ARMA is an acronym for auto regressive moving average; a stochastic process modelling time series. It adds moving average terms to the AR model.

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ARMA errors and combining explanatory variables

Currently I'm working on forecasting the employee turnover of an organisation. To do this, I'm using a time series data of the employee turnover over the past 7 years, it is an annual data. To make a ...
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Relationship between GARCH(1,1) and ARMA(1,1)

Consider an ARMA(1,1) for squared returns $y^{2}_t$: $$y^{2}_t=0.038+0.946y^{2}_{t-1}-0.855v_{t-1}+v_t,$$ where $v_t$ is residual. How to show that this model corresponds to a GARCH(1,1) for ...
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Standard errors `NaN` by `Arima` function in R

I'm working with a time series of 59 elements. I'm wondering why the R function, Arima, throws an NaN for the standard errors of ...
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18 views

ARMA to Pure AR or MA [closed]

Hello, Given that an ARMA(P,Q) process can be written as an MA or AR infinite process, I am confused as to how this would work for the attached example where P=Q=1?
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30 views

Organic vs Paid Attribution Model (Granger)

I'm wondering if there is literature or studies done on how to model organic attribution from paid user acquisition. So the context is, on our mobile app, we have paid installs that we purchase and ...
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36 views

ARMA lag selection for ARMA-GARCH models

When I read this group questions about lag selection for ARMA part of ARMA-GARCH models I found 2 different answers from moderator: The use of GARCH and ARMA GARCH estimation process in practice I ...
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Cancelling roots in ARMA(1,1) with external regressors

I am trying to find out what cancelling roots would imply for the estimators of my external regressors in my ARMA(1,1) model. Unfortunately however I'm stuck in my final step since I'm insecure about ...
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51 views

How to identify best Model for univariate time series data?

I have a time series data- 53.97 63.32 57.06 60.27 69.46 75.08 78.31 73.28 85.84 69.34 62.57 60.11 55.63 47.29 61.22 58.46 66.26 59.71 51.12 39.36 51.89 ...
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28 views

Help discussing stationarity using correlograms? ARMA/ARIMA modelling

I am currently trying to understand how to use correlograms to examine stationarity and analysis the appropriate models. Please can you advice, below I have included my ACFs and PACFs, and I am trying ...
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47 views

How to compute estimate for the first time series value using ARIMA model?

I modeled a univariate time series in R using the Arima command. One can obtain fitted values for the original series using this command by applying the function ...
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Algorithms For ARMA Estimation From Time Series Data

What are some algorithms that can be used to model time series data as an ARMA model? I do not have access to the input process, only the resulting output process as a time series. I don't want to ...
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ARMA(1,1) to AR(Infinite) Help.

Hello, Could someone please help me understand how this ARMA(1,1) model can be represented as a pure or AR infinite process? I don't understand how multiplying the two lag polynomials can generate ...
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25 views

Proper definition of AR()-ARCH() time series model

This is how I would define it, if anyone has any objections please let me know! AR(m)-ARCH(m) time series is an ARCH(m) process in which the variance at time t is conditional on the previous m times ...
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14 views

ARMA stationarity conditions

Are ARMA(p,q) model stationarity conditions are the same as for AR(p) or some additional requirements must be met?
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39 views

INARMA proces in R

I have time series of with integer value numbers. I want to test if my process is Integer Valued AR(1) pocess or similar integer valued ARMA process in R. Could somebody provide me with any ideas how ...
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how does the MA lib determine the levels for what is significant or not for ACF

I am going thru this stats course https://onlinecourses.science.psu.edu/stat510/node/48 I can follow all the steps, but have question how the lib figures out what is significant for ACF. See "ACF for ...
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20 views

Variance of predictions on AR(8) model

Given the following AR model:$$Y_{t} = 0.16Y_{t-8}-0.36Y_{t-7}+0.2Y_{t-6}-0.8Y_{t-2}+1.8Y_{t-1} + \epsilon_{t}$$ Where $\epsilon_{t}$ is white noise with zero mean and variance = $\sigma_{\epsilon}$ ...
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127 views

Given ARMA(p,q), can a ARIMA(p, 1, q) model be created?

