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I will assume a white noise is a process $(\varepsilon_t)$ with zero mean, no autocorrelation and constant variance $\sigma^2 > 0$ while a random walk is a process $(x_t)$ defined by $$x_{t+1} = x_t + \varepsilon_{t+1}$$ where $\varepsilon$ is a white noise. 1) No since $Var(x_{t+1}) = Var(x_t) + Var(\varepsilon_{t+1})$ is stricly increasing while ...

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There is a lot of ways to understand why stationarity allows to apply usual time series analysis. Here is one more. Very often, the theoretical justification of what you do in time series need to be able to identify the mean formula and the expectation: $$\frac{1}{N}\sum_{n=1}^N X_n \underset{N\rightarrow +\infty}{\longrightarrow} \mathbb{E} X,$$ where the ...

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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 ...

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The concept of 'mean reversion' is tricky in continuous time. Most people would call 'mean reverting' a process where the drift pulls back towards a long run mean, and I assume that this is what you also mean. Something like the drift of an OU process. However, in continuous time the 'pull' can be generated by the volatility. For example the process $$dX_t ... 5 I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is perfectly symmetric whereas the real one takes longer for going up and then crashes in a relatively shorter time frame. This is a known phenomenon in real financial ... 5 In terms of interpretation, an MA model simply means that the time series is a function of the error from previous periods. You might find it informative to consider plotting simple AR(1) models alongside various ARMA(1,1) to develop a more intuitive understanding. For instance, the AR(1) model (chosen as it is common for financial time series)$$x_{...

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To quote Wikipedia: In mathematics, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance, if they are present, also do not change over time and do not follow any ...

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We can talk about whether a strictly stationary or weakly stationary process might usefully describe that data. My answer to both would be yes. I also have issues with other text that people have written here. A review of mathematical definitions: A stochastic process $\{X_t\}$ is called strictly stationary if it's joint distribution function $F(X_{t}, ... 5 If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (for equity or returns based then you would need log version of this): $$S_t = S_0 + \sum \Delta S_i$$ where$\Delta S_i = S_i - S_{i-1} $is a wiener process. ... 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 O-U is continuous time mean reverting process, hence used to model stationary series. It has closed form analytic solution. This allows insight into stationary processes and act like asymptotic limiting case for calculating coefficients that matter. EDIT: You can see AR(1) below $$x_{k+1} = c + a x_k + b\varepsilon_k$$ and by substituting c=θμΔt, a=−θΔt ... 4 Regarding the relationship between white noise and a random walk, I would put it this way: a random walk is integrated white noise. [And vice versa we get a white noise when we differentiate/difference a random walk]. Or to put it in quant finance terms: white noise is like the daily changes in the S&P in points, a random walk is the S&P daily level ... 4 The point of confusion may be in thinking that a predictable price process is synonymous with a mean-reverting process while using the definitions in these papers, it's actually the opposite! In the context of these papers, a random walk would be 100% predictable: the unpredictable component of a random walk (i.e. the period specific shock which has finite ... 4 Hi: It depends on what the DGP of the original process is. Is the process trend stationary or difference stationary ? If it's trend stationary then de-trending is the way to go. If it's difference stationary, then differencing is the way to go. The two models are quite different: Trend Stationary:$y_t = \beta_{0} + \beta_1 \times t + \epsilon_t$... 3 Let's consider the following example: the process is initialized randomly with$\pm1$and then stays there forever. Seems stationary to me, but it would never cross its mean. 3 Saying that you can't analyze something as is does not make it garbage. You can't eat flour "as-is", but that doesn't mean you throw it out. In order to use "standard" analysis tools, you must first transform the series into something compatible. Some examples of such a transformation include k-th order differences or a log transformation. These ... 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 ... 3 Here is a possible explanation. Consider$X_t \sim N(0,1)$and$Y_t \sim N(1,1)$. Then$(X_t)_0^n$and$(Y_t)_0^n$are realizations from stationary time series and I would expect the null hypothesis of stationarity not to be rejected (compatibly with the size of your test). Instead, the sample$(Z_t)_1^{2n} = (X_1, \dots, X_n, Y_1, \dots, Y_n)$is drawn from ... 3 Ergodicity is connected to mixing, meaning there is one limiting distribution and it is used for time averages too. If you take a process in the real numbers that starts at a random value and then just stays at its initial point, it is stationary but not ergodic because there is not a unique distribution for time averages. 3 Just by looking at the graphs, I'd say: Unit root Constant series Seasonality AR model No AC No AC 2 You should de-trend to whatever frequency scale you are testing. I.e. 1 min means de-trend 1 min data. Merely by moving to higher frequency data, you are eliminating much of the systematic bias present at higher scales -- as 1) you have many more samples to compare (minimizing standard error) 2) At smaller intervals, the drift component also shrinks ... 2 Any data transformation to assure stationarity eliminates part of the signal in many cases the signal is not completely eliminated so you can still perform the required analyses but in some others as may the your case the signal is erased and the results seem to indicate your variables have lost predictible power although its predictive power may have been ... 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 Consider the following AR(1) process with i.i.d. normal errors that have zero mean and finite variance$\sigma^2>0$, $$x_t = (1-\rho)\mu + \rho x_{t-1} + \epsilon _t$$ Now suppose$ \rho = 1/2$and$\mu = 1$. This process does not have a unit root, and it is not mean stationary. At any point in time, the process has finite variance, although as time ... 2 I think you misunderstood the definition. Be stationary does not mean not depend of the time as you can check here. (Sorry for putting an wikipedia link here as I suppose you may have read it) Another way to think is that the law any increment of the process is given by a same function of the difference of time. More precisely$\forall ~t_2\geq t_1,$:$$\... 2 For both time-series, just plot the log returns. You will see that one is not a Random-Walk .. the S&P500 since you will get values that far beyond the normal distribution. Just watch this video by Benoit Mandelbrot (starting at 11min:54sec). Looking at both graphs, your eyes can fool you making you believe that both are generated by Random Walks... 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 This looks confused? I don't understand what you're saying in the second paragraph... Comment 1: "Best" forecast depends on what you mean by "best." Let$Y$be a random variable and$\mathcal{F}$be the information set. The "best" forecast depends on what the loss function is. If you're minimizing the expectation of squared loss: \begin{equation} \begin{... 2 Following Meucci (Risk and Asset Allocation book, page 112-113) you should use "change of yield to maturity" (simple change, not percentage) since they represent Fixed Income´s invariant. Change of yield to maturity would be the equivalent to change in price (in ln terms) for equities. 2 Let me try to write formulae to explain the differences: When$X_t=a+b\,t + c\,\xi_t$, where$\xi_t$is an iid centered and reduced noise (ie$\mathbb{E}\xi=0$and$\mathbb{E}\xi^2=1$. With$X_(t+1)-X_t=b + c\Delta\xi$, you read that you increased the amplitude of the noise$\xi$by a factor$\sqrt{2}$, you removed$a\$ and you have no more time dependent. ...

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