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Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
1
Using monthly data on the spot exchange rate
and the short term interest rate for 10 countries,
test whether uncovered interest parity holds.
Word Count: 1915 (including tables, graphs and equations)
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
2
[1] Introduction
The foreign exchange market has a huge trading volume compared to other asset classes and
its trading activity is recession proof. The Triennial FX survey by Bank of International
Settlements (BIS, 2010) shows that between April 2007 and April 2010, trading volume of
global foreign exchange market increased by 20 percent to $4.0 trillion. A large number of
researchers have tried to find the determinants of the Exchange rate changes. We also intuitively
believe that, there must be some relationship between Spot Exchange rate, Future Exchange rate
and Interest rate movements. Uncovered Interest Parity (UIP) is one of the most notable
assumptions. In essence, UIP describes that due to arbitrage opportunities are exploited
instantaneously in international finance market (exchange market) and bonds market by market
participants such that the appreciation/depreciation of Country A’s currency against Country B is
the expected result from the decrease/increase of Country A’s interest rate relative to Country B.
Ross, et al., (2008), revisited this famous assumption by linking the CIP (Covered Interest
Parity) and UFR (Unbiased Forward Rate) conditions. By assuming that there is no arbitrage
opportunity in a sense that market participant is locked in the profit and loss once he/she enters
the Forward exchange contract, then CIP states:
𝐹𝑡+1
𝑆𝑡=
1 + 𝑖𝑡∗
1 + 𝑖𝑡 [Covered Interest Parity]
Where F denotes the Forward exchange rate at time t+1, S denotes Spot exchange rate at time t.
𝑖𝑡∗ is the interest rate in foreign country and 𝑖𝑡 is the interest rate in home/base country at time t.
In addition to the CIP condition, if agents are risk neutral, then investors are happy to take either
the Forward contract with certain payoff or the Spot exchange rate in the future with uncertain
payoff today. In this case, UFR (Unbiased Forward Rate) should hold:
𝐹𝑡+1 = 𝐸 𝑆𝑡+1 [Unbiased Forward Rate]
It suggests that Forward exchange rate is equal to expected future Spot rate. Therefore
combining CIP and UFR, we can get the UIP:
𝐸(𝑆𝑡+1)
𝑆𝑡=
1 + 𝑖𝑡∗
1 + 𝑖𝑡 [Uncovered Interest Parity]
Moreover, if market participants have rational expectation, we would have:𝑆𝑡+1 = 𝐸(𝑆𝑡+1) + ∈ ,
where ∈ is white noise. We would have 𝑆𝑡+1
𝑆𝑡=
1+𝑖𝑡∗
1+𝑖𝑡 , if we take natural logs from both sides, we
would get Equation (1) (UIP Condition), as log(1 + 𝑖) ≈ 𝑖, which is tested in this paper.
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
3
[2] Empirical Framework and Analysis
[2.a. Data Description]
We estimate the UIP assumption by setting the US as the home/base country, i.e. 𝑖𝑡 denotes
the one-month interest rate (as a percentage rate per year) in the US at time t, whereas, 𝑖𝑡 denotes
the one-month interest rate in the foreign country.
In this paper, we estimate exchange rates 𝑆𝑡 from 9 different countries/regions against the
USD: Australia(AUD), Canada(CAD), Switzerland(CHF), Europe(EUR), UK(GBP), Japan(JPY),
Norway(NOK), New Zealand(NZD) and Sweden(SEK). Graph 1 and 2 below plots the monthly
Spot rates changes from the 9 countries across the whole time horizon from Jan 1976 to June
2010 and the one-month interest rates movements in 10 countries (including the US) during the
same period.
Graph 1 Graph 2
[2.b. Time Series Regression Equation]
𝑆𝑡+1 − 𝑆𝑡 = 𝑎 + 𝑏 𝑖𝑡 − 𝑖𝑡∗ + 𝜖𝑡+1 Equation (1)
The main regression equation in this paper is displayed above, in which S from Equation (1) is
expressed in natural log terms. If UIP holds, we should be able to find intercept a to be 0 and
coefficient b to be 1 statistically significant and a high R-squared.
