Analysis of Multiple Time Series Kevin Sheppard - PDF document

Analysis of Multiple Time Series Kevin Sheppard ❤tt♣✿✴✴✇✇✇✳❦❡✈✐♥s❤❡♣♣❛r❞✳❝♦♠ Oxford MFE This version: February 24, 2020 February – March, 2020

This week’s material � Vector Autoregressions � Basic examples � Properties ◮ Stationarity � Revisiting univariate ARMA processes � Forecasting ◮ Granger Causality ◮ Impulse Response functions � Cointegration ◮ Examining long-run relationships ◮ Determining whether a VAR is cointegrated ◮ Error Correction Models ◮ Testing for Cointegration ⊲ Engle-Granger Lots of revisiting univariate time series. 2 / 67

Why VAR analysis? � Stationary VARs ◮ Determine whether variables feedback into one another ◮ Improve forecasts ◮ Model the effect of a shock in one series on another ◮ Differentiate between short-run and long-run dynamics � Cointegration ◮ Link random walks ◮ Uncover long run relationships ◮ Can improve medium to long term forecasting a lot 3 / 67

VAR Defined � P th order autoregression, AR(P): y t = φ 0 + φ 1 y t − 1 + . . . + φ P y t − p + ǫ t � P th order vector autoregression, VAR(P): y t = Φ 0 + Φ 1 y t − 1 + . . . + Φ P y t − p + ǫ t where y t and ǫ t are k by 1 vectors � Bivariate VAR(1): � y 1 ,t � φ 01 � φ 11 � � y 1 ,t − 1 � ǫ 1 ,t � � � � φ 12 = + + y 2 ,t φ 02 φ 21 φ 22 y 2 ,t − 1 ǫ 2 ,t � Compactly expresses two linked models: y 1 ,t = φ 01 + φ 11 y 1 ,t − 1 + φ 12 y 2 ,t − 1 + ǫ 1 ,t y 2 ,t = φ 02 + φ 21 y 1 ,t − 1 + φ 22 y 2 ,t − 1 + ǫ 2 ,t 4 / 67

Stationarity Revisited � Stationarity is a statistically meaningful form of regularity. A stochastic process { y t } is covariance stationary if E[ y t ] = µ ∀ t σ 2 < ∞∀ t V[ y t ] = σ 2 E[( y t − µ )( y t − s − µ )] = γ s ∀ t, s � AR(1) stationarity: y t = φy t − 1 + ǫ t ◮ | φ | < 1 ◮ ǫ t is white noise � AR(P) stationarity: y t = φ 1 y t − 1 + . . . + φ P y t − P + ǫ t ◮ Roots of ( z P − φ 1 z P − 1 − φ 2 z P − 2 − . . . − φ P − 1 z − φ P ) less than 1 ◮ ǫ t is white noise � No dependence on t 5 / 67

Relationship to AR � AR(1) y t = φ 0 + φ 1 y t − 1 + ǫ t = φ 0 + φ 1 ( φ 0 + φ 1 y t − 2 + ǫ t − 1 ) + ǫ t = φ 0 + φ 1 φ 0 + φ 2 1 y t − 2 + φ 1 ǫ t − 1 + ǫ t = φ 0 + φ 1 φ 0 + φ 2 1 ( φ 0 + φ 1 y t − 3 + ǫ t − 2 ) + φ 1 ǫ t − 1 + ǫ t ∞ ∞ � � φ i φ i = φ 0 1 + 1 ǫ t − i i =0 i =0 ∞ � = (1 − φ 1 ) − 1 φ 0 + φ i 1 ǫ t − i i =0 6 / 67

Relationship to AR � VAR(1) y t = Φ 0 + Φ 1 y t − 1 + ǫ t = Φ 0 + Φ 1 ( Φ 0 + Φ 1 y t − 2 + ǫ t − 1 ) + ǫ t = Φ 0 + Φ 1 Φ 0 + Φ 2 1 y t − 2 + Φ 1 ǫ t − 1 + ǫ t = Φ 0 + Φ 1 Φ 0 + Φ 2 1 ( Φ 0 + Φ 1 y t − 3 + ǫ t − 2 ) + Φ 1 ǫ t − 1 + ǫ t ∞ ∞ � � Φ i Φ i = 1 Φ 0 + 1 ǫ t − i i =0 i =0 ∞ � = ( I k − Φ 1 ) − 1 Φ 0 + Φ i 1 ǫ t − i i =0 7 / 67

