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Autoregressive moving-average model
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<p>In the <a href="/facts/Statistics/DNPsKGYU">statistical</a> analysis of <a href="/facts/Time_series/fSXPR817">time series</a>, autoregressive–moving-average (ARMA) models are a way to describe a <a href="/facts/Stationary_stochastic_process/vozdgfEX">(weakly) stationary stochastic process</a> using <a href="/facts/AR_model/LpboAQ0W">autoregression</a> (AR) and a <a href="/facts/MA_model/NkXhnBd5">moving average</a> (MA), each with a polynomial. They are a tool for understanding a series and predicting future values. AR involves regressing the variable on its own lagged (i.e., past) values. MA involves modeling the <a href="/facts/Errors_and_residuals_in_statistics/Ef0LgAS4">error</a> as a <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> of error terms occurring contemporaneously and at various times in the past. The model is usually denoted ARMA(<i>p</i>, <i>q</i>), where <i>p</i> is the order of AR and <i>q</i> is the order of MA. </p><p>The general ARMA model was described in the 1951 thesis of <a href="/facts/Peter_Whittle_(mathematician)/eGoN2BdP">Peter Whittle</a>, <i>Hypothesis testing in time series analysis</i>, and it was popularized in the 1970 book by <a href="/facts/George_E._P._Box/ERu98hUq">George E. P. Box</a> and <a href="/facts/Gwilym_Jenkins/BAq2LLes">Gwilym Jenkins</a>. </p><p>ARMA models can be estimated by using the <a href="/facts/Box%E2%80%93Jenkins_method/M4DPdAun">Box–Jenkins method</a>. </p>