Questions tagged [autoregressive]

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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What's wrong if I fit the auto-regression with OLS?

I am doing auto-regress by usual linear regression package. e.g. $y_t=φx+ε_t$ with $x =y_{t-1}$ My reason is that, Auto-regression does assumes iid errors, same for linear regression. Linear ...
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1answer
60 views

Derivation of the distribution of $\hat{\phi}=[\hat{\phi}_1, \cdots, \hat{\phi}_p]$ in AR(p) models

Background Consider the following AR($p$) model: $$ \dot{X_t} = \phi_1 \dot X_{t-1} + \phi_2 \dot X_{t-2} + \cdots + \phi_p \dot X_{t-p} + \epsilon $$ where $\dot{X} := X - \mu = X - \mathbb{E}(X)$...
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How many lags should I include in time series prediction?

I'm wondering: how do I select the number of previous time steps to use to predict the current one? I'm just plotting the autocorrelation plot and picking previous time steps that have statistically ...
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1answer
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Difference between different autoregressive models

I am trying to understand the difference between these three different specifications of an autoregressive model for variable var in Stata: ...
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1answer
355 views

Conceptual questions: Variance of a process

Wikepedia, at Variance of Autoregressive model, mentions an expression of variance for an AR(1) process. I am learning signal processing (beginner level) and facing difficulty in understanding some ...
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1answer
182 views

Doubts in linear regression

If a linear regression model has a constant term say 1 or 0.2, for example if the original model is $y(t) = 0.2 + ay(t-1) $, then what does this constant term imply? Will it hamper the estimates if ...
3
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1answer
2k views

Backshift operator applied to a constant

This questions is two part: 1) What happens when you apply the backshift operator to a constant? For example, if I have the AR process $$(1-\phi B)(y_t-\mu)=\epsilon_t$$ does that equal $$y_t-\mu-\...
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1answer
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Problem simulating AR(2) process

OP EDIT: There where no problem with this. The problem was with the method I was using for obtaining the PACF. Apparently it doesn't work quite well in this case (I was using the scikits/tsa python ...
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1answer
580 views

Why does a AR(1) model that's mean reverting revert back to B0 as opposed to B0 + B1*B0?

In time series analysis, an AR$(1)$ model takes the form: $$x_t = \beta_0 + \beta_1 \cdot x_{t-1} + w_t,$$ where $w_t$ is the white noise term. In order for the model to be stationary and to ...
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1answer
111 views

AR model with vector valued variables (in R)

I want to estimate a vector-valued model $$\mathbf{y}_t = a\mathbf{y}_{t-1}+b\mathbf{y}_{t-2}+\cdots$$ Here, each $\mathbf{y}_t\in\mathbb{R}^n$ and the coefficients $a,b,\dotsc$ are real numbers (...
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830 views

AR(1) working correlation matrix with GEE

I'm attempting to fit a GEE model and I have a question about using the AR(1) working correlation matrix. I've read some conflicting information about this correlation matrix. In some books and ...
3
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1answer
302 views

Residual based bootstrap autoregressive series in MATLAB

I have defined the model as follows. Let $$y_1 = 0$$ and $$ y_i = \alpha + \beta y_{i-1} + \epsilon_i $$ for $i_2\ldots i_T$, where $\alpha$ and $\beta$ are the estimated coefficients and $\...
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1answer
7k views

Fit a moving average (MA) time series model to the data (R:stats::ar equivalent)

I am looking for some tools for automatic fitting of moving average (MA) time series model to my data in R. I know R:stats::ar ...
3
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1answer
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AR(1) parameter estimation

Given a time series, I'd like to estimate the parameters of an AR(1) model for it. As explained on wikipedia, there are different ways for doing that. What may be called a naive method is to compute ...
3
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1answer
45 views

Autocorrelation of an AR(1) process

I am learning about this AR process. According to the book I'm reading, the autocorrelatio function of a stationary process: $$y_t = c + \phi y_{t-1} + \varepsilon_t, \quad \quad |\phi|< 1$$ is ...
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27 views

Uniquely defined autoregression

Let $\{w_t\}, t \in \mathbb{Z}$ be a random noise. Given a sequence $\{w_t\}$, does the autoregression $x_t = x_{t-1} - 0.9 x_{t-2} + w_t$, $ t \in \mathbb{Z}$ uniquely define a sequence $\{x_t\}$? ...
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116 views

How can differencing in an ARIMA model be implicit instead of explicit?

