Estimating the liquidity of M2 components in the post-DIDMCA era.
The Depository Institutions Deregulation and Monetary Control Act of 1980 (DIDMCA), together with the Garn St. Germain Act and related Federal Reserve actions, resulted in a thorough redefinition of the money stock measures, as discussed in Anderson and Kavajecz  and related writings. The current components of M1 and M2 recognize the significant changes among existing assets that occurred throughout the 1970s; examples include the rapid growth of Money Market Mutual Funds (MMMFs) and the breakdown of distinctions between deposits of commercial banks and thrift institutions. The measures of money include assets introduced by legislation, such as NOW accounts and Money Market Deposit Accounts (MMDAs). The DIDMCA also deregulated interest rates on bank deposits, with January 1986 completing the phase-out of Regulation Q.
This paper examines the liquidity of the assets within the restructured definitions of M1 and M2. We offer empirical evidence regarding the cost of decreasing these asset holdings as well as for other bank deposits such as Individual Retirement Accounts (IRAs). The results can be used to assess how easily they can be converted into a medium of exchange. Section II features the model formulation, based upon a representative household's intertemporal optimization problem with liquidity costs.
Our study provides estimates of both relative and absolute liquidity. As Lippman and McCall  discuss, liquidity represents a fundamental characteristic of assets, yet remains elusive in its measurement. To this extent, research seeking to measure or estimate liquidity is sparse. Lippman and McCall's  criterion consists of the optimal expected amount of time to transform an asset into money. Hooker and Kohn  define liquidity as the difference between the asset price at its optimal sale relative to its immediate sale.
Adjustment cost serves as our measure for liquidity. This criterion conforms to commonly used definitions found in economics textbooks, particularly those that emphasize asset withdrawal for transactions. Mishkin [1995, 55], for example, defines liquidity as "the relative ease and speed with which an asset can be converted into a medium of exchange." The adjustment cost approach measures withdrawal costs in dollar units. It regards liquidity as an inherent characteristic of the asset and captures the total cost involved with changing asset holdings. The implicit inclusion of time costs within the formulation is consistent with Lippman and McCall .
Asymmetry in adjustment cost arises because liquidity explicitly involves the cost of reducing asset holdings. While economic units incur costs due to increasing financial assets, larger costs result from withdrawal. We use a minor modification of Pfann and Verspagen's  asymmetric adjustment cost function proposed by Dutkowsky . The function offers several attractive properties for estimating liquidity cost. It allows for asymmetry in the direction of either increases or decreases within portfolio reallocation. Further, the commonly used linear-quadratic function is nested within this equation, enabling straightforward testing for greater withdrawal cost. The asymmetric adjustment cost specification has been utilized previously by Pfann and Palm  within labor demand.
Section III reports empirical findings. We estimate the structural parameters using Generalized Method of Moments (GMM) and monthly data for the December 1983-March 1993 period. The results reveal significant evidence of asymmetry in the direction of decreasing asset holdings for all components except demand deposits. This finding supports the existence of greater cost for withdrawing as opposed to increasing holdings among interest bearing accounts. Further, the evidence points to a definite ordering of liquidity among several assets. Within this group, savings deposits rank as the most liquid, followed by small time deposits, MMMFs, and individual retirement accounts. IRAs have noticeably greater estimated withdrawal cost.
An additional result pertains to the magnitude of estimated liquidity costs of the M2 components. We find that the representative household incurs extremely low costs of decreasing savings deposits, MMMFs, and small time deposits. Estimates from splitting the sample in half suggest that these assets have become even more liquid in recent years. Their small estimated withdrawal cost indicates that in the post-DIDMCA period, many of the components of M2 can be combined to produce a satisfactory medium of exchange measure. Section IV concludes the paper.
II. HOUSEHOLD OPTIMIZATION WITH LIQUIDITY COST
The Basic Model
The formulation incorporates liquidity costs within the intertemporal optimization problem of the representative household. Although we focus on components of M2 and IRAs in the empirical testing, this approach considers the choice of all parts of consumer wealth.
There exists one monetary asset. It earns zero nominal interest but is used for consumption purchases and other transactions. The family derives utility from services that it provides. In addition, the market offers n investment assets (including various forms of debt), each with a corresponding rate of return. Given that the household is a price taker in competitive goods, labor, and asset markets, it chooses consumption, leisure, and real levels of the monetary and investment assets to maximize expected total discounted utility:
(1) [E.sub.0] [summation of] [[Beta].sup.t]u([c.sub.t], [m.sub.t], [l.sub.t]) where t = 0 to [infinity]
subject to the end-of-period wealth constraint
(2) [summation of] [k.sub.jt] where j = 1 to n + [m.sub.t] = [summation of] [r.sub.jt][k.sub.jt-1] where j = 1 to n + [i.sub.t][m.sub.t-1]
+ [w.sub.t](T - [l.sub.t]) + T[P.sub.t] - [summation of] f ([Delta][q.sub.jt]) / [p.sub.t] - [c.sub.t] where j = 1 to n.
