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23

You have two problems--and one of them is interesting. Without a noise term, the series is no longer stationary. Its value is increasing asymptotically, but definitely, toward $1:$ ARIMA applies only to stationary models--and these data are obviously not from a stationary model. That's not terribly interesting. What is interesting is that the problem ...


5

The code you have created is not even generating any (pseudo) random outputs, let alone an AR(1) process. If you would like to generate the output of a stationary Gaussian AR(1) process, you can use this function: GENERATE_NAR1 <- function(n, phi = 0, mu = 0, sigma = 1) { if (abs(phi) >= 1) { stop('Error: This is not a stationary process --- |phi| &...


3

As also pointed out in the comments, it's not an AR(p) process any more. arima function assumes the following and fits the coefficients accordingly: $$y_t=c+\phi y_{t-1}+\epsilon_t$$ It's quite normal that you don't get near the correct $\phi$. Also, after adding noise term, try to increase the sample size to get more confident estimates.


3

You asked "Is it a general thing that an ARIMA(0,0,0) becomes a MA(1) when the series is differenced?" Yes if the differencing is unwarranted ... as in this example where Y(t) is a white noise series. If Y(t)=A(t) and you difference Y you get [1-B]Y(t)= A(t)[1-1.0B] where B is the backshift operator. thus you have injected structure by differencing and ...


0

I just looked at the difficulties you seem to have with your ARIMA(0,0,1) model (or is it ARIMA(0,1,1), since you mention differencing?). I'd strongly suggest you compare your model to some very simple alternatives, which surprisingly often are quite competitive with ARIMA and other more complex methods. For instance, an overall mean is an ARIMA(0,0,0) with ...


1

There exist ways you can test seasonality. Though plots are useful you should test your assumed seasonality with cross validation. The topic of neural networks and forecasting is not as well developed as it is for other fields. DeepAR has shown some success, but with these methods come additional computational costs. You can also search for pretrained ...


1

It's not really clear from your plots, but it looks like the upper histogram is the entire 500-day data series, and the lower histogram is the 4-month restricted series. Question 1 I don't think can be answered because it seems like backwards thinking. Your model with more data is performing poorly on the test set because the test set is only representative ...


1

Assumptions for parameter estimators in an ARMA-GARCH model are made on standardized innovations, not raw ones. ACF plot of standardized innovations allows assessing whether the model assumptions are violated or not, and hence whether parameter estimates (realizations of estimators) can be trusted. Therefore, it makes sense to inspect the ACF of standardized ...


0

I would not classify all of those things as "methods", at least not in the same sense as AR and MA. A Naive forecast could be done in many ways; it is a principle, that forecast accuracy shouldn't be evaluated in a vacuum but rather should be evaluated within a forecast value added framework, comparing each forecasting step back to a "naive approach" and ...


1

Simple Models If you are looking for simple models I would test these: Linear Regression Exponential Smoothing with seasonality Seasonal ARIMA models Rule of thumb methods: seasonal naive method, seasonal mean, simple mean. These models should help you in assessing a baseline for the model performance. I believe models 1-2-3 are already implemented in ...


1

I would start by looking through the excellent free online book Forecasting: Principles and Practice (2nd ed.) by Athanasopoulos & Hyndman. It leverages the equally excellent forecast package for R, which allows you to automatically build ARIMA or Exponential Smoothing models and forecasts very easily. With only 36 data points, there is really not much ...


1

1. I am not sure why using weekly data (around 2500 data points) is giving very high error but using monthly data points (only 31 data points) was giving decent result (although not very good). What am I doing wrong here ? @IrishStat is likely right. The 53rd week issue is rough. It is usually easier to get clear seasonality with monthly data or daily ...


1

36 months of data can often be useful in forming a model. Initially, you might want to develop arima type model that incorporates trends, level shifts, and memory while dealing with unusual values. Secondly, you might investigate adding user-suggested causal variables to your model. These models are an extension and are often referred to as Dynamic ...


1

I used the 44 values . Visually there appears to be more variability at higher values . This lead the automatic analysis to suggest a log xform using AUTOBOX , a piece of software that I have helped to develop. Following is the Actual/Fit and Forecast from a useful model in log space (3,0,0)(0,0,0) The equation is here and here The model residuals are ...


0

As a general rule you don't want to do piece-fitting using previous results to feed the next stage without a simultaneous optimization. In this case after identifying the regime shift (visually obvious) , we treat the most recent set of values reflecting homogeneous structure. I took your very interesting ( to me ! ) time series which visually appeared to ...


1

I took your 981 daily values and used AUTOBOX ( a piece of forecasting software that I have helped to develop) . The original data visually suggests level shifts ( up at period 560 down at period 801 ) which was confirmed here in a useful model also containing German holiday effects AND monthly effects and here The Actual/Fit and Forecast graph is here ...


1

Answer: Every unit-root state-space model has an equivalent ARIMA(0,1,1) representation, but not every ARIMA(0,1,1) model has an equivalent state-space model representation. The following holds true for the representations of unit-root state-space models: Whatever the magnitude of the signal-to-noise ratio $\frac{\sigma_\eta^2}{\sigma_\epsilon^2}$ in the ...


