# Tag Info

19

Because of: The (extreme) dominance of noise over signal The prevalence of non-repeating patterns (many of which we know are not going to repeat) A pathetic sample size for cross-validation Regime changes due to exogenous events. These are typically in the cross-val window which makes it even worse. (GFC, financial integration, trade law changes, interest ...

10

The regression requires orthogonalization of factors. However, we need to maintain the interpretation of factors (so PCA and Factor Analysis are out). Also, we could apply an iterative method (indeed this is very common practice) but this will bias the factor loadings on the sequence of factors. Best approach is that of Klein and Chow in their paper ...

10

I don't have much to add, but wanted to address the "price of risk" question. APT is kind of "economics"-free and tries to price assets without the utility maximization required in CAPM/ICAPM. Ross's APT observes that groups of assets move together (e.g., tech stocks) and that is the risk you're bearing because the idiosyncratic risk, like the firing of ...

9

A few thoughts. Yes, your return series are autocorrelated (i.e., stocks don't exactly follow a random walk), so you should use Newey-West standard errors. If you do this as a univariate regression $$R_{i,t} = \alpha_i + \beta_i R_{j,t-1} + \epsilon_{i,t}$$ then there's almost certainly an omitted variable inside $\epsilon$ that is moving both $R_i$ and ...

7

The $R^2$s are usually close to zero for single stock regressions. The big $R^2$s that a lot of asset pricing research shows is by forming portfolios. Forming portfolios cancels a lot of the idiosyncratic returns, which has a smoothing effect. The $R^2$s should be low here, although I don't see any in the paper for you to compare. This probably means they ...

7

Regression analysis, as a minimization of the sum of squared errors, does not require normality of the error term. The requirements are that errors are homoscedastic and uncorrelated. And these are the fundamental assumptions (together with exogeneity). Then estimators are unbiased, optimal (exhibit the minimum variance within the class of unbiased ...

6

I basically agree with @John, let me expand: We want to model $y$ using a simple linear model, the most basic setup is $$y = c + \mathbf{X}\beta$$ with $y$ the $N$ observations, $c$ a constant, $\mathbf{X}$ the $N \times M$ matrix of regressors and $\beta$ a $M$-dimensional vector of coefficients. This model has $M$ parameters, the elements of $\beta$. ...

6

I believe that beta will be the covariance of the factor with the underlying asset. Is this correct? Close, it's the covariance divided by the variance of the factor. $$\beta_{f,a} = \frac{\sigma_{f,a}}{\sigma^2_f}$$ Also how is the return attributable to a specific factor calculated? Is there a single way this is done ...

6

Have you considered fitting ARIMA with exogenous regressors model? Linear regression with autocorrelated errors might be appropriate. R can do this with the arima() function via specifying the xreg argument.

6

Jennifer Bender of MSCI Barra has a paper from 2007 entitled: To Beta or Not to Beta: A Comparison of Historical Versus Fundamental Betas for Hedging Market Risk She deals specifically and exclusively with which method is superior for hedging long-only portfolios. Not surprisingly, she finds that Barra's approach is better. She tests long-only and ...

6

It appears that you are re-running the regression with each new data point. Instead, you should use an update/online formula (see an excellent answer by the famous Dr. Huber at stats.se). You can find an implementation in the R package biglm. If it doesn't have all the features you need (no windowing out of old data) you can at least adapt it and use it ...

4

If the equation satisfies all the assumptions of OLS, particularly homoscedasticity and no autocorrelation in the errors, then the expected return for the equation you laid out is $E[r_{future}|r_{history},x_{news}]=\alpha+\beta_1r_{history}+\beta_2x_{news}+\beta_3r_{history}*x_{news}$ If the unconditional expected return is zero (as is likely to be ...

4

Since you mention beta, I assume you're familiar with the capital asset pricing model (CAPM). The concept is that an asset's expected returns are linearly correlated with the market's returns. Of course, there are other ways "normalize" returns, as you put it. We can extend CAPM with Fama-French, which adds market cap and relative value to the equation. ...

4

Couple points I like to make: There exists no reliable model that can even predict future price returns (risk premiums, excess returns, whatever you want to call it) beyond a year, run as fast as you can if you hear from someone who claims he can predict risk premiums 10 years out, whether reliably or not. It makes zero sense and clearly comes from either ...

