Econ 240C
Power 17
1
Outline
• The Law of One Price
2
Law of One Price: Outline
•
•
•
•
•
Definition: slides 5-6
Applied to wheat trade: slides: slides 7-9
Time Series Notation: slides 10 - 11
Data and Traces: slides 12-14
Show that Import Price, DNSPJ, is Evolutionary:
slides 15-16
• Show that Import Price Minus Sum of Export
Price (-1) + Freight Rate (-2) is stationary: slides
18 - 26
3
Outline Cont.
• Show that logs of import & export prices are
evolutionary of order one: slides 27- 36
• Show that log of price ratio is stationary: slides
37- 49
– Speed of convergence: slides 50 - 54
• Cointegration: slides 55 – Long run equilibrium relationship between log of
import price and log of export price (with freight
and lags): slides 56 - 57
– VAR speed of adjustment model: slide 58-60
4
The Law of One Price
• The New Palgrave Dictionary of Money
and Finance
– Next slide
5
The Law of One Price
• This law is an immediate consequence of
the absence of arbitrage and, like the
absence of arbitrage, follows from
individual rationality. Departures from the
no arbitrage condition imply that there are
profit opportunities. These arise because it
would be profitable for arbitrageurs to buy
good i in the country in which it is cheaper
and transport it to the country in which it is
more expensive, and in doing so, profit in
trade.
6
Commodity Trade Issues
• Well defined product: World Wheat
Statistics
– # 2 Dark Northern Spring 14%
– Western White
– Hard Winter
• Transport costs
– US: export
• Pacific Ports
• Gulf Ports
7
Prices in $/metric ton
• Import price notation: DNSJ is Japanese
import price in $/metric ton for Dark
Northern Spring wheat
• Export price notation: DNSG is export
price for Dark Northern Spring from a Gulf
Port; DNSP is export price for Dark
Northern Spring from a Pacific Port
– Lagged one month because commodity
arbitrage takes time
8
Transport Cost in $/Metric Ton
• Freight rates are forward prices and are
lagged two months
9
Time Series
• Import Price: DNSJ
• Export Price (lagged one) Plus Freight
(lagged two): DNSGT
• Logarithm of Price Ratio: ln [DNSJ/DNSGT]
= lnDNSJ – lnDNSGT denoted lnratiodnsgjt =
ln[1 + ∆/DNSGT] ~ ∆/DNSGT, the fractional
price differential, where ∆ = DNSJ – DNSGT,
and can be positive or negative
10
Time Series
• Is the log of the export price evolutionary, of
order one?
– Ln DNSJ = lndnsj
• Is the log of the import price evolutionary, of
order one?
– Ln DNSGT = lndnsgt
• Is their difference stationary, of order zero, ie.
are they cointegrated?
• i.e. Is the log price ratio ( the fractional price
differential) stationary?
– Ln{DNSJ/DNSGT] =lnratiodnsjgt
11
# 2 Hard
Winter 13%
Date
75.09
75.1
75.11
75.12
76.01
76.02
76.03
76.04
76.05
76.06
76.07
76.08
76.09
76.1
76.11
76.12
77.01
77.02
77.03
77.04
77.05
77.06
77.07
77.08
77.09
77.1
77.11
77.12
78.01
78.02
78.03
78.04
78.05
78.06
78.07
78.08
78.09
78.1
78.11
78.12
79.01
79.02
79.03
79.04
79.05
79.06
79.07
79.08
79.09
79.1
79.11
79.12
80.01
80.02
80.03
80.04
80.05
80.06
80.07
80.08
80.09
80.1
80.11
80.12
81.01
81.02
81.03
81.04
81.05
81.06
81.07
81.08
81.09
81.1
# 2 Western
White
207
207
189
172
180
192
188
177
170
174
177
158
154
154
136
131
133
134
133
124
124
116
117
114
122
128
133
135
136
138
139
150
150
147
147
149
154
154
162
164
163
165
165
167
185
203
217
218
214
221
223
222
213
217
216
210
212
212
219
229
230
239
252
232
244