Most variables follow standard definitions. The operator [E.sub.0] denotes the rationally determined expectation conditional upon date 0 information, [Beta] is the rate of time discount with 1 [greater than] [Beta] [greater than] 0, u() refers to the quasiconcave utility function, [c.sub.t] is date t real consumption, [m.sub.t] denotes date t end-of-period real balances of the monetary asset, [l.sub.t] is date t leisure measured in hours, T denotes total available hours, [i.sub.t] is one plus the real after-tax return on the transactions asset, T[P.sub.t] refers to date t real transfer payments, and [w.sub.t] is the date t real after-tax wage rate.
The variable [k.sub.jt], for j = 1, 2, ..., n refers to the date t level of real holdings of the jth investment asset. It can be decomposed as [k.sub.jt] = [p.sub.kjt] [q.sub.it] / [p.sub.t], where [p.sub.kjt] denotes the date t price-per-share of the asset, [q.sub.jt] the number of shares, and [p.sub.t] the date t price level for the consumption good. The jth asset's one plus real, after-tax rate of return is given by [r.sub.jt].
The monotonically increasing and convex function f() denotes the nominal cost of changing holdings of an investment asset. Examination of liquidity cost requires a careful look at the argument within this function. Adjustment cost depends upon the change in the number of shares. This characteristic emphasizes that liquidity is the cost of actually withdrawing assets, as opposed to simply having real asset holdings decrease. For example, the household does not incur liquidity costs due to a ceteris paribus rise in the consumption price level. Similarly, any drop in the asset price does not contribute to the cost of withdrawal.
Our problem relates to a well-developed set of research, extensively surveyed in Barnett, Fisher, and Serletis , that seeks to estimate or measure services provided by near-monies. An early direction, pioneered by Chetty  and extended by Boughton , Husted and Rush , Sims, Takayama, and Chao , and Gauger and Schroeter , concentrates on the elasticity of substitution between near-monies and a transactions asset. Their estimated equations generally come from static optimization models. They feature a utility function or transactions services function based upon money and the near-monies.
Barnett  significantly refocuses this literature by introducing aggregation theory and statistical index number theory. He derives a monetary aggregate that satisfies explicit nonlinear aggregation conditions among imperfect substitutes. His results come from an intertemporal optimization model of household behavior under perfect certainty or risk neutrality. He also advocates the aggregate formed by the Divisia index instead of the simple sum aggregate as the proper measure of money. The author shows that the Divisia index is Diewart superlative under these conditions, i.e., it is exact for some aggregator function which can provide a second-order approximation to any linearly homogenous aggregator function.
Barnett  extends his previous work by applying monetary aggregation and index number theory to commonly used optimization models of consumer, firm, and financial intermediary behavior. Barnett  proves that the Divisia index-based quantity of money exactly tracks the theoretically correct aggregate even under uncertainty and risk aversion, if contemporaneous prices and interest rates are known. Rotemberg, Driscoll, and Poterba's  Currency Equivalent (CE) measure also arises from this basic framework, with the additional assumption of strong separability in currency within the monetary aggregator function. Barnett  states that under this restriction along with static expectations, the CE index becomes a special case of the Divisia index.
Our emphasis upon estimating structural parameters from an intertemporal optimization model with money in the utility function under risk aversion goes along with Poterba and Rotemberg  and Barnett, Hinich, and Yue . And, as we discuss below, the adjustment cost formulation could be used to extend the money aggregation research with regard to the treatment of the benchmark asset. However, our focus on asset liquidity does not match up directly with the topic of optimal monetary aggregation. We sidestep this issue by employing highly disaggregated measures for the monetary asset instead of an aggregator function, as we explain further in the data section.(1)
Returning to the model, the first order conditions for date t optimization can be expressed as:
(3) [u.sub.3]([c.sub.t], [m.sub.t], [l.sub.t]) - [w.sub.t][u.sub.1] ([c.sub.t], [m.sub.t], [l.sub.t]) = 0,
(4) [u.sub.2]([c.sub.t], [m.sub.t], [l.sub.t]) + [Beta][E.sub.t] [i.sub.t+1] [u.sub.1]([c.sub.t+1], [m.sub.t+1, [l.sub.t+1])
- [u.sub.1]([c.sub.t], [m.sub.t], [l.sub.t]) = 0,
(5) - [u.sub.1]([c.sub.t], [m.sub.t], [l.sub.t])[f[prime] ([Delta] [q.sub.jt]) / [p.sub.kjt] + 1]
+ [Beta][E.sub.t] [u.sub.1]([c.sub.t+1], [m.sub.t+1], [l.sub.t+1])
x [[r.sub.jt+1] + f[prime] ([Delta] [q.sub.jt+1]) ([p.sub.t] / [p.sub.t+1] [p.sub.kjt+1])] = 0,
for j = 1, 2, ... n. Equation (3) consists of the standard condition where the household equates the marginal utility of leisure with the marginal cost of foregoing consumption.
