1. Open Economy Macroeconomics HW 2
Juan Du & Yeqi Zhang & Danlin Li
July 24th, 2016
Contents
Part I: Variables and Data..................................................................................................................................................1
1. Variables and Ordering in Baseline Case..............................................................................................................1
2. Data Source............................................................................................................................................................1
Part II: Unit Root Test and Cointegration Test..................................................................................................................2
1. ADF Unit Root Test................................................................................................................................................2
2. Johansen Cointegration Test.................................................................................................................................2
Part III: IRF and VDC for Recursive VAR form in Level ....................................................................................................2
1. Impulse Response Function..................................................................................................................................2
2. Variance Decomposition.......................................................................................................................................4
Part IV: IRF and VDC for Recursive VAR form in First-Difference ...................................................................................4
1. Impulse Response Function..................................................................................................................................4
2. Variance Decomposition.......................................................................................................................................5
Part V: Robustness to New Ordering: {𝐆𝐲, 𝐦𝐦, 𝐡𝟏, 𝐲, 𝐫𝟏} ..............................................................................................6
Part V: Add New Variables into the Baseline Model.........................................................................................................7
Part VI: Conclusions...........................................................................................................................................................8
1. Baseline Case.........................................................................................................................................................8
2. First-Difference Case.............................................................................................................................................8
3. New Ordering Case................................................................................................................................................8
4. Adding Interest Rate into the Model ....................................................................................................................8
Appendix............................................................................................................................................................................9

2. 1
Part I: Variables and Data
1. Variables and Ordering in Baseline Case
A short-run SVAR model without exogenous variables can be written as: A(𝐼 𝐾 − 𝐴1 𝐿 − 𝐴2 𝐿2
−
⋯ − 𝐴 𝑝 𝐿 𝑝
)𝑋𝑡 = 𝐴𝜀𝑡 = 𝐵𝑒𝑡, where L is the lag operator, A, B, and 𝐴1, … , 𝐴 𝑝 are K × K matrices
of parameters, 𝜀𝑡 is a K × 1 vector of innovations with 𝜀𝑡∼ N(0; Σ) and E[𝜀𝑡 𝜀𝑡
′
] = 0 𝐾 for all s ≠
t, and 𝑒𝑡 is a K × 1 vector of orthogonalized disturbances; that is, 𝑒𝑡 ∼ N(0; 𝐼 𝐾) and E[𝑒𝑡 𝑒𝑡
′
] =
0 𝐾 for all s ≠ t.
These transformations of the innovations allow us to analyze the dynamics of the system in
terms of a change to an element of 𝑒𝑡. In a short-run SVAR model, we obtain identification by
placing restrictions on A and B. So:
𝑋 =
[
𝐺/𝑦
𝑦
𝑚𝑚
ℎ1
𝑟1 ]
𝜀 =
[
𝜀 𝑔
𝜀 𝑦
𝜀 𝑚𝑚
𝜀ℎ
𝜀 𝑟 ]
𝐴 =
[
𝐶11 0 0 0 0
𝐶21 𝐶22 0 0 0
𝐶31 𝐶32 𝐶33 0 0
𝐶41 𝐶42 𝐶43 𝐶44 0
𝐶51 𝐶52 𝐶53 𝐶54 𝐶55]
𝐵 =
[
𝐵11 0 0 0 0
0 𝐵22 0 0 0
0 0 𝐵33 0 0
0 0 0 𝐵44 0
0 0 0 0 𝐵55]
G/y is the government consumption as a share of GDP, US less Italy; y is log of GDP, US less
Italy; r1 is log of real exchange rate= s − p, where s is the log of the nominal exchange rate in
Lira per dollar, p is the log of CPI, US relative to Italy; mm is the log of money multiplier=
ln(
M
H
), US less Italy, where M is the nominal M2 money stock and H is the nominal monetary
base; h is the log of real monetary base, ln(H/P), US less Italy. We run the VAR in levels in the
baseline case, and in first differences in later case.
We learn the variable selection from Rogers (1999), but we are focused on short-run
restrictions instead of long-run ones. The share of government spending in total output is
mainly from fiscal policy, which can be thought as most exogenous, implying the zero-
restrictions in the top row. Financial market is the most endogenous and reacts quickly to
shocks so real exchange rate is in the last place, and the fifth element of row 1,2,3,4 are all zero.
GDP is determined by variables in real economy but not monetary or financial variables,
implying the zero in the second row. The assumption that the money multiplier is unaffected
by the monetary base justifies the restriction 𝐶34 = 0.
However, this ordering may have some “dubious” assumptions: MM is unaffected by MB is an
assumption for long run, and it may not hold in short run; short-run monetary policy may have
contemporaneous effect on GDP especially when we use revised GDP data which may contain
some news. We will test the robustness later by changing the ordering.
2. Data Source
Data for monetary base of Italy is from Bank of Italy. Data for all the other variables are from
Fed of St. Louis.

