During this time the moving average was introduced to remove periodic fluctuations in the time series, for example fluctuations due to seasonality. Herman Wold introduced ARMA (AutoRegressive Moving Average) models for stationary series, but was unable to derive a likelihood function to enable maximum likelihood (ML) estimation of the parameters.

It took until 1970 before this was accomplished. At that time, the classic book "Time Series Analysis" by G. E. P. Box and G. M. Jenkins came out, containing the full modeling procedure for individual series: specification, estimation, diagnostics and forecasting.

Nowadays, the so-called Box-Jenkins models are perhaps the most commonly used and many techniques used for forecasting and seasonal adjustment can be traced back to these models.

The first generalization was to accept multivariate ARMA models, among which especially VAR models (Vector AutoRegressive) have become popular. These techniques, however, are only applicable for stationary time series. However, especially economic time series often exhibit a rising trend suggesting non-stationarity, that is, a unit root.

Tests for unit roots developed mainly during the 1980:s. In the multivariate case, it was found that non-stationary time series could have a common unit root. These time series are called cointegrated time series and can be used in so called error-correction models within both long-term relationships and short-term dynamics are estimated.

ARCH and GARCH models

Another line of development in time series, originating from Box-Jenkins models, are the non-linear generalizations, mainly ARCH (AutoRegressive Conditional Heteroscedasticity) - and GARCH- (G = Generalized) models. These models allow parameterization and prediction of non-constant variance. These models have thus proved very useful for financial time series. The invention of them and the launch of the error correction model gave C. W. J Granger and R. F. Engle the Nobel Memorial Prize in Economic Sciences in 2003.

Other non-linear models impose time-varying parameters or parameters whose values changes when the process switches between different regimes. These models have proved useful for modeling many macroeconomic time series, which are widely considered to exhibit non-linear characteristics.