( 30 ) 30 4 5

1 4 1.1............................................... 4 1.............................................. 4 1..1.................................. 4 1....................................... 4 1..3..................................... 4 1..4 (......................... 5 1..5..................................... 5 1.3.............................................. 5 1.4............................... 7 1.5................................ 7 8.1 1.......................................... 8........................................... 9 3 Excel 11 3.1.................................. 11 3....................................... 11 3.3....................... 11 3.4........................................... 11 3.5........................................ 1 3.6............................................. 1 3.7.............................................. 1 3.8............................................. 1 3.9............................................... 13 4 15 4.1............................................... 15 4.1.1....................................... 15 4.1........................................ 15 4.1.3...................................... 16 4.1.4................................. 16 4.1.5 Conditional probability...................... 16 4.1.6...................................... 17 4.1.7...................................... 17 4.1.8 Bayes................................ 17 4. - -............................ 19 4..1.................................... 19 1

4........................................ 19 4..3...................................... 19 4..4................................. 19 4..5............................. 0 4..6.................................... 3 4..7 cumulative distribution function................. 3 4..8....................... 4 4..9 χ...................................... 4 4..10 Moment Generating Function................... 5 4..11...................................... 8 4..1.................................. 8 4..13 Chebyshev.................................. 30 4..14......................................... 31 4.3........................................ 3 4.3.1 Normal distribution.......................... 3 4.3. χ Chi-squared distribution........................ 3 4.3.3 F F-distribution.............................. 3 4.3.4 t t-distribution............................... 3 4.4..................................... 33 4.4.1......................................... 33 4.4....................................... 33 4.4.3......................................... 33 4.4.4.......................................... 33 4.5............................................. 34 4.5.1......................................... 34 4.5..................................... 34 4.5.3............................... 34 4.5.4......................................... 35 4.6............................................. 37 4.6.1 Testing Statistical Hypothesis................. 37 4.6................................... 39 5 45 5.1........................................ 45 5........................................... 45 5.3..................................... 45 5.4........................................... 46 5.5................................ 46 5.6.................................... 47 5.7......................................... 48 5.8 : t............................. 49

5.9.................................. 49 5.9.1......................................... 49 5.10........................................ 49 5.11.............................. 50 5.1........................................... 51 5.13........................................... 51 5.14...................................... 51 5.14.1............................... 51 5.14.......................................... 51 3

1 1.1 1. 1..1 Excel gretl 1.. 1 1..3 χ F t : 4

1..4 ( 1..5 1.3 5

HP WEB EXCEL gretl gretl Excel gretl 4 Wooldridge, Introducory Econometrics: A Modern Approach, South Westren Publishing Hill, Griffiths, and Lin, Principles of Econometrics, Wiley Stock and Watson, Introduction to Econometrics, Pearson.( ) Gujarati and Porter, Basic Econometrics, McGrow Hill 6

The Lay Tasting Tea 1.4 Excel ( ) Excel gretl R gretl R OX Scilab Octave Eviews STATA TSP Eviews TSP STATA Gauss MATLAB Excel gretl gretl 1.5 A1 A ( 3 ) R 7

.1 1 x 1, x,..., x n : n Mean n x = 1 n x i Variance σ x = 1 n ( n n ) (x i x) = 1 x i n x n Standard deviation σ x = σ x Coefficient of variation / cv x = σ x x Range Median 8

Mode Quartile 5 % 1 first quartile 50 % second quartile 75 % 3 third quartile / 0 1 x i = x i x σ x. (x 1, y 1 ), (x, y ),..., (x n, y n ) : n Covariance ( Cov(x, y) = 1 n n ) (x i x)(y i ȳ) = 1 x i y i n x ȳ n n Correlation Cor(x, y) = Cov(x, y) σ x σ y = 1 n n x i x σ x yi ȳ σ y ( 1 Cor(x, y) +1) 9

X, Y x i = X i X, y i = Y i Ȳ y i = βx i ˆβ xi y i xi y = i r = xi ˆβ = s Y r xi yi s X 3 Ŷ = ˆα + ˆβX + ˆγZ ˆβ Z Y X : r Y X.Z = r Y X r Y Z r XZ 1 rxz 1 ry Z X, Y Z r Y X.Z = r Y X Y Z Ŷ = ˆα + ˆβZ Y Z û = Y Ŷ Y Z r Y X.Z û = Y Ŷ = Y (ˆα + ˆβZ), ˆv = X ˆX = X (ˆγ + ˆδZ) rûˆv Y Ŷ R = r Y Ŷ = ( Ŷ i Ȳ ) (Yi Ȳ ) 10

