By Helmut Lütkepohl; Markus Krätzig
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Additional info for Applied time series econometrics
1 Estimation of AR Models Estimation of AR processes is particularly easy because it can be done by ordinary least squares (OLS). Therefore, it will be considered first before we comment on the estimation of more complicated models. If the deterministic 31 Univariate Time Series Analysis term is linear in the unknown parameters, it can be included in a straightforward way in the regression model used for estimation. To simplify the presentation, we assume that the deterministic term consists of a constant only, that is, µt = µ and thus yt = µ + xt ; hence, α(L)yt = α(L)µ + α(L)xt = α(1)µ + u t .
Yt−h ) are time invariant for any integer h, that is, they depend on h only and not on t. Sometimes a process satisfying this condition is described as being strictly stationary. This terminology will not be used here, but a process is simply called stationary if it has time-invariant first and second moments. , if T is the set of nonnegative integers), then it is possible that it needs some start-up period until the moments stabilize. In fact, it is conceivable that the moments reach a constant state only asymptotically.
3 Some Popular Time Series Models We have already encountered a white noise process as a specific stationary stochastic process consisting of serially uncorrelated random variables. Because most economic time series exhibit serial correlation, such a model is often insufficient for describing the DGP. There are some simple parametric models, however, that have been used frequently to describe the DGPs of economic time series. 2. In addition, we will consider moving average (MA) and mixed autoregressive moving average (ARMA) models.