Construct ARIMA forecasts from lag diagnostics

What construction is choosing

An ARIMA forecast is defined as a function of past prices and past forecast errors, so construction begins by choosing how many of each coefficient to include. In this workflow the finished object is a forecast equation built from lagged values of a level, differenced series plus lagged forecast errors, after identification, estimation, and residual checks.

TradersWeek editorial reading: treat that sequence as a gated construction habit. Do not estimate coefficients until differencing and lag tests have justified a short list. Do not keep a fit that stopped only because a revision limit was reached.

Turn a trend into a level working series

A trending series is converted to a level working series by subtracting successive observations. Differencing means repeating that subtraction if needed so a drifting series becomes a level working series before lag order is chosen. Coefficients wait until the trend is gone.

Reject a long first specification

A preliminary specification with nine autoregressive and nine moving-average terms is treated as impractical. It requires estimating 18 coefficients, repeats a prediction once per coefficient plus one for each observation, can yield several equally good coefficient sets, and can introduce division-by-zero or rounding failures.

Count terms from the correlogram

Autocorrelation coefficients compare a series with its own lagged values and are the diagnostic used to count moving-average terms. Partial autocorrelation coefficients, obtained by adjusting ordinary autocorrelations so intervening lags are held fixed, are the diagnostic used to count autoregressive terms. That lag-by-lag reading of how an ordered series relates to its own past is the autocorrelation test.

The printed lag chart of both coefficient series, with a two-standard-deviation band, is the correlogram. A lag coefficient is counted as large only when it lies beyond two standard deviations, the conventional 95 percent threshold drawn on the printed chart.

Keep the list short

The preferred specification uses one or two autoregressive terms and one or two moving-average terms. Parsimony is that construction rule. A need for more terms, a broken spike pattern, or a late spike such as a large fifth lag is read as leftover trend that further differencing should remove.

When estimation is allowed to stop

Coefficient estimation is a nonlinear iteration that stops when the coefficients barely change, when the sum of squared forecast errors barely changes, or when a revision limit is reached. A stop on the revision limit alone is treated as a reason to reject the model.

Check residuals, then apply the equation

After estimation, residual errors are inspected with a Q statistic. Any remaining trend in those errors is addressed by adding another moving-average term. The finished equation can be applied as a rolling next-interval forecast or as a multi-step path several sampling intervals ahead, and a user-chosen confidence range can be attached around the point value.

Editorial note: keep exponential smoothing beside this structured ARIMA construction as a simpler quantitative forecast that updates from ordered price, volume, or breadth observations over a defined sampling interval.