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 use GARCH model output variance

I have some difficulties to fit a GARCH model. The model input are log returns from a time series. The ouput should be the conditional variance forecast from this same time series. My problem is that when i calculate the stdev/var directly from the…
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Invertibility of MA(2) process

For MA(1) process, it is easy to show how one can convert it into AR($\infty$). However, how can we really show that MA(2), giving its characteristics roots lie outside unit circle, can have invertibility? Consider…
Jacky
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Intuitive and mathematical explanation of stationarity for AR(p)?

I'd be grateful if someone could explain me what's the link between stationarity conditions of AR(p) processes in theory and in the practical sense. I was given the following short definition about weakly stationarity: a process is weakly stationary…
Nenne
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Proving forecast errors for AR(1) h step ahead is $\epsilon_{T+h}+\beta \epsilon_{T+h-1}...\beta^{h-1}\epsilon_{T+1}$?

I'm having a little difficulty proving the corresponding forecast errors for an AR(1) model with a horizon of h. In other words, I want to show that $V_{T+h|T} = \epsilon_{T+h}+\beta \epsilon_{T+h-1}...\beta^{h-1}\epsilon_{T+1}$ given AR model $y_T…
Nikitau
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Variance of AR(2) process

I have two questions: 1) When one says an ARMA process is 'stationary,' do they mean strongly stationary or weakly stationary? 2) Is there a quick way to find the variance of a stationary AR(2) model $$y_t = \beta_1 y_{t-1} + \beta_2 y_{t-2} +…
user369210
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Moving Average and Time Series

I just recently started to learn about moving average process of order 1, however, I get confused if there are other things attached to the equation. For example: $Z_t = 8 + 2t + 5X_t$ where $X_t$ is a zero-mean stationary series with auto…
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Process in terms of another stationary process - need to find the acf.

I have a process, $Y_t = X_t - X_{t-1}$. $\{X_t\}$ is a stationary process with autoccovariance function $\gamma_{X}(k)$. I wish to find the autocovariance function of the process $\{Y_t\}$ in terms of $\gamma_{X}(k)$. I think I am correct in…
user117682
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time series: find PACF

given causal ARMA process $Xt$, find PACF: $X(t) - cX(t-1) - c^2X(t-2) = Z(t)$ $1 - cz - c^2z^2 = 0, $ $ z= \frac{-1 + \sqrt{5}}{2a}$ and $z= \frac{-1 - \sqrt{5}}{2a}$ My book (Introduction to Time Series, Brockwell & Davis) defines every…
dark blue
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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 $\gamma_y(k)$ in terms of $\gamma_x(k)$. I assume that it'd…
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If $X_t$ is a AR(2)-process, what is $Y_t = X_t - X_{t-1}$?

Q: If $X_t$ is a AR(2)-process, what is $Y_t = X_t - X_{t-1}$? $X_t$ is a AR(2)-process if it's stationary and satisfies $X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \epsilon_t$, where $\epsilon_t$ is white noise. One may elaborate…
Oskar
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Showing $x_t = \delta_t + \sum_{j=1}^tw_j$

Suppose we have the random walk with drift model $$x_t = \delta+x_{t-1}+w_t$$ for $t = 1, 2, ...$ with initial condition $x_0=0$ and where $w_t$ is white noise. The constant $\delta$ is called the drift, when $\delta$=0, the equation above is simply…
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Autocorrelation of $\frac{1}{9}(\omega_{t-1}+\omega_{t}+\omega_{t+1})$

Find the autocorrelation function of $v_t=\frac{1}{9}(\omega_{t-1}+\omega_{t}+\omega_{t+1})$ What I have tried: $$\rho(h) = \frac{\gamma(t+h, t)}{\sqrt{\gamma(t+h, t+h)\gamma(t,t)}}=\frac{\gamma(h)}{\gamma(0)}$$ I have actually got this for $h=0$,…
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COVID time series

I was asked this question I took a couple semesters back, but the prof never posted the solution and I got it wrong. The question went as "Obtain a time series of the monthly COVID cases in the US since March 2020. Fit a time series to the data…
Joey
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How can one build a time series forecasting model using non temporal features?

I am new to time series forecasting and all the articles that I read online talk about time series forecasting using a variable that is tracked over time and then forecasted after the model is trained. These examples include - sensor readings over…
Regressor
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conversion from per year to per decade for time-series data

I was wondering if this logic makes sense. I have a time series where the slope is -0.19 over 21 years, so the rate would be -0.19 per year. Now if I want a rate per decade, would it logically make sense to multiple the yearly rate by 10 to achieve…
Bubs
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