0.0
0.5
1.0
1.5
2.0
2.5
1980 1985 1990 1995 2000 2005 2010
S_AUD S_CAD S_CHF
S_EUR S_GBP S_JPY
S_NOK S_NZD S_SEK
0
5
10
15
20
25
30
1980 1985 1990 1995 2000 2005 2010
I_AUD I_CAD I_CHFI_EUR I_GBP I_JPY
I_NOK I_NZD I_SEK
I_USD
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
4
Stationary Test
To avoid spurious regression results, we first test for stationary (See Appendix [1.1]) of the
data in 9 countries. Augmented Dickey-Fuller test shows that, at 1% significant level, all
dependent variables (𝑆𝑡+1 − 𝑆𝑡 ) pass the stationary test, i.e. I(0). However, for explanatory
variables 𝑖𝑡 − 𝑖𝑡∗ , we find 7 out of 9 countries are I(0), but series CHF and EUR cannot pass the
test at 5% significant level (with P-value 0.4313 and 0.2859 respectively). ADF shows that these
two series CHF and EUR are I(1) (significant at 1% level), i.e. it is difference-stationary process
(See Appendix [1.2]).
Hence, we use Equation (3) to examine the validity of UIP for countries CHF and EUR.
Equation (3) is derived from differencing Equation (2) from (1). Using this transformation, we
can run regression on stationary data and are able to achieve the same goal (test for the validity
of UIP). If UIP holds, we would find coefficient b to be 1 and ∝ be 0 in Equation (4) below.
𝑆𝑡 − 𝑆𝑡−1 = 𝑎 + 𝑏 𝑖𝑡−1 − 𝑖𝑡−1∗ + 𝜖𝑡 Equation (2)
𝑆𝑡+1 − 𝑆𝑡 − 𝑆𝑡 − 𝑆𝑡−1 = 0 + 𝑏 𝑖𝑡 − 𝑖𝑡∗ − 𝑖𝑡−1 − 𝑖𝑡−1
∗ + 𝜖𝑡+1 − 𝜖𝑡 Equation (3)
∆ 𝑆𝑡+1 − 𝑆𝑡 =∝ +𝑏 ∆ 𝑖𝑡 − 𝑖𝑡∗ + 𝑢𝑡+1, where 𝑢𝑡+1 = 𝜖𝑡+1 − 𝜖𝑡 Equation (4)
[2.c. Time Series Regression Results]
Seven countries/regions are estimated by OLS using Equation (1). The remaining two (EUR
and CHF) are estimated using Equation (4).
Heteroscedasticity Test – White
Heteroscedastic is found significant in CHF, GBP and SEK at 1% significance level, which we
use White Robust process to correct the variance for Heteroscedasticity (See Appendix [2.1]).
For all other countries, the standard errors of the regressions residulas are found to be
Homosdecatistic at 5% sig. level.
OLS regression results of the 9 countries (with robust standard errors) can be found in
Appendix [2.2].
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
5
Serial Correlation Test
Breusch-Godfrey Serial Correlation LM Test is used to test for serial correlation in the
residuals of the regressions (with 10 lags). We notice that, at 5% significance level, they all pass
the Serial Correlation LM test except for CHF and EUR, which we know from Equation (4) that
serial correlation in residuals of these two regressions 𝑢𝑡+1 = 𝜖𝑡+1 − 𝜖𝑡 is expected. The
correlogram of residuls and squared residuals exhibit the same results (see Appendix [2.3] and
[2.3.1]).
Summary
In addition, Wald Test for joint significance of H0: a = 0 and b = 1 are estimated. At 5%
significance level, H0 is rejected in all countries. Table1 below summarize the results from the
OLS estimation.