I would just like to check my understanding for this question: Given that you have a generated ARMA(p,q) model, can an ARIMA(p,1,q) model be created? My thought was that it would be that this is ...
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Simultaneous estimation of ARMA and GARCH components

I am not allowed to comment on older questions relevant to this question, therefore I ask this myself. The question was dealt with here, here and here with a clear result: the ARMA part should be ...
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40 views

Sum of two ARMA processes that have the same innovations

Let's say $$ x_t = a + bt + \varphi e_{t-1} + e_t $$ and $$ y_t = c + y_{t-1} + \varphi e_{t-1} + e_t $$ where the $e_t$ term is white noise with zero mean. Can I just sum everything as below: ...
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51 views

Setting initial values in ARIMA

I tried to fit 2 ARMA models to a set of financial time series data with 2 sets of initial parameter values (all 0's and all 0.5's). The final estimates of the parameters are quite different. I'm not ...
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79 views

Inverting ARMA Processes

I'm just a bit confused that when you convert an ARMA process to either a pure AR or MA representation then is it always an AR or MA infinite process that we obtain? For instance an ARMA(1,1), does it ...
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29 views

Expand a power of the difference operator in terms of time series $z_t$

I am trying to use excel to plot different time series. I have the equation $(1-L)^2 * z_t$ I know that $(1-L)*z_t$ is equal to $z_t-z_{t-1}$ Can I just expand $(1-L)^2$ using basic algebra and ...
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50 views

ARMA (1,1) Variance Calculation

I really need help with this problem and would appreciate any input: When $t$ denotes the time-period, terms $\alpha$, $\phi_1$, and $\theta_1$ are constants, $a_t$ represents error-terms that are ...
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Model for prediction of binomial probabilities based on time series events with variable duration

I am new to this field, so sorry if I am not precise with the nomenclature I use. :) I am trying to develop a statistical model that will allow me to calculate the outcome probabilities of a binary ...
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ARMA(2,1) autocovariance functions

I have an ARMA(1,2) model: $$x_t = 0.6 x_{t-1} + u_t + 0.1 u_{t-1} - 0.2 u_{t-2} + 5$$ and I figured out the ($\delta$ = autocovariance functions) $\delta(0)= \beta_1\delta(1) + ...
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27 views

How to calculate a probability that a time series takes values in a given interval?

So I have a time series $X_t$, where $X_t$ is the number of sales of a product at day t. I would like to be able to calculate some probability like this $P(X_t>10)$ for $t \in [5,20]$ : this means ...
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86 views

What can go wrong using lagged terms as instrumental variables?

Can anybody give one example of when the set of all lagged $X$ can (or can't) be a good choice of IV's for $X_{t}$?
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17 views

ARIMAX with changing regressor values for t+1, t+2,

I have a question on a quite complicated rolling forecast model Objective I am trying to forecast the number of calls to a hotel. The forecasts I made are t+1 days, t+2 days, up until t+60 days. I ...
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272 views

Different AIC definitions

From Wikipedia there is a definition of Akaike's Information Criterion (AIC) as $ AIC = 2k -2 \log L $, where $k$ is the number of parameters and $\log L$ is the log-likelihood of the model. However, ...
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How to Calculate Standard Error and Prediction Intervals for ARMA Forecasts on Transformed Data?

I have been recently learning about the Box-Jenkins process for ARMA modeling, and I ran into a bit of a wall when it comes to error analysis. In a lot of my data sets, I have to apply a log ...
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100 views

Time series analysis (ACF, PACF)

I have this monthly time series with pronounced seasonality and a bit of trend: The ACF and PACF for 4 years (48 months) are: Can I suppose that the data don't need transformations like: ...
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32 views

How to configure p and q parameters in ARMA (Auto-Regressive Moving Average) model

I have a stationnary time series on which I want to execute an ARMA(Auto-Regressive Moving Average) model to predict next values of my time series. The ARMA model have two parameters: p: for ...
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80 views

ARMAX model and validation

I am new to time series and am trying to fit some time series data. I understand the general concept of ARIMA model. However, as I read more textbooks and articles from Rob Hyndman, I realized I ...
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48 views

How will ARMA be affected if a non-stationary time series is not made stationary?

