1 Econ 240C Power 17

2 Outline The Law of One Price

3 Law of One Price: Outline Definition: slides 5-6 Applied to wheat trade: slides: slides 7-9 Time Series Notation: slides Data and Traces: slides Show that Import Price, DNSPJ, is Evolutionary: slides Show that Import Price Minus Sum of Export Price (-1) + Freight Rate (-2) is stationary: slides

4 Outline Cont. Show that logs of import & export prices are evolutionary of order one: slides Show that log of price ratio is stationary: slides –Speed of convergence: slides Cointegration: slides 55 - –Long run equilibrium relationship between log of import price and log of export price (with freight and lags): slides –VAR speed of adjustment model: slide 58-60

5 The Law of One Price The New Palgrave Dictionary of Money and Finance –Next slide

6 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.

7 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

8 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

9 Transport Cost in $/Metric Ton Freight rates are forward prices and are lagged two months

10 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

11 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

12 Data

13 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%

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15 Correlogram of Import Price, DNSJ

16 Unit Root Test

17 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

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19Histogram

20Correlogram

21 Unit Root Test

22 ARONE Model

23Diagnostics

24 Correlogram of Residuals

25 Correlogram of Residuals Squared

26 ARCH-LM Test

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28 Show that Logs of Prices Are Evolutionary

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32 Conclude Log of Import Price, lndnsj, is evolutionary

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36 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

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38 Log of Price Ratio

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49 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

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51 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) = b 2 *arone(t-2) + wn(t) + b*wn(t-1) Arone(t+2) = b 2 *arone(t) + wn(t+2) + b*wn(t+1) Arone(t+u) = b u *arone(t) + wn(t+u) + b*wn(t+u- 1) + …

52 Half Life 1.1 month ~ 5 weeks Arone(t+u) = b u *arone(t) + wn(t+u) + b*wn(t+u-1) + …. E t {Arone(t+u) = b u *arone(t) + wn(t+u) + b*wn(t+u-1) + …} E t Arone(t+u) = b u *arone(t) E t Arone(t+u) /arone(t) = ½ = b u Ln [E t Arone(t+u) /arone(t)] = ln(1/2) = u*lnb /lnb= /ln 0.54 = 1.1 = u

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54 Half Life: root =0.54, Arone(t) =100 Time =uArone(t+u)bubu

55 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

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58 Error Correction VAR dlndnsj(t)= a M *e(t-1) + wn M (t) + b 11 dlndnsj(t-1) + c 12 dlndnspjt(t-1) dlndnspjt(t)= -a x *e(t-1) + wn x (t) + b 21 dlndnsj(t-1) + c 22 dlndnspjt(t-1) a M and a x 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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60 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.

61 Johansen Cointegration Test

62 Johansen Table Summary

63 Diebold, Ch.4 p. 87

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66 Null: No Cointegrating Equation Reject null at 1% level

67 Impulse response functions

68 Impulse response functions

69 Variance decomposition

70 Error Correction Model: All the way in one play

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72 Error Correction VAR, One Lag

73 VEC Cont.

74 Johansen Summary Table

75 Johansen Test: No Data Trend, Intercept

76 Impulse Response Functions Order: lndnsj, lndnspjt

77 Impulse Response Functions Order: lndnspjt, lndnsj

78 Variance Decomposition Order: lndnsj, lndnspjt

79 Table 3: log ratio of Import Price/ [Export Price (lag 1) + Freight (lag 2)] DNSGJTDNSPJTWWPJTHWGJT AR(1) MA(1)0000 Model Resnormal VEC1 lag 5 Mod. rank0,0,1,0,21,1,1,1,2 1,1,1,1,1 Rank 15%1% (3/4) 1% all

80 Table 4: log ratio of Import Price/ [Export Price (lag 1)] DNSGJDNSPJWWPJHWGJ AR(1) MA(1)0000 Model Resnormal VEC1 lag 5 Mod. rank0,0,0,0,20,0,0,0,0 0,0,1,0,2 Rank 15%