
(-1.88)
-4.2281
(-2.47)
0.6670
(4.86)
0.71
15.45
0.00
1.3692
(6.10)
0.74
15.19
0.00
Table 15: In-sample results for Chen (2004) specification are reported for the time period 1972Q1-2002Q4. Log
nominal CDN/USD exchange rates are estimated using the macroeconomic fundamentals and terms of trade variables.
T-statistics are reported in parenthesis below the coefficient estimates. The coefficient estimates that are statistically
significant at 5% are highlighted. 27
p
ε tcdn / us = α + β ft + ∑ Δft −i +ε t
i =− p
Asset approachprice flex monetary
1.2372
(6.19)
Flex-price
monetary
-1.6849
(-3.80)
Sticky price
monetary
-1.2249
(-2.18)
ptcom
0.3448
(3.00)
-0.255
(-0.25)
-0.0387
(-0.33)
pteng
-0.0257
(-0.70)
0.0799
(2.97)
0.0720
(2.78)
0.3821
(11.40)
0.1190
(4.42)
0.0951
(3.31)
α
m1tcdn − m1us
t
rt cdn − rtus
-1.493
(-1.48)
π tcdn − π tus
ytcdn − ytus
Adj- R 2
F-stats
Pr>F
-0.8456
(-2.34)
0.65
19.59
0.00
-0.2285
(-0.12)
-0.5179
(-0.30)
-1.1571
(-6.43)
0.78
15.25
0.00
-0.9148
(-3.39)
0.81
14.84
0.00
Table 16: In-sample results for Chen (2004) specification are reported for the time period 1982Q1-2002Q4. Log
nominal CDN/USD exchange rates are estimated using the macroeconomic fundamentals and terms of trade variables.
T-statistics are reported in parenthesis below the coefficient estimates. The coefficient estimates that are statistically
significant at 5% are highlighted.
28
p
ε tcdn / us = α + β ft + ∑ Δft −i +ε t
i =− p
Asset Approach
Forecast horizons
RW
The Model
Ratio
t-stats
p-value
Change of dir
t-stats
p-value
Flex-Price
Forecast horizons
RW
The Model
Ratio
t-stats
p-value
Change of dir
t-stats
p-value
Sticky-Price
Forecast horizons
RW
The Model
Ratio
t-stats
p-value
Change of dir
t-stats
p-value
ft = ⎡⎣ p
pteng m1tcdn − m1us
ytcdn − ytus ⎤⎦
t
1
2
4
12
16
20
0.00041 0.00084 0.00225 0.01044 0.01334 0.01588
0.00416 0.00507 0.00699 0.01262 0.01317 0.01384
10.243
6.066
3.111
1.209
0.988
0.872
-5.342
-5.120
-4.443
-1.535
0.118
1.342
0.00
0.00
0.00
0.13
0.91
0.18
0.317
0.179
0.209
0.224
0.269
0.269
5.970
3.795
4.175
4.363
4.924
4.924
0.00
0.00
0.00
0.00
0.00
0.00
com
t
ft = ⎡⎣ ptcom
pteng
cdn
− ytus
m1tcdn − m1us
t yt
π tcdn − π tus ⎤⎦
1
2
4
12
16
20
0.00041 0.00084 0.00225 0.01044 0.01334 0.01588
0.00166 0.00226 0.00386 0.01151 0.01340 0.01505
4.096
2.708
1.715
1.025
1.005
0.948
-6.190
-6.061
-4.541
-1.217
-0.076
0.827
0.00
0.00
0.00
0.23
0.94
0.41
0.134
0.149
0.194
0.269
0.313
0.286
3.200
3.403
3.986
4.924
5.489
4.98
0.00
0.00
0.00
0.00
0.00
0.00
ft = ⎡⎣ ptcom
pteng
cdn
m1tcdn − m1us
− ytus
t yt
π tcdn − π tus rtcdn − rtus
1
2
4
12
16
20
0.00041 0.00084 0.00225 0.01044 0.01334 0.01588
0.00123 0.00184 0.00328 0.01117 0.01344 0.01534
3.027
2.205
1.459
1.071
1.008
0.966
-6.035
-5.530
-3.435
-1.064
-0.141
0.692
0.00
0.00
0.00
0.29
0.89
0.49
0.119
0.149
0.194
0.269
0.299
0.270
2.992
3.403
3.986
4.924
5.230
4.787
0.00
0.00
0.00
0.00
0.00
0.00
Table 17: Out-of-sample results for Chen (2004) specification are reported for the time period 1981Q1-2002Q4. Log
nominal CDN/USD exchange rates are estimated using the macroeconomic fundamentals and terms of trade variables.
