Mumtaz Ahmed

ABSTRACT This study is based on examining the relationship between income and consumption series of India covering the period of 1980-2009. Data about certain indicators were obtained from the official web site of World Bank. In first step of data analysis appropriate ARMA model was determined using correlogram and information criteria as well, and applied to the consumption data only. These models (ARMA and ARIMA models) are built up from the white noise process. We use the estimated autocorrelation and partial autocorrelation functions of the series to help us select the particular model that we will estimate to help us forecast the series. Second step of data analysis was comprised of co-integration and Error Correction model. It was found that per capita Gross Domestic Product and final household consumption per capita of India are not cointegrated. It was observed that both the series are integrated at order two I (2). But second condition of co-integration was not satisfied, the residuals were not found stationary. Hence it might be possible to conclude that there is no long run relationship between consumption and GDP series of India. As we know that the series are not co-integrated so we cannot apply Error correction model, but for the sake of understanding more specifically we also applied Error Correction Model. The adjustment co-efficient was not up to the standard it was around zero, it suggest that there is no need to make adjustments. Keywords: Gross Domestic Product, Consumption, ARMA, Co-Integration, Error Correction Model 1

AUTOREGRESSIVE MOVING AVERAGE PROCESS

1. Moving Average Process

In time series analysis, the moving average (MA) model is a common approach for modeling univariate time series models. Generally Lags of error term on independent side are called moving average method. A moving average model is simply a linear combination of white noise processes, so that Yt depends on the current and previous values of a white noise disturbance term. Let ut be a white noise process i.e. E(ut) = 0 , t V(ut) = 2 , t Cov(ut , us) = 0 , t Where S = 0,1,2,3,4…………….

The qth order moving average model is denoted by MA (q). This can be expressed as: Yt = u + ut + 1 ut-1 + 2 ut-2 +………………..+ q ut-q…………………………. (1)

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Above expression denotes the MA (q) model in sigma notation. Below are the MA (1) and MA (2) models with one and two lags respectively. MA (1): Yt = u + ut + 1 ut-1 ……………………………………………… (3) Yt = u + ut + 1 ut-1 + 2 ut-2……………………………………… (4)

MA (2):

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The moving average process is denoted in lag operator form as well and it is considered easiest method of notation.

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Where Li is the ith lag of ut is being taken, that is the value that ut took i periods ago. OR it can be written as: Yt = u + (L) + ut Where: (L) = 1 + 1L + 2L2 + …………………. + qLq Not including constant we can considerably eases the complexity of algebra involved and is inconsequential for it can be achieved without loss of generality. Drift/ intercept can also be included in the equations. Properties of Moving Average Processes The distinguished properties of the moving average process of order (q) explained above are: If Yt is a MA process of order (q) i.e. MA (q): Yt = u + ut + 1 ut-1 + 2 ut-2 +………………..+ q ut-q …………..(6)

Where ut is a white noise, than Yt should satisfy the following properties. E (Yt) = u Var (Yt) = 0 = (1 + 12 + 22 + ……………………+ q2) 2

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Cov (Ys) = s = {s + s+11+ s+2 2 + ………………qq-s) 2 for s= 1, 2,……..q,,, 0 for s > q } So it is clear that a moving average process has constant mean, constant variance, and autocovariances which may be non-zero to lag q and will always be zero thereafter. 2. Autoregressive processes

An autoregressive model is a very common model for time series. Consider...