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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A closed form formula for the normalizing constant in standard normal auto-regressive series?

Let $Z_t = c_1Z_{t-1} + c_2Z_{t-2} + ... + c_nZ_{t-n} + c\epsilon_t$ where $Z_t, \epsilon_t \sim \mathtt{N}(0,1)$ are iid variables and $Z_s \sim \mathtt{N}(0,1)$ for all $s$. Given the values of $...
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1answer
258 views

Is there a convenient form for this large covariance matrix?

Consider the following bivariate vector autoregression: $$X_t=\mu +X_{t-1}A+\varepsilon_t,\ \varepsilon_t \overset{iid}{\sim} MVN(0, V),\ X_t=(X_{1,t},X_{2,t})',$$ where the assumptions on the ...
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Why don't we look at $R^2$ when fitting an autoregressive model?

$R^2$ measures explained variance. In an autoregressive model like AR(k), we are carrying out a linear regression, and as such we would have an $R^2$ and an ...
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1answer
820 views

Testing periodogram “peaks”: sine-like wave or AR/MA/ARMA noise?

I'm performing an harmonic fit to data I know (from physical constraints) come from a periodic source of the form $$\sum_j^M \sum_i^N a_{i,j}\sin(2\pi f_it)+b_{i,j}\cos(2\pi f_it)$$ using the Lomb-...
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2answers
11k views

Invertibility of AR(p) model

Notation: $\dot{Z}_t = Z_t - E(Z_t)$, so that it is centered at 0. $a_t$ stands for the residual and we assume the $a_t$ are independent and normally distributed with mean 0 and constant standard ...
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3k views

Finding variance of AR process

$\newcommand{\E}{\mathbb{E}}$How do I find the variance of an autoregressive AR(1) process $$y_t=\phi y_{t-1}+\varepsilon_{t}$$ where $\lvert {\phi}\rvert<1$ and knowing that $$y_t=\sum_{j=0}^\...
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2k views

Periodicity and seasonality of a time series

I have a time series and I have done some spectral analysis on it. When doing an autocorrelation and periodogram it shows that the time series is periodic. However when I do a Dickey-Fuller test it ...
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460 views

How to interpret the characteristic roots of moment equation of a AR(2) model?

I am learning the financial time series using the book 'Analysis of financial time series' by Ruey Tsay. In chapter 2, they introduced AR(2) models. The moment equation (which is the function between ...
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3answers
144 views

Difference between $y_t = \alpha + \beta t$ and $y_t = y_{t-1} + \beta$

Would someone mind walking me through the differences between: \begin{align} y_t &= \alpha + \beta t \\ &\& \\ y_t &= y_{t-1} + \beta \end{align} as well as between \begin{align} ...
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Does using lagged independent variables makes sense?

While it seems quite common to calculate a lagged version of the dependent variable and to use it on the right hand side of a model (e.g., autoregressive models), I have rarely seen that lagged ...
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3k views

Joint distribution of AR(1) model

Let $$y_1 \sim \mathcal{N}(\phi_0,\,\sigma^2),$$ $$y_t|y_{t-1},\, \phi_0,\,\phi_1,\,\sigma^2 \sim \mathcal{N}(\phi_0+\phi_1(y_{t-1} - \phi_0),\,\sigma^2),$$ for $t=2,3,\cdots,T$. I want to ...
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1answer
385 views

Why are exponential smoothing models not considered auto-regressive?

I've seen so far two definitions of the term "auto-regressive" model when it comes to time series modeling: The first definition is just basic AR models and their relatives such as ARMA and ARIMA, ...
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3k views

Autocorrelation and Partial Correlation plots in ARMA models

Consider the following input and its Autocorrelation and Partial Autocorrelation plots (source). What are the shaded blue areas in these plots? I often see them when studying ARMA models. What do ...
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2k views

What does R do when it plots the residuals of an AR fit?

This is a question that's been bugging me for some time. The problem is this: I'm modelling the residuals of a model $f(t,\vec{\theta})$ with (what I think is) an AR process plus a white noise process ...
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1answer
4k views

Time series regression with lagged dependent and independent variables

I have monthly data for air passengers, oil price and unemployment. I'm trying to create a model to forecast air travel demand using oil price and unemployment as explanatory variables but are facing ...
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1answer
2k views

How to build a function with the result of auto.arima in R?

