ARIMA models for time series forecasting


Notes on nonseasonal ARIMA models (pdf file)

Slides on seasonal and nonseasonal ARIMA models (pdf file)

Introduction to ARIMA: nonseasonal models
Identifying the order of differencing in an ARIMA model
Identifying the numbers of AR or MA terms in an ARIMA model
Estimation of ARIMA models
Seasonal differencing  in ARIMA models
Seasonal random walk: ARIMA(0,0,0)x(0,1,0)
Seasonal random trend: ARIMA(0,1,0)x(0,1,0)

General seasonal models: ARIMA (0,1,1)x(0,1,1) etc.
Summary of rules for identifying ARIMA models
ARIMA models with regressors
The mathematical structure of ARIMA models (pdf file)


Introduction to ARIMA: nonseasonal models

ARIMA(1,0,0) = first-order autoregressive model
ARIMA(0,1,0) = random walk
ARIMA(1,1,0) = differenced first-order autoregressive model
ARIMA(0,1,1) without constant = simple exponential smoothing
ARIMA(0,1,1) with constant = simple exponential smoothing with growth
ARIMA(0,2,1) or (0,2,2) without constant = linear exponential smoothing
A "mixed" model--ARIMA(1,1,1)
Spreadsheet implementation

ARIMA(p,d,q): ARIMA models are, in theory, the most general class of models for forecasting a time series which can be made to be “stationary” by differencing (if necessary), perhaps in conjunction with nonlinear transformations such as logging or deflating (if necessary). A random variable that is a time series is stationary if its statistical properties are all constant over time.  A stationary series has no trend, its variations around its mean have a constant amplitude, and it wiggles in a consistent fashion, i.e., its short-term random time patterns always look the same in a statistical sense.  The latter condition means that its autocorrelations (correlations with its own prior deviations from the mean) remain constant over time, or equivalently, that its power spectrum remains constant over time.  A random variable of this form can be viewed (as usual) as a combination of signal and noise, and the signal (if one is apparent) could be a pattern of fast or slow mean reversion, or sinusoidal oscillation, or rapid alternation in sign, and it could also have a seasonal component.  An ARIMA model can be viewed as a “filter” that tries to separate the signal from the noise, and the signal is then extrapolated into the future to obtain forecasts.

The ARIMA forecasting equation for a stationary time series is a linear (i.e., regression-type) equation in which the predictors consist of lags of the dependent variable and/or lags of the forecast errors.  That is:

Predicted value of Y = a constant and/or a weighted sum of one or more recent values of Y and/or a weighted sum of one or more recent values of the errors.

If the predictors consist only of lagged values of Y, it is a pure autoregressive (“self-regressed”) model, which is just a special case of a regression model and which could be fitted with standard regression software.  For example, a so-called first-order autoregressive (“AR(1)”) model for Y is a simple regression model in which the independent variable is just Y lagged by one period (LAG(Y,1) in Statgraphics or Y_LAG1 in RegressIt).   If some of the predictors are lags of the errors, an ARIMA model it is NOT a linear regression model, because there is no way to specify ”last period’s error” as an independent variable:  the errors must be computed on a period-to-period basis when the model is fitted to the data.  From a technical standpoint, the problem with using lagged errors as predictors is that the model’s predictions are not linear functions of the coefficients, even though they are linear functions of the past data.  So, coefficients in ARIMA models that include lagged errors must be estimated by nonlinear optimization methods (“hill-climbing”) rather than by just solving a system of equations.

The acronym ARIMA stands for Auto-Regressive Integrated Moving Average. Lags of the stationarized series in the forecasting equation are called "autoregressive" terms, lags of the forecast errors are called "moving average" terms, and a time series which needs to be differenced to be made stationary is said to be an "integrated" version of a stationary series. Random-walk and random-trend models, autoregressive models, and exponential smoothing models are all special cases of ARIMA models.

A nonseasonal ARIMA model is classified as an "ARIMA(p,d,q)" model, where:

To identify the appropriate ARIMA model for a time series, you begin by identifying the order(s) of differencing needing to stationarize the series and remove the gross features of seasonality, perhaps in conjunction with a variance-stabilizing transformation such as logging or deflating. If you stop at this point and predict that the differenced series is constant, you have merely fitted a random walk or random trend model. (Recall that the random walk model predicts the first difference of the series to be constant, and a random trend model predicts that the first difference of the series is a random walk, rather than a constant.) However, the best random walk or random trend model may still have autocorrelated errors, suggesting that additional factors of some kind are needed in the prediction equation.

