25 votes

How does ACF & PACF identify the order of MA and AR terms?

The quotes are from the link in the OP: Identification of an AR model is often best done with the PACF. For an AR model, the theoretical PACF “shuts off” past the order of the model. The phrase “...
Antoni Parellada's user avatar
23 votes
Accepted

Why do we care if an MA process is invertible?

Invertibility is not really a big deal because almost any Gaussian, non-invertible MA$(q)$ model can be changed to an invertible MA$(q)$ model representing the same process by changing the parameter ...
Jarle Tufto's user avatar
10 votes

Real-life examples of moving average processes

Suppose you are producing some good, stockpiling some of it and selling the rest. Your production in time period $t$ is $x_t=m+\varepsilon_t$ with $\mathbb{E}(\varepsilon_t)=0$ and your stock is $y_t$....
Richard Hardy's user avatar
9 votes

Under what circumstances is an MA process or AR process appropriate?

I can provide what I think is a compelling answer to the first part of the question ("whence MA?") but am presently pondering an equally compelling answer to the second part of the question ("whence ...
Student's user avatar
  • 139
9 votes

Moving Average, Exponential Smoothing, and Random Walk for Forecasting

Is it true that a (simple) exponential smoothing model with alpha (smoothing constant) = 1 is the same as MA(1), which is in turn the same as a random walk model? (i.e. using only the most recent ...
Richard Hardy's user avatar
9 votes
Accepted

What is the difference between first order difference and moving average?

Suppose you have a set of time-series data values $x_1,...,x_n$. For some value $k<n$ in the series, the corresponding moving average over $k$ periods up to time $t$ is: $$\bar{x}_{t}^{(k)} = \...
Ben's user avatar
  • 125k
8 votes

Intuition behind the characteristic equation of an AR or MA process

When trying to get an intuitive understanding of formal mathematical models, it is usually best to start with a simple model and then generalise later. So, with that in mind, let's start with an AR$(...
Ben's user avatar
  • 125k
8 votes

How does ACF & PACF identify the order of MA and AR terms?

Robert Nau from Duke's Fuqua School of Business gives a detailed and somewhat intuitive explanation of how ACF and PACF plots can be used to choose AR and MA orders here and here. I give a brief ...
Lino Ferreira's user avatar
8 votes
Accepted

Difference between MA and AR

A key difference which I failed to appreciate: the MA model predictions of $x_t$ include $\epsilon_{t-1}$ in its computation whereas the AR model only predicts based on $x_{t-1}$ without (explicit) ...
Jonas Lindeløv's user avatar
7 votes

Fitting a smoothed curve to a noisy data

Sounds like you just need to adjust the smoothing parameters (sometimes called bandwidth) to your liking. Either of these methods should be able to be tuned appropriately. Moving average can be ...
Underminer's user avatar
  • 4,119
7 votes
Accepted

Is there a name for a moving average when it is done not across time but some other variable?

A Moving Average Filter is a special case for a Finite Impulse Response (FIR) Filter, where equal weights are used that add up to unity. Note that in the case of time sampled data the result of the ...
Cagdas Ozgenc's user avatar
7 votes

Difference between MA and AR

You should consider the $\epsilon_t$ innovations rather than residuals. In the MA case, you average across the recent innovations, whereas in the AR case you average across the recent observations. ...
user1587692's user avatar
7 votes
Accepted

time series model with additional, time-independent regressors?

For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, ...
Skander H.'s user avatar
  • 11.9k
7 votes

How to show that an MA(2) process is strictly stationary?

There's a deeper and far more general result lurking here: a moving window operation on any strictly stationary process produces a strictly stationary process. The demonstration is just a matter of ...
whuber's user avatar
  • 323k
7 votes
Accepted

What does the I operator stand for in the context of time series modeling?

It is the identity operator, $IX_t=X_t$, and is typically used in ARIMA type formulas where you also have the backshift operator $B$ (sometimes people use $\nabla$ for the backshift), or polynomials ...
Stephan Kolassa's user avatar
7 votes

What does the I operator stand for in the context of time series modeling?

The main answer by Stephan is correct, but it is odd to use the identity operator (with the symbol $I$) in a scalar context instead of just using the number one. It would be simpler here if they just ...
Ben's user avatar
  • 125k
6 votes
Accepted

Tuning an exponential moving average to a moving window mean?

