Hot answers tagged regression
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 ...
9
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 ...
8
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 ...
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 ...
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.
\begin{equation}
\beta_{f,a} = \frac{\sigma_{f,a}}{\sigma^2_f}
\end{equation}
Also how is the return attributable to a specific factor calculated? Is there a single way this is done ...
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
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 ...
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
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
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 ...
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
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
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
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
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).
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 ...
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
The basic CAPM - which is what your regression estimates - says
$$
R_S = R_f + \beta_S (R_{Market}-R_f)
$$
where
$$
\beta_S = \frac{Cov(R_M,R_S)}{Var(R_M)}
$$
i.e. the return of a certain stock depends only on the correlation with the market portfolio.
For your pricing equation to work, you will need to have an idea about the expected market (excess) ...
2
I personally would not do that!
Your regression model has been fitted to approximate $Y(t)$ (the reality) as much as possible.
If I understand you well, you say:
at the previous period $Y(t)=55$ (Starbucks traded at 55 USD)
the last period's estimate from the regression is $\hat{Y}(t)=60$
Since $\hat{Y}(t)-Y(t) > 0$, you want to invest.
This does ...
2
The most common transformations you see for these three variables on credit desks is to compute "returns" on the credit variables. So, rather than taking the straight daily differences $\Delta s_t$ of swap spreads and $\Delta H_t$ of the high yield index (by which I assume here you mean on-the-run CDX HY), practitioners will transform to $\frac{\Delta ...
2
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 $\beta$.
...
2
See edit and comments, this response might not be applicable to the question:
When performing regression you would tend to want your regressors to be of similar type, or at the very least range. Assuming you use log return for price changes I would recommend using the untransformed interest rate. The reason for this is that they are the same type of entity, ...
1
It exists several techniques to deal with mixed-frequency data. I believe MIxed DAta Sampling is the best-known.
Eg:
bridge equation,
MIxed DAta Sampling (MIDAS) models
Mixed frequency VARs
Mixed frequency factor models
...
Here is a good document on this topic: A survey of econometric methods for mixed- frequency data
1
A cross-sectional linear factor model would regress the returns of the securities of interest on attributes that are believed to be priced in. For instance, in period 1, you could regress the return of all the S&P500 stocks against their book-to-market ratios and market capitalizations. It may be appropriate to scale the independent variables to be like ...
1
I am a bit confused about your question in that you say at one point that you want to explain the relationship between credit spreads and equity prices. Is that what you really want to know? Why? I thought you already have empirical evidence that supports the relationship between the two? Or are you after something else?
Anyway, I would actually recommend ...
1
Regarding the dividends:
In order to avoid jumps on ex-dividend date, you can make the simplifying assumption that dividends are paid continuously and adjust the returns of the assets. The size of dividends could be estimated from historical data or can be set proportionally to the asset price.
1
I suggest you look for common characteristics amongst the 70% of stocks with decent $R^2$ and the 30% with a degraded in-sample fit. If you find that all the stocks that fit your model best were in one industry and all those that didn't fit were in another industry, then it may be that your model is actually picking up an industry effect. Controlling for ...
1
Here's an answer from a purely statistical point of view:
http://www.duke.edu/~rnau/regnotes.htm#constant
And another from Cross Validated:
http://stats.stackexchange.com/questions/7948/when-is-it-ok-to-remove-the-intercept-in-lm
The lean in both cases is to include the intercept unless there is a strong theoretical reason.
A more satisfying answer would ...
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