Questions tagged [white-noise]

White noise is a random process whose "components each have a probability distribution with zero mean and finite variance, and are statistically independent".

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Can a white noise process be predicted?

I'm trying to better understand why white noise processes can't be predicted. I understand that these processes have mean zero and no correlaation between it's values at different times. I'm looking ...
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36 views

Time series forecasting - Residuals not white noise

This is my first message on CrossValidated to get some insights on an issue I am facing while trying to model properly a time series. I am relatively new to this science so please brace with me. My ...
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Covariance of a Stationary Process

Let $Y_t$ be a stationary process such that $Y_1 = a_1$ and $Y_2 = \theta a_1 + a_2$, where $\theta$ is a parameter and $a_t$ is the white noise process with mean 2 and variance $\sigma^2_a = 0.5$. ...
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Adding noise to ARIMA forecasts

I'm learning about ARIMA forecasts and I applied it on one of the example data I have. When I used model on the train set, it is giving me peaks, valleys and other irregularities. But on the test data,...
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ARMA(p,q) invertible process inequality

Let $(X_t)$ be an $ARMA(p,q)$ process such that $\phi(\cdot)$ and $\theta(\cdot)$ don't have common roots in $\Bbb C$. Let $X_t$ be invertible: $X_t = \sum\limits_{i=0}^{\infty} \pi_iX_{t-i}$ Prove ...
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39 views

Prediction error for ARMA process

Let $X(t)= \phi X_{t-12}+Z_t+\theta Z_{t-1}$ where $Z_t\sim WN(0,1)$. I need to find prediction error for projecting $X_t$ onto $H_{t-3}(X)$ (Hilbert space). So, I know that $X_t \perp P_{H_{t-3}}X_t$ ...
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Over Differencing a Time Series

Consider the simple white noise process, $Z_t = a_t$. Discuss the consequences of over-differencing by examining the ACF and AR representation of the differenced series, $W_t = Z_t - Z_{t-1}$. Answer: ...
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Time series, linear process, white noise

Let $X_n = \sum\limits_{j=0}^{\infty} a_jZ_{t-j}$, where $Z_t$ is a (weak) white noise $(0,\sigma^2)$ and $a_j \in L^2$. Prove that ACF $\gamma_X(h)\longrightarrow 0$ as $h\longrightarrow +\infty$. So:...
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36 views

Finding the variance of a process generated by white noise

Given that $a_t \sim WN(2, 0.5)$, I have generated the process defined by $$Y_1 = a_1$$ $$Y_t = \theta Y_{t - 1} + a_t$$ to be: $$Y_t = \theta^{t - 1}a_1 + \theta^{t - 2}a_2 + \cdots + \theta a_{t - 1}...
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14 views

Higher power than 2 for white noise time series?

Let $\{Yt\}$ given by $Y_{t} = Z_{t}$ With $Z_{t} \sim{N}(0,\sigma^{2})$ What are $E[Y_t^{3}]$ and $ E[Y_t^{4}]$?
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35 views

White noise that is not strictly stationary

For an exercise I was asked to provide an example of a white noise sequence that is not strictly stationary. I found in multiple sources that an example of this is $$X_t = \sin(2\pi t U) $$ where $t$ ...
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108 views

ARIMA(0,0,0) model but residuals not white noise

I have a dataset where I am trying to fit an ARIMA model to a stock return - the data set is stationary. I have used the Auto.Arima function to select appropriate AR and MA terms, and BIC selects ...
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28 views

Why the method of calculation for Cov(xt, xt-1) and Cov(et,et-1) is different?

Here et is White noise term. Now when I calculate cov(et,et-1) I substitute the values according to the given equation and look for like subscripts for e.g if I have one et-1 on the left of the comma ...
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80 views

White noise assumption in the autocorrelation proof

I followed the proof presented in Quantitative Risk Management: Concepts, Techniques and Tools by D. Duffie, S. Schaefer (proposition 4.9, pages 128-129). To arrive at the numerator for the ...
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386 views

Are colored noises correlated / uncorrelated?

