Logistic Regression and Stock Returns...

Attention conservation notice: Stock returns are non-stationary! Logistic regression, smoothing splines, generalized additive models are all interesting techniques and fun to play with...but stock returns are still non-stationary! 

Suppose instead of trying to predict tomorrow's stock return based on today's return, I just try to predict whether or not tomorrow's return will be positive.  One way to do this would be using logistic regression. The response variable will be an indicator variable that takes a value of 1 if tomorrow's return is positive, and 0 otherwise.  The name of the game is to model the probability of tomorrow's return being positive, conditional on the value of today's return.

Here is the R summary output for a logistic model of the S&P 500 data from Yahoo!Finance:

Call:
glm(formula = (tomorrow > 0) ~ today, family = binomial, data = SP500.data.v2)

Deviance Residuals:
   Min      1Q  Median      3Q     Max 
-1.647  -1.223   1.080   1.129   2.007 

Coefficients:
            Estimate Std. Error z value Pr(>|z|)   
(Intercept)   0.1106     0.0161   6.866 6.60e-12 ***
today         8.6524     1.6788   5.154 2.55e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 21458 on 15513 degrees of freedom Residual deviance: 21431 on 15512 degrees of freedom AIC: 21435

Number of Fisher Scoring iterations: 3

Both the intercept and the coefficient on today's return are highly significant.  How do we interpret the coefficients?  If today's S&P 500 return was 0.0, then the intercept represents the logistic model's prediction for the likelihood that tomorrow's S&P 500 return is positive.  However to get the predicted probability I need to transform the coefficient from the logistic scale back to the probability scale:

exp( 0.1106) / (1 + exp( 0.1106)) = 0.527611 or roughly 53%

The logistic model predicts there is a 53% chance of tomorrow's return being positive given that today's return was zero (slightly better than a coin flip!). Suppose that the S&P 500 was down 10% today (i.e., today's return is -0.10)? The predicted probability of tomorrow's return being positive is:

exp( 0.1106 + 8.6524(-0.10)) / (exp( 0.1106 + 8.6524(-0.10))) = 

0.3198024 or roughly 32%

I plot the predicted probabilities (and their confidence bands) against today's return using the logistic model for the daily returns of the S&P 500 from January 1950 through this past Friday.  The predicted probabilities increase monotonically as today's return goes from negative to positive (i.e., the lowest probability of a positive return tomorrow follows large negative returns today, and the highest probability of a positive return tomorrow follows a large positive return today.  The results of the logistic model are, I think, inconsistent with mean-reverting returns.  I would think mean reverting returns would require a higher likelihood of a positive return tomorrow if today's return where large and negative.  Note that the confidence bands are mildly asymmetric, and that they narrow considerably where the bulk of the observed returns lie.

Similar plots for the FTSE-100 from 1984-2011, and the Straits Times Index (STI) from 1987-2011. The probability of tomorrow's FTSE-100 being positive given today's return is basically a coin-flip much of the time!

The predicted probabilities from the logistic model using data from the Straits Times Index (STI) in Singapore are more similar to the predicted probabilities for the S&P 500.

I would like to have some sense of whether the above logistic models are well-specified.  How might I go about validating the above logistic regressions?  One way would be to estimate and compare the fits of some more flexible models (i.e., smoothing splines, generalized additive models, etc).  If the logistic regression is well specified, then I would expect that the more flexible models should not give significantly different predictions than the above logistic model.  Here is the plot of the predicted probabilities and the 95% confidence bands of the logistic model for the S&P 500, a smoothing spline (blue), and a generalized additive model (red).

WTF!  The smoothing spline and the GAM have very similar predicted probabilities, particularly when today's return falls within [-0.05, 0.05] (i.e., over the bulk of the observed returns).  However, visually, neither the smoothing spline nor the GAM appear to match the logistic regression well at all!  

The plot is actually a bit misleading, because your eye is immediately drawn to the huge differences in predicted probabilities for the extreme tails of today's return.  However, there are only a handful of observations in the extreme positive and negative tails of today's return (check the data rug) and thus the predicted probabilities for the smoothing spline and GAM models are unlikely to be very precise in these regions.  Better to focus on the differences in the predicted probabilities between the spline/GAM models and the logistic model in the region around today's return equals zero (where the bulk of the data is located).  In this neighborhood, the curve of predicted probabilities for all three models is increasing (consistent, I think, with some type of trend-following dynamic).  Note, however, that for the smoothing spline and GAM models, the slope of the predicted probability curve is much steeper (suggesting a more aggressive trend-following dynamic?) compared to the logistic model.