In this post - Rob Hyndman explains that: Even with that correction, the two models are not quite equivalent. In the Eviews code, the differencing is done before estimation, whereas in the R code ...
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1answer
126 views

Are all $AR(p)$ processes for which $|a_1|,…,|a_p| < 1$ stationary?

For an $AR(p)$ process $ Y_t = a_1Y_{t-1}+a_2Y_{t-2}+...+a_qY_{t-q}$ : Is having the coefficients $|a_1|,....,|a_p| < 1$ just a necessary condition for stationarity, or is it sufficient as well?
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2answers
127 views

Time Series AR1, coefficient accuracy different from linear regression

I am interested to know why running the R arima function's result has a different accuracy from my "manual" way of linear regression. ...
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2answers
617 views

Autoregression vs Sliding Window method

I'm a beginner at machine learning and have a question regarding time series. I have a data set dependent over time, with a single feature and I am trying to predict the future value of this. This far ...
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1answer
358 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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1answer
530 views

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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185 views

What is the distribution of the difference between two AR(1) processes?

I am reading a paper published in a good economics journal. An econometric model is presented in the paper. A part of the described model is not very clear to me. Please let me state a couple of ...
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1answer
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Practical issues with dynamic panel data modeling

Unfortunately for me, I've got a situation where I need to control for the lag of a dependent variable as a robustness check against an alternative interpretation of my main regression. The baseline ...
3
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1answer
37 views

Mixture of normals, dependent on past

I have the following probability model: $(X_k|\text{PastHistory}_{k-1}, \theta_0,\theta_1,\theta_2) \sim (\pi\cdot N(\theta_1+\theta_0\cdot X_{k-1},1)+(1-\pi)\cdot N(\theta_2+\theta_0\cdot X_{k-1},1))...
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342 views

Is this explanation of the Box-Jenkins approach correct?

Can someone please tell me what they think of my explanation of the Box-Jenkins approach to modeling time series? Do you have anything to add (in particular to my explanation as to the intuition ...
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2answers
889 views

Need a clear and simple auto-regressive model example

This may be hard to find, but I'd like to read a well-explained auto-regressive model example that: uses minimal math extends the discussion beyond building a model into using that model to forecast ...
3
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1answer
35 views

Statistical test for comparing means of two AR(1) time series

Say I have two time series which each follow the AR(1) model: $$ X_{t+1} = X_t + (1 - \theta_X) (\mu_X - X_t) + \epsilon_X(t) $$ $$ Y_{t+1} = Y_t + (1 - \theta_Y) (\mu_Y - Y_t) + \epsilon_Y(t) $$ ...
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1answer
59 views

Handling serial correlation in time series regression

Suppose that the time series data $(y_1, y_2,..., y_n)$ can be explained through a regression model with $k$ explanatory variables: (1) $y_t = b_0+b_1x_{1t}+b_2x_{2t}...+b_kx_{kt} + \epsilon_t,\ t=1,...
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62 views

How do the forecast intervals from an AR model behave when the time series is inherently stationary?

I'm trying to wrap my head around two contradictory intuitions behind how forecast intervals should behave when we use an AR process to model a stationary time series: (a) On one hand, since the time ...
3
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1answer
61 views

Joint AR(1) posterior distribution explicit under conjugate prior

I have encountered a problem in my textbook 'The Bayesian Choice' by Christian P. Robert. It goes something like this: $"$For a particular case of AR(1) model, $(x_t)_{1\leq t\leq T}$. Where $x_t = \...
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1answer
91 views

Why is auto.arima modeling an AR(1) process as an MA(1)?

