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Trend detection is model-dependent. We analyze this for auto-correlated temperature time series. By the comparison of two different models which both describe the same stochastic process, we can introduce the distinction between the observed trend and the intrinsic trend. We transform a model with correlated noise into the lagged dependent variable (LDV) model with white noise. Although the commonly used climate dynamical model usually contains the LDV, existing trend studies barely consider it. For the LDV model, the auto-correlation effect on trend detection not only induces the stochastic trend, but also leads to an additional trend by accumulating the intrinsic trend. The intrinsic trend exclusive of the auto-correlation effect should be more likely related to the external forcing like anthropogenic factors, which is actually the trend of main interest. By applying the LDV model to the monthly mean anomalies at the Potsdam, Hamburg, and Frankfurt stations, it is found that 87%, 78%, and 75%, respectively, of the observed trends are the intrinsic trend, which may be more relevant to anthropogenic factors; and the rest should be due to auto-correlation. Analysis of two additional Chinese stations, namely the Guangzhou and Turpan stations, demonstrates the general applicability of the LDV model for different climate zones. Our study refreshes the current understanding of the observed trend and the auto-correlation effect, which is expected to be beneficial in the exploration of the underlying mechanism of global warming. [ABSTRACT FROM AUTHOR]