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I've been working on finding the regression line that will help make predictions on the future population. The data I have is from the past 20 years and every year there has been an increase, the data consists of two variables (x and y) . I found the regression line for polynomials 2,3,4,5,and 6; to find the regression lines I found the normal equations, put them in matrix form and then solve for the coefficients with the use of summations. However in the process I realized that some of those polynomial functions won't work since it will make my numbers negative. What other type of regression can I use to model constant population growth?

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You should use Nonlinear Regression rather than polynomial. It all depends how the trends is ..... Here's an example of increasing trends for postive growth

Population $(Y) = \frac{(c)} { (1 + e^a + bX )} X$ = year. Find your starting value for parameter a,b, and c .

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In general, "future predictions" may be referred to as forecasting. In the forecasting literature, you will typically see strong opposition to the use of polynomial trend models to predict future values. Aside from predicting nonsensical (negative) values, they also invariably diverge to positive or negative infinity. These models can be good for interpolation, but do terrible jobs of predicting long run behavior without continuous and highly vacillating updating.

To obtain positive predictions (focused here on non-dynamical modeling): log-transform the outcome in models so that the exponentiated coefficients are interpreted as rates of change, and the exponentiated predictions are on the population unit-scale.

Alternately: use a generalized linear model. Quasipoisson models are used often for population dynamics as they log transform outcomes and account for the fact that large populations have proportionally greater variability in their dynamics.

Lastly, for forecasting models many have been proposed, even a simple ARMA would provide better and more extrapolations. With an ARMA, it is the forecast interval that diverges rather than the mean. This aligns with our intuition that the uncertainty of the future necessarily dominates any statistical confidence that can be garnered with data.

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