THE NECESSARY AND SUFFICIENT CONDITIONS FOR AN

EFFICIENT TIME SERIES MODEL

 

 

 

 

 

 

 

 

 

By

 

John C. Pickett

 

And

David P. Reilly

 

Robert M. McIntyre

 

 

 

 

 

 

John C. Pickett, Professor of Economics

University of Arkansas at Little Rock

Department of Economics and Finance

2801 S. University Ave.

Little Rock, AR 72204

Voice: 501.569.8878

FAX: 501.569.8871

Email: This e-mail address is being protected from spambots. You need JavaScript enabled to view it

 

 

David P. Reilly, Senior Vice-President

Automatic Forecasting Systems, Inc.

PO Box 563

Hatboro, PA 19040

Voice: 215.675.0652

FAX: 215.672.2534

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Robert M. McIntyre, Associate Professor of Industrial-Organizational Psychology

Old Dominion University

Department of Psychology

Hampton Boulevard

Norfolk, VA 23507

Voice: 757.651.1082

FAX: 757-683-5087

Email: This e-mail address is being protected from spambots. You need JavaScript enabled to view it

 

 

ABSTRACT

THE NECESSARY AND SUFFICIENT CONDITIONS FOR AN EFFICIENT TIME SERIES MODEL

 

Applied economic practitioners have a long history of forecasting economic time series. The challenge facing all forecasters is to provide forecasts that minimize the forecasting error. Reference to many introductory statistics text reveals popular--but at times misguided--practices. Most texts fail to offer an integrated discussion of causal variables, memory structure, and intervention variables.

The practitioner seeks an understanding of how to identify and estimate the most efficient model. This paper defines an efficient model as the one that satisfies the necessary and sufficient conditions.

The necessary condition focuses on including only statistically significant parameters. Think of the necessary condition as asking the question “Within the candidate forecasting model, are all parameter estimates required?” The sufficient condition asks a second question: “Have any statistically significant parameter estimates been omitted from the model?” The sufficient conditions focus on determining whether or not the assumptions underpinning time series analysis are satisfied. The paper focuses on the sufficiency assumptions. It describes each in detail and offers suggestions ("sufficiency variables") to insure the sufficiency conditions are satisfied.

 

INTRODUCTION

Applied economists create forecasting models on the basis of a summary of the characteristic patterns of a single series or on the basis of patterns that characterize a bivariate or multivariate time series.   Selection of the summary model amounts to a search for the ‘least-worse’ model. This follows from the old adage by George E. P. Box “All models are incorrect, but some are useful.”

Box’s statement is true for three reasons. First, the model is estimated from a sample realization of the population process. Hence, the analysis is an inferential search for the true forecasting model. Second, variables are never psychometrically error-free, which is to say that what the analyst thinks an observation measures may or may not be true. Third, all models are misspecified in the sense that each omits unknown, but essential independent variables.   These difficulties not withstanding, forecasters must do their job and prepare forecasts.

 

CURRENT PRACTICES

 

A review of many introductory statistics texts reveals misguided popular practices. The usual two chapters on simple and multiple regression introduce the student to the method of ordinary least squares (OLS). Included will be an extensive discussion of the methods used to calculate the estimated parameters, the summary statistics, and a discussion on how to test all estimates of the parameters. It is common practice for the author of an introductory text to discuss problems associated with multicollinearity and first-order autocorrelation. Beyond a rudimentary exposure to the latter topic, authors of an introductory text ignore any further discussion of a memory structure other than the “fix-up” imposed on OLS models for first order autocorrelation.

Following the chapters on simple and multiple regression, introductory business school statistics texts introduce the student to time series models, usually in a pejorative way. This “introduction” may discuss extrapolation techniques and seasonal components. In effect, because the basis of this introduction is OLS, these discussions of “time series techniques” can best be labeled deterministic trend extension strategies--not bona fide time series techniques.

