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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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Square of sample mean of linear processes - martingale decomposition

I've got a sample of linear processes $X_i = \sum_{l=0}^\infty c_l \eta_{i-l}$ for $i=1,\dots,n$, where $\{\eta_i\}_{i \in \mathbb Z}$ is a sequence of independent random variables with mean zero and ...
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9 views

Spurious Regression and Co-integration

I downloaded a ppt file from Spurious Regression and Co-integration On page 3 it says: "In general, regression models for non-stationary variables give spurious results. Only exception is if the ...
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4 views

Spectral density interpretation

I've been told that the above is the formula for the spectral density of a time series, and that omega stands for frequency. However, being unfamiliar with Fourier series and physics, I do not ...
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1answer
18 views

What is the probability distribution of this AR(1) function?

I'm preparing the exam for "stochastic models" and I encountered this exercise which is giving me a lot of problems: Let $X_t \sim AR(1)$, with $$X_t=-0.8X_{t-1}+ \epsilon_t, ~~~~~~~~~~\epsilon_t \...
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21 views

Geometric Brownian Motion as the limit of Binomial Tree

I know that GBM can be discretely approximated by methods such as Euler-Maruyama, and it can be shown that Binomial tree converges to GBM at the continuous time limit. However I'm having a hard time ...
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24 views

Show that sum of these two Random Variables is conditionally normal distributed (from IGARCH model)

According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed. $σ_t^2 = ...
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147 views
+100

Variance of parameter estimate using recursive least squares

I am learning about recursive least squares estimation using a forgetting factor $\lambda$ as a tool for treating time variations of model parameters and have become stuck on the following problem. ...
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8 views

Autocovariance function of $ARMA(3, 1)$ process

The ACF of a causal ARMA(p,q) process is given by the following general homogeneous equation: $$ \gamma(h) - \sum_{j = 1}^p\phi_j\gamma(h-j) = 0, \quad h \geq \max(p, q+1) $$ with initial conditions ...
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26 views

Finding statistically increasing and decreasing sub-sequences in a (noisy) vector

I have a vector of real non-negative values of length ~60. The values represent a geometric property (can be area, circumference, etc.) of an object extracted from a movie of a biological sample, and ...
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11 views

Confidence interval for a GARCH model with R[Time Series problem]

I have the following problem: Given a data file, I have to propose a good model for it, so I have started with an auto.arima() mod1 <- auto.arima(data$x) And it proposed an ARIMA(3,2), I have ...
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29 views

Match peaks of Time Series data better than troughs

I'd like some guidance on which method I can use to better match the peaks in any time series data than the troughs. For example in the following figure the dashed line is the actual sales data, and ...
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9 views

Finding an expression for the autocovariance $\gamma_k$ of a stochastic process Xt

Find an expression for the autocovariance function of the stochastic process {Xt} for general values of q1, q2, ${\alpha_i}$ and ${\beta_i}$. Where ${X_t}$ = ${\sum_{i=0}^a \alpha_i\varepsilon_i}$ + $...
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11 views

How can we study the correlation of two variables showing spatial and/or temporal autocorrelation?

Option 1: Can we eliminate the autocorrealtion of each variable, and then study the correlation? If so, how? Option 2: Whether are there methods that can directly model the correlation of the two ...
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48 views

Conditional distribution at time t+1 given information at time t is normally distributed, showing that conditional distribution of sum is also normal

According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model: A nice property of such a special random-walk IGARCH model is that the conditional ...
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1answer
30 views

backshift operator notation

Original equation: $$\begin{equation} z_t = \phi z_{t-1} + z_{t-1} - \phi z_{t-2} + \omega_t \end{equation}$$ Rewrite the equation, re-arrange terms, and factorize them: $$\begin{align} z_t ...
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1answer
31 views

Is explosive AR(1) stationary?

For simplicity, define the AR(1) model without an intercept term, that is $$ X_t := \phi X_{t-1} + w_t $$ where $w_t\sim N(0,\sigma_w^2)$ and $w_t$ is independent of $X_{t-1}$. Also assume the time ...
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34 views

Where is the error in this naive derivation of the Kalman filter?

The Kalman filter is a method of predicting the future state of a linear state space system based on the previous ones. A linear, discrete-time, stationary, state-space model is a pair of real ...
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9 views

How to deal with expected value in the context of time series?

