Hello,
I have a question regarding financial time series. I am studying the effects of Quantitative Easing on the US stock market. I have tried to fit an ARIMA model to my S&P500 returns with an exogenous variable 'Quantitative Easing' which I interpreted aswell as returns to have both variables as stationary. The result of my ARIMA model is (0,0,0). I am not sure if this is even correct. How to approach the GARCH model now?
Thank you for any help!
Dear Ondrej Michalik ,
An ARIMA model (Box-Jenkins model) approach analyzes univariate stochastic time series, i. e. error term of this time series – where the analyzed time series being analysed must be stationary to avoid spurious regressions; ie. the mean, variance and covariance of the series must all be constant over time.
In your case however, you indicated that you have an independent variable “Quantitative Easing” which you are attempting to analyse its effect on the US stock market. The fact that you integrate an independent variable implies that you are specifying and estimating an ARIMAX model; whose ARMA component determines whether time-dependent disturbances significantly follow a linear autoregressive moving average process, estimated parameters whose specification should contain the equation, eg. Yt = Zt*gamma + Ut; where Ut = rho*Ut-1 + lambda*et-1 + et (the disturbance ARMA (p, q) where rho represents the first order autocorrelation parameter, lambda denotes the first order moving average parameter and et follows IID, N(0, var) hence implying that et represents a white noise disturbance. The inclusion of the stationary independent variable, namely Quantitative Easing (X) results in formulation of an ARIMAX (transfer function) model defined by equation Yt = v(beta)Xt + alpha + Zt*gamma + Ut; where Yt is the dependent variable (US Stock Market), (Xt) is a matrix of the exogenous variable (QE), betta is a vector of the coefficient of the exogenous variable, v(betta)Xt is the transfer function (impulse response function - IRF) that permits the exogenous variable to influence the dependent variable via a distributed lag, alpha is a constant term, Zt is a matrix of the AR and MA terms, and gamma is a vector of coefficients for Zt parameters.
The information you provided does not make it clear if the ARIMAX model is what you ran first or the ARIMA analysis you conducted was exactly a univariate analysis of your dependent variable alone?
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwizsvG-hNDuAhXXUhUIHRhPC0IQFjAAegQIAhAC&url=http%3A%2F%2Fwww.diva-portal.org%2Fsmash%2Fget%2Fdiva2%3A676106%2FFULLTEXT01.pdf&usg=AOvVaw2gPd1NnEduF5Ous8IYO1MN
Check your residuals for autocorrelation (e.g Box-Pierce test). If there is no autocorrelation there, you know that ARMA(0,0) is OK. If not, try to modify the model and find the correct number of p and q. Then you apply a GARCH-type model for residuals, provided that you find ARCH effect (you can verify it via ARCH test on residuals or Box-Pierce on squared residuals). Most software allows you to fit the model in one step, i.e. the ARMAX-GARCH.
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