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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
18 views

Kalman-Filter for modified observation equation?

The Kalman-Filter yields the optimal estimate of the evolution of the hidden state space variable $\mathbf{X}_{t}$ from a sequence observations $\left\{ \mathbf{Y}_{t}\right\} _{t=1\ldots T}$ where \...
  • 1
-1 votes
0 answers
18 views

Deriving ADF unit root test form for the time series with quadratic deterministic trend

I have the following time series process $y_t $ $$\Delta y_t = \delta + \gamma t + \epsilon_t$$ where $e_t$ is white noise process with the variance of $\sigma^2$. I guess that whereas $\Delta y_t$ is ...
  • 6,225
0 votes
0 answers
8 views

Determining Values of Parameters so that Observation Equation is Stationary

Consider a system process given by $x_t=-0.9x_{t-2}+z_t$,$t=1,2,…,n$ with observation $y_t=x_t+v_t$ where ${z_t}$ and ${v_t}$ are independent white noise with variances $σ^2$ and $σ_v^2$. Assume that ...
  • 873
0 votes
0 answers
12 views

Double EMA into Single EMA

The exponential moving average (EMA) operator is defined as: $$y_t(x, \lambda) = (1-\lambda) \sum_{i=0}^\infty \lambda^i x_{t-i}$$ where $1-\lambda$ is the normalization factor, and the operator is a ...
  • 101
-1 votes
2 answers
55 views

Why is it not the case that f(t, x(t)) = f(t)?

What is the difference between: $f(t)$ and, $f(x(t), t)$ Can I write any function $f(x(t), t)$ as $f(t)$? Why or why not? Explanations in detail with supporting examples going from simple to ...
  • 25
0 votes
0 answers
25 views

How to define the uniform asymptotic negligibility assumption for moving-average process with random coeficients?

The uniform asymptotic negligibility (UAN) assumption is well know in probability theory. In my case, I have a definition of (UAN) for moving-average (MA) processes. Let $(X_n)_{n\geq 1}$ a sequence ...
  • 739
1 vote
0 answers
20 views

White-noise linear state-space representation

A discrete-time linear state-space representation written as $$x_t=A x_{t-1} +B\epsilon_t$$ $$y_t=C x_{t-1} +C\epsilon_t$$ with $E(\epsilon_t)=0$, $E(\epsilon_t\epsilon_s)=\delta_{t-s} \Sigma$, and $\...
  • 109
1 vote
0 answers
17 views

Best loss function to use to minimize mean absolute error of a portfolio

Ciao, Let $Y_1(t), \dots, Y_n(t)$ a set of timeseries. Let $P(t) = \sum_{i=1}^n Y_i(t)$. I will call this object a portfolio. Suppose you have also a set of regressor, one for each timeseries: $X_1(t),...
  • 380
0 votes
0 answers
16 views

R-squared in autoregressive model

I'm trying to figure out why $R^2$ is not a good metric for evaluating the validity of a time series. For example, for simplicity I consider AR($1$) process: $X_t = c + 0.999 \cdot X_{t-1}$, where $c&...
  • 265
0 votes
0 answers
10 views

Problem understanding a domain of time series

Assuming that a time series of random variable $X(t)$ and the domain of $X$ is $[a,b]$. $X(t)$is ralted to $X(t+1)$ through function $f=kX+y$.You can think of $X$ as a reservoir water level which is ...
  • 567
1 vote
1 answer
58 views

A quick question on AR(1) model

Consider an AR(1) model given by $$X_t=\phi_1X_{t-1}+a_t.$$ AR(1) is stationary tells that all variance and mean of $X$ at any time $t$ should be the same. However, the conditional mean $$E(X_{t+k}\...
  • 567
0 votes
2 answers
37 views

Why does AR(P) = MA(P)

I am working my way through lecture notes from an econometrics course taught at Ohio State University on time series analysis. The current set of notes I am on (https://www.asc.ohio-state.edu/de-jong....
1 vote
0 answers
19 views

Sorting data tables based on trends - With linear algebra

Assume that you have $y$ series of data. Each data set is $n$ length long. Example: $$X_1 = x_{1,1}, x_{1,2}, x_{1,3}, x_{1,4}, x_{1,5}, x_{1,6}, x_{1,7}, \dots , x_{1, n}$$ $$X_2 = x_{2,1}, x_{2,2}, ...
  • 2,624
0 votes
0 answers
12 views

Is there any reference for stationarity of nonlinear AR(1) random process?