3 Excel 3.1 3. =if(,a,b) a b 1) ) =IF(D>=60,, ) =IF(D>=80,,IF(D>=65,,IF(D>=50,, ))) 3.3 3.4 11

3.5 x = 1 n n x i s = 1 n 1 S = 1 n =average() n (x i x) =var() n (x i x) =varp() σ x = S =stdev() cor(x, y) = 1 n n (x i x)(y i ȳ) σ x σ y =correl() 3.6 Excel ( ) HP Excel 1) ) 3.7 m σ {x i } n 50 10 x i m σ = y i 50 10 {y i } n y i = 50 + x i m σ/10 x i 3.8 =frequency() Excel 1

3.9 0 9 x n ax n 1 + c (mod m) 1. (mod m) x n m ( ). a, c, m Excel =rand() [0, 1) 1. A1 =rand() return. F9 3. A1 A A1000 F9 [0,1] 0.1 1. B1 0.0. B1 B11 0.1 3. C1 =frequency($a$1:$a$1000,b1:$b$11) cf. 4. C1 C C11 5. D =C-C1 D D3 D11 6. D13 =sum(d:d11) 1000 7. D D11 8. 10 100 9. F9? Excel Add-In Excel Add-in 13

risk40t.xla : http://www.usfca.edu/ middleton/decision.htm Monte Carlo Simulation Add-In for Excel, Trial Version.40. Developed by Mike Middleton University of San Francisco [0, 1) U 1, U, U 3 X F (x) f(x) F (x) = P r{x x}, f(x) = d dx F (x) F (x) F 1 (x) F 1 (y) = inf {x : F (x) y} 0 y 1 X = F 1 (U) X F (x) P r{x x} = P r{f 1 (U) x} = P r{u F (x)} = F (x) 1989 14

4 4.1 trial sample space event 4.1.1 Ω E Ω 0 P (E) 1 P (Ω) = 1 A B = P (A B) = P (A) + P (B) A B : A B : A B = A B = A + B 4.1. A A A C P (A C ) = 1 p(a) A A C = P (A) + P (A C ) = P (Ω) = 1 P ( ) = 1 P (Ω) = 0 A B P (A) P (B) B = A + A C B P (B) P (A) = P (A C B) 0 P (A B) = P (A) + P (B) P (A B) A B = A B C + A B + A C B A = A B + A B C B = B A + B A C P (A B) P (A) P (B) = = P (A B) 15

4.1.3 1 3 4 5 6 1 3 4 5 6 7 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 1 X P (X = k) P (X = k) 0 k P (X = k) = 1. P k = P (X = k) = c k c X? 4.1.4 X = E(X) = 1 k= k P (X = k) = 1 36 + 3 36 + + 1 1 36 = 7 V ar(x) = E ( (X X) ) = 1 k= (k X) P (X = k) = = 10 36 5.833 σ = V ar(x) 5.8333.415 4.1.5 Conditional probability P (B A) = P (A B) P (A) or P (A B) = P (A) P (B A) 16

4.1.6 A, B independent P (B A) = P (B), P (A B) = P (A) A, B P (A B) = P (A) P (B). n r n r 1 A B 1 A B? 1.. 4.1.7 Ω A 1, A,... E Ω = A 1 + A + A i A j = E = E Ω = (A 1 E) + (A E) + P (E) = P (A i ) P (E A i ) i 4.1.8 Bayes Ω A 1, A,... E comment!! P (A i E) = P (A i E) P (E) = P (A i) P (E A i ) P (A i ) P (E A i ) P (A i E) = P (A i) P (E A i ) P (A i ) P (E A i ) i i P (A i ) P (E A i ) (i = 1,,...) P (A i E) P (A i ) P (A i E) 1. 3 A B C 50 % 30 % 0 % A B C 3 % 4 % 5 % (1) 1? () 1 C? 17

1 (1) # # # = A, B, C F F = (A F ) (B F ) (C F ) P (F ) = P (A F ) + P (B F ) + P (C F ) = P (A) P (F A) + P (B) P (F B) + P (C) P (F C) = 0.5 0.03 + 0.3 0.04 + 0. 0.05 = 0.037 () P (C F ) P (C F ) = P (c F ) P (F ) = P (C) P (F C) P (F ) = 0. 0.05 0.037 = 0.7. 30 % 80 % 5% 95 %? E A : : A C : P (A E) P (E A) = 0.3 P (E A C ) = 0.8 P (A) = 0.05 P (A C ) = 0.95 P (A E) P (A E) P (A E) = = P (E) P (E A) + P (E A C ) P (A) P (E A) = P (A) P (E A) + P (A C ) P (E A C ) 0.05 0.3 = 0.05 0.3 + 0.95 0.8 = 0.01935 3 U 1 U U 3 U 1 4 6 U 5 5 U 3 6 4 1 3 U 1 U 6 U 3 (1)? : 71/70 () U? : 0/53 18