3. 2
Part II: Unit Root Test and Cointegration Test
1. ADF Unit Root Test
Table 1
Tests for unit roots
Series/Tests (G/Y) y r1 r2 mm h1 h2
ADF(τ)
AIC -3.438* -0.747 -3.014** -2.950** -2.228 -2.366 -2.628
(-0.137,
-0.037)
(-0.047,
0.021)
(-0.120,
-0.025)
(-0.113,
-0.022)
(-0.086,
-0.005)
(-0.068,
-0.006)
(-0.079,
-011)
ADF denotes the Augmented Dickey–Fuller test statistic for the unit root null hypothesis versus the trend-stationary
alternative. In parentheses are 95% confidence intervals for the largest autoregressive root in the series, calculated
using the procedure of Stock (1991). A * and ** indicates rejection of the null at 10% and 5%, respectively. Lag length
was based on the Akaike Criterion.
For r1 and r2, we reject the unit root null at 5% level; for G/Y, we reject the null at 10% level, thus
we think they do not have unit root. For the rest variables, we cannot reject the null even at a 10%
significance level, so they all have a unit root.
2. Johansen Cointegration Test
Table 2
Tests of co-integrating rank
𝐻0:CI 𝜆-max. stat. 95% critical
value
Trace stat. 95% critical
valueRank=p 1 lag 5 lags 1 lag 5 lags
p=0 43.502* 55.620* 30.04 87.210* 97.881* 59.46
p≤1 29.283* 24.150* 23.80 43.707* 42.261* 39.89
p≤2 10.171 9.960 17.89 14.424 18.111 24.31
p≤3 3.986 7.716 11.44 4.253 8.152 12.53
P≤4 0.266 0.436 3.84 0.266 0.436 3.84
The system variables are: (G/Y), y, r1, mm, and h. The sample period is 1960Q1–1998Q4. Lag length in the VAR is
displayed in the top row. * indicates rejection of the null at 5% significance level.
According to both the maximum-eigenvalue test and the trace test, the null of zero co-integration
vectors and the null of at most one cointegration can be rejected for any of these lag lengths. The
results suggest estimating the VARs in level with cointegration relationships. Also, we fail to reject
the null hypothesis of at most two cointegrating equations. Thus, we conclude that there are two
cointegrating equations in the model.
Part III: IRF and VDC for Recursive VAR form in Level
1. Impulse Response Function
We choose lag length of 5, instructed by AIC. See the graphs below for IRFs of our variables:

4. 3
To see the effect of monetary shock on GDP more clearly, we draw these IRFs separately:

5. 4
We can see that a positive shock on mm has a significant potive effect on GDP from the 7th quarter
to the 19th quarter; a positive shock on real money base has a significant positive effect on GDP from
the 4st to 10th quarter. These effects begin from zero by construction.
2. Variance Decomposition
Table 3
Variance Decomposition of GDP
Horizon ℇ 𝑔
ℇ 𝑦
ℇ 𝑚𝑚
ℇℎ1 ℇ 𝑟1
The VDC of y
1 .533182 .466818 0 0 0
(.4244 .6419) (.3581 .5756)
2 .561511 .432299 .001669 .001671 .00285
(.4331 .6898) (.3050 .5600) (-.0053 .0087) (-.0111 .0151) (-.0088 .0145)
5 .589531 .299214 .003274 .030328 .077652
(.4228 .7562) (.1466 .4518) (-.0115 .0180) (-.0336 .0943) (-.0210 .0716)
20 .270278 .180477 .210818 .145568 .192859
(.0134 .5271) (-.0133 .3742) (-.0849 .5065) (-.1165 .4077) (-.0459 .4316)
24 .25205 .174548 .222136 .151374 .197713
(-.0185 .5226) (-.0270 .3761) (-.1025 .5468) (-.1376 .4447) (-.0621 .4575)
Unfortunately, although it seems that mm- and h1-shock can explain some portion of the GDP
variance, all of them are insignificant. Actually, our VDC results show the powers of mm and h1
shocks are insignificant at any horizon from quarter 1 to 24 (the first 6 years).
Part IV: IRF and VDC for Recursive VAR form in First-Difference
1. Impulse Response Function
We take the first differences of all the variables in the baseline case, and choose lag length of 4
instructed by AIC. See the graphs below for IRFs of our variables:

