49
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Is the sum of two white noise processes necessarily a white noise?
No, you need more (at least under Hayashi's definition of white noise). For example, the sum of two independent white noise processes is white noise.
Why is $a_t$ and $b_t$ white noise insufficient ...
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votes
White Noise in Statistics
TL;DR
The answer is NO, it doesn't have to be normal; YES, it can be other distributions.
Colors of the noise
Let's talk about colors of the noise.
The noise that an infant makes during the air ...
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37
votes
Is the sum of two white noise processes necessarily a white noise?
Even simpler than @MatthewGunn's answer,
Consider $b_t = -a_t$. Obviously $c_t \equiv 0$ is not white noise -- it'd be hard to call it any kind of noise.
The broader point is, if we don't know ...
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votes
Accepted
How is adding noise to training data equivalent to regularization?
Adding noise to the regressors in the training data is similar to regularization because it leads to similar results to shrinkage.
The linear regression is an interesting example. Suppose $(Y_i,X_i)_{...
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18
votes
White Noise in Statistics
White noise simply means that the sequence of samples are uncorrelated with zero mean and finite variance. There is no restriction on the distribution from which the samples are drawn. Now if the ...
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votes
How is adding noise to training data equivalent to regularization?
Overview: For linear regression, I'll show that $\ell_2$ regularization (a.k.a. ridge regression) arises from minimizing the expected squared error over random perturbations of the regressors. The ...
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7
votes
Determining whether a Time series is white noise
You want to look at an autocorrelation function (ACF) plot. If no lags are significantly correlated, then you basically have white noise or a MA(q) process aka moving average.
You can use this guide ...
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votes
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Interpretation of an I(2) process?
One interpetation is that the rate of change is random walk.
It's like a free fall where the gravitational force is stochastically changing.
If you drop the body on earth, it's moving according to ...
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7
votes
Accepted
Why does differencing White Noise induce autocorrelation of $-0.5$?
For notational simplicity, let
$$X_t \equiv \Delta Y_t.$$
By definition,
$$
\text{Corr}(X_t,X_{t-1}) = \frac{\text{Cov}(X_t,X_{t-1})}{\sqrt{\text{Var}(X_t)}\sqrt{\text{Var}(X_{t-1})}}.
$$
Here,
\begin{...
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6
votes
Accepted
Discrete white noise
Assuming that the $S(n)$ are also binary (taking on values in $\{0,1\}$) as are the $X(n)$, then I suspect that the $\epsilon(n)$ are also meant to be taking on values in $\{0,1\}$ and that $+$ in $X(...
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5
votes
Accepted
Can white noise be (losslessly) compressed?
You may need to think carefully about definitions here.
First question: when quantifying "compression", are you counting in the length of the decompression algorithm itself?
If not, then the answer ...
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votes
Is it necessary for white noise to have zero mean
White noise is called so because the defining property is that the power spectral density of the process is a constant: $S_X(f) = K, -\infty < f < \infty$. The autocorrelation function then is $...
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4
votes
Accepted
How to Interpret these ACF/PACF plots
In order to correctly interpret the acf/pacf one often needs to have an observed series that
has no pulses
has no level/step shifts
has no deterministic trends
has no seasonal pulses
has ...
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4
votes
What is the difference between white noise and "strict" white noise?
By definition, 'white noise' is serially uncorrelated. It is like hitting piano keys at random; note n+1 cannot at all be predicted by note n. This is distinguished from brown noise and pink noise, ...
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4
votes
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Are colored noises correlated / uncorrelated?
Question1: If the power spectrum is not flat, then does that mean the
colored noises are correlated?
One way to construct the power spectral density is to take the Fourier transform of the ...
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votes
Prove that a process is memoryless (simple example)
Hint: With a bit of recursive algebra it can be shown that:
$$\frac{u_t}{u_1} = x_t \prod_{i=2}^{t-1} (1+x_i),$$
which gives the recursive form:
$$x_t = \frac{u_t}{u_1 \prod_{i=2}^{t-1} (1+x_i)}.$$
...
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4
votes
Prove that a process is memoryless (simple example)
For a random walk,
$$y_t = \sum_{s=1}^{t} u_s$$
you can write
$$y_{t+1} = y_{t} + u_{t+1}$$
which shows that the distribution of $y_{t+1}$ only depends on the before last step and not the entire ...
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4
votes
Should I consider it white noise?
Purely from the data / visuals shown here it's possible it's a random walk as acf/pacf don't show any clear patterns. However see Stephan Kolassa comment on better ways to quantify/test data for ...
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votes
Does white noise guarantee that $X_{t-1}$ is uncorrelated with $u_t$?
Generally, no. The concept of white noise refers only to a single process such as $\{u_t\}$ and its internal structure (how the different elements of $\{u_t\}$ relate to each other). It does not refer ...
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3
votes
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Why white noise process and IID process are considered martingale
A stochastic process $\{X_t\}$ is called a martingale if
$$ \operatorname{E}[X_{t+1} \mid X_{t}, \ldots, X_1\} = X_t $$
That is, the expectation of the future conditional on the past is the present.
...
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votes
Does a white noise process have constant variance by definition?
There is no common definition of white noise.
You have two definitions. And there exists more definitions. For example,
white noise is a stationary stochastic process with constant spectral density (...
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votes
Correlogram q-statistics of residuals
I think the reason why the p-values are not reported is because the Q-statistic is follows a Chi-squared distribution, where the d.f. = the number of lags (e.g. at lag 2, d.f. = 2). However, for the ...
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votes
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What's the meaning of the expansion coefficient of the AR model?
Not as much an answer to your question, but to the underlying misunderstanding:
SPM does not simply set $\phi = 0.2$, it approximates the autocorrelation of an AR(1) process by a 1st-order Taylor ...
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votes
White noise test interpretation
In the Portmanteau test, the null hypothesis is that the variable follows a white noise process. And just like in any other statistical inference, a p-value that is smaller than the significance level,...
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votes
Is the sum of two white noise processes necessarily a white noise?
In electronics, white noise is defined as having a flat frequency spectrum ('white') and being random ('noise'). Noise generally can be contrasted with 'interference', one or more undesired signals ...
- 31
3
votes
Does white noise imply wide-sense stationary?
White noise has the properties that you state, but those properties are not the properties that define white noise. As Michael Chernick's comment points out, a (discrete-time) white noise process is a ...
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3
votes
Accepted
Does white noise imply wide-sense stationary?
Well, this depends on your definition of white noise. This question asks for that definition.
One answer gives:
A white noise process is a random process of random variables that are uncorrelated, ...
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votes
Where does the white noise come from in MA(q) model?
While (I think) this answer will not provide the intuition behind, it hopefully will bring some insight.
One way to see where the white noise comes from in the $MA(q)$ representation is given ...
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If the ACF of a time series is within the 95% bounds, is it white noise?
You over-modeled your data as the ar coefficient .9871 is (nearly) cancelled by the ma coefficient .9949 . Your series is probably white noise although I would need your data to confirm this as ...
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votes
Accepted
Math behind Differencing: Is White Noise Stationary?
Yes, white noise is strictly stationary here and in general, and weakly stationary if it has finite second moments (weak stationarity may depend on the precise definition of white noise, i.e. whether ...
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