Talk:Autoregressive fractionally integrated moving average

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long memory[edit]

The article should say something about the reason why such a process is long memory. Jackzhp (talk) 22:38, 1 February 2009 (UTC).

Long memory because the binomial expansion are infinite. This means it is dependent on past values. This article are mostly incomplete so users should refer to Hosking 1981 on Fractional Differencing.Meson2439 (talk) 03:20, 6 November 2009 (UTC)
The above explanation of long memory is not really right. An AR model causes X to depend on an infinite number of past shocks, and an MA model can be inverted to show that X depends on an infinite number of its own lagged values; but those are not called long memory models. Since the lede mentions long memory, and long memory is a key selling point of ARFIMA models, the lede really needs a good explanation of the long memory concept. Duoduoduo (talk) 15:46, 15 January 2013 (UTC)
Sorry for the vague explanations. From the perspective of fractional gaussian noise (fgn), an integer difference operator simply relates how the present behaviour relates to the past, so x{i} and x{i+1} has an absolute dependency. However in fractional differences, the dependency on the immediate past values are weaker but at the same time, anything that happens in the far past would likely have an impact on the future estimates dependent on the measure of d. The binomial expansion in the (0,d,0) model seeks to model this dependency mathematically, which is what I actually meant in the first response. Most book on statistics wish to say that a lot of real processes are related to its past values but the level of dependency are related to the measure d (which is related to Hausdorff dimension, H). In ARIMA, for d=0, we say that the dependence of x{i} itself are completely random; for d=1, the differences between x{i} and x{i+1} is random; for d=2, the differences between x{i} and the last two histories is random; and so the model goes for infinite values of d. However in ARIMA, implicitly (from the binomial expansion) the strength of past values goes weaker further into the past. With the introduction of the fractional d exponent, the strength of the past histories can be taken into account for a better model result. Several methods are available for estimating d using the hausdorff dimension including Hurst methods, box counting method and the workhorse of any numerical analyst, using spectral analysis. In other words, the math will explain it much better and nothing beats expanding the binomial expansion of the ARIMA for a good understanding of everything that the method implies. — Preceding unsigned comment added by (talk) 13:22, 20 February 2013 (UTC)
While it is true that the binomial expansion leading to the MA representation has an infinite number of terms, the "long memory" here exists because of the slow rate of decay of the coefficients ... the decay is as a power-law in the lag, compared to the exponential decay that occurs for, for example an AR(1) model, and as occurs for any general finite-order non-fractionally differenced ARMA model. The power-law decay in the MA coefficients leads to a power law decay in the ACF. (talk) 17:40, 20 February 2013 (UTC)

How to estimate ARFIMA?[edit]

The article needs a section on how ARFIMA models are estimated. Presumably it's basically the same as estimating an ARIMA model except that the differencing exponent is estimated -- simultaneously, or beforehand? By maximum likelihood? By searching over d and running an ARMA on (1-B)d-transformed data and then choosing the d specification that gives the best results according to some criterion? Anyone know? Duoduoduo (talk) 15:46, 15 January 2013 (UTC)