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1 Friedman[1983] Friedman ( ) Dockner[1992] closed-loop Theorem 2 Theorem 4 Dockner ( ) 31

2 Kinoshita, Suzuki and Kaiser [2002] () 1) 2) () VAR 32

3 () Mueller[1986], Mueller ed. [1990] Mueller[1986] OLS AR(1) AR(1) Geroski[1990] Gerosky Mueller ed.[1990] AR(1) Odagiri and Yamawaki[1990] Maruyama and Odagiri[2002] [1999] AR(1) Geroski [1991] AR(1) 1 ( ) VAR VARMA VAR 4 2. VAR 2.1 Sargent[1987](Chapter XI) VAR (a) p 1t = A 01t A 11 q 1t A 12 q 2t + u 1t, p 2t = A 02t A 22 q 2t A 21 q 1t + u 2t, p it t i q it t i u it t 33

4 u it MA(1) u it MA(1) A 01t A 02t (b) i ( ) C it = c 1it q it + (c 2i /2)(q it q it 1 ) 2 + s it q it. s it MA(1) 3) (c) (u it, s it ) (d) i b i (e) i t t κ ji q jt / q it q jt+ j / q it = 0 t closed-loop t open-loop i (1) V it = Σ j=0 b j i {p t+ j q it+ j C it+ j } V it / q it+ j = 0 (2) 1 b 1 c 21 q 1t+ j+1 q 1t+ j (2A 11 + A 12κ21 + c 21 + b 1 c 21 ) + c 21 q 1t+ j 1 = A 01t+ j + c 11t+ j + A 12 q 2t+ j + u 1t+ j + s 1t+ j, 2 b 2 c 22 q 2t+ j+1 q 2t+1 (2A 22 + A 21κ12 + c 22 + b 1 c 22 ) + c 22 q 2t+ j 1 = A 02t+ j + c 12t+ j + A 21 q 2t+ j + u 2t+ j + s 2t+ j. MA(1) 2.2 VAR (2) VARMA (2) t π it (i = 1, 2) (q 1t, q 2t ) 1 i (3) π it = π it + π 1 it (q 1t q 1t ) + π it 2 (q 2t q 2t ), i = 1, 2 π j it π it / q jt qit=qit, i=1,2. q 1t q 2t t i 1 G i t j q it+ j+1 = (2 G i ) q it+ j1. q it+ j+1 (2) π it (3) q 1t q 2t (4) q 1t q 1t = (π 2 2t π + 1t π 2 1t π + 2t )/Ω q 2t q 2t = (π 1 1t π + 2t π 1 2t π + 1t )/Ω π + it π it π it Ω π 1 1t π 2 2t π 2 1 1t π 2t (4) (2) 2 SVARMA structural VARMA SVARMA 34

