Homework Assignment 3
Empirical
Assigned: Monday, November 29, 2004
Due: Friday, December 10, 2004
1. Excel Problem. You are interested in finding out whether the business cycles of
the 4 Asian Tigers are more closely correlated with Japan or the USA. You are
given (in Data Table) real GDP for Hong Kong, Korea, Singapore, and Taiwan as
well as real GDP for Japan and the United States at a quarterly frequency between
the period of the first quarter of 1980 (time period = 1) and the 3rd quarter of 2003
(time period = 95). These periods will be indexed by time = 1…95. The project
will involve three steps: A) calculate potential output as the long-term trend; B)
calculate the output gap and its average volatility; C) calculate volatility and
correlations.
A) For each time series, Qt,
I.
Create a new series qt = ln(Qt).
II. Decompose the series qt = q + εt where q = β0 + β1 ∙ time + β2 ∙ time2 by
estimating the coefficients β0, β1, and β2 using the linear regression tool in
Excel. Under the tool bar, access Tools/Data Analysis/Regression. This
will bring up the regression wizard which will ask for a Y range and an X
range. For the Y range, use qt and t = 1,..95. For the X range, use time and
time2 (i.e. B2:C96). This will generate an output sheet which includes
Intercept, X Variable 1, X Variable 2. The variables to the right of
Intercept is the estimate of β0, to the right of X Variable 1 is β1, to the
right of X Variable 2 is β2.
III. Use estimates β0, β1, and β2 to calculate q = β0 + β1 ∙ time + β2 ∙ time2.
B) Use the estimate of q to estimate the output gap, εt = qt – q. To estimate
the volatility, calculate the standard deviation as the square root of the
average level of εt2.
C) Calculate the correlation between the Japanese output gap and the output
gap of the four Asian tigers. First, put the series for εtJPN in a column to the
left of the output gap of each Tiger. Then use Tools/Data
Analysis/Correlation to calculate the correlation. Which East Asian tiger
has the strongest correlation with Japan? Next, repeat the process
substituting εtUSA . Which East Asian Tiger has the closest correlation
with the US.
time
Mar-80
Jun-80
Sep-80
Dec-80
Mar-81
Jun-81
Sep-81
Dec-81
Mar-82
Jun-82
Sep-82
Dec-82
Mar-83
Jun-83
Sep-83
Dec-83
Mar-84
Jun-84
Sep-84
Dec-84
Mar-85
Jun-85
Sep-85
Dec-85
Mar-86
Jun-86
Sep-86
Dec-86
Mar-87
Jun-87
Sep-87
Dec-87
Mar-88
Jun-88
Sep-88
Dec-88
Mar-89
Jun-89
Sep-89
Dec-89
Mar-90
Jun-90
Sep-90
Dec-90
Mar-91
Jun-91
Sep-91
Dec-91
Mar-92
Jun-92
Sep-92
Dec-92
time^2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
729
784
841
900
961
1024
1089
1156
1225
1296
1369
1444
1521
1600
1681
1764
1849
1936
2025
2116
2209
2304
2401
2500
2601
2704
HK
105670.5
111605
112717.3
108245.1
117010
118834.9
119622.8
122679.4
123565.9
120939.5
122940.8
123829.4
124293.5
128312.2
131630.3
135311.9
140167.5
146849.8
145222.5
139759
149032.5
143091.8
140228.1
142380.3
150146.4
155682.9
162362.2
166890.2
170719.5
176903.5
185083.8
185149.6
186703.5
192300.8
197196.4
199737.9
199630.3
198423.3
199136.9
200457.6
201913.5
206237.6
208528.1
210301.8
213351.8
216789.8
219699.2
223851
226860.8
231146.4
235234.6
238102
JAPAN
77796.44
77717.01
78135.91
79334.02
79979.97
80173.22
80894.59
81165.64
81898.81
82885.48
82885.94
