rw_model: Random walk model

In forecast: Forecasting Functions for Time Series and Linear Models

Random walk model

Description

Fit a generalized random walk with Gaussian errors (and optional drift) to a univariate time series.

Usage

rw_model(y, lag = 1, drift = FALSE, lambda = NULL, biasadj = FALSE)

Arguments

y	a numeric vector or univariate time series of class ts
lag	Lag parameter. lag = 1 corresponds to a standard random walk (giving naive forecasts if drift = FALSE or drift forecasts if drift = TRUE), while lag = m corresponds to a seasonal random walk where m is the seasonal period (giving seasonal naive forecasts if drift = FALSE).
drift	Logical flag. If TRUE, fits a random walk with drift model.
lambda	Box-Cox transformation parameter. If lambda = "auto", then a transformation is automatically selected using BoxCox.lambda. The transformation is ignored if NULL. Otherwise, data transformed before model is estimated.
biasadj	Use adjusted back-transformed mean for Box-Cox transformations. If transformed data is used to produce forecasts and fitted values, a regular back transformation will result in median forecasts. If biasadj is TRUE, an adjustment will be made to produce mean forecasts and fitted values.

Details

The model assumes that

Y_t = Y_{t-p} + c + \varepsilon_{t}

where p is the lag parameter, c is the drift parameter, and \varepsilon_t\sim N(0,\sigma^2) are iid.

The model without drift has c=0. In the model with drift, c is estimated by the sample mean of the differences Y_t - Y_{t-p}.

If p=1, this is equivalent to an ARIMA(0,1,0) model with an optional drift coefficient. For p>1, it is equivalent to an ARIMA(0,0,0)(0,1,0)p model.

The forecasts are given by

Y_{T+h|T}= Y_{T+h-p(k+1)} + ch

where k is the integer part of (h-1)/p. For a regular random walk, p=1 and c=0, so all forecasts are equal to the last observation. Forecast standard errors allow for uncertainty in estimating the drift parameter (unlike the corresponding forecasts obtained by fitting an ARIMA model directly).

The generic accessor functions stats::fitted() and stats::residuals() extract useful features of the object returned.

Value

An object of class rw_model.

See Also

forecast.rw_model(), rwf(), naive(), snaive()

Examples

model <- rw_model(gold)
forecast(model, h = 50) |> autoplot()

forecast documentation built on March 18, 2026, 9:07 a.m.