Money demand equations

3.4.1. Estimation of the long-run money demand equations for the euro area

Each recursive step for calculating the real money gap and the monetary overhang series for both the Divisia and simple sum M3 monies to be used in the forecasting equations (3.7) and (3.8) requires new, updated estimates of the parameters of the respective long-run money demand function. There have been numerous attempts to estimate a stable money demand equation for the euro area in the previous literature. Stracca (2001) and Reimers (2002), for example, provide stable estimates for the long-run money demand equation for the Divisia M3 money. Coenen and Vega (1999), Brand and Cassola (2000), Calza et al. (2001) and Trecroci and Vega (2000), among others, provide evidence for the possibility of estimating a stable

13 The euro area wide aggregates for the GDP and its deflator are the same as in Stracca (2001)

money demand equation for the euro area wide simple sum M3 money. Table A1 in Appendix A presents a short review of the recent estimates for the income elasticity and the interest rate semi-elasticity parameters for the long-run money demand equation estimated for the euro area (for both the Divisia and the simple sum M3 money). The value of the income coefficient in particular has been under discussion, since if its value exceeds unity, the observed declining trend in the Euro area money demand could be attributed to the trend in the GDP growth. As it appears in the table, most estimates, including those of Stracca (2001) and Reimers (2002), indeed exceed unity. On the other hand, not all the studies included a formal statistical test of whether the income coefficient differs significantly from unity.

In our simulated out-of-sample exercise, the estimation of money demand equation for each recursive step of creating new series for the real money gap and the monetary overhang is conducted simply by OLS. When the Divisia M3 money was considered, our money demand specification included a constant term, the real GDP the price dual of the Divisia money along with its square. The price dual measures the opportunity cost of holding the real Divisia money balances and its square is included in the specification to capture possible non-linearities in the money demand. With this specification we again follow the practice of Stracca (2001).¹⁴

Since the variables of the money demand equations for both the Divisia and simple sum M3 monies are all I(1)-variables, estimating the money demands simply by linear regressions actually corresponds to the Engle-Granger cointegration analysis. Using linear regressions instead of the more elaborate Johansen procedure is further justified by the fact that, according to the economic theory, there should be no other cointegration relations between the variables than that defining the money demand equation.

When the full sample was considered, the money demand equation for the Divisia money was estimated to be

14In fact, including the square of the price dual in the money demand specification has been called into question by Reimers (2002) both on the grounds of some methodological considerations (see Reimers (2002, p.19.) and on empirical grounds (see Reimers (2002, p. 34). Including the square term in our specification was partly motivated by an estimation exercise with the Johansen procedure, which suggested that the square term is included in the cointegration space.

)2

Thus, in the study at hand the income elasticity of the real Divisia money also exceeds unity, which is in line with Stracca (2001) and Reimers (2002). The overall stability of the parameters of the money demand equation was examined by plotting the coefficient estimates against time (see Figure A.9 in Appendix A). The income elasticity and the squared price dual elasticity seem to be remarkably stable, while there seems to be a structural shift in the price dual elasticity and in the constant term at around 1991. Correctly estimating the value for the price dual elasticity matters only for calculating the monetary overhang series, however. Since the equilibrium value for the price dual is measured as a sample average, the product of the price dual elasticity and the equilibrium rate of the price dual (k_ii_t^* in Eq. (3.4)) becomes a part of the constant term of the equation defining the real money gap. Accordingly, the forecasts based on the real money gap series are unaffected by the value of k_ii_t^*, since it does not affect the variation of the real money gap series.¹⁵

Calza et al. (2001) argue that the correct opportunity cost variable to be used in the money demand specification for the euro area simple sum M3 money is the spread between the short-term interest rate and the own rate of interest of the M3 money. This opportunity cost variable for the simple sum M3 money is also adopted here. Thus, when the full sample was considered, the money estimated money demand equation for the simple sum M3 money of the euro area takes the form

The line graph representing the recursive estimates of the parameters of the money demand equation is presented in Figure A 10. The estimate for the income elasticity again seems to be remarkably stable and exceeds unity, just as was the case in Calza et al. (2001) referred to above, as well as in the studies e.g. by Coenen and Vega (1999) and Brand and Cassola (2000). The interest rate elasticity however, takes a value near zero, which does not sound plausible. However, the same argument applies here as in the case of the Divisia money, so

15 Likewise, the value of the constant term k does not affect the inflation forecasts based on the residuals of the money demand equations (monetary overhang series).

that the interest rate elasticity term matters only for calculating the monetary overhang series, the stability of which will be discussed later.

