Snapshot 1: The parameter estimates, their standard errors, and

-values in the fitted regression with true parameters

are shown. Due to model misspecification, the standard errors are too small, and the

-values falsely suggest the coefficient

is nonzero while the estimate for

with a

-value of about 6% is borderline.

Snapshot 2: The residual dependency plot is flat, suggesting model adequacy. Looking at this plot more carefully, we do see a nonrandom pattern, but it is less evident than in the Poincaré plot.

Snapshot 3: Comparison of the estimated and theoretical correlation functions. The parameter

is estimated by nonlinear least squares.

Snapshot 4-6: In the next 3 shapshots,

and the other settings remain the same. In this case the effect of model misspecification increases and is again detected better in the Poincaré plot than in the residual dependency plot. Both regression parameters are erroneously reported as very significant.

Residual dependency plots for checking regression fits are discussed in most regression textbooks as for example ([1, 2]).

Lagged scatterplots are sometimes called Poincaré plots ([3, 4]).

See [5] for further discussion of hidden correlation in regression.

[1] W. S. Cleveland,

*Visualizing Data*. Summit, NJ: Hobart Press, 1993.

[2] S. J. Sheather,

*A Modern Approach to Regression with R*, New York: Springer, 2009.

[3] D. Kaplan and L. Glass,

*Understanding Nonlinear Dynamics*, New York: Springer, 1995.