Here are the plots for the FTSE-100 and the Straits Times Index (STI)...

The whole point of fitting the smoothing spline and the GAM was to determine whether or not the logistic model is well specified.  As mentioned above, if the logistic model is well specified, then spline/GAM models should not be significantly better fits to the data.  We can measure goodness of fits by comparing the deviance of the logistic model with that of the GAM (I am going to ignore the smoothing spline because it doesn't technically respect the probability scale).  Thus the observed difference in deviance between the logistic model and the GAM for the S&P 500 is the deviance for the null model (i.e., Logistic model) less the deviance of the alternative (GAM model).

21430.65 - 21348.16 = 82.49624

The observed difference in deviance for the FTSE-100 and the STI are 16.53285 and 12.30237, respectively.  A smaller deviance indicates a better fit, thus, as expected, the more flexible GAM is a better fit for the S&P 500, the FTSE-100, and the STI.  Are these observed differences in deviance significant?  Let's go to the bootstrap!  Basic idea is to use the bootstrap to generate the sampling distribution for the difference in deviance between the null and alternative models, and then see how often these bootstrap replicates of the difference in deviance exceed the observed difference in deviance.  If the fraction of times that the replicated difference in deviance exceeds the observed difference is high, then it is likely that the observed improvement in fit using the GAM is the result of statistical fluctuations and is therefore not significant. 

Results? Running the above bootstrap using 1000 replications yields p-values of 0.00, 0.02, and 0.06 for the S&P 500, FTSE-100, and STI, respectively.  The p-values indicate that the improvement in fit using the GAM model on data from the S&P 500 and FTSE-100 is significant at the 5% level, and that the improvement in fit using the GAM model is significant for the STI data at the 10% level.

What does all of this mean? Well, as far as this little modeling exercise is concerning, the results of the bootstrap specification test suggest that we should use a GAM instead of a logistic model.  Here are the plots of the GAM predicted probabilities with confidence bands for the S&P 500, the FTSE-100 and the STI.  Note how wide the 95% confidence bands for the predicted probability of tomorrow's return being positive are when today's return is either really negative or really positive.  This is exactly as it should be!  There just aren't enough observed extreme returns (positive or negative) to support precise predictions.

Could this be used to construct a useful investment stratagem? I think doubtful.  Compare the GAM model's predictions for the S&P 500 above, which make use of historical data from 1950-2011, to the GAM model's predictions using data from 1993-2011 for SPY, the widely traded ETF which tracks the S&P 500 (I assume that implementing a stratagem would involve trading some ETF like SPY).  I have also included the logistic regression and its 95% confidence bands because the bootstrap specification test fails to reject the logistic model in favor of the GAM (p-value of 0.50).

The two are substantially different.  The interesting little window of trend-following behavior is now gone.  Perhaps it was a historical artifact from pre-computer trading days of yore?  The negative slope of the predicted probabilities is consistent with mean-reversion in returns.  

The underlying problem with trying to predict the behavior of stock returns, is that they are non-stationary.  It's not just that the parameters of the "data generating process" for stock returns are changing over time, the entire data generating process itself is evolving over time.  When the underlying process itself is changing, getting more historical data is unlikely to help (and in fact is likely to make predictions substantially worse!)... 

Update: Code is now available!

So you want to predict a stock market...but, which one?

Attention conservation notice: Stock returns are non-stationary time-series and as such all of the analysis below is wasted in a feeble attempt to squeeze water from the proverbial rock.

Which of the following equity markets would you expect to be more "predictable" using only information on past returns: the S&P 500, the FTSE-100, or the Straits Times index?  Personally, I had no really strong prior regarding which of these markets would be the most "predictable."  I suppose that a EMH purist (extremist?) would claim that they should all be equally unpredictable.

Some simple plots of S&P 500 logarithmic returns from 3 January 1950 - 24 August 2011 will help set the mood (all data is from Yahoo!Finance)...

