How to account for autocorrelation in a multiple linear regression?

Question

I'm trying to create a multiple linear regression with water temperature change as my response variable and four numeric explanatory variables (that influence temperature change). Each numeric variable was recorded prior to the temperature change (e.g. amount of ice added into the water).

My problem is that I do not have the statistical or R background to take into account autocorrelation, given that a previous observation would likely have an influence on a subsequent observation.

My original model is as follows, and is an MLRM with backward selection (however autocorrelation has not been taken into account):

lm(y ~ x1 + x2 + x3 + x4)

Any and all help would be greatly appreciated!

Answer 1

One of the OLS assumptions is "no perfect multicollinearity". So, if you don't have a perfect linear combination among your independent variables (IV), you do can estimate a OLS regression model.

Including a new IV that affect the DV and is uncorrelated with the other IVs is always good since it reduces error variance. However, if the new IV is somehow correlated with one or more IVs, the effect of increasing the error variance due to collinearity might be stronger.

One way to assess if the collinearity is harmful is evaluating the condition index. See example below:

x1 = rnorm(1000, 1, 1)
x2 = rnorm(1000, 2, 1)
x3 = rnorm(1000, 3, 1)
x4 = 3*x3 + rnorm(1000, 1, 0.1)

library(perturb) # colldiag
colldiag(cbind(x1, x2, x3, x4))

## Condition
## Index    Variance Decomposition Proportions
##            intercept x1    x2    x3    x4   
## 1    1.000 0.000     0.017 0.009 0.000 0.000
## 2    2.933 0.000     0.928 0.015 0.000 0.000
## 3    4.766 0.000     0.000 0.755 0.000 0.000
## 4    9.129 0.129     0.051 0.216 0.000 0.000
## 5  289.166 0.870     0.004 0.006 1.000 1.000

Belsley, Kuh, and Welsch (1980) say:

It is suggested that an appropriate means for diagnosing degrading collinearity is the following double condition:

1. A singular value judged to have a high condition index, and which is

2. High variance-decomposition proportions for two or more estimated associated with regression coefficient variances.

They suggest a condition index greater then 30 with variance-decomposition proportions greater than 0.5.

In the example above, you have a condition index of 289.166 associated with two variables (x3 and x4) with variance-decomposition proportion of 1. In this case, it would be advisable to exclude one of these variables from the model.