Quasi-Experimenaion Ch.6 005/8/7 ypo rep: The Saisical Analysis of he Simple Inerruped Time-Series Quasi-Experimen INTRODUCTION () THE PROBLEM WITH ORDINAR LEAST SQUARE REGRESSION OLS (Ordinary Leas Square) OLS AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) MODELS inerruped ime-series ARIMA Box-Jenkins 976 5000 5000 ARIMA MANOVA or GG ANOVA OLS ANOVA ( ) The Deerminisic and Sochasic Componens of a Time Series p.35 - Defining ARIMA (p, d, q) ARIMA p, d, q idenificaion ARIMA(,0,0) =.7 56% 6.0 (p.35 )
Saionariy ARIMA nonsaionary in he homogeneous sense nonsaionary in he homogeneous sense,, 3, 4, 5,, N (),,,,, ( ) No. = + θ 0 = θ 0 d Auoregressive Models p p 0 p = ARIMA(,0,0) p = ARIMA(,0,0) + a + a + φ < + < φ 3 a a ~ NID(0, σ ) Moving Average Models q φ + φ < 3 p = φ φ < φ <
ARIMA(0,0,) ARIMA(0,0,) θ = a a = a θ a θa θ < + < Mixed Models ARIMA(,0,) + a θa mixed AR MA Noise Model Idenificaion ACF PACF p.4 ACF Lag 0 3 4 5 6 7 8. Lag. 3 4 5 6 7 8 Lag.. 3 4 5 6 7 8 Lag 3... 3 4 5 6 7 8 Lag k ACF, rk r k N k = = N ( )( = + k ( ) ) for k =,, 3,.. () ACF ACF PACF, rkk ACF ule-walker
ACF PACF ACFPACF ACFPACF ACF PACF ARIMA(p,d,q) Figure 6.(ah ) Nonsaionary Processes Whie Noise Process Auoregressive Processes Moving Average Processes Mixed Processes Auocorrelaions Parial auocorrelaions lag lag p p q lag q q - p p - q ACF 4 k r j j= Q Q = N. d. ACFPACF 3. p q p q 5 4. 4 = - 5 underesimae overesimae
Esimaion of and Values ARIMA(p,d,q) - << + Diagnosis Q ACF Q ACF lag lag lag ARIMA SEASONAL MODELS Muliplicaive ARIMA Seasonal Models. Figure 6. ARIMA(p,d,q) 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 30 3 3 33 34 35 36 N-........................................ N + a
regular auocorrelaion seasonal auocorrelaion seasonal auocorrelaion P D Q S ARIMA (p, d, q) (P, D, Q) s regular seasonal ARIMA (,0,0)(,0,0) φ φ + a + φ + a + φ φφ 3 Idenifying he Seasonal Model ARIMA ACFPACF,4 4,8 seasonal regular seasonal ARIMA ARIMA (,0,0)(,0,0)S ARIMA (,0,0)(,0,0)S ARIMA (0,0,)(0,0,)S ARIMA (0,,)(0,0,)S ARIMA (0,,)(0,,)S
Auocorrelaions Parial auocorrelaions Seasonal Nonsaionary Seasonal Auoregressive Seasonal Moving Average ACF regular p+sp p p = 0 = 0 Q = lag q S q S + q Q q =, Q =, S =,,, 3, 3, 4, 5. ACFPACF ACFPACF. lag ACFPACF lag 4 ACF 3. regular PACF ACF 4. THE INTERVENTION COMPONENT ARIMA = noise = noise + inervenion p.6
Box-Jenkins ransfer funcions 6 Abrup, Consan Change = ω noise () I + I = 0 before he inervenion, < i = afer he inervenion, i prereamen : posreamen prereamen : posreamen Gradual, Consan Change δ + = noise (3) ()0 0 n δ = δ (0) + ω() = ω i = i i δ = i = δ ( ω) + ω() = δω + ω = δi + = δ ( δω + ω) + ω() = δ ω + δω + ω n n n = δ ω + δ ω +... + δ ω + δω + ω 6 i-3 i- i- i 3
ω change in level = -δ Box & Tiao(975) Hibbs(977) Abrup, Temporary Change (3)I I = 0 before he inervenion, < i = a he momen of inervenion, = i = 0 hereafer, > i n δ = δ (0) + ω() = ω i = i i δ = δ ( ω) + ω(0) = δω = i = δi + + = δ ( δω) + ω(0) i = δ ω = δ n ω n i-3 i- i- i 3 Which Transfer Funcion Should Be Used?. I (3)
..9 I 3.. () SUMMAR OF THE MODELING STRATEG Inervenion Hypohesis Tes ARIMA 7 / THREE EXAMPLES ARIMA ARIMA(0,,0)(0,,) ARIMA(0,,0) ACFPACF Deusch and Al(977) ARIMA 7
variance " unexplained" funcion ransfer = CONCLUSION 8 8 --;