Editing Linear trend estimation (section)

==Data as trend and noise==
To analyze a (time) series of data, it can be assumed that it may be represented as trend plus noise:

:<math>y_t = at + b + e_t\,</math>

where <math>a</math> and <math>b</math> are unknown constants and the <math>e</math>'s are randomly distributed [[errors and residuals|errors]]. If one can reject the null hypothesis that the errors are [[unit root|non-stationary]], then the non-stationary series <math>\{y_t\}</math> is called [[trend-stationary process|trend-stationary]]. The least-squares method assumes the errors are independently distributed with a normal distribution. If this is not the case, hypothesis tests about the unknown parameters <math>a</math> and <math>b</math> may be inaccurate. It is simplest if the <math>e</math>'s all have the same distribution, but if not (if some have [[heteroskedasticity|higher variance]], meaning that those data points are effectively less certain), then this can be taken into account during the least-squares fitting by weighting each point by the inverse of the variance of that point.

Commonly, where only a single time series exists to be analyzed, the variance of the <math>e</math>'s is estimated by fitting a trend to obtain the estimated parameter values <math>\hat a</math> and <math>\hat b,</math> thus allowing the predicted values 
:<math>\hat y =\hat at+\hat b</math>

to be subtracted from the data <math>y_t</math> (thus ''detrending'' the data), leaving the [[errors and residuals|residuals]] <math>\hat e_t</math> as the ''detrended data'', and estimating the variance of the <math>e_t</math>'s from the residuals — this is often the only way of estimating the variance of the <math>e_t</math>'s.

Once the "noise" of the series is known, the significance of the trend can be assessed by making the [[null hypothesis]] that the trend, <math>a</math>, is not different from 0. From the above discussion of trends in random data with known [[variance]], the distribution of calculated trends is to be expected from random (trendless) data. If the estimated trend, <math>\hat a</math>, is larger than the critical value for a certain [[significance level]], then the estimated trend is deemed significantly different from zero at that significance level, and the null hypothesis of a zero underlying trend is rejected.

The use of a linear trend line has been the subject of criticism, leading to a search for alternative approaches to avoid its use in model estimation. One of the alternative approaches involves [[unit root]] tests and the [[cointegration]] technique in econometric studies.

The estimated coefficient associated with a linear trend variable such as time is interpreted as a measure of the impact of a number of unknown or known but immeasurable factors on the dependent variable over one unit of time. Strictly speaking, this interpretation is applicable for the estimation time frame only. Outside of this time frame, it cannot be determined how these immeasurable factors behave both qualitatively and quantitatively. 

Research results by mathematicians, statisticians, econometricians, and economists have been published in response to those questions. For example, detailed notes on the meaning of linear time trends in the regression model are given in Cameron (2005);<ref name=":0">{{cite news|url=http://highered.mcgraw-hill.com/sites/dl/free/0077104285/160071/Chapter_7.pdf|title=Making Regression More Useful II: Dummies and Trends|access-date=June 17, 2012}}</ref> Granger, Engle, and many other econometricians have written on stationarity, unit root testing, co-integration, and related issues (a summary of some of the works in this area can be found in an information paper<ref>{{cite news|url=http://www.kva.se/Documents/Priser/Nobel/2003/sciback_ek_en_03.pdf|title=The Royal Swedish Academy of Sciences|date=8 October 2003|access-date=June 17, 2012}}</ref> by the Royal Swedish Academy of Sciences (2003)); and Ho-Trieu & Tucker (1990) have written on logarithmic time trends with results indicating linear time trends are special cases of [[cycle (sequence)|cycle]]s.

=== Noisy time series ===
It is harder to see a trend in a noisy time series. For example, if the true series is 0, 1, 2, 3, all plus some independent normally distributed "noise" ''e'' of [[standard deviation]]{{nbsp}}''E'', and a sample series of length 50 is given, then if ''E''{{nbsp}}={{nbsp}}0.1, the trend will be obvious; if ''E''{{nbsp}}={{nbsp}}100, the trend will probably be visible; but if ''E''{{nbsp}}={{nbsp}}10000, the trend will be buried in the noise.

Consider a concrete example, such as the [[global surface temperature]] record of the past 140 years as presented by the [[Intergovernmental Panel on Climate Change|IPCC]].<ref name=":1">{{cite news|url=http://www.grida.no/publications/other/ipcc_tar/?src=/climate/ipcc_tar/wg1/figspm-1.htm|title=IPCC Third Assessment Report – Climate Change 2001 – Complete online versions|access-date=June 17, 2012|url-status=dead|archive-url=https://web.archive.org/web/20091120181301/http://www.grida.no/publications/other/ipcc_tar/?src=%2Fclimate%2Fipcc_tar%2Fwg1%2Ffigspm-1.htm|archive-date=November 20, 2009}}</ref> The interannual variation is about 0.2{{nbsp}}°C, and the trend is about 0.6{{nbsp}}°C over 140 years, with 95% confidence limits of 0.2{{nbsp}}°C (by coincidence, about the same value as the interannual variation). Hence, the trend is statistically different from 0. However, as noted elsewhere,<ref name=":2" /> this time series doesn't conform to the assumptions necessary for least-squares to be valid.