Unrestricted vector autoregressive (VAR) models are widely used in empirical macroeconomics, partly because of the critique in Sims (1980) of traditional macro-econometric modelling. Unfortunately, they are subject to the curse of dimensionality: the number of parameters k grows as the square of the number of variables n, times the maximum lag s, so k=n2s. There have been various proposals for imposing restrictions on VARs to offset this difficulty: see inter alia, Doan, Litterman and Sims (1984) (who propose a `shrinkage' approach), Lütkepohl (1991), ch. 5 (who discusses various strategies for estimating subsets of VARs), and Johansen (1988) (who considers cointegration and differencing reductions). Thus, VAR modelling is a natural area for the application of PcGets.
This chapter illustrates the efficient use of batch commands to perform model selection of all the equations in a vector autoregression. Such systems can be analyzed one equation at a time, since VARs take the form:
| yt=åi=1sPiyt-i+et |
so every equation has the same set of regressors, but each variable is the regressand in turn. Once a batch file is created for one equation, it can be prepared for many equations simply by changing the line `Y =...;' (see Ch. 7).
PcGets has been developed for the reduction of linear singe-equation models. Whether there is a loss in efficiency by analyzing the equations of a VAR once at a time using single-equation model selection algorithms rather than analyzing the system has been investigated by Krolzig (2001). It can be shown that the optimality of single-equation model selection algorithms depends on the absence of instantaneous causality which implies weak exogeneity. If the variance-covariance matrix of the system is diagonal, i.e., all sij=0 for i¹j, the equations of the VAR are unrelated to each other. Thus the probability density function of yt conditional on its past Yt-1 is given by
| f(yt|Yt-1;q)=f(y1t|Yt-1;q1).....f(ynt|Yt-1;qn) |
where the parameter vectors qi of the equations i=1,...,n of the system can be varied freely. Consequently, all possible reductions of the system can be efficiently estimated by OLS, and model-selection procedures can be applied equation-by-equation without a loss in efficiency. Hence, PcGets can be used to model the system as in the single-equation examples considered so far.
Under the presence of instantaneous causality between the variables, i.e., some sij¹0 for i¹j, weak exogeneity is lost: the equations of the VAR are only seemingly unrelated to each other. Since eliminating a variable in one equation effects the others, single-equation model selection procedures are inefficient. In contrast to OLS, full information maximum likelihood (FIML) and estimated generalized least squares (EGLS) are asymptotically efficient estimation procedures for subset VARs with non-diagonal variance-covariance matrices. This has strong implications for model selection procedures.
Krolzig (2001) discusses generalizations of the PcGets algorithm for the analysis of VAR models. System procedures involve joint reductions of the system and reductions of the individual equations. In case of the reductions of the system we are interested in a system analysis of cross-equation restrictions whose acceptance would exclude a regressor from all equations of the system. For example, path reduction would search the system checking the coefficient with the lowest remaining t2-value of the system:
| (k*,j*,i*):= arg min k=1,...,n min j=1,...,n min i=1,...,stkj,i2. |
instead of checking the coefficient with the lowest remaining t-value of the k-th equation
| (j*,i*):= arg min j=1,...,n min i=1,...,stkj,i2 for k=1,...,n, |
If the coefficient akj*,i* of regressor yj,t-i in equation k is insignificant, the coefficient is restricted to zero and the equation is re-estimated by OLS. Alternatively, EGLS could be used whereby the variance-covariance matrix is taken from the reduced, but otherwise unrestricted system.
While PcGets might not be an optimal implementation of Gets for all VARs, it should still deliver reasonable results. Indeed, the Monte Carlo experiments in Krolzig (2001) show that PcGets recovers the DGP specification from a large unrestricted VAR with anticipated size, and power close to commencing from the DGP itself. He also demonstrates the feasibility of PcGets for VAR analysis of large macroeconomic data sets by an application to the US monetary system.
A convenient example is to estimate the four-equation VAR for (m-p)t, Dpt, yt and Rt, as these variables were used extensively in earlier tutorials, and have been modelled as a system by Hendry and Doornik (1994). Those authors added indicator variables for the largest residual `outliers' in the model of Hendry and Mizon (1993), respectively called Dout (zero except for unity in 1972(4), 1973(1) and 1979(2)) and Doil (zero except for unity in 1973(3), 1973(4) and 1979(3)): such interventions were attributed to the `Heath--Barber' boom and the first effects of the Thatcher government for output; and the two oil crises for inflation. We follow their VAR specification here for comparability (M1UKdum.alg creates these two dummies).
