The autoregressive (AR) model is a stochastic process modelling time series, which specifies the value of the series linearly in terms of the previous values.

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Error correction model: Interpretation, confidence bounds

I have estimated an error correction model (ECM) regressing Household Debt on GDP. I am trying to show that Household Debt and GDP are two forces in a two-equation system that will cause a cycle. In ...
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How to interpret the coefficient of an error correction model (ECM)?

I have run an error correction model. The coefficient is -0.6. I would like to know how to interpret this result.
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35 views

Writing a function n R [closed]

Hello, How Can i write this function in R? and any simulation codes about SETAR model including codes, links, books or any guide, would be appreciated.? Thanks in advance
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27 views

How do you define long and short run in an ARDL model?

I am writing up my regression analysis of an ARDL model which includes the long run equation and the short run dynamics. My reader however, would like to know what I mean by long run and short run. ...
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spss neural network forecasting with lags

I have 240 monthly data points and would like to leave 36 out of sample for neural network forecasting in SPSS. I made the covariates as lag of 1 [AR(1)] and have several questions to ask: What are ...
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1answer
23 views

AR(2) process: are leptokurtic residuals OK?

I have a time series of logarithmic returns. After inspection of the ACF and PACF plots, I tried to fit AR(2), MA(2) and ARMA(1,1) models and eventually found out that the AR(2) version can possibly ...
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1answer
16 views

Where can I find sound information about Periodic Autorregressive Multivariate models?

I am reading an article that mentions Periodic Autorregressive Multivariate Models and their noises; however, in no section have the authors explained or shown references to these models. I looked ...
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In an ARDL model is it mandatory to conduct short run and long run test? [closed]

I am wondering if I can only report on the long run analysis only?
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What are the steps in using ARDL model in regression? [closed]

I am running a regression model by using an ARDL model. Can someone outline the necessary tests I must conduct, please?
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How to reduce the residuals by autocorrelation of residuals?

I have two long matrix with Observations and Predictions, with 76 columns each. I need to reduce the residuals by means of autocorrelation of error correction. I also would like to select the best ...
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34 views

Causality and stationarity of AR models

Studying AR models, I found that there are two properties that these models can have stationarity and causality. For what concerns stationarity, I have studied that this condition is satisfied if ...
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1answer
15 views

Is it reasonable to have a zero mean Gaussian prior for the coefficients of an AR(p) process, assuming it is stable?

I wanna perform parameter estimation of an underlying AR(p) process given some data. Let's say it's stable. For example an AR(2) process is stable when the conditions $a_2 - a_1 < 1,$ $a_2 + a_1 &...
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2answers
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model two spatial auto correlation in spdep package in r

I hope to fit the spatial autoregressive model : $$ y= \gamma_1 Wy + \gamma_2 By + X\beta +\epsilon. \quad (1) $$ where $W, B$ are different weight matrices. However, every references I've found only ...
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Whitening Transformation with Autoregressive Model

I am new to the topic of whitening transformation. In financial time series studies on long memory in data, I have seen that researchers apply an AR(p) model to detrended return series in order to ...
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17 views

forecast nonstationary time series and test significance parametrs of AR model

I have a non-stationary time series, wich i want to fit with AR model, first of all i need to take difference wich make my TS stationary, then i see on PACF plot and see that difference number 4 is ...
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7 views

How the process parameters changes with the length of data aggregation?

Is there any general relationship for a process(e.g. ARMA, O-U process) applied to financial data over different time intervals. e.g.In this question there is an answer telling the O.P. to aggregate ...
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1answer
24 views

How to apply linear regression to one sensor so that it will match readings from better sensor? [closed]

I have one sensor which has the best accuracy and the other sensor which I want to calibrate using some linear regression (or something else?) - by modifying the software. How to calculate that linear ...
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54 views

What is the exact log-likelihood of an AR(2) model?