Properties of a VAR(1) and AR(1) AR(1) : y t = φ 0 + φ 1 y t − 1 + ǫ t VAR(1) : y t = Φ 0 + Φ 1 y t − 1 + ǫ t AR(1) VAR(1) ( I k − Φ 1 ) − 1 Φ 0 Mean φ 0 / (1 − φ 1 ) σ 2 / (1 − φ 2 ( I − Φ 1 ⊗ Φ 1 ) − 1 vec ( Σ ) Variance 1 ) s th Autocovariance Γ s = Φ s γ s = φ s 1 V[ y t ] 1 V[ y t ] -s th Autocovariance ′ Γ − s = V[ y t ] Φ s γ − s = φ s 1 V[ y t ] 1 Autocovariances of vector processes are not symmetric, but Γ s = Γ ′ − s � Stationarity ◮ AR(1): | φ 1 | < 1 ◮ VAR(1): | λ i | < 1 where λ i are the eigenvalues of Φ 1 8 / 67

Stock and Bond VAR � VWM from CRSP � TERM constructed from 10-year bond return minus 1-year return from FRED � February 1962 until December 2018 (683 months) � VWM t � φ 01 � φ 11 , 1 � � VWM t − 1 � ǫ 1 ,t � � � � φ 12 , 1 = + + TER M t φ 02 φ 21 , 1 φ 22 , 1 TER M t − 1 ǫ 2 ,t � Market model: V WM t = φ 01 + φ 11 , 1 V WM t − 1 + φ 12 , 1 10 Y R t − 1 + ǫ 1 ,t � Long bond model TER M t = φ 01 + φ 21 , 1 VWM t − 1 + φ 22 , 1 TER M t − 1 + ǫ 2 ,t . � Estimates � V WM t � V WM t − 1 � ǫ 1 ,t     0 . 801 0 . 059 0 . 166 � � � (0 . 122) (0 . 004) (0 . 000) =  + +    − 0 . 104 0 . 116 TERM t 0 . 232 TERM t − 1 ǫ 2 ,t (0 . 041) (0 . 002) (0 . 000) 9 / 67

Stock and Bond VAR � Estimates from VAR VWM t = 0 . 816 + 0 . 060 VWM t − 1 + 0 . 168 TER M t − 1 (0 . 000) (0 . 117) (0 . 003) TER M t = 0 . 228 − 0 . 104 VWM t − 1 + 0 . 115 TER M t − 1 (0 . 045) (0 . 000) (0 . 002) � Estimates from AR VWM t = 0 . 830 + 0 . 073 VWM t − 1 (0 . 000) (0 . 057) TER M t = 0 . 137 + 0 . 098 TER M t − 1 (0 . 224) (0 . 011) 10 / 67

Comparing AR and VAR forecasts 1-month-ahead forecasts of the VWM returns 0.2 0.1 0.0 727198 728659 730120 731581 733042 734503 735964 1-month-ahead forecasts of 10-year bond returns 0.2 0.1 0.0 -0.1 727198 728659 730120 731581 733042 734503 735964 11 / 67

Monetary Policy VAR � Standard tool in monetary policy analysis ◮ Unemployment rate (differenced) ◮ Federal Funds rate ◮ Inflation rate (differenced)       ∆ ❯◆❊▼P t ∆ ❯◆❊▼P t − 1 ǫ 1 ,t  = Φ 0 + Φ 1  +  . ǫ 2 ,t ❋❋ t ❋❋ t − 1    ∆ ■◆❋ t ∆ ■◆❋ t − 1 ǫ 3 ,t ∆ ln ❯◆❊▼P t − 1 ∆ ■◆❋ t − 1 ❋❋ t − 1 ∆ ln ❯◆❊▼P t 0 . 624 0 . 015 0 . 016 (0 . 000) (0 . 001) (0 . 267) − 0 . 816 0 . 979 − 0 . 045 ❋❋ t (0 . 000) (0 . 000) (0 . 317) ∆ ■◆❋ t − 0 . 501 − 0 . 009 − 0 . 401 (0 . 010) (0 . 626) (0 . 000) 12 / 67

