# Tag Info

19

This isn't really an answer, but it's too long to add as a comment. I've always had a real problem with the correlation/covariance of price. To me, it means nothing. I realize that it gets used (abused) in many contexts, but I just don't get anything out of it (over time, price has to generally go up, go down, or go sideways, so aren't all prices ...

19

The standard story (also told by @vonjd) is of "The Drunk and Her Dog". This is based on "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction" (1994). The story is itself based on the standard illustration for a random walk known as the "drunkard's walk". The Dickey-Fuller test is used to check for a unit root. It can be used as ...

14

Variance ratio tests have been used numerous times to show that financial asset prices do not follow a random walk. You can for example look at -Lo and MacKinlay : Stock market prices do not follow a random walk : http://press.princeton.edu/books/lo/chapt2.pdf (US Stocks) -Hoque, Kim, Pyun: A comparison of variance ratio tests of random walk: A ...

14

Yes, the weights of the first eigenvector of a covariance matrix represent the market factor and also the largest source of systematic risk (variation of returns). Why PCA? Well, PCA simply identifies the eigenvector that maximally explains the variance of the system. It turns out that this is the "market factor" - i.e. the tendency of securities to rise ...

11

Two time series $X_1$ and $X_2$ are cointegrated if a linear combination $aX_1+bX_2$ is stationary i.e. it has constant mean, standard deviation and autocorrelation function for some $a$ and $b$. In other words, the two series never stray very far from one another. Cointegration might provide a more robust measure of the linkage between two financial ...

10

Correlation is much more widely used concept and it has much more "informal" meanings. If we have only two random variables $X$ and $Y$ then correlation is simply a measure of linear dependence between the two variables: $$corr(X,Y)=\frac{cov(X,Y)}{\sqrt{var(X)var(Y)}}=\frac{EXY-EX\cdot EY}{\sqrt{var(X)var(Y)}}$$ If correlation is -1 or 1 then the two ...

10

There are a number of different tests that are generally used to compare samples to different distributions, such as Jarque-Bera, Anderson-Darling, and Kolmogorov–Smirnov (see this related question). In your case, with just the standard deviation and mean, there isn't a whole lot to say. You need to assume a distribution (e.g. normal). You would be able ...

9

From remote memory, The first question is Yes/No question. Is there any stationary, i.e. I(0), time series for different levels of combination r? This question is answered by your first table. For example, if [r=2]'s test stat is say 7 while the critical value of 99% confidence is 6.6 like your example, then I have over 99% confidence to say that all ...

9

If the means are similar, then K-means will not do a great job. I would generate new features, perhaps based on higher moments of the distribution or some other properties (auto-correlation, summary of spectral density, etc.). Using this new set of features, If you see separation of two curves when you plot draws in feature space then k-means would be an ...

8

The somewhat tongue-in-cheek blog post http://www.portfolioprobe.com/2010/10/18/american-tv-does-cointegration/ includes the example of two classes of shares on the same company. In this case you have two assets that are essentially the same but with a few details different. The buying and selling of these assets will make the prices fluctuate from each ...

8

Actually, co-skewness is represented by a rank 3 tensor, rather than a matrix. I'm going to reproduce the formulation from Bhandari and Das, Options on portfolios with higher-order moments, but I'll add and omit some details. The co-skewness tensor is $$S_{ijk} = E \left[ r_i \times r_j \times r_k \right] = \frac{1}{T} \sum_{t=1}^T r_i(t) \times r_j(t) ... 8 Some cynical but functional definitions: It's what you can't model if you're not using tick by tick data It's what proper quant pricing theory doesn't know how to model yet It's information (order book behavior) that reflects momentary fluctuations in the supply/demand of a given contract, rather than its underlying value (eg an arbitrage free price) ... 8 Treat the estimate of standard deviation as a random variable. Then you can bootstap the sample estimate and generate t-statistics and associated confidence intervals for your statistics. I describe a generic boostrap process on this post. 8 "Treshold Garch" or T-Garch models are designed to capture this asymmetry. See this exposition by U. Chicago's Ruey Tsay who has a terrific text on time-series models in "Analysis of Financial Time Series". You can use the structure of the T-Garch models to simulate data with this property. There is a package called fGarch that creates APARCH models. A ... 8 Just to be painfully clear, it only seems to make sense to consider the logarithm of returns, i.e. X=\log (1+\frac r{100}) for a simple return of r\% in an arbitrary period because this is what sums when returns are temporally aggregated. A basic property of cumulants is that cumulants of all orders are additive under convolution, for which a proof can ... 7 Why don't you try it and report back? Recall, though, that while a random walk is often a rather competitive forecast, realized data is understood to have weak dependence especially in higher moments. Having worked a bit with DieHarder, I'd suspect it to reject a number of series. But the proof is in the pudding... 7 From Quantitative Trading by Ernie Chan : "Correlation between two price series actually refers to the correlations of their returns over some time horizon (for concreteness, let's say a day). If two stocks are positively correlated, there is a good chance that their prices will move in the same direction most days. However, having a positive correlation ... 7 Let us test that x and y are co-integrated, say that x_t, y_t \sim I(1). In the Engle-Granger we test stationarity of the error term in$$y_t = \alpha + \beta x_t + u_t$$which we estimate as$$\hat u_t = y_t - \hat \alpha - \hat \beta x_t and find that $\hat \alpha =0$, $\hat \beta = 1$, and $\hat u_t = 0 \; \forall t$. So now when we Dickey-Fuller ...