The process of determining which ARIMA model is best for a given time series will be discussed in later sections of the notes, but a preview of some of types of nonseasonal ARIMA models that are commonly encountered is given below.  For those who want to know a bit more about the underlying mathematics, details are provided in the handout:  The mathematical structure of ARIMA models

ARIMA(1,0,0) = first-order autoregressive model: if the series is stationary and autocorrelated, perhaps it can be predicted as a multiple of its own previous value, plus a constant.  The forecasting equation in this case is

Ŷt  =  μ  +  ϕ1Yt-1

…which is Y regressed on itself lagged by one period. This is an “ARIMA(1,0,0)+constant” model.  The constant term is denoted by "mu" and the autoregressive coefficient is denoted by "phi", in keeping with the terminology for ARIMA models popularized by Box and Jenkins. (In the output of the Forecasting procedure in Statgraphics, this coefficient is simply denoted as the AR(1) coefficient.)  If the mean of Y is zero, then the constant term would not be included.

If the slope coefficient ϕ1 is positive and less than 1 in magnitude (it must be less than 1 in magnitude if Y is stationary), the model describes mean-reverting behavior in which next period’s value should be predicted to be ϕ1 times as far away from the mean as this period’s value.  If ϕ1  is negative, it predicts mean-reverting behavior with alternation of signs, i.e., it also predicts that Y will be below the mean next period if it is above the mean this period.

In a second-order autoregressive model (ARIMA(2,0,0)), there would be a Yt-2  term on the right as well, and so on.  Depending on the signs and magnitudes of the coefficients, an ARIMA(2,0,0) model could describe a system whose mean reversion takes place in a sinusoidally oscillating fashion, like the motion of a mass on a spring that is subjected to random shocks.

ARIMA(0,1,0) = random walk:  If the series Y is not stationary, the simplest possible model for it is a random walk model, which can be considered as a limiting case of an AR(1) model in which the autoregressive coefficient is equal to 1, i.e., a series with infinitely slow mean reversion.  The prediction equation for this model can be written as:

Ŷt  - Yt-1   =  μ

or equivalently

Ŷt  =   μ + Yt-1

...where the constant term is the average period-to-period change (i.e. the long-term drift) in Y.  This model could be fitted as a no-intercept regression model in which the first difference of Y is the dependent variable.  Since it includes (only) a nonseasonal difference and a constant term, it is classified as an "ARIMA(0,1,0) model with constant." The random-walk-without-drift model would be an ARIMA(0,1,0) model without constant

ARIMA(1,1,0) = differenced first-order autoregressive model: If the errors of a random walk model are autocorrelated, perhaps the problem can be fixed by adding one lag of the dependent variable to the prediction equation--i.e., by regressing the first difference of Y on itself lagged by one period. This would yield the following prediction equation:

Ŷt - Yt-1  =  μ  +  ϕ1(Yt-1 - Yt-2)

Ŷt  - Yt-1   =  μ

which can be rearranged to

Ŷt  =  μ  + Yt-1  +  ϕ1 (Yt-1 - Yt-2)

This is a first-order autoregressive model with one order of nonseasonal differencing and a constant term--i.e., an "ARIMA(1,1,0) plus constant" model.  

ARIMA(0,1,1) without constant = simple exponential smoothing: Another strategy for correcting autocorrelated errors in a random walk model is suggested by the simple exponential smoothing model. Recall that for some nonstationary time series (e.g., ones that exhibit noisy fluctuations around a slowly-varying mean), the random walk model does not perform as well as a moving average of past values. In other words, rather than taking the most recent observation as the forecast of the next observation, it is better to use an average of the last few observations in order to filter out the noise and more accurately estimate the local mean. The simple exponential smoothing model uses an exponentially weighted moving average of past values to achieve this effect. The prediction equation for the simple exponential smoothing model can be written in a number of mathematically equivalent ways, one of which is:

Ŷt   =  Yt-1  - θ1et-1

...where et-1 denotes the error at period t-1. Note that this resembles the prediction equation for the ARIMA(1,1,0) model, except that instead of a multiple of the lagged difference it includes a multiple of the lagged forecast error. (It also does not include a constant term--yet.) The coefficient of the lagged forecast error is denoted by the Greek letter "theta" (again following Box and Jenkins) and it is conventionally written with a negative sign for reasons of mathematical symmetry. Theta in this equation corresponds to the quantity 1-minus-alpha in the exponential smoothing formulas we studied earlier.

