Structural VAR Modeling for I(1) Data that is Not Cointegrated Assume yt = (y1t, y2t)0 be I(1) and not cointegrated. That is, y1t and y2t are both I(1) and there is no linear combination of y1t and y2t that is I(0). In this case, yt = (y1t, y2t)0 is I(0)
Introduction A convenient way of representing an economic time series yt is through the so-called trend-cycle decomposition yt = T Dt + Zt T Dt = deterministic trend Zt = For simplicity, assume T Dt = + t where (L) = 1 - 1L - - pLpand (L) = 1 + 1L + + q L
1
Choosing the Lag Length for the ADF Test
An important practical issue for the implementation of the ADF test is the specication of the lag length p.
If p is too small then the remaining serial correlation in the errors will bias the test.
If p is too
Covariance Stationary Time Series Stochastic Process: sequence of rv's ordered by time cfw_Yt = cfw_. . . , Y-1, Y0, Y1, . . . - Defn: cfw_Yt is covariance stationary if E[Yt] = for all t cov(Yt, Yt-j ) = E[(Yt - )(Yt-j - )] = j for all t and any j
Exampl
Multivariate Time Series Consider n time series variables cfw_y1t, . . . , cfw_ynt. A multivariate time series is the (n1) vector time series cfw_Yt where the ith row of cfw_Yt is cfw_yit. That is, for any time t, Yt = (y1t, . . . , ynt)0. Multivariate ti
State Space Models Defn: A state space model for an N -dimensional time series yt consists of a measurement equation relating the observed data to an m- dimensional state vector t, and a Markovian transition equation that describes the evolution of the st
Box-Jenkins Modeling Strategy for Fitting ARMA(p, q) Models 1. Transform the data, if necessary, so that the assumption of covariance stationarity is a reasonable one 2. Make an initial guess for the values of p and q 3. Estimate the parameters of the pro
Forecasting Let cfw_yt be a covariance stationary are ergodic process, e.g. an ARMA(p, q) process with Wold representation yt = +
X
Dene yt+h|t as the forecast of yt+h based on It known parameters. The forecast error is t+h|t = yt+h yt+h|t and the mean s
Introduction Consider the simple AR(1) model for t = 1, . . . , T yt = yt-1 + t, t WN(0, 2) If | < 1, then yt I(0) and yt = (L)t, (L) = such that
X X
k Lk , k = k
k=0
k=0
k| k | <
LRV = 2(1)2 = 2(1 - )-2 <
Furthermore, by the LLN and the CLT T -1 T -1
Cointegration
The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Economic theory, however, often implies equilibrium relationships between the levels of time series v