Stata 11 Stata ts (ARMA) ARCH/GARCH whitepaper mwp 3 mwp-083 arch ARCH 11 mwp-051 arch postestimation 27 mwp-056 arima ARMA 35 mwp-003 arima postestim

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2 Stata 11 Stata ts (ARMA) ARCH/GARCH whitepaper mwp 3 mwp-083 arch ARCH 11 mwp-051 arch postestimation 27 mwp-056 arima ARMA 35 mwp-003 arima postestimation 49 mwp-055 corrgram/ac/pac 56 mwp-009 dfgls DF-GLS 62 mwp-053 dfuller ADF 67 mwp-052 pperron P-P 73 mwp-054 tsset 77 mwp-002 / 83 mwp-001 StataCorp c 2011 Math c 2011 StataCorp LP Math web: master@math-koubou.jp

mwp-009 dfgls DF-GLS 62 mwp-053 dfuller ADF 67 mwp-052 pperron P-P 73 mwp-054 tsset 77 mwp-002 /

3 mwp-083 Stata arima, arch / whitepaper mwp-003, mwp-051, whitepaper VAR/VEC mwp ARCH 7. tsset 1. (1) y 1, y 2,..., y T (stochastic process) {y t } = {..., y 1, y 0, y 1,...} (1) {y t } 3 {y t} (stationary) (weakly stationary) (covariance stationary) (i) E(y t ) t (ii) V (y t ) t (iii) Cov(y t, y t s ) t s(s > 0) c Copyright Math c Copyright StataCorp LP (used with permission) 3

..} (1) {y t } 3 {y t} (stationary) (weakly stationary) (covariance stationary) (i) E(y t ) t (ii) V

4 (2) (white noise) {u t } E(u t ) = 0 (2a) V (u t ) = σ 2 (2b) Cov(u t, u s ) = 0, t s (2c) 0 (σ 2 ) 2. (1) AR(p) p (autoregressive process) *1 y t = ϕ 1 y t 1 + ϕ 2 y t ϕ p y t p + ϵ t, ϵ t WN(whitenoise) (3) (2) MA(q) q (moving average process) y t = ϵ t + θ 1 ϵ t θ q ϵ t q, ϵ t WN (4) (3) ARMA(p, q) (p, q) (autoregressive-moving average process) AR(p) MA(q) y t = ϕ 1 y t 1 + ϕ 2 y t ϕ p y t p + ϵ t + θ 1 ϵ t θ q ϵ t q, ϵ t WN (5) L ϕ(l) = (1 ϕ 1 L ϕ 2 L 2 ϕ p L p ) θ(l) = (1 + θ 1 L + θ 2 L φ q L q ) (6a) (6b) AR(p), MA(q), ARMA(p, q) AR(p) ϕ(l)y t = ϵ t (7a) MA(q) y t = θ(l)ϵ t (7b) ARMA(p, q) ϕ(l)y t = θ(l)ϵ t (7c) *1 [TS] arima p54 ϕ ρ 4

1 + + θ q ϵ t q, ϵ t WN (4) (3) ARMA(p, q) (p, q) (autoregressive-moving average process) AR(p) MA(q) y t = ϕ 1 y t 1 + ϕ 2 y t 2 + + ϕ p y t p + ϵ t + θ 1 ϵ t 1 + + θ q

5 ARCH 7. tsset 5

6 mwp-051 arch - ARCH arch ARCH, GARCH ARCH (volatility) 1. ARCH 2. ARCH/GARCH 3. ARCH/GARCH ARMA 4. EGARCH 5. PGARCH ARCH ARCH (autoregressive conditional heteroskedasticity) [TS] arch p12 (1) y t (conditional mean equation) σt 2 (conditional variance equation) A B ARCH, GARCH [TS] arch p12-13 (2) (3) arch ARCH/GARCH Example wpi1.dta ARCH/GARCH. use * 1 (wholesale price index) 1960q1 1990q4 ln wpi ln(wpi) c Copyright Math c Copyright StataCorp LP (used with permission) *1 File Example Datasets Stata 11 manual datasets Time-Series Reference Manual [TS] arch 6

ARCH ARCH (autoregressive conditional heteroskedasticity) [TS] arch p12 (1) y t (conditional mean equation) σt 2 (conditional variance equation) A B

