AUTO-REGRESSION AND DISTRIBUTED LAG MODEL SHRI PRAKASH BIMTECH OBJECTIVES OF LEARNING The following are the objectives of this module: (1) Knowing the following concepts: (i) Model and Modeling in the context of time series

modeling; (ii) Auto-regressive model- general auto-regressive model with infinite lags, and first and second order Auto-Regressive Models, (iii) Moving Average Integrated Auto-Regressive Modeling; and (2) Knowing the Concepts of (i) Distributed Lag Models, (ii) Distributed Lag Models as a

part of Auto-Regressive Models; (3) Reasons for Incorporation of Time-Lags in Time Series Models; (4)Koycks Distributed

Lag Model Specification, and Interpretation and Mark Nerloves, Shirley Almons Specification and Interpretation; and (5) finally, Problems and Procedures of Estimation of Auto-Regressive and Distributed Lag Models. 1. INTRODUCTION Social, political and economic systems are mostly resistant to change. Even if the systems change, the change is generally low and slow. There is an in-built inertia, which resists change. Most of the systems, therefore, change ever so slowly; the process of change is gradual. Gradualness of change ensures that the future values remain entrapped in the past values, though immediately preceding value remains most relevant for each succeeding value. Some Note-worthy Features (i) Socio-economic systems are whirlpools of inter-dependencies; it makes discerning and detection of causal relations intricate and complex. (ii) (iii) In time series analysis, practically all variables move with time directly or inversely. Time may be treated as a catch all variable in dynamic systems; time may be treated as a proxy of any and every variable that is subject to change. These are the traits of time series, which involves and treats time as a variable in dynamic models. Such models are considered as dynamic, since the values of the variable trace the nature and time path of change. Change characterizes only dynamic systems; static systems do not depict historically repeating or once for all change. Dynamism makes change a continuum in

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time. This trait of the systems and their values, reflected in the time series of observed values, makes forecasting of future manageable despite the complexities and uncertainty of the direction and magnitude of change. Therefore, a simple single/multiple equation model without the backing of intricate theory may efficiently perform the function of describing the observed behavior and expected change in future values of a single variable, or a set of variables in terms of its/their own values observed in the past. This belief is strengthened by the fact that the large scale simultaneous multiple equation macro econometric models have not served the purpose of forecasting so efficiently as simple single equation time series models have done. This observation may, however, not apply to returns as the base of investment in MFs and Equities; their prices and returns are generally volatile. Incidentally, Campbell et al. suggest the use of returns rather than prices since they consider returns to investment relatively non volatile and to possess some beneficial attributes. The view is misplaced since returns are directly related to prices which fluctuate a lot from one to another time period (See, Prakash and Panigrahi, 2007). Single or multiple-variable time series models involve compact specification and only a few parameters for estimation. Involvement of limited number of parameters not only makes estimation simple and easy but it also effects saving of time and effort involved in analysis. This facet depends upon the number of exogenous or endogenous variables used in the model. Time series models have the following advantages: (i) Accurate forecast of future values by the modeling of time series rather than the use of elaborate and large scale macro econometric modeling; (ii) Close approximation to the empirical observations by interpolation and experience. These two properties have made Box-Jenkins (1984) type single variable time series models popular among researchers. 2. FORECASTING AND AUTO-REGRESSION MODEL Forecasting of future values of the variable(s) under consideration is one of the important functions of time series analysis. But forecasting cannot be done without proper explanation. Explanation of the behavior of the observed and forecasting of future values of the variable(s) require time series based modeling.

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Time series is a set of data in which time enters as an independent entity. Time series models are, therefore, defined as dynamic models. We have to examine some basic questions in this context. Some of these questions are listed hereunder. 1. 2. 3. 4. 5. What is a model and what is modeling? What is a time series model? What is auto-regression (AR), or auto-regressive model (ARM)? What is the role of time in auto-regressive models? How does time enter into regression modeling?

These and some other questions relate to the basic concepts some of which are newly evolved and others are of older vintage. Answers to above questions will clear the conceptual framework of time series analysis. WORKING DEFINITION OF MODEL AND MODELING Working definitions of model and modeling are discussed here in the context of time series analysis. MODEL A model is generally conceived as the miniaturized replica of reality. Replication of reality involves the conceptual elaboration and the specification of the mechanism or the process by means of which the observed values can be generated. The time series model assumes that the observed values have been generated by the stochastic process which is embodied in the model. The process of generation of series of values in the continuum of time is stochastic Observed values are assumed to be (i) Governed by the principle or theory of probability. All observed values of the given

series constitute the part of a specified Probability Distribution Function; (PDF); and (ii) Expected Value of all probable/possible values that could have materialized at a given

point in time with a definite probability. This is explained as follows: Pit: P10 Yit : Y10 P11 P12 P13P1s P1n