Table 1:
OLS using Equation (1) [UIP Hypothesis]
Countries a p-value b p-value R-squared Wald joint test
Η0:a =0, b=1
AUD -0.00171 0.3598 -0.00039 0.4003 0.001722 Rejected
CAD -0.00046 0.6579 -0.00049 0.3796 0.001879 Rejected
GBP -0.00335 0.0659 -0.00125 0.0730 0.011457 Rejected
JPY 0.010405** 0.0000 -0.00251** 0.0001 0.039237 Rejected
NOK -0.0018 0.3025 -0.00066 0.1347 0.005437 Rejected
NZD -0.00422 0.0511 -0.00093** 0.0186 0.013399 Rejected
SEK -0.00171 0.3111 -0.00019 0.77 0.000385 Rejected
α OLS using Equation (4) Η0: α =0, b=1
CHF 3.18E-05 0.9895 -0.00782 0.0712 0.012307 Rejected
EUR -6.89E-05 0.9749 -0.00928** 0.0007 0.027541 Rejected
** Rejects the null a equals 0, b equals 0, respectively, at 1% significance level
Intercepts a are insignificant at 5% sig. level with a negative absolute value except for JPY. Most
coefficients b are also insignificant at 5% sig. level and have small negative values except for
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
6
JPY, NZD and EUR which are significant at 1% sig. level and have values of -0.00251, -0.00093
and -0.00928 respectively. For CHF and EUR, intercept α is very close to 0 and insignificant
which can be explained from Equation (3) that the intercept is cancelled out when we transform
Equation (4) by differencing Equation (1) from its lagged series Equation (2).
When Wald Test for joint significance of H0: a = 0 and b = 1; H0: ∝ = 0 and b = 1 (UIP
Hypothesis) are estimated, UIP Hypothesis are all rejected in 9 countries at 1% sig. level. The R-
squared values are very small as well. This empirical finding could be possibly due to the
“exchange rate disconnect puzzle” introduced by Obsteld and Rogoff (2000), who mention that
nominal exchange rate is disconnected from real economic variables.
[2.d. Panel Data Estimation]
It is intuitive to estimate the UIP in a panel data set that has both a cross-sectional (9 countries)
and a time series (monthly) dimension.
𝑆𝑡+1𝑗
− 𝑆𝑡𝑗
= 𝑎 + 𝑏 𝑖𝑡𝑗− 𝑖𝑡
𝑗 ∗ + 𝜖𝑡+1
𝑗 Equation (5)
Accordingly, similarly to Equation (4) we can write:
∆ 𝑆𝑡+1𝑗
− 𝑆𝑡𝑗 =∝ +𝑏 ∆ 𝑖𝑡
𝑗− 𝑖𝑡
𝑗 ∗ + 𝑢𝑡+1
𝑗, where 𝑢𝑡+1
𝑗= 𝜖𝑡+1
𝑗− 𝜖𝑡
𝑗 Equation (6)
j denotes the 9 individual country we estimated. We obtain a balanced panel data set from
Equation (6) accordingly. Hence, UIP would hold, if ∝ = 0 and b = 1 are statistically significant.
Stationary Test
Levin-Lin-Chu unit-root test and Im-Pesaran-Shin unit-root test are used to test for stationary
of the panel data set. As expected, the cross sectional time series we measure according to
Equation (6) are stationary which we cannot reject at 1% significant level (see Appendix [3.1]).
Results are summarized in Table 2 below.
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
7
Table 2
Levin-Lin-Chu P-value Im-Pesaran-Shin P-value
∆ 𝑆𝑡+1𝑗
− 𝑆𝑡𝑗 -78.4911** 0.000 -57.4518** 0.000
∆ 𝑖𝑡𝑗− 𝑖𝑡
𝑗 ∗ -45.0690** 0.000 -45.3678** 0.000
** Rejects the null that the series is non-stationary, at 1% significance level
Pooled OLS
We proceed to estimate Pooled OLS after non-stationary of the panel data set is strongly
rejected. We use robust standard errors clustered by individual (see Appendix [3.2]) when we
regress the Pooled OLS, outputs can be summarized below in Table 3.