Which components or statistics of ARMA will be affected - and how - if the stationarity condition is violated?
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114 views

Adjusting daily time series data for the seasonal component

I have a time series data containing roughly 13 000 daily observations, and the initial plan is to fit them to an ARMA model. I have adjusted the data for deterministic trend with a regression model, ...
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52 views

How to show that any Gaussian time-series is linear one?

In this paper I saw the following statement: If the time series is Gaussian (i.e., normally distributed) then the best linear forecast is in fact the best of all possible forecasts: No ...
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What nonlinear extension of ARMA and State Space Model do exist?

In ARMA model we postulate that predictions of time series can be calculated as a linear function of $N$ previous observations (AR part) and $K$ differences between the previous observations and the ...
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Am I doing this right - choosing the order of p and q for an ARMA model (time series)?

I have a time series and have to come up with the ARMA(p,q) model that fits best to the data. (I have read ARIMA model identification, which was helpful, but doesn't completely resolve my question). ...
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55 views

Estimate the best ARMAX model with one lagged independent variable (time series)?

I have two time series to work with, let's say X1 and X2. First I have to estimate the best pure ARMA model for X1; which is no problem. For that I perform the following steps: Stationarize (if ...
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2answers
86 views

How are the values of residuals (white noise) calculated in ARMA model?

I am trying to implement ARMA model in Java. I have trouble with calculating residuals (white noise) in Moving average part of the model. From the answer on this question (Fitted values of ARMA model) ...
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73 views

ARMA-GARCH model initial parameters for optimizer

I am implementing a program to fit an ARMA-GARCH model to given data. My model parameters are optimised by maximizing the Maximum Likelihood function using a nonlinear algorithm. The algorithm ...
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13 views

Deriving log likehood of AR vs MA

To derive the log-likelihood function of an MA(1) we condition on $\epsilon_0$. But when deriving the log-likelihood function function for an AR(1) we don't. Why not? Is it just because $\epsilon_0$ ...
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Is AR(1)-ARCH(1) covariance stationary?

Say I have the following model: $$ y_t = c+\phi y_{t-1} +\epsilon_t \,, \epsilon_t|\Omega_{t-1} \tilde{} WN(0,\sigma_t^2 ) $$ $$ \sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2 $$ $$ |\phi|<1 \,, ...
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165 views

How to reconstruct a stock price from ARMA/GARCH fit

In this GIST, you will find simple R code that calculates Microsoft's daily stock price based on an ARMA(1,1) + GARCH(2,2) using "rugarch" library and the data from 2008 to 2015. It compares the ...
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Autocorrelation function decay for AR(p), how to proof

How can I demonstrate that the Autocorrelation function (ACF) for an AR(p) decay exponentially? It seems so simple but neither Enders' book nor Greene's has such proof.
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38 views

Yearly restrictions on quarterly forecast in ARMA

I was thinking, is it possible to implement a quarterly forecast for one year ahead such that its sum over year equals some constant number? This problem may arise if we have, for example, some ...
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General PACF for MA(1) process [duplicate]

This following question is a sample question from my notes. I have the equation $X_t=w_t+\theta w_{t-1}$. I want to prove the general equation for PACF for the MA(1) process at lag n $\phi_{nn} = ...
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Time Series ARMA Covariance

The equation is $Z_t=\theta Z_{t−1} + a_t + \alpha a_{t−1} + \beta a_{t−2}$ $\alpha, \beta, \theta=\text{constants}$ $\text{E}(a_tZ_t)=\text{E}[a_t(\theta Z_{t−1} + a_t + \alpha a_{t−1} + \beta ...
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32 views

Difference between actual curve and the curve estimated by ARIMA

I'm trying to use ARIMA model to forecast the next point in my measurements. I have a vector with $\{x_1, x_2, \ldots ,x_t\}$ measurements. I'm trying to forecast the $x_{t+1}$ element. I'm using the ...