Another hot July, so it's time for another post on climate change.  We've all heard how el Nino and la Nina events affect temperature data in any given year, but we rarely see those events included in any kind of time series.  Usually it's just a linear time trend or some kind of smoothed trend.  So let's take a look at how those events have influenced annual temperatures per NOAA data.  I selected annual temperature data from 1980 thru 2022 along with NOAA's assessment as to whether each corresponding year was an el Nino year, a la Nina year or a neutral year.  The model is an ARMAX(1,0) type with three exogenous variables.  The dependent variable is temperature anomaly (in Celsius).  Along with a constant (which captures a "neutral" year) I've included a dummy variable for el Nino, a dummy variable for la Nina and a linear time trend.  Without further ado, here are the results:

The "phi_1" parameter is the AR(1) component of the ARMAX model.  Basically it shows that about 40% of each year's temperature persists into the next year.  Including this parameter ensures that any autocorrelation between years is scrubbed out and thereby ensuring statistical independence.  All of the coefficients have the expected sign and all are statistically significant.  A Chi-square test cannot reject the null of a normal distribution for the residuals.  And this graph shows no evidence of remaining autocorrelation in the residuals:

This graph plots actual temperatures and predicted temperatures:

Overall the fit is pretty good, which shouldn't be a surprise given that the Adjusted R-square is 0.91 (see summary output in the first slide).  

Time to play a "what if" game.  What if 2023 turns out to be a neutral?  In that case the point estimate would be an anomaly of 0.99 degrees with a 95% confidence band between 0.86 and 1.13 degrees, as shown here:

This would make 2023 the second hottest year on record, falling just short of 2016's record of 1.03 degrees.  Now, what if 2023 turns out to be an el Nino year, which seems likely?  In that case the point estimate would be an anomaly of 1.05 degrees with a 95% confidence interval between 0.92 and 1.19:

BTW, the jigsaw pattern in these graphs is due to the effects of el Nino, la Nina and neutral years.