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33

Some points to start with: i) these distributional conventions are at best approximations. They can be convenient models, but we shouldn't confuse that with the actual distribution of stock prices or returns. ii) stock prices are typically increasing (but in any case, have changing mean; the mean isn't stable). So when we're talking about the distribution ...


20

Edit: I realized the answer was lacking and have thus provided a more precise answer (see below -- or maybe above). I have edited this one for factual mistakes and am leaving it for the record. Different focus parameters: ARMA is a model for the realizations of a stochastic process imposing a specific structure of the conditional mean of the process. GARCH ...


12

ARMA Consider $y_t$ that follows an ARMA($p,q$) process. Suppose for simplicity it has zero mean and constant variance. Conditionally on information $I_{t-1}$, $y_t$ can be partitioned into a known (predetermined) part $\mu_t$ (which is the conditional mean of $y_t$ given $I_{t-1}$) and a random part $u_t$: \begin{aligned} y_t &= \mu_t + u_t; \\ \mu_t ...


11

Copying from the abstract of Engle's original paper: "These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance". Continuing with the references, as the author who introduced GARCH ...


8

You have misunderstood (I think it's pretty clearly explained at your link). Each row gives a set of weights, across the first six columns. Those do indeed sum to 1. Note that some weights are negative. The collection of rows defines the frontier.


8

They are not really different approaches in that they are solutions to different problems: one computes the sequence of filtering distributions $p(\beta_t|Y_{1:t})$, and the other the distributions based on all observations $p(\beta_t|Y_{1:T})$, for $t =1,...,T$. The smoother doesn't "hide underlying dynamics" but rather adjusts its state estimate (with ...


8

The return $Y_t$ represents the increase in the value of the stock as a percentage of its previous value. This return fluctuates a great deal in an economy, but in a properly functioning economy, it does tend to fluctuate around a small positive value. Consequently, the total stock price for a company tends to grow roughly exponentially over time. The ...


6

Yes the the series should be stationary. GARCH models are actually white noise processes with not trivial dependence structure. Classical GARCH(1,1) model is defined as $$r_t=\sigma_t\varepsilon_t,$$ with $$\sigma_t^2=\alpha_0+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2,$$ where $\varepsilon_t$ are independent standard normal variables with unit ...


6

As mentioned in the comments, the model you're looking for is Bayesian linear regression. And since we can use BLR to calculate the posterior predictive distribution $p(r_t|t, \mathcal{D}_\text{ref})$ for any time $t$, we can numerically evaluate the distribution $p(\text{CAR}|\mathcal{D}_\text{event}, \mathcal{D}_\text{ref})$. The thing is, I don't think a ...


6

The closest you'll get is probably one of Ruey S. Tsay's books; for example, Analysis of Financial Time Series. This books covers ARIMA models in Chapter 2 and then goes on to discuss other, arguably more appropriate, models for modelling financial time-series. As noted in the comments by the reference to the work of Eugene Fama, predicting stock prices is ...


6

I have never worked with recurrent networks, but from what I know, in practice, some RNN and TDNN can be used for the same purpose that you want: Predict time series values. However, they work different. It is possible with TDNN: Predict process' values Find a relationship between two processes. Some RNN, like NARX also allow you to do that, and it is ...


6

Let me save you a lot of money. There is a lot of information that you do not have, but I will give you a solution to a similar problem. For starters, let us assume that the chart you have drawn was, in fact, a real chart. For starters, there is a critical piece of information that you and most traders are lacking. The data in the chart doesn't have to ...


5

At heart, geometric means are what you want to work with because what you get back from investment is multiplicative - if you invest $1$ for two periods, getting $(1+r_1)$ and $(1+r_2)$ you end up with the product of the two single period amounts, $(1+r_1)(1+r_2)$ (since $(1+r_1)$ is available to invest after 1 period). The arithmetic mean would be what ...


5

When modelling stock prices, it is quite common to transform the original prices $P_{t}$ to logarithmic returns $r_t:=\ln(P_{t})-\ln(P_{t-1})$ and then employ a GARCH model. Logarithmic returns reflect price changes relative to price levels. If the price was fluctuating at around the same level, logarithmic returns would behave similarly to simple returns (...


5

As Stephen mentions, the confusion is between: (1) the CAPM vs. (2) the market model. Let $R^f$ denote the risk free rate. We often work with excess returns, which involves subtracting of the risk free rate. Some simple models for expected returns ``Market model" $$ R_t - R^f = \alpha + \beta\left(R^m_t - R^f \right) + \epsilon_t $$ $$ E\left[ R_t \right] ...