Let $x$ be the original time series and $x_m$ be the result of smoothing with a simple moving average with some window width. Let $f(x, \alpha)$ be a function that returns a smoothed version of $x$ ...
user20160's user avatar
  • 32.5k
6 votes

auto.arima Not Minimizing AIC

By default, auto.arima uses a stepwise search and there is no guarantee that it will find the best model. You can do a more complete search by setting ...
Rob Hyndman's user avatar
  • 56.9k
6 votes
Accepted

Writing AR(1) as a MA($\infty$) process

The usual sense in which convergence is understood in this case is in mean square: $$ E[Y_t-(\epsilon_t+\phi\epsilon_{t-1}+\phi^2\epsilon_{t-2} +\ldots+\phi^j\epsilon_{t-j})]^2=\phi^{2(j+1)} E[Y_{t-j-...
Christoph Hanck's user avatar
6 votes

Moving average process - stationarity

First see what definitions say $\{X_t\}$ is strictly stationary if for any $ t_1,t_2,...,t_n \in T$ and any $k \in T$ $$ P(X_{t_1},...,X_{t_n}) = P(X_{t_1+k},...,X_{t_n+k})$$ that is, we have ...
Stats's user avatar
  • 1,056
6 votes

Is there a name for a moving average when it is done not across time but some other variable?

Terminology can differ between fields even apparently sharing applications. Based on statistical theory and practice in several fields (time series, spatial series, any application where a response ...
Nick Cox's user avatar
  • 56.5k
6 votes
Accepted

Do non-invertible MA models imply that the effect of past observations increases with the distance?

Not a big deal - it is strongly stationary and approaches white noise The non-invertible $\text{MA}(1)$ process makes perfect sense, and it does not exhibit any particularly strange behaviour. Taking ...
Ben's user avatar
  • 125k
6 votes

Is the MA($\infty$) process with i.i.d. noise strictly stationary?

This process is always strictly stationary by definition. Recall that the process is (strictly) stationary when all $n$-variate distributions formed by selecting any pattern $(s_1,s_2,\ldots,s_{n-1})$ ...
whuber's user avatar
  • 323k
6 votes

Any difference between AR(1) model and MA(1) model in practice?

Welcome here! AR(1) and MA(1) use different input values and will not give the same results. AR(1) models $y_t = a_1 \cdot y_{t-1} + \epsilon_t$ indeed use a lagged variable of the outcome. This model ...
Arne Jonas Warnke's user avatar
5 votes
Accepted

Is the estimation of MA models unique?

Thus, for MA model, does the estimation result not highly depend on what $k$ is selected? For large $k$ we have better statistical properties, but less data for the estimation... is this correct? ...
Richard Hardy's user avatar
5 votes
Accepted

Is it good practice to use Linear Least-Squares with SMA?

Of course you can do a fit on a moving average. That is your right. But the statistical diagnostics are not reliable anymore. The reason is that the IID property required in standard OLS are violated ...
Gkhan Cebs's user avatar
5 votes

Do non-invertible MA models imply that the effect of past observations increases with the distance?

I don't think it makes sense to ask for an example "from the real world where they [non-invertible MA models] occur". All you observe is $y_1,y_2,\dots,y_n$. As I try to explain in the post you link ...
Jarle Tufto's user avatar
4 votes

What to do if ACF or PACF show significant higher lags?

[I believe this is a duplicate - and while I can find questions with this issue explained in comments, the couple that explain it correctly and fully in answers aren't really answering the same ...
Glen_b's user avatar
  • 283k
4 votes

MA(1) model - prove that correlation of order two equals to zero

Given the covariance of order two, all the terms are uncorrelated ($u_{s}$ is a white process, hence there is no correlation), therefore the covariance, respective correlation, is zero $$ Cov(Y_{t},Y_{...
HonzaB's user avatar
  • 683
4 votes
Accepted

Stationary Process versus limit of MA(q)

Proposition. Let $y_t$ be a series from $MA(q)$. Then for any $q$, there is $s$ such that $Cov(y_t,y_{t-s})=0$. Obviously, the above proposition is true. But there is no such $s$ for $AR(1)$ process ...
Julius's user avatar
  • 857

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