Stata 11 Stata VAR VEC whitepaper mwp 4 mwp-084 var VAR 14 mwp-004 varbasic VAR 25 mwp-005 svar VAR 31 mwp-007 vec intro VEC 47 mwp-008 vec VEC 75 mwp
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2 Stata 11 Stata VAR VEC whitepaper mwp 4 mwp-084 var VAR 14 mwp-004 varbasic VAR 25 mwp-005 svar VAR 31 mwp-007 vec intro VEC 47 mwp-008 vec VEC 75 mwp-063 VAR postestimation vargranger Granger 86 mwp-062 varlmar 93 mwp-059 varnorm 98 mwp-060 varsoc 104 mwp-057 varstable 109 mwp-058 varwle 114 mwp-061 VEC postestimation veclmar 120 mwp-066 vecnorm 123 mwp-067 vecrank 127 mwp-064 vecstable 132 mwp-065
3 whitepaper mwp irf IRF 136 mwp-006 fcast compute 146 mwp-068 fcast graph 151 mwp-069 StataCorp c 2011 Math c 2011 StataCorp LP Math web: master@math-koubou.jp
4 mwp-084 Stata var, vec / whitepaper mwp-004, mwp-063, whitepaper arima, arch mwp VAR VAR 4. VEC 1. VAR (1) {y t } 1 y t = (y 1t, y 2t,..., y nt ) {y t } = {..., y 1, y 0, y 1,...} (1) {y t } (i) E(y t ) µ µ t (ii) V (y t ) = E[(y t µ)(y t µ) ] t (iii) Cov(y t, y t s ) = E[(y t µ)(y t s µ) ] t s(s > 0) c Copyright Math c Copyright StataCorp LP (used with permission) 4
5 (2) VAR (VAR: vector autoregression model) AR(p) p y t = A 0 + A i y t i + u t (2) i=1 A 0 n 1 A i (i = 1, 2,..., p) n n u t n 1 u t E(u t ) = 0 (3a) V (u t ) = E(u t u t) = Σ (3b) E(u t u t s ) = 0, for s > 0 (3c) Σ (2) p VAR(p) VAR(p) (2) ( p ) I n A i z i = 0 (4) i=1 1 (3) VAR (2) VAR (SUR: seemingly unrelated regression) SUR VAR OLS (BLUE: best linear unbiased estimator) VAR var *1 [TS] var (mwp- 004 ) var p Stata preestimation varsoc [TS] varsoc (mwp-057 ) *1 VAR varbasic [TS] varbasic (mwp-005 ) 5
6 2. VAR VEC (innovation accounting) (IRF: impulse response function) (FEVD: forecast-error variance decomposition) 3. VAR 4. VEC (1) x t y t 1 I(1) β 1 x t + β 2 y t I(0) x t y t (cointegrated) β = (β 1, β 2 ) (cointegrating vector) (VECM: vector error correction model) (2) n VAR(p) y t = A 0 + p A i y t i + u t (22) i=1 {y t } y 1t, y 2t,..., y nt I(1) 1 (22) p 1 y t = A 0 + Πy t 1 + Γ i y t i + u t (23) Π = p i=1 A i I n, Γ i = p j=i+1 A j (23) Πy t 1 Πy t 1 i=1 Case 1: Π = 0 p i=1 A i = I n (23) VAR { y t } VAR {y t } 6
7 Case 2: Πy t 1 y 1t, y 2t,..., y nt 1 Π αβ (23) p 1 y t = A 0 + αβ y t 1 + Γ i y t i + u t (24) α, β n r r Π β r β y t 1 r (error correction term) α (adjustment coefficient vector) (24) VECM (24) (Granger s representation theorem) i=1 (3) (4) VEC VEC 7
8 mwp-004 var - Stata (VAR: vector autoregressive) whitepaper var VAR IRF FEVD varbasic mwp VAR 2. var 2.1 VAR var postestimation 1. VAR VAR [TS] var intro p367 3 VAR(2) x t y 1t v 1 a 11 a 12 a 13 y 1,t 1 b 11 b 12 b 13 y 2t = v 2 + a 21 a 22 a 23 y 2,t 1 + b 21 b 22 b 23 y 3t v 3 a 31 a 32 a 33 y 3,t 1 b 31 b 32 b 33 u 1t u 2t u 3t W.N.(Σ) y 1,t 2 y 2,t 2 y 3,t 2 [TS] var intro A 1 = a 11 a 12 a 13 a 21 a 22 a 23 A 2 = b 11 b 12 b 13 b 21 b 22 b 23 a 31 a 32 a 33 b 31 b 32 b 33 + u 1t u 2t u 3t (M1) c Copyright Math c Copyright StataCorp LP (used with permission) 8
