Econ 240C
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Transcript of Econ 240C
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Econ 240CEcon 240C
Power 17Power 17
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OutlineOutline
• The Law of One Price
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Law of One Price: OutlineLaw 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
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Outline Cont.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
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The Law of One PriceThe Law of One Price
• The New Palgrave Dictionary of Money and Finance– Next slide
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The Law of One PriceThe 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.
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Commodity Trade IssuesCommodity 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
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Prices in $/metric tonPrices 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
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Transport Cost in $/Metric TonTransport Cost in $/Metric Ton
• Freight rates are forward prices and are lagged two months
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Time SeriesTime 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
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Time SeriesTime 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
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# 2 Hard # 2 Western #2 DNS # 2 Hard # 2 Western # 2 DNS # 2 DNS Freight Freight FFRDate Winter 13% White 14% Winter 13% White 13% 14% Gulf Pacific
Gulf Pacific Gulf Pacific75.09 207 182 217 176 162 187 201 15.74 13.28 6.24
75.1 207 182 220 178 161 184 199 17.22 13.78 5.82 World Wheat Statistics 197875.11 189 168 198 161 145 173 186 17.22 14.76 5.22 World Wheat Statistics 197875.12 172 156 197 151 141 178 178 16.73 14.51 5.2 World Wheat Statistics 197876.01 180 157 195 151 141 178 178 12.79 14.51 4.87 World Wheat Statistics 197876.02 192 164 201 166 151 186 186 14.76 13.53 4.77 World Wheat Statistics 197876.03 188 162 196 168 146 182 181 15.74 14.51 4.84 World Wheat Statistics 197876.04 177 148 185 156 138 162 175 15.74 13.78 4.82 World Wheat Statistics 197876.05 170 142 189 148 132 164 178 14.76 15.01 5.29 World Wheat Statistics 197876.06 174 147 193 154 134 173 181 15.01 15.01 5.48 World Wheat Statistics 197876.07 177 147 193 158 134 166 178 16.73 15.01 5.31 World Wheat Statistics 197876.08 158 140 174 137 126 142 156 15.74 15.25 5.29 World Wheat Statistics 197876.09 154 137 155 133 123 138 146 15.99 15.25 5.25 World Wheat Statistics 1978
76.1 154 127 146 119 112 131 134 16.24 15.25 5.03 World Wheat Statistics 197876.11 136 122 144 117 111 129 131 16.97 15.25 4.95 World Wheat Statistics 197876.12 131 115 135 115 105 136 125 16.24 14.27 4.65 World Wheat Statistics 197877.01 133 119 138 118 108 137 126 13.78 14.76 4.61 World Wheat Statistics 197877.02 134 121 141 118 111 140 130 13.78 14.76 4.68 World Wheat Statistics 197877.03 133 121 141 114 110 136 131 13.78 14.02 4.69 World Wheat Statistics 197877.04 124 121 141 109 110 122 129 13.78 14.02 4.73 World Wheat Statistics 197877.05 124 119 135 101 110 115 125 14.37 14.02 5.35 World Wheat Statistics 197877.06 116 112 129 97 104 107 117 13.78 14.02 5.39 World Wheat Statistics 197877.07 117 118 125 101 108 104 115 13.78 13.78 5.42 World Wheat Statistics 197877.08 114 120 125 98 108 102 115 14.27 13.78 5.9 World Wheat Statistics 197877.09 122 118 130 104 105 112 119 13.78 13.53 6.14 World Wheat Statistics 1978
77.1 128 115 134 108 105 117 123 14.02 13.53 6.47 World Wheat Statistics 197877.11 133 121 140 117 109 125 127 14.51 13.53 6.51 World Wheat Statistics 197877.12 135 124 140 118 111 134 127 15.01 14.76 6.56 World Wheat Statistics 197878.01 136 130 143 120 117 137 131 15.25 14.76 6.7 World Wheat Statistics 198178.02 138 136 146 122 124 137 134 14.51 13.53 6.78 World Wheat Statistics 198178.03 139 138 142 127 128 138 134 14.76 13.53 6.79 World Wheat Statistics 198178.04 150 148 142 135 136 145 140 14.76 13.53 6.89 World Wheat Statistics 198178.05 150 146 152 129 135 144 143 19.68 13.78 7.36 World Wheat Statistics 198178.06 147 146 152 130 136 139 143 20.66 14.02 7.6 World Wheat Statistics 198178.07 147 155 154 130 141 132 139 18.7 14.27 7.81 World Wheat Statistics 198178.08 149 150 149 130 139 131 138 19.14 14.27 8.04 World Wheat Statistics 198178.09 154 152 152 134 141 138 142 22.63 14.51 8.45 World Wheat Statistics 1981
