Drift Rate
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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, stochastic drift is the change of the average value of a stochastic (random) process. A related concept is the drift rate, which is the rate at which the average changes. For example, a process that counts the number of heads in a series of n fair
coin toss A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by a ...
es has a drift rate of 1/2 per toss. This is in contrast to the random fluctuations about this average value. The stochastic mean of that coin-toss process is 1/2 and the drift rate of the stochastic mean is 0, assuming 1 = heads and 0 = tails.


Stochastic drifts in population studies

Longitudinal studies A longitudinal study (or longitudinal survey, or panel study) is a research design that involves repeated observations of the same variables (e.g., people) over long periods of time (i.e., uses longitudinal data). It is often a type of observation ...
of secular events are frequently conceptualized as consisting of a trend component fitted by a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
, a cyclical component often fitted by an analysis based on
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
s or on a
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
, and a random component (stochastic drift) to be removed. In the course of the
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
, identification of cyclical and stochastic drift components is often attempted by alternating autocorrelation analysis and differencing of the trend. Autocorrelation analysis helps to identify the correct phase of the fitted model while the successive differencing transforms the stochastic drift component into
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...
. Stochastic drift can also occur in
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, s ...
where it is known as
genetic drift Genetic drift, also known as random genetic drift, allelic drift or the Wright effect, is the change in the Allele frequency, frequency of an existing gene variant (allele) in a population due to random chance. Genetic drift may cause gene va ...
. A ''finite'' population of randomly reproducing organisms would experience changes from generation to generation in the frequencies of the different genotypes. This may lead to the fixation of one of the genotypes, and even the emergence of a new species. In sufficiently small populations, drift can also neutralize the effect of deterministic
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the Heredity, heritable traits characteristic of a population over generation ...
on the population.


Stochastic drift in economics and finance

Time series variables in economics and finance — for example,
stock price A share price is the price of a single share of a number of saleable equity shares of a company. In layman's terms, the stock price is the highest amount someone is willing to pay for the stock, or the lowest amount that it can be bought for. B ...
s,
gross domestic product Gross domestic product (GDP) is a monetary measure of the total market value of all the final goods and services produced and rendered in a specific time period by a country or countries. GDP is often used to measure the economic performanc ...
, etc. — generally evolve stochastically and frequently are
non-stationary In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ...
. They are typically modelled as either trend-stationary or
difference stationary In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is ...
. A trend stationary process evolves according to :y_t = f(t) + e_t where ''t'' is time, ''f'' is a deterministic function, and ''e''''t'' is a zero-long-run-mean stationary
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. In this case the stochastic term is stationary and hence there is no stochastic drift, though the time series itself may drift with no fixed long-run mean due to the deterministic component ''f''(''t'') not having a fixed long-run mean. This non-stochastic drift can be removed from the data by regressing y_t on t using a functional form coinciding with that of ''f'', and retaining the stationary residuals. In contrast, a unit root (difference stationary) process evolves according to :y_t = y_ + c + u_t where u_t is a zero-long-run-mean stationary random variable; here ''c'' is a non-stochastic drift parameter: even in the absence of the random shocks ''u''''t'', the mean of ''y'' would change by ''c'' per period. In this case the non-stationarity can be removed from the data by first differencing, and the differenced variable z_t = y_t - y_ will have a long-run mean of ''c'' and hence no drift. But even in the absence of the parameter ''c'' (that is, even if ''c''=0), this unit root process exhibits drift, and specifically stochastic drift, due to the presence of the stationary random shocks ''u''''t'': a once-occurring non-zero value of ''u'' is incorporated into the same period's ''y'', which one period later becomes the one-period-lagged value of ''y'' and hence affects the new period's ''y'' value, which itself in the next period becomes the lagged ''y'' and affects the next ''y'' value, and so forth forever. So after the initial shock hits ''y'', its value is incorporated forever into the mean of ''y'', so we have stochastic drift. Again this drift can be removed by first differencing ''y'' to obtain ''z'' which does not drift. In the context of
monetary policy Monetary policy is the policy adopted by the monetary authority of a nation to affect monetary and other financial conditions to accomplish broader objectives like high employment and price stability (normally interpreted as a low and stable rat ...
, one policy question is whether a central bank should attempt to achieve a fixed growth rate of the
price level The general price level is a hypothetical measure of overall prices for some set of goods and services (the consumer basket), in an economy or monetary union during a given interval (generally one day), normalized relative to some base set. ...
from its current level in each time period, or whether to target a return of the price level to a predetermined growth path. In the latter case no price level drift is allowed away from the predetermined path, while in the former case any stochastic change to the price level permanently affects the expected values of the price level at each time along its future path. In either case the price level has drift in the sense of a rising
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
, but the cases differ according to the type of non-stationarity: difference stationarity in the former case, but trend stationarity in the latter case.


See also

*
Secular variation The secular variation of a time series is its long-term, non-periodic variation (see '' Decomposition of time series''). Whether a variation is perceived as secular or not depends on the available timescale: a variation that is secular over a times ...
*
Decomposition of time series The decomposition of time series is a statistical task that deconstructs a time series into several components, each representing one of the underlying categories of patterns. There are two principal types of decomposition, which are outlined belo ...


References

* Krus, D.J., & Ko, H.O. (1983) Algorithm for autocorrelation analysis of secular trends. ''Educational and Psychological Measurement,'' 43, 821–828. {{usurped,
(Request reprint).
} * Krus, D. J., & Jacobsen, J. L. (1983) Through a glass, clearly? A computer program for generalized adaptive filtering. ''Educational and Psychological Measurement,'' 43, 149–154 Time series Mathematical finance