Y11 Y12 Y13 Y1sY1n

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In the above series of values, Yis refers to one value of Y that could have materialized and the probability of its materialization is depicted by Pis. Expected value of Y is given below: E(Y) = P1sYis Expected value is average value of the stochastic series of values. Above specification makes time series values to emanate from a stochastic process. MODELING As against the concept of model, modeling refers to the process/mechanism, or the method of formulation of the model. Each model encompasses a specific conceptual framework, assumptions and the process of generation of the series of values. General process of modeling also needs the formulation of causal relation that may be an essential component of the process of generation of the series of values. Conceptual framework is drawn from probability theory. For example, Markov Chain theory, or Classical Probability theory, or Baysesian Probability Theory may furnish the conceptual framework. Time series models are invariably stochastic in nature: (i) If the same stochastic process underlies the generation of several sets of values, observed over a given period of time, each set will be different;

(ii)

Observed values of every set will be governed by the same laws of probability distribution that characterizes the stochastic process of generation of values at each point in time of the period under consideration.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3. AUTO REGRESSIVE MODELING Auto-Regressive (AR) Models are mainly a part of Time Series Forecasting Modeling. It implies explanation of behavior of the variables is secondary rather than primary function of time series modeling; forecasting is the basic function of time series modeling, though forecasting may involve explanation as well. The term Auto-Regression is used to describe the stochastic process of generation of a set of observed values of one or several4

variables in which the current value of the variable under consideration and its random errors are treated as a function of their own lagged value(s). GENERAL AUTO-REGRESSION MODEL A general Auto Regressive Model is a single variable model of infinite lags, which stipulates the current value of variable, Yt, and its random error, Ut to depend upon their own past values, Yt:Yt-s; and Ut: Ut-s; t=0,1,.,T, t stands for time. In other words, an auto-regressive model treats current value of the variable and its errors to be the function of all their past values alone. Such models do not involve any exogenous variable(s), Xt, as the determinant of the dependent variable, Yt. It may, therefore, be inferred that auto-regression models are devoid of any theoretical backing. At the most, such models may be considered to have very weak theoretical backing. If, however, such models are treated as difference equation models, which these models actually are, then, the models may be taken to be encompassed in Cobweb Theorem. Besides, there is a theory of time being treated as a variable. The theory is embodied in the explanation of the four components of time series: seasonal, cyclical, trend and residual/random. Seasonal variation relate to behavior of demand in different seasons of the year. Such fluctuation in demand and price are short lived since the duration of seasons is a couple of weeks/months.

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Cyclical fluctuations are of relatively longer duration. There are several competing theories of business cycles. So, explanation of cyclical or historically repeating changes has a theoretical base. Random fluctuations arise from the influence of random factors. But random factors do not have any definite theory for explanation. Behavior of trend may be examined on the basis of alternative hypotheses which may be derived from empirics. Therefore, above inference of absence of theory of time series analysis is erroneous. General Auto-Regression Model The general auto-regression model is specified as hereunder: Yt=+ tY0+ 1Yt-1+ 2Yt-2+.. sYt-s++Ut+V1Ut-1+V2Ut-2+...VsUt-s (1) No specific value is assigned to t. neither it is required, nor is it feasible in principle to exactly specify the origin of the process of generation of time series. Relation 1 is defined as an auto-regression model of infinite lags. If the length of the lag is specified, model will become auto-regression model with finite lags. Infinite lag model possesses certain mathematical properties, which give advantage in empirical research. Length of the lag appropriate for a given time series is a matter of empirics, though treatment of time as an independent entity is based on theoretical or a priori considerations.3.1 FIRST ORDER AUTO-REGRESSIVE MODEL A first order auto-regressive model is the simplest of this genre of models. Such models treat current value to depend upon its immediately preceding value alone.6

Such models are embedded in the concepts of seasonal and cyclical behavior of the series. In such cases, preceding value of the variable exercises more decisive influence than the remote values of the variable. First order auto-regressive model is a Markov Process Model. First Order Markov Process Model is a special case of General Markov Process. Thrust of the argument is that Markov Chain Probability Theory is the underlying base of such models. The first order auto-regression model is outlined below: Yt =0+ 1 Yt-1 +

Ut .(2)

This model contains only one lagged value of Y and U. 3.2 AUTO -REGRESSIVE PROCESS AND GENERAL FORM General form of first order auto-regressive model is auto-regressive process based model. This model is outlined below: Yt - =U (Yt-1 ) + Ut (3)

Rearrangement of terms of 3 will give the following model: Yt =(1- U) +U Yt-1 + Ut (4)

The term Auto-Regression applies not only to stochastic process of generation of observed values of Y at different points in time, but it also applies to the stochastic process of generation of error term Ut. As a first order random error, Ut is also specified by auto-regression model of first order: Ut =V Ut-1+t .(4a) V is coefficient of auto-covariance, or auto-correlation of Ut at time t. Auto-correlation coefficient, V, is stipulated to lie in the range: -1< V