5 (5) π 1t+ j+1 = const + α 1 π 2t+ j+1 + β 11 π 1t+ j + β 12 π 1t+ j 1 + γ 11 π 2t+ j + γ 12 π 2t+ j 1 + MA(1) π 2t+ j+1 = const + α 2 π 1t+ j+1 + β 21 π 2t+ j + β 22 π 2t+ j 1 + γ 21 π 1t+ j + γ 22 π 1t+ j 1 + MA(1) SVARMA (5) MA(1) (u it ) (s it ) MA(1) SVARMA π 1t+ j+1 π 2t+ j+1 VARMA (6) π 1t+ j+1 = v 10 + v 11 π 1t+ j+ v 12 π 1t+ j 1 + v 13 π 2t+ j + v 14 π 2t+ j 1 + ε 1t+ j+1 + θ 1 ε 1t+ j, π 2t+ j+1 = v 20 + v 21 π 2t+ j + v 22 π 2t+ j 1 + v 23 π 1t+ j + v 24 π 1t+ j 1 + ε 2t+ j+1 + θ 2 ε 2t+ j, ε it+ j+1 + θ i ε it+ j MA(1) 10 v in i=1,2. n=0,1,,4 1 1 b 1 b 2 b b 1 = b 2 v i2 = 1/b, v i4 = 0 (i = 1, 2) (7) Π 1t+ j+1 = v 10 + v 11 π 1t+ j + v 13 π 2t+ j + ε 1t+ j+1, Π 2t+ j+1 = v 20 + v 21 π 2t+ j + v 23 π 1t+ j + ε 2t+ j+1, Π it+ j+1 π it+ j+1 v i2 π it+ j 1 θ 1 ε it+ j, ε it i, i, d, Gaussian 1 v 10 H 20 /H 21 (A 02 c 12 )/(b 2 c 22 H 21 ) H 20 /(b 2 H 21 ) + A 21 H 10 /(b 2 c 22 H 21 ) + (2A 22 + c 22 + b 2 c22 + A 21 κ 12 )H 20 /(b 2 c 22 H 21 ) (1/(H 21 H))[H 22 {H 21 (H 10 + (A 01 c 11 )/(b 1 c 21 ) + H 10 /b 1 (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )H 10 /(b 1 c 21 ) A 12 H 20 /(b 1 c 21 )) H 11 (H 20 +(A 02 c 12 )/(b 2 c 22 )+ H 20 /b 2 A 21 H 10 /(b 1 c 21 )) H 11 (H 20 + (A 02 c 12 )/(b 2 c 22 ) + H 20 /b 2 A 21 H 10 /(b 2 c 22 ) (2A 22 + c 22 + b 2 c 22 + A 21 κ 12 )H 20 /(b 2 c 22 ))}], v 11 A 21 H 11 /(b 2 c 22 H 21 ) + (2A 22 + c 22 + b 2 c 22 + A 21 κ 12 )H 21 /(b 2 c 22 H 21 ) (1/H 21 H)[H 22 {H 21 ( (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )H11/(b 1 c 21 ) A 12 H 21 /(b 1 c 21 )) H 11 ( A 21 H 11 /(b 2 c 22 ) (2A 22 + c 22 + b 2 c22 + A 21 κ 12 )H 21 /(b 2 c 22 ))}], v 12 1/b 2 H 22 H 11 (1/b 1 1/b 2 )/H, v 13 A 21 H 12 /(b 2 c 22 H 21 )+(2A 22 +c 22 +b 2 c22+ A 21 κ 12 )H 22 /(b 2 c 22 H 21 ) (1/H 21 H)[H 22 {H 21 ( (2A 11 +c 21 +b 1 c 21 + A 12 κ 21 )H 12 /(b 1 c 21 ) A 12 H 22 /(b 1 c 21 )) H 11 ( A 21 H 12 /(b 2 c 22 ) (2A 22 + c 22 + b 2 c22 + A 21 κ 12 )H22/(b 2 c 22 ))}], v 14 H 22 /(b 2 H 21 ) H 22 (H 12 H 21 /b 1 H 11 H 22 /b 2 )/(H 21 H), v 20 (1/H)[H 21 {H 10 + (A 01 c 11 )/(b 1 c 21 ) + H 10 /b 1 (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )H 10 /(b 1 c 21 ) A 12 H 20 /(b 1 c 21 )} H 11 {H 20 + (A 02 c 12 )/(b 2 c 22 ) + H 20 /b 2 A 21 H 10 /(b 2 c 22 ) (2A 22 + c 22 + b 2 c22 + A 21 κ 12 )H 20 /(b 2 c 22 )}], v 21 (1/H)[H 21 { (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )H 12 /(b 1 c 21 ) A 12 H 22 /(b 1 c 21 )} H 11 { A 21 H 12 /(b 2 c 22 ) (2A 22 + c 22 + b 2 c22 + A 21 κ 12 )H 22 /(b 2 c 22 )}], v 22 (H 21 H 12 /b 1 H 11 H 22 /b 2 )/H, 35

