Advantages and Disadvantages

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The Similarities
▪ Both are extrapolative methods.
▪ Both are adaptive. ▪ Both can model trends and seasonal patterns.
▪ Both are practical approaches when large numbers of forecasts must be prepared (i.e. model selection and fitting can be automated).
▪ In certain cases, both give the same forecasts.

The Points of Departure
▪ ARIMA models are based on autocorrelations while exponential smoothing is based on a structural view of the data that can include level, trend, seasonality and events.
▪ ARIMA models are linear, exponential smoothing includes both linear and nonlinear models.
▪ Exponential Smoothing attempts to estimate the trend as part of the modeling process. ARIMA attempts to eliminate the trend before modeling the autocorrelations.

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Historically Box-Jenkins techniques have not enjoyed widespread use in corporate America. This is largely due to the difficulty of model identification using the strategy originally proposed by Box and Jenkins. This strategy required considerable statistical skills and was extremely subjective.
Today's automatic identification strategies routinely outperform human experts. This is confirmed by testing with the M-competition data, results from the M-2 and M-3 competitions, and a study published in the American Statistician.
Business oriented software incorporating automatic model identification has brought ARIMA into more widespread use.

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Box-Jenkins Lens on the World 1. IDENTIFICATION: The first step is to identify the model, i.e. to select p, d and q. One can use the traditional Box-Jenkins approach or use an automatic algorithm.

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The Box-Jenkins text describes a model identification procedure based upon visual examination of the autocorrelation and partial autocorrelation functions.
This highly subjective procedure is difficult, time consuming and often leads to sub-optimal models. In short, the procedure is severely flawed.
The automatic identification procedure in Forecast Pro has been compared to the manual procedure many times and has always substantially outperformed the manual procedure in terms of forecast accuracy. These comparisons include a study performed by Spyros Makridakis using the Makridakis 111 series and a study published in the American Statistician. ▪ It is good practice to examine forecasts visually.

Manual Identification
▪ Accuracy can be measured via out-of-sample or hold-out sample performance.

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The principal advantages of ARIMA are its optimality (under certain assumptions) and the comprehensiveness of the family of models. In the M-2 study, on an aggregate basis, 45% of all data was best suited for ARIMA models.
There are more disadvantages than advantages, but the advantages may still outweigh them.
▪ Manual ARIMA identification is difficult and time consuming. Many of the models have no structural interpretation.
▪ ARIMA may be difficult to explain to others.
▪ Identification and estimation can be badly distorted by outlier effects.
▪ Models that perform similarly on the historical data may yield quite different forecasts.
On balance, we feel that adoption of ARIMA forecasting by corporations depends upon automatic identification strategies that outperform experts and that outperform exponential smoothing.

▪ Intervention Model
ARIMAX in which the inputs consist of step and impulse functions. These can account for such effects as competitors entering a market, strikes, etc.
These intervention effects can be processed dynamically in a variety of ways. The dependence of the current value Y t on lagged values is called an autoregressive (AR) process.

Extensions of the ARIMA Model
The AR(p) Autoregressive Process: The AR process relates the current value to its own previous p values "p" is called the order or memory of the AR process.
Order AR Process Abbreviation Low order AR models are intuitively appealing as descriptions of nature. Much of classical physics can be written as low order differential equations.

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The relation of the current value Y t to previous forecasting errors is called a moving average process.
MA(q) Moving Average process: The MA process relates the current value to the previous q errors. "q" is called the order or memory of the moving average process.
Order MA Process Abbreviation MA models are not intuitively appealing as descriptions of nature. Instead, they tend to occur as a result of the aggregation of errors, as when we aggregate individual products into groups, stores into chains, months into quarters, etc.

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An ARIMA(p,d,q) model can include any combination of differencing, autoregressive, and moving average components. For example:

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There are several equivalent forms for the BIC. The most convenient is BIC = s 2 T (n/T) where s 2 is the estimated within-sample error, n is the number of fitted parameters and T is the number of sample data points.
The BIC is a model selection criterion that balances a reward for goodness-of-fit with a penalty for model complexity. The BIC is used to compare different models from the same model family. The model that minimizes the BIC is likely to provide the most accurate forecasts. Since it is scaled to the standard forecast error, the BIC can be very loosely interpreted as an estimate of out-ofsample forecast error.