5

Let's say we have 25 portfolios $i=1, \ldots, 25$. Consider the time-series regressions for each portfolio $i$. $$R_{it} - R^f_t = \alpha_i + \beta_i \left( R^m_t - R^f_t \right) + \epsilon_{it}$$ If all the right hand side variables in your time-series regression are tradeable[1], then the $\alpha_i$ in your time series regression are equivalent to the ...


5

This tries to answer the original question and not get into Marcos's paper etc. If you think that the level of a variable ( say log price ) has information, then differencing the series ( to obtain returns ), throws out information. If you don't think that the level has information, then differencing is fine. Engle and Granger in their 1987 econometric ...


5

Stock prices can be thought of being a cumulative sum of mean-independent increments due to economic (and other types of) shocks. This is per definition a process with a unit root: $$ X_t=X_{t-1}+\varepsilon_t=(X_{t-2}+\varepsilon_{t-1})+\varepsilon_t=\dots=\sum_{\tau=0}^t\varepsilon_\tau. $$ (After the first equality, the coefficient in front of $X_{t-1}$ ...


5

It looks like you are supposing the covariance matrix of $(X_1,X_2,\ldots,X_N)$ is $$\Sigma = \sigma^2\pmatrix{1 & \rho_1 & \rho_2 & \cdots & \rho_{N-1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{N-2}\\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{N-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \rho_{N-...


4

The ARMA and GARCH processes are very similar in their presentation. The dividing line between the two is very thin since we get GARCH when an ARMA process is assumed for the error variance.


4

e.g. if I start with 100 $ and if my stock then goes up +10%, and then from 110 it goes down -10%, the mean of the return would be 0, This is not necessarily about arithmetic or geometric mean. This is also about simple or continuous return. Consider this: $100(1+0.1)(1-0.1)=100(1-0.01)=99$ $100 e^{0.1}e^{-0.1}=100$ In the first case I assumed that ...


4

$\hat{L}$ is the likelihood: it's a number that comes from a Maximum Likelihood Estimation (MLE), which estimates the value of your model's parameters. The Likelihood $L(\theta | X)$ says how likely the parameter values in vector $\theta$ fit the data in $X$. It is derived from the Bayes theorem: $$L(\theta | X) P(X) = P(\theta) P(X | \theta) \quad \...


4

Bailey and Marcos López de Prado designed a method do exactly that. They use the fact that Sharpe Ratio's are asymptotically normal distributed, even if the returns are not. here gamme_3 and gamma_4 are the skewness and kurtosis of the returns. They use this expression to derive the Probabilistic Sharpe Ratio. SR^* is the value of the sharpe ratio under ...


4

Later edit: I give what seems to be a better solution here. Note that the paper uses a different parameterization from the form given in the question. As Yves noted in comments, it uses $-a$ in place of your $a$ (both are common parameterizations; the only difficulty may be when it is unclear which parameterization is being used). If you convert answers ...


4

If I recall correctly, it has something to do with a constant called Jensen's alpha and the extension to something called a multi-factor model. During my classes (and also wikipedia), the CAPM was stated as follows: \begin{align} \mathbb{E}[R_i] = R_{f} + \hat{\beta_{1}}(\mathbb{E}[R_{m}] - R_{f}) \end{align} When taking excess returns, one would simply ...


4

If you sort the t_id column in the test dataset you will see that it goes from 1-36000. I would assume that it refers to "trade id". The way financial time series forecasting works is the you usually take lagged values of features at time t-1 and use them to predict the target value at time t, thus I would assume that all the features from 1-21 are lagged ...


4

Besides using Pearson correlation, you can also use rank correlation such as Spearman or Kendall correlation. You can also display the scatterplot of the ranks whose distribution (called the empirical copula) is an estimator of the underlying copula encoding the `true' dependence between your time series. In pseudo Python (rather transparent in R): n = len(...


4

The probability that a gaussian random variable is positive is maximized by maximizing the quantity $\frac{\mu}{\sigma}$. For an affine combination of two gaussians we have $\mu = x \mu_1 + (1-x) \mu_2$ and $\sigma^2 = x^2 \sigma_1^2 + (1-x)^2 \sigma_2^2$, so we would like to maximize $$\frac{x \mu_1 + (1-x)}{\sqrt{x^2 \sigma_1^2 + (1-x)^2 \sigma_2^2}}$$ ...


4

Since my similar question was flagged as duplicate (good debate in the comments!), I came across Simon Kuttruf's explanation on Medium: for integer orders of differencing only a (small) finite set of past values is reflected in the resulting differenced series: the preceding value in first order differencing, two preceding values for second order ...


4

I agree with @1muflon1(+1) about systematic removal of outliers, for no reason other than size. That is almost never useful. (Of course, outliers obviously due to data entry errors or equipment failure should be removed: A basketball player listed at 9' 6" tall, reports of a temperature -60$^o$F in Hawaii, etc.) With sample sizes is the thousands, t tests ...


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