In a linear model including ANOVA one can test a trend (e.g., linearity, quadratic effect, etc.) among the ordered effects (regression coefficients or factor levels) through assigning proper weights. For example, suppose that we have a time series regression model
where $Y$ is a vector of length $n$, $X_i$ is an explanatory variable of length $n$ ($i=1,2,3,4$), and $\epsilon$ is the residual vector. The linear and quadratic trends among four equally-spaced (and ordered) effects $\beta_1, \beta_2, \beta_3, \beta_4$ can be tested with weights of -3, -1, 1, 3, and 1, -1, -1, 1 respectively. When those effects are not equally-spaced, such weights can be obtained through the coefficients of orthogonal polynomials.
My understanding is that typically the exponential trend among some sequential effects can be tested through log-transforming the response variable. However, the situation I'm dealing is not directly about the response variable, but the effect estimates (or regression coefficients, e.g., 30 ordered effects similar to a linear model above). So the challenge I'm facing here is, how to find the weights when testing for an exponential trend, when the effects are equally- or unequally-spaced?
P.S. There are two time scales involved here that may cause some confusions. At the local time level, each explanatory variable $X_i$ in the time series regression model codes for a causal event that lasts for a short period of time. At the higher level of time, there is the sequence of those causal events and their estimated effects $\beta_1$, $\beta_2$, .... It is the exponential trend or saturation effect among those estimated effects that is of interest to me. For a linear trend, I could perform a general linear hypothesis testing with appropriate weights for those effects. However, I'm not so sure whether it's feasible to do the same for an exponential trend (or saturation effect).