Financial Econometrics Lecture Notes 4 University of Piraeus Antypas Antonios

Vector Autoregression Analysis ARIMA(p, d, q) models are useful for univariate time series analysis When we have several time series, we need to take into account the interdependence between them which creates endogeneity Endogeneity is present when some of the independent variables in a regression are correlated to the error term Endogeneity causes serious misspecification biases What causes endogeneity? Omitted Variables Measurement Errors Simultaneity – While we consider one variable as exogenous, in reality this variable might be influenced simultaneously by the depended variable

Vector Autoregression Analysis Simultaneity is a common feature in economics and finance e.g. Gross Domestic Product depends on Investments, but Investments also depends on the Gross Domestic Product. The VAR model is a very useful starting point in the analysis of the interrelationships between the different time series Time series in Vector Autoregression Analysis have to be stationary Representation of a bivariate (with two variables) Vector AutoRegression Model of order p – VAR(p)

Vector Autoregression Analysis One drawback of VAR(p) models is over parameterization. Observe that in the above example we have to estimate 2(1+2p) coefficients On the other hand, VAR(p) models provide us important features Testing Causality Impulse Response Analysis

Vector Autoregression Analysis How to estimate a VAR(p) model in gretl Go Model – Time series – Vector Autoregression

Vector Autoregression Analysis How to estimate a VAR(p) model in gretl Select endogenous variables For each endogenous variable an equation will be included in the system Select exogenous variables Exogenous variables will appear in the above equations as explanatory factors Select lag order Look slide no 8 to see how this number can be determined Select to include a constant to avoid error term issues

Vector Autoregression Analysis How to estimate a VAR(p) model in gretl VAR(p) output is similar to OLS, main difference being that it includes results for more than one equations

Vector Autoregression Analysis How to choose lag p of VAR Go to Model – Time Series – VAR lag selection

Vector Autoregression Analysis How to choose lag p of VAR Same tool as estimating VAR(p). This time the output however is information criteria for all VAR(p) models for p=1 to the one defined. Select your preferable criteria for model selection and the optimal level of p is demonstrated by the asterisk (*)

Vector Autoregression Analysis Granger Causality Testing We aim to decide whether x causes y, by examining how much of the current y can be explained by past values of y lagged values of x and to see whether adding lagged values of x can improve the explanation The y is said to be Granger-caused by x, if x helps in the prediction of y, or equivalently if the coefficients on the lagged x’s are statistically jointly significant (using the F statistic) Null Hypothesis: y does not Granger cause x Null Hypothesis: x does not Granger cause y It is possible to have one-way causation: only x Granger causes y or only y Granger causes x, but also two-way causation is frequently the case; x Granger causes y and y Granger causes x

Vector Autoregression Analysis Granger Causality Testing How to perform Granger Causality Test in gretl gretl includes Granger Causality test in the output of an estimate VAR(p) model Is the section titled F-tests of zero restrictions If Probability < 0.05 then we reject the null of no Granger Causality. In this example we reject only the second Null, therefore we can infer that there is one-way Granger causality

Vector Autoregression Analysis Impulse Response Analysis An exogenous shock to one variable not only directly affects this specific variable but is also transmitted to all of the other endogenous variables through the dynamic (lag) structure of the VAR. An impulse response function traces the effect of a one standard deviation shock to one of the innovations on current and future values of the endogenous variables. Consider the representation of a bivariate VAR(1) model A shock on u2,t will affect xt directly through the specification of xt. But xt will then affect yt+1 in two ways. First it will appear in the specification of yt+1 (why?) and in addition the shock on u2,t will might also affect u1,t because of a non-zero covariance σ12 between u1,t and u2,t. This will affect yt which will also appear in yt+1

Vector Autoregression Analysis Impulse Response Analysis The Impulse Response Analysis allows us to observe the dynamic behavior of the endogenous variables included in our VAR(p) model How to perform Impulse Response Analysis in gretl After estimating the desirable VAR(p) model, go Graphs -> Impulse Responses (combined)

Vector Autoregression Analysis Impulse Response Analysis A shock on x, directly moves x to a higher level and then x returns to its initial status no-monotically after some time The shock on x does not affect contemporaneously y but after one period. Then y starts to return to its first status no-monotically

Vector Analysis and non stationary series In order to estimate a VAR(p) model, series must be stationary If the series are non stationary we can transform them to stationary by differentiating. The differencing operation used to achieve a stationarity involves a loss of potential information about long-run movements There is one case where this information can be measured and used in econometric modeling. We call this case cointegration and is applied when the series of interest are integrated of the same order, with order larger to zero