9 u t (W.N.) Σ (M1) σ 11 σ 12 σ 13 Σ = σ 21 σ 22 σ 23 σ 12 = σ 21, σ 13 = σ 31, σ 23 = σ 32 σ 31 σ 32 σ 33 3 VAR(2) a ij 9 b ij 9 σ ij 6 24 (M1) VAR (SUR: seemingly unrelated regression) SUR VAR OLS (BLUE: best linear unbiased estimator) u t OLS 2. var 2.1 VAR VAR Example lutkepohl2.dta 3 VAR. use * 1 (Quarterly SA West German macro data, Bil DM, from Lutkepohl 1993 Table E.1) (M1) y 1, y 2, y 3 3 dln inv ln(inv) inv dln inc ln(inc) inc dln consump ln(consump) consump inv *1 File Example Datasets Stata 11 manual datasets Time-Series Reference Manual [TS] var 9
10 . list qtr inv ln inv dln inv in 1/8, separator(4) * 2 qtr inv ln_inv dln_inv q q q q q q q q q1 1982q4 1978q4 var VAR(2) varsoc [TS] varsoc (mwp-057 ) Statistics Multivariate time series Vector autoregression (VAR) Model : Dependent variables: dln inv dln inc dln consump Lags: Include lags 1 to: 2 1 var Model *2 Data Describe data List data 10
11 by/if/in : If: qtr <= tq(1978q4) 2 var by/if/in. var dln_inv dln_inc dln_consump if qtr <= tq(1978q4), lags(1/2) Vector autoregression Sample: 1960q4 1978q4 No. of obs = 73 Log likelihood = AIC = FPE = 2.18e 11 HQIC = Det(Sigma_ml) = 1.23e 11 SBIC = Equation Parms RMSE R sq chi2 P>chi2 dln_inv dln_inc dln_consump
12 Coef. Std. Err. z P> z [95% Conf. Interval] dln_inv dln_inv L L dln_inc L L dln_consump L L _cons dln_inc dln_inv L L dln_inc L L dln_consump L L _cons dln_consump dln_inv L L dln_inc L L dln_consump L L _cons (M1) a 11 a 12 a a 21 a 22 a 23 = a 31 a 32 a b 11 b 12 b b 21 b 22 b 23 = b 31 b 32 b v v 2 = v
13 p dln inv a 11 b 11, a 12, b 12, a 13, b 13, v 1 0 VAR 2.3 Σ e(sigma). matrix list e(sigma) * 3. matrix list e(sigma) symmetric e(sigma)[3,3] dln_inv dln_inc dln_consump dln_inv dln_inc dln_consump var postestimation var postestimation # 1 varsoc varlmar varstable varnorm Granger vargranger lag-exclusion varwle 2 predict fcast compute/graph 3 IRF IRF/DM/FEVD irf *3 ereturn list 13
14 mwp-008 vec intro - VECM Stata (VECM: vector error-correction models) whitepaper VECM whitepaper 1. Granger VECM VECM 3.4 Johansen 3.5 Postestimation VECM 1. Granger VECM E(y t ) = µ y t t k Cov(y t, y t k ) = E[(y t µ)(y t k µ)] = γ k (weak stationary) (covariance stationary) y t y t = y t y t 1 (unit root process) y t I(1) d 1 d d (integrated process) I(d) I(0) c Copyright Math c Copyright StataCorp LP (used with permission) 14
![y t I(1) c y t I(0) c y t (cointegration) y t (cointegrated) c (cointegrating vector) Granger (Granger representation theorem) Granger [TS] vec intro p451 (7) K VAR(p) I(1) y t [TS] vec intro p451](/docs-images/91/106698632/images/15-0.jpg "(8) VECM y t r Π K K αβ K r β β y t I(0) r α (adjustment parameters) 2.")
