Working through magnitudes, quadratic, and BS: Climate change

The following exercise is developed by the case study on climate change from the book “Real Stats.”

Climate change may be one of the most important long-term challenges facing humankind. We’d really like to know if temperatures have been increasing and, if so, at what rate. Figure 7.5 shows global temperature since 1880. Panel (a) plots global average temperature by year over time. Temperature is measured in deviation from average pre-industrial temperatures. The more positive the value, the more temperature has increased. Clearly there is an upward trend.

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How would you characterize this trend?

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Answer

One can tell from the first graph (panel A) that the relationship can be characterized as linear or non-linear. Either way has a decent representation of the data. However, when comparing linear vs. quadratic, it seems that the fit if better in the quadratic model.

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How would you write the regressions that characterize these relationships?

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Answer

A first linear version would be $Temperature_i=\beta_0+\beta_1Year_i+\epsilon_i$

a non-linear relationship would look like: $Temperature_i=\beta_0+\beta_1Year_i+\beta_2Year^2+\nu_i$

Let’s say you have two ways of modeling these relationships and estimating the regression. The results you would get are the ones below:

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What is the first model indicating? What are we learning from the first model?

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Answer

The first model (linear) is saying, overall as time has gone by we have seen increases in temperature.

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Has climate change accelerated? If so, by how much? Think about how you would answer this question before reading. Maybe jot down some ideas on how you would approach it.

This is a very general question, and although it would make a “bad” didactical question, these are the types of questions that people may ask you, and without much guidance, you’ll have to come up with an interpretation. Let’s go step by step.

Suppose you wanted to use the first model (model a) to answer this question. Has climate change accelerated? What does that mean? Well, it means how fast the temperature has risen over the years. If things had accelerated, the passage of time today would have increased the temperature faster than in the past. What does that mean in terms of our framework? Think about it for a second.

This refers back to the marginal effect of time and how the marginal effect of time changes as time goes on. Once you have connected those dots, your brain should immediately go into the “taking a derivative” framework. Let’s pick a random year, say 1900, and then a future year, say 2010. So now, using the model, let’s figure out if temperature changes have increased over time. Using the first model, what’s the marginal effect of time (years) in 1900?

\frac{\delta Temperature}{\delta Year}=0.006

What about the marginal effect of time in the year 2010?

\frac{\delta Temperature}{\delta Year}=0.006

So it’s the same! According to this model, one additional year increases temperature by the same amount, meaning no acceleration. You probably could have figured that out because this is linear. It is good to highlight that a model using “real” data could provide these results, and someone could have concluded that there is “no acceleration.” Is that the truth? How could someone have a model with real data and produce something that doesn’t seem right?

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Practicing calling BS: Well, here is this person's issue: They are not explicitly testing if there is no acceleration; all they are doing is choosing a model that does not allow them to test if there is acceleration, and that’s not a great model to test if there is acceleration or not. This is a case in which model choice matters for answering particular questions.

Let’s move into model two. Following the same logic, we take some derivatives:

\frac{\delta Temperature}{\delta Year}=-0.166+0.000044\times2\times Year \\ At\ year=1900 \\ \frac{\delta Temperature}{\delta Year}=-0.166+0.000044\times2\times 1900 \\ \frac{\delta Temperature}{\delta Year}=0.0012

Now we figure out the change for 2010:

\frac{\delta Temperature}{\delta Year}=-0.166+0.000044\times2\times Year \\ At\ year=2010 \\ \frac{\delta Temperature}{\delta Year}=-0.166+0.000044\times2\times 2010 \\ \frac{\delta Temperature}{\delta Year}=0.01088

Using this model then, we notice that the rate of change between these two years has increased from 0.0012 to 0.01088. This means we do have evidence for acceleration! Now we have to answer the question about “how much”? Here is one way to approach: you can say, “It has increased by 0.01088-0.0012=0.00968”. This is right, but it doesn’t mean much to an average reader. Both numbers seem small. That’s a fine “student” answer, but I’m not trying to train you to be a good student. I’m trying to train you to be a translational person. As I’ve tried to communicate in class, I want to push you to consider translating this into something more meaningful. So, what framework do you want to use? Here is where I recommend using frameworks we learned on economic significance. A simple one is doing the following:

\frac{0.01088}{0.0012}=9.067

I divide the 2010 marginal effect over the 1900 marginal effect. Why? Well, this will give us a number on how much larger the rate of change is in 2010 vs. 1900. By doing this, we can use the number 9 to come up with a colloquial answer to the “how much.” We could say, “Since the 1900s, we have nonuple the rate at which temperature has increased.” If you want to go ever more colloquial, you could say, “Since the 1900s, we have almost 10x the rate of change at which temperature is increasing” or “Currently, we have close to ten times the increase in temperature as we had in the 1900s”. You can play with those statements.

Notice that you can keep doing this exercise for a different set of years. What about the last 50 years? Or ten years? Etc. This is a great way to start tackling this very broad question. It was general enough that it allowed us flexibility, but we used frameworks learned in class that guided us and helped us give a tangible answer to the question.