Time series plot of the logarithmic returns exhibits clustered volatility (i.e., heteroskedasticity and serial correlation).  Kernel density plot is leptokurtic (i.e., excess kurtosis, which leads to heavier tails, than the best-fit Gaussian).  More on the scatter plot below...I will not bore you with the other plots (suffice it to say that they all exhibit the same properties).

A good place to start is a simple test of the random walk hypothesis.  If share prices follow a random walk with a drift, then tomorrow's share price should be the sum of today's share price and the drift term (and some statistical white noise).   Thus if we regress tomorrow's return on today's return, the coefficient on today's return should be statistically insignificant.

R_t+1 = μ + β R_t + ε _t+1

H₀: β = 0

Here is a scatter plot of FTSE-100 returns from 2 April 1984 - 24 August 2011.  The above regression specification is plotted in red, and the random walk null hypothesis is plotted in grey.  The two are virtually identical for the FTSE-100.

Although OLS parameter estimates are consistent in the presence of arbitrary heteroskedasticity and serial correlation,  OLS estimates of the standard errors are not.  To make sure my inferences are valid, I use HAC standard errors.  For the S&P 500 and the FTSE-100, using HAC standard errors, the coefficient on today's return is statistically insignificant.  However, even with HAC standard errors, the coefficient on today's return remains positive and significant for the Straits Times Index.  What this says to me is that predicting tomorrow's return using a linear model is pretty damned useless (unless you invest in Singaporean equities!).  But why should we limit ourselves to a linear model?  We shouldn't.

Above, I made the implicit assumption that the relationship between today's return and tomorrow's return is linear.  Is this assumption supported by the data?  One way we can test this assumption is to calculate the observed difference in the mean-squared error of the linear specification and that of a general non-linear specification.  One would expect that a general non-linear model would have a lower mean-squared error because it is more flexible, so we need to check whether this difference is statistically significant.  One way to do this is to generate a sampling distribution of this difference using the bootstrap, and then see how often we would observe a difference between the linear and non-linear specification as large as what we actually did observe.  If such a large difference is very rare, then this is evidence that the linear model is mis-specified.

Now, some care needs to be taken in generating the bootstrap replicates given that the time-series exhibits such complicated dependencies.  For my implementation,  I generate the simulated response variable by resampling the the residuals from the linear null model.

Linear or Non-Linear? That is the question...

For computational reasons, I use a smoothing spline to estimate my non-linear specification, using generalized cross-validation to choose the curvature penalty (you could easily use some other estimation method such as kernel regression in place of the smoothing spline).  Even though the smoothing spline has the lower MSE in all three cases, it is only significantly lower for the S&P 500.  The p-value for the bootstrap specification test (sketched above) is roughly 0.03, indicating that the linear specification can be rejected (p-values are 0.18 and 0.13 for the FTSE and the STI, respectively).

I plot the smoothing spline and 95% confidence bands (these are basic confidence intervals generated via a non-parametric bootstrap that re-samples the residuals of the smoothing spline) for both the S&P 500 and the Strait Times Index.  Since we are interested in prediction I plot the splines over a uniform grid of m=500 points (instead of simply evaluating and plotting the splines at the observed values of today's return).  Given that the smoothing spline is not significantly superior to the linear model for STI returns, I include the both the smoothing spline (blue) and the OLS regression (red) specifications in the scatter plot of STI returns.  Do these confidence bands seem too "tight" to anyone else?  Sample sizes are pretty large (over 15,000 obs. for the S&P 500)...

Could we use the above models to craft a profitable investment strategy?  Note that the OLS regressions for both the S&P 500 and the Straits Times Index have positive slope.  This is because, the bulk of the data (say for returns in [-0.02, 0.02]), exhibits mild "trend following" behavior.  Using the estimated smoothing splines, if today's S&P 500 return was -0.01 (or down 1%), 95% of the time tomorrow's return should be between -0.00156 (or down 0.156%) and -0.000657 (or down 0.0657%).  Similarly, if today's S&P 500 return was 0.01 (or up 1%), tomorrow's return should be between 0.000978 (or up 0.0978%) and 0.00188 (or up 0.188%).  Taking advantage of such minute expected returns would require, I think, either a large bankroll, or lots of leverage (or both!).