7 . list if n <= 4 n >= ( N - 3), separator(4) * 2 wpi t ln_wpi q q q q q q q q D.ln wpi ln wpi 1. twoway (line D.ln wpi t), yline(0) * 3 postestimation estat archlm [R] regress postestimation time series Engle LM (Lagrange Multiplier test) ARCH D.ln wpi *2 Data Describe data List data *3 Graphics Twoway graph (scatter, line, etc.) 7

755313 D.ln wpi ln wpi 1. twoway (line D.

. regress D.ln wpi * 4. regress D.ln_wpi Source SS df MS Number of obs = 123 F( 0, 122) = 0.00 Model 0 0. Prob > F =. Residual.02521709 122.000206697 R squared = 0.0000 Adj R squared = 0.0000 Total.

8 . regress D.ln wpi * 4. regress D.ln_wpi Source SS df MS Number of obs = 123 F( 0, 122) = 0.00 Model 0 0. Prob > F =. Residual R squared = Adj R squared = Total Root MSE = D.ln_wpi Coef. Std. Err. t P> t [95% Conf. Interval] _cons estat estat archlm Statistics Postestimation Reports and statistics estat : Reports and statistics: Test for ARCH effects in the residuals (archlm) Specify a list of lag orders to be tested: 1 1 estat archlm. estat archlm, lags(1) LM test for autoregressive conditional heteroskedasticity (ARCH) lags(p) chi2 df Prob > chi H0: no ARCH effects vs. H1: ARCH(p) disturbance p ARCH *4 Statistics Linear models and related Linear regression 8

0133878 estat estat archlm Statistics Postestimation Reports and statistics estat : Reports and statistics: Test for ARCH effects in the residuals (archlm) Specify a list of lag orders to be tested:

9 ARCH GARCH(1, 1) D.ln wpi y t ARCH(1) { yt = x t β + ϵ t (M1) σt 2 = γ 0 + γ 1 ϵ 2 t 1 GARCH(1, 1) { yt = x t β + ϵ t σ 2 t = γ 0 + γ 1 ϵ 2 t 1 + δ 1 σ 2 t 1 (M2) GARCH(1, 1) ARCH GARCH [TS] arch p24-25 Statistics Time series ARCH/GARCH ARCH and GARCH models Model : Dependent variable: D.ln wpi Specify maximum lags: ARCH maximum lag: 1 GARCH maximum lag: 1 2 arch - Model 9

10 . arch D.ln_wpi, arch(1/1) garch(1/1) (setting optimization to BHHH) Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = (switching optimization to BFGS) Iteration 5: log likelihood = Iteration 6: log likelihood = Iteration 7: log likelihood = Iteration 8: log likelihood = Iteration 9: log likelihood = Iteration 10: log likelihood = ARCH family regression Sample: 1960q2 1990q4 Number of obs = 123 Distribution: Gaussian Wald chi2(.) =. Log likelihood = Prob > chi2 =. OPG D.ln_wpi Coef. Std. Err. z P> z [95% Conf. Interval] ln_wpi ARCH _cons arch L garch L _cons e arch (M2) β 0 = γ 0 = γ 1 = δ 1 = γ 1 p

11099 Iteration 7: log likelihood = 373.18939 Iteration 8: log likelihood = 373.23277 Iteration 9: log likelihood = 373.23394 Iteration 10: log likelihood = 373.

11 3. ARCH/GARCH ARMA 4. EGARCH 5. PGARCH 6. 11

12 mwp-003 arima - arima (ARMA: autoregressive moving-average) arima ARMA ARMAX 1. ARMA 2. arima 2.1 ARIMA(1,1,1) ARMAX 1. ARMA ARMA [TS] arima Introduction p54 ARMA(1, 1) µ t ρµ t 1 θϵ t 1 t ϵ t 1 µ t, ϵ t ρ θ p q ARMA(p, q) ARMA(p, q) µ t = y t x t β (M1) µ t = ρ 1 µ t ρ p µ t p + θ 1 ϵ t θ q ϵ t q + ϵ t (M2) (M2) (M1) L c Copyright Math c Copyright StataCorp LP (used with permission) 12

ARMA ARMA [TS] arima Introduction p54 ARMA(1, 1) µ t ρµ t 1 θϵ t 1 t ϵ t 1 µ t, ϵ t ρ θ p q ARMA(p, q)

