Date x1 x2 x3 r1 r2 r3 rb
1 2005-01-01 1235.89 902.17 1528.41 0 0 3.500 9.500
2 2005-02-01 1227.77 929.38 1549.25 0 0 3.510 9.553
3 2005-03-01 1248.90 947.40 1585.89 0 0 3.165 9.028
4 2005-04-01 1243.45 947.19 1584.36 0 0 3.729 9.267
5 2005-05-01 1231.78 945.22 1603.69 0 0 3.070 9.127
6 2005-06-01 1221.64 963.80 1627.26 0 0 3.216 9.058\[ \newcommand{\vy}{\mathbf{y}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vb}{\mathbf{b}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vone}{\mathbf{1}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\mX}{\mathbf{X}} \newcommand{\mZ}{\mathbf{Z}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mJ}{\mathbf{J}} \newcommand{\mSigma}{\boldsymbol{\Sigma}} \newcommand{\E}{\mathrm{E}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\vr}{\mathbf{r}} \newcommand{\vs}{\mathbf{s}} \newcommand{\vpi}{\boldsymbol{\pi}} \]
1 Abstract
This vignette shows the package at work: the data it expects, the Divisia monetary aggregate of Barnett (1980) with its user costs and weights, the other methods, and the plain-language reading of a result. The example data are simulated; the vignette “Theory” gives the formulas and “Validation” shows that they reproduce published values.
2 The data
dmai() takes a data frame with one row per date: a date column, the quantities of the components and their rates of return. By default the columns are named Date, x1, x2, … and r1, r2, …, as in version 0.5.0; date, x and r name other columns. money_sim holds simulated monthly data from 2005 to 2024: eight components (currency, demand deposits, savings deposits, four maturities of time deposits and foreign currency deposits), their rates in percent and a benchmark rate rb, a long bond yield.
3 The Divisia aggregate
The services of money are priced by the user cost: the interest given up by holding an asset instead of a benchmark asset that yields no monetary services. Currency pays no interest, so its user cost is the whole benchmark rate; a time deposit that pays nearly the benchmark rate costs little to hold and is held mostly as a store of value.
fit <-
dmai(
.data = money_sim
, method = "Barnett"
, logbase = NULL
, date = "Date"
, x = NULL
, r = NULL
, user_cost = NULL
, benchmark = "rb"
, percent = TRUE
, base = 100
, as_dt = FALSE
)
fitDivisia (Barnett) index of 8 components, 240 dates
==================================================
Dates: 2005-01-01 to 2024-12-01 (240); components: x1, x2, x3, x4, x5, x6, x7, x8
Benchmark rate: column rb (upper envelope with the component rates)
Growth (DMAI, log, percent): mean 0.8896, sd 0.5902, last 0.6266
Index: 100 at the first date, 838.3 at the last; simple sum: 1,106
Date DMAI index value
<Date> <num> <num> <num>
1: 2005-01-01 NA 100.0 5413
2: 2005-02-01 0.8504 100.9 5459
3: 2005-03-01 1.7329 102.6 5555
4: 2005-04-01 -0.2494 102.4 5541
5: 2005-05-01 0.3359 102.7 5560
---
236: 2024-08-01 0.1823 804.2 43533
237: 2024-09-01 0.5280 808.5 43764
238: 2024-10-01 0.9856 816.5 44197
239: 2024-11-01 2.0063 833.0 45093
240: 2024-12-01 0.6266 838.3 45376
Sources: Barnett (1978); Barnett (1980); Tornqvist (1936); Theil (1967). Full
references: dmai_cite(x).DMAI is the growth from the previous date in percent (the log difference times 100), index the aggregate with 100 at the first date and value the aggregate in the units of the components, equal to the simple sum at the first date. With as_dt = TRUE (the default) dmai() returns only that table, as version 0.5.0 did.
plot(
x = fit
, which = "index"
)
The Divisia aggregate grows more slowly than the simple sum here because the interest-bearing deposits grow fastest and they carry small weights. The weights over time:
plot(
x = fit
, which = "shares"
)
4 Reading the result
dmai_interpret(
x = fit
, top = 3
)Divisia (Barnett) monetary aggregate
From 2005-01-01 to 2024-12-01 the Divisia (Barnett) aggregate grew 11.3% a
year, against 12.8% for the simple sum (1.6 points less).
