Chickens vs. Eggs: Replicating Thurman and Fisher (1988) by Arianto A. Patunru Department of Economics, University of Indonesia 2004

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1 Chicens vs. Eggs: Relicaing Thurman and Fisher (988) by Ariano A. Paunru Dearmen of Economics, Universiy of Indonesia Inroducion This exercise lays ou he rocedure for esing Granger Causaliy as discussed in he celebraed aer of Thurman and Fisher (American Journal of Agriculural Economics, 988) eniled Chicens, Eggs, and Causaliy, or Which Came Firs?. This is insired by Roger Koener of Universiy of Illinois. 2. Granger Causaliy, Coinegraion and Uni Roos 2.. Daa and Variables The daa used in his ar was originally rovided by Thurman and adjused by Koener. I is available from R websie. I consiss of annual ime series for he U.S. egg roducion in millions of dozens and U.S.D.A esimae of he U.S. chicen oulaion. Table : Descriive saisics Variables No. of Arih. Sd. Min Max Median Obs. mean Eggs Chicens Year Tesing he Granger Causaliy: Which Came Firs, Chicen or Egg? The following general equaions wih =, 2, 3 and 4 were used for esing Granger- Causaliy o reroduce he comarable es saisics wih Thurman and Fisher (988) s wor. Eggs = µ + α Eggs + β Chicens (2.2.) i = = Chicens = µ + α Chicens + β Eggs (2.2.2) i = = Did he Chicen Come Firs? To es wheher Chicens do no Granger-cause Eggs, we firs esimaed he four varians (wih differen no. of lags in he RHS equaion) of equaion Then we carried ou he F-es as follows

2 Ho: β =... = β = 0 (Chicens do no Granger cause Eggs) Ha: A leas one of β is no zero (Chicens Granger cause Eggs) The resuls of F-es under Ho are resened as follows. Table 2. F-es Resuls under Ho: Chicens do no Granger cause Eggs = no. of lags df F-saisic -value Adj. R 2 of he regression (, 50) (2, 47) (3, 44) (4, 4) We see ha he above F-saisic resuls failed o rejec our null hyohesis a 5% level in all four varians of regression model Did he Eggs Come Firs? To es wheher Eggs do no Granger cause Chicens, we firs esimaed he four varians (wih differen no. of lags in he RHS equaion) of equaion Then we carried ou he F-es as follows: Ho: β =... = β = 0 (Eggs do no Granger cause Chicens) Ha: A leas one of β is no zero (Eggs Granger cause Chicens) The F-es resuls resened as follows: Table 3. F-es Resuls under Ho: Eggs do no Granger cause Chicens = no. of lags df F-saisic -value Adj. R 2 of he regression (, 50) (2, 47) (3, 44) (4, 4) Our F-es resul above rovided he emirical evidence agains our null hyohesis Eggs do no Granger cause chicens. In summary, our conclusion from Granger causaliy es resuls found o be consisen wih he resuls shown by Thurman and Fisher (988) desie he difference beween heir and our calculaed F-es saisics. The difference may be due o he difference in he number of observaions. Observing he aerns of arial residuals may hel o exlain furher he raher sriing resuls above. We grah hese aerns as follows: 2

3 Grah. Parial Residual Plos Chicen Egg Egg() Parial Residual Plo A Chicen() Parial Residual Plo B Chicen Egg Egg(2) Parial Residual Plo C Chicen(2) Parial Residual Plo D However, we see ha he arial residual los above do no seem o suor he hyohesis of Eggs Grainger-cause chicens. Therefore, we need o roceed on conducing necessary ess on he series used. 3. Tes for Uni Roos In order o be able o confirm our conjecure in he revious secion, we need o conduc ess for uni roos on he wo series. Before doing so, we observe he aern of he series agains ime as follows: Grah 2. Chicen Series chic year 3

