Stepwise regression is generally an unwise approach, even though commonly used and published. For supported inference, evaluating model support and model fit, the canned ArcGIS tools are not a good platform for specifying a statistical model. A one-size fits all approach to fitting regression models is never a good idea.
The message regarding the foibles of stepwise regression is currently being pushed by some of the current greats in statistics/biometrics (Harrell, Bolker, Bivand, Cressie, Hastie, Friedman, …). I believe that these limitations apply equally to stepwise AIC because it is a forward selection procedure. Keep in mind that the original intent behind AIC was hypothesis testing, using competing models, not feature selection.
Some of the identified issues with stepwise approaches are:
- The r-square values are inflated in relation to the sample population
- The test-statistics do not retain the correct distribution (F,
chi-squared) which results in a deflation of p-values and standard
- Bonferroni corrections are overly conservative
- Regression coefficients are upwards biased and marginal variables
are more likely to be included if coefficient is
overestimated and excluded if underestimated.
- Removing "insignificant" variables sets their coefficients to zero,
which may be implausible.
The general advice is to fit a full model, only remove parameters if α > 0.5, if the sign (+/-) is not sensible or if a coefficient of 0 is plausible. One should account for colinearity by combining variables (via mathematical function or PCA reduction).
If one really needs to reduce parameter space (eg., for estimation purposes) then a step down (backwards selection) approach is highly recommend. Backwards elimination methods can be implemented in R using the “fastbw” function in the "rms" library. As recommend by @whuber, I would highly recommend exploring a LASSO regression. If you are comfortable letting the data fit your model, you could implement a Bayesian Model Averaging approach. A Metropolis Hastings algorithm is a good choice when the parameter space is large. The R packages "BMA" and "BMS" have fairly simple implementations for Bayesian Model Selection.
Here is a simple method, implemented in R, for selecting parameter(s) with a p < 0.05 using a generalized linear model. This could be used to remove spurious independent variables. It would be easy to adapt this for an Ordinary Least Squares but, generally, in this case, I would use a maximum likelihood. I would think of this as more of a grab-bag approach than stepwise. However, I would evaluate the parameter(s) coefficents before applying an automatic selection procedure such as this.
Create some example data with one highly significant variable (x4)
x1 <- rnorm(10); x2 <- rnorm(10); x3 <- rnorm(10)
y <- rnorm(10)
x4 <- y + 5
( dat <- as.data.frame(cbind(x1,x2,x3,x4,y)) )
Specify a Generalized Linear Model (GLM)
model <- glm(y ~ x1 + x2 + x3 + x4, data = dat)
Use p-value from t-test to select supported parameter(s)
( sel.x <- summary(model)$coeff[-1,4] < 0.05 )
( sig.x <- names(sel.x)[sel.x == TRUE] )
Specify new GLM with selected parameters
sig.model <- glm(reformulate(sig.x, response="y"), data = dat)