### Replicating Goyal/Welch (2008) 2013-02-02

# Some theory

I created this blog because I replicate quite a few finance papers with R and I thought that some of you might profit from some of my work. Note, however, that I consider myself a mediocre R user! I'm still one of those guys that uses Stackoverflow mainly for asking questions, not for answering them...so I'm sure that my approach isn't always the most efficient one and if you have any comments on how to improve my code, let me know!

OK, let's start with Goyal/Welch (2008): A Comprehensive Look at The Empirical Performance of Equity Premium Prediction, Review of Financial Studies. You find the paper and the data on Amit Goyal's webpage. That's what I love about finance research nowadays...data is available from the authors, so all you have to do is to download it and you can play around with it yourself.

What's this paper all about? Well, there is a huge literature on "return predictability": you take a predictor, such as the dividend-price ratio or the consumption-wealth ratio, and you try to forecast future excess returns with it. Note that the very idea of return predictability is a paradigm change from the early days, when the random-walk hypothesis implied stock returns that are close to unpredictable. Nowadays, economists argue that stock returns have to be predictable. Why? One big reason is that we have a changing price-dividend ratio over time. If future dividend growth and expected returns were i.i.d., the price-dividend ratio would have to be constant. Thus a changing price-dividend ratio means that one of the two must be forecastable; and research indicates that returns should be the more relevant part here. To read more about that topic, check out chapter 20 in Cochrane (2005, Asset Pricing) or his presidential address to the AFA.

As already mentioned, there are now tons of papers that deal with return predictability. Goyal/Welch point out that the literature has become so big, with many different predictors, methods, and data sets, that it is quite tough to absorb. Hence, the goal of their article (taken from Goyal/Welch, 2008, p. 1456)

is to comprehensively re-examine the empirical evidence as of early 2006, evaluating each variable using the same methods (mostly, but not only, in linear models), time-periods, and estimation frequencies. The evidence suggests that most models are unstable or even spurious. Most models are no longer signiﬁcant even insample (IS), and the few models that still are usually fail simple regression diagnostics. Most models have performed poorly for over 30 years IS. For many models, any earlier apparent statistical signiﬁcance was often based exclusively on years up to and especially on the years of the Oil Shock of 1973–1975. Most models have poor out-of-sample (OOS) performance, but not in a way that merely suggests lower power than IS tests. They predict poorly late in the sample, not early in the sample. (For many variables, we have difﬁculty ﬁnding robust statistical signiﬁcance even when they are examined only during their most favorable contiguous OOS sub-period.) Finally, the OOS performance is not only a useful model diagnostic for the IS regressions but also interesting in itself for an investor who had sought to use these models for market-timing. Our evidence suggests that the models would not have helped such an investor.

Therefore, although it is possible to search for, to occasionally stumble upon, and then to defend some seemingly statistically signiﬁcant models, we interpret our results to suggest that a healthy skepticism is appropriate when it comes to predicting the equity premium, at least as of early 2006. The models do not seem robust.

# Replication

In my opinion, the most interesting part about this paper is that they don't bother to look at any regression coefficients. So the results are very different from what you usually see in such papers. Basically, they just look at the mean squared errors (MSE) (from Wikipedia: In statistics, the mean squared error (MSE) of an estimator is one of many ways to quantify the difference between values implied by an estimator and the true values of the quantity being estimated. MSE is a risk function, corresponding to the expected value of the squared error loss or quadratic loss. MSE measures the average of the squares of the errors.) and transform the MSE to several useful statistics:

$$
\begin{aligned}
R^2 &= 1 - \frac{MSE*A}{MSE*N} \
\Delta RMSE &= \sqrt{MSE*N} - \sqrt{MSE*A}\
\end{aligned}
$$

where $h$ is the degree of overlap ($h=1$ for no overlap). Note that $MSE*N=E[e*N^2]$ and $MSE*A=E[e*A^2]$, where $e*N$ denote the vector of rolling OOS errors from the historical mean model and $e*A$ denote the vector of rolling OOS from the OLS model. Now, this is very important so let's think about this for a while: many scholars within the return predictability literature argue that a bunch of variables forecast equity premiums. Goyal/Welch (2008) now use the same test with the same time period for all those variables and compare it to the simplest of all forecasting techniques: the simple historic average.

Note, however, that these statistics only apply to the OOS tests. For the IS tests, $R^2$ is just the R squared you know from a linear regression, and $\overline{R}^2$ is the adjusted one.