4. - - discrete : continuous : 4..1 r 0 π 0 x x 0 π x = πr? 0 πr r πr + dx P {πr x π + dx} = 1 πr dx dx 0 P {x = πr} 0 1 0 4.. X Ω = {x 0, x 1,..., x n,...} 1 {p k ; k = 0, 1,,...} p k 0 p k = 1 p k = P (X = x k ) (k = 0, 1,,...) k {p k } X 4..3 X Ω = R(set of real number) f(x) f(x) 0, f(x)dx = 1 a < b P (a < X < b) = Ω b a f(x)dx f(x) X probability density function 4..4 X {p k } ϕ(x) x ϕ(k) p k X ϕ(x) E {ϕ(x)} k ϕ(x) = x E{X} = k k p k = X or µ 1 1 1 19

ϕ(x) = ax + b E{X} = ae{x} + b ϕ(x) = (x µ) E { (X µ) } = k (X µ) (k µ) p k = V ar(x) σ = V ar(x) i.e. V ar(x) = σ σ V ar(ax + b) = a V ar(x) V ar(x) = E(X ) (E(X)) X f(x) ϕ(x) x ϕ(x)f(x)dx X ϕ(x) E {ϕ(x)} Ω X X = E(X) = Ω x f(x)dx (= µ) σ = V ar(x) = E ( (x µ) ) = σ = V ar(x) Ω E{aX + b} = ae{x} + b V ar(ax + b) = a V ar(x) V ar(x) = E(X ) (E(X)) (x µ) f(x)dx 4..5 1. X p k = A λk k! k = 0, 1,,... (a) A (b) E(X) V ar(x). e x = k=0 x k k! (a) p k = A k=0 k=0 A = e λ λ k k! = A eλ = 1 e 0

(b) E(X) = E(X ) = k e k=0 k e k=0 k! = λ λ k 1 e λ (k 1)! = λ e λ = λ k=1 k! = λ λ k 1 eλ (1 + k 1) (k 1)! k=1 λ k 1 (k 1)! + λ e λ λ k (k )! = λ + λ λ λk = λ e λ k=1 λ λk V ar(x) = E(X ) (E(X)) = λ k= : Poisson. X p k = n C k p k (1 p) n k k = 0, 1,,... E(X) = k k=0 n 1 = np k 1=0 n! k!(n k)! pk (1 p) n k (n 1)! (k 1)! ((n 1) (k 1))! pk 1 (1 p) (n 1) (k 1) = np (1 + (1 p)) n 1 = np E(X ) = k n! k!(n k)! pk (1 p) n k k=0 n 1 = np = np = np k 1=0 n 1 k 1=0 n 1 k 1=0 k (n 1)! (k 1)! ((n 1) (k 1))! pk 1 (1 p) (n 1) (k 1) (1 + k 1) n 1 + n(n 1)p (n 1)! (k 1)! ((n 1) (k 1))! pk 1 (1 p) (n 1) (k 1) (n 1)! (k 1)! ((n 1) (k 1))! pk 1 (1 p) (n 1) (k 1) k =0 = np + n(n 1)p V ar(x) = np + n(n 1)p (np) = np(1 p) (n )! (k )! ((n ) (k ))! pk (1 p) (n ) (k ) 1. X f(x) = 1 (a < x < b) b a 0 others 1

X (a, b) Uniform distribution X Unif(a, b). X E(X) = E(X ) = b x a b a x 1 b a dx = a + b 1 b a dx = 1 3 (a + ab + b ) σ = V ar(x) = a + ab + b ( ) a + b = 3 σ = b a 3 { λ e λx (0 < x < + ) f(x) = (λ 0) 0 others (b a) 1 X λ exponential distribution E(X) = 1 λ V ar(x) = 1 λ 3. X f(x) = 1 ) (x µ) exp ( πσ σ ( < x <, < µ <, 0 < σ < ) X µ σ X N(µ, σ) E(X µ) = 1 ( + (x µ) exp 1 ( ) ) x µ dx = 0 πσ σ E(X) = µ V ar(x) = E ( (x µ) ) = 1 ( + ( ) ) (x µ) 1 x µ exp dx πσ σ = 1 { ( + (x µ) σ exp 1 ( ) )} x µ dx πσ σ ( = {[ σ 1 (x µ) exp 1 ( ) )] x µ + πσ σ ( + +σ exp 1 ( ) ) } x µ dx σ ( = σ 1 + exp 1 ( ) ) x µ dx = σ πσ σ f(x)

(a) x = µ (b) x = µ ± σ (c) lim f(x) = 0 x ± 4..6 X N(µ, σ ) z = x µ Z N(0, 1 ) σ ( 1 E(Z) = E σ X µ ) = 1 σ σ µ µ σ = 0 ( 1 V ar(z) = V ar σ X µ ) ( ) 1 = V ar(x) = 1 σ σ σ σ = 1 0 1 f(y ) = P (z Y ) = P (x µ + σy ) = µ+σy 1 e 1 ( x µ σ ) dx πσ x µ = t x = t = x = µ + σy t = Y dx = σdt σ Y 1 F (Y ) = P (z Y ) = e t dt π Z N(0, 1 ) x µ = t σ 1 πσ e 1 ( x µ σ ) 1 πσ e t dx dt = 1 e t π X f(x) y = h(x) Y = h(x) h (x) 0 g(x) = f(h 1 (y)) dx dy 4..7 cumulative distribution function x f(x) F (x) F (x) = x 3 f(t)dt