78.1 154 152 152 139 141 143 144 22.14 15.25 8.96 World Wheat Statistics 198178.11 162 156 163 151 141 149 148 22.14 16.24 9.76 World Wheat Statistics 198178.12 164 159 166 138 140 153 150 22.14 17.47 10.03 World Wheat Statistics 198179.01 163 157 166 140 140 153 149 21.89 17.47 10.07 World Wheat Statistics 198179.02 165 153 166 144 137 153 151 21.89 17.47 10.06 World Wheat Statistics 198179.03 165 155 168 143 139 150 155 24.11 18.2 10.09 World Wheat Statistics 198179.04 167 157 168 141 139 146 152 24.11 18.2 10.01 World Wheat Statistics 198179.05 185 167 185 146 147 155 161 24.11 19.19 10.24 World Wheat Statistics 198179.06 203 193 212 164 166 174 184 28.54 27.55 10.29 World Wheat Statistics 198179.07 217 210 228 179 178 189 193 36.41 29.52 10.47 World Wheat Statistics 198179.08 218 200 214 173 171 180 185 34.44 28.63 10.94 World Wheat Statistics 198179.09 214 197 219 180 163 185 187 34.44 28.78 11.43 World Wheat Statistics 1981
79.1 221 194 223 183 161 188 189 34.44 29.47 13.77 World Wheat Statistics 198179.11 223 194 221 183 158 189 186 34.44 29.47 13.18 World Wheat Statistics 198179.12 222 194 220 185 157 194 187 34.44 29.52 13.78 World Wheat Statistics 198180.01 213 194 215 179 161 191 181 34.44 29.52 13.82 World Wheat Statistics 198180.02 217 196 219 176 163 189 184 34.44 29.03 14.13 World Wheat Statistics 198180.03 216 194 220 169 158 180 182 34.44 30.01 17.19 World Wheat Statistics 198180.04 210 190 211 158 155 166 176 43.3 30.01 17.61 World Wheat Statistics 198180.05 212 182 221 163 148 176 188 43.3 32.47 10.98 World Wheat Statistics 198180.06 212 180 224 159 148 174 190 37.39 32.47 9.47 World Wheat Statistics 198180.07 219 190 240 168 156 195 207 37.15 31.83 9.03 World Wheat Statistics 198180.08 229 192 239 175 158 192 205 37.88 31.83 9.61 World Wheat Statistics 198180.09 230 193 244 183 162 197 212 37.88 31.83 10.87 World Wheat Statistics 1981
80.1 239 203 249 192 171 204 218 37.88 31.83 12.81 World Wheat Statistics 198180.11 252 214 251 200 179 212 218 39.85 26.08 15.85 World Wheat Statistics 198180.12 232 211 245 188 171 212 206 39.85 29.77 18.9 World Wheat Statistics 198181.01 244 210 246 193 172 218 210 39.85 30.01 19.08 World Wheat Statistics 198281.02 234 210 242 185 172 216 207 39.85 30.01 15.93 World Wheat Statistics 198281.03 225 200 237 177 166 196 203 36.41 29.77 14.7 World Wheat Statistics 198281.04 225 201 249 183 169 191 214 36.41 29.77 15.72 World Wheat Statistics 198281.05 214 197 244 176 166 187 213 34.44 29.77 18.52 World Wheat Statistics 198281.06 220 189 232 172 161 183 202 36.16 29.77 19.1 World Wheat Statistics 198281.07 217 192 221 173 160 177 192 36.16 28.78 19.04 World Wheat Statistics 198281.08 216 194 219 174 161 173 186 31.49 28.29 17.82 World Wheat Statistics 198281.09 212 193 217 175 159 176 183 22.14 28.29 15.87 World Wheat Statistics 1982
81.1 216 197 216 175 164 180 184 24.6 28.29 15.08 World Wheat Statistics 198281.11 0 0 0 0 0 0 0 0 0 0
Data
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Table Three: Pacific DNSTable 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%
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Trace of Import Price and Pacific Export Price (lagged one) Plus Pacific Freight Rate (lagged two), DNS, $/Metric Ton
0
50
100
150
200
250
300
Date
$/M
etr
ic T
on
DNSJ
DNSPJT
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Correlogram of Import Price, DNSJCorrelogram of Import Price, DNSJ
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Unit Root TestUnit Root Test
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Dark Northern Spring JapanDark 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
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Trace of Import Price Minus the Sum of Pacific Export Price (lag1) Plus Pacific Freight Rate (lag 2), DNS, $/Metric Ton
-30
-20
-10
0
10
20
30
40
Date
$/M
etr
ic T
on
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0
2
4
6
8
10
12
14
-20 -10 0 10 20 30
Series: JAPMMINXSample 1975:11 1981:10Observations 72