Playing around with auto.arima to see how effective it is at model selection. I first simulated an $AR(1)$ process with $X_{t+1} = 0.9 X_t + \epsilon_t$ ...
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1answer
77 views

Parameters in Autoregressive representation of an ARCH model

Suppose we have a $0$ mean time serie representing stock index returns about a title, $r$. I also know it follows an $ARCH(p)$ model with parameters $\omega$ and $\alpha$, specified in the following ...
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1answer
138 views

Is this an Autoregression or OLS?

Say I have a time series $y_t = \frac{1}{n} \sum_{i=1}^n x_t^{(i)}$. For example, $y_t$ can be the returns of the S&P 500 index at time $t$ and the $x^{(i)}_t$ is return of the $i^{th}$ company in ...
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2answers
656 views

An autoregressive process has to be always gaussian?

It is well known that if a generic autoregressive process of $n$ order, $AR(n)$, has a gaussian white noise error term ("innovations"), then it is gaussian too. So I presume that if the error term is ...
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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, won't they also be collinear, making e.g. autoregressive ...
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1answer
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Fitting model AR(1) with R

I've sampled 100 variables from a Gauss distribution with mean 0 and standard deviation 1. > set.seed(1) > wn=rnorm(100) Then I've fitted an AR(1) model ...
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1answer
181 views

How do I quantify the decay in the initial condition of an AR process?

I'm working on basic code that generates data from AR and VAR processes; the code generates enough observations to dampen the effect of the initial conditions. For example, if I want to generate 30 ...
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1answer
176 views

How to identify relationship between response time series(Yt) & input time series(Xt) only in terms of Yt-1 & Xt?

I have a response time series(Y) & Input time series Xt & Zt. My only objective is to identify functional form Yt=f(Yt-1,Xt,Zt) where f(Yt-1,Xt,Zt) contains only lags of Yt , Xt & Zt as ...
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1answer
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Meaning of output of function “ar” in R

How should I read the output of the function ar in R. For example, take this VAR model: ...
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1answer
710 views

Residuals in double seasonal exponential smoothing

I have a time series with muliple seasonal cycles, which are 24 and 168 hours for my case. I would like to use Double Seasonal Exponential Smoothing method to forecast, which was published by James W. ...
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2answers
480 views

Evaluation of Autocorr/Part Autocorr values

I am practicing MA and AR modelling by using autocorrelation and partial autocorrelation values. My data is in the image below; I can see that only at lag 12 there is a value that might be considered ...
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1answer
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Different results of Engle's Lagrange multiplier test for conditional heteroscedasticity from SAS and FinTS

To fit a simple AR(5) model, I use SAS PROC AUTOREG. I called the option ARCHTEST=(QLM) which provides Engle’s Lagrange ...
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2answers
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Stationary ARMA model as infinite AR or MA process

How can a stationary, invertible ARMA(1,1) process be represented as either an infinite order AR or infinite order MA process?
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1answer
54 views

When to use AR and when to use MA model?

When to use an AR model and when to use an MA model to model time-series data. What aspects of data are modelled by the AR process which can't be done by MA and vice-versa?
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1answer
72 views

First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$ where $$ \mu = 0.85 \\ \phi=0.59 $$ and $$ W_t = X_t - X_{t-1} $$ Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
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Do the assumptions for linear regression apply to AR(p) models?

If we have a stationary time series and we want to model it as an AR(p) process, what conditions must hold besides the stationarity itself? Are they the same a the assumptions for linear regression: ...
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134 views

Dealing with autocorrelation using Generalized Least Squares

I have a time series data set where the auto correlation of the residuals follow an exponential decay. I was wondering how I should deal with this? I would like to fit a linear model and have tried ...
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Thomas Sargent's intuition as to why every covariance stationary series has an infinite-order Wold representation

In his classic book "Time Series Analysis", James Hamilton references Thomas Sargent (["Dynamic Macroeconomic Theory"], 1987, pp. 286-290) as a "nice sketch of the intuition behind this result [Wold ...
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simulate autoregressive data that is also multivariate normal

I am trying to simulate data that is correlated to varying degrees. However, the data itself will have a degree of autocorrelation as well. I can get the first part of the problem with mvrnorm ...