4 Grah 3. Egg Series 5836 egg year The grahs above show ha here was no sysemaic aern of chicen series, while here seems o be a relaively sysemaic aern of egg series, es. afer he firs quarer of he observaions. However, we should no rely solely on such rough grahs. Therefore, we need o es he non-saionariy formally. Tha is, we emloyed he augmened Dicy- Fuller (ADF) ess on he boh series. 3. Augmened Dicy-Fuller Equaions for Chicens The hree equaions below reresen our esing model for (i) random wal behavior, (ii) random wal wih drif, and (iii) random wal wih drif and rend, resecively. Chicens Chicens Chicens = ( ρ ) Chicens Chicens (3..) = ( ρ ) Chicens Chicens = Consan + (3..2) = Consan + = ( ρ ) Chicens + γrend Chicens = (3..3) 3.2 Augmened Dicy-Fuller equaions for Eggs Similarly, he equaions for esing he egg series are: Eggs Eggs = ( ρ ) Eggs Eggs (3.2.) = ( ρ ) Eggs Eggs = Consan + (3.2.2) = 4

5 Eggs ( ρ ) Eggs + γrend Eggs = Consan + (3.2.3) Before esimaing he ADF equaions from 3.. o 3.2.3, we firs deermine he value of. For his urose we arbirarily seleced =, 2, 3 and 4 and esimaed four varians of hose equaions. Then we carried ou he F-es o es he join Ho: δ =... = δ = 0 agains Ha: A leas one of he δ is no zero. In case ha he F-es rovided he emirical evidence agains Ho, we hen comued he Schwarz s Informaion Crierion (SIC) using equaion below for hose esimaed equaions. = SIC j = log σˆ 2 + ( / n) log n (3.2.4) j j where n is he number of observaions, is he number of arameers, and residual sum of squares esimaed from OLS divided by n. 2 ˆσ is he Our resuls for SIC for differen lag srucure in equaion 3.. o are summarized as follows: Table 5. SIC values for varians of Equaions (3..) o (3.2.3) Equaion SIC values corresonding o ADF equaions No. of No consan, No rend lags no rend () Chicen Eggs Wih consan and rend As a decision rule we chose number of lags ha minimized he SIC for each equaion. If we follow his rule for SIC value we should use = in ADF equaions for Chicen and Eggs (secion 3. and 3.2) for he urose of esing uni roos. However, for edagogical urose, we esimaed ADF equaions for all four differen values, ranging from o Hyohesis Tesing for Nonsaionariy We firs esimaed he ADF equaions from 3.. o wih =, 2, 3, 4 and esed he following hyohesis in each equaion o deermine wheher here is a uni roo in he given ime series. Ho: ρ = (There is uni roo in he series) Ha: ρ < (There is no uni roo in he series) The es resuls and corresonding es saisics are resened as follows: 5

6 Table:6 Augmened Dicey-Fuller Tess for Chicen Series Ho: Uni-roo resens Ha: Uni-roo does no resen Cons Tes % Criical 5% Criical Lag Trend Saisic Value Value -value Rejec Ho Z() c No c No No Z(2) c No c No No Z(3) c No c No No Z(4) c No c No No Based on he resuls resened in Table 6, we failed o rejec our Ho, hence confirmed ha ime series observaions for Chicens is nonsaionary. Table: 7 Augmened Dicey-Fuller Tess for Eggs Series Ho: Uni-roo resens Ha: Uni-roo does no resen Cons Tes % Criical 5% Criical Lag Trend Saisic Value Value -value Rejec Ho Z() c No c No No Z(2) c No c No No Z(3) c No c No No Z(4) c No c No No In he same fashion, based on he resuls resened in Table 7, we also failed o rejec he associaed Ho, hence confirmed ha ime series observaions for Eggs is also nonsaionary. Thus we confirmed ha boh Chicens and Eggs series are nonsaionary. Tha is, hey exhibi he resence of uni-roo or I() rocess. This imlies a violaion o he classical iid condiions for residuals in equaion (2..) and (2..2). In oher words, our rior conclusion ha Eggs Grainger-cause chicens is somehow weaened, in he sense ha our F-saisics in Table 2 migh have been oversaed. To re-chec he resul we can imose firs difference on boh series in order o mae hem follow I(0) rocess. Therefore, our new models for causaliy ess are: 6

