# Reason to worry if the emp. residual distribution is more dense around zero compared to a theoretical normal?

My goal was to evaluate, if a marketing scheme did benefit or not. I observed data about the price ($P_t$) of a specific product over time.

Since my dependend variable is only defined on $[0,\infty)$ I usually should use a link-function of some sort but ignoring this, as many emp. studies do, I used the framework of GAM (generalized additive model) to fit the influence of time ($T$) (given by the yearday) and number of sales ($S_t$) in a more flexible way since I expected that both dependencies are non-linear (or at least, it is unlikely that they can be modeled in a linear sense).

The dummy-variable ($D$), which indicates if the marketing scheme did occur, is also included in the model.

After fitting a model like:

$P_t = f_1(S_t,D) + f_2(T) + e_t$

using P-Splines for $f_1$ and a cyclic-spline for $f_2$ I check my residuals $\hat{e_t}$ for the underlying normal assumption.

This is what I get:

It obviously tells me, that the variance of my residuals, i.e., $\hat{\sigma}_e$ is less then expected if I assume that my residuals $\hat{e_t}$ are actually normal.

So my question is, if this underdispersion in the GAM-context is in some way problematic? If so, what problems do I get if my goal is to figure out if there is a significant difference in $f_1(S_t,D=0)$ compared with $f_1(S_t,D=1)$?

• I guess the "empirical" in your graph should correspond to "blue" and to "red" again. Then, it looks like you may have a symmetric around zero distribution (=no skewness) that exhibits excess kurtosis. As a start, you can see this post stats.stackexchange.com/questions/11821/… – Alecos Papadopoulos Jan 17 '14 at 14:00
• Oh thats true. I did not noticed that the titel said red in both cases. The empirical is indeed the blue one. This might seem a silly question but the GAM-estimator, i.e., using splines, kind of looks like the OLS-Estimator. So I would, without explicitly checking it, assume, that the Gauss-Markov-Theorem also applies here. So as long as my model is correctly specified and I've homoscedastic errors I should be fine. If not I should use some sort of heteroscedastic corrected version of the standard errors? In the context of GAM, what should be used? Bootstrap? – Druss2k Jan 17 '14 at 15:26
• I forgot to mention: Since $f_1$ corresponds to the decreasing marginal revenue relationship (the number of sales increases if the prices goes down), I'm really interested in the area beneath $f_1(S_t,D=0)$ compared to the area beneath $f_(S_t,D=1)$. To be confident, that the difference, $f_(S_t,D=1)-f_1(S_t,D=0)$ is non negative, I would need to use the confidence bands as margins. For instance, as a worst case scenario, I could use the lower margin of $f_1(S_t,D=0)$ and compare it with the lower margin of $f_1(S_t,D=1)$. Bottom line, I would need some sort of consistent confidence bands. – Druss2k Jan 17 '14 at 15:49
• On what basis do you say that the variance is smaller than you'd expect from a normal? Just from the image? (That looks both more peaked and heavier tailed; that is, it looks leptokurtotic) – Glen_b Jan 18 '14 at 5:28