The fourth equation reflects the choice of the monetary asset. It is identical to the condition within the intertemporal substitution model of labor supply with money in the utility function, as in Dutkowsky and Dunsky . Positive marginal utility of transactions services ensures that the household will demand money. Feenstra  demonstrate's that under plausible conditions, a functional equivalence exists between real money balances in the utility function and a liquidity cost function describing the reduction of money balances. Croushore  shows that the functional equivalence can also result from a shopping time model. Equation (5) describes the condition involving the jth investment asset, j = 1,2,...,n. This set of Euler equations are separable across the various nonmonetary wealth components.
The group of conditions in equation (5) shows how the adjustment cost formulation could be applied to extend the monetary aggregation literature. Within that context, our model features multiple benchmark assets. As discussed in Barnett, Fisher, and Serletis , and Rotemberg, Driscoll, and Poterba , the benchmark asset does not provide any monetary services and is used to transfer wealth intertemporally. Models in this area typically feature a single asset of this type.
The adjustment cost formulation permits interior solutions where the household chooses to hold a number of different benchmark assets. The set of equations in (5) reveal that in the absence of liquidity cost, the investment asset with the highest interest rate would dominate the rest. In our model, the household at date t can choose to hold a portfolio of nonmonetary assets, based upon their respective interest rates, contemporaneous marginal adjustment costs, and expected marginal costs of adjustment at date t+1.
A related characteristic from our formulation involves the ability of the household to use all or part of the benchmark asset(s) at some point for consumption or other purchases. All investment assets can potentially be converted into a medium of exchange. The household incurs withdrawal costs by doing so. Adding to wealth in the form of deposits or asset purchases also results in costs.
The marginal cost of adjusting asset balances introduces additional dynamics to the portfolio allocation decision. In this way, the liquidity cost formulation complicates the examination of how closely the Divisia index tracks the optimal measure of money. The intertemporal optimization model with adjustment cost also blurs any simple relationship between interest rate differentials across assets and differences in their liquidity. The asset balance resulting from the household's withdrawal at date t becomes the initial balance within their date t+l decision.
The Operational Model
We focus on the subset of equation (5) that consists of MMMFs and deposits offered by banks. For this group, [p.sub.kjt] = $1.00 and [q.sub.jt] = [p.sub.t][k.sub.jt] [equivalent to] [M.sub.jt]. With these assets, the number of "shares" equals their nominal value, although not in dollar units (i.e. "the number of dollar bills").(2)
Following the intertemporal substitution literature, we specify the utility function as:
(6) [Mathematical Expression Omitted].
The above utility function offers a number of desirable features, particularly for applied work. Using equation (6) within an intertemporal substitution model with money, Dutkowsky and Dunsky  obtain estimates of the structural parameters which are significant and conform to a priori predictions.
We assume that the liquidity cost function for the jth investment asset at date t (M.sub.jt]) takes the form:
(7) [Mathematical Expression Omitted]
for j = 1, 2, ..., n. Equation (7) is the Pfann and Verspagen  asymmetric adjustment cost function with a minor modification proposed by Dutkowsky . Dutkowsky's adjustment consists of subtracting [Mathematical Expression Omitted] within the quadratic term, for more straightforward parameter interpretation.
To examine the roles of the coefficients within the asymmetric adjustment cost function, consider the third order Taylor series approximation of equation (7) around [Delta][M.sub.jt] = 0:
(8) [Mathematical Expression Omitted].
The [[Gamma].sub.j] parameter measures the symmetric part of adjustment cost, corresponding to its role in the linear-quadratic function. It would be expected to have a positive sign. The [[Mu].sub.j] parameter measures effects due to asymmetry. If households face greater costs for withdrawing as opposed to increasing deposits, the model predicts a negative sign for this parameter. Furthermore, less liquid assets should have larger absolute magnitudes of [[Gamma].sub.j] and [[Mu].sub.j], for j = 1, 2, ..., n.
When [[Mu].sub.j] = 0, equation (7) reduces to the symmetric linear-quadratic form. Dutkowsky  shows that equation (7) is convex everywhere when [Mathematical Expression Omitted]. For [Mathematical Expression Omitted] and [[Mu].sub.j] [less than] 0, a well-behaved adjustment cost function exists within the region [Mathematical Expression Omitted]. This interval includes all the negative values of [Delta][M.sub.jt], zero, and a range of positive values.
III. EMPIRICAL RESULTS
Data and Variable Definitions
The sample consists of seasonally adjusted monthly data spanning December 1983 - March 1993, 112 observations in all. Even with the necessary lags, this period falls well beyond the notable change in asset composition due to the introduction in early 1983 of MMDAs and Super NOW accounts. All data except for bank deposit interest rates come from the Citibase tape; we obtained the latter data from the Federal Reserve. The Appendix provides further detail regarding data and variable construction.
We report findings from two estimated models, based upon the measure of the household's real balances of the monetary asset. Both measures of nominal money within the [m.sub.t] variable represent levels of aggregation at or below M1 as defined in the pre-DIDMCA period. This gets around problems associated with proper aggregation of near-monies, as suggested by Barnett [1980, 39].