234
225
225
214
220
217
216
212
216
182
182
168
156
157
164
162
148
142
147
147
140
137
127
122
115
119
121
121
121
119
112
118
120
118
115
121
124
130
136
138
148
146
146
155
150
152
152
156
159
157
153
155
157
167
193
210
200
197
194
194
194
194
196
194
190
182
180
190
192
193
203
214
211
210
210
200
201
197
189
192
194
193
197
#2 DNS
14%
217
220
198
197
195
201
196
185
189
193
193
174
155
146
144
135
138
141
141
141
135
129
125
125
130
134
140
140
143
146
142
142
152
152
154
149
152
152
163
166
166
166
168
168
185
212
228
214
219
223
221
220
215
219
220
211
221
224
240
239
244
249
251
245
246
242
237
249
244
232
221
219
217
216
# 2 Hard
# 2 Western # 2 DNS
# 2 DNS Freight
Freight
Winter 13% White
13%
14%
Gulf
Pacific
Gulf
Pacific
Gulf
Pacific
176
162
187
201
15.74
178
161
184
199
17.22
161
145
173
186
17.22
151
141
178
178
16.73
151
141
178
178
12.79
166
151
186
186
14.76
168
146
182
181
15.74
156
138
162
175
15.74
148
132
164
178
14.76
154
134
173
181
15.01
158
134
166
178
16.73
137
126
142
156
15.74
133
123
138
146
15.99
119
112
131
134
16.24
117
111
129
131
16.97
115
105
136
125
16.24
118
108
137
126
13.78
118
111
140
130
13.78
114
110
136
131
13.78
109
110
122
129
13.78
101
110
115
125
14.37
97
104
107
117
13.78
101
108
104
115
13.78
98
108
102
115
14.27
104
105
112
119
13.78
108
105
117
123
14.02
117
109
125
127
14.51
118
111
134
127
15.01
120
117
137
131
15.25
122
124
137
134
14.51
127
128
138
134
14.76
135
136
145
140
14.76
129
135
144
143
19.68
130
136
139
143
20.66
130
141
132
139
18.7
130
139
131
138
19.14
134
141
138
142
22.63
139
141
143
144
22.14
151
141
149
148
22.14
138
140
153
150
22.14
140
140
153
149
21.89
144
137
153
151
21.89
143
139
150
155
24.11
141
139
146
152
24.11
146
147
155
161
24.11
164
166
174
184
28.54
179
178
189
193
36.41
173
171
180
185
34.44
180
163
185
187
34.44
183
161
188
189
34.44
183
158
189
186
34.44
185
157
194
187
34.44
179
161
191
181
34.44
176
163
189
184
34.44
169
158
180
182
34.44
158
155
166
176
43.3
163
148
176
188
43.3
159
148
174
190
37.39
168
156
195
207
37.15
175
158
192
205
37.88
183
162
197
212
37.88
192
171
204
218
37.88
200
179
212
218
39.85
188
171
212
206
39.85
193
172
218
210
39.85
185
172
216
207
39.85
177
166
196
203
36.41
183
169
191
214
36.41
176
166
187
213
34.44
172
161
183
202
36.16
173
160
177
192
36.16
174
161
173
186
31.49
175
159
176
183
22.14
175
164
180
184
24.6
FFR
13.28
13.78
14.76
14.51
14.51
13.53
14.51
13.78
15.01
15.01
15.01
15.25
15.25
15.25
15.25
14.27
14.76
14.76
14.02
14.02
14.02
14.02
13.78
13.78
13.53
13.53
13.53
14.76
14.76
13.53
13.53
13.53
13.78
14.02
14.27
14.27
14.51
15.25
16.24
17.47
17.47
17.47
18.2
18.2
19.19
27.55
29.52
28.63
28.78
29.47
29.47
29.52
29.52
29.03
30.01
30.01
32.47
32.47
31.83
31.83
31.83
31.83
26.08
29.77
30.01
30.01
29.77
29.77
29.77
29.77
28.78
28.29
28.29
28.29
6.24
5.82
5.22
5.2
4.87
4.77
4.84
4.82
5.29
5.48
5.31
5.29
5.25
5.03
4.95
4.65
4.61
4.68
4.69
4.73
5.35
5.39
5.42
5.9
6.14
6.47
6.51
6.56
6.7
6.78
6.79
6.89
7.36
7.6
7.81
8.04
8.45
8.96
9.76
10.03
10.07
10.06
10.09
10.01
10.24
10.29
10.47
10.94
11.43
13.77
13.18
13.78
13.82
14.13
17.19
17.61
10.98
9.47
9.03
9.61
10.87
12.81
15.85
18.9
19.08
15.93
14.7
15.72
18.52
19.1
19.04
17.82
15.87
15.08
World
World
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Wheat
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Wheat
Wheat
Wheat
