# Questions tagged [time-series]

This tag is used for question related to time series models such as AR, ARMA, ARCH, GARCH and their properties and techniques used for inference.

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### Kalman-Filter for modified observation equation?

The Kalman-Filter yields the optimal estimate of the evolution of the hidden state space variable $\mathbf{X}_{t}$ from a sequence observations $\left\{ \mathbf{Y}_{t}\right\} _{t=1\ldots T}$ where \...
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### Deriving ADF unit root test form for the time series with quadratic deterministic trend

I have the following time series process $y_t$ $$\Delta y_t = \delta + \gamma t + \epsilon_t$$ where $e_t$ is white noise process with the variance of $\sigma^2$. I guess that whereas $\Delta y_t$ is ...
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### Determining Values of Parameters so that Observation Equation is Stationary

Consider a system process given by $x_t=-0.9x_{t-2}+z_t$,$t=1,2,…,n$ with observation $y_t=x_t+v_t$ where ${z_t}$ and ${v_t}$ are independent white noise with variances $σ^2$ and $σ_v^2$. Assume that ...
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### Double EMA into Single EMA

The exponential moving average (EMA) operator is defined as: $$y_t(x, \lambda) = (1-\lambda) \sum_{i=0}^\infty \lambda^i x_{t-i}$$ where $1-\lambda$ is the normalization factor, and the operator is a ...
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### Why is it not the case that f(t, x(t)) = f(t)?

What is the difference between: $f(t)$ and, $f(x(t), t)$ Can I write any function $f(x(t), t)$ as $f(t)$? Why or why not? Explanations in detail with supporting examples going from simple to ...
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### How to define the uniform asymptotic negligibility assumption for moving-average process with random coeficients?

The uniform asymptotic negligibility (UAN) assumption is well know in probability theory. In my case, I have a definition of (UAN) for moving-average (MA) processes. Let $(X_n)_{n\geq 1}$ a sequence ...
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1 vote
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### Prove medoids or points closer to cluster centroid have good data representation

I have some time series data points $x_1, x_2, x_3..,x_n$ $\in \mathcal{R}^{d}$. I have extracted the characteristics of those data points (peak, mean, min...) and clustered them based on the ...
26 views

### Double moving average smoothing time series

I´m reading: Regression modeling with actuarial and financial applications by Frees, and in page 274 he talks about a double smoothing procedure for a times series with linear trend: Suppose that we ...
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1 vote
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### Kalman filter with contemporaneous variable

In standard Kalman filter calculations. Namely, the prediction step is given by $\hat{x}_{k|k-1} = F_k\,\hat{x}_{k-1|k-1} + B_k\,u_k,$ $P_{k|k-1} = F_k\,P_{k-1|k-1}\,F_k^\top + Q_k$ Under this setup, ...
1 vote
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### Showing $x_t = \delta_t + \sum_{j=1}^tw_j$

Suppose we have the random walk with drift model $$x_t = \delta+x_{t-1}+w_t$$ for $t = 1, 2, ...$ with initial condition $x_0=0$ and where $w_t$ is white noise. The constant $\delta$ is called the ...
1 vote
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### Time series: ARMA characteristic polynomials have common roots

I have a question regarding the idea that if the roots of the characteristic polynomials of a time series (say some ARMA process) lie outside the unit circle, then the series will be invertible/causal ...
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### Time series analysis on ACF and PACF plots

So I have a non stationary time series that is hourly, daily and monthly recorded for a year and I have the ACF and PACF plots for the serie. ACF and PACF I applied the the Ljung-Box test and for ...
27 views

### Rice formula on time series

I have a time-series which is a mean-reverting spread with a long-run mean (equilibrium) close to zero. I am looking to use the Rice formula to get the level crossings rate, e.g. how many times the ...
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### Practical correlation metric for a large number of vectors

I am dealing with a timeseries consisting of input flow sampled every 5 minutes over 441 days. My aim is to find any possible correlation from data coming from: The same day of the week The same ...
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### Clarify answer for asymtotic of median smoothing

Calculating algorithmic complexity for median smoothing in Time Series I came up with the question same as this. Quote it here: A time series with T observations is given. Median smoothing with width ...
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### Why do GLS and ML estimators coincide for the estimation of a VAR(p) model?

When estimating the coefficients in a VAR(p) model (assuming normality), the coefficient estimators using GLS and MLE coincide. Could anyone explain why this is the case?
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### ARMA-GARCH model Gaussian assumption

I have some questions concerning a ARMA(p,q)-GARCH(a,b) process, specified by $$y_t=\mu+\sum\limits_{i=1}^{p}\phi_i(y_{t-i}-\mu)+\sum\limits_{i=1}^{q}\theta_j \varepsilon_{t-j}+\varepsilon_t,$$ \...
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