P (a < x < b) = F (b) F (a) lim F (x) = 0 x lim F (x) = 1 x + F (x) = d dx x f(t)dt = f(x) 4..8 x x N(0, 1) y = x G(y) = P r{x y} = P r{ y x y ( ) 1 y} = exp t dt = F ( y) 1 0 π y g(y) = G (y) = 1 f( y) = 1 y 1 ( exp y ) (0 < y < + ) y π 0 otherwise 4..9 χ 0 1 S n = x 1 + x + + x n n χ S n = x 1 + x + + x n χ (n) X χ (n) E(X) = n V ar(x) = n X χ (n) f(x) = 1 ( n ) x n Γ n 1 e x 0 < x < + 4

Γ( ) Gamma Function Γ(p + 1) = + 0 Γ(p) = + x p lim x + e x = 0 0 x p 1 e x dx (p 0) x p e x dx = [ x p e x] + 0 + p + Γ(p + 1) = pγ(p) p x = y ( ) 1 Γ = Γ(1) = 1 Γ(n + 1) = nγ(n) = n(n 1)Γ(n 1) = = n! + 0 x 1 e x dx = + 0 0 x p 1 e x dx = pγ(p) e y dy = π + e y dy = π? φ(t) = (1 t) n t < 1 χ 4..10 Moment Generating Function X = x i p k = P (X = x k ) {p k } φ(z) = k z k p k φ(1) = k p k = 1 φ (z) = k kz k 1 p k, φ (z) = k k(k 1)z k p k,... z = 1 φ (1) = k kp k = E(X) φ (1) = k k(k 1)p k = k k p k k kp k = E(X ) E(X) 5

1. X B(n, p) φ(z) = n z k nc k p k (1 p) n k = (pz + (1 p)) n k=0 φ (z) = np (pz + (1 p)) n 1 φ (pz + (1 p)) n E(X) = φ (1) = np V ar(x) = E(X ) (E(X)) = (φ (1) + φ (1)) (φ (1)) = n(n 1)p + np n p = np(1 p). X P oisson(λ) φ(z) = z k e λ λk k! = eλ(z 1) k=0 X f(x) φ X = E(e tx ) = X φ(0) = φ (t) = φ (t) = + + + e tx f(x)dx = f(x)dx = 1 k=0 xe tx f(x)dx, φ (0) = x e tx f(x)dx, φ (0) = t k x k f(x)dx = k! + + k=0 xf(x)dx = E(x) x f(x)dx = E(X ) t k k! E(Xk ).. φ (n) (t) = + x n e tn f(x)dx, φ (n) (0) = + x n f(x)dx = E(X n ) 1. 6

X N(µ, σ ) MGF φ(t) = 1 ( + exp(tx) exp 1 ( ) ) x µ ) dx = = exp (µt + σ πσ σ t ) φ (t) = (µ + σ t) exp (µt + σ t ) ) φ (t) = σ exp (µt + σ t + (µ + σ t) exp (µt + σ t φ (0) = µ = E(X) φ (0) = σ + µ = E(X ) X 1, X,..., X n φ X1 (t), φ X (t),..., φ Xn (t) c 1 X 1 + c X + + c n X n φ(t) φ c1 X 1 +c X + +c n X n (t) = φ X1 (c 1 t)φ X (c t) φ Xn (c n t) X i N(µ i, σ i ) X 1, X X 1 ± X φ(t) φ X1 ±X (t) = φ X1 (t)φ X (±t) = exp (µ 1 t + σ 1 = exp ((µ 1 ± µ )t + σ 1 ) + σ t ) (±µ t + σ ) t X 1 ± X N(µ 1 ± µ, σ 1 + σ ) X i N(µ i, σ i ), i.i.d X 1 + X + + X n N(µ 1 + µ + + µ n, σ 1 + σ + + σ n ) X 1, X,..., X n µ 1 = µ = = µ n = µ σ 1 = σ = = σ n = σ X 1 + X + + X n n N ) (µ, σ n µ σ n 0 n 7

φ(t) = + + + exp (t(c 1 x 1 + c x + + c n x n )) f(x 1, x,..., x n )dx 1 dx dx n X 1, X,..., X n f(x 1, x,..., x n ) = f(x 1 )f(x ) f(x n ) φ(t) = φ 1 (c 1 t)φ (c t) φ n (c n t) 4..11 µ σ Π n x 1, x,..., x n x = (x 1 + x + + x n )/n x µ σ/ n n N(0, 1) z = x µ σ/ n = x 1 µ σ n + x µ σ n + + x n µ σ n x 1 µ σ/ n = u u φ 1(t) φ u (t) = E(e ut ) = 1 + E(u)t + t! E(u ) + t3 3! E(u3 ) + ( = 1 + t 1! nσ E ( (x µ) ) ) ( + t3 3! ( ) == 1 + t 1 n + O n 3/ = 1 + 1 n x j µ σ n φ(t) = (φ u (t)) n = 1 (σ n) 3 E ( (x 3 µ) 3) ( t + ε(n) ), lim ε(n) = 0 n ) + z φ(t) ( ) n 1 + t + ε(n) = n ( 1 + t + ε(n) n ) n t + ε(n) t + ε(n) e t 4..1 Gauss u M u M 8

