Urban Accessibility

The use of VAR in macroeconomics has generated much empirical evidence, giving fundamental support to many economic theories (see Blanchard and Watson (1986) and Bernanke (1983) among others). Vector autoregression (VAR) models were introduced by Sims (1980) to model the joint dynamics and causal relations among a set of macroeconomic variables.The VAR model is a multi-equation system where all the variables are treated as endogenous. There is thus one equation for each variable as dependent variable. Each equation has lagged values of all the included variables as dependent variables, including the dependent variable itself. Since there are no contemporaneous variables included as explanatory, right-hand side variables, the model is a reduced form. Thus all the equations have the same form since they share the same right-hand side variables.

Multivariate simultaneous equations models were used extensively for macroeconometric analysis when Sims (1980) advocated vector autoregressive (VAR) models as alternatives. At that time longer and more frequently observed macroeconomic time series called for models which described the dynamic structure of the variables. VAR models lend themselves for this purpose. They typically treat all variables as a priori endogenous. Thereby they account for Sims’ critique that the exogeneity assumptions for some of the variables in simultaneous equations models are ad hoc and often not backed by fully developed theories. Restrictions, including exogeneity of some of the variables, may be imposed on VAR models based on statistical procedures. VAR models are natural tools for forecasting. Their setup is such that

58 current values of a set of variables are partly explained by past values of the variables involved. They can also be used for economic analysis, however, because they describe the joint generation mechanism of the variables involved. Traditionally VAR models are designed for stationary variables without time trends. Trending behavior can be captured by including deterministic polynomial terms. In the 1980s the discovery of the importance of stochastic trends in economic variables and the development of the concept of cointegration by Granger (1981), Engle and Granger (1987), Johansen (1995) and others have shown that stochastic trends can also be captured by VAR models. If there are trends in some of the variables it may be desirable to separate the long-run relations from the short-run dynamics of the generation process of a set of variables. Vector error correction models offer a convenient framework for separating longrun and short-run components of the data generation process (DGP). The advantage of levels VAR models over vector error correction models is that they can also be used when the cointegration structure is unknown.

VAR models are the best method for investigating shock transmission among variables because they provide information on impulse responses (Adrangi and Allender (1998).

Zellner and Palm (1974), Zellner (1979), and Palm (1983) show that any linear structural model can be written as a VAR model. Therefore, a VAR model serves as a flexible approximation to the reduced form of any wide variety of simultaneous structural models.

Vector Auto Regressive (VAR) models have been much used in empirical studies of macroeconomic issues since they were launched for such purposes by Sims (1980). This first study related to the estimation of a six-variable dynamic system namely GNP, money supply, unemployment rate, wages, price level and import price based on an alternative style of macro-econometrics without using theoretical perspectives. He suggests that it should be feasible to estimate large scale macro-models as unrestricted reduced forms, treating all variable as endogenous (Sims, 1980). Sims also criticized the way that the classical

59 simultaneous equations models are identified as well as questioned about the exogenous assumptions for some variables not necessary backing by theoretical framework. In contrast, VAR model overcomes this problem by treating all variables as endogenous variables.

Basically, the form of a VAR model treats all variables symmetrically without making reference to the issue of dependence versus independence or of them as endogenous variables and estimating dynamic systems without using theoretical perspectives. This methodology is one of the most successful, flexible and easy to analyze the multivariate time series (Sims, 1980). It is the extension of the univariate autoregressive model to dynamic multivariate time series and proven to be useful for explain the dynamic behavior of economic and financial time series. They are now widely used in all kinds of empirical macroeconomic studies, from relatively theoretical exercises such as data description and forecasting, to tests of fully specified economic models. In brief, VAR is an econometrics tool that shows the dynamic interrelationship between stationary variables. Thus, VAR is used when the variables are either stationary, or stationary and not cointegrated. When the variables are non-stationary and not cointegrated, a VAR in first differences are used in order to determine the interrelation between them. However, if the variables are non-stationary and cointegrated, VEC model is estimated. VAR is a model which consists only of endogenous variables and allows for the variables to depend not only on its own lags. Consider a case of bivariate VAR which consists of two variables, and , which each dependent variable depends on the combination of their lags, k, and error terms:

(3.4) where is a white noise disturbance with

60 Moreover, there are two techniques from VAR employed in order to show the statistically significant impacts of each variables on the future values, for example whether the changes of a variable have a positive or negative effect on other variables in the system, namely the VAR’s impulse responses and variance decompositions. In determining both techniques, ordering of the variables plays a very important role.

Impulse responses show how the shocks to any single variable affect the dependent variable in the VAR. More specifically, impulse responses record the size of the impact inflicted by single shocks to the errors to the VAR system. Moreover, impulse responses will be generated afterwards for the total of n variables in the system. Impulse responses are achieved by writing VAR as Vector Moving Average (VMA).

Another way to explain the effects of the shocks is to analyze the variance decompositions.

Variance decompositions analysis is slightly different with impulse responses in term of how the shocks are applied. It records the effect on dependent variable due to its own shocks against shocks to other variables in the system. Moreover, variance decompositions analysis focuses not only on the movement of the dependent variable, but also on the forecast error variance produced by the shocks which helps to show the sources of the volatility.