Stationary Time Series, Conditional Heteroscedasticity, Random Walk, Test for a Unit Root, Endogenity, Causality and IV Estimation Chapter 1 Financial Econometrics Michael Hauser WS18/19 1 / 67 Content I Notation I Conditional heteroscedasticity, GARCH, estimation, forecasting I Random walk, test for unit roots I Granger causality I Assumptions of the regression model I Endogeneity, exogeneity I Instrumental variable, IV, estimator I Testing for exogeneity: Hausman test and Hausman-Wu test I Appendix: OLS, Weak stationarity, Wold representation, ACF, AR(1), MA(1), tests for WN, prediction of an AR(1) 2 / 67 Notation X, Π ::: matrices x, β, ::: column vectors x, β, : : : real values, single variables i; j; t ::: indices L ::: lag operator. We do not use the backward shift operator B. WN, RW ::: white noise, random walk asy ::: asymptotically df ::: degrees of freedom e.g. ::: exempli gratia, for example i.e. ::: id est, that is i.g. ::: in general lhs, rhs ::: left, right hand side rv ::: random variable wrt ::: with respect to 3 / 67 Conditional heteroscedasticity GARCH models 4 / 67 Conditional heteroscedasticity A stylized fact of stock returns and other financial return series are periods of high volatility followed by periods of low volatility. The increase and decrease of this volatility pattern can be captured by GARCH models. The idea is that the approach of new information increases the uncertainty in the market and so the variance. After some while the market participants find to a new consensus and the variance decreases. We assume - for the beginning - the (log)returns, rt , are WN 2 rt = µ + t with t ∼ N(0; σt ) 2 2 σt = E(t jIt−1) 2 2 2 σt is the predictor for t given the information at period (t − 1). σt is the 2 2 2 2 2 conditional variance of t given It−1.It−1 = ft−1; t−2; : : : ; σt−1; σt−2;:::g. 5 / 67 Conditional heteroscedasticity Volatility is commonly measured either by 2 2 I t the ’local’ variance (rt contains the same information), or I jt j, the modulus of t . 6 / 67 Generalized autoregressive conditional heteroscedasticity, GARCH The generalized autoregressive conditional heteroscedasticity, GARCH, model of 2 2 order (1; 1), GARCH(1,1), uses as set of information It−1 = ft−1; σt−1g 2 2 2 σt = a0 + a1t−1 + b1σt−1 The unconditional variance of t is then 2 2 2 2 2 2 2 [We set E(σt ) = E(t ) = σ . So σ = a0 + a1σ + b1σ and we solve wrt σ .] 2 a0 E(t ) = 1 − (a1 + b1) The unconditional variance is constant and exists if a1 + b1 < 1 2 Further as σt is a variance and so positive, a0; a1 and b1 have to be positive. a0; a1; b1 > 0 7 / 67 GARCH(r; s) A GARCH model of order (r; s), GARCH(r; s), r ≥ 0; s > 0, is s r s r 2 X 2 X 2 X X σt = a0 + aj t−j + bj σt−j with aj + bj < 1; aj ; bj > 0 1 1 1 1 An autoregressive conditional heteroscedasticity, ARCH, model of order s is a GARCH(0; s). E.g. ARCH(1) is 2 2 σt = a0 + a1t−1 with a1 < 1; a0; a1 > 0 Comment 1: 2 2 A GARCH(1,0), σt = a0 + b1σt−1, is not useful as it describes a deterministic 2 decay once a starting value for some past σt−j is given. Comment 2: The order s refers to the number of aj coefficients, r to the bj coefficients. 8 / 67 IGARCH(1,1) An empirically relevant version is the integrated GARCH, IGARCH(1,1) model, 2 2 2 σt = a0 + a1t−1 + b1σt−1 with a1 + b1 = 1; a0; a1; b1 > 0 where the conditional variance exists, but not the unconditional one. 9 / 67 GARCH(1,1): ARMA and ARCH representations We can define an innovation in the variance as 2 2 2 2 νt = t − E(t jIt−1) = t − σt 2 2 2 Replacing σt by σt = t − νt in the GARCH(1,1) model we obtain 2 2 t = a0 + (a1 + b1)t−1 + νt − b1νt−1 2 2 This model is an ARMA(1,1) for t . So the ACF of rt can be inspected for getting an impression of the dynamics. The model is stationary, if (a1 + b1) < 1. 