Mean 0.181250Median -0.370000Maximum 32.80000Minimum -19.01000Std. Dev. 8.761377Skewness 0.923579Kurtosis 5.306363
Jarque-Bera 26.19391Probability 0.000002
Japan DNS Export price Minus Sum of Import Price (-1) +freight(-2)
HistogramHistogram
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CorrelogramCorrelogram
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Unit Root TestUnit Root Test
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ARONE ModelARONE Model
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DiagnosticsDiagnostics
-20
-10
0
10
20
30
-20
0
20
40
1976 1977 1978 1979 1980 1981
Residual Actual Fitted
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Correlogram of ResidualsCorrelogram of Residuals
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Correlogram of Residuals SquaredCorrelogram of Residuals Squared
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ARCH-LM TestARCH-LM Test
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Trace of Logarithms of Import Price and the Pacific Export Price (lag1) Plus Pacific Freight Rate (lag 2), DNS
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
Apr-75 Aug-76 Jan-78 May-79 Oct-80 Feb-82
Date
ln
lndnsj
lndnspjt
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Show that Logs of Prices Are Show that Logs of Prices Are EvolutionaryEvolutionary
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0
2
4
6
8
10
12
14
4.9 5.0 5.1 5.2 5.3 5.4 5.5
Series: LNDNSJSample 1975:09 1981:10Observations 74
Mean 5.209513Median 5.262690Maximum 5.525453Minimum 4.828314Std. Dev. 0.219149Skewness -0.176652Kurtosis 1.557133
Jarque-Bera 6.803957Probability 0.033307
Log of the Import Price, Dark Northern Spring
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ConcludeConclude
• Log of Import Price, lndnsj, is evolutionary
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0
2
4
6
8
10
4.9 5.0 5.1 5.2 5.3 5.4 5.5
Series: LNDNSPJTSample 1975:11 1981:10Observations 72
Mean 5.207131Median 5.258248Maximum 5.520781Minimum 4.858106Std. Dev. 0.202780Skewness -0.112281Kurtosis 1.619657
Jarque-Bera 5.867330Probability 0.053202
Log of Pacific Export Price (lagged One) Plus Pacific Freight (lagged 2)
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ConcludeConclude• 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
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-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
76 77 78 79 80 81
DLNSJ
4.8
5.0
5.2
5.4
5.6
76 77 78 79 80 81
LNDNSJ
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
76 77 78 79 80 81
DLNSPJT
4.8
5.0
5.2
5.4
5.6
76 77 78 79 80 81
LNDNSPJT
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Log of Price RatioLog of Price Ratio
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Log of Ratio of Import Price to the Sum of the Pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Date
ln
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0
2
4
6
8
10
12
-0.10 -0.05 0.00 0.05 0.10 0.15
Series: LNRATIODNSPJTSample 1975:11 1981:10Observations 72
Mean -0.002541Median -0.001870Maximum 0.168084Minimum -0.103687Std. Dev. 0.046206Skewness 0.612068Kurtosis 5.041199
Jarque-Bera 16.99500Probability 0.000204
Log of Ratio of Import Price to the Sum of pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS
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0
2
4
6
8
10
12
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 1975:12 1981:10Observations 71
Mean 4.47E-15Median -0.002010Maximum 0.123133Minimum -0.095299Std. Dev. 0.038372Skewness 0.409841Kurtosis 4.159879
Jarque-Bera 5.967543Probability 0.050602
Residulas from ARONE Model for lnratiodnspjt
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ConclusionsConclusions• 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
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Log of Ratio of Import Price to the Sum of the Pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Date
ln
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How fast does any price differential How fast does any price differential get arbitraged to zero?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+u-1) + …
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Half Life 1.1 month ~ 5 weeksHalf 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
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ARONE Model for Dark Northern Spring,Pacific, Root 0.54