Gauss X = u M X P r(x X x + dx) = f(x)dx f(x) < x < + + f(x)dx = 1 u n M 1, M,..., M n X i = M i u X i P r(x 1 X 1 x 1 + dx,..., x n X n x n + dx) = Πf(x i )(dx) n Πf(x i ) = Πf(M i u) u M 1, M,..., M n u ū ( ) ū u ) Gauss ū = (M 1 + +M n )/n Gauss d f(mi u) = 0 f (x 1 ) du f(x 1 ) + f (x ) f(x ) + + f (x n ) f(x n ) = 0 f (x) f(x) = F (x) F (x 1) + F (x ) + + F (x n ) = 0 (ū) (M 1, M,, M n ) (M 1 ū)+(m ū)+ +(M n ū) = 0 (x 1 + x + + x n = 0) dxn dx i = 1(i = 1,,, n 1) F (x 1 )+F (x )+ +F (x n ) = 0 x i F (x i )+F (x n ) dxn dx i = 0 F (x i ) = F (x n )(i = 1,,, n 1) F (x 1 ) = F (x ) = = F (x n ) F (x) x F (x) = c 1 F (x) = c 1 x + c F (x 1 ) + F (x ) + + F (x n ) = 0 c = 0 f (x) f(x) = c 1x f(x) = ke c1 x c 1 < 0 c 1 = h k f(x) = h e h x h > 0 π 9

4..13 Chebyshev X µ σ k P { X µ kσ} 1 k σ = V ar(x) = = µ kσ µ kσ + (x µ) f(x)dx (x µ) f(x)dx + (x µ) f(x)dx + ( µ kσ k σ f(x)dx + P { X µ kσ} 1 k µ+kσ µ kσ + µ+kσ + µ+kσ (x µ) f(x)dx + (x µ) f(x)dx + µ+kσ (x µ) f(x)dx ) f(x)dx = k σ P { X µ kσ} 1 Chebyshev x 1, x,..., x n f 1, f,..., f n x σ x i x λσ (for arbitary positiveλ > 1) N (1 1λ ) N = n f i σ Nσ = n f i (x i x) I = {i; x i x λσ} Nσ f i N N ( λ = N 1 1 ) λ f i (x i x) λ σ f i f i < N σ N i I i I i I i I X Chebyshev Chebyshev { X X B(n, p) ε lim P x } n n p < ε 30

X B(n, p) µ = np σ = np(1 p) Chebyshev { P x np < k } np(1 p) 1 1 { k x } p(1 p) P n p < k 1 1 n k p(1 p) ε = k k p(1 p) { x } = n ε 1 P n n p p(1 p) < ε 1 { ε n lim P x } n n p < ε = 1 ε n x p n 1 4..14 1. a X 0, x < 0 F (x) = x a, 0 x 1 1,, 1 < x (a) a = 1 3 (b) (c) a > 0. mode f(x) x X f(x) = { cx e x d, 0 < x < + 0, x 0 x = 3 mode c d 3. χ f(x) f(x)dx = 1 α α X 5 χ mode median mean 4. : χ X 5 χ P (X < c) = 0.90 c 31

4.3 4.3.1 Normal distribution X N(µ, σ ) Z = X µ σ N(0, 1 ) X N(µ, σ ) cx N(cµ, c σ ) X 1 N(µ 1, σ 1 ) X N(µ, σ ) X 1, X : indep. X i N(µ i, σ i ) X i : i.i.d. X 1 + X N(µ 1 + µ, σ 1 + σ ) ( n n X i N µ i, x=1 x=1 n x=1 σ i ) 4.3. χ Chi-squared distribution X N(0, 1 ) X χ (1) X i N(0, 1 ) X i : i.i.d. n S n = X i χ (n) x=1 ( ) X µ X N(µ, σ ) χ (1) σ X i N(µ i, σ i ) n ( ) Xi µ i S n = χ (n) X i : i.i.d. σ i 4.3.3 F F-distribution X 1 χ (n 1 ) X χ (n ) X 1, X : indep. X 1 n 1 X n F (n 1, n ) 4.3.4 t t-distribution X 1 N(0, 1) X χ (n ) X 1, X : indep. X 1 X n t(n) X F (1, n) X t(n) 3

