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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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Bartlett’s formula for variance

I have an ACF plot done on python where I’ve got a shaded blue region which is of a curved shape. I am trying to explain why this is the case and so far I’ve been able to establish that it’s because ...
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12 views

Mean function and the auto-covariance function for {Yt}.

Struggling with understanding some of the basics for calculations around auto-covarance functions with the following question: Suppose that ${X_t}$ is a stationary time series with mean $µ=0$ and ...
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11 views

Guess date from numeric string [on hold]

I have a list of numeric strings such as: enter image description here Is there any way I can guess if these are in any way associated with a date?
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25 views

OLS estimator of AR($1$) is biased

Suppose that we have a sample $X_0,X_1,\ldots,X_n$ from the AR($1$) model given by $$ X_t=\phi X_{t-1}+\varepsilon_t $$ for $t\in\mathbb Z$, where $|\phi|<1$ and $\{\varepsilon_t\}_{t\in\mathbb Z}$ ...
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2answers
51 views

Decomposing Sums of Random Variables

Suppose I have $M$ random variables, and a number of realizations of each variable. Each RV has the probability mass function: $$\rho_{X_i}(x) = \begin{cases} p_i, & x = 1\\ 1-p_i, & x = 0\\ 0,...
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14 views

Second-Order stationarity condition for complex-valued autoregressive process

Let $\{c_n\}$ be a complex-valued discrete autoregressive process of order $p$, $\mathsf{AR}(p)$, such that: \begin{equation} \label{cn} c_n = \sum\nolimits_{i=1}^{p}\rho_i c_{n-i} + w_n, \quad n \in (...
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22 views

Symbol for the set of all timestamps?

Is there any convention to denote the set of all timestamps such as $\mathbb{T}$? By timestamp I mean date and time that could indicate events in an irregular time series. I am just asking because I ...
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7 views

Lag selection for daily data under AIC

I have daily fx equity data and am applying a VAR model. How many lags should be ideally used for daily time-series data?
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9 views

why are the eigenvectors in cointegration test taken as estimates for cointegrating vectors?

I am currently studying cointegration using Tsay's Multivariate Time Series Analysis. when explaining the Cointegration Test(Johansen Test), the textbook explained the details of how the Likelihood ...
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1answer
23 views

Conditional expectation $\operatorname E[\varepsilon_s\varepsilon_t\mid X_1,\ldots,X_{n-1}]$

Suppose that $\{X_t\}_{t\in\mathbb Z}$ is an AR($1$) process given by $$ X_t=\mu+\phi X_{t-1}+\varepsilon_t $$ for $t\in\mathbb Z$, where $\mu\in\mathbb R$, $|\phi|<1$ and $\{\varepsilon_t\}_{t\in\...
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Autoregressive process with random walk perturbation.

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $...
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6 views

Prediction of time series based on lagged correlations

I have several questions. I will split the text up in one high-level description of the goal of my exercise, a detailed description of my potential solution and finally my actual questions. Please ...
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1answer
19 views

What does $\rho_k = 0$ mean?

I'm reading Introductory Time Series with R in a section where the correlogram is discussed. I'm confused by one of the statements: If $\rho_k = 0$, the sampling distribution of $r_k$ is ...
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11 views

Find variance and autocorrelations for AR(3) model

$r_t -.01 = .4(r_{t-1} - .01) + .3(r_{t-2}-.01) + .23(r_{t-3} - .01) + a_t$ I found $E[r_t] = .01$ and that $r_t$ is weakly stationary. Now, $Var(r_t) = E[r_t^2] - E[r_t]^2$ so I just need to find $...
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14 views

Mathematical concept of time series modeling

I am trying to learn about time series (self learner), and I read about the mathematical (or statistical) concept behind analysing time series, however I am still confused about some notions. The ...
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1answer
43 views

Smoothness of a time series: relationship between ARMA model and signal derivatives

I have a discrete stochastic function $f(t_k)$ (in my case, it's a time signal related to atmospheric parameters). I know this function to have a certain smoothness. One way I can specify its ...
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13 views

k order backward difference reduce k degree polynomial

$X_t$ is a k degree polynomial at t. How to deduce that $\triangledown ^kX_t$ is a constant?
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1answer
31 views

Shift in Time Domain by altering Fourier-Coefficients

I want to apply a circular shift to a time-series, by changing the phase of the fourier-coefficients. But I run into trouble. In the current state the true sequence starts in the middle of the time-...
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1answer
30 views

Interpretation of the p-value and the test-statistic W of the Shapiro.test in R

Shapiro-Wilk normality test data: Part1 W = 0.14846, p-value = 6.478e-16 Shapiro-Wilk normality test data: Part2 W = 0.47978, p-value < 2.2e-16 Shapiro-Wilk normality test data: Part3 ...
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24 views

Time series question about autoregressive process

I am stuck on this time series question. I've included a photo of the question and the work I've done so far. I think I have a mistake in part (a) because part (b) isn't working out correctly. My ...
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1answer
19 views

How is the second-order difference equation derived?

