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$\alpha=0$ does not imply constant volatility. Consider just a simple Garch(1,1): $$\sigma^2_t = \omega + \alpha \eta_t^2 + \beta \sigma^2_{t-1}$$ Note that: $$\sigma^2_t = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2- \sigma^2_{t-1})$$ Now add $\eta_{t+1}^2$ to both sides: $$\eta_{t+1}^2 = \omega + (\alpha + \beta) \eta_t^2 - \beta ... 3 Since you're asking on a quant finance forum, the mathematical approach would be Decide on a model that the stock price follows, and Compute the expected value of the price, conditional on the most recent price. A famous model, made ubiquitous by Black, Scholes and Merton, is a geometric Brownian motion. Under this model, the stock price S_T at time ... 2 Interesting question, as All the answers (including mine) could not be generalized unfortunately. As far as I am concerned, I use a univariate EGARCH for risk modelling purposes (Filtered Historical Simulation (FHS), etc.). 1 - EGARCH, merely because GARCH models do not take into account so-called leverage effects, which is crucial to me for skewed and ... 2 v should be the total number of parameters (constants + AR + MA + GARCH + ARCH). I disagree with @kiwiakos, the student t(df) distribution is used because we are using standard errors which are estimates of standard deviations (and not true standard deviations) to compute the statistic. That is the reason why we use student t test eventhose the ... 2 Before you start asking about the number of dof, how do you know that the finite sample distribution of parameters is student-t? I don't think it is. In linear regression they are student-t because of linearity and under assumption for the residual distribution. In Garch you can just say that if you estimate using max-likelihood then asymptotically (not ... 2 Did you try solving for w_k?$$\bar{r}_t = \sum_{k=0}^p w_k r_{t-k}\bar R = W R$$Since you probably have t>>k, you can solve for W using OLS$$\bar R = W R +\varepsilon$$-- UPDATE You can try applying Kalman filter. Here, your state evolution is$$r_t=\mu+\varepsilon_t$$. You introduce new vector x_t=(r_t, r_{t-1}, \dots, ... 1 I can't comment yet on the topic due to my reputation level (so I will throw an answer up) but having just done my MFE capstone research on EVT implementation for VaR. According to my advisor who was a director of a quant research group at Citi before returning to academia, not many people are doing this. My research was to start collecting data comparing ... 1 1- It seems to me there is a problem in the original code the variable b should be defined as b= sqrt(1 + 3*lamda^2 - a^2) 2- The likelihood is defined just after equation 8. in the paper. You have to take into account the  \frac{1}{\sigma} term (in  \frac{1}{\sigma} \times g(..)  , ie to scale the densitie) . So the - 0.5*log(h(t)) refers to this ... 1 Here is an MLE I built that uses logistic mapping. %MLE iterator: for cxm = 1:cxmax for cxth = 1:wx; %thx %Incr. theta within asymptotic min and max. thi1 = thA1(cxth,1); mint = thA1(cxth,2); maxt = thA1(cxth,3); thix = -log((maxt - mint)/(thi1 - mint) - 1); %Logistic inverse. if rand > 0.5; signx = -1; ... 1 If \log{(|R_t|)} is your first term, I'm not sure why this is a matrix. Modulus (determinant herein) applied to a matrix R_t gives a scalar. If your implementation in python produces a matrix, that's likely because modulus is treated as an element-wise abs() function for each element of a matrix. It may be easier and faster to use rugarch (univariate ... 1 The \lambda value used in the original paper is arbitrary, but you can estimate that by assuming (in the simplest case) 2 assets and running the following model: \sigma^2_{12,t+1} = \lambda$$*$$\sigma^2_{12,t-1}$$+$$(1-\lambda)$$r_{1,t}$$*$$r_{2,t}$; given$r_{1,t}$and$r_{2,t}$respectively as the returns for the asset 1 and 2 and ... 1 Yes, these are the fundamental building blocks for a money making strategy. To partially solve the issues you mention (small/low positive means/profits with large standard errors), you can investigate on many assets simultaneously. The idea is to take the advantage of Central Limit Theorem. Assuming the signal for each asset are i.i.d., and each signal ... 1 Approach 1 is parametric regression, whereas approach 2 is non-parametric regression. How are they related: non-parametric regression models the entire distribution of all possible function forms, and then do the integration to calculate a single value E[Y|X]. It is function-form free. In contrast, parametric linear regression ASSUMES that the function ... 1 You can apply the Kolmogorov-Smirnov test. I simply quote from the entry: "The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples." There is an R-implementation too. 1 Thanks @Aksakal for suggesting Kalman Filter. Here I provide more details. We will view it as a state-space model: $$\begin{split} z_t &= A_t z_{t-1} + B_t u_t + \epsilon_t, \\ y_t &= C_t z_t + D_t u_t + \delta_t, \\ \epsilon_t &\sim \mathcal{N}(0, Q_t),\ \delta_t \sim \mathcal{N}(0, R_t), \end{split}$$ where$z_t\$ is the latent variable, ...