Spline model with a constant I'm working on a reproduction of a model and I am running into some issues with the wording of the model and implementing it into R. The model that I am working with is an Additive AutoRegressive model, nonlinear in this case. The paper mentions that the final model that they chose was a "Spline model with a constant and with two lags, where a B-spline base of order 3 and 25 knots was chosen." The package that they used to generate this model was the "mgcv" package.
That's the background now here's what I have so far. I have the following dataset,




y
x.1
x.2




0.7
NA
NA


0.4
0.7
NA


-0.2
0.4
0.7


0.5
-0.2
0.4




Where I know the values of y, x.1 is the first time lag, and x.1 is a second time lag. I can put this data into R and create a model with the inputs that are listed as a spline model using the GAM function from the R package mgcv. Where I define k=25 knots, this can be changed to 2 since I only provide a small dataset, bs="bs" defines the B-spline method and s() is simply a smoothing function placement for both lags.
test2<- gam(y~ s(x.1, bs="bs", k=25)+s(x.2, bs="bs", k=25), 
            data=full_set_1)
test2

I then get the following output when I run the full dataset, which is what I expect however I'm having a bit of an issue in not understanding how do I define the B-spline base order of 3? And secondly, I'm also not certain by the wording of "Spline model with a constant" and how would I implement a constant to my model below?
Family: gaussian 
Link function: identity 

Formula:
y ~ s(x.1, bs = "bs", k = 25, fx = FALSE) + s(x.2, bs = "bs", 
    k = 25, fx = FALSE)

Estimated degrees of freedom:
23.6 22.4  total = 46.99 

GCV score: 0.01041977     

Thank you for the help.

I presume the "constant" just means the model intercept or constant term. The intercept/constant is implied in R's formula notation, but you can state it explicitly via
m <- gam(y ~ 1 + s(x.1) + s(x.2), data = df, method = "REML")

but it isn't necessary to specify it explicitly as it is always included unless you suppress it with 0 + or - 1 added to the formula.
As for using cubic (order 3) B splines, this is the default, but you can specify this via the m argument. The default is m = c(3,2) which is a cubic B spline basis with 2nd order derivative penalty.
