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Our aim is to study the theory
data from the IMF analysing v
percentage annual growth of
study stationarized variables.
To do this analysis, we start b
and prices running the regressi
Prices t=beta1+ beta2 money t
Hence for each country we obt
Bolvia
reg prices money
Source | SS
-------------+-------------
Model | 134755251
Residual | 1257373.81
-------------+-------------
Total | 136012625
quantity of money in the long-run an
ariables like M3 and IPC consumer i
oney and prices. We decided to anal
studying the relation between annu
n:
ut
in the following results:
df MS Numbe
---------------- F( 1
1 134755251 Prob
52 24180.2656 R-squ
---------------- Adj R
53 2566275.94 Root
d for that we took the
dex and we took the
yse yearly variables to
al % change in money
r of obs = 54
, 52) = 5572.94
> F = 0.0000
ared = 0.9908
-squared = 0.9906
MSE = 155.5
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------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | 1.656262 .0221864 74.65 0.000 1.611742 1.700783
_cons | -46.48212 21.57874 -2.15 0.036 -89.78302 -3.181215
Conclusions for this especific country:
The regressor money is statistically significant since we obtain a p-value of 0.000
99% of the variation of the inflation is explained linearly by the variable annual% change in
money.
19561957195819591960196119621963196419651966196719681969197019711972197319741975197619771978197919801981198219831984
1985
1986198719881989199019911992199319941995199619971998199920002001200220032004200520062007200820090
.2
.4
.6
.8
1
Leverage
0 .2 .4 .6 .8Normalized residual squared
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Take the observation of year 1985, 1984
Reggressing model with dummys for Hyperinflationary period
(1984-1985)
To see whether there is a structural change in the model during the hyperinflationary times
I will introduce a Dummy in the model:
Dummy=1 if t=hyperinflationary times (Bolivia case(1984,1985) t=29, t=30)
Dummy=0 if not
Model with dummy
Pricest=beta1+ beta2*moneyt+ beta3*dummyt+ beta4 dummy*money+ ut
If Dummy=1
Model: prices t=(beta1+beta3)+ (beta2+beta4)money t + u t
If dummy=0
0
5000
10000
15000
0 2000 4000 6000 8000money
prices Fitted values
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Model: prices t= beta1 + beta2 money t + ut
Under H0: Beta3=beta4=0 (there is no structural change in the model)
Codes in stata:
generate obs=_n
. gen dummy=0
. replace dummy=1 if obs==29
(1 real change made)
. replace dummy=1 if obs==30
(1 real change made)
gen dmoney=money*dummy
reg prices money dummy dmoney
Source | SS df MS Number of obs = 54
-------------+------------------------------ F( 3, 50) =46859.71
Model | 135964266 3 45321422 Prob > F = 0.0000
Residual | 48358.6224 50 967.172447 R-squared = 0.9996
-------------+------------------------------ Adj R-squared = 0.9996
Total | 136012625 53 2566275.94 Root MSE = 31.099
------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | .9679052 .0822909 11.76 0.000 .802619 1.133191
dummy | -1403.398 40.24342 -34.87 0.000 -1484.229 -1322.567
dmoney | .9152123 .0826704 11.07 0.000 .749164 1.081261
_cons | -6.038745 5.229494 -1.15 0.254 -16.54249 4.465003
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By introducing the dummy variable we conclude that during hyperinflationary periods, a
unit change in annual % in money is reflected in a 1.99 percentual points in inflation,
reflecting that there is some exogenous shock, apart from changes in prices, that are
causing this change in the value of inflation. When the dummy takes the value zero
meaning that we are in non-hyperinflationary period, this relation is closer to one-to-one.