3.4.2. Time series properties of the monetary indicator variables

Line graphs of the five different monetary indicator series for the Divisia money are shown in Figures A.11 – A.14 in Appendix A, while Figures A.15 – A.18 show them for the simple sum M3 money. The results of the formal unit root tests for all the monetary indicator series are reported in Table A.4. Beginning with the nominal growth rates of the monies, according to the ADF and the PP tests, the growth rate of the nominal Divisia money seems to be I(0) process. Inspection of Figure A.1 however suggests the series rather to be trend-stationary or even following stochastic trend. Since the linear trend also turned out to be statistically significant in the unit root tests, I ended up with removing the trend –whether it ultimately is stochastic or deterministic in nature- by differencing the series. Likewise, according to the Figure A.1 the growth rate of the simple sum M3 money behaves much like the growth of the Divisia money in the sample period. Since now the ADF- test also suggests non-stationarity, the trend was also removed from the simple sum M3 series by differencing.

Figures A.11 – A.13 and A.15 – A.17 show that, as might be expected, the real money gap series are somewhat sensitive to the underlying estimate of the potential output. According to the unit root tests then, all the real money gap series for both the Divisia and the the simple sum M3 monies seem to be I(1) processes. Beginning with the real money gap series for the Divisia money, a closer inspection of Figure (A.11) suggests that in the case of the series based on the potential output estimates from HP filtering, the non-stationarity suggested by the unit root tests may follow from the structural break at the end of the series. Thus, the stationarity of the series was re-examined ignoring the last three years of the data.

Accordingly, the tests now indicated stationarity. Thus, in the forecasting exercise the specification (3.7) was used for this series.

In the case of the the real money gap series based on the ECB estimate of the potential output, Figure (A.12) in fact suggests a deterministic trend in the data. For the real money gap series based on the OECD estimate of potential output, in turn, the possibility of a stochastic trend in the data is also supported by inspection of the line-graph of the series (Figure (A.13)),

according to which the series contains relatively large upswings and downswings. The volatility of the series also seems to be largest of all the three real money gap measures considered.¹⁶ Thus, the real money gap series based on the ECB and the OECD estimates of the potetential output were considered as non-stationary series and specification (3.8) was used in the forecasting exercise for these series. On purely a priori grounds, the real money gap series are in fact expected to be I(0) variables, since the gap reflects only deviations of the real money stock from its equilibrium level and this gap should be closed in the long-run.

Because of the short sample period, modelling the series as I(1) processes can, however, be considered as a good approximation of the data generating process during this period.

The monetary overhang series for the Divisia money is shown in Figure A.14. Testing the stationarity of the monetary overhang series in fact means simply that our Engle-Granger type of cointegration analysis is completed. Since the monetary overhang series consists of residuals of the money demand equation, the series should be stationary or otherwise no stable cointegration relationship between the variables would exist at all. According to Figure A.14 the series looks stationary even in our short sample period considered, except an obvious structural break in the end of the sample. When the full sample was considered, both the ADF and PP tests suggested non-stationarity, but when the last four observations were excluded, the tests also suggested stationarity.

The real money gap series for the simple sum M3 real money gaps are plotted in Figures A.15 – A.17. The unit root tests of Table A.4 suggest non-stationarity as in the cases of the corresponding Divisia money series, and according to the figures, there are at least as good reasons for considering the real money gap series for the simple sum M3 as non-stationary as in the case of the Divisia M3 based series. The HP-filter based real money gap series is, however, again included in the forecast equation in levels (specification 3.7) to make easier the comparison of the performance of this series with its Divisia money counterpart¹⁷. The line graph of the monetary overhang for the simple sum M3 series, in turn, is plotted in Figure A.18. A notable feature in the figure is the large structural break between 1992 – 1994, although the series otherwise looks as stationary. The unit root tests mostly reported

16 Note that because of some measurement issues, the constant terms of the real money gap and the monetary overhang series are probably not correctly estimated. The “wrong” value for the constant term does not affect the results of our forecasting exercise, but there is now no sense in offering any interpretation for the absolute values of the real money gap and monetary overhang series.