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

Before I try to answer your question we need to establish a difference between what one wants to analyse. It is true that before modern time-series methodologies were developed, researches used "correlation" between prices as a means of analysis. However, since a Price (at a specific moment in time) is 1 value, it makes no sense to compare 2 prices with ...

7

Nick Higham happens to have given a talk on this very subject this summer; he continues to actively work to improve nearest correlation matrix algorithms. You can see his talk and notes here: http://mxm.mxmfb.com/rsps/ct/c/629/r/90368/l/48110

7

The term has a different meaning to different people. to econometricians, microstructure noise is a disturbance that makes high frequency estimates of some parameters (e.g. realized volatility) very unstable. Generally this strand of the literature professes agnosticism as to the its origin; to market microstructure researchers, microstructure noise is a ...

7

There are rigorous econometric definitions, as has already been eluded to by others. For practical purposes, microstructure noise is a component of a price process that exhibits mean reversion on some (possibly time-varying) frequency. This reversion is particularly attractive to liquidity provisioners, who seek to profit from this noise component (along ...

7

These patterns are of course well-known enough to have been "priced in" to the financial markets. Jump diffusions are a classic way to capture the phenomenon, and often have closed-form option pricing formulas associated with them. The implied option skew, for example, gets a lot flatter when you use a JD model. Jump diffusions are often combined with ...

7

Attilio Meucci does some very interesting things with PCA. See e.g. his paper on managing diversification which makes heavy use of it (and explains it very intuitively along the way): Managing Diversification by Attilio Meucci

7

The clearest and most intuitive article I have seen so far is Kritzman et al., Regime Shifts: Implications for Dynamic Strategies in FAJ (May / June 2012) It not only shows how you can use HMM for financial modelling but it also goes through the actual estimation algorithm (Baum-Welch) step-by-step and even gives full MATLAB-code. From the abstract: ...

6

Correlation between two financial time series should be calculated as correlation of the returns (or log returns for prices). There is absolutely no relationship between correlation of the returns and cointegration. Two correlated time series can be cointegrated or not cointegrated. Two cointegrated time series can be correlated or not correlated. ...

6

Two integrated series $X_t$ and $Y_t$ are cointegrated if their linear combination (some, not any) $\alpha X_t+\beta Y_t$ is stationary. If you have $P(X_t=Y_t)=1$ for all $t$, then $P(\alpha X_t+\beta Y_t=(\alpha+\beta) X_t)=1$. So according to definition of cointegration $(\alpha+\beta) X_t$ should be stationary, which is identical to $X_t$ being ...

6

The short answer is that I don't know, but your question gives some hints about how to find out. The key thing for me is that you want a minimum variance portfolio. I don't think you should be thinking about some abstract mathematical operation that is "best", but rather look over a few mathematical operations and see which seems to work best for your ...

6

Here is my code: require(xts) require(urca) # Load data gld <- read.csv("~/Downloads/CBA.csv", stringsAsFactors = FALSE) gdx <- read.csv("~/Downloads/WBC.csv", stringsAsFactors = FALSE) # Convert to xts gld <- xts(gld[, 4], as.POSIXct(gld[, 1], tz = "GMT", format = "%Y-%m-%d", tzone = "GMT")) gdx <- xts(gdx[, 4], as.POSIXct(gdx[, 1], tz = ...

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