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Advanced Econometrics Topic 3: Multivariate Time Series II Wendun Wang Erasmus University Rotterdam and Tinbergen Institute Spring 2015 Wendun Wang (ESE) Advanced Econometrics: MvTS Spring 2015 1 / 22

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Review Review Last class What is multivariate time series? Why do we need to use multivariate time series - Provide better forecast - Examine causal relationship between economic variables - Simplify modelling, such as avoiding outliers, structural shifts Wendun Wang (ESE) Advanced Econometrics: MvTS Spring 2015 2 / 22
Review Review Last class General notation of a K -variable VAR( p ) model y t = c + A 1 y t 1 + . . . + A p y t p + u t , where y t = ( y 1 ,t , . . . , y K,t ) ( K × 1 vector) c = ( c 1 , . . . , c K ) is a K × 1 fixed vector of intercept u t = ( u 1 ,t , . . . , u K,t ) is a K × 1 vector of error A i are fixed coefficient matrices ( K × K ) Wendun Wang (ESE) Advanced Econometrics: MvTS Spring 2015 3 / 22

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Review Review Last class Definition (Stability) A VAR(1) process is stable if all eigenvalues of A 1 have modulus < 1 Figure: Bivariate VAR processes Stable process Unstable process Wendun Wang (ESE) Advanced Econometrics: MvTS Spring 2015 4 / 22
Review Review Last class How to check the stability of multivariate time series For a VAR(1) model, we can check whether the solutions of z of the following equation all have modulus > 1 det( I K A 1 z ) = 0 For a VAR( p ) model, we can check whether the solutions of the following equation all have modulus > 1 det( I K A 1 z . . . A p z p ) = 0 Wendun Wang (ESE) Advanced Econometrics: MvTS Spring 2015 5 / 22

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Review Stationarity Stationarity Definition (Stationarity) The VAR( p ) process is stationary if its first and second moments are time invariant, i.e. E ( y t ) = μ for all t E [( y t μ )( y t h μ ) ] = Γ y ( h ) = Γ y ( h ) for all t and h The condition means that all y t have constant finite mean and autocovariances that does not depend on t but just on the time period h between y t and y t h . A stable VAR( p ) process is stationary, and therefore the stability condition is also referred to as stationarity condition But unstability does not mean nonstationary Wendun Wang (ESE) Advanced Econometrics: MvTS Spring 2015 6 / 22
Estimation Empirical Modelling Several steps to construct a VAR model for empirical research 1. Estimate the parameters of VAR( p ) for p = 1 , . . . , M 2. Select the lag order p based on some criterion/tests 3. Investigate the properties of the estimated residuals from VAR( p ) 4. Use the estimated VAR( p ) to answer empirical questions Causality between variables Forecasting Impulse response analysis . . .

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