The previous articles on time series analysis showed how to perform Autoregressive Integrated Moving Average (ARIMA) on GDP of India for the period 1996 – 2016 using STATA. The underlining feature of ARIMA is that it studies the behaviour of univariate time series like GDP over a specified time period. Based on that, it recommends an ARIMA equation. This equation is then used for forecasting GDP for further years. However ARIMA is insufficient in defining an econometrics model with more than one variable. For instance, for finding the effect of Gross Fixed Capital Formation (GFC) and Private Final Consumption (PFC) on GDP, ARIMA is not the correct approach. That is where multivariate time series using VAR is useful.
Equation of Vector Auto-Regression (VAR)
In multivariate time series, the prominent method of regression analysis is Vector Auto-Regression (VAR). VAR should be better understood in parts for clarity. Firstly, the term ‘auto-regression’ is used due to the appearance of lagged value of dependent variables on the right side. Secondly, the term ‘vector’ is used when dealing with vector of two or more variables. The resultant equation will be as follows:
In the above VAR equation, simultaneous achievement of all three variables are inter-related. Popular inter-relation between variables are used to reach the assumption of above equation. Since GFC and PFC play a role in calculation of GDP, the simultaneity between these variables are universal.
Performing VAR analysis in STATA
To proceed with VAR analysis in STATA, it is important to recognize all the steps, assumptions and important tests to be performed.
Steps and assumptions
|1. Lag selection of Variables||It is noticeable in the above equation (Fig 1) that the variables are interrelated with lagged values of other variables. However, it is unclear for how many lags the variables are interrelated.
Therefore, to begin VAR, first it is imperative to recognize the exact level of lags at which variables are inter-connected or endogenously obtained.
|2. Stationarity||In the previous articles the time series data showed that GDP is non-stationary. Resultantly apply first differencing. The same case could also happen for GFC and PFC. Therefore, the second step would be to check and assure stationarity in data.|
|3. Test for Co-integration||In case of co-Integration, suppose there are two or more than two non-stationary variables for regression. While estiamting residuals from regression, the residuals turns out to be stationary. That means, two or more than two non-stationary series may result in a stationary series. This is known as co-integration. The implication of co-integration is that, two variables have a long term casuality and in long run, the variables might converge towards an equilibrium value. Equilibrim value is steady, means have equal means and variance or ‘stationary’. Therefore, before initaiting VAR, it is imperative to know if the prersent model contains any co-integration or equalibrium state. co-integration or co-integrated variables indicates long term association among two or more than non-stationary variables|
|1. If Co-integration is not present = We apply VAR.||VAR technique where variables are endogenous and dependent on lagged values of other variables.|
|2. If co-integration is present = we apply Vector Error Correction Model (VECM).
|VECM model takes into account the long term and short term causality dynamics. It offers a possibility to apply VAR to integrated multivariate time series|
|3. VECM diagnostic, tests and forecasting||After constructing the VECM model, review further the assumptions of autocorrelation and normality. After that, perform forecasting.|
|4. ARCH (Autoregressive Conditionally Heteroscedastic Model)||Time series models incorporating the effects on volatility.|
|5. Extensions of ARCH||GARCH (Generalized Autoregressive Conditional Heteroskedasticity) and T-GARCH (Threshold- Generalized Autoregressive Conditional Heteroskedasticity)|