Abstract
The resurgence of interest in money demand functions in single countries over the last couple of years (see, for example, Hall, Henry, and Wilcox (1990), Johansen and Juselius (1990), and Muscatelli and Papi (1990)), has been accompanied by a growing interest in the extent to which a similar approach can be taken in respect of the demand for a European aggregate money supply (see Bekx and Tullio (1987), Angeloni, Cottarelli, and Levy (1991), and Monticelli and Strauss-Kahn (1991)). “Economic and Monetary Integration and the Aggregate Demand for Money in the EMS,” by Kremers and Lane (1990) was one of the first papers to suggest that a stable exchange rate mechanism (ERM)-wide money demand function does indeed exist. In view of the important policy implications that could follow from the existence of such a stable function, it is essential that any results from which this conclusion might be drawn should be subject to rigorous testing—even if this means using the benefit of “hindsight” and data not available at the time of the original investigation.
The resurgence of interest in money demand functions in single countries over the last couple of years (see, for example, Hall, Henry, and Wilcox (1990), Johansen and Juselius (1990), and Muscatelli and Papi (1990)), has been accompanied by a growing interest in the extent to which a similar approach can be taken in respect of the demand for a European aggregate money supply (see Bekx and Tullio (1987), Angeloni, Cottarelli, and Levy (1991), and Monticelli and Strauss-Kahn (1991)). “Economic and Monetary Integration and the Aggregate Demand for Money in the EMS,” by Kremers and Lane (1990) was one of the first papers to suggest that a stable exchange rate mechanism (ERM)-wide money demand function does indeed exist. In view of the important policy implications that could follow from the existence of such a stable function, it is essential that any results from which this conclusion might be drawn should be subject to rigorous testing—even if this means using the benefit of “hindsight” and data not available at the time of the original investigation.
Kremers and Lane present an econometric model that explains the demand for ERM-wide narrow money in terms of income, interest rates, inflation, and exchange rates against the U.S. dollar, each of which is aggregated or averaged across countries as appropriate. On the basis of their results, the authors conclude that “even at the present stage of economic and monetary integration, a European central bank might be able to implement monetary control more effectively than the individual national central banks” (p. 777). This note tests the robustness of Kremers and Lane’s results by attempting to replicate them, with a view to subjecting them to alternative statistical tests as set out in MacKinnon (1990) and by testing their validity over an updated estimation period.
The broad conclusion is that while the model has some validity within the period over which h was estimated (1978 to 1987), it breaks down when asked to explain the demand for money over an updated period (1978 to 1990).
I. Summary of Kremers and Lane Model
The aggregation methods are described in some detail in the paper, but the main points concerning the variables in Kremers and Lane’s model are
| Variable | Symbol | Description |
|---|---|---|
| Money | m | National Ml, aggregated at purchasing power parity exchange rates for 1985 |
| Income | y | Real GDP/GNP, aggregated as for money |
| Interest rates | rs and rl | Three-month and “long” rates, average based on ECU weights |
| Prices | p | Consumer price index, average based on national shares in aggregate income |
| Inflation | infl | Price change over the previous year |
| Exchange rate | ecu | Dollars per ECU, adjusted to exclude the pound sterling, the Luxembourg franc, and, for the extended period, the peseta. |
| Variable | Symbol | Description |
|---|---|---|
| Money | m | National Ml, aggregated at purchasing power parity exchange rates for 1985 |
| Income | y | Real GDP/GNP, aggregated as for money |
| Interest rates | rs and rl | Three-month and “long” rates, average based on ECU weights |
| Prices | p | Consumer price index, average based on national shares in aggregate income |
| Inflation | infl | Price change over the previous year |
| Exchange rate | ecu | Dollars per ECU, adjusted to exclude the pound sterling, the Luxembourg franc, and, for the extended period, the peseta. |
All data are quarterly, and seasonally adjusted where appropriate.
Estimation Methods
Kremers and Lane used the standard Engle-Granger two-step method. This approach is a valid way of proceeding only if the data have certain characteristics. In particular, the variables in the first-step equation must be integrated of order one (I(1)), and the residuals from the first equation must be integrated of order zero. Kremers and Lane test for these characteristics using tests set out in Sargan and Bhargava (1983).