Table 3 - Pooled OLS - robust standard errors
∝ p-value b p-value R-squared [UIP Hypothesis]
0.0000355 0.074 -0.0029506* 0.022 0.0048 Rejected
* Rejects the null a equals 0, b equals 0, respectively, at 5% significance level
Pooled OLS estimate does not indicate the validity of UIP. b is significantly different from 0,
but has a negative value, again fails the UIP assumption. Our finding is consistent with findings
from other literatures, for example, Maynard and Philips (2001), pp.673, argues that “UIP
condition must be reversed in sign.”
Fixed Effect and Random Effect models
Furthermore, Fixed Effect model (allows individual effect) and Random Effect model (does
not allow individual effect) are estimated. We run the pooled OLS regression first and test for
AR(1) serial correlation in residual as expected from Equation (6). By using Wooldridge test for
autocorrelation in panel data (in Stata, written by Drukker(2003) according to
Wooldridge(2002)), we strongly reject no AR(1) serial correlation at 5% significant level (See
Appendix [3.1]). Therefore, we correct for the presence of AR(1) serial correlation by running
the regression taking into account the AR(1) disturbances in the two models we estimate (See
Appendix [3.3]), which are summarized in below:
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
8
Table 4
Fixed Effect Estimation [UIP Hypothesis]
α p-value b p-value α =0, b=1
0.0001006 0.915 -0.0031456** 0.000 Rejected
Random Effect Estimation
α p-value b p-value α =0, b=1
0.0001881 0.920 -0.0031054** 0.000 Rejected
** Rejects the null that the estimate is 0 at 1% level
Empirical findings from Fixed Effect and Random Effect estimations suggest statistical
insignificant intercepts a and significant slope coefficients b (very small negative value) at 1%
sig. level. UIP Hypothesis is again rejected in both models. We notice that both models show
similar results and intuitively we use Hausman test (see Appendix [3.5]) to distinguish which
model is superior in our panel data estimation. According to Hausman (1978) test, we cannot
reject (P-value: 0.6219) the null that unobserved effect (individual effect) is uncorrelated with
every explanatory variable in both cross-sectional (9 countries) and time series (monthly)
dimensions (at 5% sig level). So Hausman test recommends Random Effect model. However,
Johnston and DiNardo (2007) mention that it is common in applied research that two estimators
are not significantly different from each other. One possible reason is that the variance in the
change in explanatory variables is not large enough to distinguish the two estimators.
[3] Conclusion
Empirical findings in our paper fail to accept the validity of the Uncovered Interest
Parity (UIP) in individual OLS on time series data including 9 countries (Australia,
Canada, Switzerland, Europe, Great Britain, Japan, Norway, New Zealand and
Sweden) and pooled OLS, Random Effect model and Fixed Effect model on panel
data. We conclude that all models we use cannot resolve the UIP puzzle. All results
show that the coefficient b has a negative value across different significance level