Also, what is the best way to trade the S&P 500 "market portfolio" in a cost efficient manner on a daily basis?  Exchange Traded Funds like the SPY might be an option.  Here is a plot of the smoothing spline fit to the data for SPY over the entire history of the fund (trading began on 29 January 1993).  I include the OLS regression fit because the slope is negative and significant (even using HAC standard errors) and because the spline fit is not a significant improvement over the linear model (p-value of 0.53 using my bootstrap specification test).

This plot of the SPY data from 1993 onwards looks quite a bit different, and the behavior of returns seems quite a bit different than the corresponding plot of S&P 500 data from 1950 onwards (despite the fact that SPY tracks S&P 500 almost identically from 1993 forward).  If the parameters of the underlying data generating process (DGP) driving stock returns remained constant over time, then one would expect to find that subsets of the data behaved similarly to the entire sample.  Maybe there are simply structural breaks in the data? Or, remember what I said about stock returns being non-stationary!

What about the FTSE-100?  Is all hope lost in trying to predict FTSE returns using past data? I may try predicting whether or not tomorrow's return will be positive given today's observed return using logistic regression in a future post (I am currently having difficulties getting R to give me the correct predicted values for the logistic models!  It worked fine yesterday!)

Update: Code is now available! 

Heteroskedasticity simulations...

Finally getting back into my routine after a very successful summer school in Jerusalem.   This blog post is about part of a graduate econometrics lab on violations of the classical linear regression model that I have been working on lately.  Basic idea of the lab, which I borrowed from Prof. Mark Schaffer, is to expose students to Monte Carlo simulation and violations of the classical linear model simultaneously by having them construct simulations that demonstrate what happens to standard hypothesis tests when the DGP exhibits one of the following issues:

1. Heteroskedasticity
2. Autocorrelation
3. Endogenous regressor
4. Non-stationary regressor

To keep this post reasonably short, I am only going to focus on the section of the lab that compares various techniques for dealing with heteroskedasticity via Monte Carlo simulations.

The DGPs:
I specify the two data generating processes as follows.  For DGP #1 x can only take positive values:

• x = abs(rnorm(n, mean=0, sd=1))
• e = rnorm(n, mean=0, sd=1)
• u = e*x (implies that the variance of u increases with the square of x!)
• y = x + u 

The first pane in the plot below is y versus x for n=1000 with the OLS regression line in red and the true regression function in grey.  If the heteroskedasticity isn't obvious enough, the second pane is a plot of the squared OLS residuals against the regressor x.  The true variance of the disturbance increase with the square of x (i.e., the grey curve).  Note that the true variance process looks roughly linear (even though it actually is not!).  The fact that a linear approximation to the true variance process looks like it would work pretty well for DGP #1 plays a role later.

For DGP #2 the regressor, x, is allowed to take both positive and negative values (i.e., x=rnorm(n, mean=0, sd=1)).  Everything else is the same as in DGP #1.  Here is the analogous plot for DGP #2:

Note that the even though the true variance process is non-linear in both DGP #1 and DGP #2, it can be (reasonably) well approximated by a linear function in DGP #1.  However, a linear approximation to the true variance function is an EPIC FAIL for DGP #2!  This plays a huge role in the breakdown of FGLS documented below.

Comparison #1 (OLS, OLS w/ HAC Std. Errors, and WLS): Assume that you know the true form of the variance function (i.e., you have ritually slaughtered the requisite small woodland creature, dried its bones, and left them for 3.5 days to be scattered by the wind at which point you were able to use the pattern created to divine the true variance function...alternatively, if you live in Greece, you could simply consult the Oracle).

After having done all of this, the first simulation sets the sample size to n=100 and regresses y on x (where x is given by DGP # 1) 5000 times while capturing the parameter estimate for the coefficient on x using both OLS and WLS (which incorporates the optimal weights given the true variance function) and the various flavours of the standard errors.  Note the HAC stands for Heteroskedastic and Autocorrelation Consistent standard errors.  These standard errors are used frequently in econometrics to adjust the standard errors in the presence of heteroskedasticity and autocorrelation (HAC standard errors are rumored to be consistent in the presence of arbitrary forms of heteroskedasticity and autocorrelation...although I confess to not having read the original journal articles).