6. 5
We have known from Part II that there are cointegration relationships in our system so we should
run VAR in levels. If we run it in the first differences, there will be an over-differencing problem,
where we may lose the long-run relationships among the variables.
The IRFs of GDP illustrate this point: We can see that monetary multiplier has no significant effect
on GDP, while money base has a significantly positive effect from quarter 6 to 7. The effects of
monetary shock on GDP is covered because of the differencing.
Another difference between level and first difference is that the IRF confidence intervals for first-
differenced variables converge with time while those for leveled variables does not.
2. Variance Decomposition
Table 4
Variance Decomposition of first-differenced GDP
Horizon ℇ 𝑔
ℇ 𝑦
ℇ 𝑚𝑚
ℇℎ1 ℇ 𝑟1
The VDC of ∆y
1 .486279 .513721 0 0 0
(.3720 .6006) (.3994 .6280)
2 .486171 .509603 .002677 .00006 .00149
(.3714 .6010) (.3954 .6238) (-.0105 .0158) (-.0024 .0025) (-.0103 .0133)
5 .446259 .446557 .003274 .027076 .069313
(.3310 .5615) (.3413 .5518) (-.0130 .0346) (-.0206 .0748) (.0030 .1356)
20 .43117 .423779 .022324 .049878 .072849
(.3135 .5489) (.3174 .5302) (-.0132 .0579) (-.0110 .1107) (.0047 .1410)
24 .431081 .423723 .022447 .049891 .072858
(.3134 .5488) (.3173 .5301) (-.0133 .0582) (-.0110 .1108) (.0047 .1410)

7. 6
It still shows that the explanatory powers of monetary shocks are insignificant at any horizon.
GDP variance is mainly explained by government expenditure and itself (almost half and half).
However, in this case, real exchange rate variance also accounts for part of GDP variance, and
the portion increases with horizon. On a 24-quarter horizon, it can explain 7.3% of GDP
variance.
Part V: Robustness to New Ordering: {
𝐆
𝐲
, 𝐦𝐦, 𝐡𝟏, 𝐲, 𝐫𝟏}
We arrange GDP to the fourth place because our GDP data is revised data instead of real time data,
which may add news about monetary policies when they are released. So it is plausible to put GDP
behind mm and h1.
Now the IRFs are:
A positive mm shock has a significantly positive effect on GDP during quarter 6 to 15; a positive h1
shock has a significantly positive effect on GDP during quarter 4 to 15. These responses are similar
to our previous ordering. Notice that the responses do not exactly begin from zero because of the
new ordering, but it still insignificant.
Table 5
Variance Decomposition of GDP in New Ordering
Horizon ℇ 𝑔
ℇ 𝑦
ℇ 𝑚𝑚
ℇℎ1 ℇ 𝑟1
The VDC of y
1 .533182 .462783 .00374 .000295 0
(.4244 .6419) (.3546 .5710) (-.0095 .0170) (-.0034 .0040)
2 .561511 .424407 .006958 .002853 .00285
(.4331 .6899) (.2988 .5500) (-.0144 .0311) (-.0120 .0177) (-.0183 .0335)
5 .589531 .291719 .008282 .032816 .077652
(.4228 .7562) (.1432 .4403) (-.0203 .0369) (-.0364 .1021) (-.0210 .1763)