which is consistent with findings from voluminous empirical literatures.
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
9
Appendix (complete version of Eviews output is available on request)
(Eviews outputs and outputs summary)
[1] Stationary Test – ADF test Augmented Dickey-Fuller test statistic
(5% ADF test stat: -2.868; 1%: -3.446 𝑆𝑡+1 − 𝑆𝑡 p-value 𝑖𝑡 − 𝑖𝑡
∗ p-value
Null Hypothesis: unit root exists
Exogenous: Constant; Lag Length: 0
(Automatic - based on SIC, maxlag=17)
AUD -19.12886** 0.000 -3.010237* 0.0348
CAD -20.94159** 0.000 -3.426791* 0.0106
CHF -19.93280** 0.000 -1.698397 0.4313
EUR -19.76744** 0.000 -2.002378 0.2859
GBP -18.43164** 0.000 -4.187651* 0.0008
JPY -19.46875** 0.000 -3.585440* 0.0065
NOK -18.93116** 0.000 -3.761899* 0.0036
NZD -9.399106** 0.000 -3.003397* 0.0354
SEK -18.23383** 0.000 -3.651351* 0.0052
**Rejects the null that series is non-stationary, at 1% sig. level; *at 5% sig level
[1.1 ADF test – I (1)] CHF I(1)
Null Hypothesis: DDI_CHF has a unit root
Exogenous: Constant
Lag Length: 4 (Automatic - based on SIC, maxlag=17) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -13.01161 0.0000
Test critical values: 1% level -3.446162
5% level -2.868405
10% level -2.570492
EUR I(1)
Null Hypothesis: DDI_EUR has a unit root
Exogenous: Constant
Lag Length: 8 (Automatic - based on SIC, maxlag=17) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -6.832881 0.0000
Test critical values: 1% level -3.446321
5% level -2.868475
10% level -2.570530
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
10
[2]Regression and residuals tests [2.1] Heteroskedasticity Test - White
Countries F-statistic Prob. F(2,410)
AUD 2.413461 0.0908
CAD 0.361601 0.6968
CHF 5.743049** 0.0035
EUR 2.004996 0.1360
GBP 5.607478** 0.0040
JPY 1.652777 0.1928
NOK 1.325663 0.2668
NZD 2.479638 0.0850
SEK 9.614281** 0.0001
**Rejects the null that residuals exhibit Homosdecasity, at 1% sig. level e.g. CHF
Heteroskedasticity Test: White F-statistic 5.743049 Prob. F(2,409) 0.0035
Obs*R-squared 11.25429 Prob. Chi-Square(2) 0.0036
Scaled explained SS 23.13748 Prob. Chi-Square(2) 0.0000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 05/17/11 Time: 04:34
Sample: 1976M03 2010M06
Included observations: 412 Variable Coefficient Std. Error t-Statistic Prob. C 0.002248 0.000249 9.043456 0.0000
DDI_CHF -0.000491 0.000311 -1.580557 0.1148
DDI_CHF^2 0.000236 0.000102 2.316651 0.0210 R-squared 0.027316 Mean dependent var 0.002409
Adjusted R-squared 0.022560 S.D. dependent var 0.004914
S.E. of regression 0.004858 Akaike info criterion -7.808966
Sum squared resid 0.009654 Schwarz criterion -7.779687
Log likelihood 1611.647 Hannan-Quinn criter. -7.797385
F-statistic 5.743049 Durbin-Watson stat 1.091168
Prob(F-statistic) 0.003469
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
11
[2.2] OLS Regression with robust Standard Error Dependent Variable: S
Method: Least Squares; Date: 05/15/11 Time: 22:54
Sample (adjusted): 1976M02 2010M06
Included observations: 413 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -0.001708 0.001863 -0.916781 0.3598
DI_AUD -0.000386 0.000459 -0.841928 0.4003
