Cointegration Analysis of Commodity Prices: Much Ado about the Wrong Thing? Mindy L. Mallory and Sergio H. Lence September 17, 2010

Cointegration Analysis of Commodity Prices: Much Ado about the Wrong Thing? Mindy L. Mallory and Sergio H. Lence September 17, 2010

Cointegration Analysis, Commodity Prices What is cointegration analysis?

9 8 7 6 5 4 3 Ethanol Corn 2 1 0

Cointegration Analysis, Commodity Prices How do we use cointegration analysis? Spatial Arbitrage Law of One Price Trade Conditions Purchasing Power Parity Farm-Retail, Retail-Farm Price Transmission

Cointegration Analysis, Commodity Prices Two Basic Approaches Error Correction Type Models Include an equilibrium term in a VAR type model Residual Based Models Test for unit root in the residuals of a regression of one variable on the others

Cointegration Analysis, Commodity Prices Residual Based Models Test for unit root in the residuals of a regression of one variable on the others x = β + β x + βx + ε 1t 0 1 2t 3t t

Cointegration Analysis, Commodity Prices Residual Based Models Test for unit root in the residuals of a regression of one variable on the others x = β + β x + βx + ε 1t 0 1 2t 3t t p = β + β p + βp + ε corn ethanol natural gas t 0 1 t t t

Cointegration Analysis, Commodity Prices Residual Based Models Test for unit root in the residuals of a regression of one variable on the others x = β + β x + βx + ε 1t 0 1 2t 3t t p = β + β p + β p + ε corn ethanol natural gas t 0 1 t t t

Cointegration Analysis, Commodity Prices Error Correction Type Models Include an equilibrium term in a VAR type model k 1 αβ t ' t 1 i= 1 i t i t X = X + Γ X + e

Cointegration Analysis, Commodity Prices Error Correction Type Models c c c p t p t p t 1 p = αβ ' p + Γ p + e ng ng ng p t p t p t 1 eth eth eth t t t 1 t

Cointegration Analysis, Commodity Prices Error Correction Type Models c c c p t p t p t 1 p = αβ ' p + Γ p + e ng ng ng p t p t p t 1 eth eth eth t t t 1 t Johansen test: estimates the rank of the matrix αβ

Cointegration Analysis, Commodity Prices Previous Research Unit Root testing (therefore residual based tests) Negative Moving Average (NMA) in the error term small sample size distortion and low power Schwert 1989 notes that DF type tests do not converge to stated asymptotic distribution when NMA present

Cointegration Analysis, Commodity Prices Previous Research Error Correction Johansen framework Also note poor small sample performance when NMA is present Cheung and Lai, Toda, Johansen (2002), Ahn and Reinsel, and Reimers Engle (2000)

Cointegration Analysis, Commodity Prices Our study: How prevalent are NMA errors in commodity price data? How bad is the size distortion-low power? How much data does one need to overcome the small sample issues?

Cointegration Analysis, Commodity Prices How prevalent are NMA errors in commodity price data?

Commodity Model: φ 1 φ 2 φ 3 φ 4 θ 1 θ 2 θ 3 θ 4 R 2 Alfalfa meal 1.73 *** 0.99 *** 1.70 *** 0.96 *** 0.087 (160.34) ( 92.25) ( 71.71) (40.51) Barley 0.91 *** 0.63 *** 0.13 ** 0.69 *** 0.37 *** 0.145 (6.17) (-7.56) (2.35) ( 4.66) (4.08) Butter 1.32 *** 0.95 *** 1.30 *** 0.99 *** 0.050 ( (60.02) ( 43.31) 129.25) (99.11) Chicken, 0.23 0.23 0.043 whole (0.94) ( 0.09) Chicken 0.72 0.67 0.003 breast ( 1.49) (1.31) Chicken Legs 1.60 *** 0.87 *** 1.58 *** 0.81 0.058 (18.90) ( 11.14) ( 16.00) (8.09) Corn 0.13 0.10 0.17 *** 0.10 0.11 0.048 ( 0.27) (0.22) (3.42) ( 0.95) ( 1.62) Corn gluten 1.04 *** 0.06 0.45 ** 0.14 ** 1.02 *** 0.21 0.68 *** 0.106 feed (6.14) (0.19) ( 2.06) ( 2.29) ( 6.15) ( 0.74) (4.13) Corn gluten 0.87 *** 0.07 0.004 0.10 0.97 *** 0.062 meal (13.54) ( 0.98) (0.058) (1.65) ( 31.40) Corn meal 0.35 0.54 *** 0.037 ( 1.51) (2.57)

