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If the random vectors $X, Y$ are independent, then, as functions of independent random variables are independent (see Functions of Independent Random Variables), $X_i$ and $Y_j$ are independent, since selecting one given component from a vector is a function. But the covariance of independent random variables are zero, that gives you the result.
In your ...
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While it is mentioned in a number of regression texts, the plot you have mentioned here does not seem particularly useful to me. A far better alternative is the added variable plot, which correctly represents the relationship between an individual explanatory variable and the response variable conditional on other explanatory variables. For the explanatory ...
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I think you are doing it right...
"I am not sure how to express the covariance matrix": but you specify variance to be 1, right? This is what you want? You just need to draw each time conditionally on drawing $X's$ in the right order: first draw first 10 $X's$, then the other five conditional on first five $X's$, and lastly draw $Y's$ condtional on 1, 5, 7,...
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(NB: I started to write this before the question was edited; I don't comment here on the "empirical aspect", i.e., what is done in the literature, but rather on what "should" be done.)
This is a good question, and I don't think there is a good answer.
What I think is the following. Generally in statistics things are less "black and white" and it is much ...
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I think that you can use regression to analyze your data. In essence, you have described a regression with a single outcome, y (home price), with three predictor variables x1-x3 (proportion restaurant, proportion government, and proportion agriculture, respectively). You further have multiple home price observations in each of your sites, necessitating some ...
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But my question is, does the NN take all series in consideration when
creating the forecasts for one series?
Well, that depends your exact model. A typical choice might be a neural network $f$, with the model defined as: $p(x^k_t| x_{<t}; \theta) = \mathcal{N}(\mu^k, \sigma^k = f(x_{<t}, \theta))$ where $t$ indexes time, $k$ indexes over your 4 ...
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I think Li-Mak (Li & Mak, 1994) (univariate) and Ling-Li (Ling & Li, 1997) (multivariate) tests are suitable candidates for error diagnostics in univariate and multivariate GARCH models, respectively. Unlike some other (popular) tests*, they account for the fact that standardized residuals from GARCH models are not equal to true standardized ...
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If you take cross correlation as measure for distance between curves
For the curves in the Andrews plot
$$f_x(t) = x_1 \frac{1}{\sqrt{2}} + x_2 \sin(t) + x_3 \cos(t) + x_4 \sin(2t)+ ...$$
The cross covariance
$$\int_{-\pi}^{\pi} f_x(t) f_y(t) dt$$
can be simplified as separate products (because the trigonometric functions are orthogonal)
$$ x_1 y_1 \...
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It's better to operate on log domain in such cases. There are implementations (employing factorizations like Cholesky, LU) available in common languages:
python: numpy, tensorflow
R: msos, lme4
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Dynamic Time Warping might be a good choice if you are explicitly trying to use one time series to predict another.
This method works by means of determining similarity between two sequences which vary, typically in either time or speed.
From looking at your time series, it would not appear (at least visually) that there is a lag between the two time ...
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