Particularly r squared, p value, and t-statistic.
I looked in my CFA books, googled… just need a source of definitions that I can give to someone who knows nothing at all about statistics. I understand them but don’t know it well enough, inside and out to give a simple definition without fear of mixing or misinterpreting something. Or maybe I’m just horrible at articulating it but the person who is asking doesn’t get it. Does such a source exist? Thanks in advance!
R squared tells you how much smaller your range of errors are by using your model versus just guessing the mean of whatever you are trying to explain. So if I have a regression that predicts returns with an R^2 = 0.5, then the range of errors is about 50% smaller than they would be if you just guessed the historical average return. It’s not quite that, because “errors” is actually the square of the deviations rather than the absolute deviation, but, conceptualy, that’s what’s going on.
The t-statistic is a number that tells you if a particular average (or other statistic you solve for) is worth paying attention to (note that not all parameters have a corresponding t-statistic, but sometimes there are analogs that go with a different distribution; betas and constants tend to have t-statistics, as well as simple difference-of-means tests). If the absolute value is larger than a critical value (typically about 2), then you are reasonably confident that whatever you measured is not the result of random noise (though there are a lot of assumptions that can go wrong here). The exact critical value depends on how much confidence you need to have and how large your data set is. Greater required confidence tends to go with higher values.
The p-level is a short cut that can be used to determine if the statistic explains something in your model. It tells you what level of confidence would make that t-statistic (and whatever it’s measuring) significant or worth paying attention to. The confidence level associated with p is (1-p), unless you are testing for a one sided deviation, in which case the confidence is (1-p/2). In practice, if your output has a p-value, one generally doesn’t really need to pay much attention to t (traditionally people calculated t values and looked up the significance in a table, hence the tradition of showing them; modern computers are able to calculate p-values more easily, so really those are all you need these days).
You might be able to shorten that up a bit, but it’s a start.
Thank you for your post, I couldn’t have said it any better myself, and think if I shortened it, it would lose meaning.
I’m getting questions like, “is high t-stat/p-value/r-squared, good or bad?” I saw it depends then go on to explain what they are first. Totally loses them. They want 3rd grade level explanation but I don’t think that’s possible.
You can say "Generally, high R^2 is good, high (absolute value) t-statistics are good, and low p-scores are good, but it depends on the context. Also, the statistical effects may be significant, but the actual influence may still be small (which may be good or bad, depending on the context). e.g. taller people may be able to jump farther on average, but it’s only an extra inch or two, whereas athletic ability ends up being more important, making differences of feet.
Finally, high R^2 can be misleading if you haven’t been careful about the model. For example, two things that trend over time (in any direction) will have a high R^2 if you don’t prepare the data correctly, yet not have any meaningful relationship."
Since a lot of this stuff is dependent on the context, you may have to ask these people what the context is.
From a research point of view, finding significant relationships is generally a good thing, because it suggests you know something that will help you be able to make better estimations and predictions, and thus manage risk and reward better. But sometimes you are trying to eliminate effects (like making a portfolio market neutral, or something), so it depends on context
It’s been a while since I worked with these, but, for a 3rd grader, I would probably say:
R^2 is how much of a variable’s change is driven by another variable. So, if variable 1 increases 100% and variable 2 has an R^2 of 0.8, then variable 2 probably drove 80% of that 100%.
In the above example, we assumed that 80% of the change was driven by variable 2. A p value is basically the probability that you’re wrong about you’re assumption.
Before using my examples, you should wait to see if someone smarter than me says I’m way off base.
That’s a nice way of describing the p-value. Not a bad way to describe R^2, either, although you’d want to point out that “driven” implies more causality than may be warranted. It’s more along the lines of “if you know variable X, you can improve your estimate of Y.”
Here’s another insight into R. R is essentially the correlation between the regression prediction and the actual value (which is why it can be zero, but not negative, unlike a normal correlation). For a bivariate model (one dependent variable, one independent one), R is equal to the correlation. For a multivariate model, R is a kind of “collective correlation” after incorporating the information from all variables (though the F test is the more kosher joint test). For some people, that’s enough, but correlations are sometimes tricky to interpret, so it doesn’t always work.
BChad, what about ARIMA Model for a 3rd grader?
If you rearrange ARIMA, it spells MARIA. So basically it’s like praying the rosary, except you’re using lagged values of the dependent variable instead of beads.
Using the driving test to explain Type 1 and Type 2 errors:
Null hypothesis: You suck at driving and shouldn’t be on the roads.
Alternative hypothesis: You’re good enough to drive.
Type 1 Error: The test was so easy that a bad driver like you who shouldn’t be on the roads passed.
Type 2 Error: The test was so hard that an excellent driver like you failed.
I explained the type 1 type 2 error in terms of a trial:
Null: Not guilty
Type 1: Innocent man gets locked up
Type 2: Gulity man goes free
I think in terms of false positives and false negatives.
Type I : False Positive - you think it’s true (the relationship) but it’s not.
Type Ii : False Negative - you think it’s not true, but it is.
Then I remember that the type 1 error is what we are guarding against when we choose a critical t stat or p value.
x% R^2 might be bad in one context and fantastic in another. It depends too much on the model to be of much use in explaining something to a layperson, I think.
That being said, I think confidence intervals are, by far, the most confusing statistical concept that you see discussed in the media.
Layman’s definition of Autoregressive Conditional Heteroskedasticity please…
Conditional heteoskedasticity in this context means that the variance of the model changes over time. Autoregressive describes how that variance changes over time. In particular, the variance today is a function of the (squared) errors/residuals (not variances) from previous periods. GARCH adjusts ARCH so that variance today is a function of the (squared) errors AND variances from previous periods.
Note that ARCH/GARCH are different from stochastic volatility. In GARCH/ARCH the changes in volatility over time are driven by the errors of the return equation. In stochastic volatility, the volatility has its own error process. This subtle difference means that GARCH/ARCH is pretty easy to estimate with Maximum Likelihood, while stochastic volatility requires more sophisticated methods since the error is not observed.