# Moving-average model

Moving average modelmoving averagemoving average processMoving-averageMA(''q'')moving average regression modelmoving-average (MA) model
In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series.wikipedia
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### Autoregressive model

autoregressiveautoregressionAutoregressive process
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure.
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.

### Autoregressive–moving-average model

ARMAautoregressive moving average modelautoregressive moving average
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure.
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).

### Time series

time series analysistime-seriestime-series analysis
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure. In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series.
The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model).

### Autoregressive integrated moving average

ARIMAAutoregressive integrated moving average modelAutoregressive integrated moving average (ARIMA)
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure.
Non-seasonal ARIMA models are generally denoted ARIMA(p,d,q) where parameters p, d, and q are non-negative integers, p is the order (number of time lags) of the autoregressive model, d is the degree of differencing (the number of times the data have had past values subtracted), and q is the order of the moving-average model.

### Stationary process

stationarynon-stationarystationarity
Contrary to the AR model, the finite MA model is always stationary.
Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.

### Moving average

exponential moving averagesimple moving averageWeighted moving average
The moving-average model should not be confused with the moving average, a distinct concept despite some similarities.
In a moving average regression model, a variable of interest is assumed to be a weighted moving average of unobserved independent error terms; the weights in the moving average are parameters to be estimated.

### Autocorrelation

autocorrelation functionserial correlationautocorrelated
The autocorrelation function (ACF) of an MA(q) process is zero at lag q + 1 and greater.
Unit root processes, trend stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.

### White noise

whitenoisestatic
where μ is the mean of the series, the θ 1, ..., θ q are the parameters of the model and the ε t, ε t−1,..., ε t−q are white noise error terms.
In this case the noise process is often modeled as a moving average process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.

### Lag operator

backshift operatorlagLag or backshift operator
This can be equivalently written in terms of the backshift operator B as

### Box–Jenkins method

Box–Jenkins Box–Jenkins approachBox–Jenkins analysis
Sometimes the ACF and partial autocorrelation function (PACF) will suggest that an MA model would be a better model choice and sometimes both AR and MA terms should be used in the same model (see Box–Jenkins method#Identify p and q).
The autocorrelation function of a MA(q) process becomes zero at lag q + 1 and greater, so we examine the sample autocorrelation function to see where it essentially becomes zero.

### Univariate

In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series.

### Linear prediction

linearlySignal predictioncoefficient
The moving-average model specifies that the output variable depends linearly on the current and various past values of a stochastic (imperfectly predictable) term.

### Stochastic

stochasticsstochastic musicstochasticity
The moving-average model specifies that the output variable depends linearly on the current and various past values of a stochastic (imperfectly predictable) term.

### Linear regression

regression coefficientmultiple linear regressionregression
Thus, a moving-average model is conceptually a linear regression of the current value of the series against current and previous (observed) white noise error terms or random shocks.

### Normal distribution

normally distributedGaussian distributionnormal
The random shocks at each point are assumed to be mutually independent and to come from the same distribution, typically a normal distribution, with location at zero and constant scale.

### Finite impulse response

FIRFIR filterFinite Impulse Response (FIR)
The moving-average model is essentially a finite impulse response filter applied to white noise, with some additional interpretation placed on it.

### Vector autoregression

VARvector autoregressive modelstructural VAR estimation
Second, in the MA model a shock affects X values only for the current period and q periods into the future; in contrast, in the AR model a shock affects X values infinitely far into the future, because affects X_t, which affects X_{t+1}, which affects X_{t+2}, and so on forever (see Vector autoregression#Impulse response).

### Curve fitting

nominalbest-fitbest fit
This means that iterative non-linear fitting procedures need to be used in place of linear least squares.

### Partial autocorrelation function

Partial autocorrelationPACF
Sometimes the ACF and partial autocorrelation function (PACF) will suggest that an MA model would be a better model choice and sometimes both AR and MA terms should be used in the same model (see Box–Jenkins method#Identify p and q).

### Urban traffic modeling and analysis

To begin with, statical methods are based on auto-regression and moving average methods.

### Weight function

weighted sumweightedweights
Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable.

### List of statistics articles

List of statistical topicsList of statistics topicsIndex of statistics articles

### Autoregressive conditional heteroskedasticity

GARCHARCHARCH model
Exponentially weighted moving average (EWMA) is an alternative model in a separate class of exponential smoothing models.