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Before getting to my answer, I think I should point out that there is a mismatch between your question title and the body of the question. Bootstrapping time series is in general a very wide topic that must grapple with the various nuances of the particular model under consideration. When applied to the specific case of cointegrated time series, there are ...


5

Writing the model using backshift operator polynomials, you have $$ y_t = \theta_3(B)\mu_t \tag{1} $$ and $$ \phi_2(B)\mu_t = \epsilon_t. \tag{2} $$ Applying $\phi_2(B)$ to both sides of (1) and using (2) yields $$ \phi_2(B)y_t = \theta_3(B)\phi_2(B)\mu_t = \theta_3(B)\epsilon_t $$ which shows that $y_t$ is ARMA(2,3) with autocovariance function that can be ...


4

Of course you can look back in time--but you cannot look forward. That distinguishes the past from the future. The following brief, elementary account uncovers the basic underlying concepts and reveals how "time's arrow" is modeled in statistical applications. The backshift operator isn't limited to stochastic processes: it applies to arbitrary sequences ...


4

Evidently $\operatorname{Var}(y_t)$ will involve covariances of the $\mu$ series at lags $0,$ $1,$ $2,$ and $3.$ A comment to the question you reference directs us to a thread that explains how to find those covariances. You should obtain the solution $$\eqalign{ &\operatorname{Var}(\mu_t,\mu_t) &= \gamma_0 = \left(\frac{1-\phi_2}{1+\phi_2}\right)\...


3

Yes, it makes complete sense in certain cases to examine the density of time-series, especially after mean-centring the original series. This relates directly to the concept of density forecasting and providing probabilistic forecasts of our continuous variables in the form of predictive densities functions. A famous reference on the matter is Gneiting et al....


1

I would suggest using your knowledge to effect an initial weeding of the low values and then use a robust nonlinear regression method to estimate the curve. Tests with synthetic data indicate this can work extremely well. Step 1 is to form a rolling maximum of the monthly data. The initial weeding removes all values less than some small fraction of the ...


1

They are two main options to explore, when we want to show the mean of an existing time-series (and not extrapolate further). Either use a decomposition technique like STL where we assume the presence of both a seasonal and a smooth trend component or employ a fully non-parametric approach using a LOESS smoother directly where we "only smooth" our time-...


1

We need to first clarify things here. The original derivation of Kalman filter is optimal for causal predictions. That means you predict at time $t$ given observations until time $t$. Now for the maximum likelihood (ML) inference of parameters, assuming that these parameters are shared across time, during inference of hidden state variables you need to use ...


1

From what I have seen, people will work with the "iid definitions" of (strict) propriety and pretty much work "pointwise", even in time series forecasting contexts. For examples, see papers of Gneiting and colleagues, who routinely illustrates his papers with meteorological forecasts. I am certainly no expert in meteorological forecasting, but judging from ...


1

It's not that "we don't care". Mathematically, it is not possible to test the null hypothesis $\alpha > 1$, i.e. the model is explosive, against the alternative, say, $\alpha \leq 1$. One can characterize the distribution of, say, the Dickey-Fuller-type $t$-statistic for each $\alpha > 1$. The limit distribution is a Cauchy distribution. However, the ...


1

Your normalization process doesn't look correct to me, you first need to normalize the training data, then use these normalization parameters to scale the validation and test sets. Also, are you sure you need to normalize the outputs? I'm not saying that these are the cause of your problem but maybe you can give it a try. Judging from the trend of your ...


1

Rather than asking how to approach this this issue in "this way" , I suggest that you follow the standard approach of model identification and model diagnostic checking culminating in a model with statistically significant parameters and an error process that can't be proven to be non-gaussian. I suggest that you follow the tried and trusted paradigm here ...


1

Impute values for missing holiday values (or any values) by taking an average of the day before and the day after. This is rudimentary just to get going and can ultimately be modified/corrected via Intervention Detection procedures. Set your seasonality to 5 and jointly identify arima and latent deterministic effects like daily effects , pulses , step-level ...


1

If you archive the models and then simply create an updated forecast after new observations are observed, this is a very effective way of dealing with tons of time series. You can then rotate new model development (and storage) for specified subsets ..e.g. the state of New York .


1

Well, you have to decide which model you want to assume behind your discrete data-points! If you simply draw linear lines between your points, then averaging the discrete data-points is almost exactly the same as calculating the area/width. (because the 2 outer most data-points would have half weight) So it's the method of fitting that makes the ...


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