Econometric methods courses introduce students to “intermediate” statistical methods. The majority of modern econometric texts used in these courses take students on a rather complicated journey that requires them to confront and best a beast named matrix algebra, all the while fighting little side skirmishes with simultaneous equations, asymptotic distributions, covariance-stationary processes, unit roots, cointegration, and general method of moments. However satisfying to erudite mathematical authors, complicated, and confusingly described statistical techniques do not serve the needs of students whose vocation is that of practitioner.   Our experience indicates that what practitioners seek is sound understanding of how to identify and estimate a valid and reliable forecasting model. Unfortunately--perhaps because of introductory texts are the only accessible source to practitioners—they seem to rely on erroneous but apparently simple techniques. Typical forecasting practices used by practitioners involve attempts at pre-specifying a forecasting model’s form, the use of OLS software to estimate the model, the application of dummy variables to account for seasonal variations and outliers, and the use of Hildreth-Lu techniques for dealing with the problem of first order autocorrelation. Practitioners identify the final model by extensive hands-on manipulation of the data, and numerous iterations before it is identified. In addition to using inappropriate model-building strategies, the practitioner suffers the burden of a tedious iterative process.

The time series techniques discussed below focus on the dynamic consequences of events occurring over time. Observations in a time series will be at equally spaced intervals through time, such as hourly, daily, monthly, quarterly, or annually. The critically important distinction between time series data and cross-sectional data is that observations in cross-sectional data set are assumed to be randomly sampled from some population. Put another way, cross-sectional sample data are assumed to be independent of one another. As such, a relatively high value for one data point would tell us nothing at all about whether another value is likely to be high or low. In contrast, time series analysts do not expect independence among time series data. Instead, the analysts expect there to be dependence (covariation) among observations collected over time, particularly among observations that occur at relatively close time intervals. In other words, 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. Since time series data do not behave as a random cross-sectional sample of data, they require special statistical methods--Time Series Analysis.   The latter provides the basis for circumventing the non-independence in time series data. In fact, beyond dealing with the non-independence of time series data as a nuisance to be controlled, this set of techniques actually exploits the non-independence while OLS techniques do not.

Time series methods should be used to analyze time series data.   OLS techniques should only be used to analyze cross-sectional data set. This argument is based on the fact that time series observations do not comprise a random sample drawn from a population and cannot satisfy the underlying assumption that the observations are independently distributed.

BEST PRACTICE METHOD

 

The following provides a description of how a time series model can be selected from among all possible models. The rapid growth in microcomputer power and powerful software innovations coupled with the emerging finality of modeling techniques offer the analyst an opportunity to adopt the “best practices” method of estimating the final model. Here, ”best practices” refers to the set of methods implied by cutting edge knowledge on efficient model identification and estimation.

Before beginning our discussion of the best practices, several points must be made, some of which have already been mentioned above. First, most data analyzed by forecasting practitioners are of the time series variety. As was pointed out above, OLS techniques, developed in the early 20^th Century, were designed to analyze cross-sectional data. Since time series techniques were not codified until the 1960s, analysts adopted OLS techniques from the cross-sectional domain and applied them within the time series domain. Second, OLS techniques require the analyst to pre-identify the functional form. There are thousands of candidate models, and time constraints prevent the analyst from considering all the possible OLS models. Third, most OLS techniques require the parameters to be linear. While non-linear estimation techniques are available, most analysts do not begin with nonlinear methods. In contrast, time series techniques assume non-linearity in parameters from the outset.   Linearity in parameters is a subset of the more general nonlinearity assumption. Fourth, OLS technique can be used to estimate autoregressive time series models but cannot be used to estimate moving average models. This is especially important deficiency. Moving average models assume that an observation time t is, at least in part, determined by residuals of observations occurring prior to time t. The effects of omitted variables are technically embedded in the residuals of the past. Therefore, to omit the moving average class of models is to mishandle models with omitted variables. Fifth, current OLS techniques focus on testing each estimated parameter to determine if all are statistically significant followed by performing a selected number of ‘fix-up’ routines if heteroscedasticity, autocorrelation, and seasonality appear to be present.

AN EFFICIENT METHOD

 

Model selection withusing OLS techniques entails maximizing adjusted R² or minimizing the forecasting error. The theme of this paper is that there are necessary and sufficient conditions that should be used to select the best forecasting model.   Think of the necessary and sufficient conditions that insure the selection of the efficient model in the same way as a mathematician uses them to establish a local maximum or minimum of a function.

NECESSARY CONDITION

 

The necessary condition of an efficient forecasting model is that it contains only essential parameter estimates. One should think of the necessary condition as asking the question “Within the candidate forecasting model, are all parameter estimates required?” This question is implicitly answered by determining whether parameter estimates are statistically significantly different from zero.