I am trying to study a nonlinear AR(1) process as below: $$ X_{t+1} = X_{t}-\beta\sqrt{X_{t}}\sqrt{D_t}+D_t $$ where $\beta \in (0,1)$ and $D_t$ can be some i.i.d noise $\mathbb{E}(D_t)=\mu$. Is there ...
1 vote
1 answer
32 views

If moving average is Gaussian, so are the innovations

Let $$ X(k) = \sum_{j = 0}^\infty a_j\, \varepsilon(k - j) , \quad k \in \mathbb{N}, $$ where $\sum_{j = 0}^\infty a_j^2 < \infty$ and $\big(\varepsilon(j) \big)_{j \in \mathbb{Z}}$ is a sequence ...
  • 227
0 votes
0 answers
17 views

Contradiction in Stationarity of AR(1) under the condition that |Φ| < 1

Something must have gone wrong with my reasoning. Given a zero-mean AR(1) process $\{X_t\}_{t \in T}$: $X_t = \phi X_{t-1} + \varepsilon_{t}$; $\{\varepsilon_t\}_{t \in T}$ is $WN(0, \sigma^2)$ There ...
  • 135
0 votes
0 answers
17 views

Conditions for weak stationarity of AR(1)

I don't know if it is my lack of understanding, or it is a misconception in time series analysis. Here is a definition of an AR(1) process $\{X_t\}_{t \in T}$: $X_t = \omega + \psi X_{t-1} + \...
  • 135
0 votes
0 answers
20 views

Prove medoids or points closer to cluster centroid have good data representation

I have some time series data points $x_1, x_2, x_3..,x_n$ $\in \mathcal{R}^{d}$. I have extracted the characteristics of those data points (peak, mean, min...) and clustered them based on the ...
0 votes
1 answer
26 views

Double moving average smoothing time series

I´m reading: Regression modeling with actuarial and financial applications by Frees, and in page 274 he talks about a double smoothing procedure for a times series with linear trend: Suppose that we ...
  • 2,867
1 vote
0 answers
20 views

order for double sum of Fourier coefficients

Define $e_{st}=\int_{-\lambda}^\lambda e^{i(t-s)x}dx$ for some $\lambda\in(0,1)$. Do we have the result that $\sum_{s=1}^n\sum_{t=n+1}^{2n}e_{st}=o(\sqrt{n})$ or $\sum_{s=1}^n\sum_{t=n+1}^{2n}e_{st}=o(...
  • 282
3 votes
0 answers
63 views

A problem of convergence of stochastic processes

Let $X = (X_t)_{t \in \mathbb{Z}} \sim P$ and $Y = (Y_t)_{t \in \mathbb{Z}}\sim Q$ be two stochastic processes. In order to define the Mallows metric, for all $m\in \mathbb{N}$, let $\mathcal{M}_m$ be ...
  • 739
0 votes
1 answer
54 views

Determine order of an ARIMA process

My problem: ${X_t}$ is a stationary process where ${X_t={\phi}X_{t-1}+Z_{t}+Z_{t-2} }$ with $Z_{t}$ being the error term aka white noise(0,$\sigma^2$). We are given the process ${Y_t=Y_{t-1}+X_{t}-{\...
0 votes
1 answer
58 views

Kalman filter with contemporaneous variable

In standard Kalman filter calculations. Namely, the prediction step is given by $\hat{x}_{k|k-1} = F_k\,\hat{x}_{k-1|k-1} + B_k\,u_k, $ $P_{k|k-1} = F_k\,P_{k-1|k-1}\,F_k^\top + Q_k$ Under this setup, ...
1 vote
0 answers
16 views

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 ...
1 vote
0 answers
21 views

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)}{\...
0 votes
0 answers
32 views

Mean of moving average

Consider the time series: $$x_t = \beta_1+\beta_2t + w_t$$ Show that the mean of the moving average $$v_t = \frac{1}{2q+1}\sum_{j=-q}^q x_{t-j}$$ is $\beta_1+\beta_2t$, and give a simplified ...
1 vote
0 answers
22 views

python forecasting building LSTM

I came across these two pages - page 1 and page 2 which use LSTM for forecasting. Thing that confused me is how/if they are using past Y variable values to predict future Y variable values - for ...
0 votes
0 answers
34 views

Expectation of infinite moving average process

Let $$ X(t) = \sum_{k = -\infty}^\infty a_j \varepsilon_{t-j}, \quad t \in \mathbb{R}, $$ where $\varepsilon_{j}$ are iid zero mean, finite variance random variables. In my opinion, $\mathbb{E}X(t) = ...
  • 227
0 votes
0 answers
16 views

How to verify the correctness of forecast?

I would like to forecast the car rental (count time series). Given hourly integer valued car rentals for a month's period from 24th september to 24th October. I need to forecast car rental demand ...
0 votes
0 answers
20 views

How to build a poisson regression model using tscount package?

I am quite the beginner in Poisson regression and am trying to build a model through the tsglm function from the tscount package. The data I have is the following: ...
0 votes
0 answers
49 views

How to show that a process is not ergodic?

I'm trying to show that a certain process is not ergodic, but as I don't have much experience, I would first like to learn how to show simple cases. We know that if a discrete stochastic process is i....
  • 739
0 votes
0 answers
53 views

Is this Gram Schmidt?

Begin with a matrix $\mathbf{X}$ consisting of a time series of $M$ observations of an $N$-dimensional variable, \begin{align} \mathbf{X}=\left[\mathbf{x}(1),\mathbf{x}(2),\dots,\mathbf{x}(N_t)\right]....
  • 109
0 votes
0 answers
5 views

What kind of model could I used to forecast residual so that I could do ARIMA multi-step forecast?