83478.87
83411.48
83858.81
84532.33
84711.38
85252.6
86493.15
87629.46
87606.52
89859.04
90736.9
91406.93
92597.12
92702.88
93454.76
94393.18
94964.55
95013.57
96335.7
98056.53
100243.3
102041.2
103183.7
104848.2
106145.3
109010.5
107858
109255.8
112073.4
112199.7
115157.4
116636
117078.1
118995.9
118656.3
118873.2
119873.4
120832
119490.8
120208.3
120401.8
KOREA
28442.11
28924.36
28871.74
28716.21
29334.18
29935.83
30559
32100.28
31318.12
31999.02
32730.91
34689
34575.24
35639.51
36891.74
37789.57
38591.01
39013.04
39489.51
40152.86
40675.36
41347.3
42014.24
43296.72
44264.52
45719.69
47496.6
48117.58
49689.02
51423.43
52301.23
52793.48
56310.61
55633.03
57396.95
58583.08
58704.32
59255.74
60733.55
62950.48
64189.38
64980.59
66714.3
67657.64
70012.52
71452.19
72484.99
73869.47
75138.74
75797.58
75664.63
76985.7
SINGAPORE
TAIWAN USA
9145.37 588766.6
2725.3
9359.704 593741.6
2729.3
9587.786 598985.1
2786.6
9854.373 612708.9
2916.9
9979.895 624255.1
3052.7
10317.1 631998.6
3085.9
10562.58 638794.4
3178.7
10781.39 647127.6
3196.4
10881.38 647812.7
3186.8
11070.46
653454
3242.7
11240.72 658629.2
3276.2
11425.87 673101.5
3314.4
11675.17 681475.4
3382.9
11944.18 702182.9
3484.1
12212.05 726659.7
3589.3
12578.76 744837.7
3690.4
12932.47 769049.5
3809.6
13077.08 787815.2
3908.6
13214.59
799088
3978.2
13226.3 802377.7
4036.3
13320.29 810468.1
4119.5
12954.23 823694.9
4178.4
12805.23 829831.8
4261.3
12629.34
850352
4321.8
12868.99 889964.9
4385.6
13077.96 904521.2
4425.7
13270.19 933266.2
4493.9
13576.19 970633.4
4546.1
13845.29 1003965
4613.8
14222.38 1034890
4690
14673.96 1059995
4767.8
15184.76 1070805
4886.3
15358.78 1087840
4951.9
15976.98 1109450
5062.8
16400.11 1141123
5146.6
16748.23 1157879
5253.7
16831.76 1179446
5367.1
17876.78 1211441
5454.1
17930.52 1231244
5531.9
18240.29 1245130
5584.3
18988.87 1261080
5716.4
19173.57 1259763
5797.7
19410.4 1288711
5849.4
19712.81 1320294
5848.8
20196.23 1338514
5888
20439.94 1364749
5964.3
20832.51 1397463
6035.6
21021.25 1416657
6095.8
21306.47 1447005
6196.1
21646.4 1470504
6290.1
22221.4 1493313
6380.5
22822.71 1519314
6484.3
2. Internet Problem Construct a trade weighted exchange rate index for China. The
following chart shows the top 5 trading partners for China according to total trade.
Follow the following steps.
Japan
USA
Hong Kong
Korea
Taiwan
Exports
59453.99
92510.15
76323.6
20104.85
9013.773
Imports
74204.07
33882.96
11138.96
43160.53
49364.17
Trade
133658.1
126393.1
87462.55
63265.38
58377.95
A. Use these economies share of trade to construct weights. Let TRADE be
Traden
the sum of trade for all 5 countries. Then for country n, wn 
.
TRADE
B. Download the monthly US dollar exchange rates from Jan 2000 to
November 2004 with China, Japan, Hong Kong, Korea, and Taiwan from
http://fx.sauder.ubc.ca/data.html . Hint: Retrieve as a Tab spreadsheet.
Save the worksheet to a txt file. Open the text file with Excel.
C. Each of the above series should be in the form of number of local currency
units per US dollar. The first series (Renminbi per US$) is already in the
form of Renminbi per foreign currency unit). Convert each of the other
series to renminbi per foreign currency unit format. Do this by dividing
each series by the number of renminbi per US dollar series.