1

You asked 2 questions ... 1)Could you please help me to see if my prediction method is accurate and how to handle errors in R. I wouldn't think so because you didn't fully extract a sufficient equation as per Help me about using ARIMA forecasting rainfall 2) Could you please help me run the code in R to predict rainfall according to the above data can'...


1

I believe that our problem is that we are jumping directly to ARIMA model without trying the traditional models. for this reason, you can find the model is not giving the needed results. In your case, I tested your data, I found that there is a seasonality every 12 months which is clear for you, but also I found that a simple moving average of 3 terms ...


0

The problem with using a simple arima approach is that model identification is done when it is based upon the tacit assumption that there is no deterministic structure in the data and model parameters are invariant over time. Your rainfall data (216 monthly values) has two significant seasonal dummies ( October and November) and a few pulses ( 6 of them all ...


1

My suggestion is that you consider approach than ARIMA if necessary, which is typically used for forecasting. Anomaly detection appears to be the goal here. There are several options available in R and SAS (where I have experience). You didn't mention a particular language or software so I won't comment on all the ones I'm aware of for now, though I will ...


2

For anyone who is wondering about this question still, i will clarify - Volatility clustering does not at all imply that the series is non-stationary. It would suggest that there is a shifting conditional variance regime - which may still satisfy constancy of the unconditional distribution. The GARCH(1,1) model of Bollerslev is not weakly stationary when $\...


0

A key difference which I failed to appreciate: the MA model predictions of $x_t$ include $\epsilon_{t-1}$ in its computation whereas the AR model only predicts based on $x_{t-1}$ without (explicit) regard as to whether the prediction at $t-1$ was under- or over-estimating $x_{t-1}$. Fleshing out what @user1587692 highlighted, the MA model averages across ...


0

I would suggest reducing the 72 to say 24 or 12 as the Ljung-Box test has not passed the test of time ( that's a joke ! ) as it's assymptotic properties have been called into serious question. See Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey and elsehwere ..... Time to tell your instructor and the authors of your textbook the news !


2

The estimated values of the autoregressive parameters are already reported in the output you provided: Coefficients: ar1 ar2 ar3 ar4 ar5 ar6 intercept 3.3058 -4.5471 3.0440 -0.6789 -0.3044 0.1651 0e+00 The associated standard errors are listed immediately below these estimated values: ar1 ...


0

A finite AR model can be expressed as an MA model and vice-versa , If one has an ar(1) model with coefficient .333333333 then the models are (nearly) identical . Consider the case for an ar(1) with coefficient of .3 x(t)=ϵ(t)+.3*x(t−1) since x(t-1)=e(t-1)+.3*x(t-2) we can substitute for x(t-1) and get x(t)=ϵ(t)+.3* [ e(t-1) + .3*x(t-2) ] x(t)=ϵ(t)+.3*...


3

You should consider the $\epsilon_t$ innovations rather than residuals. In the MA case, you average across the recent innovations, whereas in the AR case you average across the recent observations. Even if the models are stationary and have no deterministic terms, the innovations and the observations are different. For example, suppose the $\epsilon_t$ are ...


2

A principled approach is to make probabilistic predictions. This gives a probability distribution over possible values at each timepoint, rather than a simple point prediction. The methods you described can provide this. The model should capture the ordinary behavior of the system, so it should ideally be fit to data containing no anomalies. Then, a point ...


2

To properly answer your question regarding automatic model differences requires a little bit of history https://autobox.com/pdfs/econometrics.pdf to explain some different approaches to ARIMA model identification. Model identification has and always will be an iterative process much like peeling an onion where clues are found and followed and possibly ...


1

The 5 predictors future values may be the cause of your "increasing forecast" OR a trend coefficient tat is unwarranted . How are you specifying the values for these (X) series into the future ? Additionally your arima model might be questionable as it has redundant arima structure . Also note that AUTOBOX used differences of all series as triggered by the ...


0

1) Unnecessary (excessive) differencing can inject structure 2) Identifiable contemporary and lag structure does not require pre-differencing 3) one can mix and match the nature of the predictors. https://autobox.com/pdfs/SARMAX.pdf


1

Your equation is [y(t)-6.8840][1-.9916B]= +ϵ(t) or y(t)= .0084*6.8840 + .9916*y(t-1) y(t)= .0578 + .9916*y(t-1) What has you confused is for your stationary model the constant that is estimated is a Left-Hand side constant not the right-hand side that you were (normallly !) expecting. Your model in my opinion should be changed to a first difference ...


0

There is a mistake in the ARMA(1,1) model. Look at Alternative construction of ARMA(1,1) process where the correct ARMA(1,1) was derived.


5

Answer: There is a mistake in the formula for $\theta$. The correct computation must align autocovariances of the MA components of two representations. The correct formula is $$ \theta = \frac{\sqrt{\xi^2-4} -\xi}{2}$$ where $\xi:= \phi + \frac{\sigma^2_v+\sigma^2_w}{\phi \sigma^2_v}$. Substituting the chosen values for $\phi,\sigma_v,\sigma_w$ gives $\...


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