When a lagged forecast error is included in the prediction equation as shown above, it is referred to as a "moving average" (MA) term. The simple exponential smoothing model is therefore a first-order moving average (MA(1)) model with one order of nonseasonal differencing and no constant term --i.e., an "ARIMA(0,1,1) model without constant." This means that in Statgraphics (or any other statistical software that supports ARIMA models) you can actually fit a simple exponential smoothing by specifying it as an ARIMA(0,1,1) model without constant, and the estimated MA(1) coefficient corresponds to "1-minus-alpha" in the SES formula.

What’s the best way to correct for autocorrelation: adding AR terms or adding MA terms?  In the previous two models discussed above, the problem of autocorrelated errors in a random walk model was fixed in two different ways:  by adding a lagged value of the differenced series to the equation or adding a lagged value of the forecast error.  Which approach is best?  A “rule of thumb” for this situation, which will be discussed in more detail later on, is that positive autocorrelation is usually best treated by adding an AR term to the model and negative autocorrelation is usually best treated by adding an MA term.  In business and economic time series, negative autocorrelation often arises as an artifact of differencing.  (In general, differencing reduces positive autocorrelation and may even cause a switch from positive to negative autocorrelation.)  So, the ARIMA(0,1,1) model, in which differencing is accompanied by an MA term, is more often used than an ARIMA(1,1,0) model.

ARIMA(0,1,1) with constant = simple exponential smoothing with growth: By implementing the SES model as an ARIMA model, you actually gain some flexibility. First of all, the estimated MA(1) coefficient is allowed to be negative: this corresponds to a smoothing factor larger than 1 in an SES model, which is usually not allowed by the SES model-fitting procedure. Second, you have the option of including a constant term in the ARIMA model if you wish, in order to estimate an average non-zero trend. The ARIMA(0,1,1) model with constant has the prediction equation:

Ŷt   =  μ  + Yt-1  - θ1et-1

The one-period-ahead forecasts from this model are qualitatively similar to those of the SES model, except that the trajectory of the long-term forecasts is typically a sloping line (whose slope is equal to mu) rather than a horizontal line.

ARIMA(0,2,1) or (0,2,2) without constant = linear exponential smoothing: Linear exponential smoothing models are ARIMA models which use two nonseasonal differences in conjunction with MA terms. The second difference of a series Y is not simply the difference between Y and itself lagged by two periods, but rather it is the first difference of the first difference--i.e., the change-in-the-change of Y at period t. Thus, the second difference of Y at period t is equal to (Yt - Yt-1)  - (Yt-1 - Yt-2)  =  Yt - 2Yt-1 + Yt-2. A second difference of a discrete function is analogous to a second derivative of a continuous function: it measures the "acceleration" or "curvature" in the function at a given point in time.

The ARIMA(0,2,2) model without constant predicts that the second difference of the series equals a linear function of the last two forecast errors:

Ŷt - 2Yt-1  + Yt-2  =  - θ1et-1 - θ2et-2

which can be rearranged as:

Ŷt   =  2 Yt-1  - Yt-2 - θ1et-1 - θ2et-2

where θ1 and θ2 are the MA(1) and MA(2) coefficients. This is a general linear exponential smoothing model, essentially the same as Holt’s model, and Brown’s model is a special case.   It uses exponentially weighted moving averages to estimate both a local level and a local trend in the series.  The long-term forecasts from this model converge to a straight line whose slope depends on the average trend observed toward the end of the series.

A "mixed" model--ARIMA(1,1,1): The features of autoregressive and moving average models can be "mixed" in the same model. For example, an ARIMA(1,1,1) model with constant would have the prediction equation:

Ŷt  =  μ + Yt-1  +  ϕ1 (Yt-1 - Yt-2 ) - θ1et-1

Normally, though, we will try to stick to "unmixed" models with either only-AR or only-MA terms, because including both kinds of terms in the same model sometimes leads to overfitting of the data and non-uniqueness of the coefficients.

Spreadsheet implementation: ARIMA models such as those described above are easy to implement on a spreadsheet. The prediction equation is simply a linear equation that refers to past values of original time series and past values of the errors. Thus, you can set up an ARIMA forecasting spreadsheet by storing the data in column A, the forecasting formula in column B, and the errors (data minus forecasts) in column C. The forecasting formula in a typical cell in column B would simply be a linear expression referring to values in preceding rows of columns A and C, multiplied by the appropriate AR or MA coefficients stored in cells elsewhere on the spreadsheet.

Go to next topic: Identifying the order of differencing