1 Econ 240C Power 17. 2 Outline The Law of One Price.

11

Econ 240CEcon 240C

Power 17Power 17

22

OutlineOutline

• The Law of One Price

33

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

44

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

55

The Law of One PriceThe Law of One Price

• The New Palgrave Dictionary of Money and Finance– Next slide

66

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.

77

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

88

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

99

Transport Cost in $/Metric TonTransport Cost in $/Metric Ton

• Freight rates are forward prices and are lagged two months

1010

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

1111

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

1212

# 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






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

1313

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%

1414

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

1515

Correlogram of Import Price, DNSJCorrelogram of Import Price, DNSJ

1616

Unit Root TestUnit Root Test

1717

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

1818

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

1919

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

2020

CorrelogramCorrelogram

2121

Unit Root TestUnit Root Test

2222

ARONE ModelARONE Model

2323

DiagnosticsDiagnostics

-20

-10

0

10

20

30

-20

0

20

40

1976 1977 1978 1979 1980 1981

Residual Actual Fitted

2424

Correlogram of ResidualsCorrelogram of Residuals

2525

Correlogram of Residuals SquaredCorrelogram of Residuals Squared

2626

ARCH-LM TestARCH-LM Test

2727

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

2828

Show that Logs of Prices Are Show that Logs of Prices Are EvolutionaryEvolutionary

2929

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


Log of the Import Price, Dark Northern Spring

3030

3131

3232

ConcludeConclude

• Log of Import Price, lndnsj, is evolutionary

3333

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


Log of Pacific Export Price (lagged One) Plus Pacific Freight (lagged 2)

3434

3535

3636

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

3737

-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

3838

Log of Price RatioLog of Price Ratio

3939

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

4040

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


Log of Ratio of Import Price to the Sum of pacific Export Price (lag1) Plus Pacific Freight (lag 2), DNS

4141

4242

4343

4444

4545

4646

4747

4848

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


Residulas from ARONE Model for lnratiodnspjt

4949

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

5050

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

5151

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) + …

5252

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

5353

ARONE Model for Dark Northern Spring,Pacific, Root 0.54

1

10

100

0 1 2 3 4 5 6

Months

Va

lue

50

5454

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

5555

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

5656

5757

5858

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)

5959

6060

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.

6161

Johansen Cointegration TestJohansen Cointegration Test

6262

Johansen Table SummaryJohansen Table Summary

6363

Diebold, Ch.4p. 87

6464

6565

6666

Null: No Cointegrating EquationNull: No Cointegrating Equation

Reject null at 1% level

6767

Impulse response functionsImpulse response functions

6868

-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


Impulse response functionsImpulse response functions

6969

Variance decompositionVariance decomposition

7070

Error Correction Model: All the way Error Correction Model: All the way in one playin one play

7171

7272

Error Correction VAR, One LagError Correction VAR, One Lag

7373

VEC Cont.VEC Cont.

7474

Johansen Summary TableJohansen Summary Table

7575

Johansen Test: No Data Trend, InterceptJohansen Test: No Data Trend, Intercept

7676

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


7777

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


7878

Variance DecompositionVariance DecompositionOrder: lndnsj, lndnspjtOrder: lndnsj, lndnspjt

7979

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

8080

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%

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1 Econ 240C Power 17. 2 Outline The Law of One Price.

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