“Has there been any warming yet?”

This question should have been the touchstone where every policy analyst should have started from the beginning. The corresponding scientific question should have started with, “Has the observed warming been statistically significant? Astonishingly, that did not happen.

This article examines lower troposphere temperature anomalies in conjunction with carbon dioxide levels using classical regression methods. These techniques are accessible to anyone with no more than a minor in undergraduate statistics. The technical question to be asked and answered is, has the observed warming to date been statistically significant?

**Conclusion.** At first blush, the empirical evidence appears to support the assertion that there has been warming. However, after conventional model diagnostics and reformulation, the statistical significance disappears completely and we must conclude that the observed warming does not meet any reasonable criterion of statistical significance. The observed warming could easily be the result of simple chance.

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Why examine temperatures of the lower troposphere when it is the surface temperatures we experience?

- There is no reason to expect any substantive temperature divergence between the surface and the lower troposphere that lies directly above it.
- Temperatures for the lower troposphere have been consistently and regularly measured by satellites over the past thirty-five years. This is very high quality data.

**CO2 and Trend Models.** Let’s start with a picture. To anyone who has investigated global warming, this image should be familiar.

The trend line certainly looks convincing.

Next, let’s look at the regression of temperature anomaly on carbon dioxide levels.

The regression looks strong. 42% R-Squared. With a t-statistic of 17.69 and a significance <0.0001, the coefficient for CO2 certainly looks significant.

**Diagnostics.** What’s left to check? Model assumptions. Are the model’s errors (aka residuals) normally distributed, constant (homoskedastic) and independent?

To cut to the chase, the Durbin-Watson statistic of 0.49 tells us that there is something seriously amiss with this CO2 model. The D-W falls well below the lower bound of 1.65 for D-W from a standard table for >=100 observations. Therefore, we are forced to consider the implications of significant auto-correlation in the model’s residuals.

A check of the ACF (autocorrelation function) and PACF (partial autocorrelation function) on the residuals strongly confirms significant and substantial, even profound, violations of residuals independence at both the first and second lags. The t-statistic for the PACF for the first lag is 15.8 and the second is 5.1. Both have positive signs.

What does this mean? Estimates or inferences that depend on error variance are suspect, at best. That includes any tests of statistical significance. The errors are not independently and identically distributed (iid). We often push the limits on statistical assumptions for normality and constant variance, but not independence.

There is a related point to consider. Carbon dioxide and temperature have both been increasing over this time interval. So, they are correlated. However, does CO2 level do a better job than a trivial time trend model? If CO2 were a useful explanatory variable, we would expect it to perform at least a little better than a trivial trend model.

Does CO2 do any better than case number? No. Model testing shows that the standard error for the CO2 model is 0.1727 while the standard error for a trivial trend (case number) model is 0.1728. Note: lower is better. An improvement of 0.0001 is no difference. This is a tell to experienced modelers that the CO2’s correlation relationship with temperature is spurious.

**Can we even answer the question? **All is not lost, of course. We can still move forward and develop a model to learn whether the warming has been statistically significant. In general, how do we reformulate models when our error terms are riven with autocorrelation? We dust off our Box-Jenkins text and try out an ARIMA model.

Cutting through a pleasant afternoon of model exploration, this parsimonious ARIMA(1,1,0) model emerged as adequate for our purposes.

ARIMA(1,1,0) is a simple change model. The middle number ‘1’ means that this is a first difference (simple change) model. The first ‘1’ means that there is a single autoregressive term. That is to say, each observation is closely related to the previous. The ‘0’ means that there is no moving average (MA) term.

In the output table, the ‘Overall Constant’ can also be referenced as ‘drift’. This term corresponds to the trend coefficient in the simple trend model. The (AR)P(1) term is the autoregression coefficient. In mathematical form, this model says:

- dX(t) = Drift + AR * dX(t-1) + A(t,0,SE)
- dX(t) = 0.0017 – 0.3359 * dX(t-1) + A(t,0,0.1131)

Where,

- dX(t) is the change for time period t.
- Drift is the amount of expected change for every time increment. In this model, that would be 0.0017 degrees per month or 0.20 degrees per decade.
- dX(t-1) references the previous change
- AR is the coefficient that adjusts the previous change. The -0.3359 value indicates strong reversion or rebound.
- A(…) refers to white noise for time period t with mean of zero and standard error of 0.1131.

Compare the much improved ARIMA standard error of 0.1131 with standard error of 0.1728 for the CO2 and simple trend models. The ARIMA model passes model diagnostics for normality, independence and constant variance.

**Where’s the Warming?** The ‘Overall Constant’ is the drift term that corresponds to global warming. If there were warming, this term would show significance. In this model, the monthly change is 0.0017 degrees (0.20/decade) with a standard error of 0.0135. The t-ratio is the ratio of the constant to the standard error.

What does the t-ratio tell us? From the t-statistic we infer the likelihood that a result came about as the consequence of chance. A t-ratio greater than 1.96 (~2.0) indicates a likelihood of <.05 (<5%) that a result was the result of randomness. This is called p-value. Interpretation of t gets more involved when there are fewer than thirty observations.

In academia, the lowest standard for a t-ratio to be considered significant has traditionally been 1.96. In the business world, I have used terms in models with t-ratios as low as 1.2.

The t-ratio for the drift term for this reformulation is (0.0017/0.0135) = 0.1224. That is to say, the drift is not significantly different from zero. No self-respecting analyst who wanted to keep their job would ever consider retaining a term in a model with a t-ratio of 0.1224.

**Does this mean that there has been no global warming?** Not at all. What this does mean, however, is that in the thirty six years since we started taking temperature measurements from satellites, there has been no statistically significant warming in the lower troposphere. This is not even a close call. The observed warming could very very easily be the mere consequence of random variation. That is to say, nothing out of the ordinary with respect to lower troposphere temperature changes has occurred.

Nevertheless, this flatly contradicts the models put forward by warming activists. Over the past thirty-six years, carbon dioxide levels in the atmosphere have increased by nearly twenty percent. If that change hasn’t produced statistically significant changes in temperature, then their models lack validity. Their predictions have zero basis in the empirical evidence.

Yes, it really is that simple.

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**Data Notes.** The data used was acquired from two sources. Anyone can recreate this analysis with these data tables.

Carbon dioxide levels were downloaded from the Earth System Research Laboratory.

http://www.esrl.noaa.gov/gmd/ccgg/trends/index.html

ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_weekly_mlo.txt

Lower Troposphere temperature anomaly records were obtained from National Space Science and Technology Center, hosted at the University of Alabama, Huntsville.

http://www.nsstc.uah.edu/nsstc/

http://www.nsstc.uah.edu/data/msu/t2lt/uahncdc_lt_5.6.txt

The CO2 data is in weekly form while the satellite temperature anomaly data is aggregated monthly by the NSSTC. The carbon dioxide data is labeled ‘CO2 molefrac’, while the temperature data is labeled ‘Globe’. To get the weekly CO2 data to mesh with the monthly temperature data, the average was taken of the weeks over the month. While there were several missing weeks, there were no entire months with missing data. There was no additional processing or transformation of the data.