The first estimated model features real per capita currency held by the public - nominal currency deflated by the consumption price deflator and population. It considers six investment assets, all in nominal per capita units following equation (5): demand deposits (plus travelers checks), other checkable deposits, savings deposits (which include MMDAs), MMMFs, small time deposits, and IRAs (plus Keough accounts).(3)
The second estimation has real per capita currency plus demand deposits as the monetary asset. This specification assumes that households regard currency and non-interest bearing checking accounts as perfect substitutes. The investment components consist of the remaining assets. We do not consider other checkable deposits as part of the transactions asset. NOW accounts do have unrestricted checking privileges based upon frequency and amount. But the unregulated interest on these deposits may be compensating consumers for other restrictions, such as high minimum balances.
Bank interest rates are measured by rates for commercial banks. The rate for Super NOW accounts serves as the interest rate on other checkable deposits. The rate on savings deposits consists of the MMDA rate spliced in April 1986 to the rate on total savings deposits.4 We construct the rate on small time deposits as a weighted average of rates for time deposits of maturities three month, six month, one year, 1-21/2 years, and over 2 1/2 years. The rate on the over 2 1/2 year maturity time deposits serves as the interest rate on IRAs. This assumes that the household chooses to invest this relatively illiquid asset in the longest maturity bond possible. The rate on 6 month commercial paper proxies the rate for MMMFs. Both currency and demand deposits pay zero nominal interest.
The interest rate data reflect deregulation. Since other checkable deposits include Super NOW accounts and MMDAs fall within savings deposits, the rates exceed Regulation Q ceilings even prior to January 1986. The mean interest rates in ascending order are other checkable deposits, savings deposits, MMMFs, small time deposits, and IRAs.
We assume that households earn an additional implicit nominal return for their deposits within banks. These accounts provide increased safety as well as deposit insurance coverage. We assign an implicit return of 1.25% for deposits offered by banks. This "safety return" approximately equals the spread between the mean interest rates on MMMFs and savings deposits. All the estimation results are robust with respect to alternative reasonable choices of implicit returns (including zero). We use these measures of nominal return to form [r.sub.jt], the one plus real, aftertax return on the jth asset; details are provided in the Appendix.
The remaining variables follow standard choices and correspond to Dutkowsky and Dunsky . The authors show that their measures for the real, after-tax wage rate and asset returns conform to the consumption plus savings equals disposable income identity for the representative household. They contend that the more carefully constructed data accounts in part for their supportive empirical results.
The consumption measure in this study represents the only other deviation from the usual choice of variables. Total consumption, rather than nondurables and services, serves as consumption. We find surprisingly inferior results when we use nondurables and services. As one reason for this unusual sensitivity, a substantial part of household withdrawal of investment assets may be to purchase durable goods.
We jointly estimate by GMM the system comprised of equation (4) and the subset of equation (5) consisting of the assets listed above. We estimate all the Euler equations divided through by [u.sub.1]([c.sub.t], [m.sub.t], [l.sub.t]). This transformation puts all the estimated equations in terms of stationary variables, based upon Augmented Dickey-Fuller unit root tests. As Ogaki  discusses, consistent GMM estimation requires stationarity (see footnote 1 for further details).
Stationarity considerations are the reason that we exclude equation (3) from the estimation. Transforming that Euler equation as discussed above results in an equation linear in [c.sub.t] and [w.sub.t][l.sub.t]. However, this equation fails to meet stationarity requirements with the variables in levels. Augmented Dickey-Fuller tests cannot reject the null hypothesis of unit roots for [c.sub.lt] and [w.sub.t][l.sub.t], although the tests reject unit roots for the first difference of each variable. The Engle and Yoo test fails to reject the null hypothesis of non-cointegration. We estimated the above system with a first-differenced version of equation (3), as in Dutkowsky and Dunsky , and obtained similar results.
The computer programs use PROC IML on SAS, following the example in Gallant . The procedure forms the weighting matrix for GMM estimation using the Parzen kernel, with our choice of the bandwidth parameter based upon sample size (see Gallant 1987 for further explanation). The instrument set contains 14 elements: a constant; [c.sub.t]/[c.sub.t-1]; [l.sub.t]/[l.sub.t-1]; [m.sub.t-1]; [r.sub.st] / ([c.sub.t]/[c.sub.t = 1]) with [r.sub.st] the rate of return on stock; [i.sub.t] = 1/([p.sub.t]/[p.sub.t-1]), the real return on currency as well as demand deposits; date t changes in nominal per capita holdings for all monetary variables except for currency; and date t real after-tax returns on savings deposits and small time deposits. Augmented Dickey-Fuller tests indicate stationarity.
In practical terms, this set of instruments would appear to satisfy the exogeneity and relevance conditions associated with instrumental variables estimation, as discussed in Hall, Rudebusch, and Wilcox  and related work. The instruments are within the household's date t information set, implying that rational expectations-based orthogonality conditions critical to GMM estimation should hold. At the same time, with the exception of [r.sub.st]/([c.sub.t]/[c.sub.t-1]), they consist of variables which either appear in the Euler equations or lags of date t+1 variables in the model. This feature suggests that the instruments should be reasonably correlated with the endogenous variables. We investigate instrument quality after reporting our results. Hall et al. , in particular, argues against using relevance statistics for pre-estimation screening.