C -0.000456 0.001029 -0.443195 0.6579
DI_CAD -0.000494 0.000562 -0.879512 0.3796
C 3.18E-05 0.002418 0.013151 0.9895
DDI_CHF -0.007815 0.004320 -1.809115 0.0712
White heteroskedasticity-consistent standard errors & covariance;
Included observations: 412 after adjustments
C -6.89E-05 0.002190 -0.031490 0.9749
DDI_EUR -0.009280 0.002723 -3.407567 0.0007
Included observations: 412 after adjustments
C -0.003346 0.001815 -1.843925 0.0659
DI_GBP -0.001245 0.000693 -1.797630 0.0730
White heteroskedasticity-consistent standard errors & covariance
C 0.010405 0.002436 4.271105 0.0000
DI_JPY -0.002509 0.000612 -4.096947 0.0001
C -0.001797 0.001741 -1.032304 0.3025
DI_NOK -0.000658 0.000439 -1.498905 0.1347
C -0.004219 0.002157 -1.956014 0.0511
DI_NZD -0.000927 0.000392 -2.362622 0.0186
C -0.001712 0.001689 -1.014115 0.3111
DI_SEK -0.000188 0.000643 -0.292572 0.7700
White heteroskedasticity-consistent standard errors & covariance
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
12
e.g. CHF:
Dependent Variable: DDLOGS_CHF
Method: Least Squares
Date: 05/15/11 Time: 23:36
Sample (adjusted): 1976M03 2010M06
Included observations: 412 after adjustments
White heteroskedasticity-consistent standard errors & covariance Variable Coefficient Std. Error t-Statistic Prob. C 3.18E-05 0.002418 0.013151 0.9895
DDI_CHF -0.007815 0.004320 -1.809115 0.0712 R-squared 0.016554 Mean dependent var 9.76E-05
Adjusted R-squared 0.014156 S.D. dependent var 0.049551
S.E. of regression 0.049199 Akaike info criterion -3.181061
Sum squared resid 0.992405 Schwarz criterion -3.161542
Log likelihood 657.2986 Hannan-Quinn criter. -3.173340
F-statistic 6.901540 Durbin-Watson stat 3.042441
Prob(F-statistic) 0.008935
[2.3] Breusch-Godfrey Serial Correlation LM Test: Countries F-statistic Prob. F(10,401)
AUD 0.424557 0.9346
CAD 1.683549 0.0824
CHF 37.67349** 0.0000 EUR 31.20954** 0.0000
GBP 0.915795 0.5184 JPY 1.464996 0.1501
NOK 1.068878 0.3851 NZD 1.795441 0.0595
SEK 1.545964 0.1208
**Rejects the null that residual has no Serial Correlation, at 1% sig. level e.g. CHF
Breusch-Godfrey Serial Correlation LM Test: F-statistic 37.67349 Prob. F(10,400) 0.0000
Obs*R-squared 199.8298 Prob. Chi-Square(10) 0.0000
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Date: 05/17/11 Time: 04:36
Sample: 1976M03 2010M06
Included observations: 412
Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C -0.000899 0.001763 -0.510207 0.6102
DDI_CHF 0.000583 0.002189 0.266377 0.7901
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
13
RESID(-1) -0.904699 0.049616 -18.23409 0.0000
RESID(-2) -0.787641 0.066626 -11.82176 0.0000
RESID(-3) -0.665453 0.075609 -8.801223 0.0000
RESID(-4) -0.668714 0.080348 -8.322710 0.0000
RESID(-5) -0.544315 0.082494 -6.598249 0.0000
RESID(-6) -0.554731 0.082742 -6.704386 0.0000
RESID(-7) -0.391971 0.080494 -4.869554 0.0000
RESID(-8) -0.344374 0.076005 -4.530970 0.0000
RESID(-9) -0.173158 0.067479 -2.566114 0.0106
RESID(-10) -0.149579 0.050057 -2.988164 0.0030 R-squared 0.485024 Mean dependent var -2.02E-19
Adjusted R-squared 0.470862 S.D. dependent var 0.049139
S.E. of regression 0.035744 Akaike info criterion -3.796152
