Parameters and Relations of Stochastically Lagged and Disaggregative Time Series

A model is constructed for three related time series: an inflow, a stock, and an outflow.
The inflow and outflow time series are related via a truly stochastic distributed lag and
the three time series are viewed as cumulations of a random number of units of randomly
determined size. Among the interesting results are the serial correlations of the stock and
outflow time series viewed singly and the cross and cross serial correlations between outflow
in t and stock at t -1, outflow in t, and inflow in t -i, as well as between stock in t
and inflow in t -i. These relations and others make it possible to evaluate the comparative
advantage of lead series in forecasting lag series and as such may be of methodological value
for the evaluation of stabilization policies. The model may be viewed as a description of
several inflow-stock-outflow phenomena: trade credit and consumer credit processes,
demand deposits of commercial banks, population and labor force, the formation and
decay of aggregates of capital projects, and some aspects of the income-expenditure process.
1. INTRODUCTION
THISPAPER develops a model of a stochastic process that gives rise to three time
series : an inflow, a stock, and an outflow. These series come into being as sums of
subcomponents of differing magnitudes, where the number of subcomponents in
the sums differs both between series cross sectionally, and within series over time.
The inflow time series is simply the sum of units of unequal size which come into
being in period t. The stock time series is the sum of units of unequal size which
have come into being in periods preceding t and have not yet vanished in t. The
outflow series is the sum of the unequal magnitudes of all those units which came
into being in periods preceding t and which cease to exist in t. Additionally, the
period between the time a unit comes into being and the time it vanishes is a random
variable. Time series that are formed in such a fashion are encountered in both
microeconomic and macroeconomic phenomena. Credit sales, receivables and
collections of a firm, as well as new loans made, loans outstanding, and loans repaid
of a commercial bank, provide microeconomic examples. Income: cash balances
and expenditures for a group of individuals, new capital investments, capital
stock and depreciated capital for a group of firms, as well as accessions to the labor
force, labor force outstanding, and secessions from the labor force (retirements and
deaths), may serve as macroeconomic examples. Often there arises the need to
evaluate the variances of such time series, their cross correlations as well as their
serial and cross serial correlations. Such a need cannot be satisfied by currently
available distributed lag models since the lag in available models is deterministic
and the approach aggregative.' Such a need, however, may be satisfied by the model
developed in this paper, a model which links inflow, stock, and outflow time series
via a stochastic lag applicable to subcomponents of the series