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Wheat
Wheat
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Statistics
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Data
12
Table Three: Pacific DNS
• Log of Ratio of Import Price to Pacific Export
Price (lag 1) Plus Pacific Freight (lag2)
– Stationary
• No unit root
• AR (1) Model: root 0.54, normal residual
• Log of Import Price and Log of Pacific Export
Price (lag 1) Plus Pacific Freight (lag 2)
– Cointegrated
• VEC: one lag
• Rank: 1, 1, 1, 1, 2
• Data: no trend; Integrating Equation: Intercept, no trend, rank
one, 1%
13
ov
-7
Fe 5
b7
M 6
ay
-7
Au 6
g7
N 6
ov
-7
Fe 6
b7
M 7
ay
-7
Au 7
g7
N 7
ov
-7
Fe 7
b7
M 8
ay
-7
Au 8
g7
N 8
ov
-7
Fe 8
b7
M 9
ay
-7
Au 9
g7
N 9
ov
-7
Fe 9
b8
M 0
ay
-8
Au 0
g8
N 0
ov
-8
Fe 0
b8
M 1
ay
-8
Au 1
g81
N
$/Metric Ton
Trace of Import Price and Pacific Export Price (lagged one) Plus Pacific Freight Rate (lagged two), DNS,
$/Metric Ton
300
DNSJ
250
DNSPJT
200
150
100
50
0
Date
14
Correlogram of Import Price, DNSJ
15
Unit Root Test
16
Dark Northern Spring Japan
• Export Price is Evolutionary
• The Import Price Minus the Sum of the
Export Price (-1) + Freight Rate (-2) is
Stationary
• So the Import Price and the Export Price
Never Wander Off from each other, i.e.
they are cointegrated
17
N
ov
-
-10
75
b7
M 6
ay
-7
Au 6
g7
N 6
ov
-7
Fe 6
b7
M 7
ay
-7
Au 7
g7
N 7
ov
-7
Fe 7
b7
M 8
ay
-7
Au 8
g7
N 8
ov
-7
Fe 8
b7
M 9
ay
-7
Au 9
g7
N 9
ov
-7
Fe 9
b8
M 0
ay
-8
Au 0
g8
N 0
ov
-8
Fe 0
b8
M 1
ay
-8
Au 1
g81
Fe
$/Metric Ton
Trace of Import Price Minus the Sum of Pacific Export Price (lag1) Plus Pacific Freight Rate (lag 2), DNS,
$/Metric Ton
40
30
20
10
0
-20
-30
Date
18
Histogram
Japan DNS Export price Minus Sum of Import Price (-1) +freight(-2)
14
Series: JAPMMINX
Sample 1975:11 1981:10
Observations 72
12
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
8
6
4
2
Jarque-Bera
Probability
0.181250
-0.370000
32.80000
-19.01000
8.761377
0.923579
5.306363
26.19391
0.000002
0
-20
-10
0
10
20
30
19
Correlogram
20
Unit Root Test
21
ARONE Model
22
Diagnostics
40
20
30
0
20
-20
10
0
-10
-20
1976
1977
1978
Residual
1979
Actual
1980
1981
Fitted
23
Correlogram of Residuals
24
Correlogram of Residuals Squared
25
ARCH-LM Test
26
Trace of Logarithms of Import Price and the Pacific Export Price (lag1) Plus Pacific Freight Rate (lag 2),
DNS
5.6
5.5
5.4
lndnsj
lndnspjt
5.3
ln
5.2
5.1
5
4.9
4.8
4.7
Apr-75
Aug-76
Jan-78
May-79
Oct-80
Feb-82
Date
27
Show that Logs of Prices Are
Evolutionary
28
Log of the Import Price, Dark Northern Spring
14
Series: LNDNSJ
Sample 1975:09 1981:10
Observations 74
12
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
8
6
4
2
Jarque-Bera
Probability
5.209513
5.262690
5.525453
4.828314
0.219149
-0.176652
1.557133
6.803957
0.033307
0
4.9
5.0
5.1
5.2
5.3
5.4
5.5
29
30
31
Conclude
• Log of Import Price, lndnsj, is evolutionary
32
Log of Pacific Export Price (lagged One) Plus Pacific Freight (lagged 2)
10
Series: LNDNSPJT
Sample 1975:11 1981:10
Observations 72
8
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
6
4
5.207131
5.258248
5.520781
4.858106
0.202780
-0.112281
1.619657
2
Jarque-Bera
Probability
5.867330
0.053202
0
4.9
5.0
5.1
5.2
5.3
5.4
5.5
33
34
35
Conclude
• Both the log of the import price and the log
of the Pacific export price (lagged one)
plus the Pacific Freight Rate (lagged two)
are evolutionary, of order one.