For example, in this MA(2) model, $y_t = u_t + \phi u_{t-2}$ $u_t$ is identically, independently, normally distributed with a mean of 0 and a variance of $\sigma^2$. (Does variance matter here?) I ...
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12 views

Factorised form of Autoregressive Polynomial

I'm new to Time Series Analysis. I've read that when inverting autoregressive characteristic polynomial of arbitrary finite degree, we need to write it in its factorized form: $$\phi_p(x) = \prod_{i=1}...
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1answer
13 views

Stationary linear random fields

Murry Rosenblatt in his 2000 book "Gaussian and Non-Gaussian Linear Time Series and Random Fields" introduces stationary linear random fields with d-dimensional vector indices thus: \begin{equation} ...
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83 views

Reference request in optimal stopping

I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
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26 views

detect anomaly in time series

I have data on 5 agriculture fields over the same period of time. At some point, there is a sudden change in behavior for all 5 time series, which correspond to the harvest of the field. My task is ...
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23 views

Differenced Time Series Analysis

I have a question regarding time series analysis. Consider I have hourly measurement of O_3 gasses from a given place. I have measurements for approx. a whole month, and there is no missing data. If ...
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2answers
28 views

non-constant acceleration, how to find time and distance?

A roulette ball spins around a rim. What is the time ($t$) and distance ($d$) in which the velocity of the ball $= v$ (whatever number)? The Deceleration of the ball from $4$ spins was plotted on a ...
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Does there exist a closed-form solution to the Kalman filter problem with auto-regressive measurement errors?

Suppose the state variable evolves as $x_t = \rho \cdot x_{t-1} + u_t$ where $u_t$ is mean-zero, normally distributed iid noise. Suppose the observation is $y_t = x_t + \epsilon_t$, where, ...
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48 views

Aggregate and interpolate overlapping time-series data

I'm trying to aggregate counter data from two different types of measurements. The first type of measure gives an exact value of the counter on a given day. ...
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24 views

Finding joint expectations

I am having trouble trying to understand this question and how to proceed. It would be helpful if I can get some insight for it. Thank you Question: Let $X_1, X_2, X_3$ be jointly gaussian with ...
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15 views

Autocovariance function of $MA(q)$

I'm trying to compute the autocovariance function $\gamma_{X}$, where: $$X_t = Z_t + \sum_{k=1}^q \theta_k Z_{t-k} =: (F_{{\alpha}_{\theta}}Z)_t, \; \forall t\in \mathbb{Z},\text{ with :} $$ $$ Z\sim ...
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3answers
99 views

Binary variables in time series: integer linear programming

I'm working on a problem and I can't seem to find an easy solution to it. It's about an optimization problem, concerning a time series. I have a binary variable $\alpha_t$ for $t \in [0, 24[$. I ...
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14 views

Solving a difference equation via z-transforms

I'm trying to figure out if there is a "nice" solution to this difference equation $$\alpha p_t = p_{t-1} + \beta x_t(x_t - x_{t-1}).$$ Using a $z$ transform, I get $$P_z(\alpha - z^{-1}) = \beta ...
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17 views

Why is a specific conditional density function in the state space model context assumed to be normal?

A linear, discrete-time, stationary, state-space model is a pair of real valued stochastic processes $\{X_t \}_{t \in \mathbb{N}},\{Y_t\}_{t \in \mathbb{N}}$ that obey the recursive equations $$ \...
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1answer
19 views

The autoregressive model is a particularity of class general ARIMA?

I'm social science student and recently started studyng time series. My question is if the autoregressive model is a particularity of class general ARIMA? Thanks.
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9 views

How to find Cross Correlation of two series over time containing periodic trends?

