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I fit a multiple GLS model with time series as a response and explanatory variables. I previously removed the seasonality and trends by using . I found that the correlation structure of the residuals was AR(3), so I totally removed the autocorrelation from the residuals.The coefficient of the explanatory variables that finally remained in the model are all different to zero (p-values <0.01).

My question is how should I interpret the coefficient considering that I fit the model with transformed variables? How should I interpret positive and negative coefficients?

My variables are centered in zero with positive and negative values so, for instance, if my coefficient is negative and the value of my explanatory variable is negative, then the response variable will be positive. Should I interpret this as a positive effect? or Should I interpret it the other way around if my coefficient is positive?

Is there a way to translate the interpretation in terms of the real scale of variables?

Any help will be appreciated

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  • $\begingroup$ A positive coefficient means that the variables tend to move in the same direction and this is always called a "positive effect", irrespective of whether the values of the variables involved are negative or positive, in any combination. Analogously for a negative coefficient. The effect does not change character or name when you move to the real scale of the variables, although it may be "swamped" by the trend, season, drift, and persistence components. Given that you have taken out all these features of your variables, try to describe verbally what exactly is left that you are measuring. $\endgroup$ – Alecos Papadopoulos Sep 10 '13 at 12:50
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    $\begingroup$ It appears from your description you are performing a sequence of steps rather than an integrated approach which is the preferred method. Furthermore the desire to "interpret coefficients" in a multivariate model where data has been simply observed can be quite tricky do to possible collinearity in the input series. $\endgroup$ – IrishStat Sep 18 '13 at 11:53

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