Findings:
- Largest growth at 2009-03-01, 2021-12-01, 2024-01-01 (2.49, 2.24, 2.17);
smallest at 2007-06-01, 2012-09-01, 2018-12-01 (-0.71, -0.67, -0.61).
- At the last date the largest weights are x3 (28.1%), x1 (19.5%), x2
(18.8%).
- The weights moved most for x1 (32.3 to 19.5%), x4 (4.3 to 9.9%), x6 (3.7 to
8.5%).
- The simple sum overstates money growth: it gives the same weight to
interest-bearing deposits, which are held largely as savings, as to
currency and demand deposits.
Recommendations:
- Compare the methods with dmai_compare(); check how the result depends on
the benchmark rate.
- dmai_moments() shows when the components grow at very different rates.
Sources:
- Barnett, W. A. (1978). The user cost of money. Economics Letters, 1(2),
145-149. doi:10.1016/0165-1765(78)90051-4
- Barnett, W. A. (1980). Economic monetary aggregates: an application of
index number and aggregation theory. Journal of Econometrics, 14(1), 11-48.
doi:10.1016/0304-4076(80)90070-6
- Tornqvist, L. (1936). The Bank of Finland's consumption price index. Bank
of Finland Monthly Bulletin, 10, 1-8.
- Theil, H. (1967). Economics and Information Theory. North-Holland,
Amsterdam.5 The other methods
dmai_compare() computes every method on the same data:
cmp <-
dmai_compare(
.data = money_sim
, methods = c("Barnett", "Hancock", "Fisher", "SimpleSum", "CE")
, logbase = NULL
, date = "Date"
, x = NULL
, r = NULL
, user_cost = NULL
, benchmark = "rb"
, percent = TRUE
, base = 100
)
cmp5 monetary aggregates of 8 components, 240 dates
================================================
method mean_growth annual_growth sd_growth last_index
<fctr> <num> <num> <num> <num>
1: Barnett 0.8896 11.27 0.5902 838.3
2: Hancock 0.9044 11.39 0.5954 856.4
3: Fisher 0.8896 11.27 0.5902 838.2
4: SimpleSum 1.0055 12.82 0.5726 1105.9
5: CE 0.8347 10.54 2.6930 735.2
Sources: Barnett (1978); Barnett (1980); Hancock (1985); Fisher (1922); Diewert
(1976); Rotemberg et al. (1995). Full references: dmai_cite(x).
plot(
x = cmp
, which = "index"
)
The Divisia and Fisher indexes nearly coincide: both are superlative indexes (Diewert 1976). The Hancock variant uses simple instead of log growth rates and drifts above them; the currency equivalent index moves with the rates.
6 User costs and second moments
dmai_user_cost(
.data = money_sim
, date = "Date"
, x = NULL
, r = NULL
, benchmark = "rb"
, percent = TRUE
)User costs of 8 components, 240 dates
=====================================
Averages over the dates:
component mean_rate mean_user_cost mean_share last_share
<fctr> <num> <num> <num> <num>
1: x1 0.000 0.08124 0.22914 0.19497
2: x2 0.000 0.08124 0.20366 0.18827
3: x3 3.050 0.05360 0.30779 0.28085
4: x4 4.903 0.03662 0.07486 0.09851
5: x5 5.414 0.03193 0.05076 0.06799
6: x6 5.921 0.02727 0.06106 0.08527
7: x7 6.647 0.02059 0.04325 0.06416
8: x8 1.316 0.06916 0.02949 0.01998
Sources: Barnett (1978). Full references: dmai_cite(x).
m <-
dmai_moments(
.data = fit
)
plot(x = m)
7 Your own data
A CSV file with one row per date can be read with data.table::fread(); give the column names with date, x and r, and the benchmark with benchmark (a column, a number or one value per date). Without a benchmark the largest component rate is used at each date, as in version 0.5.0; that component then has a user cost of zero and no weight, so a benchmark above every component rate is the better choice. dmai_app() does all of this without code.
8 References
Barnett, W. A. (1978). The user cost of money. Economics Letters, 1(2), 145–149.
Barnett, W. A. (1980). Economic monetary aggregates: an application of index number and aggregation theory. Journal of Econometrics, 14(1), 11–48.
Diewert, W. E. (1976). Exact and superlative index numbers. Journal of Econometrics, 4(2), 115–145.