4

Generally we use models that go so far out in a comparative sense, not as an absolute decision. You are definitely do some good reading but I believe you are thinking about these models in the wrong way. I think (and correct me if I'm wrong) you are looking at creating or finding the perfect "crystal ball" model that will predict returns/risk premiums etc. ...

4

Note that you can understand the $\Delta$ as an "operator" acting on $r$. So just act on $r$ twice: $$\Delta^2 r_t = r_t - 2 r_{t-1} + r_{t-2}.$$ In fact if you write the $r$ as a vector, $r = (r_1, r_2, \ldots, r_N)$, then $\Delta$ is an $N\times N$ matrix with elements $\Delta_{i,j} = \delta_{i,j} - \delta_{i-1,j}$. The AR(2) model can be written as ...

3

What you are talking about is called regression using fractional polynomials and it has its merits. The canonical reference is this one: Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling by Royston and Altman (1994) From the abstract: The relationship between a response variable and one or more ...

3

If $\sum_{i=1}^k \alpha_i<1$, then you could just leave the remainder of the portfolio in cash. If $\sum_{i=1}^k \alpha_i>1$, that means you will have to take on some leverage in order to minimize tracking error. If you have a leverage constraint, then you can run this as a quadratic program with bounds on your coefficients. A regression should give ...

3

Well vix is a measure of volatitity which would make it an estimate of a second moment for S&P 500 so you might try an arch/garch in the mean type model on S&P. A good starting place for a project like this is to just do Vector Autoregressions on industry groups that you think might be related and see what comes up. N+30 is a long way in the ...

3

Van Belle describes a basic correction for autocorrelation in a t-test, although it may be hard to wedge it into the regression t-test. For the 1-sample t-test of the mean, the correction is to multiply the t-statistic by $\sqrt{\frac{1 - \rho}{1 + \rho}}$, where $\rho$ is the 1-period autocorrelation (or estimate thereof).

3

If you have a series of observations of the return as a vector, $\mathbf{r}$ with corresponding observations of the factor returns in matrix $Z$, then the least squares estimate of the vector of betas is $$\hat{\beta} = \left(X'X\right)^{-1} X'\mathbf{r},$$ where $X$ is the matrix with $Z$ and a column of all ones (for the intercept term). The last ...

3

This question was ultimately answered on Cross Validated Here are a couple of articles that deal with this subject: Britten-Jones and Neuberger, Improved inference and estimation in regression with overlapping observations Harri & Brorsen, The Overlapping Data Problem

3

The Newey-West procedure is meant to adjust the covariance matrix of the parameters to account for autocorrelation and heteroskedasticity. It is typically used in financial applications when one estimates the alpha (a parameter in a regression model) of a portfolio or strategy. One would adjust the standard errors using the Newey-West procedure in order to ...

3

The following is a good way to judge the quality of fits for a model. http://en.wikipedia.org/wiki/Akaike_information_criterion

3

In full generality this is a very difficult question. The closest you will get to a general framework is Vapnik-Chervonenkis theory. You can read about this in Chapter 7.9 of "The elements of statistical learning" by Hastie, Tibshirani and Friedman which can be downloaded from their website . But be warned that this is a theoretical approach. Often more ...

3

Definitely time series analysis. What you essentially want to do is some form of impact analysis. this can be done naturally using multivariate time series models like Vector Auto Regression models. Also when working with data to model liquidity you might want to use some specialized procedures like GARCH and ACD. Further there are methods to model non ...

3

I was going to comment but it turned out to be quite elaborate. My experience with certain AI/ML methods is that they're not deterministic. Take RBM for instance, a very wide-spread paradigm. To train such a machine you have two approaches, backpropagation or Kullback-Leibler divergence. Both require you to initialise the machine to a random state. And ...

3

Yes, this is an issue. There will be datamining bias. The best practice is to hold enough of your data out-of-sample to test your models. For example, if you have 10 years of data, use the first 5 years to come up with your models. You could then rank them from best to worst, based on whatever metric you prefer. Then use the second 5 years of data to test ...

2

When trying to predict returns, I think you should never look at in-sample statistics like R-squared. Only look at out-of-sample prediction results. Cross validation is a useful tool in at least the initial phase of modelling. In addition to over-enthusiasm, in-sample statistics easily lead to overfitting: ...

2

It sounds like all you need is to run a logistic regression, with the sign of $Y$ as your dependent variable instead of $Y$ itself. This will only give weight to the sign of the variable, and not to the magnitude. Once you have reformulated your question in more general terms (sign and magnitude of $Y$, rather than direction and volatility), you may be ...

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