Let, $x$ be a random variable (r.v) that is white Gaussian, has a flat power spectrum. $y$ can be any colored noise. I think another term for uncorrelated is i.i.d (identically and independently ...
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670 views

Probability density function for white Gaussian noise

in many signal processing text books and lectures we find that if we assume that the noise is white Gaussian then the probability density function itself takes the Gaussian form (see here for example) ...
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Applying fitted model of input on output time-series in pre-whitening

I have two different time series (input time series and output time series)for doing cross correlation. When I fit input time series for pre-whitening, it has good fit for ARIMA(0,1,1). So for pre-...
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108 views

Ljung-Box test to determine white noise of time-series (not residual)

I know we use Ljung-Box test to determine if the residuals are white noise or not. Can we use same test to determine if the time-series in itself is white - noise? This is to by-pass acf-pacf plotting,...
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41 views

Show that $x_{t},y_{t}$ are jointly stationary, and interpretation of CAcovF, $\gamma_{XY}(h)$ not being symmetric for lags $h$

Consider two white noise processes $(w_{t})_{t}$~$WN(0,\sigma_{w}^{2})$ and $(u_{t})_{t}$~$WN(0,\sigma_{u}^{2})$ that are also independent of each other such that $y_{t}=w_{t}-\theta w_{t-1}+u_{t}$ ...
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65 views

Second-order moments of a White Noise process

One of my textbooks on time-series analysis claims that Dependency in the second moments of the residuals contradicts the assumption of a constant, time-invariant variance. Thus [the residual] is not ...
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192 views

Math behind Differencing: Is White Noise Stationary?

I'm just starting to learn the math behind stationarity and differencing, so I apologize if this is a silly question. Lets say I have a non-stationary time series process (pure random walk) defined by:...
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Vector AR/ARMA Model [closed]

For a vector AR/ARMA model in practice, if there are k different time series in the vector, there are k corresponding Gaussian white noises as well. Is it realistic to assume that those k white noises ...
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272 views

Is an independent process always a white noise process?

In econometrics, an independent process means that all values are independent of each other, but does this also mean that all independent processes are white noise processes? and is the reverse true?
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85 views

Variance of a white noise process

How do I find the variance of this? $$y_t=M + \sum_{i=0}^n 0.5^i\varepsilon_{t-2i}$$ M is the mean.
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164 views

Is error term in MA model in univariate time series the same as white noise

I what to know if the error term referred to in moving average model of time series the same as white noise? which is usually define in r as ...
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169 views

In a VAR model, why are white noises correlated with each other in the reduced form model

With my understanding, we have 2 different models when studying VAR processes (Vector Autoregression). We have $$ \begin{bmatrix} y_{t} \\ x_{t} \end{bmatrix} = \begin{bmatrix} c_{1,0} \\ c_{2,0} ...
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62 views

Interpreting Ljung-Box white-noise test p-value

Good evening all, I am having some trouble understanding the Ljung-Box white-noise test p-value from SAS Forecast. So there are lags where the p-value exceeds 0.05, meaning that we fail to reject the ...
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207 views

Is an ARIMA(0,0,0) with non-zero mean equivalent to white noise?

Is an ARIMA with a non-zero mean equivalent to white noise? If not, how should the mean be interpreted?
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67 views

Forecasting with ARIMA models

I want to do a rolling window forecast on a time series but it seems the series is white noise ARIMA (0,0,0) with non-zero mean. But when I difference the dataset and model it with an ARIMA(0,1,1) I ...
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265 views

What is the difference between white noise and “strict” white noise?

The normwhn.test package in R has a function to test if data is white noise. The documentation of the function states the following : Performs an Univariate Test ...
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73 views

Statsmodel Ljung-Box test for non-zero mean

I have few time series, which I would like to model using ARIMA. But plotting ACF and PACF indicates these series are probably white noise. I am using Ljung-Box test to confirm it. I am implementing ...
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121 views

In ARMA model, the white noise is uncorrelated, but why there are autocorrelation existing between them?