8. 7
The VDC results still show the monetary shocks have no significant explanatory power to GDP
variance. The magnitude and significance of the portions are all similar to those in the baseline case.
Part V: Add New Variables into the Baseline Model
We think the interest rate differential and the inflation differential may be useful to add to our
model, because interest rate is usually used as an endogenous variable representing monetary
shock.
We put interest differential in the last place because it is contemporaneously affected by monetary
policy. Here the AIC still leads us to choose lag 5.
The IRFs show that mm has significantly positive effect on GDP during quarter 11 to 17; h1 has a
significantly positive effect on GDP from quarter 5 to 7; interest differential has a significantly
negative effect on GDP during quarter 3 to 8.
As we can see, adding interest rate into the model shortens the effect of monetary multiplier and
money base. Since interest rate is directly affected by shocks on monetary multiplier and money
base, it is difficult to know the effect of monetary shock holding the interest rate “constant”.
Table 6
Variance Decomposition of GDP after Adding Interest Rate
Horizon ℇ 𝑔
ℇ 𝑦
ℇ 𝑚𝑚
ℇℎ1
ℇ𝑖 ℇ 𝑟1
The VDC of y
1 .524368 .475632 0 0 0 0
(.4145 .6342) (.3658 .5855)
2 .54726 .449665 .000816 .001264 .00002 .000975
20 .270278 .147548 .237827 .151488 .192859
(.0134 .5271) (-.0132 .3083) (-.1099 .5601) (-.1180 .4210) (-.0459 .4316)
24 .25205 .140335 .250238 .159665 .197713
(-.0185 .5226) (-.0249 .3055) (-.1337 .5538) (-.1392 .4586) (-.0621 .4575)

9. 8
(.4167 .6778) (.3197 .5797) (-.0040 .0057) (-.0067 -.0092) (-.0010 .0010) (-.0058 .0077)
5 .559025 .308696 .001331 .023978 .064211 .042759
(.3853 .7327) (.1539 .4635) (-.0023 .0050) (-.0326 .0806) (-.0294 .1578) (-.0246 .1101))
20 .275164 .181865 .155182 .147708 .112256 .127824
(-.0157 .5661) (-.0168 .3805) (-.1027 .4131) (-.1164 .4119) (-.0835 .3080) (-.0688 .3245)
24 .270748 .170798 .161647 .158616 .100789 .137403
(-.0462 .5877) (-.0335 .3751) (-.1202 .4435) (-.1358 .4530) (-.0917 .2932) (-.0781 .3529)
It still shows that adding interest rate do not affect the explanatory power of monetary shocks in
explaining GDP variance. Even interest rate itself has no significant power to explain GDP variance.
Part VI: Conclusions
1. Baseline Case
1) IRFs show a positive monetary multiplier shock has a significantly potive effect on GDP
from the 7th quarter to the 19th quarter; a positive shock on real money base has a
significantly positive effect on GDP from the 4st to 10th quarter.
2) VDC results show monetary shocks has no significant power to explain GDP variance.
2. First-Difference Case
1) IRFs show the effect of monetary shocks are weakened. The reason is that the over-
differencing problem covers the long-run relationship between variables.
2) VDC results are similar with those in baseline case, except that the real exchange rate shows
some explanatory power at long horizon.
3. New Ordering Case
Placing GDP behind monetary variables has little effects on our results, implying robustness of
our baseline model.
4. Adding Interest Rate into the Model
1) IRFs show the effect of monetary shocks are shortened. The reason might be that interest
rate and monetary variables are directly and highly correlated, so it is hard to control one
variable constant and study the other one’s effect.
2) VDC results are similar with those in baseline case. New added variable has no explanatory
power.

10. 9
Appendix
/*This is the STATA code for Open Economy Macroeconomics Course, HW2
Editted by Juan Du, 07/24/2016
*/
tsset t
*---------------------------Part I: Unit Root Test------------------------------
tsline g2y //have a trend
*choose lags for DF test
varsoc D.g2y, maxlag(8) exog(t L.g2y)
*we should choose lag 3
dfuller g2y, lags(3) trend regress
*have a unit root on 5% level (but no unit root on 10% level)
tsline y //no trend?
varsoc D.y, maxlag(8) exog(t L.y)
*AIC tell us to choose lag 1, while BIC let us choose lag 0
gen Dy=D.y
foreach i of numlist 1/2 {
reg D.y t L.y L(1/`i').Dy if t>3
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)
//coefficient for L.Dy is insignificant, so we choose lag zero. i.e. the standard DF test
dfuller y, lags(0) regress
*have a unit root on all levels
tsline r1 //no trend
varsoc D.r1, maxlag(8) exog(t L.r1)
*AIC tell us to choose lag 3, while BIC let us choose lag 1
gen Dr1=D.r1
foreach i of numlist 1/3 {
reg D.r1 t L.r1 L(1/`i').Dr1 if t>4
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)
*coefficients for lag 1 and lag 3 are all significant
dfuller r1, lags(1) regress
*have a unit root on 5% level (but no unit root on 10% level)
dfuller r1, lags(3) regress
*no unit root
tsline r2 //no trend
varsoc D.r2, maxlag(8) exog(t L.r2)
*AIC tell us to choose lag 3, while BIC let us choose lag 1
gen Dr2=D.r2
foreach i of numlist 1/3 {
reg D.r2 t L.r2 L(1/`i').Dr2 if t>4
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)