Let's say we have the following AR(2) model: $y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+e_t, \; e_t\sim N(0,\sigma^2_e)$ with T observations in total. Working out the conditional log-likelihood is ...
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Obtaining $T$ residuals from AR($p$) model

I have my estimates for an AR(3). To obtain the residuals I'm supposed to use $$Y_t-\hat\phi_0-\hat\phi_1Y_{t-1}-\hat\phi_2Y_{t-2}-\hat\phi_3Y_{t-3},$$ where the $Y$'s are from the dataset. If I ...
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25 views

ARDL bounds testing: model specification and selection

I am performing ARDL bounds testing. In particular, my variables are GDP, renewable and non-renewable electricity, carbon emissions and 2 other control variables, capital and labour. I have seen that ...
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14 views

Correlation between sequential binary choices

I have an experiment in which subjects perform a binary choice for each question. I want to explore the effect of n-1 choice on the nth choice. I think it has something to do with autocorrelation but ...
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1answer
24 views

References on ARDL model

Please suggest books/references on ARDL model and ARDL bounds test approach to study.
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39 views

Autoregressive model for time series with structural breaks

I'm using a structural break model (threshold model or regime switching model) to examine the dynamics of a time series. The ADF test shows that the series has a unit root. Right now I'm regressing $y$...
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Do I still need ergodicity if I have multiple/infinite time series of the same data generating process?

The main reason we need ergodicity (and therefore stationarity) is, as Shalizi puts it: The ergodic theorem is important, because it tells us that a single long time series becomes ...
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Showing that a process is autoregressive

Consider a sequence $(Z_t)$ of i.i.d. standard normal variables and real numbers $\alpha > 0$ and $\theta \in (0,\frac{1}{3\sqrt{3}})$. Let $X_t = \sigma_tZ_t$ for $\sigma_t^2$ defined by $$\...
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A 'semi random' AR model

Let $\phi$ and $\psi$ be two sets of AR model parameters. Let $(Y_n, 0\leq n\leq N)$ be a time sequence, and let $T\subset [0,1,...,N]$ be a set of times. The time series is defined as follows: $$\...
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1answer
35 views

Linear least squares regression with a smoothness penalty vs linear regression with ARIMA errors

I am about to choose between the two options mentioned in the title and I am not really sure what to pick. As a first option, we have classical linear regression plus a smoothness penalty, i.e., \...
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60 views

Stata: Predicted values in autoregressive system

I'm trying to replicate the results from Yagan (2016, pp 8-11). There, the following autoregressive system is run: The author then runs this system based on data until 2007. Then, he computes ...
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Intuition behind the characteristic equation of an AR or MA process

Ok, so I've just started learning Time Series Analysis. We can write an MA(q) process as Yt = θ(L) ϵt and an AR(p) process as ϵt = φ(L) Yt in terms of the lag operator. Then, with no explanation (...
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54 views

Incorporating autocorrelation into forecasts

I have a time series $x_{t}$ which is an AR(1) process with a constant term, e.g. $ x_{t} = c + \phi x_{t-1} + \epsilon_{t} $ How can I incorporate information about the autocorrelation of $x_{t}$ ...
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30 views

Conflicting cointegration results due to different lags in Johansen procedure

I have been using two different models for cointegration: Johansen's test and ARDL (autoregressive distributed lag). I guess this example could be extendent for other cointegration models as well. I ...
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1answer
33 views

How to implement a multiple regression for AR models (time series)?

Let's say I have the following model: So I have an AR model of order 3, and I want to estimate A1, A2, and A3. I understand how regression normally works for two variables x and y. Also, after ...
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Threshold autoregressive model(TAR) best lags

What is procedure of threshold auto regressive model(TAR) for selecting lags. Suppose that this two-regimes fitted model: This is an example from Analysis of Financial Time Series book. How this ...
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How to forecast with a regression model with ARIMA errors?

I have obtained the following coefficients after using the auto.arima function in R: $y_t = 0.87 x_t - 0.51x_{t-1} + n_t$ where $n_t = 0.83n_{t-1} + e_t$ My question is, how to obtain $\hat{n}_{t-1}$...
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Can a Markov chain be approximated with an AR process?

In some MCMC literature/source code, a Markov chain is often approximated with an AR(1) process. There is some theory to suggest that such an approximation is somewhat valid for a finite state space, ...
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What is $\hat{\sigma}^2$ for AR(1) non-causal case?