Sum squared resid 0.511065 Schwarz criterion -3.679035
Log likelihood 794.0073 Hannan-Quinn criter. -3.749826
F-statistic 34.24863 Durbin-Watson stat 1.996650
Prob(F-statistic) 0.000000
[2.3.1] Correlogram of residuals and squared rediduals
CHF
Sample: 1976M03 2010M06
Included observations: 412
Autocorrelation Partial Correlation AC PAC Q-Stat Prob ****|. | ****|. | 1 -0.529 -0.529 116.12 0.000
.|. | **|. | 2 0.049 -0.320 117.14 0.000 .|. | *|. | 3 0.024 -0.171 117.38 0.000 *|. | **|. | 4 -0.090 -0.223 120.73 0.000 .|* | *|. | 5 0.106 -0.089 125.44 0.000 *|. | **|. | 6 -0.137 -0.211 133.29 0.000 .|* | *|. | 7 0.127 -0.089 140.09 0.000 *|. | *|. | 8 -0.111 -0.187 145.31 0.000 .|* | .|. | 9 0.125 -0.029 151.88 0.000 *|. | *|. | 10 -0.111 -0.138 157.10 0.000
Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|*** | .|*** | 1 0.449 0.449 83.575 0.000 .|. | **|. | 2 0.036 -0.207 84.123 0.000 .|. | .|. | 3 -0.026 0.061 84.410 0.000 .|. | .|. | 4 -0.020 -0.032 84.580 0.000 .|. | .|. | 5 0.027 0.057 84.893 0.000 .|. | .|. | 6 0.045 0.005 85.751 0.000 .|. | .|. | 7 0.029 0.009 86.104 0.000 .|. | .|. | 8 0.035 0.032 86.632 0.000 .|* | .|* | 9 0.113 0.112 92.021 0.000 .|* | .|. | 10 0.106 0.004 96.829 0.000
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
14
EUR
Date: 05/16/11 Time: 05:21 Sample: 1976M03 2010M06 Included observations: 412
Autocorrelation Partial Correlation AC PAC Q-Stat Prob ****|. | ****|. | 1 -0.522 -0.522 112.98 0.000
.|. | **|. | 2 0.050 -0.305 114.04 0.000 .|. | *|. | 3 0.019 -0.160 114.19 0.000 *|. | **|. | 4 -0.084 -0.206 117.13 0.000 .|* | *|. | 5 0.084 -0.101 120.10 0.000 *|. | *|. | 6 -0.097 -0.175 124.03 0.000 .|. | *|. | 7 0.065 -0.121 125.79 0.000 .|. | *|. | 8 -0.032 -0.134 126.23 0.000 .|. | .|. | 9 0.046 -0.053 127.14 0.000 .|. | *|. | 10 -0.052 -0.107 128.30 0.000
Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|*** | .|*** | 1 0.473 0.473 92.818 0.000 .|* | *|. | 2 0.163 -0.078 103.85 0.000 .|. | .|. | 3 0.028 -0.024 104.17 0.000 .|. | .|. | 4 -0.024 -0.021 104.41 0.000 .|. | .|. | 5 -0.004 0.028 104.42 0.000 .|. | .|. | 6 0.010 0.004 104.46 0.000 .|. | .|. | 7 -0.000 -0.014 104.46 0.000 .|. | .|. | 8 -0.012 -0.010 104.52 0.000 .|. | .|. | 9 0.021 0.044 104.71 0.000 .|. | .|. | 10 0.032 0.007 105.13 0.000
[2.4] Wald Test C(1)=0, C(2)=1 F- Test Statistic df Probability
AUD 3249498** (2, 411) 0.0000
CAD 1905624** (2, 411) 0.0000
CHF 27879.24** (2, 410) 0.0000 EUR 68669.04** (2, 410) 0.0000
GBP 1621953** (2, 411) 0.0000 JPY 3005953** (2, 411) 0.0000
NOK 3715154** (2, 411) 0.0000
NZD 5576160** (2, 411) 0.0000 SEK 1493832** (2, 411) 0.0000
**Rejects the null that a=0, b=1 at 1% sig. level
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
15
e.g. CHF
Wald Test:
Equation: EQ_CHF Test Statistic Value df Probability F-statistic NA (2, 410) NA
Chi-square NA 2 NA Restriction variance cannot be computed. Restrictions may
not be unique.
Null Hypothesis: C(1)=0, C(1)=1
Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. C(1) 3.18E-05 0.002418
-1 + C(1) -0.999968 0.002418
Restrictions are linear in coefficients.