4.4 4.4.1 n p k P (X = k) ( n ) P (X = k) = p k (1 p) n k k ( ) n n! k = k!(n k)! n, p X Bin(n, p) E(X) = np V ar(x) = np(1 p) Excel B4 n = 1, p = 0.5 n B9 p B10 4.4. P (X = k) = e λ λ k (k = 0, 1,,...) k! X X P oisson(λ) E(X) = V ar(x) = λ 1/λ λ Excel λ = 15 λ B6 4.4.3 µ σ X N(µ, σ ) f(x) = 1 e 1 ( x µ σ ) πσ 4.4.4 X N(µ, σ ) Z = X µ σ 0 1 Excel 1 0 1 B4 0 B5 1? 3 4 33

4.5 4.5.1 statistical inference statistical estimation test of statistical hypothesis 4.5. point estimation point estimator interval estimation 4.5.3 unbiasedness θ ˆθ E(ˆθ) = θ ˆθ θ unbiased estimator consistency θ ˆθ ϵ lim P ( ˆθ θ} ε) = 0 n ˆθ θ consistent estimator efficiency θ θ 1, θ θ 1 θ θ 1 θ µ σ n x 1, x,..., x n x n = 1 n (x 1 + x + + x n ) s = 1 n ( (x1 x) + (x x) + + (x n x) ) E( x n ) = 1 n nµ = µ E(s ) = n 1 n σ 34

x n n n 1 s = 1 n n 1 (x i x) x n µ σ n ( ) x n µ p σ/ n > k < 1 k ε = k σ n P ( x n µ > ε) < σ nε lim P ( x n µ > ε) = 0 n x n. 4.5.4 µ σ x i N(µ, σ ) x N(µ, σ n x = 1 n (x 1 + x + + x n ) S = 1 ( (x1 x) + (x x) + + (x n x) ) n 1 x µ (n 1)S ) σ N(0, 1) σ n χ (n 1) t = x µ σ/ n (n 1)S σ (n 1) = x µ S n t(n 1) P ( t t α ) = α α t α x µ S 1 α n S S 1 α confidence coefficient x t α n µ x + t α n 1 α confidence interval (n 1)S σ χ (n 1) 35

1 α P (z χ L ) = α P (z χ U ) = α χ L, χ U χ L (n 1)S σ (1 α) (n 1)S χ U 1. 1 χ U σ σ (n 1)S χ L 1 10cm 90 % 95 % 3cm. 90 % 95 1.64 10 µ 1.64 3 10 1.64 3 µ 10 + 1.64 3 5.08 µ 14.9 10 1.96 3 µ 10 + 1.96 3 4.1 µ 15.88. 10 1 10 10cm. 90 % 95 1.64 10 µ 3/ 1.64 10 10 1.64 3/ 10 µ 10 + 1.64 3/ 10 8, 44 µ 11.56 10 1.96 3/ 10 µ 10 + 1.96 3/ 10 8.14 µ 11.86 3. 7 {360, 410, 350, 380, 40, 400, 410} g 90 % 95 % 36

. 7 390g (4400/6)g 4400/6 = 10.4g 7 390 µ 4400/6 7 t(7 1) 390 1.94 10.4 µ 390 + 1.94 10.4 370.1 µ 409.9 95 % 364.9 µ 415.1 99 % 35.0 µ 48.0 (xi x) = 4400 (n 1)S 4400 14.45 σ 4400 1.37 304 σ 3557 4.6 4.6.1 Testing Statistical Hypothesis T N T p N 1 p 1 n = 1 T alternative hypothesis H 1 : p > 1 n = 1 T X X 3 T X 9 10 37

T X x P {X = x} = n C x p x (1 p) x (x = 0, 1,,..., n) p H 0 : p = 1 null hypothesis H 0! n = 1 T 1 H 0 T N 1 T 1 P {X = 1} = 1 C 1 ( 1 ) 1 ( ) 1 0 = 0.00044414 0.05 % H 0 10,000 3 H 0 H 0 reject n = 1 T 10 1.6 % T significant level Fisher 5 % 1 % 5 % T 10 0.019 % H 0 H 0 H 0 H 1 H 0 critical region H 1 : p > 1 x 10, {10, 11, 1} H 1 : p < 1 x, {0, 1, } 5 % D : T 10 P (D H 0 ) = 0.0161 + 0.009 + 0.000 = 0.019 H 0 H 1 38

H 0 D Type 1 error α D H 0 Type error H 0 : H 0 : H 0 : H 0 : One-tailed test Two-tailed test 4.6. 1. : N 1oz 5 {0.90oz, 0.95oz, 1.00oz, 1.05oz, 0.85oz} 1.00oz 0.05oz. : 0.95 : 0.05 H 0 : µ = 1.00 H 1 : µ 1.00 5 % x µ σ = 0.95 1.00 0.05 =.4 n 5 5 % 1.96 1.00oz. : t 1 1.00oz S S = 1 n 1 n (x i x) σ S (n 1) t x µ S n t(n 1) 39

