What is in a linear time trend?
Posted: Fri May 13, 2011 12:26 am
y = a + b.T where T is a time variable.
What do you think about this passage I un Wikipedia
Although this linear trend line appears most often in statistical and econometric models, it also receives many valid critics that lead to the search for more creative approaches to avoid its use in model estimation. One of the novel approaches involve unit root tests and cointegration technique in econometric studies. In fact,when a linear time trend (represented by the variable t*, t*+1, t*+2,...,t*+n; with t* being the time base, and n the number of observations), the estimated coefficient associated with this linear time trend variable is interpreted as a measure of the impact of a number of unknown or known but unmeasurable factors on the dependent variable over one unit of time. Strictly speaking, that interpretation is applicable for the estimation time frame only. Outside that time frame, one does not know how those unmeasurable factors behave both qualitatively and quantitatively. Furthermore, the linearity of the time trend poses many questions: (i) why should it be linear? (ii) if the trend is non-linear then under what conditions its inclusion does not influence the magnitude as well as the statistical significance of the estimates of other parameters in the model? (iii) the commonly accepted law of nature, especially in economic models, is "what goes up must come down one day, and the reverse is also true" so when the inclusion of a linear time trend in a model obviously violates this law when n tends to infinity ? (iv) problem when both dependent and explanatory variables move with a linear time trend thus a spurious relationship exists in the model; and (v) collinearity problem due to high correlation between the linear time trend and non-trend explanatory variables leading to serious bias in estimates of model parameters.
Research results of mathematicians, statisticians, econometricians, economists have been published in journals in response to those questions (eg. the work of John Blatt (mathematical meaning of a linear time trend), C. Granger and many other econometricians (on unit root testing, co-integration and related issues), Ho-Trieu & Tucker (on logarithmic time trend which is non-linear with results alluding to a proof rejecting the existence of linear trend, and a linear trend is just a misnomer of a special form of a cyclical trend when the periodicity of the cycle is large relative to the time interval over which data is collected for model estimation (please see http://ideas.repec.org/a/ags/remaae/12288.html for further details).
What do you think about this passage I un Wikipedia
Although this linear trend line appears most often in statistical and econometric models, it also receives many valid critics that lead to the search for more creative approaches to avoid its use in model estimation. One of the novel approaches involve unit root tests and cointegration technique in econometric studies. In fact,when a linear time trend (represented by the variable t*, t*+1, t*+2,...,t*+n; with t* being the time base, and n the number of observations), the estimated coefficient associated with this linear time trend variable is interpreted as a measure of the impact of a number of unknown or known but unmeasurable factors on the dependent variable over one unit of time. Strictly speaking, that interpretation is applicable for the estimation time frame only. Outside that time frame, one does not know how those unmeasurable factors behave both qualitatively and quantitatively. Furthermore, the linearity of the time trend poses many questions: (i) why should it be linear? (ii) if the trend is non-linear then under what conditions its inclusion does not influence the magnitude as well as the statistical significance of the estimates of other parameters in the model? (iii) the commonly accepted law of nature, especially in economic models, is "what goes up must come down one day, and the reverse is also true" so when the inclusion of a linear time trend in a model obviously violates this law when n tends to infinity ? (iv) problem when both dependent and explanatory variables move with a linear time trend thus a spurious relationship exists in the model; and (v) collinearity problem due to high correlation between the linear time trend and non-trend explanatory variables leading to serious bias in estimates of model parameters.
Research results of mathematicians, statisticians, econometricians, economists have been published in journals in response to those questions (eg. the work of John Blatt (mathematical meaning of a linear time trend), C. Granger and many other econometricians (on unit root testing, co-integration and related issues), Ho-Trieu & Tucker (on logarithmic time trend which is non-linear with results alluding to a proof rejecting the existence of linear trend, and a linear trend is just a misnomer of a special form of a cyclical trend when the periodicity of the cycle is large relative to the time interval over which data is collected for model estimation (please see http://ideas.repec.org/a/ags/remaae/12288.html for further details).