4. Univariate Time Series Modelling

Paz Moral and Pilar González
December 10, 2003

Data in economics are frequently collected in form of time series. A time series is a set of observations ordered in time and dependent of each other. We may find time series data in a wide variety of fields: macroeconomics, finance, demographics, etc. The intrinsic nature of a time series is that its observations are ordered in time and the modelling strategies of time series must take into account this property. This does not occur with cross-section data where the sequence of data points does not matter. Due to this order in time, it is likely that the value of a variable $ y$ at moment $ t$ reflects the past history of the series, that is, the observations of a time series are likely to be correlated. Since the observations are measurements of the same variable, it is usually said that $ y$ is correlated with itself, that is, it is autocorrelated.

Time Series Analysis is the set of statistical methodologies that analyze this kind of data. The main tool in Time Series Analysis is a model that should reproduce the past behavior of the series, exploiting its autocorrelation structure. The objectives of Time Series Analysis are basically two: to describe the regularity patterns present in the data and to forecast future observations. Since a pure time series model does not include explanatory variables, these forecasts of future observations are simply extrapolations of the observed series at the end of the sample. If we consider a single variable in our study, we shall construct what is called a univariate time series model. But if two of more variables are available, the possibility of dynamic interactions among them may be important. We can think, for instance, in economic variables such as consumption, investment and income that influence each other. In this case, multivariate time series models can be constructed to take into account these relations among variables (Lütkepohl; 1991).

This chapter will focus on which is called Univariate Time Series Analysis, that is, building a model and forecasting one variable in terms of its past observations. It is centered in time series models based on the theory of linear stochastic processes. A good survey on nonlinear formulations of time series models may be found in Granger and Teräsvirta (1993) among others. The chapter starts with a brief introduction of some basic ideas about the main characteristics of a time series that have to be considered when building a time series model (section 4.1). Section 4.2 presents the general class of nonseasonal linear models, denoted by $ ARMA$ models that can be shown to be able to approximate most stationary processes. Since few time series in economics and business are stationary, section 4.3 presents models capable of reproducing nonstationary behavior. Focus is set on $ ARIMA$ models, obtained by assuming that a series can be represented by an $ ARMA$ stationary model after differencing. In section 4.4, the theory to obtain Minimum Mean Squared Error forecasts is presented. Model building and selection strategy for $ ARIMA$ models is explained in section 4.5 along with an economic application analyzing the European Union G.D.P. series. Finally, in section 4.6 the issue of regression models with time series data is brought up briefly.