Kremers and Lane’s Equations
Kremers and Lane report the following equation for the first step:
(In this and succeeding equations, t-statistics are in parentheses.)
This equation satisfies the condition that the variables be I(1) and the residuals just pass a test of being I(0).
This equation appears reasonable from an economics perspective. An increase in the short-term interest rate tends to reduce the demand for money as asset holders switch from money to interest-bearing assets. Lagged inflation appears as a proxy for expected inflation, increases in which cause a movement away from money (as it declines in value in real terms) and into goods. That lagged, rather than current, inflation is included does not present a problem on theoretical grounds; if the sample size had been significantly greater, current and lagged values would have given approximately the same coefficient. However, it does point toward some small sample problems that might also lead to instability in the estimated coefficients and tests.
There are, however, some problems with the statistical results. Kremers and Lane present a battery of tests in their paper, but most of these, along with the reported t-statistics, are invalid from a statistical perspective when used in the context of the first step of this estimation process,1 although they may still provide a useful guide to modeling at a practical level. In particular, they assert that “the parameter estimates appear to be stable” (p. 795), an assertion that cannot (in a strict statistical sense) be supported by the tests that are used. They also impose a unit coefficient on income on the basis of a test performed on an unrestricted equation; this test, too, is a useful guide but not a true statistical test. As will be shown below, relaxing this restriction may provide a useful step forward.
In Kremers and Lane’s second step, they estimated the following equation:
(A “D” indicates that the variable is in first-difference form.)
Equation (2) was tested for stability, and this test was passed easily, as were all of the others that Kremers and Lane reported for this equation. The tests do not suffer from the problems associated with those used for equation (1), and these results, along with other supporting evidence presented in the paper, suggest that the equation performs reasonably well on the basis of econometric criteria.
An interesting feature of this equation is that changes in short rates have to wait for three quarters before delivering their effect via the term D(rs(–3)), by which time these changes will already have had an effect because they themselves help to determine the equilibrium level (which appears in the equation as part of the lagged deviation term, ecm). If one were to accept the hypothesis (which cannot be rejected on the basis of these results) that the coefficient on ecm is actually unity, then a change in short rates would have the following effects. First, the equilibrium level of money holdings would change (as in equation (1)); this would have no contemporaneous effect on actual money, but in the following quarter asset holders would adjust their money balances completely to the new equilibrium level. Second, six months later, money holdings would move away from equilibrium (the D(rs(–3)) term in the second equation having its turn), and in the subsequent quarter they would move back again. (The story would be almost the same if the coefficient were 0.95, as estimated.)
It may be in this specific sample period that on a few occasions, money changed significantly three quarters after a change in short rates, and, therefore, that the best representation of the relationship in this period includes a third lag. But this should not be taken as an indication of underlying behavior that would be likely to be repeated in the future, particularly, if the implied behavior is difficult to rationalize. There thus appears to be something in this data period that might not be reflected in a larger sample and leads to some doubt about the stability of the reported results.
II. Re-estimation of Kremers and Lane’s Equations
The first step in this process was to construct data as close as possible to that used by Kremers and Lane. To this end the authors provide an unusually detailed description of their sources and construction methods, making it possible to produce data that, on the basis of comparison with the figures in Kremers and Lane, were almost identical to the original, except for some small discrepancies in the case of nominal money. The difficulty here was that while the general source (International Financial Statistics (IFS)) contains both seasonally adjusted and unadjusted Ml series, Kremers and Lane preferred to adjust the data for each country themselves. In this note, adjusted data from IFS were used, and resulted in an ERM-wide aggregate similar to Kremers and Lane’s. Another possible reason for differences between the results presented here and those of Kremers and Lane may be data revisions.
In order to test the robustness of Kremers and Lane’s model to an extension of the estimation period, it was first necessary to establish that this data would produce a model reasonably simitar to Kremers and Lane’s over their sample period or, more important, that they would produce a model that was consistent with what one would infer from Kremers and Lane’s. The re-estimated model for the levels equation was
(West-adjusted t-statistics in parentheses.2)
This equation contains much the same features as Kremers and Lane’s. As with theirs, the variables are all I(1), and the residuals are (just) I(0) (see Table 1). Although the tests for a structural break are invalid, two tests failed to indicate a break in 1984, the same period for which Kremers and Lane tested. Similarly, there was no evidence of autocorrelation in the residuals.