6 v 23 [H 21 { (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )H 11 /(b 1 c 21 ) A 12 H 21 /(b 1 c 21 )} H 11 { A 21 H 11 /(b 2 c 22 ) (2A 22 + c 22 + b 2 c 22 + A 21 κ 12 )H 21 /(b 2 c 22 )}], v 24 H 21 H 11 (1/b 1 1/b 2 )/H, H H 11 H 22 H 12 H 21, H 10 [e π2 {e π2 q 1 ( c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 22 )/(q 2 p 2 2 )) + e π2 q 2 (A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p 2 ) π 2 }]/{ c 12 /(q 2 p 2 )+A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/(q 2 p 2 2 )}+D/E D { 0.5c 22 (1 G 2 )/p 2 + A 22 ( c c 22 (1 G 2 )q 2 )/p 2 2 }[ e π2 { c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/(q 2 p 2 2 )}{e π1 q 2 A 12 ( c c 21 (1 G 1 )q 1 )/p 2 1 +e π1 q 1 (A 11 ( c 11 o.5c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 ) π 1 }+e π1 {A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{e π2 q 1 ( c 12 /(q 2 p 2 )+ A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 22 )/(q 2 p 2 2 )) + e π2 q 2 (A 22 ( c c 22 (1 G 2 )q 2 /p c 22 (1 G 2 )/p 2 ) π 2 )}] E { c 12 /(p 2 q 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/q 2 p 2 2 }[ e π1 π2 A 12 { c c 21 (1 G 1 )q 1 }(1/p 2 1 ){ c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/q 2 p 2 2 } e π1 π2 {A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p2}], H 11 [ 0.5c 22 (1 G 2 )/p 2 + A 22 { c c 22 (1 G 2 )q 2 }]( e π2 )[ c 12 /(q 2 p 2 ) + A 21 { c c 22 (1 G 2 )q 2 2 }/(q 2 p 2 2 )]/F F [ c 12 /(q 2 p 2 )+A 21 { c 12 q 1 0.5c 22 (1 G 2 )q 2 2 }/q2p2 2 ](e π1 π2 )[A 12 { c c 21 (1 G 1 )q 1 }/p 2 1 ][ c 12 /(q 2 p 2 )+ A 21 { c 12 q 1 0.5c 22 (1 G 2 )q 2 2 }/q 2 p 2 2 ] (e π1 π2 )[A 11 { c 11 q 1 0.5c 22 (1 G 1 )q 2 2 }/p c 21 (1 G 1 )/p 1 ][A 22 { c c 22 (1 G 2 )q 2 }/p c 22 (1 G 2 /p 2 ], H 12 e π2 /[ c 12 /(q 2 p 2 ) + A 21 { c 12 q 1 0.5c 22 (1 G 2 )q 2 2 }/(q 2 p 2 2 )}] + [ 0.5c 22 (1 G 2 )/p 2 + A 22 { c c 22 (1 G 2 )q 2 }/p 2 2 ]( e π1 )[A 11 { c c 21 (1 G 1 )q 1 }/p c 21 (1 G 1 )/p 1 ]/L, L [ c 12 /(q 2 p 2 ) + A 21 { c 12 q 1 0.5c 22 (1 G 2 )q 2 2 }/(q 2 p2 2 )](e π1 π2 )[A 12 { c 11 q 1 0.5c 21 (1 G 1 )p 2 1 }/p 2 1 { c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )} {A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p 2 }], H 20 [ e π2 { c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/(q 2 p 2 2 )}{e π1 q 2 (A 12 ( c c 21 (1 G 1 )q 1 )/p 2 1 ) + e π1 q 1 (A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 ) π 1 }+e π1 {A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{e π2 q 1 ( c 12 /(q 2 p 2 ) + A21( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/(q 2 p 2 2 )) + e π2 q 2 (A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p 2 ) π 2 }]/M, M (e π1 π2 )A 12 { c c 21 (1 G 1 )q 1 }/p 2 1 { c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/q 2 p 2 2 } (e π1 π2 ){A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p 2 }, H 21 e π2 { c 12 /(q 2 p 2 ) + A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/q 2 p 2 2 }/N, N (e π1 π2 ){A 12 ( c c 21 (1 G 1 )q 1 )/p 1 2 }{ c 12 /(q 2 p 2 ) + A 21 ( c 21 q 1 0.5c 22 (1 G 2 )q 2 2 )/(q 2 p 2 2 }} (e π1 π2 ) {A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p 2 }, 36

7 H 22 e π1 {A 11 ( c (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }/R, R (e π1 π2 )A 12 { c c 21 (1 G 1 )q 1 }(1/p 2 1 ){ c 12 /(q 2 p 2 )+A 21 ( c 12 q 1 0.5c 22 (1 G 2 )q 2 2 )/(q 2 p 2 2 )} (e π1 π2 ) {A 11 ( c c 21 (1 G 1 )q 1 )/p c 21 (1 G 1 )/p 1 }{A 22 ( c c 22 (1 G 2 )q 2 )/p c 22 (1 G 2 )/p 2 }, 3. 2 ( ) ) CAPM 1971 = (60 ) + β 60 β β 1977 = 10 +β ( ) β 1 ( ) () (5) v i j (6) v 10 v 20 37