Commodity Model: φ 1 φ 2 φ 3 φ 4 θ 1 θ 2 θ 3 θ 4 R 2 Cottonseed 0.23 0.18 0.06 0.13 * 0.15 0.143 meal ( 0.51) ( 1.03) ( 0.55) ( 1.91) ( 0.33) Crude oil 1.27 *** 0.23 ** 0.18 *** 0.90 *** 0.244 (15.88) ( 2.23) ( 2.69) ( 16.06) Diesel 1.10 *** 0.1 0.12 * 0.93 *** 0.073 (13.55) ( 1.11) ( 1.91) ( 16.52) Eggs 1.62 *** 0.81 0.11 1.79 *** 1.11 ** 0.08 0.123 (5.59) ( 1.60) ( 0.39) ( 6.14) (2.22) ( 0.27) Feeder cattle 0.90 *** 0.98 *** 0.87 *** 0.98 *** 0.040 ( 55.08) ( 60.26) (55.42) (63.59) Gasoline 1.90 *** 1.53 *** 0.58 *** 0.25 *** 1.73 *** 1.00 *** 0.198 (31.53) ( 11.99) (4.46) ( 3.94) (64.41) (64.41) Ground beef 0.02 0.17 0.033 ( 0.07) ( 0.58) High fructose 1.67 *** 1.81 *** 1.50 *** 0.87 *** 1.55 *** 1.56 *** 1.39 *** 0.82 *** 0.266 corn syrup (28.04) ( 15.98) (13.91) ( 16.17) ( 24.84) (15.01) ( 14.47) (15.02)

Commodity Model: φ 1 φ 2 φ 3 φ 4 θ 1 θ 2 θ 3 θ 4 R 2 Lean hogs 0.47 ** 0.02 0.12 ** 0.52 ** 0.026 (2.21) ( 0.46) ( 2.22) ( 2.43) Live cattle 0.11 0.04 0.06 0.20 *** 0.19 *** 0.085 (0.52) ( 0.21) ( 1.34) ( 4.32) ( 2.98) Meat and 0.99 *** 0.34 *** 0.30 *** 0.13 ** 0.90 *** 0.077 bone meal ( 9.18) ( 4.46) ( 3.38) ( 2.33) (9.31) Milk, 2.59 *** 3.11 *** 1.94 *** 0.62 *** 2.12 *** 2.27 *** 1.43 *** 0.57 *** 0.520 farmgate (23.35) ( 11.76) (7.82) ( 6.87) ( 19.78) (11.17) ( 8.90) (8.99) Milk, 0.07 0.11 0.21 ** 0.56 0.214 wholesale ( 0.15) (0.44) ( 1.94) (1.07) Oats 0.04 0.29 0.058 ( 0.15) (1.18) Pork bellies 0.80 *** 0.15 *** 0.09 * 0.96 *** 0.046 (15.86) (2.63) ( 1.91) ( 40.57) Potatoes 1.38 *** 0.49 *** 0.96 *** 0.256 (24.31) ( 8.92) ( 36.63) Sorghum 0.37 0.23 0.063 (0.21) (1.24)

Commodity Model: φ 1 φ 2 φ 3 φ 4 θ 1 θ 2 θ 3 θ 4 R 2 Soybean 0.85 *** 0.09 0.12 ** 0.94 *** 0.043 (15.94) (1.59) ( 2.46) ( 30.63) Soy meal 0.13 0.06 0.20 *** 0.12 0.041 (0.58) (1.43) ( 4.43) ( 0.52) Soy oil 0.11 0.23 ** 0.17 *** 0.36 0.137 (0.35) (2.31) ( 2.97) ( 1.12) Sugar 0.97 *** 0.95 *** 0.51 *** 0.01 0.43 *** 0.89 *** 0.541 (15.49) ( 13.83) (8.66) ( 0.19) ( 11.33) (26.50) Urea 0.69 *** 0.83 *** 0.34 *** 0.39 0.95 *** 0.64 *** 0.85 *** 0.222 ( 9.99) (8.00) (4.11) ( 6.83) (16.80) ( 6.55) ( 15.26) Wheat 0.00 0.31 0.088 (0.003) (1.86) * Wheat bran 1.06 *** 0.17 0.67 *** 1.28 *** 0.03 0.88 *** 0.27 *** 0.160 (4.64) (0.42) ( 2.95) ( 5.72) ( 0.08) (3.22) ( 4.34) Wheat flour 1.00 *** 1.08 *** 0.09 0.05 ( 156) (19.61) (1.63)

