Provide conditions under which the mean function is linear but has a negative coefficient

Depending on your dependent/outcome variable, a negative value for your constant/intercept should not be a cause for concern. This simply means that the expected value on your dependent variable will be less than 0 when all independent/predictor variables are set to 0. For some dependent variables, this would be expected. For example, if the mean value of your dependent variable is negative, it would be no surprise whatsoever that the constant is negative; in fact, if you got a positive value for the constant in this situation, it might be cause for concern (depending on your independent variables).Even if your dependent variable is typically/always positive (i.e., has a positive mean value), it wouldn't necessarily be surprising to have a negative constant. For example, consider an independent variable that has a strongly positive relationship to a dependent variable. The values of the dependent variable are positive and have a range from 1-5, and the values of the independent variable are positive and have a range from 100-110. In this case, it would not be surprising if the regression line crossed the x-axis somewhere between x=0 and x=100 (i.e., from the first quadrant to the fourth quadrant), which would result in a negative value for the constant.The bottom line is that you need to have a good sense of your model and the variables within it, and a negative value on the constant should not generally be a cause for concern. Typically, it is the overall relationships between the variables that will be of the most importance in a linear regression model, not the value of the constant.