13 ARMA x t ARMA(1, 1) ARMA(2, 2) (1) ARMA(1, 1) ARMA(p, q) (1 ρ 1 L)(y t β 0 ) = (1 + θ 1 L)ϵ t y t β 0 ρ 1 (y t 1 β 0 ) = ϵ t + θ 1 ϵ t 1 y t β 0 = ρ 1 (y t 1 β 0 ) + θ 1 ϵ t 1 + ϵ t (M3) ARMA(1, 1) (2) ARMA(2, 2) ARMA(2, 2) y t β 0 = ρ 1 (y t 1 β 0 ) + ρ 2 (y t 2 β 0 ) + θ 1 ϵ t 1 + θ 2 ϵ t 2 + ϵ t (M4) ARIMA(p, d, q) I integrated d ARMA(p, q) 2. arima 2.1 ARIMA(1,1,1) (wholesale price index) Example wpi1.dta. use * q1 1990q4. list wpi t if n <= 4 n >= ( N - 3), separator(4) * 2 *1 File Example Datasets Stata 11 manual datasets Time-Series Reference Manual [TS] arima *2 Data Describe data List data 13

integrated d ARMA(p, q) 2. arima 2.1 ARIMA(1,1,1) (wholesale price index) Example wpi1.dta. use http://www.stata-press.com/data/r11/wpi1 * 1 1960q1 1990q4.

14 wpi t q q q q q q q q4 wpi D.wpi wpi 1. twoway (line wpi t), title(wpi) * 3. twoway (line D.wpi t), yline(0) title(d.wpi) wpi D.wpi ARMA arima arima(p,d,q) ar(), ma() 2 AR 1 p MA 1 q 1 4 numlist ar(), ma() arima(p,d,q) d arima(1,1,1) D.wpi wpi *3 Graphics Twoway graph (scatter, line, etc.) 14

15 arima(p,d,q) arima wpi Statistics Time series ARIMA and ARMAX models Model : Dependent variable: wpi ARIMA(p,d,q) specification: p = d = q = 1 1 arima Model. arima wpi, arima(1,1,1) (setting optimization to BHHH) Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = (switching optimization to BFGS) Iteration 5: log likelihood = Iteration 6: log likelihood = Iteration 7: log likelihood = Iteration 8: log likelihood = ARIMA regression Sample: 1960q2 1990q4 Number of obs = 123 Wald chi2(2) = Log likelihood = Prob > chi2 =

41838 Iteration 3: log likelihood = 135.36691 Iteration 4: log likelihood = 135.35892 (switching optimization to BFGS) Iteration 5: log likelihood = 135.

OPG D.wpi Coef. Std. Err. z P> z [95% Conf. Interval] wpi ARMA _cons.7498197.3340968 2.24 0.025.0950019 1.404637 ar L1..8742288.0545435 16.03 0.000.7673256.981132 ma L1..4120458.1000284 4.12 0.000.6080979.

16 OPG D.wpi Coef. Std. Err. z P> z [95% Conf. Interval] wpi ARMA _cons ar L ma L /sigma (M3) wpi t = ( wpi t ) ϵ t 1 + ϵ t sigma ϵ ar(), ma() arima D.wpi Statistics Time series ARIMA and ARMAX models Model : Dependent variable: D.wpi Supply list of ARMA lags: List of AR lags: 1 List of MA lags: 1 2 arima Model 16

7971829 (M3) wpi t 0.750 = 0.874 ( wpi t 1 0.750) 0.412 ϵ t 1 + ϵ t sigma 0.725 ϵ ar(), ma() arima D.

17 . arima D.wpi, ar(1) ma(1) ARIMA(1, 1, 1) [TS] arima postestimation (mwp-055 ) ARMAX 17

AR(1) y t = φy t 1 + ɛ t, ɛ t N(0, σ 2 ) 1. Mean of y t given y t 1, y t 2, E(y t y t 1, y t 2, ) = φy t 1 2. Variance of y t given y t 1, y t

87 6.1 AR(1) y t = φy t 1 + ɛ t, ɛ t N(0, σ 2 ) 1. Mean of y t given y t 1, y t 2, E(y t y t 1, y t 2, ) = φy t 1 2. Variance of y t given y t 1, y t 2, V(y t y t 1, y t 2, ) = σ 2 3. Thus, y t y t 1,