APPENDIX IV: Numerical analysis of the identifiability of the SVAR model 80
2.4 The identification restrictions
To avoid the under-identification problem, the number of the zero restrictions should be at least (n * (n-1)) / 2, where n is the number of variables in the model. Because the final model actually includes 56 instead of 55 zero restrictions, it’s exact identifiability had to be checked by numerical methods (see Appendix IV). According to the results, the model seems to be exactly identified.
11 See e.g. Leeper and al. (1996) pp. 3 – 5.
12 See e.g. Litterman and Weiss (1985) and Sims (1986)
13 See e.g. Eichenbaum and Evans (1995) and Grilli and Roubini (1996)
Table 2.1. Identifying restrictions
r r * M M * e R R * P P * Y Y *
r 1 1 1 0 1 1 0 1 0 1 0
r * 0 1 0 1 1 0 1 0 1 0 1
M 1 0 1 1 1 1 0 1 0 1 0
M 0 * 1 1 1 1 0 1 0 1 0 1
e 1 1 1 1 1 1 1 1 1 1 1
R 1 1 1 0 1 1 1 0 0 0 0
R * 1 1 0 1 1 1 1 0 0 0 0
P 0 0 1 0 1 0 0 1 0 1 0
P * 0 0 0 1 1 0 0 0 1 0 1
Y 0 0 0 0 0 1 0 1 0 1 1
Y * 0 0 0 0 0 0 1 0 1 0 1
The rows of the table refer to the coefficients of the contemporaneous values of the 11 explanatory variables (shown in columns) in the 11 VAR equations. If the coefficient is unrestricted, it is denoted by 1 while 0 refer to a zero restriction. r = call money rate, r*= federal funds’ rate, M = the growth rate of German M3, M*= the growth rate of US M3, e = the nominal DEM/USD exchange rate, R= the German long term government bond rate, R*= the US long term government bond rate, P= the German inflation, P*= the US inflation, Y= the German industrial production, Y*= the US industrial production.
The first two rows of the matrix correspond to the reaction functions of the central banks of the Fed and the Bundesbank, with short-term interest rates as their policy instruments14. The reaction functions can be interpreted as variants of a Taylor-rule type policy rule, although in the specification of Table 2.1 the central banks react to larger sets of variables compared with the original Taylor rule, which has only inflation and output gaps as its arguments15. The specification here corresponds to an assumption that the policymaker considers a broad array of information when forming expectations of inflation and output, in addition to the own histories of these variables.
14 The identification of the reaction functions has been influenced by the empirical results of Clarida and al. (1998), who estimated reaction functions for Germany, Japan, the US, France, Italy and UK. Their specification of the reaction function was also consistent with many theoretical studies about central bank behavior with quadratic loss functions over inflation and output.
15 Since the SVAR modeling focuses on the unexpected part of the monetary policy, the first two equations of the model could best be characterized as central bank reaction functions in surprises, as opposed to the Taylor rule in the strict sense of the word, which is more concerned of the systematic part of the monetary policy. See Clarida (2001), p. 3.
Because the model is estimated with monthly data, the reaction function should only consist of variables from which there are real-time information available to the central bank within one month’s time. In practice, the problem boils down to the question whether to include contemporaneous values of outputs and prices themselves in the reaction functions or not. According to previous research, it is not clear how much formal or informal information the monetary authority can collect about the contemporaneous values of those variables16. Partly motivated by the need to avoid too large a number of zero restrictions, the contemporaneous values of the domestic output and inflation are, however, included in the reaction functions.17
The money stock, long-term interest rate, output and inflation of the respective foreign country are omitted from the reaction functions of both central banks simply because these variables are assumed not to be of interest to the policymakers. The specifications of the reaction functions differ from each other in that the Bundesbank is expected also to look at the foreign short-term interest rate while the Federal Reserve is not. The domestic long-term interest rates are included to the reaction functions because the long rates may reflect information about expected inflation.
The third and the fourth row characterize the money demand equations of the countries.
In both countries the change in demand for nominal money is affected by the contemporaneous values of output, inflation, foreign money growth, nominal exchange rate and nominal short and long term interest rates18 of the respective country19. This specification of money demand is close to the standard specifications where the demand for real money depends on the short-term nominal interest rate and output. The inclusion of the foreign money in the money demand equations is theoretically motivated by the
16 According to Eichenbaum (1998) for instance, the assumption that the Federal Reserve looks at the current output and price level when setting the monetary policy stance, seems at least as plausible as assuming that it does not.
17 For a more detailed discussion of the reasonableness of this kind of restrictions for the central banks’
information set, see Sims et al. (1998), p. 25, or Christiano, Eichenbaum and Evans (1998), p. 18.
18 It is assumed in this specification of the money demand that the two alternative assets to the agents are cash and bonds. Hence, the opportunity cost of holding cash balances is the nominal rather than real interest rate.
19 The real balances are implicitly defined in terms of consumer prices. Hence, the coefficient of the nominal exchange rate is not restricted to zero in the money demand equations.
need to take into account the currency substitution and the growth rate of world money for the money demands of the countries20.
The row describing the determination of the nominal DEM/USD exchange rate (e), is an
“information market equation”, which contains all the variables contemporaneously.
The motivation behind this kind of specification for the determination of the exchange rate is based on the idea that in efficient exchange markets the exchange rate can respond quickly to all relevant information in both countries. The specification is inspired by the examples of Cushman and Zha (1997) and Sims (1996) and the equation reflects the existence of disturbances to the private sectors of the countries that cannot be defined in terms of the sticky non-financial variables included in the model.
The next two rows describe the determination of the long-term interest rates. The inclusion of the long-term interest rates as variables to VAR is motivated by three considerations: Firstly, there are some a priori reasons to believe that for many long-term consumption and investment decisions, the long-long-term interest rate should be a variable of more interest than the short run rate (see e.g. Taylor (1995), p. 17). Secondly, as has been argued by Grilli and Roubini (1996), the movements in long-term interest rates might be capturing agents’ expectations about long-term inflationary trends.
Finally, according to the empirical evidence provided by Bagliano and Favero (1998), the contemporaneous long-term interest rate may be relevant variables in the policymaker’s reaction function.21 The long-term interest rates of both countries are assumed to respond contemporaneously to movements in all other interest rate variables of the model, in the exchange rate and in the money supply growth of the home country.
Just like the nominal exchange rate, the long-term interest rates are assumed to be determined in the capital markets, reflecting a broad array of information of different variables affecting the investor’s expectations.
20 For a discussion about the foreign money supply for the monetary autonomies of the US and the ERM countries, see Heimonen 1999, p. 2. For a more general discussion about the international currency substitution, see McKinnon (1982).
21 There is actually one problem in including the long-term interest rate as a variable in the VAR model designed to study monetary transmission which is related to distinguishing the structural shocks to long-term rates from structural shocks to short-long-term rates. This identification problem is due to the feedbacks between interest rates of the different maturities.
Rows 8 –11 constitute the sluggish sectors of the two economies. The variables in this block are assumed not to respond contemporaneously to monetary innovations, but instead, to be predetermined relative to the monetary policy shocks.22 According to the rows 8 and 9, inflation is driven by the contemporaneous domestic output levels, changes in the growth rates of money and the exchange rate. The last two rows of Table 3.1 refer to the outputs (industrial productions) of both countries (YandY*). The outputs are supposed to be affected contemporaneously by the domestic long term interest rates and inflation rates. In addition, the German output is supposed to instantly react to changes in the US output, but not the other way round.