Vector Analysis and non stationary series Cointegration: An illustration A drunk man takes his dog out for walk without a strap. Then both of them will walk randomly in the field. We can not infer something for their future position

Vector Analysis and non stationary series Cointegration: An illustration If the drunk man attach a strap on the dog, this restricts the dog from walking completely random since his walk depends on the walking path of his boss

Vector Analysis and non stationary series Cointegration: An illustration In this example, our series of series is the path of the drunk man and the dog. Since they are not walking around a straight line we can infer that their path is not stationary. Instead their paths contain a unit root Non stationarity (unit root) is present in both paths for both cases (that is when the dog is not tied to his boss and when he is not) The difference between these two cases, is that when the dog is tied to his boss, there is a power (the strap) that prevents the difference between these two paths to diverge

Vector Analysis and non stationary series Cointegration: An illustration From an economic point of view we are interested on answering: Can we test for the existence of this attractor? If it exists, how can be introduced into our econometric modeling? Formal Definition of Cointegration: If the series under examination are integrated of the same order I(k) and there is a linear combination of these series that produce a stationary series I(0), then the series are said to be cointegrated This constructed I(0 ) series works as a proxy of the power that drives the random walks and prohibits them to diverge in the long run We can use this constructed series in our models

Vector Analysis and non stationary series Testing for cointegration between two series Engle-Granger procedure Step 1 Estimate the well known linear regression with these series, including a constant Step 2 Save the residuals of the regression and test them for the Null Hypothesis of a Unit Root – (Alternatively, the Null Hypothesis is No Cointegration) If the Null Hypothesis is rejected then we have cointegration

Vector Analysis and non stationary series Testing for cointegration between two series Engle-Granger procedure gretl has the entire Engle-Granger procedure automated

Vector Analysis and non stationary series Testing for cointegration with more than two series Johansen’s method Two major advantages with respect to Engle-Granger procedure: Testing for number of cointegrating vectors when N>2 Joint procedure: testing and maximum likelihood estimation of the vector error correction model and long run equilibrium relations. Johansen’s method reports two statistics Trace Test Maximum Eigenvalue Test

Vector Analysis and non stationary series Testing for cointegration with more than two series Johansen’s method in gretl

Vector Analysis and non stationary series Testing for cointegration with more than two series Johansen’s method in gretl Optimal p from VAR estimation of series Stationary version IMPORTANT: Series tested for Cointegration can not be stationary

Vector Analysis and non stationary series Testing for cointegration with more than two series Johansen’s method in gretl If Prob > 0.05 then we do not reject the Null We see that we reject the Null of at least 0 Cointegrating relationships but not reject for at least 1. Therefore, we can conclude that we have 1 Cointegrating relationship

Vector Analysis and non stationary series Econometric Modeling with cointegrated series If we test our series and find them cointegrated, then we can estimate a Vector Error Correction Model - VEC(p) model VEC(p) models are a straight extension of the VAR(p) models, where we add as an explanatory variable in the VAR(p) specification the error correction mechanism that drives the cointegrated series ( as approximated by the residuals from the Engle-Granger procedure ) E.g. in the case of two cointegrated series a VEC(2) model takes the following form: I(0)

Vector Analysis and non stationary series Econometric Modeling with cointegrated series Note that even though for detecting cointegration we require our series to be cointegrated of the same order, in the VEC(p) model we use first differences - stationary series – as in the case of the VAR(p) Estimating a VEC(p) model in gretl Same procedure as in the estimation of a VAR(p) model with the difference that we choose VECM...

Vector Analysis and non stationary series Econometric Modeling with cointegrated series How to decide if a series should be included in a VAR(p) or a VEC(p) model For the VAR(p) If Granger Causality Tests suggest that a series does not cause or is not caused by other series, that is all the coefficients regarding this series are found not significant jointly we should drop this series For the VEC(p) If Granger Causality Tests suggest that the stationary series does not cause or is not caused by other series, that is all the coefficients regarding this series are found not significant jointly AND The non stationary series is not significant in the Engle – Granger equation then we should drop this series

Vector Analysis and non stationary series Guideline for modeling multivariate time series Collect Data Test for Non Stationarity (Unit Root) -> Find the order of Integration of all series If all series are stationary then estimate a VAR(p) If some series are non stationary and integrated of the same order Check for cointegration If series are cointegrated proceed by estimating a VEC(p) model If they are not cointegrated, transform them to one lower level of integration by differentiating and start the procedure from the beginning Finally, if you find no evidence of cointegration, repeating the differentiating procedure will result in stationary series and you proceed with estimating a VAR(p) with these series