Let's get started with the spurious regression then, which I have seen implemented in the real world far too often. Here we simply build a linear model and examine the results:

    > fit.lm <- lm(Temp ~ CO2, data = climate)

    > summary(fit.lm)

    Call:
    lm(formula = Temp ~ CO2, data = climate)

    Residuals:
         Min       1Q  Median      3Q     Max 
    -0.36411 -0.08986 0.00011 0.09475 0.28763 

    Coefficients:
                  Estimate  Std. Error  t value     Pr(>|t|) 
    (Intercept) -2.430e-01   2.357e-02   -10.31   <2e-16 ***
    CO2          7.548e-05   5.047e-06    14.96   <2e-16 ***
    ---
    Signif. codes: 
    0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    Residual standard error: 0.1299 on 93 degrees of freedom
    Multiple R-squared: 0.7063, Adjusted R-squared: 0.7032 
    F-statistic: 223.7 on 1 and 93 DF, p-value: < 2.2e-16

Notice how everything is significant, and we have an adjusted R-squared of 0.7. OK, they are highly correlated but this is all meaningless as discussed by Granger and Newbold (1974). Again, I have seen results like these presented in meetings with many people with advanced degrees, and I had to be the bad guy and challenge the results.

We can plot the serial correlation, starting with a time series plot of the residuals, which produce a clear pattern:

    > plot.ts(fit.lm$residuals)

The output of the preceding code is as follows: 

Then we create an ACF plot showing significant autocorrelation out to lag 10:

    > acf(fit.lm$residuals)

You can test for autocorrelation by performing the Durbin-Watson test. The null hypothesis in the test is no autocorrelation exists:

    > dwtest(fit.lm)

    Durbin-Watson test

    data: fit.lm
    DW = 0.77425, p-value = 4.468e-12
    alternative hypothesis: true autocorrelation is greater than 0

From examining the plots, it comes as no surprise that we can safely reject the null hypothesis of no autocorrelation. The simple way to deal with autocorrelation is to incorporate lagged variables of the dependent time series and/or to make all the data stationary. We will do that next using vector autoregression to identify the appropriate lag structure to incorporate in our causality efforts.