McCain & McCleary (1979) The Statistical Analysis of the Simple Interrupted Time-Series Quasi-Experiment
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1 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 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 )
2 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 <
3 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 Lag Lag Lag Lag k ACF, rk r k N k = = N ( )( = + k ( ) ) for k =,, 3,.. () ACF ACF PACF, rkk ACF ule-walker
4 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 underesimae overesimae
5 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) N N + a
6 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
7 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
8 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
9 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)
10 ..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
11 variance " unexplained" funcion ransfer = CONCLUSION ;
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