1
10
100
0 1 2 3 4 5 6
Months
Va
lue
50
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Half Life: root =0.54, Arone(t) =100Half 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
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CointegrationCointegration• 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
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Error Correction VARError 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)
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Significant speed of adjustment Parameter. If lndnsj is greater than the fitted value, i.e.Greater than c + b*lndnspjt, theResidual e(t) is positive, and nextPeriod, lndnspjt will increase toClose the gap.
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Johansen Cointegration TestJohansen Cointegration Test
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Johansen Table SummaryJohansen Table Summary
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Diebold, Ch.4p. 87
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Null: No Cointegrating EquationNull: No Cointegrating Equation
Reject null at 1% level
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Impulse response functionsImpulse response functions
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-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSJ to DLNDNSJ
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSJ to DLNDNSPJT
-0.005
0.000
0.005
0.010
0.015
0.020
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSPJT to DLNDNSJ
-0.005
0.000
0.005
0.010
0.015
0.020
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSPJT to DLNDNSPJT
Response to One S.D. Innovations ± 2 S.E.
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSJ to DLNDNSJ
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSJ to DLNDNSPJT
-0.005
0.000
0.005
0.010
0.015
0.020
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSPJT to DLNDNSJ
-0.005
0.000
0.005
0.010
0.015
0.020
1 2 3 4 5 6 7 8 9 10
Response of DLNDNSPJT to DLNDNSPJT
Response to One S.D. Innovations ± 2 S.E.
Impulse response functionsImpulse response functions
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Variance decompositionVariance decomposition
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Error Correction Model: All the way Error Correction Model: All the way in one playin one play
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Error Correction VAR, One LagError Correction VAR, One Lag
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VEC Cont.VEC Cont.
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Johansen Summary TableJohansen Summary Table
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Johansen Test: No Data Trend, InterceptJohansen Test: No Data Trend, Intercept
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Impulse Response FunctionsImpulse Response FunctionsOrder: lndnsj, lndnspjtOrder: lndnsj, lndnspjt
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSJ to LNDNSJ
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSJ to LNDNSPJT
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSPJT to LNDNSJ
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSPJT to LNDNSPJT
Response to One S.D. Innovations ± 2 S.E.
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Impulse Response FunctionsImpulse Response FunctionsOrder: lndnspjt, lndnsjOrder: lndnspjt, lndnsj
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSJ to LNDNSJ
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSJ to LNDNSPJT
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSPJT to LNDNSJ
-0.02
0.00
0.02
0.04
0.06
0.08
1 2 3 4 5 6 7 8 9 10
Response of LNDNSPJT to LNDNSPJT
Response to One S.D. Innovations ± 2 S.E.
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Variance DecompositionVariance DecompositionOrder: lndnsj, lndnspjtOrder: lndnsj, lndnspjt
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Table 3: log ratio of Import Price/ Table 3: log ratio of Import Price/ [Export Price (lag 1) + Freight (lag 2)][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
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Table 4: log ratio of Import Price/ Table 4: log ratio of Import Price/ [Export Price (lag 1)][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%