Formulate the equation for (m-p)t as dependent variable on yt, Dpt and Rt with 2 lags for each, and delete the contemporaneous values of yt, Dpt and Rt so every regressor is lagged, and add Dout and Doil to obtain (omitting the expert... commands, as we used the default Liberal strategy):
module("PcGets");
package("PcGets");
system
{
Y = mp;
X = Constant, mp_1, mp_2, y_1, y_2, Dp_1, Dp_2, R_1, R_2, Dout, Doil;
}
setdetectoutliers(0);
set0lagorder(1);
set0topdown(1);
set0bottomup(1);
setsplitsample(1);
setstrategy("lib",1);
setreporting(0);
estimate("GETS", 1963, 4, 1989, 2);
This is executed in a flash. Save the batch file, and also that for the specific model (as a PcFiml batch -- from the Test menu):
Now change `Y = mp;' to `Y = Dp;' and rerun; then to `Y = y;', and finally repeat for `Y = R;'. Alternative you could testimate all equations in one go by running the batch file M1UKVAR.FL:
module("PcGets");
package("PcGets");
usedata("M1UK.in7");
setdetectoutliers(0);
set0lagorder(1);
set0topdown(1);
set0bottomup(1);
setsplitsample(1);
setstrategy("lib",1);
setreporting(0);
system
{
Y = mp;
X = Constant, mp_1, mp_2, y_1, y_2, Dp_1, Dp_2, R_1, R_2, Dout, Doil;
}
estimate("GETS", 1963, 4, 1989, 2);
pcfimlbatch;
system
{
Y = Dp;
X = Constant, mp_1, mp_2, y_1, y_2, Dp_1, Dp_2, R_1, R_2, Dout, Doil;
}
estimate("GETS", 1963, 4, 1989, 2);
pcfimlbatch;
system
{
Y = y;
X = Constant, mp_1, mp_2, y_1, y_2, Dp_1, Dp_2, R_1, R_2, Dout, Doil;
}
estimate("GETS", 1963, 4, 1989, 2);
pcfimlbatch;
system
{
Y = R;
X = Constant, mp_1, mp_2, y_1, y_2, Dp_1, Dp_2, R_1, R_2, Dout, Doil;
}
estimate("GETS", 1963, 4, 1989, 2);
pcfimlbatch;
To collect the PcFiml batch results for the specific models, we use the search facilities of GiveWin. Enter:
in the search window at the top-left of the GiveWin screen as shown:
and use the up arrow (assuming you are at the end of the Results window). Page up once each time and copy and paste the `system' outcomes into a new text file, such as for (m-p)t (we edited out the quote marks):
//PcGive batch file created by PcGets
module("PcGive");
package("PcGive");
usedata("M1UK.in7");
system
{
Y = mp;
Z = Constant,mp_1,mp_2,y_1,y_2,Dp_1,Dp_2,
R_1,R_2,Dout,Doil;
}
model
{
mp = Constant,mp_1,mp_2,y_1,Dp_1,R_1,Doil;
}
estimate("FIML",1963,4,1989,2, 0, 0);
Collecting the individual-equation specifications delivers a model of the VAR that parsimoniously encompasses the unrestricted formulation and delivers a batch file for (say) PcGive's system procedures (as in M1UKFIML.FL):
module("PcGive");
package("PcGive");
usedata("M1UK.in7");
system
{
Y = mp,Dp,y,R;
Z = Constant,mp_1,mp_2,y_1,y_2,Dp_1,Dp_2,R_1,R_2,Dout,Doil;
}
model
{
mp = Constant,mp_1,mp_2,y_1,Dp_1,R_1,Doil;
Dp = Dp_1,Dp_2,Doil;
y = y_1,y_2,R_1,Dout;
R = R_1,Doil;
}
estimate("FIML",1963,4,1989,2, 0, 0);
This collects all the relevant information on the subset VAR. Retaining every variable which matters in at least one equation shows that only R_2 has been eliminated from the system. Thus, the system implied by the specific models happens to be almost the original VAR, although the individual equations are all much more parsimonious.
Finally, we ran the resulting batch file in PcGive, and obtained an LR test of over-identifying restrictions of c2(28)= 29.463 [0.3893] against the unrestricted VAR.
This concludes our tutorials. If you have made it this far, you can now manage on your own. We hope you enjoyed modelling empirical evidence using PcGets and found it accurate, powerful, and fast, yet easy to use. We recommend reading Chapter 10 on the econometrics of model selection, and Chapters 9 and 11 on the theory of reduction -- the precursor to Gets -- and why earlier criticisms are not germane.
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