Thanks for viewing my question. I am just curious about how exactly ARIMA multi-step forecast works when MA terms exist, since we do not actually observe the real residual. One-step forecast is easier ...
  • 1
1 vote
0 answers
23 views

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 ...
  • 844
0 votes
0 answers
14 views

Can an autoregressive modell match behaviour of simple exponential smoothing?

As I understand it, simple exponential smoothing has the following form: $y^*_t = \alpha \cdot y_{t} + (1-\alpha) \cdot y^*_{t-1}$ Which for $y^*_2$ looks like this: $y^*_2 = \alpha y_2 + \alpha (1-\...
  • 1
0 votes
0 answers
8 views

Car rental time series forecasting

I have the time series of car rentals/demand in a location. The time axis is every hour for 3 months. The y-axis is number of car rentals/demand. I want to do prediction for future hours given this ...
0 votes
0 answers
7 views

How to interpret this ACF plot of my residuals?

I am working with a SARIMAX model. This is the ACF plot from my worst forecast (highest MAPE). I interpret this ACF plot as "It is clear the model has not explained all of the seasonality in the ...
2 votes
2 answers
73 views

Why is asymmetry in percentage change a problem when analyzing time series?

When analyzing time series, a benefit of log difference, $log(y_t) - log(y_{t-12})$, is that it is symmetric, unlike percentage change, $\frac{y_t-y_{t-12}}{y_{t-12}}$. My question is —— why is ...
  • 27
2 votes
1 answer
18 views

Summability Condition for Linear Process

I have been studying linear processes in time series of the following form: $X_t= \sum_{j=0}^{\infty}c_j\epsilon_{t-j}$ In this case, the following implication (about the covariance function) holds: $ ...
0 votes
0 answers
40 views

Can you modeling complicated dynamics without using differential/difference equations?

Let's imagine there is a phenomenon I want to understand. I have a few multivariate time series about the phenomenon but not a lot. I don't know how the variables are related to each other but from ...
0 votes
0 answers
38 views

(Stochastic) Time Series Reconstruction

Assume a linear stochastic dynamical system $$ \begin{align} \mathrm{d}x&=-J_{1,1}x+J_{1,2}y+\sigma_{1}\,\mathrm{d}W_{1}\\\mathrm{d}y&=-J_{2,1}x-J_{2,2}y-\sigma_{1}\,\mathrm{d}W_{1}+\sigma_{2}\...
  • 1
2 votes
1 answer
34 views

Time series: ARMA characteristic polynomials have common roots

I have a question regarding the idea that if the roots of the characteristic polynomials of a time series (say some ARMA process) lie outside the unit circle, then the series will be invertible/causal ...
  • 39
0 votes
0 answers
18 views

Time series analysis on ACF and PACF plots

So I have a non stationary time series that is hourly, daily and monthly recorded for a year and I have the ACF and PACF plots for the serie. ACF and PACF I applied the the Ljung-Box test and for ...
0 votes
0 answers
27 views

Rice formula on time series

I have a time-series which is a mean-reverting spread with a long-run mean (equilibrium) close to zero. I am looking to use the Rice formula to get the level crossings rate, e.g. how many times the ...
0 votes
0 answers
12 views

Practical correlation metric for a large number of vectors

I am dealing with a timeseries consisting of input flow sampled every 5 minutes over 441 days. My aim is to find any possible correlation from data coming from: The same day of the week The same ...
  • 50
0 votes
0 answers
10 views

Clarify answer for asymtotic of median smoothing

Calculating algorithmic complexity for median smoothing in Time Series I came up with the question same as this. Quote it here: A time series with T observations is given. Median smoothing with width ...
  • 111
0 votes
0 answers
20 views

Why do GLS and ML estimators coincide for the estimation of a VAR(p) model?

When estimating the coefficients in a VAR(p) model (assuming normality), the coefficient estimators using GLS and MLE coincide. Could anyone explain why this is the case?
0 votes
0 answers
16 views

ARMA-GARCH model Gaussian assumption

I have some questions concerning a ARMA(p,q)-GARCH(a,b) process, specified by $$y_t=\mu+\sum\limits_{i=1}^{p}\phi_i(y_{t-i}-\mu)+\sum\limits_{i=1}^{q}\theta_j \varepsilon_{t-j}+\varepsilon_t,$$ $$\...
0 votes
0 answers
19 views

time-shifted signals correlation

imagine you have the 2 following signals: s=[0,0,0,1,1,1,0,0,0,3,3,3,0,0,0] p=[1,1,1] The correlations for each shift (considering only lags for which the shifted ...
2 votes
1 answer
54 views

Explanation/Reference request for necessary and sufficent conditions for polynomial roots to lie inside unit circle

In the book Applied Econometric Time Series by Walter Enders (third edition, page 30) there is a discussion about the characteristic polynomial of the homogeneous part of an n-th order difference (...

1
2 3 4 5
19