Notice R2 is almost 100% showing the introduction of the new variables increased the
explicative power of the model over inflation
Testing the structural change
The RSSE of the F-statistic corresponds to the SSE of the original model, since under H0
there is no structural change on the model (which corresponds to the original model)
The RSSE=1.257.373,81 and SSE=48.358,6224
Since F-value> Fcritic, we reject the null hypothesis and then we conclude with a level of
significance of 5% that there is a statistically significant structural change in the model
during the hyperinflationary period
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Argentina
Model
Prices t=beta1+ beta2*Money t+ ut
reg prices money
Regression with 49 observations
Source | SS df MS Number of obs = 49
-------------+------------------------------ F( 1, 47) = 755.31
Model | 13552690.4 1 13552690.4 Prob > F = 0.0000
Residual | 843334.13 47 17943.2794 R-squared = 0.9414
-------------+------------------------------ Adj R-squared = 0.9402
Total | 14396024.5 48 299917.178 Root MSE = 133.95
------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | 1.468127 .0534197 27.48 0.000 1.36066 1.575593
_cons | -37.97795 21.0049 -1.81 0.077 -80.23435 4.278448
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------------------------------------------------------------------------------
Comments:
Relation of price and money=1.46
By P- value=0 we can see that we reject the null hypothesis (H0: beta2=0)
concluding with a level of significance of 5% that money is statistically
significant when explaining the evolution of prices
R2=0,94 = everything else constant, 94% of the variability of prices can be
explained by the aggregate variable M3.
Stata code for graphic:
summarize leverage
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
leverage | 49 200.0563 531.364 -66.45961 3243.579
code: twoway( scatter prices money)(lfit prices money)
lvr2plot, mlabel(year)
0
100
0
2000
3000
0 500 1000 1500 2000 2500money
prices Fitted values
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.
Reggressing model with dummys for Hyperinflationary period
(1989-1990)
To see whether there is a structural change in the model during the
hyperinflationary times I will introduce a Dummy in the model:
Dummy=1 if t=hyperinflationary times (Argentina case(1989,1990) t=29,
t=30)
Dummy=0 if not
Model with dummy
Pricest=beta1+ beta2*moneyt+ beta3*dummyt+ beta4 dummy*money+ ut
If Dummy=1
Model: prices t=(beta1+beta3)+ (beta2+beta4)money t + u t
If dummy=0
Model: prices t= beta1 + beta2 money t + ut
Under H0: Beta3=beta4=0 (there is no structural change in the model)
1961196219631964196519661967196819691970197119721973197419751976197719781979198019811982 19831984
198519861987 1988
1989
1990
19911992199319941995199619971998199920002001200220032004200520062007200820090
.2
.4
.6
.8
Leverage
0 .2 .4 .6Normalized residual squared
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Stata codes
. gen obs=_n
. gen dummy=0
. replace dummy=1 if obs==29
(1 real change made)
. replace dummy=1 if obs==30
(1 real change made)
. gen dmoney=money*dummy
. reg prices money dummy dmoney
Source | SS df MS Number of obs = 49
-------------+------------------------------ F( 3, 45) = 2176.84
Model | 14297504.4 3 4765834.8 Prob > F = 0.0000
Residual | 98520.1186 45 2189.33597 R-squared = 0.9932
-------------+------------------------------ Adj R-squared = 0.9927
Total | 14396024.5 48 299917.178 Root MSE = 46.79
------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | 1.045071 .0487685 21.43 0.000 .9468459 1.143295
dummy | 1582.43 102.8682 15.38 0.000 1375.243 1789.617
dmoney | -.3712628 .075946 -4.89 0.000 -.5242259 -.2182996
_cons | -8.715363 8.334931 -1.05 0.301 -25.50278 8.07205
------------------------------------------------------------------------------
By introducing the dummy variable we conclude that during hyperinflationary periods, a
unit change in annual % in money is reflected in a 0,6 percentual points in inflation,
reflecting that there is some exogenous shock apart from changes in prices which are
causing this change in the value of inflation. When the dummy takes the value zero, this
relation is closer to one-to-one (beta2=1,045)
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Notice R2 is almost 100% showing the introduction of the new variables increased the
explicative power of the model over inflation
Testing the structural change
The RSSE=843.334,13 and SSE=98.520,1186
Since F-value> Fcritic, we reject the null hypothesis and then we
conclude with a level of significance of 5% that there is a
statistically significant structural change in the model during the
hyperinflationary period
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Nicaragua
Full regresin
Source | SS
-------------+-------------
Model | 162370961
Residual | 9313436.64
-------------+-------------
Total | 171684398
---------------------------
prices | Coef.