GMM estimates of the utility function parameters and the rate of time discount appear in Table I. All the estimated structural parameters fall within plausible ranges. Estimates for the consumption parameter are near unity and significantly different from zero. The positive and significant estimates for the [Phi] coefficient support the existence of money in the utility function. The estimated 0 coefficient is positive but not significantly different from zero. The parameter's insignificance indicates that the utility function equation (6) reduces to a Cobb-Douglas with constant returns to scale. Compared to Dutkowsky and Dunsky's  estimates, the model with liquidity costs generates larger coefficients for [Alpha] and smaller [Theta] estimates which are insignificant. Besides the model specification, the different results may be due to the sample period.(5)
Both estimated liquidity cost models produce credible coefficients for the discount parameter. Estimates for the [Beta] parameter estimated annualized rates of time discount equal to 1.70% and 2.68%. J statistics for testing overidentifying restrictions indicate that the null hypothesis of orthogonality in either estimated model cannot be rejected at any reasonable level of significance.
Table II reports estimates of the liquidity cost parameters for the model with currency in the utility function. All the asymmetry parameters ([u.sub.j]) are highly significant. With the exception of demand deposits, they all have negative sign. These results indicate that apart from non-interest bearing checking accounts, households incur greater costs for withdrawing as opposed to increasing deposits.
The estimated symmetry coefficients ([[Gamma].sub.j]) are considerably smaller and less precisely estimated [TABULAR DATA FOR TABLE 1 OMITTED] than the parameters for asymmetry (see notes to Table II). They vary in sign and significance across the different assets. The coefficient has a positive sign and is significantly different from zero for small time deposits and IRAs. The estimate for other checkable deposits is also positive, but insignificant from zero. The results put forth negatively signed estimates for demand deposits, MMMFs, and savings deposits. The estimate for savings deposits is the only estimate significant from zero within this group.
Comparing the size of the estimated [Gamma] coefficients along with the absolute magnitudes of the [Mu] estimates across the different assets enables us to gauge their relative liquidity. IRAs have the largest coefficients based upon these criteria. Its property of being the most illiquid asset, as indicated by the empirical results, matches stylized facts concerning the substantial penalties for early withdrawal. The same comparison reveals that of the assets other than demand deposits, savings deposits rank as the most liquid. They have the smallest T coefficient and their [Mu] estimate ranks lowest in absolute value.
The findings generate an ambiguous ordering of liquidity among other checkable deposits, MMMFs, and small time deposits. The assets have similar asymmetry estimates, with MMMFs the largest in absolute magnitude. On the other hand, small time deposits have the highest positive symmetry parameter for liquidity cost.
The results suggested by the estimates for other checkable deposits deviate somewhat from prior thought. While the estimates conform to theoretically predicted signs, the estimated asymmetry coefficient is larger than that of savings deposits and even small time deposits. Institutional restrictions on these accounts, such as high minimum balances, may be responsible for the relatively lower liquidity indicated by the findings.
The last two columns report estimated withdrawal costs, computed from the asymmetric adjustment cost function given by equation (7). As expected, IRAs generate the largest cost, approximately 5.24% for a $50 withdrawal. Estimated liquidity cost with IRAs is sharply higher in percentage terms for a $100 decrease. IRAs feature noticeably greater estimated costs for withdrawal than the other assets.
Among the remaining components other than demand deposits, the estimated withdrawal [TABULAR DATA FOR TABLE II OMITTED] costs rank savings deposits as the most liquid, followed by small time deposits, other checkable deposits, and MMMFs. The larger cost for decreasing holdings of MMMFs as opposed to time deposits can be viewed as surprising. The result suggests that, despite the asset's limited checkability along with penalties for early withdrawal of time deposits, households incur greater time and other transactions costs from MMMFs. One possible reason is that MMMFs cannot be readily accessed from a nearby bank.
The findings indicate small withdrawal costs for all the M2 assets in our estimated model. The estimation generates negligible costs for the representative household decreasing asset holdings by $50. Withdrawal costs are greater in percentage terms for the $100 decrease. Still, they each fall below 2% of the desired withdrawal, with savings and time deposits under 1%. The low transactions costs indicate that all the assets other than IRAs can be converted easily into a medium of exchange.
For demand deposits, the estimated coefficients are opposite in sign from their counterparts. This finding may reflect the role of non-interest bearing checking accounts as a receptor for withdrawals. Households might convert holdings of other assets into demand deposits in order to pay for consumption purchases. In this way, the "liquidity costs" of demand deposits may mirror those of the other investment assets.6 The findings also indicate that estimated withdrawal costs for demand deposits are extremely small in absolute magnitude. These overall results suggest that demand deposits play more of a pure transactions role.