15 y t I(1) c y t I(0) c y t (cointegration) y t (cointegrated) c (cointegrating vector) Granger (Granger representation theorem) Granger [TS] vec intro p451 (7) K VAR(p) I(1) y t [TS] vec intro p451 (8) VECM y t r Π K K αβ K r β β y t I(0) r α (adjustment parameters) 2. vec vec Model 5 1 vec - Model [TS] vec intro p451 (8) p452 (9) (11) µ, ρ, γ, τ 5 [TS] vec intro Trends in the Johansen VECM framework p452 15
16 3. VECM Example txhprice.dta VEC. use * 1 4 Austin, Dallas, Houston, San Antonio t austin dallas houston sa m m m m m m m m *1 File Example Datasets Stata 11 manual datasets Time-Series Reference Manual [TS] vec intro 16
17 Dallas Houston var vec varsoc [TS] vec intro p451 (7) (8) VECM VAR p 1 vec VAR p dallas, houston varsoc Statistics Multivariate time series VEC diagnostics and tests Lag-order selection statistics (preestimation) Main : Dependent variables: dallas houston Options: Maximum lag order: 4 17
18 4 varsoc - Main. varsoc dallas houston Selection order criteria Sample: 1990m5 2003m12 Number of obs = 164 lag LL LR df p FPE AIC HQIC SBIC e * e 06* * * * e e Endogenous: dallas houston Exogenous: _cons (LR: likelihood ratio) LR FPE *
19 . varsoc dallas houston austin sa. varsoc dallas houston austin sa Selection order criteria Sample: 1990m5 2003m12 Number of obs = 164 lag LL LR df p FPE AIC HQIC SBIC e e * e * * e 11* * e Endogenous: dallas houston austin sa Exogenous: _cons 3.2 VECM vec 1 vecrank Statistics Multivariate time series Cointegrating rank of a VECM Model : Dependent variables: dallas houston 5 vecrank - Model. vecrank dallas houston, trend(constant) Johansen tests for cointegration Trend: constant Number of obs = 166 Sample: 1990m3 2003m12 Lags = 2 5% maximum trace critical rank parms LL eigenvalue statistic value *
20 Johansen * vecrank 2. vecrank austin dallas houston sa, lag(3). vecrank dallas houston austin sa, trend(constant) lags(3) Johansen tests for cointegration Trend: constant Number of obs = 165 Sample: 1990m4 2003m12 Lags = 3 5% maximum trace critical rank parms LL eigenvalue statistic value * VECM /2 VEC y t = (dallas, houston) ( ) ( ) ( ) y1t v1 α1 (β1 ) ( ) ( ) ( ) ( ) y = + β 1,t 1 γ11 γ y1,t 1 ε1t + (M1) γ 21 γ 22 y 2,t 1 ε 2t y 2t v 2 α 2 y 2,t 1 Statistics Multivariate time series Vector error-correction model (VECM) Model : Dependent variables: dallas houston Number of cointegrating equations (rank): 1 Maximum lag to be included in underlying VAR model: 2 Trend specification: constant * 2 *2 20
21 6 vec - Model. vec dallas houston, trend(constant) Vector error correction model Sample: 1990m3 2003m12 No. of obs = 166 AIC = Log likelihood = HQIC = Det(Sigma_ml) = 2.50e 06 SBIC = Equation Parms RMSE R sq chi2 P>chi2 D_dallas D_houston
22 Coef. Std. Err. z P> z [95% Conf. Interval] D_dallas _ce1 L dallas LD houston LD _cons D_houston _ce1 L dallas LD houston LD _cons Cointegrating equations Equation Parms chi2 P>chi2 _ce Identification: beta is exactly identified _ce1 Johansen normalization restriction imposed beta Coef. Std. Err. z P> z [95% Conf. Interval] dallas houston _cons vec ce cointegrating equations β 2 VECM(1) ( ) ˆβ1 = ˆβ 2 ( ) y 1t y 2t (M2) I(0) 22
23 (adjustment coefficients) α ) ( ) ) ( ) ) (ˆα (ˆv (ˆγ11 ˆγ = = 12 = ˆα ˆv ˆγ 21 ˆγ 22 ( ) (M2) VECM y 3. predict ce, ce equation( ce1). twoway (line ce t), yscale(range( )) yline(0) ylabel(-0.4(0.2)0.4) 3.4 Johansen 3.5 Postestimation (1) predict VECM predict 23
