Xt=A1 Xt-1 +A2 Xt-2+………………, Ak Xt-k+ t 3) Where Xt is (n*1) and each of Ai is (n*n) matrix of parameters. This type of VAR model has been advocated mostly by Sims (1980) as a way to estimate dynamic relationships among jointly endogenous variables without imposing strong priori restrictions. The system is in the reduced form with each variable in Xt regressing on only lagged values of both itself and all the other variables in the system. Thus, Ordinary Least Squares (OLS) is an efficient way to estimate each equation since the right-hand side of each equation comprises a common set of (lagged and thus predetermined) regressors. A vector error correction model (VECM) can be obtained by reformulating equation (3). Thus,
The Granger representation theorem states that if the coefficient matrix has reduced rank r<n, then there exists n*r matrices and , each with rank r such that = and 'Xt is stationary. r is the number of cointegrating relations (the cointegrating rank) and each column of is the cointegrating vector. The elements of are known as the adjustment parameters in the vector error correct model. The Johansen method is to estimate the matrix in an unrestricted form, then test whether we can reject the restrictions implied by the reduced rank of . The cointegrating vector is solved as the eigevectors associated with the r largest statistically significant eigenvalues is tested using two test statistics, namely, “Maximum eigenvalue statistic” ( l max) and “trace statistic” ( l trace) . l trace tests null hypothesis that the number of cointegrating vector is less than or equal to r against an unspecified alternative. If l trace = 0, when all the l I = 0, so it is a joint test. In the similar manner, l tests the null hypothesis that the number of cointegrating max vectors is r against an alternative of r+1. The test statistics for cointegration are as follows.
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Where, T is the sample size and l r+1 is an estimated eigenvalue.
Graphical View of Stationarity and Non-stationarity of Time Series Data Non-stationarity occurs when a series exhibits no affinity for a mean value. Many macroeconomic series exhibit non-stationarity due to an upward drift over time. When estimating electricity demand it is first necessary to derive a stationary series. If non-stationarity is existent in the raw series, a technique such as first difference of the original is employed to obtain a stationary series. ACs and PACs for the sample period are examined to check whether the series is stationary or non-stationary. Accordingly the following figures represent the plots of the original and first difference of the raw data. The plot of the original data reveals the series under consideration to be nonstationary which is shown in figures 1, 3, 5, and 7. The first difference of the original, non-stationary data, plotted in figures 2, 4, 6, and 8 is stationary.
Figure 1: Real GDP of Nepal (1980-2002)
Figure 2: Real GDP of Nepal (Plots of First Difference)
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Cointegration is a phenomenon that each component xi,t, where i = 1, 2, …..k of a vector time series process, xt is a unit root process, possibly with drift, but certain linear combinations of the xi,t's are stationary. Consider a vector xt of n potentially endogenous variables within which specification of data generating process is possible. Again consider xt as an unrestricted vector auto regressive (VAR) model involving up to k lags.
Where T is the sample size and l i = l 1, l 2, ……… ën are the estimated n-r smallest eigenvalues.
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analysing time series data in the log-run.
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