ACF (autocorrelation function)

The ACF consists of the autocorrelations for lags 1, 2, 3, ... N. Generally, the ACF is displayed as a correlogram, i.e. a bar chart of the autocorrelations arranged by lag.

ARIMA model

(AutoRegressive Integrated Moving Average) model. A family of sophisticated statistical models used by Box and Jenkins to describe the autocorrelations of a time series data. The symbol ARIMA(p,d,q) indicates a model involving p autoregressive terms and q moving average terms, applied to data that have been differenced d times. The Box-Jenkins technique involves (1) Identification of a particular ARIMA model to represent historic data; (2) Estimation of ARIMA model coefficients, (3) Statistical validation of the model; and (4) Preparation of forecasts.

Autocorrelation

The correlation of a variable and itself N periods later, and hence a measure of predictability.

Base

The forecast base is the time point from which forecasts are prepared.

BIC (Bayes information criterion )

A model selection criterion proposed by Schwarz [1978]. Within a model family (e.g. exponential smoothing or Box-Jenkins), the model that minimizes the BIC is likely to provide the most accurate forecasts. Since models with many parameters often fit the historical data well, but forecast poorly, the BIC balances a reward for goodness-of-fit with a penalty for model complexity. If your current model yields the lowest BIC out of the models you have tested, Forecast Pro marks it with “Best thus far.”

Box‑Cox power transform

Logarithmic or power transform of the data. Used to reduce or eliminate dependence of the local range of a time series on its local mean.

Box-Jenkins

Strictly speaking, the statistical technique developed by Box and Jenkins to fit ARIMA models to time series data. More loosely, the term refers to the ARIMA models themselves.

Confidence limits

A forecast is generally produced along with its upper and lower confidence limits. Each confidence limit is associated with a certain percentile. If the upper confidence limit is calculated for 97.5% and the lower for 2.5%, then actual values should fall above the upper confidence limit 2.5% of the time, and below the lower confidence limit 2.5% of the time. These are often called the 95% confidence limits to indicate that the actual value should fall inside the confidence band 95% of the time. In practice, confidence limits tend to overstate accuracy. You can set the confidence limit percentiles in Configure.

Dependent variable

The variable you want to forecast. Strictly speaking this term only applies to regression modeling, where there are independent variables as well, but it is sometimes convenient to use it for the variable in univariate models as well.

Differencing

To difference a time series variable is to replace each value (except for the first) by its difference from the previous value. The seasonal difference replaces each value (except for those in the first year) by its difference from the value one year previously.

Durbin-Watson test

This statistic checks for autocorrelation in the first lag of the residual errors. It should be about 2.0 for a perfect model. Forecast Pro computes the Durbin-Watson d-statistic, which is, strictly speaking, applicable only for regressions that include a constant intercept term, but do not include lagged dependent variables.

Exogenous variable

An exogenous variable is an explanatory variable that can be treated as a time series of ordinary numbers. Practically speaking, independent variable means the same thing.

Exponential smoothing

A robust forecasting method that extrapolates smoothed estimates of level, trend, and seasonality of a time series.

Fit set

The historic data set used to fit the parameters of a model, and as the base of extrapolation for the forecasts.

Forecast error

Standard error of the within-sample forecasts, computed by running the forecast model through the historic data. Used as an estimate of the one-step forecast error.

Forecast horizon

Number of periods you wish to forecast.

Forecast scenario

A forecast scenario extends the historic series of independent variables into the future. Dynamic regression forecasts are dependent on the forecast scenario.

Lag

The time difference between a time series value and a previous value from the same series.

Ljung-Box test

Checks for autocorrelation in the first several lags of the residual errors. If the Ljung-Box test is significant for a correlational model (Box-Jenkins or dynamic Regression) then the model needs improvement. The test is significant if its probability is > .99, in which case it is marked with two asterisks in the standard diagnostic output.

Local level

See local mean.

Local mean

The average level of a time series in the general neighborhood of a given point in time. Sometimes called the local level.

Local trend

The average rate of increase of a time series in the general neighborhood of a given point in time.

MAD

Mean Absolute Deviation. This measure of goodness-of-fit is calculated as the average of the absolute values of the errors. It is an important statistic in rolling simulation analysis.

MAPE

Mean Absolute Percentage Error. A statistic used to measure within sample goodness-of-fit and out-of-sample forecast performance. It is calculated as the average of the unsigned percentage errors.

Model

A forecasting model is an equation, or set of equations, that the forecaster uses to represent and extrapolate features in the data.

Model complexity

Model complexity is measured by the number of parameters that must be fitted to the historic data. Overfitting, i.e., using too many parameters, leads to models that forecast poorly. The BIC can help to find the model that properly trades off goodness‑of‑fit in the historic fitting set, and its model complexity.

Multivariate

Involving more than one variable at a time. Dynamic regression is a multivariate technique.

Residual error

The difference between a predicted value and a true value in the fitting set, i.e. the fitted error.

Robust

A robust method is insensitive to moderate deviations from the underlying statistical assumptions.

Root mean squared error (RMSE)

A statistic that is used as an indication of model fit. It is calculated by taking the square root of the average of the squared residual errors.

Seasonality

Periodic patterns of behavior of the series. For instance, retail sales exhibit seasonality of period 12 months. Usually the forecaster must take seasonality explicitly into account during the model fitting process.

SKU

Stock Keeping Unit.

Stochastic

A process is said to be stochastic when its future cannot be predicted exactly from its past. In a stochastic process, new uncertainty enters at each point in time.

Univariate

Involving only one variable at a time. Exponential smoothing and Box‑Jenkins are univariate techniques.