Plotting the kernel densities of the OLS estimator and the WLS estimator shows that while both estimators appear to be unbiased (consistent with theory), only the WLS estimator takes advantage of the known variance function (OLS ignores this information) and thus the variance of the WLS estimator is much smaller.

Next, let's consider the size properties of the standard hypothesis test (i.e., type I error).  How often would you expect to reject the null hypothesis of β=1 if your use the standard test statistic and critical values of -1.96 and +1.96 were valid? (Answer: 5% of 5,000 repetitions = 250 times). How many rejects do we actually observe in this simulation?

• OLS: ~ 1200!
• OLS w/ HAC standard errors: ~ 450 
• WLS: ~ 250

It would seem that only the WLS has the expected amount of type I error.  Not surprisingly, OLS is in bad shape.  To get even more information about the size properties, let's look at histograms of the p-values associated with the above tests.  If the test statistics were unbiased, then we should expect to find uniformly distributed p-values...

...and with OLS w/ HAC standard errors, and WLS we do find more or less uniformly distributed p-values.  OLS w/ HAC exhibits some very slight distortions...but these are nothing compared with OLS!

Finally, let's look at the power of the standard hypothesis test (i.e., type II error). Below I plot the power curves using the various flavors of standard errors.  What are we looking for in this plot? Two things.  First, a hypothesis test with good size properties should obtain a minimum value of 0.05 (for a two-tailed test) at the true value of β=1.  Second, the power curve should be relatively narrow.  The wider the power curve, the lower is the probability of correctly rejecting a false null hypothesis (i.e., the higher is the type II error).

Unsurprising conclusion of this comparison: if you are going to invest the 3.5 days necessary to obtain the true variance function, then you should use it when estimating your regression.

Comparison #2 (OLS, OLS w/ HAC Std. Errors, FGLS, and WLS (npreg): Now suppose that you do not know the true form of the variance function but wish to estimate it from the data.  This simulation sets the sample size to n=100 and regresses y on x (where x is given by DGP # 1) 5000 times while capturing the parameter estimate for the coefficient on x using OLS, FGLS, and WLS (npreg) and the various flavours of the standard errors.  Feasible Generalised Least Squares (FGLS) estimates the true variance function from the data by linearly regressing the log of the squared OLS residuals on x and using the fitted values as the estimate of the variance function. WLS (npreg) estimates the variance function using the fitted values from a non-parametric regression of the log of the squared OLS residuals on x (see Cosma Shalizi's lecture 5 notes for details).

While all three estimators appear to be unbiased, the FGLS and WLS (npreg) estimators estimate the unknown variance function (OLS does not) and thus the variance of these estimators is much smaller.

Again, consider the size and power of the standard hypothesis. How many rejections of the null hypothesis of β=1 do we actually observe?

• OLS: ~1200!
• OLS w/ HAC standard errors: ~ 400 
• WLS (npreg): ~ 600 
• FGLS: ~ 300 

FGLS would seem to perform the best (in a relative sense), however all four procedures have excessive type I error.  They all reject a true null hypothesis too frequently.  I was expecting the size properties of the WLS (npreg) estimator to be superior to those of the FGLS estimator.  Could ye old trade-off between bias and variance.  Given that the true variance process is roughly linear for DGP #1 the bias introduced by using FGLS should be pretty small.  I also just used the default bandwidth for npreg() in my R code.  Perhaps I should have spent more time trying to optimize the bandwidth (if default bandwidth too high, then npreg() is overfitting, which should lead to higher variance in the WLS (npreg) estimator).  Also, perhaps if I had implemented in iterative procedure to estimate the variance function using the non-parametric regression technique, WLS (npreg) would outperform FGLS.  Any thoughts/ideas from readers would be welcome.

Histograms of the p-values:

In this case only with FGLS we do find more or less uniformly distributed p-values.  OLS w/ HAC se exhibits some very slight distortions...but these are nothing compared with OLS!  As touched on above, WLS (npreg) also exhibits surprisingly large size distortions.

Plot the power curve:

Again, hypothesis tests using OLS w/ HAC and FGLS have pretty good size properties, but the hypothesis tests using FGLS have superior power properties.  If you have reason to suspect that the true variance function is roughly linear, then it looks like FGLS is the way to go (although the difference between the two procedures might become negligible as sample size increases). 