I learnt the ARMA model allows the autocorrelation between noise. but I check the definition of white noise is a sequence of uncorrelated, fixed mean random variables. how can the uncorrelated random ...
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151 views

Time Series assumptions for iid $\epsilon$

thanks for reading my post. I know its fundamental and rather easy qns but I'm seriously struggling. Please help me, thank you very much! Let $\boldsymbol{X}$ have a distribution with mean $\mu$ and ...
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528 views

Proof of white noise process

I have the process $y_t=e_t+ae_{t-1}e_{t-2}$ where $e_t$ is iid with mean of $0$ and variance $\sigma^2$ How do I go about mathematically proving that this is a white noise process?
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Autocovariance of White Noise

I'm studying time series, I don't understand the description below. The white noise series $w_t$ has $E(w_t) = 0$ and $$\gamma_w(s, t) = \operatorname{cov}(w_s, w_t)= \begin{cases} \sigma_w^2 & ...
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258 views

What is the relation between the serial correlation and white noise series?

I am a bit confused about why would I use the Ljung-Box test in order to determine whether a series is a white noise ? I know that the Ljung-Box test studies whether we have serial correlation. I ...
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117 views

Outliers Kalman Filtering

This might not be the right place to ask this questions, but I figured it's more of a machine learning question. I am also asking on the pyro forum for brevity. I'm working with the simple extended ...
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1answer
67 views

What happens when a time series is multiplied by an iid error

Let's say say I have a standard AR(1) process, apart from the fact that is multiplied by the $ε_i$~$iid (0,1)$. Would this affect the independence of the the AR(1) series? My intuition is no ...
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357 views

How to compute error terms in moving average time series model? [duplicate]

Currently I am studying time series Moving Average model MA(q) $$X_t -\mu= \epsilon_t + \theta_1\epsilon_{t-1} + \theta_2 \epsilon_{t-2} + ... + \theta_q \epsilon_q$$ where $\theta_1,...,\theta_q$ are ...
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52 views

Including regressors to improve forecasts on white noise

I am conducting some time series forecasts using quite limited data, 13 years annually. Basically, I am trying to forecast companies emission totals using historical values. The historical data ...
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366 views

Add Noise to a Dataset

I've generated a dataset of 100 elements from a 3-variate Gaussian distribution with parameters $\mu = 0$ and $\Sigma = \begin{pmatrix}1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_1 \\ \...
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255 views

Wold Decomposition Theorem and Moving Average model - Error Terms

I'm stuck with Wold's decomposition theorem in time series analysis. The theorem says that every stationary time series can be written as a sum of two components, one being entirely deterministic (...
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Do coefficients in moving average process add to 1?

I'm studying weakly stationary stochastic processes, and I'm confused by the title of the "moving average" representation of such a process. Suppose that $y_t$ is a weakly stationary stochastic ...
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Parameters of ARMA model

In my professor's notes, it is written that if the variable $y$ is explained with an ARMA($p$,$q$) model, then $y_t$ (i.e. $y$ at time t) depends on the most recent $p$ lags of its own value and the ...
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538 views

Understanding the infinite sum of random variables

I am doing a course on time series analysis, and am struggling with this definition: We call a weakly stationary process $\{X_t\}$ invertible with respect to a white noise $\{\epsilon_t\}$ if ...
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230 views

If the ACF of a time series is within the 95% bounds, is it white noise?

I have a detrended series where the ACF and PACF has lags all within the 95% confidence bounds. This would suggest the series is a White Noise. However, fitting it to an ARMA model (in R) gives the ...
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477 views

How to determine mse of estimate from correlation matrix of estimate error?

I have a model of an information transmission system Y = XH + N, where X is a diagonal matrix with the transmitted "symbols" (known), H is a column vector which distorts the transmitted symbols and N ...
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Can the sum of several time-series be a white noise process, when the individual time series are not?

Intuitively, I think that it is possible for a sum of time series to be white noise, when the individual time series are not. Reason I am asking, is because I want to know if it's useful to ...
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39 views

Continuous white noise

I have been reading about topics that include statistics, and slowly start to get a grasp. However sometimes it seems to me that I dont get the simplest concepts. White noise seems to have several ...
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68 views

How to determine the line which a time-series fluctuating around

Let $X_t=1+3t+0.5X_{t-1}+ \epsilon_t$ be a trend-stationary model, where $\epsilon$ is a white noise, which has zero expected value and standard deviation. Which line is the time series fluctuating ...