D. Calculate the growth rate of China’s exchange rate with each trading
S
partner, n as gtSn  ln( t ,n
) for t = February 2000 to November 2004.
St 1,n
E. Use the weights in 2A to calculate a weighted average growth rate, g tSW .
F. Calculate the average of the growth rate of the trade weighted exchange
rate, g. Calculate the deviation t  gtSW  g from the average from each
period. Calculate the square root of the average of the square of this
deviation t 2 as the standard deviation.
G. Set the January 2000 trade weighted exchange rate equal to S0 = 1. Then
for each following period as St = exp( g tSW ) ∙St-1.
Practice Problems
1. Rational Expectations and Monetary Policy Shocks
Demand follows the quantity theory of money
t  t   qt  q    t
where mt is (logged) money supply, vt is velocity, pt is the price level, and qt is the
output level. The supply curve is given by
 t  REt 1  t     qt  q 
Assume a constant potential output. To simplify algebra, normalize potential output, q
= 0. Monetary policy follows a random walk.
a. Monetary policy follows a random walk so that the growth rate of money is
unpredictable white noise: t   t . This implies that te  REt 1 ( t )  0 . Assume
that velocity is constant,  t =0. Calculate the expected price level. Calculate actual
levels of output as a function of εt.
b. Assume that velocity follows the process,  t   t 1   tv where ρ < 1. This
implies that  te  REt 1 ( t )   t 1 . Assume that monetary policy systematically
targets velocity, t    t 1  te    t 1 . Calculate output and prices as
functions of  t 1 and  tv . What effect does the choice of λhave on output?
2. Okun’s Law and Time Consistency
Each % point increase in output above potential output results in θ% extra inflation.
 t  REt 1  t     qt  q 
Each % point increase in output above potential output reduces unemployment below
the natural rate by ψ%.


urt  ur NR   qt  q   t  REt 1  t   

ur  ur NR 
 t
The government is interested in low inflation and pushing unemployment toward
some ta1rget level ur*. The government sets inflation policy to minimize
2
2
a  t   b  urt  ur *
subject to  t  REt 1  t   

urt  ur NR  taking expectations as given.


b. Calculate the optimal inflation policy, πP, as a function of ur*, urNR, and
REt 1  t  .
c. What will equilibrium inflation and unemployment be when   REt 1  t    P ?
d. If the government can choose a target unemployment rate, ur*, in order to have
zero inflation, what would it be?
3. Adaptive Expectations and Dynamics
Construct a small model of the economy that includes A) a Planned expenditure
curve; B) a Taylor rule; and C) an expectations augmented Philips curve
A. The planned expenditure curve sets demand as a negative function of the real
interest rate and an exogenous demand shock, αt.
qt  t  b  rt
B. The Taylor rule sets the real interest rate as an increasing function of the
inflation rate
rt  r  d   t
C. The SRAS curve sets inflation acceleration as a function of the output gap.
 t   t 1    qt  q 
Calibrate the parameters of the model equal to r = .1, q = 1, b = .5 , d = .5 and θ = .5.
a. Assume that at time t-1, the economy was in a steady state where  t 1   t 2 and
 t 1 =1.10. Solve for output, qt-1, the interest rate, rt-1, and inflation πt-1.
b. Assume that unexpectedly  t =1.10 drops to 1.05. Using the answer from 3a for
πt-1, solve for the new real interest rate output, qt, the interest rate, rt, and inflation
πt.
c. Using πt from 3b, solve for qt+1 and πt+1. Using πt+1 solve for qt+2 and πt+2. Using
πt+2 solve for qt+3 and πt+3. Graph the dynamic response of inflation and output to
this demand shock.
d. In the far future, inflation will again reach a constant level. Solve for the constant
level of inflation when  t =1.10.
e. Re solve 3a and 3b for qt-1 and qt under the assumption that the central bank sets
an aggressive response to inflation, d = 2. How does an aggressive response to
inflation affect the response of output to a demand shock? Explain