Interpreting Estimates � Variable scale affects cross-parameter estimates ◮ Not an issue in ARMA analysis � Standardizing data can improve interpretation when scales differ ∆ ln ❯◆❊▼P t − 1 ∆ ■◆❋ t − 1 ❋❋ t − 1 ∆ ln ❯◆❊▼P t 0 . 624 0 . 153 0 . 053 (0 . 000) (0 . 001) (0 . 267) − 0 . 080 0 . 979 − 0 . 015 ❋❋ t (0 . 000) (0 . 000) (0 . 317) ∆ ■◆❋ t − 0 . 151 − 0 . 028 − 0 . 401 (0 . 010) (0 . 626) (0 . 000) � Other important measures – statistical significance, persistence, model selection – are unaffected by standardization 13 / 67

VAR(P) is really a VAR(1) � Companion form: y t = Φ 0 + Φ 1 y t − 1 + Φ 2 y t − 2 + . . . + Φ P y t − P + ǫ t � Reform into a single VAR(1) where µ = E [ y t ] = ( I − Φ 1 − . . . − Φ P ) − 1 Φ 0 z t = Υz t − 1 + ξ t   . . . Φ 1 Φ 2 Φ 3 Φ P − 1 Φ P   y t − µ . . . I k 0 0 0 0   y t − 1 − µ     . . . 0 I k 0 0 0 z t =  , Υ =  .    .     . . . . . . . . . . . . .    . . . . . .   y t − P +1 − µ . . . 0 0 0 I k 0 ◮ All results can be directly applied to the companion form. ◮ Can also be used to transform AR(P) into VAR(1) 14 / 67

Revisiting Univariate Forecasting � Consider standard AR(1) y t = φ 0 + φ 1 y t − 1 + ǫ t � Optimal 1-step ahead forecast: E t [ y t +1 ] = E t [ φ 0 ] + E t [ φ 1 y t ] + E t [ ǫ t +1 ] = φ 0 + φ 1 y t + 0 � Optimal 2-step ahead forecast: E t [ y t +2 ] = E t [ φ 0 ] + E t [ φ 1 y t +1 ] + E t [ ǫ t +2 ] = φ 0 + φ 1 E t [ y t +1 ] + 0 = φ 0 + φ 1 ( φ 0 + φ 1 y t ) = φ 0 + φ 1 φ 0 + φ 2 1 y t � Optimal h -step ahead forecast: h − 1 � φ i 1 φ 0 + φ h E t [ y t + h ] = 1 y t i =0 15 / 67

Forecasting with VARs � Identical to univariate case y t = Φ 0 + Φ 1 y t − 1 + ǫ t � Optimal 1-step ahead forecast: E t [ y t +1 ] = E t [ Φ 0 ] + E t [ Φ 1 y t ] + E t [ ǫ t +1 ] = Φ 0 + Φ 1 y t + 0 � Optimal h-step ahead forecast: E t [ y t + h ] = Φ 0 + Φ 1 Φ 0 + . . . + Φ h − 1 Φ 0 + Φ h 1 y t 1 h − 1 � Φ i 1 Φ 0 + Φ h = 1 y t i =0 � Higher order forecast can be recursively computed E t [ y t + h ] = Φ 0 + Φ 1 E t [ y t + h − 1 ] + . . . + Φ P E t [ y t + h − P ] 16 / 67

What makes a good forecast? � Forecast residuals e t + h | t = y t + h − ˆ ˆ y t + h | t � Residuals are not white noise � Can contain an MA( h − 1 ) component ◮ Forecast error for y t +1 − ˆ y t +1 | t − h +1 was not known at time t . � Plot your residuals � Residual ACF � Mincer-Zarnowitz regressions � Three period procedure ◮ Training sample: Used to build model ◮ Validation sample: Used to refine model ◮ Evaluation sample: Ultimate test, ideally 1 shot 17 / 67

Multi-step Forecasting � Two methods � Iterative method ◮ Build model for 1-step ahead forecasts y t = Φ 0 + Φ 1 y t − 1 + ǫ t ◮ Iterate forecast out to period h h − 1 � Φ i 1 Φ 0 + Φ h y t + h | t = ˆ 1 y t i =0 ◮ Makes efficient use of information ◮ Imposes a lot of structure on the problem � Direct Method ◮ Build model for h -step ahead forecasts y t = Φ 0 + Φ h y t − h + ǫ t ◮ Directly forecast using a pseudo 1-step ahead method y t + h | t = Φ 0 + Φ h y t ˆ ◮ Robust to some nonlinearities 18 / 67