-------------+-------------
money | .8866074
_cons | 125.0878
df MS Numbe
---------------- F( 1
1 162370961 Prob
34 273924.607 R-squ
---------------- Adj R
35 4905268.51 Root
--------------------------------
Std. Err. t P>|t| [9
--------------------------------
.036416 24.35 0.000 .8
91.14757 1.37 0.179 -60
r of obs = 36
, 34) = 592.76
> F = 0.0000
ared = 0.9458
-squared = 0.9442
MSE = 523.38
------------------
5% Conf. Interval]
------------------
126011 .9606137
.14633 310.322
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0
5000
100
00
0 5000 10000 15000money
prices Fitted values
197319741975197619771978197919801981198219831984198519861987
1988
1989
1990
1991199219931994199519961997199819992000200120022003200420052006200720080
.2
.4
.6
.8
Leverage
0 .2 .4 .6Normalized residual squared
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Reggressing model with dummys for Hyperinflationary period (1988-
1991)
To see whether there is a structural change in the model during the hyperinflationary times I will introduce a
Dummy in the model:
Dummy=1 if t=hyperinflationary times (Nicaragua case(1988-1991) t=16-19)
Dummy=0 if not
Model with dummy
Pricest=beta1+ beta2*moneyt+ beta3*dummyt+ beta4 dummy*money+ ut
If Dummy=1
Model: prices t=(beta1+beta3)+ (beta2+beta4)money t + u t
If dummy=0
Model: prices t= beta1 + beta2 money t + ut
Under H0: Beta3=beta4=0 (there is no structural change in the model)
By looking at the leverage graphic we can see the the influential observations are the ones for the years of
1988,1989,1990,1991 which are composed either by high leverage observations wither by outliers.
Stata codes
. gen obs=_n
. gen dummy=0
. replace dummy=1 if obs==16
(1 real change made)
. replace dummy=1 if obs==17
(1 real change made).
. replace dummy=1 if obs==18
(1 real change made)
. replace dummy=1 if obs==19
(1 real change made)
gen dmoney=money*dummy
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. reg prices money dummy dmoney
Source | SS df MS Number of obs = 36
-------------+------------------------------ F( 3, 32) = 2429.82
Model | 170934013 3 56978004.5 Prob > F = 0.0000
Residual | 750384.37 32 23449.5116 R-squared = 0.9956
-------------+------------------------------ Adj R-squared = 0.9952
Total | 171684398 35 4905268.51 Root MSE = 153.13
------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | 1.860181 .2740634 6.79 0.000 1.301932 2.41843
dummy | 2575.911 135.2056 19.05 0.000 2300.506 2851.316
dmoney | -1.236711 .2746246 -4.50 0.000 -1.796103 -.6773195
_cons | -29.30189 30.83229 -0.95 0.349 -92.10521 33.50144
By introducing the dummy variable we conclude that during hyperinflationary periods, a
unit change in annual % in money is reflected in a 0,6 percentual points change in inflation,
reflecting that there is some exogenous shock apart from changes in prices which are
causing this change in the value of inflation. When the dummy takes the value zero, this
relation is closer to one-to-one.
Notice R2 is almost 100% showing the introduction of the new variables increased the
explicative power of the model over inflation
Testing the structural change
The RSSE=9.313.436,64 and SSE=750.384,37
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Since F-value> Fcritic,
conclude with a level
statistically significan
hyperinflationary period
Israel
Model with all the 54 obser
Source | SS
-------------+-------------
Model | 217697.428
Residual | 39795.8832
we reject the null hypoth
of significance of 5% t
structural change in the
vations
df MS Numbe
---------------- F( 1
1 217697.428 Prob
52 765.305445 R-squ
sis and then we
hat there is a
model during the
r of obs = 54
, 52) = 284.46
> F = 0.0000
ared = 0.8454
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-------------+------------------------------ Adj R-squared = 0.8425
Total | 257493.311 53 4858.36436 Root MSE = 27.664
------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | .8276106 .0490701 16.87 0.000 .7291443 .926077
_cons | -3.095239 4.353302 -0.71 0.480 -11.83078 5.6403
------------------------------------------------------------------------------
0
100
200
300
400
0 100 200 300 400 500money
prices Fitted values
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Stata codes
. gen obs=_n
. gen dummy=0
. replace dummy=1 if obs==29
(1 real change made)
. replace dummy=1 if obs==30
(1 real change made).
