The data available in the form of time series {y_t: t = 1, 2, …, n} are the bivariate data, where t is the independent variable and the variable quantity y is the dependent variable. From this data we have to obtain the linear trend that suits to the time series. According to linear regression model, we have to obtain the linear trend model y_t = α + βt + u_t, where t = 1, 2, …, n and u_t is an error variable. According to the least squares method, we determine the values of the constants α and β in such a manner that the sum of the squares of error variable, i.e., Σe² = Σ(y_t – α – βt)² is minimised. If ‘a’ and ‘b’ denote the estimated values of a and p respectively then ‘α’ and ‘β’ can be obtained by the following formulae:

\(b =\frac{ nΣty−(Σt)(Σy)}{nΣt^2−(Σt)^2}\) and a = ȳ – bt̄

where, ȳ = \(\frac{Σy}{n}\); t̄ = \(\frac{Σt}{n}\); n = No. of observations
Thus, the fitting of trend line is given by ŷ_t = a + bt. From this trend line we can forecast the variable quantity y of the time series for the values of t following the time duration after t = n.

We can obtain the estimates of trend values for all the terms of the time series by this method. If for the given time series there is no linear trend, this method is not useful.