– To be of order one, not higher, their
differences should be stationary, i.e. of order
zero.
– Unit root tests show this is the case
36
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
-0.05
-0.10
-0.10
-0.15
-0.15
76
77
78
79
80
81
76
77
78
DLNSJ
79
80
81
80
81
DLNSPJT
5.6
5.6
5.4
5.4
5.2
5.2
5.0
5.0
4.8
4.8
76
77
78
79
LNDNSJ
80
81
76
77
78
79
LNDNSPJT
37
Log of Price Ratio
38
ov
-7
Fe 5
b7
M 6
ay
-7
Au 6
g7
N 6
ov
-7
Fe 6
b7
M 7
ay
-7
Au 7
g7
N 7
ov
-7
Fe 7
b7
M 8
ay
-7
Au 8
g7
N 8
ov
-7
Fe 8
b7
M 9
ay
-7
Au 9
g7
N 9
ov
-7
Fe 9
b8
M 0
ay
-8
Au 0
g8
N 0
ov
-8
Fe 0
b8
M 1
ay
-8
Au 1
g81
N
ln
Log of Ratio of Import Price to the Sum of the Pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
Date
39
Log of Ratio of Import Price to the Sum of pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS
12
Series: LNRATIODNSPJT
Sample 1975:11 1981:10
Observations 72
10
8
6
4
2
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.002541
-0.001870
0.168084
-0.103687
0.046206
0.612068
5.041199
Jarque-Bera
Probability
16.99500
0.000204
0
-0.10
-0.05
0.00
0.05
0.10
0.15
40
41
42
43
44
45
46
47
Residulas from ARONE Model for lnratiodnspjt
12
Series: Residuals
Sample 1975:12 1981:10
Observations 71
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
8
6
4
2
0
-0.10
Jarque-Bera
Probability
-0.05
0.00
0.05
4.47E-15
-0.002010
0.123133
-0.095299
0.038372
0.409841
4.159879
5.967543
0.050602
0.10
48
Conclusions
• Log of ratio of import price to the export
price (lagged one) plus freight rate (lagged
two) is stationary and is modeled as an
autoregressive process of the first order
with mean zero and root 0.54
49
ov
-7
Fe 5
b7
M 6
ay
-7
Au 6
g7
N 6
ov
-7
Fe 6
b7
M 7
ay
-7
Au 7
g7
N 7
ov
-7
Fe 7
b7
M 8
ay
-7
Au 8
g7
N 8
ov
-7
Fe 8
b7
M 9
ay
-7
Au 9
g7
N 9
ov
-7
Fe 9
b8
M 0
ay
-8
Au 0
g8
N 0
ov
-8
Fe 0
b8
M 1
ay
-8
Au 1
g81
N
ln
Log of Ratio of Import Price to the Sum of the Pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
Date
50
How fast does any price differential
get arbitraged to zero?
•
•
•
•
•
•
Arone(t) = b*arone(t-1) + wn(t)
Arone(t-1) = b* arone(t-2) + wn(t-1)
Arone(t) = b[b* arone(t-2) + wn(t-1)] + wn(t)
Arone(t) = b2 *arone(t-2) + wn(t) + b*wn(t-1)
Arone(t+2) = b2 *arone(t) + wn(t+2) + b*wn(t+1)
Arone(t+u) = bu *arone(t) + wn(t+u) + b*wn(t+u1) + …
51
Half Life 1.1 month ~ 5 weeks
• Arone(t+u) = bu *arone(t) + wn(t+u) +
b*wn(t+u-1) + ….
• Et {Arone(t+u) = bu *arone(t) + wn(t+u) +
b*wn(t+u-1) + …}
• Et Arone(t+u) = bu *arone(t)
• Et Arone(t+u) /arone(t) = ½ = bu
• Ln [Et Arone(t+u) /arone(t)] = ln(1/2) = u*lnb
• - 0.693/lnb= -0.693/ln 0.54 = 1.1 = u
52
ARONE Model for Dark Northern Spring,Pacific, Root 0.54
100
Value
50
10
1
0
1
2
3
4
5
6
Months
53
Half Life: root =0.54, Arone(t) =100
Time =u
Arone(t+u)
bu
0
100
1
1
54
0.54
2
29.16
0.2916
3
15.7464
0.157464
54
Cointegration
• Logs of export price and import price
(lagged with freight lagged) are of order
one.
