BOX JENKINS (ARIMA) METHODOLOGY ARIMA MODELS Specialized class of linear filtering techniques that uses current and past values of the dependent variable and residuals to produce accurate short-term forecasts. BOX JENKINS METHOD An iterative approach of identifying a potential model from a general class of ARIMA models for a stationary time series.

AUTOREGRESSIVE MODELS AR = AutoRegressive The explanatory variables in the model are time-lagged values of the forecast variable.

MOVING AVERAGE MODELS MA = Moving Average Not the same as moving average methods. Here we are using weighted averages of the error series.

ARMA MODELS AR models can be effectively coupled with MA models to form a general and useful class of tie series models called autoregressive moving average (ARMA) models. These can only be used when data are stationary. Stationary data is has a constant mean and constant variance. These models can be used with data that are non-stationary in the mean by differencing the series.

ARIMA (p, d, q) MODELS AR: p = order of the autoregressive process I: d = degree of differencing involved MA: q = order of the moving average process

BASIC MODELS ARIMA (0, 0, 0) ― WHITE NOISE ARIMA (0, 1, 0) ― RANDOM WALK ARIMA (1, 0, 0) ― AUTOREGRESSIVE MODEL (order 1) ARIMA (0, 0, 1) ― MOVING AVERAGE MODEL (order 1) ARIMA (1, 0, 1) ― SIMPLE MIXED MODEL

RANDOM WALK ARIMA (0, 1, 0) or where B is the backward shift operator

AUTOREGRESSIVE MODEL (order 1) ARIMA (1, 0, 0) or AR(1) or where is autoregressive parameter and B is the backward shift

MOVING AVERAGE MODEL (order 1) ARIMA (0, 0, 1) or MA(1) or where is moving average parameter and B is the backward shift operator

SIMPLE MIXED MODEL ARIMA (1, 0, 1)

BOX-JENKINS PARADIGM IDENTIFICATION ↓    ESTIMATION   ↓    data preparation model selection           ↓    ESTIMATION   ↓    DIAGNOSTIC CHECKING   IF NOT OK    ↓ OK FORECAST

IDENTIFICATION STAGE Step1: Make series stationary. Difference data to stabilize mean If trend is linear, first order differencing will suffice If trend is nonlinear, second order differencing is required (take differences of the 1st differenced data) Transform data to stabilize variance Transform data using log or square root Adds another level of complexity to the process

IDENTIFICATION STAGE Step2: Identify an appropriate model. Compare the ACF of the series with the theoretical ACF for various ARIMA models. Compare the PACF of the series with the theoretical PACF for various ARIMA models PACF = partial autocorrelation measures the degree of relationship between current value and some earlier value of the series when the effects of other time lags are removed.

THEORETICALLY AR (p)  (i) autocorrelations tail off to zero exponentially (ii) p significant partial autocorrelations MA (q)  (i) q significant autocorrelations (ii) partial autocorrelations tail off to zero exponentially

IDENTIFICATION Plot the data. If necessary transform data to achieve stationarity in the variance (log or power transformation). Check time series plot, ACF and PACF for nonstationarity in the mean. If nonstationary in the mean, take first differences. When stationarity has been achieved, examine ACF and PACF to identify plausible model by comparing to theoretical patterns. Note: Because we are dealing with real data, and randomness is present, the ACF and PACF will rarely follow the underlying theoretical process exactly.

ESTIMATION Study sampling statistics of current solution a) check for significance of parameters by looking at t-test results (p-values less than typical α = .05 level indicate significance). b) can overfit to verify the model solution (additional terms will be insignificant) c) decide between competing models by checking for a minimum MS or AIC (the latter is the Akaike’s Information Criterion, however it is not provided in Minitab)

DIAGNOSTIC CHECKING Study residuals to verify that they are random a) check ACF of residuals (no significant spikes should be present) b) check PACF of residuals (no significant spikes should be present) c) test for individual significant autocorrelations and use Box-Pierce Q statistic to test for nonrandomness in a set of autocorrelations (all p-values should be > 0.05) d) if residuals are not white noise, go back to identification to improve model selection

BOX-PIERCE Q STATISTIC Null Hypothesis: The set of autocorrelations out to lag k is equivalent to the null set (implies the residuals are random). Accepting the null hypothesis indicates that the model is adequate as there is no pattern left in the residuals. Consequently we want p-values greater than the typical alpha level of 0.05 as we want to accept the null here.