Estimates from the model with currency plus demand deposits in the utility function appear in Table III. For the most part, we obtain findings similar to those of the previous model. The one area in which the results are inferior involves other checkable deposits. In particular, we obtain a larger estimated asymmetry coefficient for this asset. It ranks second in absolute magnitude only to IRAs. [TABULAR DATA FOR TABLE III OMITTED] Other checkable deposits also generate the highest withdrawal costs within the M2 components, although they remain small in size.
The only other major difference pertains to the estimates for IRAs. Estimated coefficients from this model indicate that IRAs have greater liquidity, although the asset remains the most illiquid of the group. Estimated withdrawal costs for IRAs are lower than those of the model with currency in the utility function, but still noticeably higher than the M2 components. This estimated model generates a more plausible estimated conversion cost of $12.54 for a $100 decrease.
Our credible results are somewhat surprising in light of a growing literature on the poor performance of the GMM estimator in empirical applications.(7) As Fuhrer, Moore, and Schuh , Pagan , and others discuss, finite sample properties of GMM estimation seem to hinge upon the quality of the instruments. This contention matches the general view regarding bias that can arise from using poorly correlated instruments, as shown by Bound, Jaeger, and Baker . In this regard, studies like Hall et al. , Pagan , and Shea  argue for the importance of examining instrument relevance.
We investigate instrument quality with the Shea  partial [R.sup.2] statistic, also used by Fuhrer et al. . The statistic is computed for each of our instruments within each Euler equation. Here it provides an informative but crude look at instrument quality, given that the statistic (as well as most other instrument relevance measures) applies to linear, single equation models. In reporting our findings, we focus on the model with currency in the utility function; the other model generates very similar results.
The interest rates have the largest partial correlations. The median partial [R.sup.2] for this group equals 0.67, and all are above 0.40. The asset change variables ([Delta][M.sub.jt+1] and [Delta][M.sub.jt]) exhibit reasonably high partial correlation, for the most part. For the date t variables, the [R.sup.2] exceeds 0.28 in all cases except for IRAs ([R.sup.2] = 0.14), with a median of 0.40. The median partial correlation for the date t+1 variables equals 0.27, with all statistics greater than 0.20 except for demand deposits (0.19) and IRAs (0.07). The [c.sub.t+1]/[c.sub.t] and [m.sub.t+1]/[m.sub.t] have median partial correlations of 0.22 and 0.23. The [R.sup.2] for these variables varies little within each Euler equation. The [l.sub.t+1]/[l.sub.t] variable performs the worst. The median [R.sup.2] equals 0.08, with statistics ranging from 0.06 to 0.09.
Except for the leisure and IRA variables, nearly all the partial correlations fall within or exceed the "fairly high" category referred to by Fuhrer et al. . The nature of the Euler equations in our model suggest reason to be encouraged beyond the above findings. Each adjustment cost parameter appears in two places within its corresponding Euler equation. Consequently, unlike the linear model, the GMM procedure draws information to estimate the parameter from more than one variable. This might account in part for the relatively precise estimates of these coefficients, particularly the asymmetry parameters. In addition, the instrument quality statistic for the ratios of future to current consumption, money, and leisure in this model may be misleading. These variables as well as the utility function parameters appear in all the Euler equations. Theory-based cross-equation restrictions might improve estimator precision and reduce instrument related bias.
On a more institutional issue, many believe that the almost continuous pace of financial innovation over time has brought forth changes in asset liquidity. We investigate this contention by testing for structural change, based upon splitting the sample in half. The Wald test proposed by Andrews and Fair  rejects the null hypothesis of structural stability at the five percent level for each model. Estimates of each subsample reveal that the asymmetry parameters primarily account for the test findings.(8)
Table IV reports the asymmetry estimates for the model with currency in the utility function (the model with currency plus demand deposits generates very similar findings). With the exception of demand deposits, all the coefficients are negative and highly significant for each subsample. Although the estimates for other checkable deposits remain relatively large, the results for each period generally follow the hierarchy of liquidity indicated by the previous results.
On the other hand, the findings show uniformly smaller asymmetry coefficients for the second half of the sample. MMMFs exhibit the most noticeable reduction. These estimates provide evidence indicating greater liquidity among this set of assets in recent years. The estimated withdrawal costs reflect these results. From Table IV, observe that all the interest bearing assets have lower conversion costs for a $100 withdrawal in the more recent sample, with MMMFs showing the largest decrease.(9)
This paper implements asymmetric adjustment cost to estimate the liquidity of several types of bank deposits and MMMFs. This specification of liquidity cost is applied to the intertemporal optimization problem of the representative household. GMM estimation of the structural parameters generates estimates which, by and large, conform to theoretical predictions.
Despite the generally favorable results, several empirical limitations apply. While using the Board of Governors data enables us to investigate liquidity among a number of different components of M2, the aggregates contain holdings of firms as well as households. In addition, even the relatively small monthly time unit does not pick up many changes taking place among asset holdings. Aggregation in the reported money stock measures tends to cancel out opposite movements among different individuals. Finally, previous work indicates that the GMM procedure demands very high quality instruments. The data may be incapable of discerning fine differences in liquidity.