MSE ratios those are significantly less than one according to the Diebold and Mariano (1995) t-statistics at 5% are
highlighted as well as the significant change of direction statistics.
⎤⎦
29
p
ε tcdn / us = α + β ft + ∑ Δft −i +ε t
i =− p
Asset Approach
Forecast horizons
RW
The Model
Ratio
t-stats
p-value
Change of dir
t-stats
p-value
Flex-Price
Forecast horizons
RW
The Model
Ratio
t-stats
p-value
Change of dir
t-stats
p-value
Sticky-Price
Forecast horizons
RW
The Model
Ratio
t-stats
p-value
Change of dir
t-stats
p-value
ft = ⎡⎣ p
pteng m1tcdn − m1us
ytcdn − ytus ⎤⎦
t
1
2
4
12
16
20
0.01065
0.00042 0.00097 0.00225 0.00621 0.00834
0.00212 0.00290 0.00506 0.01010 0.01422 0.02161
5.083
2.998
2.250
1.767
1.705
1.572
-4.478
-4.379
-3.697
-3.902
-4.356
-4.268
0.00
0.00
0.00
0.00
0.00
0.00
0.304
0.185
0.077
0.296
0.370
0.333
3.102
2.431 11.907
3.309
3.911
3.606
0.01
0.02
0.00
0.00
0.00
0.00
com
t
ft = ⎡⎣ ptcom
pteng
cdn
− ytus
m1tcdn − m1us
t yt
π tcdn − π tus ⎤⎦
1
2
4
12
16
20
0.00042 0.00097 0.00225 0.00621 0.00834 0.01065
0.00105 0.00154 0.00294 0.00851 0.01082 0.01742
2.527
1.593
1.310
1.370
1.298
1.266
-2.719
-2.059
-2.012
-2.522
-3.305
-3.382
0.01
0.05
0.05
0.02
0.00
0.00
0.222
0.259
0.296
0.370
0.333
0.304
2.726
3.017
3.309
3.911
3.606
3.104
0.01
0.01
0.00
0.00
0.00
0.01
ft = ⎡⎣ ptcom
pteng
cdn
m1tcdn − m1us
− ytus
t yt
π tcdn − π tus rtcdn − rtus
1
2
4
12
16
20
0.00042 0.00097 0.00225 0.00621 0.00834 0.01065
0.00102 0.00177 0.00300 0.00785 0.00991 0.01158
2.451
1.829
1.335
1.264
1.189
1.087
-3.294
-3.124
-2.121
-2.542
-2.516
-2.325
0.00
0.00
0.04
0.02
0.02
0.03
0.148
0.222
0.296
0.333
0.333
0.333
2.126
2.726
3.309
3.606
3.606
3.606
0.04
0.01
0.00
0.00
0.00
0.00
Table 18: Out-of-sample results for Chen (2004) specification are reported for the time period 1991Q1-2002Q4. Log
nominal CDN/USD exchange rates are estimated using the macroeconomic fundamentals and terms of trade variables.
MSE ratios those are significantly less than one according to the Diebold and Mariano (1995) t-statistics at 5% are
highlighted as well as the significant change of direction statistics.
⎤⎦
30
Δε tcdn / us = α + βΔft + ε t
ft = ⎡⎣ ytcdn − ytus rt cdn − rtus
π tcdn − π tus m1tcdn − m1ust
parameter estimate t-value p-value Adj- R 2
0.11
0.0052
3.02
0.00
α
us
rt cdn
−1 − rt −1
0.3431
1.88
0.06
− m1
-0.0339
-0.70
0.48
cdn
t
y
us
t
−y
-0.6212
-3.92
0.00
π
cdn
t
−π
us
t
0.5397
1.59
0.11
cdn
t
m1
us
t
⎤⎦
F-stats
4.72
Pr>F
0.00
Table 19: In-sample results for first-differences specification are reported for the time period 1972Q1-2002Q4.
Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log differences in real
output levels, money stocks and differences in inflation rates and short-term real interest rates. The coefficient
estimates that are statistically significant at 5% are highlighted.
Δε tcdn / us = α + βΔft + ε t
ft = ⎡⎣ ytcdn − ytus rt cdn − rtus
π tcdn − π tus m1tcdn − m1ust
parameter estimate t-value p-value Adj- R 2
0.0029
1.39
0.17
0.22
α
cdn
t −1
r
− rtus−1
m1tcdn − m1us
t
0.6871
2.82
0.01
-0.0056
-0.09
0.93
us
t
-0.9807
-3.84
0.00
us
t
0.9432
2.15
0.03
cdn
t
y
−y
π
cdn
t
−π
⎤⎦
F-stats
6.71
Table 20: In-sample results for first-differences specification are reported for the time period 1983Q1-2002Q4.
Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log differences in real
output levels, money stocks and differences in inflation rates and short-term real interest rates. The coefficient
estimates that are statistically significant at 5% are highlighted.
31
Pr>F
0.00
Δε tcdn / us = α + βΔft + ε t
Forecast horizon
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
ft = ⎡⎣ ytcdn − ytus rt cdn − rtus
π tcdn − π tus m1tcdn − m1ust
1
2
4
12
16
0.00041 0.00038 0.00038 0.00040 0.00039
0.00048 0.00050 0.00044 0.00052 0.00049
1.189
1.320
1.167
1.303
1.252
-1.415
-2.234
0.760
-1.987
-1.795
0.16
0.02
0.45
0.05
0.07
0.523
-0.440
2.562
-0.335
0.0715
0.60
0.66
0.01
0.74
0.94
0.045
0.075
0.075
0.060
0.075
1.759
2.307
2.307
2.047
2.307
0.08
0.02
0.02
0.04
0.02
⎤⎦
20
0.00042
0.00059
1.420
-3.758
0.00
-1.770
0.08
0.048
1.761
0.08
Table 21: Out-of-sample results for first-differences specification are reported for the out-of-sample period 1981Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log differences in
real output levels, money stocks and differences in inflation rates and short-term real interest rates. The MSE ratio
estimates that are statistically significant according to Clark and West (2006) t-statistics at 5% are highlighted as well
as Change or Direction estimates that are significant at 5%.
Δε tcdn / us = α + βΔft + ε t
Forecast horizon
RW
The Model
MSE Ratio
t-stats
p-value
CW t-stats
p-value
Change of direction
t-stats
p-value
ft = ⎡⎣ ytcdn − ytus rt cdn − rtus
π tcdn − π tus m1tcdn − m1ust
1
2
4
12
16
0.00042 0.00043 0.00039 0.00033 0.00037
0.00050 0.00058 0.00047 0.00041 0.00049
1.193
1.365
1.207
1.244
1.335
-0.763
-1.504
0.451
-0.919
-1.081
0.45
0.14
0.66
0.37
0.29
0.366
-0.463
1.594
0.437
-0.070
0.72
0.65
0.12
0.67
0.94
0.111
0
0.074
0.111
0.074
1.803
0
1.442
1.803
1.442
0.08
1
0.16
0.08
0.16
⎤⎦
20
0.00043
0.00063
1.469
-2.211
0.04
-1.163
0.26
0.043
1
0.33
Table 22: Out-of-sample results for first-differences specification are reported for the out-of-sample period 1991Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log differences in
real output levels, money stocks and differences in inflation rates and short-term real interest rates. The MSE ratio
estimates that are statistically significant according to Clark and West (2006) t-statistics at 5% are highlighted as well
as Change or Direction estimates that are significant at 5%.
32
Δε tcdn / us = α + βΔft + ε t
ft = ⎡⎣ ytcdn − ytus rt cdn − rtus
parameter estimate
0.0053
α
ptcom
−1
eng
t −1
p
−r
cdn
t −1
us
t −1
r
ptcom
pteng π tcdn − π tus
t-value
3.05
p-value
0.00
0.0216
0.79
0.43
0.0017
0.13
0.90
0.3538
1.93
0.06
cdn
t
− m1tus
-0.0351
-0.71
0.48
cdn
t
y
us
t
−y
-0.6484
-3.85
0.00
π
cdn
t
−π
us
t
0.5465
1.60
0.11
m1
m1tcdn − m1us
t
Adj- R 2
0.10
⎤⎦
F-stats
3.23
Pr>F
0.01
Table 23: In-sample results for first-differences specification are reported for the time period 1972Q1-2002Q4.
Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log energy and commodity
price indices, log differences in real output levels, money stocks and differences in inflation rates and short-term real
interest rates. The coefficient estimates that are statistically significant at 5% are highlighted.
Δε tcdn / us = α + βΔft + ε t
ft = ⎡⎣ ytcdn − ytus rt cdn − rtus
parameter estimate
-0.0077
α
ptcom
−1
eng
t −1
p
cdn
t −1
r
−r
us
t −1
ptcom
pteng π tcdn − π tus
t-value
-1.14
p-value
0.2583
0.0430
0.35
0.7304
0.6672
0.49
0.6268
1.026
2.94
0.0043
cdn
t
− m1
-0.00027
-2.19
0.0319
cdn
t
y
us
t
−y
-0.00014
-1.74
0.0858
π
cdn
t
− π tus
0.43089
0.86
0.3931
m1
us
t
m1tcdn − m1us
t
Adj- R 2
0.12
⎤⎦
F-stats
2.84
Pr>F
0.0151
Table 24: In-sample results for first-differences specification are reported for the time period 1983Q1-2002Q4.
Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log energy and commodity
price indices, log differences in real output levels, money stocks and differences in inflation rates and short-term real
interest rates. The coefficient estimates that are statistically significant at 5% are highlighted.
33
Δε tcdn / us = α + βΔft + ε t
ft = ⎡⎣ ytcdn − ytus rtcdn − rtus ptcom pteng π tcdn − π tus
1
2
4
12
Forecast horizon
0.00041 0.00039 0.00038 0.00041
RW
0.00051 0.00051 0.00045 0.00055
The Model
1.243
1.332
1.177
1.343
MSE Ratio
-1.713
-2.282
0.985
-1.684
t-stats
0.09
0.03
0.33
0.10
p-value
0.275
-0.280
2.891
-0.334
CW t-stats
0.75
0.78
0.01
0.74
p-value
0.106
0.108
0.111
0.109
Change of direction
2.777
2.779
2.784
2.571
t-stats
0.07
0.07
0.01
0.01
p-value
⎤⎦
20
0.00042
0.00055
1.293
-2.324
0.02
-0.413
0.68
0.085
2.069
0.04
m1tcdn − m1us
t
16
0.00039
0.00055
1.404
-2.356
0.02
-0.819
0.42
0.118
2.582
0.01
Table 25: Out-of-sample results for first-differences specification are reported for the out-of-sample period 1991Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log energy and
commodity price indices, log differences in real output levels, money stocks and differences in inflation rates and shortterm real interest rates. The MSE ratio estimates that are statistically significant according to Clark and West (2006) tstatistics at 5% are highlighted as well as Change or Direction estimates that are significant at 5%.
Δε tcdn / us = α + βΔft + ε t
ft = ⎡⎣ ytcdn − ytus rtcdn − rtus ptcom pteng π tcdn − π tus
1
2
4
12
Forecast horizon
0.00042 0.00043 0.00039 0.00033
RW
0.00051 0.00060 0.00050 0.00045
The Model
1.218
1.407
1.300
1.378
MSE Ratio
-0.828
-1.573
0.281
-1.118
t-stats
0.42
0.13
0.78
0.27
p-value
0.501
-0.350
1.610
0.280
CW t-stats
0.62
0.73
0.12
0.78
p-value
0.074
0
0.037
0.074
Change of direction
1.442
0
1
1.442
t-stats
0.16
1
0.33
0.16
p-value
⎤⎦
20
0.00043
0.00068
1.577
-2.464
0.02
-1.300
0.21
0.043
1
0.33
m1tcdn − m1us
t
16
0.00037
0.00051
1.385
-1.141
0.26
-0.002
0.99
0.037
1
0.33
Table 26: Out-of-sample results for first-differences specification are reported for the out-of-sample period 1991Q12002Q4. Quarterly differences in log nominal CDN/USD exchange rate series are estimated using the log energy and
commodity price indices, log differences in real output levels, money stocks and differences in inflation rates and shortterm real interest rates. The MSE ratio estimates that are statistically significant according to Clark and West (2006) tstatistics at 5% are highlighted as well as Change or Direction estimates that are significant at 5%.
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