By definition of rate constant itself (for example think of the Arrhenius equation k =A×exp( -Ea/RT) ), it cannot be negative.

I think the problem lies in your experimental design, i.e. the design matrix you are using to estimate slope and intercept using least squares method, the problem is called " Multicollinearity " and can occur in your case if the unit vector column is nearly collinear to your experimental variable X column vector, this usually happens if the chosen X values, i.e the design points in input variable space are very close to one another.

I suggest you look up the concept of Multicollinearity from any standard text on Regression analysis or online available material, and determine the condition number of your current design matrix, a condition number of 100 or more is an indication of significant multicollinearity in design, there are several online tools also that gives a calculated condition number once the design matrix is input into the software.

Suppose your model is Y= c + m*X, and your experimental X values are x1,x2,....xp then the design matrix is the p by 2 matrix X = [ 1 x1 ; 1 x2 ; .... 1 xp ] , where the terms separated by ; are rows of the matrix.

I suggest you go with an experimental design with more spread in X values, or better use the orthogonal polynomial regression approach, this basically provides a set of equispaced points in X space, by centering and scaling, so that the resulting X column is orthogonal to the intercept column and hence no multicollinearity, the table of orthogonal polynomials , for any order polynomial model e.g linear, quardratic or higher order can be found online or in standard texts on regression modeling.

Debopam Ghosh Thank you very much! I will look in to the multicollinearity part.

no, pseudo second order reaction rate constant never negative .

First, what you call "pseudo-first order constant" is a constant in an empirical hyperbolic function - a constant that has no physical meaning. Second, the hyperbolic function was, is, and will be non-linear regardless of the transform used. Third, it may be negative provided that errors were made in kinetic tests and/or numerical calculations were incorrectly performed. However, I would advise against publishing such nonsensical results - which unfortunately happens.