. 0.0065 x µ S n = 0.95 1.00 0.0065 5 = 1.414 5 %.78 3. : N 100 6.5 %. p H 0 : p = 0.05 i x i { 1: x i = 0: E(x i ) = 1 p + 0 (1 p) = p V ar(x i ) = (1 p) p + (0 p) (1 p) = p(1 p) x = 1 n E( x) = p V ar( x) = n x i p(1 p) n N x p p(1 p) x p p(1 p) = n N(0, 1) 0.06 0.05 0.05(1 0.05) =.4 5 % n 100 4. : N or t Π 1, Π Π 1 Π n 1 n {x 1 1, x 1,..., x 1 n 1 }, {x 1, x,..., x n } x 1, x µ i σ 1 40

x 1 N (µ 1, σ 1 ) n 1 x N (µ, σ ) H 0 : µ 1 = µ n z = x 1 x σ 1 + σ n 1 n N(0, 1) σ 1 = σ 3 z = x 1 x 1 σ + 1 n 1 n S S 1 n 1 = n 1 + n n (x i x) + (y j ȳ) σ z t(n 1 + n ) σ 1 σ 4 5. : N Π i (i = 1, ) P p i Π i n i P R i H 0 : p 1 = p ( R i N p i, p ) i(1 p i ) n i n i 3 R 1 n 1 R n N j=1 ( p 1 p, p 1(1 p 1 ) + p ) (1 p ) n 1 n 3 σ 1 = σ 4-1981 41

z = R 1 R n 1 n p(1 p) p(1 p) + n 1 n N(0, 1), p = p 1 = p = R 1 + R n 1 + n 6. : χ 1 5 1oz 1.00oz 0.05oz 0.05oz σ σ 0 H 0 : σ = σ 0 Π(µ, σ ) n {x 1, x,..., x n } n (x i x) σ χ (n 1) H 0 : σ = σ 0 n (x i x) n 1 χ 1 { 0.90oz, 0.95oz, 1.00oz, 1.05oz, 0.85oz} x = 0.95 z = 5 n=1 σ 1 0.05 ((0.90 0.95) + + (0.85 0.95) ) = 10.0 4 χ 5 % χ.5 %.5 % 0.48 11.14 7. : χ x x m x p 1, p,..., p m x n m n 1, n,..., n m n j = n np 1, np,..., np m x H 0 : H 1 : p 1 = p 10, p = p 0,..., p m = p m0 H 0 4 m p j0 = 1 j=1 j=1

H 0 z = m (n i np i0 ) t(m 1) np i0 5 z 300 1 3 4 5 6 44 51 45 56 47 57 50 50 50 50 50 50 H 0 : p 1 = 1 6, p = 1 6,..., p 6 = 1 6 z = 6 (n i np i0 ) (44 50) (51 50) (57 50) = + + + = 3.1 np i0 50 50 50 5 % z 5 χ 5 % 11.07 8. 100 M F B 30 10 40 0.4 G 0 40 60 0.6 50 50 100 0.5 0.5 0.5 0.4 = 0. 0.5 0.6 = 0.3 100 0 30 0 0 40 30 30 60 50 50 5 pp.390 394 43

H 0 : H 0 : p MB = p M p B, p MG = p M p G, p F B = p F p B, p F G = p F p G 7 z = (30 0) 0 + (0 30) 30 + (10 0) 0 + (40 30) 30 = 16.67 4 3 = 1 χ 5 % 5 % 3.84 m n 1 n sum 1 x 11 x 1 x 1n n 1 x 1 x x n n........ m x m1 x m x mn n m sum x 1 x x n n z = m n j=1 ( x ij n ni n n j n n ni n n j n ) χ ((m 1)(n 1)) (m 1)(n 1) χ (m 1) (n 1) 44

5 5.1 x y y = f(x) x y 1 6 y = a + bx log y = α + β log x a α b β 7 x y y = f(x) u 5. 5.3 1. 0 : E(u i ) = 0. : E ( (u i ) ) = σ i = σ 3. 0 : E(u i, u j ) = 0 (i j) 4. 5. 6. 6 x y 1 7 x y y = Ax β 45

5.4 Y i = α + βx i + u i (i = 1,,..., T ) ((x 1, y 1 ), (x, y ),..., (x T, y T )) y i = ˆα + ˆβx i e i = y i ˆα ˆβx i i = 1,,..., T e i = (y i ˆα ˆβx i ) = φ(ˆα, ˆβ) ˆα, ˆβ α, β φ(ˆα, ˆβ) ˆα, ˆβ ( 8 ) 9 y i = T ˆα + ˆβ x i x i y i = ˆα x i + ˆβ α = β = ˆα = ȳ ˆβ x (x i x)(y i ȳ) ˆβ = (x i x) x i 1 T x x i x y i (xj x) x i x y i (xj x) y 1 1 5.5 OLS y i = α + βx i + u i ˆβ = β + x i x u i (xj x) 8 9 ( x, ȳ) 46