11. 10
*coefficients for lag 1 and lag 3 are all significant
dfuller r2, lags(1) regress
*have a unit root on 5% level (but no unit root on 10% level)
dfuller r2, lags(3) regress
*no unit root
tsline mm //trend
varsoc D.mm, maxlag(8) exog(t L.mm)
*AIC tell us to choose lag 8, while BIC let us choose lag 5
gen Dmm=D.mm
foreach i of numlist 1/8 {
reg D.mm t L.mm L(1/`i').Dmm if t>9
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)
*coefficients are all significant
dfuller mm, lags(8) trend regress
*have a unit root
dfuller mm, lags(5) trend regress
*have a unit root
tsline h1 //trend
varsoc D.h1, maxlag(8) exog(t L.h1)
*AIC tell us to choose lag 3, while BIC let us choose lag 1
gen Dh1=D.h1
foreach i of numlist 1/3 {
reg D.h1 t L.h1 L(1/`i').Dh1 if t>4
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)
*coefficients are all significant
dfuller h1, lags(3) trend regress
*have a unit root
dfuller h1, lags(1) trend regress
*have a unit root
tsline h2 //trend
varsoc D.h2, maxlag(8) exog(t L.h2)
*AIC tell us to choose lag 8, while BIC let us choose lag 1
gen Dh2=D.h2
foreach i of numlist 1/8 {
reg D.h2 t L.h2 L(1/`i').Dh2 if t>9
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)
*coefficients are all significant
dfuller h2, lags(8) trend regress
*have a unit root
dfuller h2, lags(1) trend regress
*have a unit root
foreach i of numlist 1/8{

12. 11
qui var g2y y r1 mm h1, lag(1/`i')
estimates store modL`i'
}
esttab modL*, se stats(N r2 aic bic)
varsoc g2y mm h1 y r1, maxlag(8)
*AIC:lag 5; BIC:lag 1
*________________________Part II: Cointegration Test____________________________
vecrank g2y mm h1 y r1, lags(5) trend(none) max
vecrank g2y mm h1 y r1, lags(1) trend(none) max
*results show there are two cointegrated equations. So the cointegration rank should be 5-2=3
*__________________________Part III: baseline case_____________________________
var g2y y mm h1 r1, lags(1/5)
irf create order1, step(24) set(baseline)
irf graph oirf, irf(order1)
irf graph oirf, irf(order1) impulse(mm) response(y)
irf graph oirf, irf(order1) impulse(h1) response(y)
irf table fevd, irf(order1) impulse(g2y y mm h1 r1) response(y)
*__________________________Part IV: first difference case_____________________________
varsoc Dg Dy Dmm Dh1 Dr1, maxlag(8)
*AIC:4, BIC:0
var Dg Dy Dmm Dh1 Dr1, lags(1/4)
irf create order1, step(24) set(difference)
irf graph oirf, irf(order1) impulse(Dmm) response(Dy)
irf graph oirf, irf(order1) impulse(Dh1) response(Dy)
irf table fevd, irf(order1) impulse(Dg Dy Dmm Dh1 Dr1) response(Dy)
*__________________________Part V: New order_____________________________
var g2y mm h1 y r1, lags(1/5)
irf create order2, step(24) set(baseline)
irf graph oirf, irf(order2)
irf graph oirf, irf(order2) impulse(mm) response(y)
irf graph oirf, irf(order2) impulse(h1) response(y)
irf table fevd, irf(order2) impulse(g2y y mm h1 r1) response(y)
*_________________________Part VI: Add new Variables__________________________
varsoc g2y y mm h1 i r1, maxlag(8)
*AIC:5, BIC:1
var g2y y mm h1 i r1, lags(1/5)
irf create order1, step(24) set(addi)
irf graph oirf, irf(order1) impulse(mm) response(y)
irf graph oirf, irf(order1) impulse(h1) response(y)
irf graph oirf, irf(order1) impulse(i) response(y)
irf table fevd, irf(order1) impulse(g2y y mm h1 i r1) response(y)