Test Results for Kremers and Lane Sample Equations
All critical values for cointegration tests are from McKinnon (1990).
Test Results for Kremers and Lane Sample Equations
| Equation | Test | Result | |
|---|---|---|---|
| (3) | Cointegration | –4.55 | |
| 5 percent critical value | –4.41a | ||
| (4) | Data to 1984:4 | ||
| Serial correlation LM(4) | 9.45 | ||
| 5 percent critical value | 9.49 | ||
| F(4,14) | 2.44 | ||
| 5 percent critical value | 3.11 | ||
| Data to 1987:4 | |||
| Serial correlation LM(4) | 14.02 | ||
| 5 percent critical value | 9.49 | ||
| F(4,26) | 4.34 | ||
| 5 percent critical value | 2.75 | ||
| Predictive failure post-1984:4 LM(13) | 19.29 | ||
| 5 percent critical value | 22.36 | ||
| F(13,18) | 1.48 | ||
| 5 percent critical value | 2.34 | ||
| Structural break (Chow) 1984:4 LM(5) | 13.43 | ||
| 5 percent critical value | 11.07 | ||
| F(5,26) | 2.69 | ||
| 5 percent critical value | 2.53 | ||
All critical values for cointegration tests are from McKinnon (1990).
Test Results for Kremers and Lane Sample Equations
| Equation | Test | Result | |
|---|---|---|---|
| (3) | Cointegration | –4.55 | |
| 5 percent critical value | –4.41a | ||
| (4) | Data to 1984:4 | ||
| Serial correlation LM(4) | 9.45 | ||
| 5 percent critical value | 9.49 | ||
| F(4,14) | 2.44 | ||
| 5 percent critical value | 3.11 | ||
| Data to 1987:4 | |||
| Serial correlation LM(4) | 14.02 | ||
| 5 percent critical value | 9.49 | ||
| F(4,26) | 4.34 | ||
| 5 percent critical value | 2.75 | ||
| Predictive failure post-1984:4 LM(13) | 19.29 | ||
| 5 percent critical value | 22.36 | ||
| F(13,18) | 1.48 | ||
| 5 percent critical value | 2.34 | ||
| Structural break (Chow) 1984:4 LM(5) | 13.43 | ||
| 5 percent critical value | 11.07 | ||
| F(5,26) | 2.69 | ||
| 5 percent critical value | 2.53 | ||
All critical values for cointegration tests are from McKinnon (1990).
The dynamic equation, re-estimated over Kremers and Lane’s sample period, produced
Although this equation is reasonably similar to that of Kremers and Lane, it should be noted that the longer-sample equation failed a test for a structural break in 1984 (contrary to the result obtained by Kremers and Lane), although it passed a predictive failure test based on the same period (as did Kremers and Lane’s equation). Contrary to Kremers and Lane’s results, there appeared to be some residual autocorrelation in equation (4).
III. Estimation Over an Extended Sample
There are two general reasons for testing the reported model over a longer sample. First, the stability of models of this type must be called into question as the ERM economies become more closely integrated. Second, while the specific equations obtained by Kremers and Lane may be of interest from an historical perspective, the primary interest is in the state of the world as it is now; if the underlying structure that one is trying to measure is unchanged in the extended period, the longer sample should provide more accurate estimates, and if it has changed, estimates over the most recent data are most relevant.
The specific reasons for suspecting that there might be some instability in a longer sample are, as noted earlier: (1) the presence of lagged inflation in the reported long-run equation; (2) the dynamic response to interest rate changes in the short-term equation; (3) the presence of serial correlation in the re-estimated short-run equation; and (4) the failure of the re-estimated short-run equation to pass a structural break test. Re-estimation of the levels equation yielded
The main result from the larger sample was that the data no longer appear to have the characteristics necessary for the Engle-Granger method to be valid; that is, although the variables are stilt I(1), the residuals are not I(0) (see Table 2). This applies both to the residuals from an equation estimated over the longer sample and to long-sample residuals constructed using the estimated coefficients from Kremers and Lane’s equation (1) above.
Test Results for Extended Sample Equations
All critical values for cointegration tests are from McKinnon (1990).
