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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Equivalent estimation of ARMAX models of time series

Suppose I have an ARMAX model as follows: \label{eq:ARMAX} \begin{split} \ln{\epsilon_t}^2=\phi_0+&\sum_{i=1}^{p_1} \phi_i \ln{\epsilon_{t-i}^2}+\sum_{j=1}^{p_2} \phi_{p_1+j} ...
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Normal distribution comparing n variables, are there different solutions?

I am looking for the probability that the variable in year $y_i$ is the largest in a time series. The mean follows a normal distribution $N(μ_i, σ)$ in which $μ_i = α + ßy_i$, $α$ and $ß$ are ...
50 views

Convergence of weighted sum to Brownian Motion

Let $\{\varepsilon_t\}_{t = 1}^T$ be a sequence of iid random variables such that $\varepsilon_t \sim N(0, \sigma^2)$ and $\sigma^2 > 0$. Then it is known that (see 17.3.6 in James Hamilton's Time ...
1 vote
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GLMs where response variable is calculated from multiple data points in a time series?

Hopefully this isn't too broad of a topic: I'm a student research assistant working with matched gene expression counts data in a time series. As a simplified example, say I have two sets of time ...
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How can I coefficients for a model of n linear differential equations in n unknowns [migrated]

I have (a lot of) time series data, to which I wish to fit a system of n linear differential equations in n unknowns, where by "fit" I mean to find coefficients on the unknowns in a way that ...
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1 vote
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Sum of autocorrelation coefficients

This is a follow-up to this thread: Proof that sum over autocorrelations is -1/2 I am posting a new thread as that was posted 6 years ago. In that threat the stackexhange author (Kuhlambo) lists some ...
10 views

ARIMA(p, d, q) with d > 0 is non-stationary

My textbook says: An ARIMA(p,d,q) process with $d > 0$ is not stationary and therefore has no stationary variance. Is this just to say that once we have decided to model a process with an ARIMA ...
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1 vote
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What does it really mean to take correlation between time series? [closed]

I have a conceptual problem when we extend the correlation to time series. I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from ...
27 views

EM algorithm for Markov switching models.

Consider the model $y_t = F_{S_t} x_t + \varepsilon_{S_t}$ and $x_t = A_{S_t} x_{t-1} + \nu_{S_t}$, where $\varepsilon_{S_t}, \nu_{S_t} \sim N(0, R_{S_t})$ and $N(0, Q_{S_t})$ and $S_t$ is Markov ...
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Variance and autocovariance or process $W_t = Y_t-Y_{t-1}$ with $Y_t$ process AR(1)

I need to find the variance and autocovariance or process $W_t = Y_t-Y_{t-}$ with $Y_t = c + \phi_1Y_{t-1} + E_t$ an AR(1) process with $-1 < \phi_1 < 1$ I end up with the following calculation :...
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General state space form for ARIMA(p,d,q)(P,D,Q)

I cannot for the life of me find a resource which gives the general state space form for an ARIMA(p,d,q)(P,D,Q) model. I am reading Time Series Analysis by State Space Methods, by Durbin & Koopman,...
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Spectral density of $\{Y_t\}$ where $Y_t - \alpha Y_{t-1} = X_t + W_t$, where $\{X_t\}$ is an AR process and $\{W_t\}$ is white noise

This is a question from Chapter 4 of Time Series: Theory and Methods by Brockwell and Davis (1991). Question: Let $\{Z_t\}$ and $\{W_t\}$ be white noise processes with mean zero and variance $\sigma^2$...
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Finding the spectral measure of a of a weakly stationary process

For these exercices, I am asked to find the spectral measure of their processes if they are weakly stationary. However, I do not understand how to do so. https://i.sstatic.net/vUQIB.png For exemple, ...
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1 vote
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Let ${Y_t}$ be a stationary process with mean zero and let $a$ and $b$ be constants. Prove $X_t$ is a stationary

Let ${Y_t}$ be a stationary process with mean zero and let $a$ and $b$ be constants. Show that $X_t = Y_t - Y_{t-1} - Y_{t-12} + Y_{t-13}$ is stationary. I encountered this issue when working on a ...
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1 vote
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Cointegration - different units, order of variables. [closed]

If two series $\{x_y\}$ and $\{y_t\}$ are not stationary but their a linear combination of them, say $u_t = \beta x_t - y_t$, is a stationary process, then we say $\{x_t\}$ and $\{y_t\}$ are ...
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Prediction interval for AR(1) forecast

This link (and others, e.g. slides 43 and 46 of this) say that: Where all the coefficients in the model are point estimates, we could calculate the MSE to generate distributions for the distribution ...
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How to take the conjugate of a single point in a purely real discrete time series (e.g. ECG). As seen in the auto-Wigner Distribution

Im trying to implement the Wigner distribution and I'm stuck on how to take the conjugate of a single point in a function. I can understand finding the conjugate of a function, and the conjugate of a ...
18 views

Fractional differintegral for the reconstruction of time series

How do economists and climatologists reconstruct their time series from the available data? I know in particular that economists have built models to extimate the GDP per capita between the year 0 and ...
115 views

Simple algebra of integral [closed]

I'm studying economics and I'm having trouble with math calculations. In my economics textbook, the following equation comes out, and for me it's hard to understand how this relationship is ...
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For any chaotic system such as a chaotic attractor does there exist a higher dimensional representation in which the system is no longer chaotic? [closed]

For a given n dimensional chaotic system such as a chaotic attractor or really a time dependent chaotic dynamic system does there exist a higher dimensional representation where the behavior is in ...
1 vote
The autoregressive process $\phi(B)X_t = Z_t$ is called causal if the autoregressive characteristic function $1 - \phi_1z - \cdot \cdot \cdot - \phi_pz ^ {p}$ has no zeroes inside of the unit circle, ...