I'm reading through some documentation on timeseries. The first-order difference for timeseries is given as: $y'_{t} = y_{t} - y_{t-1}$ The second-order difference for timeseries is given as: $y''...
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9 views

Prove a time series to be NOT identically independent distributed

I am trying to prove that this time series (given that $X_{t}$ and $M_{t}$ are iid and independent of each other) $$ Y_{t} = X_{t}(1-X_{t-1})M_{t} $$ is not i.i.d, so my understanding is that I need ...
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16 views

Biased estimates of Hurst exponent in R/S analysis

I've used the standard R/S algorithm for estimating the Hurst exponent in Mathematica*, and tested it on fBm and fGn for $H\in\{0.05,0.1,\ldots,0.95\}$, generating 1000 time series for each $H$. The ...
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0answers
9 views

Deriving AR(1) Time Series Model with repeated substitutions

I am simply trying to go from formula (2.10) to (2.11) in Analysis of Financial Time Series from Ruey Tsay. What I am not understanding is how the substitution results in 2.11 do not include the mean,...
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1answer
21 views

Finding the ACF and PACF

Q: Find the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the following AR(2) process up to and including lag 3: I am trying to understand how to find the ACF and ...
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29 views

Optimization of a function of time-series data.

Consider a non-linear function $f(x_t; \theta)$. I have datapoints $\{x_t, y_t\}$ and I am trying to optimize $\theta$ for some cost function $C(y^\prime_t,y_t)$ where $y^\prime_t = f(x_t;\theta)$. I ...
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1answer
16 views

Partial permutation of time sequence data that keep order of events

Suppose you have sequence S of N elements that are descending ordered by time. How many ways can you take K element subsets from S preserving time descending ordering? example for sequence S={A,B,C,D,...
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1answer
22 views

Forecasting of stationary time series: $WN(0,\sigma^2)$

I am trying to solve the next problem: Let the time series $X_n, n ∈ \mathbb Z$ be a $WN(0, σ^2)$. Find an optimal (in mean square sense) predictor for $X_{n+1}$ if you can observe: 1) $X_n$, 2) $X_{...
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25 views

Check for stationarity $X(t) = V * \cos(wt+U)$, V and U are indep. rand. variables

Can somebody help me please to figure out how to solve this problem: Are the following process stationary? $X(t) = V \cos(wt+U), t\epsilon R^1$ U and V are independent variables, $U$ ~ $Uniform(-...
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9 views

What is the big difference of Time series and Cross sectional data observation?

First of all, What I know is that the i.d.d cannot be applied at each observation and another observation is not the same because at each time point that data point can be any possible value at that ...
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11 views

How do I find the expectation of a non-stationary auto-regressive time series, with absorbing states?

Apologies for not knowing Latex! Consider the following recursive function: $$ y(t+1) = y(t) (1+r) + R - e(t) $$ Where $r,R$ are known constants, $r>0$, and $e(t)$ is distributed as a truncated ...
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1answer
24 views

Determine stationarity of time series containing sin of white noise [closed]

Could someone help me determine the stationarity of the the following time series Y? $ Z_t $ represents white noise with variance $ \sigma^2 $. $ Y_t = \sin(Z_t) + Z^2_t - Z_{t-1}$ I have tried ...
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16 views

energy density spectrum vs energy spectral density

I am doing a project on ocean wave simulation and there is a formula I am trying to test. It is called the random coefficient scheme and it is meant to simulate a random time series. One part of the ...
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10 views

Convergence result for approximation error - stationary AR(1) and finite order MA

I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
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10 views

Metrics for seasonal time series forecasting accuracy

A time series $Y=\{1,0,1,0,1,0,1,0\} $ is forecasted using two different algorithms. The results are $A=\{0,1,0,1,0,1,0,1\} $ and $ B=\{0,0,0,0,1,1,1,1\}$. While my application would perform much ...
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13 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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0answers
10 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
24 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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30 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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41 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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1answer
190 views

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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0answers
14 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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33 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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12 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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0answers
32 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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0answers
12 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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12 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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51 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
41 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
45 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 ...