Sims (1980) criticized the simultaneous equation literature for the ad hoc restrictions needed for identification and for the ad hoc classification of exogenous and endogenous variables in the system, see Chapter 11. Instead, Sims (1980) suggested Vector Autoregression (VAR) models for forecasting macro time-series. VAR assumes that all the variables are endogenous. For example, consider the following three macro-series: money supply, interest rate, and output. VAR models this vector of three endogenous variables as an autoregressive function of their lagged values. VAR models can include some exogenous variables like trends and seasonal dummies, but the whole point is that it does not have to classify variables as endogenous or exogenous. If we allow 5 lags on each endogenous variable, each equation will have 16 parameters to estimate if we include a constant. For example, the money supply equation will be a function of 5 lags on money, 5 lags on the interest rate and 5 lags on output. Since the parameters are different for each equation the total number of parameters in this unrestricted VAR is 3 x 16 = 48
parameters. This degrees of freedom problem becomes more serious as the number of lags m and number of equations g increase. In fact, the number of parameters to be estimated becomes g + mg2. With small samples, individual parameters may not be estimated precisely. So, only simple VAR models, can be considered for a short sample. Since this system of equations has the same set of variables in each equation SUR on the system is equivalent to OLS on each equation, see Chapter 10. Under normality of the disturbances, MLE as well as Likelihood Ratio tests can be performed. One important application of LR tests in the context of VAR is its use in determining the choice of lags to be used. In this case, one obtains the log-likelihood for the restricted model with m lags and the unrestricted model with q > m lags. This LR test will be asymptotically distributed as X2q-m)g2. Once again, the sample size T should be large enough to estimate the large number of parameters (qg2 + g) for the unrestricted model.
One can of course impose restrictions to reduce the number of parameters to be estimated, but this reintroduces the problem of ad hoc restrictions which VAR was supposed to cure in the first place. Bayesian VAR procedures claim success with forecasting, see Litterman (1986), but again these models have been criticized because they are devoid of economic theory.
VAR models have also been used to test the hypothesis that some variables do not Granger cause some other variables.1 For a two-equation VAR, as long as this VAR is correctly specified and no variables are omitted, one can test, for example, that y1 does not Granger cause y2. This hypothesis cannot be rejected if all the m lagged values of y1 are insignificant in the equation for y2. This is a simple F-test for the joint significance of the lagged coefficients of y1 in the y2 equation. This is asymptotically distributed as Fm T-(2m+1). The problem with the Granger test for non-causality is that it may be sensitive to the number of lags m, see Gujarati (1995). For an extensive analysis of nonstationary VAR models as well as testing and estimation of cointegrating relationships in VAR models, see Hamilton (1994) and Liitkepohl (2001).
If xt = xt-1 + ut where ut is IID(0, a2), then xt is a random walk. Some stock market analysts believe that stock prices follow a random walk, i. e., the price of a stock today is equal to its price yesterday plus a random shock. This is a nonstationary time-series. Any shock to the price of this stock is permanent and does not die out like an AR(1) process. In fact, if the initial price of the stock is xo = Ц, then
x1 = Ц + u1, x2 = Ц + u1 + u2,…, and xt = ц + ^ij=1 Uj
with E(xt) = Ц and var(xt) = ta2 since u ~ IID(0, a2). Therefore, the variance of xt is dependent on t and xt is not covariance-stationary. In fact, as t ^ x>, so does var(xt). However, first differencing xt we get ut which is stationary. Figure 14.8 plots the graph of a random walk of size T = 250 generated as xt = xt-1 + et with et ~ IIN(0,4). Figure 14.9 shows that the autocorrelation function of this random walk process is persistent as s increases. Note that a random walk is an AR(1) model xt = pxt-1 + ut with p = 1. Therefore, a test for nonstationarity is a test for p = 1 or a test for a unit root.
Using the lag operator L we can write the random walk as (1 — L)xt = ut and in general, any autoregressive model in xt can be written as A(L)xt = ut where A(L) is a polynomial in L. If A(L) has (1 — L) as one of its roots, then xt has a unit root.
