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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: \begin{equation}\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} ...
entropy's user avatar
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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 ...
rasmus wiuff's user avatar
2 votes
1 answer
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 ...
Wittgenstein's Poker's user avatar
1 vote
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21 views

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 ...
lineardepression's user avatar
-1 votes
0 answers
11 views

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 ...
andrewH's user avatar
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21 views

Time series projection problem, the stationarity is unnecessary here?

Problem (Time-Methods-Peter-J-Brockwell problem 2.8): Suppose $\{X_t,t=1,2,\cdots\}$ is a stationary process with mean zero. Show that $$ P_{\bar{sp}\{1,X_1,X_2,\cdots,X_n\}}X_{n+1}=P_{\bar{sp}\{X_1,...
onsdriver's user avatar
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8 views

What is the spectral density matrix, in the context of vector stochastic processes

I cannot seem to find any good/easy to read resources on the spectral theory, and in particular for multivariate stochastic processes. I want to know: Any resources explaining spectral theory ...
Dylan Dijk's user avatar
1 vote
0 answers
38 views

Using Gradient Descent for coefficient estimation in ARMA model

I'm trying to implement an ARMA model from scratch using gradient descent with adam optimizer to estimate its coefficients . I know it might not be the ideal solution. But the thing that I'm mostly ...
LNTR's user avatar
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20 views

Truncated Fourier series of Benth and Benth

I am currently working on this paper https://www.duo.uio.no/bitstream/handle/10852/10566/pm12-05.pdf?sequence=1 of Benth and Benth, and they use a truncated Fourier series to fit the seasonal part of ...
Valentin's user avatar
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13 views

Problem with understanding the expected value $E(y_t)$ in definition of a stationary time series.

I am trying to understand the conditions for time series $\{y_t\}$ to be stationary, i.e.: $E(y_t)=\mu$ is constant for all $t >0,$ $Var(y_t) = V$ is constant for all $t >0$ and $cov(y_t, y_s)$ ...
Brzoskwinia's user avatar
3 votes
2 answers
151 views

The "turning-point fraction" of a random sample from a discrete distribution must have expectation less than 2/3?

A sequence of reals $x_1,...,x_n$ is said to have a turning point at index-value $i$ ($1\lt i\lt n$) iff $x_{i-1}\lt x_{i}\gt x_{i+1}$ or $x_{i-1}\gt x_{i}\lt x_{i+1}$. The number of turning points ...
r.e.s.'s user avatar
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19 views

Conditional Expectation Notation in ARCH Model

I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd ...
Newbie's user avatar
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1 vote
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Independence of time in one-step error

Let $\langle V, \langle\cdot, \cdot \rangle \rangle$ be an inner product space. We say $\varphi:\mathbb{Z}\to V$ is stationary if there exists a function $\gamma:\mathbb{Z}\to \mathbb{C}$ such that $$\...
André Armatowski's user avatar
0 votes
1 answer
29 views

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 ...
Bazool's user avatar
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2 votes
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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 ...
tealing123's user avatar
1 vote
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69 views

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 ...
Curious student's user avatar
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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 ...
openspace's user avatar
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26 views

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 :...
Ippotis TheKing's user avatar
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17 views

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,...
bosco98's user avatar
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Approximating the half-life of a shock to a system?

I found the following statement in here regarding the effect of twice lagged differences of CO2 ($\Delta C$) in the atmosphere on the once lagged values, i.e. $$\Delta C_{\text{ @ }t=-1}= 0.83 \times ...
JAP's user avatar
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6 votes
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90 views

Show that $X_{t}:=\alpha X_{t-1}+\epsilon_{t}$ is strictly stationary for $|\alpha|<1$ and $\epsilon_{t}$ i.i.d$~\sim N(0,\sigma^{2})$.

The title can be shortened to "prove that $AR(1)$ processes are strictly stationary when $|\alpha|<1$". This has been discussed many times on MSE and Cross Validated, but I found no ...
JacobsonRadical's user avatar
1 vote
0 answers
39 views

reference for time-inhomogeneous random walk

I have a problem with a Markov chain that is time-inhomogeneous. Consider a Markov chain $(X_t)$ in discrete time $t=0, 1, \dotsc$ on the integers. At each time $t$, the process can go up by 1, down ...
Stephan W's user avatar
1 vote
0 answers
31 views

Conditional mean of an ARMA($p,q$) process

This question is about a paragraph in The Analysis of Time Series: An Introduction with R (7th Edition) by Chatfield and Xing. I quote Section 12.1, p. 135: In particular, suppose that $X_t$ follows ...
Balkys's user avatar
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27 views

Degrees of Freedom in PCA

Suppose we are doing PCA over a historic time series of temperatures. The feature for the PCA to be explained is the time when the temperature was observed every hour. Let’s say for the sake of ...
Identicon's user avatar
3 votes
1 answer
55 views

Find conditional MLE of AR time series

I was given a model $r_{t} = ϕ_{0} + ϕ_{2}r_{t-2} + ϵ_{t}$ with $\epsilon_t \sim N(0,\sigma^2)$ and have to derive the likelihood of $(r_{3}, r_{4}, . . . , r_{T})$ conditional on $(r_{1}, r_{2})$ and ...
Tin Dao's user avatar
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Non-negative autoregressive model

I am trying to fit a linear 1-order autoregressive model to some multivariate time-series data. The model I am using is of the form $$x_t = Ax_{t-1}+\xi_{t-1}$$ and I am solving it in R using the mAr ...
citizenfour's user avatar
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0 answers
35 views

Optimization Models and the chicken-and-egg problem

I have started working with optimization models to minimize the total generation costs of electricity in a system. The basic idea in this model is to minimize the overall system cost given a certain ...
Eve Chanatasig's user avatar
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0 answers
12 views

