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Results tagged with time-series
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user 52295
A temporal sequence of events measured at discrete points in time.
1
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150
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Show that $Y_t$ and $Y_{t+h}$ are independent if $X_t$ is Gaussian
If $Y_t=\sum_{i=0}^qa_iX_{t-i}$ where $X_{t-i}$ is Gaussian with mean $\mu$ and variance $\sigma^2$, how do I show that $Y_t$ and $Y_{t+h}$ are independent (for $|h|>q$) using the joint pdf. I know th …
- 221
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91
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Find best linear predictor of $X_2$ given $1, X_1$
I'm having a problem calculating the best linear predictor of a time series. I'm using the book Brockwell-Davis 2016 - Introduction to Time Series and Forecasts. First let me write down one notational …
- 221
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68
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Is the mean of a stationary timeseries the same everywhere?
Say for example I have the white noise process $Y_t\sim\text{WN}(\mu,\sigma^2)$. Is it true that $\mathbb{E}[Y_t]=\mathbb{E}[Y_{t-h}]$, where $h\in\mathbb{N}?$
- 221
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163
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Question about slides in lecture note: What if we can't assume $\mu=0?$
The question popped up when I was reading these lecture notes online. Consider the MA$(1)$ process given by $X_t=W_t+bW_{t-1}$ where $W_t$ is white noise distributed with constant variance $\sigma_W^2 …
- 221
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82
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Show that $\text{Cov}[X_r,X_s]=\text{Cov}[X_{r+h},X_{s+h}]$ for $X_t=a+bZ_t+cZ_{t-2}.$
Problem: Let $\{Zt\}$ be a sequence of independent normal random variables, each with mean $0$ and variance $\sigma^2$, and let $a$,
$b$, and $c$ be constants. Is $X_t=a+bZ_t+cZ_{t-2}$ a (weakly)
sta …
- 221
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199
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Why is $Z_t$ uncorrelated with $X_{t-1}$ in $X_t=\theta X_{t-1}+Z_t$?
In a solution to the problem below, the teaching assistant solves it by calculating $\mathbb{E}[X_t^2]$ and ends up with also having to calculate $\mathbb{E}[X_{t-1}Z_t]$ after expanding the square. T …
- 221
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Show that $\text{Cov}[Z_t,Z_{t+h}]=\text{Cov}[Z_s,Z_{s+h}].$
Problem: If $X\sim\text{WN}(\mu,\sigma^2).$ Let then $Z$ be the process defined by \begin{equation}
Z_t=\sum_{i=0}^na_iX_{t-i} \end{equation} for some coefficients $a_1,...,a_n\in\mathbb{R}$ wit …
- 221