Cointegration Analysis, Commodity Prices How prevalent are NMA errors in commodity price data? 20 out of 35 have NMA significantly diff from zero at 5% 26 out of 35 have NMA point estimates

Cointegration Analysis, Commodity Prices Monte Carlo Investigation (following Banerjee et al 1986 and Haug 1996) X X = ε, ε = ρε + u 1, t 2, t t t t 1 t X + X =Ψ, Ψ =Ψ + v, v = w + θ w 1, t 2, t t t t 1 t t t t 1 2 u iid t 0 σ, u σ uw N 2 w = t 0 σuw σ w

Cointegration Analysis, Commodity Prices 2. Monte Carlo Investigation k 1 X = αβ ' X + Γ X + e t t 1 i= 1 i t i t

Table 4: Empirical Daily Data from a Bivariate System. Parameterization b Sample Length Empirical Size Based on Standard 5% Asymptotic Critical Value b Actual 5% Critical Value Trace Test Max Test Trace Test Max Test θ = 0, k = 2 1 month 0.18 0.12 24.38 20.24 3 months 0.08 0.04 19.58 16.15 6 months 0.06 0.03 18.49 15.35 2 years 0.05 0.03 18.22 15.34 4 years 0.05 0.03 18.27 15.17 10 years 0.05 0.03 18.44 15.33 20 years 0.05 0.03 18.16 15.00 100 years 0.05 0.03 18.22 15.11 θ = 0.8, k = 2 1 month 0.20 0.14 25.07 20.89 3 months 0.07 0.04 19.33 16.02 6 months 0.06 0.03 18.63 15.44 2 years 0.05 0.03 18.11 14.95 4 years 0.04 0.02 17.72 14.69 10 years 0.04 0.02 17.74 14.88 20 years 0.04 0.02 17.83 14.84 100 years 0.04 0.02 17.68 14.62

Table 4: Empirical Daily Data from a Bivariate System. Parameterization b Sample Length Empirical Size Based on Standard 5% Asymptotic Critical Value b Actual 5% Critical Value Trace Test Max Test Trace Test Max Test θ = 0.8, k = 2 1 month 0.22 0.15 25.13 20.92 3 months 0.39 0.30 28.37 25.06 6 months 0.60 0.52 35.48 32.03 2 years 0.73 0.67 53.01 49.91 4 years 0.75 0.70 58.54 55.68 10 years 0.77 0.71 63.71 61.04 20 years 0.77 0.72 66.14 62.99 100 years 0.77 0.72 68.08 65.29

Table 6. Size-Adjusted Finite-Sample Power under the Alternative of Cointegration for Daily Data from a Bivariate System. Persistence a Sample θ = 0, k = 2 b θ = 0.8, k = 2 b θ = 0.8, k = 2 b θ = 0.8, k = 4 b Length Trace Trace Max Trace Max Trace Max Test Max Test Test Test Test Test Test Test ρ = 0.90 1 month 0.05 0.05 0.05 0.06 0.05 0.05 0.05 0.05 3 months 0.07 0.07 0.07 0.07 0.07 0.04 0.07 0.05 6 months 0.16 0.14 0.15 0.15 0.09 0.04 0.10 0.04 2 years 0.99 0.99 0.99 1 0.22 0.03 0.57 0.32 4 years 1 1 1 1 0.71 0.25 1 1 10 years 1 1 1 1 1 1 1 1 20 years 1 1 1 1 1 1 1 1 100 years 1 1 1 1 1 1 1 1 ρ = 0.95 1 month 0.05 0.05 0.06 0.06 0.05 0.05 0.05 0.05 3 months 0.05 0.05 0.06 0.06 0.06 0.04 0.05 0.05 6 months 0.09 0.08 0.08 0.08 0.05 0.04 0.06 0.04 2 years 0.55 0.54 0.54 0.60 0.09 0.03 0.18 0.05 4 years 1 1 1 1 0.17 0.02 0.61 0.32 10 years 1 1 1 1 0.90 0.62 1 1 20 years 1 1 1 1 1 1 1 1 100 years 1 1 1 1 1 1 1 1