Comparison #3 (OLS, OLS w/ HAC Std. Errors, FGLS, and WLS (npreg) with DGP #2: Now suppose that you do not know the true form of the variance function but wish to estimate it from the data.  This simulation sets the sample size to n=100 and regresses y on x (where x is given by DGP # 2) 5000 times while capturing the parameter estimate for the coefficient on x using OLS, FGLS, and WLS (npreg) and the various flavours of the standard errors.   Recall that the true variance process for DGP #2 is poorly approximated by a linear specification.  Thus my prior was that FGLS should be an EPIC FAIL, but the WLS (npreg) would save the day! I turned out to be half right...

While all three estimators appear to be unbiased, the WLS (npreg) estimator is considerably more precise than either the OLS or FGLS estimators. Why is FGLS doing so poorly? The true variance function for DGP #2 is sufficiently non-linear that the bias introduced by using FGLS to estimate the variance function produces and EPIC FAIL!  Note that the kernel density of FGLS tracks more or less with the density of the OLS estimator!

Consider the size and power of the standard hypothesis. How many rejections of the null hypothesis of β=1 do we actually observe?

• OLS: ~1200!
• OLS w/ HAC standard errors: ~ 350
• WLS (npreg): ~ 900 ?!
• FGLS: ~ 1400!!

FGLS is an epic fail.  OLS with HAC standard error doesn't do too poorly.  What is going on with WLS (npreg)!  I was hoping that it would shine through in this most realistic of cases!  I must be over-fitting and the sample size is pretty small (only n=100).  Extending the sample size might help.  I am really perplexed by this...
 
Histograms of the p-values:

Only OLS with HAC standard errors has roughly uniform p-values.  The rest exhibit significant distortions.

Finally, the power curve:

OLS with HAC would seem to win again.  Its power properties are not great, but should improve with sample size.

As always R-code is available upon request.  If anyone has thoughts on what is going on with my implementation of WLS (using non-parametric regression to get the optimal weights), definitely feel free to comment!

More on Smoothing Splines...

Chalk yesterday's post up as a learning experience...following a very helpful comment left on yesterday's blog post, I made two changes to my code:

1. I am now calculate the confidence bands by bootstrapping the residuals (instead of the data points themselves).  The confidence bands yesterday were huge.  The reason why the confidence bands were so wide was because the bootstrap method I implemented scrambled the dataset during each iteration, which destroyed the time series nature of the underlying data.  Bootstrapping over the smoothing spline's residuals should preserve the times series nature of the data (which should narrow the confidence bands considerably). 
2. I am now using generalized cross validation (GCV) to pick the penalty for the curvature( for those following at home that is setting cv=FALSE using the function smooth.spline().

Here are the relevant plots using daily, weekly, and monthly returns:

Daily Returns:

Weekly Returns:

Monthly Returns:

Both the daily and monthly plots exhibit significant asymmetry between the left and right tails of the data.  Although I still wonder about the sensitivity of the results to certain outliers (particularly with the daily data).  Recall the EMH implies that returns should follow random walk with a drift, and the red-dotted line represents the EMH null prediction (ignoring the drift...feel free to mentally shift the dotted-red line up or down as you see fit). 

My (not very enlightening) interpretation of the above plots is that EMH works pretty well for the "body" of the data (all three plots are roughly "flat" where most of the data lies), but there is something fundamentally different governing the dynamics for large returns (and that whatever is governing the dynamics must effect large negative returns differently than large positive returns).  Note that the robust asymmetry is much harder (maybe impossible) to detect using a linear model. 

Today's Distraction...

For the past month or so I have been working my way through Cosma Shalizi's excellent course on data analysis.  Today I distracted myself from my own research by playing with Smoothing Splines (Cosma's Lecture 11).  Everything below was done in R.

First, I grabbed some historical data for the S&P 500 from Yahoo finance using the get.hist.quote() function from the tseries library.  I pulled down daily, weekly, and monthly data starting on 3 January 1950 and ending 31 Dec 2010 (start and end dates are for daily data).  I then constructed S&P 500 returns by taking the first-difference of the logarithm of the S&P 500 Adjusted Closing Price.   Here are (probably familiar) time series plots of the daily returns and a density plot...

