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Results tagged with time-series
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user 309015
Time series are data observed over time (either in continuous time or at discrete time periods).
2
votes
1
answer
494
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Linear process autocovariance function converges to zero as h goes to infinity
Assume that time series $(X_t)$ is given by:
\begin{equation}
X_t = \sum_{i=0}^{\infty} c_i \varepsilon_{t - i},
\end{equation}
where $(\varepsilon_t)$ is a weak white noise $\text{WN}(0, \sigma^2)$ a …
1
vote
Accepted
Linear process autocovariance function converges to zero as h goes to infinity
I found the solution on my own, so I share it with you.
We know that
\begin{equation}
\gamma_X(h) = \sigma^2\sum_{i = 0}^\infty c_ic_{i+h}.
\end{equation}
To test the convergence consider
\begin{equat …
1
vote
Accepted
Forecasting $X_{t+2}$ for causal AR(p)
Consider $X_{t+2}$ term:
\begin{equation}
\begin{split}
X_{t+2} & = \sum_{i=1}^p \psi_iX_{t+2-i} \,+Z_{t+2} = \psi_1X_{t+1} + \sum_{i=2}^p\psi_iX_{t+2-i} \,+Z_{t+2} \\
& = \psi_1\left(\sum_{i=1}^p \ps …
1
vote
Accepted
Prediction error for ARMA process
Remark: Mind, that we can solve this only is $(X_t)$ is casual. A sufficient condition for that is $|\varphi|<1$. Using the representation with a lag operator $B$
\begin{equation}
(1-\varphi B^{12})X_ …