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10

Looking at transaction prices, they would occur at the market bid if the active part is a seller, and at the ask if the active part is a buyer. With a random flow of sellers and buyers, the price will bounce between the bid and ask prices, creating a negative autocorrelation in returns. This penomenon is known as the bid-ask bounce, and has been discussed ...

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The correct answer has some intuition though it doesn't generalize to continuous time very easily: Think about the paper below like this: $Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)$ The generalization is slightly hard because the dynamics of $\mu$ and $\sigma^2$ could be dependent for arbitrary returns. You can use a GMM estimator to derive the asymptotic ...

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The answer is that it depends. In addition to the Lo paper above, there are a number of excellent references that go into depth about annualizing or time scaling non-i.i.d. returns, one of which is Roger Kauffman, "Long-Term Risk Management", 2005 which can be found at http://www.rogerkaufmann.ch/all-Budapest.pdf. There are some well known cases where the ...

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In terms of interpretation, an $MA$ model simply means that the time series is a function of the error from previous periods. You might find it informative to consider plotting simple $AR(1)$ models alongside various $ARMA(1,1)$ to develop a more intuitive understanding. For instance, the $AR(1)$ model (chosen as it is common for financial time series) $$x_{... 4 Autocorrelation is the correlation of a series with itself. Suppose X = {X_1, X_2, X_3, ...} is your time series. Then the autocorrelation between X_t amd X_s is:$$ \frac{E[(X_t-\mu_t)(X_s-\mu_s)]}{\sigma_t \sigma_s} $$This can be simplified quite a lot if the series you have is stationary (a common assumption), in which case the autocorrelation ... 4 You need to compute the autocorrelation of the log returns r_t, not of the prices, p_t. The relationship of the log return series to the price series is$$ r_t = \log \frac{p_t}{p_{t-1}} $$The price series is obviously very autocorrelated, since today's price is yesterday's price plus small delta. 4 I think @zer0hedge has constructed a clever example by which to demonstrate what is implied by the stylized fact by which volatility begets volatility. It is correct to conclude volatility bursts are a type of absolute autocorrelation. All volatility bursts display characteristics of autocorrelation of absolute returns, but will all types of autocorrelation ... 3 TL;DR: the test statistic's distribution is N(0,1) A bit more information about the Automatic Variance Ratio Test: H_0: {\Delta}r_t is serially uncorrelated (where {\Delta}r_t=r_t-r_{t-1}) H_1: {\Delta}r_t is serially correlated The test statistic is VR=\sqrt{T/l}[\hat{VR}(l)-1]/\sqrt{2} \quad {\xrightarrow{d}} \quad N(0,1) The d over ... 3 What is the mathematical basis to say that u^{2}_{t}/\sigma_{t}^{2} will exhibit little auto-correlation in the series? Let's r_{t} be a series of returns and let's assume (Assumption I) it follows a covariance stationary process defined as : r_{t}=\sigma_{t} z_{t} where z_{t} is i.i.d with E_{t}(z_{t})=0 and Var_{t}(z_{t})=1 ; Then  ... 3 I'm not a time series expert but one idea occurs to me: look at the distribution of the increments if Z(t). If the w are stochastic , that distribution should have fat tails relative to the distribution that is generating the Levy process. 3 You should check for autocorrelation. However, its presence does not necessarily mean your model will produce inaccurate figures. The ARCH family of models were developed to help analyze the volatility of a time-series. This data is assumed to display a degree of heteroskedasticity. Using the GARCH model, small amounts of auto-correlation (not of practical ... 3 Hi: Subtract k from z_t and add k to z_{t-k}. Then you have cov(z_{t-k,} z_{t}) which by definition is \gamma_{-k}. But, by stationarity, this has to be equal to cov(z_{t}, z_{t-k})= \gamma_{k} because the covariance is only a function of the lag difference. 3 Just by looking at the graphs, I'd say: Unit root Constant series Seasonality AR model No AC No AC 3 EWMA (and other sort of moving averages) introduces positive autocorrelation into otherwise uncorrelated returns. The fitted values of EWMA are linear combinations of past returns, and the constituent elements of these combinations overlap. Therefore, positive autocorrelation arises. If you have autocorrelated returns to begin with, they would in all ... 3 In its simplest terms, imagine you were just using the yield curve as your single predictor of recessions. Suppose (horribly simplistically) that curve inversions tend to signal downturns in 12-18 months time. The curve 12-18 months ago is thus a relevant variable for whether the economy is going into recession or not today. It might also be the case that ... 3 One option is just to fill them in - interest rates don't usually jump around, so interpolating from surrounding data would be unsurprising. If you want to know what effect that is having, by all means mark those you're filling in and duplicate the analysis with a shift to all those filled in rates. If you have an index rather than a set of term rates, then ... 