• Their difference is of order zero
• The long run relationship:
– Lndnsj = c + b*lndnspjt + e
– Where the residual is an estimate of price
differential over time
55
56
57
Error Correction VAR
dlndnsj(t)= aM*e(t-1) + wnM(t) + b11 dlndnsj(t-1) + c12 dlndnspjt(t-1)
dlndnspjt(t)= -ax*e(t-1) + wnx(t) + b21dlndnsj(t-1) + c22dlndnspjt(t-1)
aM and ax are speed of adjustment parameters of fractional
change in import and export prices to the fractional price
differential, i.e. e(t-1)
58
59
Significant speed of adjustment
Parameter. If lndnsj is
greater than the fitted value, i.e.
Greater than c + b*lndnspjt, the
Residual e(t) is positive, and next
Period, lndnspjt will increase to
Close the gap.
60
Johansen Cointegration Test
61
Johansen Table Summary
62
Diebold, Ch.4
p. 87
63
64
65
Null: No Cointegrating Equation
Reject null at 1% level
66
Impulse response functions
67
Impulse response functions
Res pons e to One S.D. Innov ations ± 2 S.E.
Response of DLNDNSJ to DLNDNSJ
Res pons e to One S.D. Innov ations ± 2 S.E.
Response of DLNDNSJ to DLNDNSPJT
0.05
Response of DLNDNSJ to DLNDNSJ
Response of DLNDNSJ to DLNDNSPJT
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.00
0.00
-0.01
-0.01
-0.02
-0.02
1
2
3
4
5
6
7
8
9
10
1
Response of DLNDNSPJT to DLNDNSJ
2
3
4
5
6
7
8
9
10
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.00
0.00
-0.01
-0.01
-0.02
-0.02
1
Response of DLNDNSPJT to DLNDNSPJT
2
3
4
5
6
7
8
9
10
1
Response of DLNDNSPJT to DLNDNSJ
0.020
0.020
0.020
0.015
0.015
0.015
0.015
0.010
0.010
0.010
0.010
0.005
0.005
0.005
0.005
0.000
0.000
0.000
0.000
-0.005
1
2
3
4
5
6
7
8
9
10
-0.005
1
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
Response of DLNDNSPJT to DLNDNSPJT
0.020
-0.005
2
-0.005
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
68
9
10
Variance decomposition
69
Error Correction Model: All the way
in one play
70
71
Error Correction VAR, One Lag
72
VEC Cont.
73
Johansen Summary Table
74
Johansen Test: No Data Trend, Intercept
75
Impulse Response Functions
Order: lndnsj, lndnspjt
Response to One S.D. Innovations ± 2 S.E.
Response of LNDNSJ to LNDNSJ
Response of LNDNSJ to LNDNSPJT
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
-0.02
-0.02
1
2
3
4
5
6
7
8
9
10
1
Response of LNDNSPJT to LNDNSJ
2
3
4
5
6
7
8
9
10
Response of LNDNSPJT to LNDNSPJT
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
-0.02
-0.02
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
76
Impulse Response Functions
Order: lndnspjt, lndnsj
Response to One S.D. Innovations ± 2 S.E.
Response of LNDNSJ to LNDNSJ
Response of LNDNSJ to LNDNSPJT
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
-0.02
-0.02
1
2
3
4
5
6
7
8
9
10
1
Response of LNDNSPJT to LNDNSJ
2
3
4
5
6
7
8
9
10
Response of LNDNSPJT to LNDNSPJT
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
-0.02
-0.02
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
77
Variance Decomposition
Order: lndnsj, lndnspjt
78
Table 3: log ratio of Import Price/
[Export Price (lag 1) + Freight (lag 2)]
DNSGJT DNSPJT
WWPJT HWGJT
AR(1)
0.63
0.54
0.49
0.39
MA(1)
0
0
0
0
Model Res
normal
normal
normal
normal
VEC
1 lag
1 lag
1 lag
1 lag
5 Mod. rank
0,0,1,0,2
1,1,1,1,2
1,1,1,1,2 1,1,1,1,1
Rank 1
5%
1% (3/4)
1% (3/4) 1% all
79
Table 4: log ratio of Import Price/
[Export Price (lag 1)]
DNSGJ
DNSPJ
WWPJ
HWGJ
AR(1)
0.74
0.71
0.73
0.64
MA(1)
0
0
0
0
Model Res
normal
normal
normal
normal
VEC
1 lag
1 lag
1 lag
1 lag
5 Mod. rank
0,0,0,0,2
0,0,0,0,0
0,0,0,0,0 0,0,1,0,2
Rank 1
5%
80