Figure 14.8 Random Walk Process
Subtracting xt-1 from both sides of the AR(1) model we get
Axt = (p – 1)xt-i + ut = Sxt-i + ut (14.3)
where 8 = p — 1 and Axt = xt — xt-1 is the first-difference of xt. A test for Ho; p = 1 can be obtained by regressing Axt on xt-1 and testing that Ho; 8 = 0. Since ut is stationary then if 8 = 0, Axt = ut and xt is difference stationary, i. e., it becomes stationary after differencing it once. In this case, the original undifferenced series xt is said to be integrated of order 1 or I(1). If we need to difference xt twice to make it stationary, then xt is I(2). A stationary process is by definition I(0). Dickey and Fuller (1979) showed that the usual regression t-statistic for Ho; 8 = 0 from (14.3) does not have a t-distribution under Ho. In fact, this t-statistic has a non-standard distribution, see Bierens (2001) for a simple derivation of these results. Dickey and Fuller tabulated the critical values of the t-statistic = (p — 1)/s. e.(p) = 8/s. e.(8) using Monte Carlo experiments. These tables have been extended by MacKinnon (1991). If t exceeds the critical values, we reject Ho that p = 1 which also means that we do not reject the hypothesis of stationarity of the time-series. Non-rejection of Ho; p = 1 means that we do not reject the presence of a unit root and hence the nonstationarity of the time-series. Note that non-rejection of Ho may also be a non-rejection of p = 0.99. More formally stated, a weakness of unit root tests in general is that they have low power discriminating between a unit root process and a borderline stationary process. In practice, the Dickey-Fuller test has been applied in the following three forms:
Axt = 8xt-i + ut (14.4)
Axt = a + 8xt-1 + ut (14.5)
Axt = a + /3t + 8xt-1 + ut (14.6)
where t is a time trend. The null hypothesis of the existence of a unit root is Ho; 8 = 0. This is the same for (14.4), (14.5) and (14.6), but the critical values for the corresponding t-statistics
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
Figure 14.9 Correlogram of a Random Walk Process
are different in each case. Standard time-series software like EViews give the proper critical values for the Dickey-Fuller statistic. For alternative unit root tests, see Phillips and Perron (1988) and Bierens and Guo (1993). In practice, one should run (14.6) if the series is trended with drift and (14.5) if it is trended without drift. Not including a constant or trend as in (14.4) is unlikely for economic data. The Box-Jenkins approach differences the series and looks at the sample correlogram of the differenced series. The Dickey-Fuller test is a more formal test for the existence of a unit root. Maddala (1992, p. 585) warns the reader to perform both the visual inspection and the unit root test before deciding on whether the time-series process is nonstationary.
If the disturbance term ut follows a stationary AR(1) process, then the augmented Dickey – Fuller test runs the following modified version of (14.6) by including one additional regressor, Axt-i:
Axt = a + et + Sxt-i + XAxt-i + et (14.7)
In this case, the t-statistic for 8 = 0 is a unit root test allowing for first-order serial correlation. This augmented Dickey-Fuller test in (14.7) has the same asymptotic distribution as the corresponding Dickey-Fuller test in (14.6) and the same critical values can be used. Similarly, if ut follows a stationary AR(p) process, this amounts to adding p extra regressors in (14.6) consisting of Axt-1, Axt-2,.Axt-p and testing that the coefficient of xt-1 is zero. In practice, one does not know the process generating the serial correlation in ut and the general practice is to include as many lags of Axt as is necessary to render the et term in (14.7) serially uncorrelated. More lags may be needed if the disturbance term contains Moving Average terms, since a MA term can be thought of as an infinite autoregressive process, see Ng and Perron (1995) for an extensive Monte Carlo on the selection of the truncation lag. Two other important complications when doing unit root tests are: (i) structural breaks in the time-series, like the oil embargo of 1973, tend to bias the standard unit root tests against rejecting the null hypothesis of a unit root, see Perron (1989). (ii) Seasonally adjusted data also tend to bias the standard unit root tests against rejecting the null hypothesis of a unit root, see Ghysels and Perron (1992). For this reason, Davidson and MacKinnon (1993, p. 714) suggest using seasonally unadjusted data whenever available.
For the trended consumption series with drift, the Augmented Dickey-Fuller test using EViews yields the following regression:
ACt = 665.60 + 30.57 t – 0.072 C-i+ 0.449 ACt-i +residuals
(1.80) (1.60) (1.42) (3.17) ( . )
where the numbers in parentheses are the usually reported t-statistics. The null hypothesis is that the coefficient of Ct-i in this regression is zero. Table 14.1 gives the Dickey-Fuller t- statistic (-1.42) and the corresponding 5% critical value (-3.508) tabulated by MacKinnon (1996). Note that @TREND(1959) is the time-trend starting at 1959. The Schwarz criterion found the optimal number of lags of ACt-i to be included in this regression is one. Since the p-value is 0.84, we do not reject the null hypothesis of the existence of a unit root. We conclude that Ct is nonstationary. This confirms our finding from the sample correlogram of Ct given in Figure 14.2.
Table 14.1 Dickey-Fuller Test
One can check whether the first-differenced series is stationary by performing a unit root test on the first-differenced model. Let Ct = ACt, then run the following regression:
ACt = 213.77 – 0.533 Ct_i + residuals
(3.59) (4.13) ( ‘9)
the coefficient of Ct_1 has a f-statistic of -4.13 which is smaller than the 5% critical value of -2.925. In other words, we reject the null hypothesis of unit root for the first-differenced series ACt. The same conclusion would have been reached if a linear trend was included besides the constant. We conclude that Ct is I(1).
So far, all tests for unit root have the hypothesis of nonstationarity as the null with the alternative being that the series is stationary. Two unit roots tests with stationarity as the null and nonstationarity as the alternative are given by Kwaitowski et al. (1992) and Leybourne and McCabe (1994). The first test known as KPSS is an analog of the Phillips-Perron test whereas the Leybourne-McCabe test is an analog of the augmented Dickey-Fuller test. Reversing the null may lead to confirmation of stationarity or nonstationarity or may yield conflicting decisions.