Note that stock returns exhibit clustered volatility and are negatively skewed with significantly heavier tails than one would expect if returns were Gaussian.  On a side note (related to my current research), a couple of years ago some researchers at the Santa Fe Institute (specifically Stefan Thurner, J. Farmer, and John Geanakoplos) published a paper titled "Leverage Causes Fat Tails and Clustered Volatility."  Their model also predicts that returns should be negatively skewed (a point I think they should have included in the title).

Back to learning about splines! Is today's S&P 500 return useful in predicting tomorrow's S&P 500 return?  For the null hypothesis, I take a strongish form of the Efficient Market Hypothesis (EMH):

• Ho: Stock prices follow a random walk with a drift (i.e., returns should be mean zero white noise)

For alternative hypotheses, I use a brutally simple parametric model, and then what ever functional form the smoothing spline finds

• HA,1: r_t+1 = β₀ + β₁ r_t + ε_t
• HA,2: Whatever the smoothing spline kicks out

To test the null against the parametric alternative, we simply need to test the joint restriction that β₀=β₁=0.  Presumably to test the smoothing spline, we need to calculate some 95% confidence bands for the fitted spline and then look to see if the confidence bands contain the curve (really a line) predicted by the null hypothesis.  This is my first time using splines, so if anyone out there knows whether a better way (or a book) about how to do hypothesis testing with smoothing splines, I would be interested in hearing from you.

Here is a scatter plot of tomorrow's return against today's return.  I fit a simple linear regression to the data and plotted the curve in gray.  While both the slope and intercept terms are very significant (p-values essentially zero for both), it is worth noting that the standard confidence intervals are not valid (much too narrow) given the blatant violation of Gauss-Markov assumptions for the regression.  More work needs to be done before we can take this as evidence against the EMH null (since this post is about smoothing splines I am going to simply state that I would be surprised if, after calculating appropriate standard errors (either using bootstrapping, or some type of heteroskedastic robust standard errors, etc), the parameter results were still significant...but maybe!)

The smoothing spline is in orange.  I used the smooth.spline() function in R to fit the spline (using leave-one-out cross-validation to pick the optimal penalty for the curvature).  

If stock prices reflect all relevant information about the value of the stock, then one would expect that today's return should be pretty useless in predicting tomorrow's return (thus under the null the true regression line should be the dotted red line in the above scatter).

But what about the smoothing spline?  A few things:

1. While the regression line is positively sloped, the smoothing spline is negatively sloped for larger negative and large  positive values of today's return.
2. The asymmetry.  The slope of the smoothing spline is more negative for large negative values of today's return (compared with the slope of the smoothing spline for large positive values of today's return).
3. Outliers:  The October 1987 stock market crash looms large in the data. How sensitive is the estimated smoothing spline to these 1-2 observations?
4. Is the asymmetry of the smoothing spline a statistical artifact?
5. Most importantly, despite the dramatic appearance, is the smoothing spline significantly different than the dotted red line?

I am going to focus on point 5 for the rest of the post.  I calculated 95% confidence bands for the smoothing spline using a bootstrap re-sampling of the data points.  Basically, I re-sampled (with replacement) the stock return data creating a new synthetic data set, fit a smoothing spline to this new data set, and then repeated the process a bunch of times to build up a distribution that I could use to create the confidence bands.

For the most part, the dotted-red line lies entirely within the 95% confidence band for the smoothing spline (if you squint you can kind of see a small portion of the dotted-red line that lies outside the bands).  So despite the dramatic appearance of the smoothing spline I would say that we can not statistically distinguish it from the dotted-red line.

I was curious to see what the above plot might look like if I used weekly and monthly S&P 500 returns instead of daily returns...

Weekly Returns:

Monthly Returns:

I was surprised at how different the daily, weekly, and monthly smoothing splines turned out to be...still in all three cases the 95% confidence bands for the smoothing spline contain (almost completely) the red-dotted line.  I will have a think as to why they are so different, and perhaps follow up with another post.  My R code will be posted as soon as I have time to get my Google Code page up and running...until then feel free to email me (or leave email in a comment) and I will send it to you.

Update: As pointed out in a comment below, EMH predicts stock returns should follow a random walk with a drift...which implies that the dotted-red line doesn't necessarily need to have a zero intercept.  One would hope that the drift is slightly positive!