# Tag Info

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Just about every introductory Econometrics class teaches that the violations of BLUE ("Best Linear Unbiased Estimator" -- the properties of linear least squares) are invalid standard errors in the case in the heteroscedasticity, so while your parameter estimates are still valid ("unbiased") your inference may be off invalid estimates (!!) in the presence ...

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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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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_{... 5 Apparently yes, (I haven't verified the math but have no reason to doubt it). For this simple case you can find a closed form in the following paper: Jeff A. BILMES: What HMM can do The closed form is given on part 4.4 of the paper but the whole thing is worth reading as it clearly shows the main properties of these models. You can also note that ... 5 Thanks gappy for your precise response. However the answer to this auto-correlation is much more important than an academic discussion of which portfolio performance ratio is best. Auto-correlation distorts max draw-down calculations raising the question of whether the (positive) auto-correlation will continue in the future producing large draw-downs, or ... 5 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 Simple...because you are interested in deviations from a metric, and not whether it deviates above or below. The very definition of volatility is a "measure of deviation". Squaring returns or using the absolute values just eases the calculation to arrive at a deviation measure. Otherwise volatility would have to be calculated in other ways as positive and ... 4 To simplify, consider the errors rather than the returns. The variance is effectively the average of the squared errors, while absolute deviation is the average of the absolute errors. So plotting the squared errors or absolute errors over time could give an indication of whether the variance or absolute deviation is constant over time. Since variance is ... 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 W.l.o.g we use the initial condition S(0)=1 and define \gamma:=\mu-\frac{\sigma^2}{2}. Hence we have the dynamics$$S_t=e^{\gamma t +\sigma W_t}$$By definition Cov(X,Y)=E((X-E(X))(Y-E(Y))=E(XY)-E(X)E(Y), where the last equality follows from the linearity of the expectation. Note \gamma t+\sigma W_t is normal distributed with mean \gamma t and ... 4 This is a partial explanation in that trading strategies with longer horizons have higher information ratios, t-statistics, slope coefficients, and R^2 in general. In other words, if information ratios for both strategies are identical then the longer-term trading strategy is already worse. John Cochrane illustrates how longer horizons have higher t-stats ... 4 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 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 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 ... 3 Autocorrelation is usually a problem when you are doing the analysis of your error terms. When you build a model, you expect that the error term will have non significant autocorrelation. It is simple to understand: If your error term still have autocorrelations it certainly means that you are missing some information that could be introduced in your model. ... 3 Have you considered a Monte Carlo simulation on your returns? Then you could look at the distribution of Maximum Drawdowns. 3 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 ...

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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)

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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 ...

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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 ...

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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 ...

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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 $... 1 The direction of the relationship cannot be determined from just this information (a set of correlation coefficients). You need to estimate a model of volume based on lagged volume and lagged returns, checking if the lagged return terms are significant. Then as a second step you estimate a model of returns that includes past returns and past volumes and see ... 1 The high serial correlation you are getting in the first case is a spurious correlation. The correct way to do it is with returns. The price series has a unit root. You need to take diff(log(prices))) in order to have a stationary time series, on which you can then estimate autocorrelations, auto regressive coefficients, etc. properly. This was shown by ... 1 It is not entirely clear what you're after, since Method 1 from the question is a statistical model, while Method 2 is a statistical test. From the initial question, I'm going to make the assumption that what you're actually after is some number that summarises "momentum" on a given day. If this is the case, I would weakly prefer the Ljung-Box test ... 1 you hypothesize that your data is generated by the following process:$y_t=\phi_0+\sum_{k=1}^P\phi_ky_{t-k}+\varepsilon_t$, where$\phi_k$are your autocorrelation coefficients, and$\varepsilon_T$- random errors. Next, you estimate your$\phi_t$using one of the methods of estimation of autoregressive processes AR(P) of order P, e.g. see AR(P), there's no ... 1 1.) Autocorrelation is the correlation of a time series against the lagged version of itself. 2). First autocorrelation is the correlation of the time series against the lag(1) version of itself. Let's look at the example below Period_Numbers = [1,2,3,4,5,6,7,8,9,10] Time_Series = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100] First Autocorrelation is ... 1 You are probably computing autocorrelation in the prices. If you compute autocorrelation between the returns or log returns then you will not see the results you are getting. This is because: Tomorrow's price will always be influenced by lagged prices and the series will not look weak stationary if you plot it. The direct differencing doesn't help either ... 1 Check your calculations, gold prices are indeed auto-correlated. acf(diff(log(OilGold$price_gold))) will yield no auto-correlation in gold log-returns.

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I think Matt Wolf has missed the point. Equity risk premium prediction models actually tend to perform better over longer horizons than shorter horizons. Why this is the case is disputable, however, I think it largely reflects the mean-reverting qualities of equity markets. With regard to the OP's point about over-lapping time periods, it is OK to us ...

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