Assume that $|\phi|>1$ and {$X_k$} is the stationary solution of the non-causal AR(1) equations, $$X_k=\phi X_{k-1}+Z_k$$ where {$Z_k$} is white noise with mean $0$ and variance $\sigma^2$. Show ...
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Stationarity in the Almon lag model

I have a quick question regarding the Almon approach (Shirley Almon) as presented in chapter 17 of Gujarati's Basic Econometrics. In an example given in the textbook, they use non-stationary data ...
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In AR(1), why is $X_i|X_{i-1}=x_{i-1}\sim N(\alpha x_{i-1},\sigma^2)$?

The AR(1) model starting with $X_0=0$ is $$X_i=\alpha X_{i-1}+\epsilon_i, \ i=1,...,n, \ -1<\alpha<1$$ where $\epsilon_i\sim N(0,\sigma^2)$ are independent error terms. Why then is $$X_i|X_{i-...
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Non stationary solutions for stationary ARMA equations

By "stationary" I mean "weakly stationary". Consider a "stationary" AR(1) equation: $$X_t=\varphi X_{t-1}+\varepsilon_t,$$ where $t\in\mathbb{Z}$ are discrete time moments, $\varepsilon_t$ a zero-...
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What to do if ACF or PACF show significant higher lags?

I have monthly climate data for 90 years. I assembled the best model I could (added sensible parameters to minimize AIC), and then tried various ARMA correlation structures (using ...
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How to get short run effect in ARDL? [closed]

If an independent variable $\Delta x_t$ has three lags in an autoregressive distributed lag (ARDL) model with estimated effects 0.123$\Delta x_{t-1}$ ($p$-value=0.001), 0.850$\Delta x_{t-2}$ ($p$-...
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Should multicollinearity problem be looked into while doing cointegration?

Multicollinearity and Cointegration is not the same thing however, if the series actually move together in the long-run i.e. are cointegrated wont they also be collinear making e.g. Autoregressive ...
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1answer
45 views

Derive bias when AR(1) is approximated by MA(1)

Consider the MA(1) process: $$ y_t = \varepsilon_t + \theta_1 \varepsilon_{t-1} $$ where $\varepsilon$ is a white noise process with $\mathbb{E}(\varepsilon_t) = 0$ and $\operatorname{Var}(\...
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In the midasr package in r, how is the AR* model different than the AR model?

At the end of the ?midas_r documention example what is the fourth parameter option in the mls() of the lagged dependent variable "*" doing that is different that the "regular" AR(1) model above? I've ...
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Modelling auto-correlated binary time series

What are the usual approach to modelling binary time series? Is there a paper or a text book where this is treated? I think of a binary process with strong auto-correlation. Something like the sign of ...
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1answer
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Solving for a difference equation for $s_{t}$

Given $f_{t}=u_{t} - \bar{P}$ and the law of motion for $u_{t} = \rho u_{t-1} + \epsilon_{t}$, where $0<\rho<1$, $\epsilon_{t}$ is mean-zero iid and can be interpreted as a domestic price level ...
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Coeffs change using Yule-Walker vs. ML autoregression correction

When estimating a small-sample time series regression (in SAS, using PROC AUTOREG), I was surprised that the coefficients changed so much when changing from Yule-Walker to maximum likelihood. The ...
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Distribution of two correlated multivariate normal distributions

Let $\varepsilon_{s}^{FR}$ follow a Gaussian Markov process so that ${\varepsilon_{s}^{FR} = \rho \varepsilon_{s-1}^{FR} + \xi_{s}^{FR}, \: \xi_{s}^{FR} \sim \mathrm{i.i.d.} \: N(0,\sigma^{2}_{\xi^{...
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Can we use Threshold Auto-regressive Regression (TAR) for continous inputs and binary output?

Can we use Threshold Auto-regressive Regression (TAR) for continuous inputs and binary output? Is it appropriate for classification modeling? Output is is one if ...
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Relationship between T, the Adjustment Rate, and Power for an Autoregressive Unit Root

So I have a question related to autoregressive unit roots. This picture shows a graph for c and n with power = .9 in blue and power = .8 in red. Any ideas what could be used to fit this graph or ...