Test Results for Extended Sample Equations
| Equation | Test | Result | |
|---|---|---|---|
| (5) | Cointegration | –3.6 | |
| 5 percent critical value | –4.3a | ||
| (6) | Cointegration | –4.77 | |
| 5 percent critical value | –4.71 | ||
All critical values for cointegration tests are from McKinnon (1990).
Test Results for Extended Sample Equations
| Equation | Test | Result | |
|---|---|---|---|
| (5) | Cointegration | –3.6 | |
| 5 percent critical value | –4.3a | ||
| (6) | Cointegration | –4.77 | |
| 5 percent critical value | –4.71 | ||
All critical values for cointegration tests are from McKinnon (1990).
This result implies rather more than the inappropriateness of an estimation method; the residuals not being I(0) implies that there is no tendency for the ratio of real money to income to move toward any equilibrium level determined by these variables. Essentially, the equation breaks down over the longer period. This is not to say that two separate equations for different subperiods do not exist, but one can be fairly sure that the Kremers and Lane equation is not valid over the sample as a whole.
Although the problems noted above are considerable, even if they could be solved or refuted in some way, one would still have to deal with the problem of the (now) positive coefficient on the short-term interest rate. Even if this equation did have I(0) residuals, one’s normal prior belief about this coefficient could not allow its acceptance for operational purposes.
In an attempt to track down the cause of the breakdown, the long-run equation was estimated recursively—that is, by repeatedly adding one observation to the sample, and the estimated coefficients plotted as time series. The results are shown in Figures 1 to 4, in which the vertical lines indicate the end of the Kremers and Lane sample, the central line shows the coefficient estimate, and the two others, two standard deviations on either side.
Recursive Coefficient: Constant
(Plus/minus two standard errors)
Citation: IMF Staff Papers 1992, 003; 10.5089/9781451973174.024.A010
Recursive Coefficient: Short Rate
(Plus/minus two standard errors)
Citation: IMF Staff Papers 1992, 003; 10.5089/9781451973174.024.A010
Recursive Coefficient: Inflation
(Plus/minus two standard errors)
Citation: IMF Staff Papers 1992, 003; 10.5089/9781451973174.024.A010
Recursive Coefficient: ECU
(Plus/minus two standard errors)
Citation: IMF Staff Papers 1992, 003; 10.5089/9781451973174.024.A010
Recursive estimates take some time to “settle down,” but it appears that all four of the Kremers and Lane coefficients were showing signs of having converged on reasonable values by the end of their sample period, and remained at these values until 1989. However, for every coefficient other than that for the ECU, there is clearly a change in 1989. Thus, these figures tend to confirm the impression that even if the Kremers and Lane model was valid for their sample, it was not for the extended sample.
Relaxing the Imposed Coefficient on Income
When the income coefficient in the long-run equation was freely estimated, a relationship was found (equation (6)) that just cointegrated, although the result is so close that estimation on a further extended sample would be necessary to convince the more skeptical reader (see Table 2):
Furthermore, the coefficient on the interest rate has the correct sign and much of the instability in the coefficients suggested in Figures 1 to 4 is removed. However, as Figure 5 shows, it seems that this instability is in fact taken up by the income coefficient itself. Thus, while the relaxation of this constraint suggests a possible way forward, there is still some way to go before an equation on which policy could be based is found.
Recursive Coefficient: Income
(Plus/minus two standard errors)
Citation: IMF Staff Papers 1992, 003; 10.5089/9781451973174.024.A010
IV. Conclusion
The model estimated by Kremers and Lane exhibits some encouraging features “in-sample,” but there are signs, even within the sample, that it may not be as stable over a longer period. Re-estimation using a data set that produced economically similar results for the Kremers and Lane sample seems to confirm these doubts over an extended sample, as the long-run equation ceases to cointegrate and recursive estimation reveals strong indications of parameter instability from 1989 onward.
Although it may be the case that revisions of the data can explain this apparent instability, the similarity of the results from the two data sets over the common sample suggests that there may be a more fundamental cause. Unless it can be established that these revisions are responsible, the inevitable conclusion is that further research will be necessary if a stable ERM-wide function is to be found. Relaxing the unit constraint on income appears to improve the results and may point the way for further research. The timing of the change in the estimated coefficients—that is, late 1989—suggests that some investigation of the effects of German unification might prove profitable.
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