8 Chen 3 ADF F ADF F 4 (6) VAR F VAR F VAR, VAMA 1 3 univariate AR, univariate ARMA VAR VAMA univariate AR, univariate ARMA (5) (5) (5) VARπ =MA VAR 2 L( ) 2 2 MA MA(1) 1 2 π (5) (5) π =(VAR) 1 MA, (VAR) 1 VAR L 4 VAR (5) univariate ARMA(4,3) 4.1 3) Lumsdaine and Papell [1997] 1 Zivot and Andrews[1992] ) ADF T(ρ µ 1) T ρ µ ADF ADF 1 2 (k) t 1.6 5) 4 i T(ρ µ 1) T(ρ µ 1)/(1 Σ k π ik ) 2 38

9 2 T(ρ µ 1) (1) 1 (2) 2 1% 2.5% 5% 10% 1% 2.5% 5% 10% ADF ADF T(ρ µ 1) = (k = 0) % (k = 0) 4.7% (k = 1) 0.05% ADF 8.40 (k = 0) % (k = 1) 0.4% (k = 1) 0.07% Chen Chen Chen and Tiao[1990] ARMA ARIMA ARMA ARIMA Chen 2 1 Chen

10 ADF F ADF ADF π it = +ρπ it 1 + α 1 π it 1 + α 2 π it 2 + α 3 π it 3 + ε it ε it i.i.d.gaussian π ip p = 3 ADF α i F F F OLS F α VAR F (7) VAR b = 0.9 Π 1t+ j+1 (6) MA(1) MA univaliate ARMA(4, 3) Π 1t+ j+1 = v 11 π 1t+ j + v 13 π 2t+ j + ε 1t+ j+1, Π 2t+ j+1 = v 21 π 2t+ j + v 23 π 1t+ j + ε 2t+ j+1, VAR F 1968 v v v v 21 v 23 = 0 F 1994 v 23 F Chen AR(2) ( ) 40

11 6) 5. (6) (7) ARMA(4,3) MA(3) i.i.d. ARMA(4,3) (6) univariate ARMA(4,3) b 1 = b 2 = 0.9 π it+ j+1 v i2 π it+ j 1 v i2 = 1/0.9, 1 (6) (6) (7) 1 2 VAR OLS SUR (7) (7) t t (61,62,63 64 ) π it ARMA(4,3) i.i.d. MA(3) (-1/0.9)π it 2 π it Π it v (5.8549) ( ) (1.7512) v (2.5751) v (2.8921) 73 v (1.9858) 73 v ( ) 90 v (4.4542) SBIC R v (9.4862) v (1.3804) ( ) ( ) ( ) 90 v (2.8879) 98 v ( ) 66 v ( ) 70 v ( ) 95 v ( ) 98 v (3.2457) SBIC R v i0 v i1 v i3 v i0 v i1 v i3 3 v i0, v i1, v i3 v i0, v i1, v i3 1 (8) v i j = v i j, i = 1, 2, j = 0, 1, 3, 3 (6) (7) v i2 = 1/0.9 v i4 = 0 v i j ( j = 0, 1, 3) 41

12 t π i = π it π i 3 1 π i p t q t (9) t i = {pit q it c 1it q it c 2it (1 G i ) 2 q 2 it /2}/(p it q it ), i = 1, 2, 1 q i+t t j q it+ j+1 = (2 G i )q it+ j (2) q 1t = (A 02 c 12 )/(A 21 G 1 ) [b 2 c 22 {1 + G 2 2 /b 2 G 2 (2A 22 + c 22 + b 2 c 22 + A 21 κ 12 )/(b 2 c 22 )}][ A 21 (A 01 c 11 )G 1 /(b 1 b 2 c 21 c 22 ) (A 02 c 12 ){1 + G 2 1 /b 1 G 1 (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )}/(b 2 c 22 )]/S, q 2t = A 21 G 1 [ A 21 (A 01 c 11 )G 1 /(b 1 b 2 c 21 c 22 ) (A 02 c 12 ){1 + G 2 1 /b 1 G 1 (2A 11 + c 21 + b 1 c 21 + A 12 κ 21 )/(b 1 c 21 )}/(b 2 c 22 )]/S, S A 21 G 1 [A 12 A 21 G 1 G 2 /(b 1 b 2 c 21 c 22 ) {1+G 2 2 /b 2 G 2 (2A 22 +c 22 +b 2 c 22 + A 21 κ 12 )/(b 2 c 22 )}{1+G 2 1 /b 1 G 1 (2A 11 + c 21 + b 1 c 21 + A 12 κ 12 )/(b 1 c 21 )}], i q it = (x it )/p it (10) (x it ) = p it q it, i = 1, 2, A 01, A 02, A 11, A 12, A 21, A 22, c 11, c 12, c 21, c 22, p 1t, p 2t, κ 21, κ 12 ) 10 (8),(9),(10) (a) 12 c 11 = c 12 = 1 12 (8),(9),(10) 2 A 11 = 1, A 12 = 1, A 21 = 1, A 22 = 1, c 21 = 1, c 22 = 1, p 1t = 100, p 2t = 100, κ 21 = 0, κ 12 = 0 Gauss=Newton A(A 11, A 12, A 21, A 22 ) = {0 10 }, c 2 (c 21, c 22 ) = {0 10 }, κ(κ 21, κ 12 ) = {0 10 }, p (p 1t, p 2t ) = {1, 100, 1000} (11) A i1 A i2, c 2i 0, c 2i 0, p it 0, A 11 A 12 A 21 A 22 () A 11 A 21 /H A 11 < A 12 42