Incorporating the absolute difference and the relative difference in a single metric

I have a series of 12 connected 'zones' that have a specific numerical value at any given point in time, either negative or positive. This amounts to time series data. I have some forecasted values ...
Tom's user avatar
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1 answer
52 views

Understanding correlation in the context of a time series, simulation and brownian motion

I have the doubt of calulating and meaning of correlation. I know it is from my incapacity to grasp a concept, specially regarding time series but would appreciate any comments on it. I think I ...
Curious student's user avatar
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0 answers
21 views

Solve VAR(2) for the n-step ahead forecast

I'm trying to find for this VARX*(2) $$x_t=a_0+a_1t+F_1x_{t-1}+F_2x_{t-2}+\Theta_0d_t+\Theta_1d_{t-1}+\Theta_2d_{t-2}+\varepsilon_t$$ an explicit form for $x_{T+n}$, i.e. solve it as an equation for ...
Meh Mech's user avatar
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0 answers
45 views

Distribution of AR(1) process with unit roots - why does one need the Functional Central Limit Theorem?

I was reading Chapter 17 of Hamilton's "Time Series Analysis" about univariate processes with unit roots. In particular, I am looking at the AR(1) process $y(t) = y(t-1) + \epsilon(t)$ with $...
lostmathematician's user avatar
1 vote
1 answer
41 views

Could someone please help me figure out the p value of this ARIMA(p,d,q) model?

The exercise is the following: (b) (4 points) Consider the following model $$ \begin{array}{r} X_t=c+X_{t-1}+\phi_1\left(X_{t-1}-X_{t-2}\right)+\phi_2\left(X_{t-2}-X_{t-3}\right) \\ +\phi_3\left(X_{t-...
Ray Robson's user avatar
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0 answers
40 views

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 $\sigma^2$...
Balkys's user avatar
  • 749
0 votes
0 answers
19 views

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, ...
Raidriar's user avatar
4 votes
2 answers
184 views

Is an AR(1) process with Bernoulli errors mixing or ergodic?

Before the $\text{AR}(1)$ model, first look at a simpler example $$y_t=\rho^t y_0+\epsilon_t$$ where $0<\rho<1$ and $\epsilon_t\overset{\text{i.i.d.}}{\sim} \text{Bernoulli} \left(\frac{1}{2} \...
Jack's user avatar
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0 answers
26 views

Derivation of Double (Brown) Exponential Smoothing

I am learning the double (Brown) exponential smoothing. Comparing to the simple exponential smoothing, the Brown exponential smoothing smooths the output sequence $Y_t$ twice, such that \begin{align} ...
Stephen Ge's user avatar
1 vote
0 answers
49 views

Calculate $Var\left(\frac{X_1+\cdots+X_n}{n}\right)$ and estimate $\sigma_{\bar{X}}$

Suppose that in a sample of size $n = 100$ from an AR(1) process with mean $\mu$ $$X_t - \mu = \phi (X_{t-1}-\mu) + Z_t$$ where ${Z_t} \sim \operatorname{WhiteNoise}(0, \sigma^2), \sigma^2 = 2, \phi = ...
dienhosp3's user avatar
  • 573
0 votes
1 answer
74 views

Proving $S_m=\sum\limits_{j=1}^m{\theta^j X_{n-j}}$ converges in mean square as $m\to \infty$

Suppose that $\{X_t, t = 0, \pm1,\dots\}$ is is stationary and that $|\theta|<1$. Show that for each fixed $n$, the sequence $S_m=\sum\limits_{j=1}^m{\theta^j X_{n-j}}$ converges in mean square as $...
dienhosp3's user avatar
  • 573
0 votes
0 answers
24 views

Generic Chaining/Dudley's integral for supremum of average of indicator random variables in sup norm metric space?

I want to use generic chaining/ Dudley's integral to bind the below stochastic process \mathbb E\sup_{tin T}\frac{1}{n}\sum_{i=1}^nX_{i,t} where X_{i,t} takes value either 1 or 0 (binary random ...
Ankita Ghosh's user avatar
1 vote
0 answers
38 views

Time series in Hilbert spaces and expectation of a projection

Let $X_{t}$ be a stochastic process. For the realizations or random variables the natural metric is the standard deviation and the covariance as the inner product. For simplicity sake, lets assume ...
Fledermaus's user avatar
1 vote
0 answers
22 views

Linear Combination of Two Stationary Time Series - Autocovariance Function?

I have an autocovariance function $\gamma_x(k)$ for a process $x_t$ which is stationary. Now, I have another process $y_t = x_t - x_{t-1}$. I'd like to express the autocovariance function $\gamma_y(k)...
SG1931241's user avatar
1 vote
1 answer
232 views

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 ...
dienhosp3's user avatar
  • 573
1 vote
1 answer
66 views

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 ...
Szymon83's user avatar
0 votes
0 answers
66 views

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 ...
Cyclopropane's user avatar
0 votes
0 answers
15 views

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 ...
Taha Abbasi Hashemi's user avatar
0 votes
0 answers
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 ...
Beppe's user avatar
  • 1
3 votes
1 answer
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 ...
guest's user avatar
  • 143
0 votes
1 answer
51 views

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 ...
Matthew Wander's user avatar
1 vote
0 answers
86 views

Non-Causal Autoregressive Time Series

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, ...
Mamad Fasih's user avatar
0 votes
2 answers
157 views

How to prove that a stock's price deviation from its moving average is a mean reversion process?

If you build a moving average from stock price values, then the price graph fluctuates around this moving average. How can one mathematically prove that a price's deviation from its moving average is ...
Roustem Akhiarov's user avatar

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