2 2 2 2 Recursive substitution of lags of σt = a0 + a1t−1 + b1σt−1 in σt gives an infinite ARCH representation with a geometric decay 1 2 a0 X j−1 2 σt = + a1 b1 t−j 1 − b1 1 10 / 67 Variants of the GARCH model I t-GARCH: As the unconditional distribution of rt can be seen as a mixture of normal distributions with different variances, GARCH is able to model fat tails. However empirically, the reduction in the kurtosis of rt by a normal GARCH model is often not sufficient. A simple solution is to consider an already fat tailed distribution for t instead of a normal one. Candidates are e.g. the t-distribution with df > 2, or the GED (generalized error distr) with tail parameter κ > 0. I ARCH-in-mean, ARCH-M: If market participants are risk avers, they want a higher average return in uncertain periods than in normal periods. So the mean return should be 2 higher when σt is high. 2 2 rt = µ0 + µ1σt + t with µ1 > 0; t ∼ N(0; σt ) 11 / 67 Variants of the GARCH model I asymmetric GARCH, threshold GARCH, GJR model: As the assumption that both good and bad new information has the same absolute (symmetric) effect might not hold. A useful variant is 2 2 2 2 σt = a0 + a1t−1 + b1σt−1 + γDt−1t−1 where the dummy variable D indicates a positive shock. So Dt−1 = 1, if t−1 > 0 Dt−1 = 0, if t−1 ≤ 0 2 2 The contribution of t−1 to σ is (a1 + γ), if t−1 > 0 a1, if t−1 ≤ 0 If γ < 0, negative shocks have a larger impact on future volatility than positive shocks. (GJR stands for Glosten, Jagannathan, Runkle.) 12 / 67 Variants of the GARCH model I Exponential GARCH, EGARCH: 2 By modeling the log of the conditional variance, log(σt ), the EGARCH guarantees positive variances, independent on the choice of the parameters. It can be formulated also in an asymmetric way (γ 6= 0). j j (σ2) = + t−1 + (σ2 ) + γ t−1 log t a0 a1 2 b1 log t−1 2 σt−1 σt−1 If γ < 0, positive shocks generate less volatility than negative shocks. 13 / 67 Estimation of the GARCH(r; s) model Say rt can be described more generally as 0 rt = xt θ + t 0 0 0 with vector xt of some third variables, xt = (x1t ;:::; xmt ) , possibly including lagged rt ’s, seasonal dummies, etc. t with conditional heteroscedasticity can be written as t = σt ξt with ξt iid N(0; 1) 2 2 2 2 2 2 σt has the form σt = a0 + a1t−1 + ::: + as t−s + b1σt−1 + ::: + br σt−r . So the conditional density of rt jxt ; It−1 is given by 1 1 2 ( j ; ) = (− t ); = ( ; ) + ;:::; fN rt xt It−1 q exp 2 t max r s 1 n 2 2 σt 2πσt 2 2 Comments: V(σt ξt jxt ; It−1) = σt V(ξt jxt ; It−1) = σt , 2 2 starting values σ1; : : : ; σr appropriately chosen, 2 2 t = t (θ), σt = σt (a0; a1;:::; as; b1;:::; br ) 14 / 67 Estimation of the GARCH model The ML, maximum likelihood, estimator of the parameter vector θ; a0; a1;:::; as, b1;:::; br is given by n Y max fN(rt jxt ; It−1) θ;a0;a1;:::;as;b1;:::;br max(r;s)+1 as the ’s are uncorrelated. The estimates are asy normal distributed. Standard t-tests, etc. apply. Note the requirement of uncorrelated ’s. In case autocorrelation in the returns rt is ignored, the model is misspecified. And, you will detect ARCH effects although they might not exist. So in a first step, model the level of rt by ARMA, e.g., and then fit GARCH models to the residuals. 2 Remark: If rt is autocorrelated, so also rt is autocorrelated i.g. 15 / 67 Forecasting a GARCH(1,1) process 2 2 2 2 Using σ = a0=(1 − (a1 + b1)) in σt = a0 + a1t−1 + b1σt−1 we rewrite the model as 2 2 2 2 2 2 σt − σ = a1[t−1 − σ ] + b1[σt−1 − σ ] 2 So the 1-step ahead forecast E(t+1jIt ) is 2 2 2 2 2 2 2 2 σt+1jt = (σt+1) = E(t+1jIt ) = σ + a1[t − σ ] + b1[σt − σ ] 2 2 with the 1-step ahead forecast error t+1 − σt+1jt = νt+1. The h-step ahead forecast is, h ≥ 2, 2 2 2 [replacing both t ; σt by their (h − 1)-step ahead forecast σt+h−1jt ] 2 2 2 2 σt+hjt = σ + (a1 + b1)[σt+h−1jt − σ ] = 2 h−1 2 2 2 2 = σ + (a1 + b1) [a1(t − σ ) + b1(σt − σ )] 16 / 67 Random walk and unit root test Random walk, I(1), Dickey Fuller test 17 / 67 Random walk, RW A process yt = αyt−1 + t with α = 1 is called random walk. yt = yt−1 + t with t WN 2 Taking the variances on both sides gives V(yt ) = V(yt−1) + σ . This has only a 2 solution V(y), if σ = 0. So no unconditional variance of yt exists. Pt Starting the process at t = 0 its explicit form is yt = y0 + 1 j .