14.1 Introduction

In Chapter 13 we discussed time-series analysis as it stood about two decades ago. The main emphasis was on transforming the data to achieve stationarity and then estimating ARMA models. The differencing operation used to achieve a stationarity involves a loss of potential information about long-run movements.

In the present chapter we discuss three major developments during the last two decades, mainly to handle nonstationary time series: vector autoregressions (VARs), unit roots, and cointegration. The Box-Jenkins method of differencing the time series after a visual inspection of the correlogram has been formalized in the tests for unit roots. The VAR model is a very useful starting point in the analysis of the interrelationships between the different time series. The literature on unit roots studies nonstationary time series which are stationary in first differences. The theory of cointegration explains how to study the interrelationships between the long-term trends in the variables, trends that are differenced away in the Box-Jenkins methods. In the following sections we discuss the VAR models, unit roots, and cointegration, UnUke previous chapters, in this chapter we use matrix notation at some places.

14.2 Vector Autoregressions

[n previous sections we discussed the analysis of a single time series. When we have several time series, we need to take into account the interdependence between them. One way of doing this is to estimate a simultaneous equations model as discussed in Chapter 9 but with lags in all the variables. Such a model is called a dynamic simultaneous equations model. However, this formulation involves two steps: first, we have to classify the variables into two categories, endogenous and exogenous and second, we have to impose some constraints on the parameters to achieve identification. Sims argues that both these steps involve many arbitrary decisions and suggests as an alternative, the vector autoregression (VAR) approach. This is just a multiple time-series generalization of the AR model. The VAR model is easy to estimate because we can use the OLS method.

Consider two economic time series y,, and ,. If we are considering the relationship between money growth and inflation rate, then y„ and , are, re-

C. A. Sims, "Macroeconomics and Reality," Econometnca, Vol. 48, January 1980, pp. 1-48. -The multiple time-series analog of the ARMA model is the VARMA model. But given the complexities in the estimation of MA models discussed earlier, it is clear that estimation of VARMA model is still more complicated for our purpose. The VARMA model was introduced by G. C. Tsiao and G. E. P. Box, "Modelling Multiple Time Series with AppUcations," Journal of the American Statistical Association, Vol. 76, 1981, pp. 802-816.

14.2 VECTOR AUTOREGRESSIONS

spectively, the rate of growth of money supply and the GNP deflator. The VAR model with only one lag in each variable (suppressing constants) would be

1, = „ 1., , + ,2 2.,-1 + e„ = 2 1.,-1 + 22 2,,-1 + £2,

(14.1)

We can also add lagged values of some "exogenous" variables z„ but then we again face the problem of classifying variables as endogenous and exogenous. In practice there would often be more than two endogenous variables and often more than one lag. In this case with endogenous variables and p lags, we can write the VAR model in matrix notation as

y, = A,y,„, + • • • + , + e,

where y, and its lagged values, and e, are : x 1 vectors and A„ . . . , A are matrices of constants to be estimated.

To fix ideas, lets go back to the two-equations system (14.1). We can write the system in terms of the lag operator L as

1 - ai,£ -aiiL

1 -

This gives the solution

- ai,L ""«12- -a2\L 1 - a22l

1 - 22 Cti2L

OLjiF 1 - a,,L

(14.2)

where

= (1 = 1

= (1

- a„L)(l - 022) - (a,2L)(a2,L) (a,, -1- a22)I- -b (a„a22 - a,2a2i)L X,jL)(1 - KjL) say

where \, and Xj are the roots of the equation

\ - (a„ + a22)\ -t- (a,,a22 - aj2a2i) = 0

In order that we have a convergent expansion for yj, and 2, in terms of z„ and 82, we should have \ < 1 and \2l < 1- Following the results in the appendix to Chapter 7 we note that this is the same condition that the roots of ]A - \I) = 0 are less than one in absolute value, where A is the matrix of the lag coefficients

«21

«12

«22

Once the condition for stability is satisfied, we can express y,, (and yj,) as functions of the current and lagged values of e,, and ej,. These are known as the

impulse response functions. They show the current and lagged effects over time of changes in e,, and 62, on y„ and 2,. For instance, in the simple model of money (y„) and prices ( 2,), we have, from (14,2),

y„ = [(1 - - + a,2LE2,J

and expanding - in powers of L and gathering the expressions with the same powers of L, we get

= e„ + a„e,,, , + (a?, + a,2a2,)e,,, 2 + • • •

+ oi,2e2j , + a,2(a„ + a22)E2,-2 + • • •

with a similar expression for 2,. Thus a price shock in period t has no effect on money until period (/ + 1), and vice versa for the effect of a money shock on prices. Of course, these lags are a consequence of the one period lags in the VAR model (14.1).

14.3 Problems with VAR Models in Practice

We have considered only a simple model with two variables and only one lag for each. In practice, since we are not considering any moving-average errors, the autoregressions would probably have to have more lags to be useful for prediction. Otherwise, univariate ARMA models would do better. Suppose that we consider say six lags for each variable and we have a small system with four variables. Then each equation would have 24 parameters to be estimated and we thus have 96 parameters to estimate overall. This overparameterization is one of the major problems with VAR models. The unrestricted VAR models have not been found very useful for forecasting and other extensions using some restrictions on the parameters of the VAR models have been suggested. One such model that has been found particularly useful in prediction is the Bayesian vector autoregression (BVAR). In BVAR we assign some prior distributions for the coefficients in the vector autoregressions. (See Section 2.5 of Chapter 2 for prior distributions.) In each equation, the coefficient of the own lagged variable has a prior mean 1, all others have prior means 0, with the variance of the prior decreasing as the lag length increases. For instance, with two variables y„ and 2, and 4 lags for each, the first equation will be

= , ., , + 2 =J u-J + 2 / ,,- + „

The prior for , will have mean 1 and variance with X < 1. The priors for aj, ttj, a4 will have means 0 and variance X, X\ X", respectively. The priors for p,,

The RATS computer program is the one commonly used for the estimation of VAR and BVAR models.