Measuring Short Run and Long Run Relationship Between Gdp Per Capita and Consumption Per Capita of India free essay sample

By Rizwan Mushtaq Under supervision of 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). We will write a custom essay sample on Measuring Short Run and Long Run Relationship Between Gdp Per Capita and Consumption Per Capita of India or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page 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 ARMA assumes that the time series is stationary-fluctuates more or less uniformly around a time-invariant mean. Non-stationary series need to be differenced one or more times to achieve stationarity. ARMA models are considered inappropriate for impact analysis or for data that incorporates random shocks†. More specifically an ARMA (pq) process is a combination of AR (p) and MA (q) models. Such a model states the current values of some series y depends linearity on its own previous values plus a combination of current and previous values of a white noise error term. The model could be written as: Keeping the effect of (Yt-1, Yt-2, Yt-3, Yt-4) fixed. ACF and PACF patterns for possible ARMA (p,q) models are as follows: AR(Process) MA(Process) ACF PACF ACF PACF Geometrically Number of non-zero It is significant at and It declines declines points = order of AR up to order of MA geometrically process, it takes non- process zero value up to order of AR ARMA (p,q) Process ACF Declines geometrically PACF Declines geometrically This methodology used sometimes and have certain flaws and issues. If both ACF and PACF declines geometrically we got ARMA procedures, just see the graphs and decide. BOX-JENKINS APPROACH They provide a methodology to fit an ARMA model to any given data series. It tells how to fit your ARMA model, there approach involves three steps: i. ii. iii. Identification Estimation Diagnostic Step 1: Identification Determining the order of ARMA model. This is done by plotting both ACF and PACF overtime. It tells us what order should we keep. Step 2: Estimation In this ste p we estimate the parameters of the model specified in Step I, using OLS and Maximum Likelihood method, depending on the model. Step 3: Diagnostic In this step model checking takes place. Box and Jenkins suggested two types of diagnostics 1) Over fitting (deliberately fitting a larger model than that is required) 2) Residuals diagnostic (Checking residuals for independence using Ljung-box test). Drawbacks in Box and Jenkins Approach Most of the time plot of ACF and PACF do not provide a clear picture. They do not match with deciding criteria; neither has MA nor AR process. So where we have messy real data we are unable to know which model is to use, and interpretation is very hard in this case. 7 Solution to This Problem Solution to this problem is to use the information criteria. Several criteria are available in literature but the most important criteria are discussed here. 1) Akaike’s Information Criteria AIC 2) Schwarz’s Bayesian Criteria SBIC 3) Hannan-Quinn Criteria AIC = ln(? ^2) + 2k/T SBIC = ln(? ^2) + k/T * lnT HQIC = ln(? ^2) + 2k/T * ln(lnT)) Where ? ^2 = RSS/T-K T= No. of observations, K=No. of regressors HQIC When plots are difficult to interpret and decide. We use information criteria; SBIC is considered the best one. The minimum value of SBIC is acceptable. CO-INTEGRATION 1. Integration To understand co-integration, it is essential to discuss integration first. A series is said to be cointegrated of order (1), if it becomes stationary after taking the first difference. The original series will called integrated at I (1) if it attains staionarity at second difference the series will called integrated at order two which can be written as I (2). And if the series become stationary at order (p) time the original series will be I (p). 8 2. Co-Integration After brief explanation of integration, now it is palpable to interpret co-integration. If two variables that are I (1) are linearity combined, then the combination will also be I (1). Two and more series (Xt, Yt) are said to be co-integrated if, i. i. They have same order of integration The residuals obtained from regressing Y on X are stationary. These two conditions must be fulfilled otherwise series will not considered as co-integrated. Engle and Granger, Procedure of Co-Integration Engle and Granger, proposed a Procedure for Co-Integration in (1987). X ? I (1): X is integrated of order (1) Y ? I (1): Y is integrated of order (1) Series X and Y are said to be co-integrated at order One I (1). They are actually non-stationary at level and become stationary at first difference. The combination of series X and Y will also be integrated at order one, it can be expressed as: Z = ? X + ? 2Y Z ? I (0) This process involves four steps: 9 Step I: Test the variables (x, y) for their order of integration using ADF. a) If both (x, y) are integrated of order (0) i. e. both are stationary at level than there is no need to test X, Y ? I (0). b) If both variables (X Y) are integrated of different order, than their will be no cointegration. c) If both variables (X Y) are integrated of same order, than proceeds to step II. Step II: Estimate long run (possible co-integration) equation if, X Y ? I (1). Here one thing should be noted that 95% of the economic series become stationary at order (1). If X Y ? I (0). Than estimate the following equation and get residuals Yt = ? 1+ ? 2 Xt + ? t Step III: Check the order of integration of residuals i. e. residuals are tested for stationary using ADF. It is important here to note that stationarity of residuals is tested by estimating the model without intercept and without time trend. So, estimate the following model. ? ? 10 Note: estimate this model and test the null hypothesis, also note that we have to use different critical values which are more negative than the usual Dickey-Fuller critical values, use critical values proposed by Engle and Granger. Step IV: In step 4 we estimate Error Correction Model (ECM). It gives us both short run and long-run impacts of X on Y, and also provides the adjustment co-efficient. Which is the co-efficient of lagged values of error term i. e. et-1. ERROR CORRECTION MODEL Error Correction Model (ECM) simply corrects the error. Here one thing is important to discuss that if variables X Y are co-integrated than the residuals (et) obtained from regression of Y on X will be stationary. It might be expressed in this way: et ? I(0) So, we can express the relationship between X and Y in the form of an Error Correction Model as: ?Yt = b1 + ? Xt + ? t-1+ Vt†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦ (10) Where, b1 = is short run impact of x on y. Vt = is the error term. And ? is the co-efficient of et. It is also called adjustment co-efficient, feedbacks and adjustment effect. If ? = 1 than 100% of adjustment taking place. If ? = 0. 5 than 50% of adju stment taking 11 place, and If ? ? 0 than there is no need to make adjustments. Basically Error Correction Model provides us both short run and long run impacts of X on Y. EMPIRICAL ANALYSIS ARMA 1-Identification Figure: 1 Correlogram Consumption Step I: As we know that the first step of ARMA is identification, it is done through correlogram. Figure: 1 Correlogram consumption denotes the typical processes from the ARMA family with their so called characteristics autocorrelation and partial autocorrelation. These described function of autocorrelation are not derive from relevant formula, rather are estimated using underlying simulated observations with disturbance drawn from a normal distribution. Figure: 1 articulates that the autocorrelation and partial autocorrelation functions are significant at lag 1, while the autocorrelation function declines geometrically, and is significant until lag 3. Plot of the 12 onsumption series (see appendix figure 1) also shows increasing trend which represents that the series is integrated, and we need to proceed with taking logarithms and first differences of the series. Step II: We now in step two because of above behavior of consumption series which we observe through correlogram. Here we take the log of consumption series and then first difference of said series. Below are the comman ds that are used to do so: genr lcons=log(cons)†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (i) genr dlcons=lcons-lcons(-1)†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (ii) We get correlogram of newly created dlcons (log-differenced) series, which is portrayed here: Figure: 2