24 Statistics Postestimation Predictions, residuals, etc. Main : New variable name: ce1 Produce: Prediction of cointegrating equation: Equation to predict: ce1 8 predict - Main. predict ce1, ce equation(_ce1) ce2. predict ce2, ce equation( ce2). predict ce2, ce equation(_ce2) 24
25 ce1, ce2. twoway (line ce1 t), yscale(range( )) yline(0) ylabel(-0.4(0.2)0.4) > title(" ce1") * 3. twoway (line ce2 t), yscale(range( )) yline(0) ylabel(-0.4(0.2)0.4) > title(" ce2") ce *3 Graphics Twoway graph (scatter, line, etc.) 25
26 (2) vecstable (3) veclmar (4) vecnorm VECM 26
27 mwp-006 irf - IRF irf var, svar, vec irf 10 3 irf set IRF irf create IRF irf graph whitepaper VAR VEC [TS] vecintro (mwp-008 ) 1. IRF 2. Cholesky ordering 3. DM 4. FEVD 1. IRF (IRF: impulse response function) 1 VAR(p) [TS] irf create p172 (1) Σ IRF IRF (orthogonalized IRF) (Cholesky decomposition) Σ [TS] irf create p c Copyright Math c Copyright StataCorp LP (used with permission) 27
28 (1) VAR IRF VAR Example lutkepohl2.dta. use * 1 (Quarterly SA West German macro data, Bil DM, from Lutkepohl 1993 Table E.1) dln inv, dln inc, dln consump 3 VAR(2) mwp-004. var dln inv dln inc dln consump if qtr<=tq(1978q4), lags(1/2) dfk * 2. var dln_inv dln_inc dln_consump if qtr <= tq(1978q4), lags(1/2) dfk Vector autoregression Sample: 1960q4 1978q4 No. of obs = 73 Log likelihood = AIC = FPE = 2.18e 11 HQIC = Det(Sigma_ml) = 1.23e 11 SBIC = Equation Parms RMSE R sq chi2 P>chi2 dln_inv dln_inc dln_consump Coef. Std. Err. z P> z [95% Conf. Interval] dln_inv dln_inv L L dln_inc L L dln_consump L L _cons *1 File Example Datasets Stata 11 manual datasets Time-Series Reference Manual [TS] irf *2 Statistics Multivariate time series Vector autoregression (VAR) 28
29 dln_inc dln_inv L L dln_inc L L dln_consump L L _cons dln_consump dln_inv L L dln_inc L L dln_consump L L _cons (2) IRF IRF FEVD.irf / irf001.irf irf set Statistics Multivariate time series Manage IRF results and files Set active IRF file irf set : Set active IRF file: irf001 Replace any existing file with an empty file: 1 irf set 29
30 . irf set "irf001", replace (file irf001.irf created) (file irf001.irf now active) irf001.irf Stata (3) IRF IRF FEVD irf create order1 order1 2 Statistics Multivariate time series IRF and FEVD analysis Obtain IRFs, dynamic-multiplier functions, and FEVDs Main : Create IRF and store as name: order1 Forecast horizon: 8 2 irf create - Main 30
31 . irf create order1 (file irf001.irf updated) irf001.irf order1 (4) IRF order1 OIRF 3 VAR 3 3 = 9 9 dln inc dln consump Statistics Multivariate time series IRF and FEVD analysis Graphs by impulse or response Main : Statistics to graph: Orthogonalized impulse-response functions (oirf) IRF result sets: order1 Impulse variables: dln inc Response variables: dln consump 3 irf graph - Main. irf graph oirf, irf(order1) impulse(dln_inc) response(dln_consump) 31
32 2. Cholesky ordering 3. DM 4. FEVD 32
TS002
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218 6 219 6.11: (EQS) /EQUATIONS V1 = 30*V999 + 1F1 + E1; V2 = 25*V999 +.54*F1 + E2; V3 = 16*V999 + 1.46*F1 + E3; V4 = 10*V999 + 1F2 + E4; V5 = 19*V999 + 1.29*F2 + E5; V6 = 17*V999 + 2.22*F2 + E6; CALIS.
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1,a) 1,b) 1/f β Generation Method of Animation from Pictures with Natural Flicker Abstract: Some methods to create animation automatically from one picture have been proposed. There is a method that gives
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基礎数学I
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