## #StackBounty: #time-series #confidence-interval #standard-error #differences Standard Error of the cumulative value for time series

### Bounty: 50

I have two time series, as in the picture below. The data was gathered experimentally. A practical example could be a measured mass flow rate, where I measure the mass flow rate over a certain time period with changing boundary conditions and I am interested in the total mass of fluid consumed during this period.

The depicted intervals represent the Standard error of the mean and were determined through ten repeated measurements of the entire cycle. For each time step, the interval equals:
$$SEM= frac{sigma}{sqrt{10}}$$
The ten replications naturally constitute ten identifiable time series.

I now have three questions :

(1). How can I assess, at a specific time step, if the difference between the means of the time series is statistically significant ? t-test ?

(2). How can I determine the SEM of the cumulated value for each time series ?

(3). How can I assess if the difference between the cumulated values of both time series is statistically significant ?

Does it make sense to create an “interval” of the cumulative value of both time series by cumulating the upper bound values and lower bound values ?

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## #StackBounty: #time-series #mathematical-statistics #autoregressive #moving-average Why do we care if an MA process is invertible?

### Bounty: 50

I am having trouble understanding why we care if an MA process is invertible or not.

Please correct me if I’m wrong, but I can understand why we care whether or not an AR process is casual, ie if we can “re-write it,” so to speak, as the sum of some parameter and white noise – ie a moving average process. If so, we can easily see that the AR process is causal.

However, I’m having trouble understanding why we care whether or not we can represent an MA process as an AR process by showing that it is invertible. I don’t really understand why we care.

Any insight would be great.

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## #StackBounty: #time-series #forecasting #entropy #information-theory #spectral-analysis Interpretation of spectral entropy of a timeser…

### Bounty: 50

The `tsfeatures` package for R has an `entropy()` function. The vignette for the package describes it as:

The spectral entropy is the Shannon entropy $$-int_{pi}^{pi} hat{f}(lambda)loghat{f}(lambda) dlambda$$

where $$hat{f}(lambda)$$ is an estimate of the spectral density of the data. This measures the “forecastability” of a time series, where low values indicate a high signal-to-noise ratio, and large values occur when a series is difficult to forecast.

The documentation for the `ForeCA::spectral_entropy()` which is used by `entropy()` function suggests the density is calculated such that
$$int_{-pi}^{pi} f_x(lambda) dlambda = 1$$.

I’m wondering what the most accurate interpretation of this calculated quantity is (I’m sure there’s a good reason why ‘forecastability’ is in quotation marks). I’ve got a suspicion it’s based on a narrow interpretation of what is forecastable.

1. Is it correct in saying that the spectral density is obtained using a (discrete?) Fourier transform on the time-series?

2. Would it be more accurate to say that this calculation measures ‘forecastability’ by testing whether the time series is a linear combination of signals at different frequencies and at different levels of power?

3. What other assumptions are made? Surely this measure has very limited ability to account for path dependence and other complex/non-linear behavior which may or may not be forecastable given a sophisticated understanding and model?

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## #StackBounty: #time-series #bayesian #econometrics #mcmc Can double dipping be reasonable?

### Bounty: 50

I found a paper where the authors used bayesian methods to estimate asymmetric effects in impulse response functions. In short the estimation procedure is:

1. Calculate a VAR and Impulse responses (no matter what identification strategy).
2. Express this IRF´s as a a set of gaussian basis function. (This reduces the number of parameter)
3. Use this estimates as the initial guess (=prior?) of a Metropolis-Hastings Algorithm.

All steps use the same data.

I’m a bit confused if it makes sense to extract the prior information from the same data where the MCMC algorithm will be used in the next step? I learned that “double dipping” is a problem in bayesian statistics. Since it is a relatively well-known paper, I assume that there is an explanation for this point, but I don’t get it.

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## #StackBounty: #data-mining #time-series #predictive-modeling #statistics #prediction How can i find trend time for my articles?

### Bounty: 50

our article is time-based, that means is my article search more in a specific time.
as you can see in under chart this article search more in specific period time.

if my dataset looks like this(it can be a json,csv,xlsx) :

How can I find the trend time of my article (time between the red line in image 1)

I need to know that period of time to be ready for next year.

(I need excel solution for that, but if you can solve this problem
with other ways and programming language it’s ok)

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## #StackBounty: #time-series #arima #simulation #garch #finance Using ARMA-GARCH models to simulate foreign exchange prices

### Bounty: 50

I’ve fitted an ARIMA(1,1,1)-GARCH(1,1) model to the time series of AUD/USD exchange rate log prices sampled at one-minute intervals over the course of several years, giving me over two million data points on which to estimate the model. For clarity, this was an ARMA-GARCH model fitted to log returns due to the first-order integration of log prices. The original AUD/USD time series looks like this:

I then attempted to simulate a time series based on the fitted model, giving me the following:

I both expect and desire the simulated time series to be different from the original series, but I wasn’t expecting there to be such a significant difference. In essence, I want the simulated series to behave or broadly look like the original.

This is the R code I used to estimate the model and simulate the series:

``````library(rugarch)
rows <- nrow(data)
data <- (log(data[2:rows,])-log(data[1:(rows-1),]))
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)), mean.model = list(armaOrder = c(1, 1), include.mean = TRUE), distribution.model = "std")
fit <- ugarchfit(spec = spec, data = data, solver = "hybrid")
sim <- ugarchsim(fit, n.sim = rows)
prices <- exp(diffinv(fitted(sim)))
plot(seq(1, nrow(prices), 1), prices, type="l")
``````

And this is the estimation output:

``````*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,1)
Distribution    : std

Optimal Parameters
------------------------------------
Estimate  Std. Error     t value Pr(>|t|)
mu      0.000000    0.000000   -1.755016 0.079257
ar1    -0.009243    0.035624   -0.259456 0.795283
ma1    -0.010114    0.036277   -0.278786 0.780409
omega   0.000000    0.000000    0.011062 0.991174
alpha1  0.050000    0.000045 1099.877416 0.000000
beta1   0.900000    0.000207 4341.655345 0.000000
shape   4.000000    0.003722 1074.724738 0.000000

Robust Standard Errors:
Estimate  Std. Error   t value Pr(>|t|)
mu      0.000000    0.000002 -0.048475 0.961338
ar1    -0.009243    0.493738 -0.018720 0.985064
ma1    -0.010114    0.498011 -0.020308 0.983798
omega   0.000000    0.000010  0.000004 0.999997
alpha1  0.050000    0.159015  0.314436 0.753190
beta1   0.900000    0.456020  1.973598 0.048427
shape   4.000000    2.460678  1.625568 0.104042

LogLikelihood : 16340000
``````

I’d greatly appreciate any guidance on how to improve my modelling and simulation, or any insights into errors I might have made. It appears as though the model residual is not being used as the noise term in my simulation attempt, though I’m not sure how to incorporate it.

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