Considering the data in the series is real time in nature and there are periodic trends within the series , how to do a Cross-Correlation in real time so that each periodic trend can be identified?
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17 views

Relation between Vector Auto-regressive models and correlation matrix

I am generating a multivariate time series using Vector Autoregressive Models- $$X(t) = AX(t-1) + \epsilon$$ where $X \in R^{n \times 1}$, $A \in R^{n \times n}$ and $\epsilon \in R^{n \times 1}$ is a ...
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1answer
96 views

Approximating the tangent vector in a phase space (or state space) reconstruction

I am investigating an application of differential geometry in experimental dynamical systems. Given a 1D time series (e.g., one that has been experimentally obtained), $x(t)$, I am considering the ...
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18 views

Statistics Time series model problem

$X_t = 2X_{t−1} − 1.7X_{t−2} + 0.7X_{t−3} + w_t − 0.1w_{t−1}$ Find the characteristic equations, ARIMA order, and $E[X_t]$ Also for $E [\nabla X_t]$ and determine if $\nabla X_t$ is stationary and ...
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1answer
47 views

Statistic ARMA model

$Xt = 5+0.8(X_{t−1} −5)+0.8(X_{t−2} −5)−0.6(X_{t−3} −5)−0.5e_{t−1} + 0.25e_{t−2} +e_t$ Write the process using characteristic equations then determine ARIMA order and E[Xt]. Finally, find the values ...
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1answer
31 views

Statistics--Time Series Problem

$X_t$ = $3/2X_{t-1} - 1/2X_{t-2} + 1/2e_t - e_{t-1}$ Write the model into an ARMA form, determine if it is stationary if it is invertible and determine p and q. If it can be reduced, write the ...
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1answer
25 views

How to convert values from one range to another with in-equal slopes and real time data?

There are two series , Sa and Sb. Sa ranges always within 0-1 . Sb ranges variably sometimes from 130-145, 2017-2077 and many more etc.. The data points are real time in nature. For each second ...
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10 views

Is there any theoretical result on how to stabilize a polynomial by changing its coefficients?

The stability of a general $n$ order polynomial is associated with the following statement: if all the roots of the following equation falls in unit circle on the complex plane, then the system is ...
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12 views

Conditional densities and known past values

Let there be a random variable with the following properties $Y_{t} = \mu + \beta Y_{t-1} + \epsilon_{t} \quad\text{such that} \quad \epsilon_{t}\sim \mathcal{N}(0,\sigma^{2})$ Estimate the ...
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8 views

Error in time series forecasting

If I have hourly input data and I want to produce an output forecast of half hourly granularity, I must interpolate the hourly input data - but how can I calculate the average error attributed to my ...
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131 views

How to transform maximum distances between two graphs into cross-over and vice-versa?

Considering there are have two series v1,v2 over time which ranges between 0 and 1. Two graphs are plotted below in colour Blue and Green and the pattern between the graphs is shown in the diagram.How ...
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43 views

Bias adjustment for the Box-Cox back-transformation

I'm learning time series analysis and I don't understand why the back-transform of Box-Cox transformation outputs the median instead of the mean of the forecast distribution. The family of Box-Cox ...
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10 views

How to assess the convergence of a time series towards a periodic signal and assess the sufficiency of the sample size?

I have a sampled temporal signal, result of a transient fluid simulation. At the beginning the flow is being established, so the first few seconds of signal should be ignored (question : is there a ...
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34 views

upper bound of an inequality

Let $X_t = \theta_1\epsilon_{t-1} + ... + \theta_q\epsilon_{t-q} + \epsilon_t$ where $\epsilon_t$ is a white noise with zero mean variance $\sigma^2$. $X_t$ is said to be invertible if $\epsilon_t = ...
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Calculating Coefficients for MA and AR representations of ARMA Models

Was working on a problem - how to find the coefficients for a MA and AR representation of given ARMA models. Specifications in the image. Thanks! Image of Problem
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1answer
28 views

Calculating Mean, Variance and Autocovariance Functions of Time Series

I'm having trouble finding the mean, variance and autocovariance functions of a time series function. Looked around and couldn't find a problem like this. Image attached Vsubt = 1/q * summation Xsub(...
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30 views

Classifying Customers by Purchase Periodicity

I want to classify retail customers into groups reflecting the periodicity of their purchasing behaviors: e.g. weekly, monthly, seasonal, annual, etc. I thought this question was trivial, but now find ...
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15 views

Sample autocorrelation function as $n\rightarrow\infty$

Let $\{X_1,\cdots,X_n\}$ be observed values of a time series at times $1,\cdots,n$, and let $\hat{\rho}(h)$ be the sample autocorrelation function at lag $h$ which is defined as \begin{align*} \hat{\...