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
16
[3 Panel Data]
(Stata output)
[3.1] Stationary Unit Root Test
Adjusted t* -78.4911 0.0000 Unadjusted t -74.3863 Statistic p-value LR variance: Bartlett kernel, 23.00 lags average (chosen by LLC)ADF regressions: 1 lag
Time trend: Not includedPanel means: IncludedAR parameter: Common Asymptotics: N/T -> 0
Ha: Panels are stationary Number of periods = 412Ho: Panels contain unit roots Number of panels = 9 Levin-Lin-Chu unit-root test for FX
Adjusted t* -45.0690 0.0000 Unadjusted t -44.3663 Statistic p-value LR variance: Bartlett kernel, 23.00 lags average (chosen by LLC)ADF regressions: 1 lag
Time trend: Not includedPanel means: IncludedAR parameter: Common Asymptotics: N/T -> 0
Ha: Panels are stationary Number of periods = 412Ho: Panels contain unit roots Number of panels = 9 Levin-Lin-Chu unit-root test for IntDif
Z-t-tilde-bar -57.4518 0.0000 t-tilde-bar -17.5366 t-bar -35.0699 -2.150 -1.970 -1.880 Statistic p-value 1% 5% 10% Fixed-N exact critical values ADF regressions: No lags included
Time trend: Not includedPanel means: Included sequentiallyAR parameter: Panel-specific Asymptotics: T,N -> Infinity
Ha: Some panels are stationary Number of periods = 412Ho: All panels contain unit roots Number of panels = 9 Im-Pesaran-Shin unit-root test for FX
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
17
[3.2] Pooled OLS with robust standard errors clustered by individual
[3.3] Autocorrelation test for Panel Data
Z-t-tilde-bar -45.3678 0.0000 t-tilde-bar -14.1680 t-bar -19.9820 -2.150 -1.970 -1.880 Statistic p-value 1% 5% 10% Fixed-N exact critical values ADF regressions: No lags included
Time trend: Not includedPanel means: Included sequentiallyAR parameter: Panel-specific Asymptotics: T,N -> Infinity
Ha: Some panels are stationary Number of periods = 412Ho: All panels contain unit roots Number of panels = 9 Im-Pesaran-Shin unit-root test for IntDif
_cons .0000355 .0000173 2.05 0.074 -4.40e-06 .0000753 IntDif -.0029506 .0010451 -2.82 0.022 -.0053607 -.0005405 FX Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust (Std. Err. adjusted for 9 clusters in Countries)
Root MSE = .043 R-squared = 0.0048 Prob > F = 0.0224 F( 1, 8) = 7.97Linear regression Number of obs = 3708
Prob > F = 0.0000 F( 1, 8) = 437.025H0: no first-order autocorrelationWooldridge test for autocorrelation in panel data
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
18
[3.4] Fixed Effect and Random Effect with corrected AR(1) Residuals
F test that all u_i=0: F(8,3689) = 0.00 Prob > F = 1.0000 rho_fov 2.462e-06 (fraction of variance because of u_i) sigma_e .05703201 sigma_u .00008949 rho_ar -.50135856 _cons .0001006 .0009377 0.11 0.915 -.0017379 .0019391 IntDif -.0031456 .0008365 -3.76 0.000 -.0047857 -.0015056 FX Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = -0.0028 Prob > F = 0.0002 F(1,3689) = 14.14
overall = 0.0049 max = 411 between = 0.0218 avg = 411.0R-sq: within = 0.0038 Obs per group: min = 411
Group variable: Countries Number of groups = 9FE (within) regression with AR(1) disturbances Number of obs = 3699
theta 0 rho_fov 0 (fraction of variance due to u_i) sigma_e .056964 sigma_u 0 rho_ar -.50135856 (estimated autocorrelation coefficient) _cons .0001881 .0018697 0.10 0.920 -.0034765 .0038527 IntDif -.0031054 .0008324 -3.73 0.000 -.004737 -.0014739 FX Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, Xb) = 0 (assumed) Prob > chi2 = 0.0010 Wald chi2(2) = 13.92
overall = 0.0048 max = 412 between = 0.0126 avg = 412.0R-sq: within = 0.0048 Obs per group: min = 412
Group variable: Countries Number of groups = 9RE GLS regression with AR(1) disturbances Number of obs = 3708
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
19
[3.5] Hausman Test
(Without robust standard error but corrected for AR (1) in residuals)
Prob>chi2 = 0.6219 = 0.24 chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
Test: Ho: difference in coefficients not systematic
B = inconsistent under Ha, efficient under Ho; obtained from xtregar b = consistent under Ho and Ha; obtained from xtregar IntDif -.0031449 -.0031042 -.0000407 .0000824 FE1 RE1 Difference S.E. (b) (B) (b-B) sqrt(diag(V_b-V_B)) Coefficients
Diploma in Economics: Econometrics Project, May 2011 Candidate Number: 3290A
(Clive) Xuli Xiao
20
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