[TABULAR DATA FOR TABLE IV OMITTED]
Nonetheless, a number of interesting findings emerge. The estimation results point to significantly higher costs for withdrawing rather than adding to these assets. And for the most part, the findings conform to perceived notions regarding the relative liquidity of the various M2 components. Our model formulation can be applied to study the liquidity of other financial assets as well. Another promising direction for extension would be to apply the adjustment cost formulation to examine monetary aggregation.
While the results make no explicit statement about the theoretically correct definition of money, they support the role of the non-M1 components of M2 in forming a satisfactory medium of exchange measure. We obtain minimal estimated withdrawal costs for other checkable deposits, savings deposits, MMMFs, and small time deposits. At the same time, the estimates indicate substantially greater costs if households wish to remove funds from IRAs. In this way, the empirical findings accurately separate IRAs from the other set of assets.
They also point to the significant effects of financial innovation regarding bank deposits and MMMFs. The innovations created assets that households can convert inexpensively and increased the liquidity of traditional M2 components. Our findings suggest that these assets have become even more liquid in recent years. The results indicate that in the post-DIDMCA era these interest bearing near-monies have become very near to currency and demand deposits.
Data Definitions and Sources
All data are seasonally adjusted; we use PROC X11 on SAS to seasonally adjust series that came without adjustment. Since we adhere to the wealth identity under the monthly time frame, we deannualize all the seasonally adjusted annualized NIPA quarterly series by dividing them by 12. The Citibase tape provides all data except for bank interest rates.
Data are converted to real per capita units using the price deflator [p.sub.t] multiplied by population PO[P.sub.t]. The price deflator is the ratio of current dollar total consumption (GMC in Citibase) to consumption in 1982 dollars (GMCQ). The noninstitutional population age 16 or over including armed forces (PO16) measures PO[P.sub.t]. The [c.sub.t] variable is consumption in 1982 dollars (GMCQ) divided by PO[P.sub.t].
Per capita labor hours for the month, [n.sub.t], are calculated as
(A1) [n.sub.t] = [([HOURS.sub.t] x [EMP.sub.t] x 365)
/(12 x 7)]/PO[P.sub.t],
where HOURS denotes average hours of work per week for the household (LHCH), EMP refers to the total employed labor force (LHEMR), and POP is the population measure defined above. The numerical constant [365 days/year]/[(12 months/year) x 7 days/week)] converts the numerator in equation (A1) into units of total hours/month. The leisure variable/t is formed by subtracting [n.sub.t] from 730 [(24 hours/day) x (365 days/year)]/[12 months/year].
We approximate the marginal tax rate with an average tax rate ([[Tau].sub.t]) in constructing the after-tax wage rate and interest rates. The tax rate comes from the disposable income identity
(A2) [YD.sub.t] = (1 - [[Tau].sub.t] [WS.sub.t] + [OLY.sub.t]
+ (1 - [[Tau].sub.1]) [INTE.sub.t] + [TP.sub.t];
where YD is nominal disposable income (GMYD); WS denotes the sum of nominal wages and salaries (GMW) and proprietary income (GMPRO); OLY refers to other labor income (GMPOL); INTE denotes the sum of total interest (GMPINT), dividends (GMDIV), and rental income (GMPREN) less interest paid on consumer debt (GMCINT); [p.sub.t] is the consumption deflator defined above; and TP refers to nominal transfer payments (GMPT). We follow IRS tax laws by observing that other labor income, which consists primarily of labor benefits, as well as transfer payments are tax exempt. Solving (A2) for (1 - [[Tau].sub.t]) yields the average tax rate measure.
The after-tax real wage rate is computed as
(A3) [w.sub.t] = [(1 - [[Tau].sub.t]) [WS.sub.t] + [OLY.sub.t]] / ([n.sub.t][p.sub.t]),
with all variables defined above.
We use the following assets: currency held by the public (FMSCU), demand deposits (FMSD) plus travelers checks (FMSTC), other checkable deposits (FMCD), savings deposits (FMSV), money market mutual funds (FMMG), small time deposits (FMST), and total individual retirement accounts and Keough accounts (FZMIK). We convert the investment assets to nominal per capita variables. The monetary asset (the asset which appears in the utility function) is measured in real per capita terms.
Bank interest rates are all measured by rates for commercial banks. The rate for Super NOW accounts serves as the interest rate on other checkable deposits. The rate on savings deposits in the estimation consists of the MMDA rate spliced in April 1986 to the rate on total savings deposits. We construct the rate on small time deposits as a weighted average of rates for Time Deposits of maturities three month, six month, one year, 1-2 1/2 years, and over 2 1/2 years. We deannualize all interest rates into monthly rates in the estimation.
The time deposit rate on the over 2 1/2 year maturity serves as the interest rate on IRAs. The rate on six month commercial paper (FYCP) proxies the rate for MMMFs. The nominal interest rate equals zero for both currency and demand deposits.
