How to Select a Most Efficient 'Ols' Model for a Time Series Data

Ordinary Least Square (OLS) models are often used for time series data, though they are most appropriated for cross-sectional data ... provides a check list of conditions that must be satisfied for an OLS model to be most efficient ... also, gives sufficiency variables that can be used to overcome various problems in the model. Practicing forecasters seek techniques that maximize forecasting accuracy and minimize forecast error. Their usual challenge is to make forecasts of the next period on the basis of time series data, which has a monthly, quarterly or annual data. Despite the tomes of econometricians' ponderous recommendations residing in (sometimes dusty) university libraries, it is not unusual for many practitioners to resort to an "old friend" - the ordinary least squares (OLS) model. In this article, we will show how to use our old OLS friend for optimum results. We also provide ways of identifying and estimating the most efficient OLS model, the model that minimizes the forecasting error. OLS models, developed in the early 20m Century, were designed to analyze cross-sectional data, not time series data. A fundamental assumption required of the user of OLS models is that target data are randomly sampled from a population, and as such are independent of each other. There are occasions when practicing forecasters may use data that meet this critical assumption. For example, they may examine cross-sectional data collected from across geographical regions, age groups, income levels in a given time period. In a cross-sectional dataset, knowledge of a value for one observation would tell us nothing about the value of another observation. However, when forecast practitioners use time series data to make their predictions, they face something quite different from cross-sectional data properties. Time series data are collected at equally spaced intervals through time. In contrast to cross-sectional data, knowledge of an observation collected at time 1 may well provide information with regard to the value of another observation collected at time 1 plus 1. In other words, time series data cannot be treated as randomly selected observations from a population. Time series observations collected at proximal time periods tend to be more similar than observations collected from two distant time periods. It is as though the data have time-driven "memory"! The OLS user in this circumstance must develop an efficient OLS model-a modification of the OLS model-that circumvents the violations of the assumption of independence. SUFFICIENCY VARIABLES There are eight assumptions, described in the next section, which must hold for a forecast model to be sufficient. If one of the underlying sufficiency assumptions is not satisfied, then a deficiency exists within the model that may be remedied by incorporating the corresponding "sufficiency variable" in the model. Table 1 gives a list of sufficiency variables along with their description. It is conceivable that a model may fail to meet more than one of the sufficiency assumptions. In such cases, more than one sufficiency variables may be required. OLS MODEL SUFFICIENCY CONDITIONS An OLS model must satisfy the following conditions (assumptions): 1. All the variables contained in a model are statistically significant. It means that each of the model parameter estimates is statistically significant and is greater than zero. Furthermore, the model does not omit any essential variables. These conditions are so basic that they cannot be addressed by means of sufficiency variables listed in Table 1. 2. This condition pertains to residuals. Residuals represent the difference between the actual and fitted value of each observation in the data series. The mean of residuals should not deviate significantly from zero in any subset of the time series. In other words, if the data comprising the time series data are broken down into different subsets, the mean of residuals of each subset should not deviate significantly from zero. …