2 I would advice you not to do any overlapping analysis. The results will be hard to interpret and misleading. I have seen many "practioners" looking at histograms of overlapping returns. They saw interesting patterns and found funny explanations - which were simply wrong. If you are new to econometrics then correction methods (do there exist helpful ... 2 1.Is it correct, that the coefficients are now different to the coefficients of the arima output? It seems right that the ARMA coefficients are different. Indeed, in the second model, the GARCH component will capture fluctuations that the ARMA component will not have to capture, resulting in different ARMA parameter estimates. 2.This is the acf of the ... 2 1.) Autocorrelation is the correlation of a time series against the lagged version of itself. 2). First autocorrelation is the correlation of the time series against the lag(1) version of itself. Let's look at the example below Period_Numbers = [1,2,3,4,5,6,7,8,9,10] Time_Series = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100] First Autocorrelation is ... 2 If you are using Spatial Econometrics toolbox in Matlab you could use the lrratio function which implements a sequence of such tests beginning at a maximum lag (specified by the user) down to a minimum lag (also specified by the user). (more info in http://fmwww.bc.edu/ec-p/software/matlab/mbook.pdf) 2 I can offer my opinion in response to your first two questions: 1.) Unfortunately, this is one of the problems with numbers; the answer is that if the observation is outside of the confidence interval by even a millionth of a percent, it is significant. If it is below by even the smallest amount, it is not significant. Changing your significance level or ... 2 The direction of the relationship cannot be determined from just this information (a set of correlation coefficients). You need to estimate a model of volume based on lagged volume and lagged returns, checking if the lagged return terms are significant. Then as a second step you estimate a model of returns that includes past returns and past volumes and see ... 2 The high serial correlation you are getting in the first case is a spurious correlation. The correct way to do it is with returns. The price series has a unit root. You need to take diff(log(prices))) in order to have a stationary time series, on which you can then estimate autocorrelations, auto regressive coefficients, etc. properly. This was shown by ... 2 Your code basically implements the assumption that you cited: The volatility of return processes is not constant with respect to time. Whether it's a single burst or some kind of a fancy function \sigma_t is not important here. The fact is that your volatility is time varying. You may call it piece-wise constant, but it still is characterized as time ... 2 Such volatility pattern is a well-known stylized fact of financial time series (see Cont, Rama. Empirical properties of asset returns: stylized facts and statistical issues. (2001): 223-236 for more details) which is called volatility clustering. Qualitatively, it means that high returns are likely to be followed by high returns, the same applying for low ... 2 You are correct in that the series is not stationary. The ADF test isn't designed to test for stationarity outside the center of location. You are not going to be able to use the square root rule to extrapolate because you have significant autocorrelation of the variances. I do have a suggestion on your problem by noting that returns are not data. Prices ... 2 If you are predicting the return from time "i" to time "i+l" then you cannot use any information beyond time "i" to train your model. As it appears you are getting returns from "i-5" to "i" and assuming that this same relationship will hold into the future from day "i" to "i+5". In theory there is nothing glaringly bad about this approach, but I would ... 2 Note that the function f only depends on |t-u|, meaning it is actually symmetric: f(x)=f(-x). Doing the change of variable \tau:=t-u:$$\begin{align} \int_0^Tdu\int_0^Tf(t-u)dt &=\int_0^Tdu\int_{-u}^{T-u}f(\tau)d\tau \\ &=\int_0^Tdu\left(\int_0^{T-u}f(\tau)d\tau+\int_0^uf(\tau)d\tau\right) \\ &=\int_0^T{du \left(\int_0^T{f(\tau) \textbf{1}...

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Let $n$ be the number of stocks (here $n=3$) Let $T$ be the number of sequential returns to generate (for example $T=12$ if you want to generate a year's worth of monthly returns) Let $M$ be the number of alternative scenarios to generate (for example $M=1000$ to generate 1000 different outcomes) Then, Step 1. You generate a $n \times T$ matrix RETS of ...

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in method 1 you did not use the correct definition of the covariance. For two random variables $X$ and $Y$ we have that $$Cov(X, Y) = E[XY] - E[X]E[Y].$$ Also, we can use that the covariance is linear in both arguments (this is the bilinearity of the covariance). Therefore \begin{align} \gamma_X(h) &= Cov(X_{t+h}, X_t) = Cov(a+ bZ_{t + h} + cZ_{t + h -...

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