If [i.sub.jt] denotes the nominal rate of interest on the jth asset in monthly terms (the annual percentage rate divided by 1200), the corresponding real aftertax return within the Euler equation ([r.sub.jt]) is given by:
(A4) [r.sub.jt] = [(1 - [[Tau].sub.t])[i.sub.jt] + s + 1] / ([p.sub.t]/[p.sub.t-1]),
where s = 1.25/1200 denotes the implicit safety return for bank deposits, expressed in monthly terms. We set the tax rate equal to zero when constructing the return on IRAs.
Following Dutkowsky and Dunsky , we construct the real after-tax rate of return on stock, [r.sub.st], as:
(A5) [r.sub.st] = [(1 - [[Tau].sub.t])[i.sub.st] + [p.sub.kt] / [p.sub.kt-1]] / ([p.sub.t]/[p.sub.t-1]),
The interest rate portion [i.sub.st] is the dividend yield on Standard and Peers Common Stock (FSDXP), divided by 1200 to convert into monthly decimal units. The index of prices of Standard and Peers common stocks (FSPCOM) serves as our measure of [p.sub.kt], the asset price.
This paper benefitted substantially from the comments of two anonymous referees. We are grateful to Adrian Pagan and John Shea for sending us copies of their papers. We also thank seminar participants at Clarkson University for their comments along with Gretchen Schmidt, Brian Reid, and Sean Collins of the Board of Governors for sending us the data on bank interest rates and answering questions regarding the composition of the different types of bank deposits. The views expressed here do not necessarily conform to those of the Federal Reserve System or the General Accounting Office.
1. One reason that we did not pursue the problem of optimal monetary aggregation more vigorously is that we could not find an aggregator function which results in Euler equations with all stationary variables. As Ogaki  states, all variables must be stationarity (or possibly cointegrated, within a linear model) for consistent GMM estimation. Augmented Dickey-Fuller tests reject the null hypothesis of unit roots for the levels of interest rates. But the tests indicate that the wealth components used in this study are integrated of order one.
2. Hooker and Kohn's  emphasis on asset prices in their liquidity definition imply that their criterion cannot be applied easily to this set of assets. We can incorporate their criterion for bonds or equity in our formulation by adding the capital gain or loss onto the withdrawal cost.
3. We refer to these assets as investment assets throughout the paper for convenience, even though several of them can serve directly as a medium of exchange for some purchases. A more accurate but cumbersome term might be, "less than pure transactions assets." The sign of the estimated asymmetry parameter offers some evidence regarding whether an asset plays more of an investment or pure transactions role, as we discuss when reporting our results. We combine travelers checks with demand deposits since both assets pay zero nominal interest. Estimations with travelers checks added onto other checkable deposits do not generate substantially different results. The IRA measure contains retirement accounts in MMMFs as well as commercial banks and thrifts. We construct the rate of return measure for IRAs based upon bank interest rates, since the percentage of deposits in mutual funds is small.
4. Splicing was necessary since the Federal Reserve began publishing the interest rate series for total savings deposits in April 1986 and stopped reporting MMDA rates in September 1991. The rate for other checkable deposits consists of the rate on Super NOW accounts up to December 1985 and NOW accounts afterward, coinciding with the Federal Reserve changing their reporting of these rates. In both cases, the data does not exhibit a notable break around the splice.
5. Some further investigation indicates that the sample period seems to be more responsible for the different estimates of the structural parameters. Estimating the Dutkowsky-Dunsky Intertemporal Substitution model with real per capita currency plus demand deposits in the utility function using this data yields (standard errors are in parentheses) [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted].
6. Another explanation involves the use of monetary aggregate data in estimating models of household decisions. Firms hold a considerable amount of these assets, particularly demand deposits. Their actions could reflect behavior not put forth by the theory.
7. In addition to the studies that we mention, The Journal of Business and Economic Statistics devotes practically its entire July 1996 issue to the subject.
8. based upon individual testing of the corresponding parameters between sample periods, MMMFs have the only significantly different estimates among the [Mathematical Expression Omitted] parameters. The time discount coefficients are the only other estimates that yielded a significant test result.
9. A nonintuitive result from Table IV involves the magnitude of the split sample asymmetry estimates relative to the corresponding estimate from the full sample estimation. In four of the six cases, the estimation generates larger coefficients for each subsample than the estimate using the entire set of data. This might be due to the model's overall nonlinearity as well as the different weighting matrices in the estimations. In particular, the sample size in the split sample estimations calls for a bandwidth of two, as opposed to the bandwidth of three for the full sample.
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Dutkowsky: Associate Professor, Department of Economics, Maxwell School of Citizenship and Public Affairs Syracuse University, New York Phone 1-315-443-1918, Fax 1-315-443-3717 E-mail firstname.lastname@example.org
Dunsky: Economist, U.S. General Accounting Office Washington, D.C., Phone 1-202-512-9110 Fax 1-202-512-9096, E-mail email@example.com
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|Title Annotation:||Depository Institutions Deregulation and Monetary Control Act of 1980|
|Author:||Dutkowsky, Donald H.; Dunsky, Robert M.|
|Date:||Apr 1, 1998|
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