u i OLS 10 u i ( ) xi x ω i = (xj x) ˆβ = β + E( ˆβ) = E ( ω i u i ˆβ β + ) ω i u i = β + V ar( ˆβ) ( = E ( ˆβ β) ) = ω i E(u i ) = β ω i E(u i ) + i j ω i ω j E(u i u j ) = σ (xi x) 0 ˆβ β Best Linear Unbiased Estimator OLS BLUE 5.6 β OLS ˆβ y 1, y,..., y T β β β = d i y i d 1, d,..., d T β E(β ) = d i E(y i ) = α d i x i = β d 1, d,..., d T d i = 0, d i x i = 0 10 α 47

( T ) V ar(β ) = V ar d i y i = d i V ar(y i ) = σ T (d i ω i + ω i ) T = σ ( (di ω i ) + ω ) T i + (d i ω i )ω i σ ω i = V ar( ˆβ) d i = ω i β ((d i ω i )ω i ) = d i x i x d i ( T ) = 0 (x i x) (x i x) (x i x) 5.7 { ˆβ β z = ( T ) σ (x i x) 1 H 0 : β = 0 H 1 : β 0 ˆβ = ( T ) σ (x i x) 1 N(0, 1) σ z σ σ ˆσ 11 t = z ˆσ ˆσ = 1 T σ T «P 0.5 11ˆσ (x i x) ˆβ = (y i ˆα ˆβx i ) ˆσ σ χ (T ) ˆβ ( T ) ˆσ (x i x) 48 1 t(t )

5.8 : t t 1 5.9 5.9.1 (x i, y i ) x i ŷ i ŷ i = ˆα + ˆβx i e i = y i ŷ i (y i ȳ) = (ŷ i ȳ) + y 1 e i R = / = 1 - / R = (ŷ i ȳ) = 1 (y i ȳ) e i (y i ȳ) R 0 1 1 R 13 R R ( Σei ) R /(T k 1) = 1 (Σ(y i ȳ) Σ = T ) /(T 1) T k 5.10 1 t - 13 R = 1 49

5.11 3 OLS BLUE d.w. y i = α + βx i + u i u i = ρu i 1 + ϵ i H 0 : ρ = 0, H 1 : ρ 0 d.w. = (e i e i 1 ) i= e i ρ ρ = e i e i 1 i= e i e i e i ρ d.w. (1 ρ) i= 0 1-1 d.w.= d.w.=0 d.w.=4 d.w. d.w. d.w. d.w. d.w. 14 14 50

5.1 y i = αz i + βx i + u i z i = 1 α z i x i z i λx i y i = αz i + βx i + u i (λα + β)x i + u i λα + β α, β 5.13 y i = α + βx i + u i V ar(u i ) = σ i σ i OLS 5.14 5.14.1 y i = β 0 + β 1 x 1i + β x i + + β k x ki + ε i (i = 1,,..., n) 1 1. k(n). ε 1, ε,..., ε n N(0, σ ) 3. t d.w. 5.14. 1. : t { H 0 : β j = β j0 H 1 : β j β j0 51

t. : F { ˆβ j β j s.d.of ˆβ j t (n (k + 1)) H 0 : β 1 + β = 1, β 3 = β 30, β 4 = β 5 H 1 : H 0 F RSS(H 0 ) := RSS(H 1 ) := F = (RSS(H 0) RSS(H 1 ))/p F (p, n k 1) p RSS(H 1 )/(n k 1) α % F α (p, n k 1) F : Y = AK α L β or log Y = Ã + α log K + β log L α + β = 1 { H 0 : α + β = 1 H 1 : α + β 1 RSS(H 0 ) : log Y/L = Ã + α log K/L β : 1 α β : β = 1 ᾱ V ar(ˆβ) = V ar(ˆα) n = 7 log Y/L = 1.069 +0.637 log K/L (0.13) (0.0754) R = 0.943, R = 0.941, RSS = 0.85574 β = 1 α = 1 0.637 = 0.363 β = 0.0754 log Y = 1.069 +0.637 log K +0.363 log L t (8.1) (8.45) (4.81) R = 0.943, R = 0.941 log Y = 1.171 +0.603 log K +0.376 log L t (3.58) (4.79) (4.40) R = 0.944, R = 0.939, RSS = 0.85163 F = (RSS(H 0) RSS(H 1 ))/p (RSS(H 1 )/(n k 1)) = (0.85574 0.85163)/1 0.85163/(7 1) = 0.116 F (1, 4) F F 0.05 (1, 4) = 4.6 5

:? : Y = AK α L β : Y = A α L β A A α α β β log Y = β 0 + β 1 log K + β log L + γ 0 β 0 D + γ 1 β 1 ((log K)D) + γ β ((log L)D) { H 0 : γ 0 = γ 1 = γ = 0 H 1 : H 0 53