A simple example may be helpful to understand the necessary condition and make the transition to the sufficient condition. Consider a model:

Y_t = S_t +   A_t

 

where:

 

Y_t is the dependent series,

 

S_t is the signal referring to a set of parameter estimates initially hypothesized to be of importance, and

 

A_t is the error series.

 

The necessary condition points to the requirement that only essential (“statistically significant”) parameter estimates are included within S_t.   Within OLS multiple-regression procedures, the well-known procedure of stepwise regression exemplifies a type of necessity testing.

The sufficient condition focuses on a second question: “Have any statistically significant parameter estimates been omitted from the model?” In other words, does A_t contain any non-random “structure” that can be identified and moved into S_t? This question is the downfall of most forecasters using OLS techniques.   Furthermore, answering the question requires the analyst to determine if a statistically significant lag structure is present within the errors or if there is any other “untapped” information embedded in the dependent and independent variables. Answering this question is complicated by the presence of outliers, which can mask the relationship between the dependent and independent variables and the structure of the errors.

SUFFICIENT CONDITIONS

 

The sufficient conditions focus on determining whether or not the assumptions underpinning time series analysis--in particular, assumptions regarding the behavior of A_t--are satisfied. The analyst begins with a set of assumptions about the residuals and constructs a set of mathematical calculations so that when the final model is determined, the residuals from the model meet these assumptions. If none of the assumptions isare satisfied, then the model is completely inefficient. Stated another way, if the sufficient conditions are not satisfied, then the second question-- Have any statistically significant parameter estimates been omitted from the model?--is answered in the affirmative.

SUFFICIENCY VARIABLES

 

There are seven conditions described below which serve as the basic assumptions that must hold for a model to be sufficient. Each of them can be addressed by the creation of a "sufficiency variable." If one of the underlying sufficiency assumptions is not satisfied, then a deficiency exists that may be remedied by determining whether one of the sufficiency variables can be included in the prospective model (see Table 1 for a list of sufficiency variables below). It is conceivable that a series of residuals may fail to meet more than one of the sufficiency assumptions. In such cases, more than one sufficient variable may need to be added to the model to satisfy the sufficiency requirement. The sufficiency variables may include the following:

_____________________________________________________________________________________________

Table 1. List of Sufficiency Conditions
Sufficiency Variable Type	Sufficiency Condition Description
1	Lagged values of Y
2	Lagged values of each X and lead values for each what?
3	Intervention variable(s) representing a pulse(s).
4	Intervention variable(s) representing a seasonal pulse(s).
5	Intervention variable(s) representing a level shift(s)
6	Intervention variable(s) representing local time trend(s).
7	Moving average term(s) representing lagged values of the error terms

____________________________________________________________________________________________

The following is a brief discussion along with an example of each of the sufficiency conditions and the sufficiency variables designed to address it.   Note that the graphs presented are only single instances portraying the failure of the sufficiency assumption.

Sufficiency Condition 1. E(e_i) = 0

This condition implies that the mean of the residuals should not deviate from zero in any subset of the time series. The analyst must therefore investigate all possible subsets and not limit the test to the overall mean of the residuals. If the mean of a subset of the residuals deviates significantly from zero, then type 3 - 6 sufficient variables (listed above) may be required. Figure 1 shows the mean of the residuals increasing at observation 30.   The difference in the means of the two subsets is statistically significantly different from zero, and in this case, visually obvious.

 

Sufficiency Condition 2. s_i² = k

The second sufficiency condition implies that the residuals have constant variance throughout the series.

Here again, the analyst must test all possible pairs of subsets.   If the variance of the residuals is not constant, then a number of remedies are possible. A weighted regression, a power transformation, or generalized autoregressive conditional heteroskedasticity (GARCH) techniques may be used to ensure the variance is constant. Figure 2 shows a plot of the residuals, where the variance increases, beginning with observation 60. Notice that not only is there a difference in the variance but the residuals are clearly autocorrelated. Oftentimes Often times a violation manifests different symptoms.        