gen dmoney=money*dummy
. reg prices money dummy dmoney
Source | SS df MS Number of obs = 54
-------------+------------------------------ F( 3, 50) = 453.11
Model | 248358.083 3 82786.0276 Prob > F = 0.0000
Residual | 9135.22809 50 182.704562 R-squared = 0.9645
-------------+------------------------------ Adj R-squared = 0.9624
Total | 257493.311 53 4858.36436 Root MSE = 13.517
------------------------------------------------------------------------------
prices | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | .813224 .0481041 16.91 0.000 .716604 .909844
dummy | 275.4169 21.39669 12.87 0.000 232.4404 318.3934
dmoney | -.6105901 .0737808 -8.28 0.000 -.7587832 -.4623969
_cons | -4.980728 2.462908 -2.02 0.049 -9.927624 -.0338329
------------------------------------------------------------------------------
By introducing the dummy variable we conclude that during hyperinflationary periods, a
unit change in annual % in money is reflected in a 0,2 percentual points change in inflation,
reflecting that there is some exogenous shock apart from changes in prices which are
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causing this change in the value of inflation. When the dummy takes the value zero, this
relation is closer to one-to-one.
Notice R2 is almost 100% showing the introduction of the new variables increased the
explicative power of the model over inflation
Testing the structural change
The RSSE=39.795,8832 and SSE= 9135.22809
Since F-value> Fcritic, we reject the null hypothesis and then we conclude with a level of
significance of 5% that there is a statistically significant structural change in the model
during the hyperinflationary period.
Pannel Analysis
In the last part of the problem set we want to see whether the four countries analysed
previously influence the overall relation between money and prices for the 18 countries.
To check this we will use a panel data which makes this analysis over countries and over
years.
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For this purpose we need to construct a proper data with all the countries with the same
period where the variables were observed.
Then we will proceed to our econometric work:
We will estimate the following model:
Pricesit= beta1+ beta2*moneyit+ beya3*dummyit+ beta4dummoneyit+ uit
Where i represent the countries and t the time period
We construct our dummy by defining:
Dummyit=1 of country i was subject to hyperinflation
Dummyit=0 if country i was not subject to hyperinflation
Our null hypothesis will be that:
H0:beta3=beta4=0 (the hyperinflationary countries do not influence the relationship
between annual percentage change in money and prices over the 18 countries under
analysis)
Stata codes
Gen dummoney=money*dummy
xtreg prices money dummy dummoney, re i(code)
Random-effects GLS regression Number of obs = 486
Group variable (i): code Number of groups = 18
R-sq: within = 0.8836 Obs per group: min = 27
between = 0.9836 avg = 27.0
overall = 0.8928 max = 27
Random effects u_i ~ Gaussian Wald chi2(3) = 4016.05
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
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prices | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
money | .7527448 .6407409 1.17 0.240 -.5030844 2.008574
dummy | 51.03512 32.60992 1.57 0.118 -12.87915 114.9494
dummoney | .2863953 .6409639 0.45 0.655 -.9698708 1.542661
_cons | -1.263624 17.59736 -0.07 0.943 -35.75381 33.22656
-------------+----------------------------------------------------------------
sigma_u | 0
sigma_e | 278.86117
rho | 0 (fraction of variance due to u_i)
Since beta2+beta4= 1.03 we can conclude that there is almost a perfect one-to-one relation
between Money and prices within these 4 countries.
By doing the wald test we obtain a Wald-value=4016.05 for which prob(Wald-value>wald
critic)=0.0000 and hence we conclude that there is a statistically significant change in the
model including hyperinflationary countries or not.
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