Mathematics Stack Exchange
Mathematics Stack Exchange

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.

A time-series model is one which postulates a relationship amongst a num- ber of temporal sequences or time series. An example is provided by the simple regression model

$$y(t) = x(t) \beta + \epsilon(t)$$

or more commonly,

$$y(t) = \sum_{i=1}^p \phi_i y(t-i)+ \sum_{i=1}^k \beta_i y(t-i)+\sum_{i=1}^q \mu_i \epsilon(t-i)$$

965 questions
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How to compute lag operator in time series

i'm trying to understand time series and lag operators. However there is a lot of stuff about it on the internet (and too on stack overflow, but not what I'm looking for), I cannot understand, how to compute Lag (sometimes called Backshift) operator…
joettriscik
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Finding autocovariance of AR(2)

Could someone explain why I didn't get the same answer? This is an AR(2), $$X_t={\phi_0}+{\phi_1}{X_{t-1}}+{\phi_2}{X_{t-2}}+{Z_t} \hspace{6 mm} \text{where } Z_t \sim WN(0, \sigma^2)$$ $\begin{align*} \gamma(0) &= \rm{cov}(X_t, X_t)\\ …
user13985
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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 want to use such a symbol in a formalization of a…
Jarno
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Delay embedding for an irregularly-sampled time series

I'm interested in exploring some time series from the point of view of a delay embedding and Taken's theorem. All the examples I have seen of time delay embedding involve regularly (evenly) sampled time series where lagged versions of the observed…
Gavin Simpson
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What is a causal process in time series analysis?

Hi can someone please explain to me what Causal process means for time series process? I know the formulas that process has to satisfy to be causal but I want to get a better intuition, thanks!!
butterbetter
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Stationarity of an AR(1) process

Suppose we have a AR(1) process $X_t=\theta X_{t-1}+Z_t$ with $t\in\mathbb{Z}$ and $\theta\in\mathbb{R}$ and $Z_t$ white noise. I already know how to derive the fact that if $|\theta|>1$ or $|\theta|<1$ then there exists a stationary solution. Also…
user155670
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Best Linear Predictor

If $X$ has $0$ mean and second moment $\sigma^2_X<\infty$ and is observed with some error $Z$, which is White noise independent of $X$ with second moment $\sigma^2_Z$ (again finite) i.e $Y = X + Z$ How do I find the best linear predictor of $X$ and…
elbarto
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How to find expectation of independent white noise?

$x_t=w_tw_{t-1}$, where $w_t$~$N(0,\sigma^2)$, $w_t$ uncorrelated. $y_t=x^{2}_{t}$ $$E[y_t]=(\sigma^2)^2$$ Compute autocovariance $\gamma(h)$ of $y_t$, when $h=1$. $$cov(y_t, y_{t+1})=cov(x^{2}_{t}, x^{2}_{t-1})$$ $$=cov(w_tw_{t-1},…
user13985
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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 model with positive difference, we have already…
tealing123
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Forecasting Future Revenue Data For ROI Calculation

I have some daily revenue data and I am trying to calculate the return on investment (ROI) by predicting what the expected revenue 'should be' and comparing it to what the company actually made. I used a moving average time series method on the…
Dino Abraham
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Why would this statement be true? "Every iid process is strictly stationary, but it may not be weakly stationary."

After learning the basic definitions of stationarity and iid, there was a question on my exercise: "Every iid process is strictly stationary, but it may not be weakly stationary." On the definition of weak stationarity, it doesn't usually highlight…
Sehyun Cho
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If $X_t$ is an ARIMA(1,1,1) process, what is $Y_t = Y_{t-1} + X_t$?

Q: If $X_t$ is an ARIMA(1,1,1) process, what is $Y_t = Y_{t-1} + X_t$? Attempted solution: $X_t$ is an ARIMA(1,1,1), i.e $\nabla X_t = X_t - X_{t-1} = Z_t $ where $Z_t$ is a casual ARMA(1,1) process and satisfies $(1-\phi_1 B)Z_t =…
Oskar
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Closed-form expression for h-step forecast for AR(2) process

I'm trying to derive $\mathbb E[y_{t+h} \mid y_t]$ where $y_t$ is an AR(2) process $$ y_t = \phi_1 y_{t-1} + \phi_2y_{t-2} + c + \varepsilon_t $$ As per usual I'm recursively applying $\mathbb E[y_{t+h} \mid y_t] = \mathbb E[\mathbb E[y_{t+h} \mid…
user126540
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Book or a source to study time series analysis

Can anyone plz suggest me a book for time series analysis. I am a beginner in this topic, but there is a course in my college and for that I need a book which gives all information in detail. I am doing major in business and studying time series in…
ARPIT GAUR
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Spooky correlation between two independent random walks

Let $x$ and $y$ are two arrays of independent random numbers with Gaussian normal distribution. $X$ and $Y$ are their accumulated values at each step, $X_i = \sum_{k=0}^i x_k$ $Y_i = \sum_{k=0}^i y_k$ Even though there is no correlations between $x$…
wang1908
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