Sufficiency Condition 3. Cov(e_i,e_j) = 0

 

This condition requires that residuals not be autocorrelated for all lags. With regard to tests for autocorrelation, readers should be aware that the Durbin-Watson statistic tests for the existence of first-order autocorrelation only. The analyst must, therefore, examine autocorrelation of residuals for all possible lags. If the residuals are autocorrelated, then type 1, 2, or 7 sufficient variables from Table 1 may be required. Figure 3 shows a plot of the residuals with a pattern that clearly evidences autocorrelation of residuals.

 

 

Sufficiency Condition 4: e_i @ (NID)

This condition states that the residuals are normally and independently distributed. Failure of this so-called independence assumption is closely related to the failure of assumption 3 above. If the residuals are not independent, then type 1, 2, or 7 sufficient variables may be required. If the residuals are not normally distributed, then either a power transform of the dependent variable or a type 1 sufficient variable may cure the deficiency. Figure 4 shows a plot of the residuals that do not meet this condition. A histogram of the first 16 observations would have a different shape than the histogram of the last 24 observations.

 

 

 

Sufficiency Condition 5: e_i ¹ ¦(X_{i - t})

 

This condition states that the residuals are not a function of lagged values of X.   If the residuals are a function of lagged values X, then the analyst has omitted a statistically significant lag structure on X. This deficiency may be remedied by a type 2 sufficient variable. Figure 5 shows a characteristic pattern of data that fail to meet this condition. A simple regression of the residuals on the lagged values of X would show the estimated parameter to be statistically significant.

 

Sufficiency Condition 6: X_i ¹ ¦(e_{i - t})

This condition states that the X values in a series are not a function of the lagged residuals.

 

If the Xs are a function of the lagged residuals, then the one-way causal model is the incorrect functional form. Failure of this assumption is frequently observed when modeling large macroeconomic systems. In these models, the dependent and independent variables are all interdependent, which requires multiple time series techniques (vector time series). Figure 6 shows a characteristic pattern where this condition is not met. The parameter estimated from a simple regression between X and the lagged residuals would be statistically significant. However, there are many cases where a series is affected by “future values” such as when customers forestall purchasing products because they are aware of price changes that will occur. This is a “lead effect “ and can be modeled without loss of generality.

 

 

Sufficiency Condition 7. The distribution of the residuals is invariant over time.

One subset of the series data should have the same covariance structure as another subset. In Figure 7 the autocorrelation function for the first 10 observations is different from the last nine observations. The autoregressive parameter for the first 10 observations would be positive while it would be negative for the last nine. Hence the parameter would not be consistent over time. If the covariance structure is not the same, then the data exhibit time-variant parameters. Corrections for the failure of this assumption are not found in the seven sufficient variables. Rather, modeling a change in parameters while theoretically interesting can lead to “death by model” or a model with too many parameters. It is quite reasonable to use “Occam’s Razor,” and simply focus on the most recent homogenous set. It is fair to say that modeling a realization that effectively are samples of more than one process is technically beyond the state of the art at this time. If this sufficiency condition is not met, then the analyst identifies when the change occurs and estimates the model within the set of most recent observations. Figure 7 shows a characteristic pattern. Clearly the slope of the line fitted for the first 10 observations would be positive while the slope of the line fitted for the last nine observations would be negative.

 

 

SUMMARY

 

Practitioners devote significant resources to forecasting economic time series. Traditional OLS methods are not appropriate techniques for forecasting time series. This follows because observations on a time series are not a random sample from a population. Rather, time series observations have dependencies among them, and, as a result, violate the underlying assumption of independence. The present paper's primary focus is to review the necessary and sufficient conditions that the final model must satisfy before it can be declared the efficient model.   Think of the necessary condition as a checklist to identify the statistically significant parameters that must be included in the efficient model. The sufficiency condition ensures that the data and their residuals satisfy the underlying assumptions.   If one or more of the underlying assumptions are not satisfied, then one or more of the above sufficient variables are necessary for the final model to be deemed efficient.

The "best practices method" described in this paper offers the analyst two insights.   First, it provides a paradigm for focusing on the necessary and sufficient conditions to ensure the final model is efficient. Second, it encourages the analyst to identify an efficient model from among all possible model specifications. Referring to the previous model,

Y_t =   S_t + A_t

 

the feedback resulting from the necessary and sufficiency tests indicate that   information embedded in A_t should be moved to S_t by adding one or more of the seven sufficiency variables.

 

REFERENCES

 

An extensive list of applicable references may be found at:

 

www.autobox.com/reference.html.