Figure 11: Johansen s Trace and Max Statistics for Varying Sample Lengths, Based on Twenty Years of Simulated Monthly Data Under the Null Hypothesis of No Cointegration NMA in the Error Term.

Figure 10: Johansen s Trace and Max Statistics for Varying Sample Lengths, Based on Twenty Years of Simulated Monthly Data Under the Null Hypothesis of No Cointegration No NMA in the Error Term.

Figure 1: Johansen s Trace and Max Statistics for Varying Sample Lengths of Crack Spread Price Data.

Figure 2: Johansen s Trace and Max Statistics for Varying Sample Lengths of Wheat, Flour, and Bran Price Data.

Figure 3: Johansen s Trace and Max Statistics for Varying Sample Lengths of Soybean Crush Price Data.

Figure 4: Johansen s Trace and Max Statistics for Varying Sample Lengths of Cattle Crush Price Data.

Figure 5: Johansen s Trace and Max Statistics for Varying Sample Lengths of Lean Hogs and Pork Bellies Price Data.

Thank You! Presented in the Fall 2010 seminar series of the Department of Agricultural and Consumer Economics University of Illinois at Urbana-Champaign SEPTEMBER 17, 2010 Mindy L. Mallory and Sergio H. Lence 2010

Commodity Group H 0 a Trace Test Maximum Eigenvalue Test Test Critical Value b Test Critical Value b Statistic 5% 1% Statistic 5% 1% Chicken: whole, breast, r 0 56.49 29.68 35.65 48.57 20.97 25.52 and legs r 1 7.91 15.41 20.04 5.22 14.07 18.63 r 2 2.70 3.76 6.65 2.70 3.76 6.65 Corn gluten feed and r 0 19.69 15.41 20.04 16.76 14.07 18.63 corn gluten meal r 1 2.93 3.76 6.65 2.93 3.76 6.65 Crack spread r 0 109.79 29.68 35.65 64.93 20.97 25.52 r 1 44.87 15.41 20.04 39.09 14.07 18.63 r 2 5.77 3.76 6.65 5.77 3.76 6.65 Corn, sorghum, barley, r 0 80.61 47.21 54.46 40.14 27.07 32.24 and oats r 1 40.47 29.68 35.65 23.32 20.97 25.52 r 2 17.15 15.41 20.04 10.73 14.07 18.63 r 3 6.41 3.76 6.65 6.42 3.76 6.65

Commodity Group H 0 a Trace Test Maximum Eigenvalue Test Test Critical Value b Test Critical Value b Statistic 5% 1% Statistic 5% 1% Lean hogs and r 0 42.46 15.41 20.04 27.11 14.07 18.63 pork bellies r 1 15.35 3.76 6.65 15.35 3.76 6.65 Live cattle, feeder r 0 92.15 29.68 35.65 71.37 20.97 25.52 cattle, and corn r 1 20.78 15.41 20.04 15.74 14.07 18.63 r 2 5.05 3.76 6.65 5.05 3.76 6.65 Milk and butter r 0 45.20 15.41 20.04 43.85 14.07 18.63 r 1 1.35 3.76 6.65 1.35 3.76 6.65 Soybean, soybean oil, r 0 84.35 29.68 35.65 52.04 20.97 25.52 and soybean meal r 1 32.31 15.41 20.04 24.19 14.07 18.63 r 2 8.11 3.76 6.65 8.11 3.76 6.65 Wheat, flour, and bran r 0 100.43 29.68 35.65 63.18 20.97 25.52 r 1 37.25 15.41 20.04 34.45 14.07 18.63 r 2 2.78 3.76 6.65 2.77 3.76 6.65

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 15 20 25 30