13 A A A A c c p 1t p 2t κ κ closed-loop b i 2 b i 0.9 b i Federal Reserve Board s Quarterly Model b i c 11 = c 12 = 1 A 0i MA 3 Worthington[1990] 2 ()

14 (et) 1 y t y t = y t 1 + e t, 225 y t y t (y 0 ) 225 y t ADF T(ρ µ 1) T(ρ µ 1) 7 t t = x t t Sales t t D () T T () 1 ln(sales t x t ) = (ln(sales t 1 ) x t 1 ) (ln(sales t 2 ) x t 2 ), x t = D (73 1) D(98 1) T(62 = 1) T (73 = 1) T (90 = 1) T (98 = 1), ln(sales t x t ) = (ln(sales t 1 ) x t 1 ) ln(sales t 2 ) x t 2 ), x t = D (66 1) D (95 1) T(62 = 1) T(66 = 1) T (73 = 1)) T(90 = 1) T(95 = 1) (1999) ( 10 ) NTT. Bresnahan, T. F. (1982) The Oligopoly Solution Concept is Identified, Economic Letters 10, pp Chen, C. and G. C. Tiao (1990) Random Level-Shift Time Series Models, ARIMA Approximation, and Level- Shift Detection, Journal of Business and Economic Statistics, vol.8, no.1, Engelbert, J. D. (1992) A Dynamic Theory of Conjectural Variations, The Journal of Industrial Economics, Vol.XL, No.4, December Friedman, J. W. (1983) Oligopoly and the Theory of Games, North-Holland, Amsterdam. Geroski, P. A. (1990) Modeling Persistent Profitability, in Muller, D. C., ed. The Dynamic of Company Profits: An International Comparison. (1991), Market Dynamics And Entry, Brackwell: Cambridge, MA. Kinoshita, J., N. Suzuki and H. M. Kaiser (2002) Explaining Pricing Conduct in a Product-Differentiated Oligopolistic Market: An Empirical Application of a price Conjectural Variations Model, Agribusiness, vol.18, No.4. Lumsdaine, R. L. and D. H. Papell (1997) Multiple Trend Breaks and the Unit-Root Hypothesis, The Review of Economics and Statistics, vol.79, Maruyama, N. and H. Odagiri (2002) Does the Persistence of Profits Persists?: A Study of Company Profits in Japan , International Journal of Industrial Organization, vol.20, Muller, D. C., ed. (1986) Profits in the Long Run, Cambridge University Press: Cambriege, MA. (1990) The Dynamic of Company Profits: An International Comparison, WZB-Publication: Wissenschaftszentrum, Berlin. Odagiri, H. and H. Yamawaki (1990) The Persistence of Profits in Japan, in Mueller, D. C.,ed. The Dynamic of 44

15 Company Profits: An International Comparison. Roberts, M. L. and Samuelson (1988) An Empirical Analysis of Dynamic, Nonprice Competition in an Oligopolistic Industry, Rand journal of Economics, Vol.19, No.2. Seldon, B. J., S. Banerjee and R. G. Boyd (1993) Advertizing Conjectures and the Nature of Advertizing Competition in an Oligopoly, Managerial and Decision Economics, Vol.14, Sergeant, T. J. (1987) Macroeconomic Theory, Academic Press, San Diego:CA. Worthington, P. R. (1990) Strategic Investment and Conjectural Variations, International Journal of Industrial Organization, Vol.8, Zivot, E. and D. W. K. Andrews (1992) Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit- Root Hypothesis, Journal of Business and Economic Statistics, vol.10, no.3. 45

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