7 Eggs α i Eggs + β Chicens + = = = µ + ε (3.3.) Chicens α i Chicens + β Eggs + = = = µ + ε (3.3.2) The associaed resuls are resened in he following able: Table 8. F-es Resuls under Ho: Chicens do no Granger cause Eggs (Saionary) Adj. R 2 of he = no. of lags df F-saisic -value regression (,49) (2,46) (3,43) (4,40) Table 9. F-es Resuls under Ho: Eggs do no Granger cause Chicens (Saionary) Adj. R 2 of he = no. of lags df F-saisic -value regression (,49) (2,46) (3,43) (4,40) I urns ou ha he resuls in Table 8 and Table 9 rovide suor o our conclusion ha Eggs Grainger-cause Chicens. In addiion, afer correcing for nonsaionariy, we found ha even he model wih lag-one of eggs in chicens-eggs equaion is significan Tess for Coinegraion We confirmed from Table 6 and 7 ha boh variables Chicens and Eggs are nonsaionary meaning ha hey showed he resence of uni roos. To es wheher hose series are coinegraed, we firs observe he relaionshis beween he residuals and ime and beween he residual wih is lag as follows: 7

8 Grah 4. Residual from Chicen Equaion Agains Time Residuals year Residual vs Time Grah 5. Residual vs Lagged Residual Residuals L.residual Residual vs Lagged Residual There are no many inferences we can ge from Grah 4. On he oher hand, Grah 5 suggess a wea osiive relaion beween residual and is lag. Nex, we esimaed he following long run equilibrium equaion (3.4.) for Chicen-Egg rocesses. Chicens = β + β 2Eggs + ν 2 and ~ iidn ( 0, σ ) ν (3.4.) I T Then we esimaed he following augmened Engle-Graenger (AEG) equaions for esing he resence of uni roos in residuals of equaion (3.4). ( ρ ) ˆ ν ˆ ν ε ˆ ν = + (3.4.2) = ( ρ ) ˆ ν ˆ ν ε ˆ ν = Consan + + (3.4.3) = The rocedure is similar o ADF. The only difference here is ha we imose he es on residual series. This is why his es is also called residual-based es. 8

9 ( ρ ) ˆ ν + γ ˆ ν ˆ ν = Consan + rend (3.4.4) We esed he following hyohesis for he resence of uni roos in he above hree AEG equaions. Ho: ρ = (There is uni roo in he esimaed residuals of equaion 3.4.) Ha: ρ < (There is no uni roo in he esimaed residuals of equaion 3.4.) = Our Dicy-Fuller es saisics are resened as follows: Table 0. Augmened Dicey-Fuller Tess for Residuals Series Ho: Uni-roo resens Ha: Uni-roo does no resen Cons Tes % Criical 5% Criical Lag Trend Saisic Value Value -value Rejec Ho Z(0) c No c No Yes (5%) Z() c No c No No Z(2) c No c No Yes (5%) Z(3) c No c No Yes (5%) Z(4) c No c No Yes (5%) This imlies ha esimaed residuals from equaion 3.4. exhibis nonsaionary. We also found ha boh esimaed arameers for models wih rend and drif were saisically insignifican a 5% level in all hree equaions from Wih he exceion of cases wihou boh consan and rend, in general, we failed o rejec he null. This means, he eggs and chicens series were no coinegraed. 4. Noes You can do all he above rocedure on Saa in a rey sraighforward way. Beer ye, his is robably a good way o sar using R. Koener has made he daa available in R. 9