I use: fit = auto.arima(Y, xreg=X) in R to get ARIMA(1,0,0), result as follows: ...
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1answer
263 views

How to interpret the expression of MA(1) as AR($\infty$)

When AR(1) is expressed as MA($\infty$), I can interpret it as: let's say my wage this year depends only on last year's wage and a random shock (my boss' mood). But last year's wage also depends on ...
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1answer
315 views

Sum of autocovariances for AR(p) model

Suppose I have the following $AR(p)$ model. $$X_t = \sum_{i=1}^{p} \phi_i X_{t-i} + \epsilon_t\,, $$ where $\epsilon_t$ has mean 0 variance $\sigma^2$. I am not interested in fitting this model, but ...
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2answers
69 views

Why are we not concerned about the distribution of the $x_t$ in an AR(1) model?

I am trying to investigate the reasons why we don't bother about the distribution of the $x_t$ in an autoregressive model. Why do we concern ourselves about the distribution of $e_t$? And why are ...
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1answer
440 views

Interpreting the coefficient of a non-binary defined dummy variable

I'm reading an economics paper, where the author is using dummy variables to test for political effects on variables, such as GDP and unemployment. The model is a simple autoregressive model with ...
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191 views

Are Auto-Associative Regression Trees Distinct from Auto-Regressive Trees?

After some reading in the field I was confused as to whether these two models are distinct or really the same. I'm just looking for a simple yes/no with a brief explanation. Note that Auto-...
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816 views

How do I Estimate Joint Entropy Using a Histogram?

I am trying to estimate the entropy for two time series, defined by random variables $X$ and $Y$, each distributed according to an unknown PDF which is to be estimated empirically (using a histogram ...
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504 views

Estimation of mean of AR(1) process

It is well known that for the basic AR(1) model $X_k=\phi X_{k-1}+\varepsilon_k$ where $\vert\phi\vert<1$ and $\varepsilon\propto\mathrm{WN}(0,\sigma^2)$ the sample mean $\bar{X}_n$ (after being ...
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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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2answers
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Cross correlation influenced by self auto correlation

I have two stationary time series ts1, ts2, I wanna find the cross correlation ($\textrm{CCF}$) between them. As a result, it ...
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1answer
3k views

How do I interpret lagsarlm output from R's spdep?

I've run lagsarlm on my dataset, using a mixed model and using a row-standardized adjacency matrix. I have results that I think are good, but would am not sure how ...
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1answer
4k views

Determining the amount of lag in an autoregressive model

I have done a lot of work in regression (time-invariant) but I am just now studying forecasting. My question is about determining the amount of lag to use in an autoregressive model. I assume that ...
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1answer
3k views

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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Breusch-Godfrey Test and the length of the lag, p

I'll use Breusch-Godfrey (BG) test to test correlation of an AR(1) model. In order to perform a BG test, the simple regression model is first fitted by ordinary least squares to obtain a set of sample ...
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2answers
2k views

Steps to perform time series analysis

I'm trying estimate an autoregressive model with an exogenous variable. It's about the impact of changes in oil prices on the economy. I'm planning on regressing gdp growth rate on its own lagged ...
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1answer
634 views

Determining the lag order of an autoregression model

I'm trying to fit my time series data, which have 37 entries, with an autoregression model. The data look like: I just loop over 1 to 18, each of which is used as the order of AR model. A plot of the ...
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1answer
81 views

How to correct for order bias in responses to psychometrics questions?

I'm looking at the data from Individual Aesthetic Preferences for Faces Are Shaped Mostly by Environments, Not Genes by Germine et. al., where members of twin pairs were presented faces in an order ...
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1answer
291 views

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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1answer
746 views

How to plot spectra of an AR(2) process

I am stuggling with this problem and was hoping to find some guidance to answer it. Let $y_t=\phi_1y_{t-1}+\phi_2y_{t-2}+\epsilon_t$, with $\epsilon_t\sim N(0,1)$. Now, I want to plot the spectra ...
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44 views

Autocorrelation in Elo ratings

FiveThirtyEight uses the following formula for their NFL Elo ratings: $$ R_i^{k+1} = R_i^k + K \cdot M(z) \cdot A(x) \cdot (S_{ij} - \sigma(x)) $$ where $z$ is the game's margin of victory, $x=R_i^k - ...
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130 views

Stationary Distribution of Multiplicative Autoregressive Model

I know for the additive autoregressive model the stationary distribution of $\{X_t\}$ can be found, if it exists, in the following way: \begin{align} X_t &= \alpha X_{t-1} + \epsilon_t\\ \...
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Identification Problem in Minimum Distance Estimation

I have the following problem with a system of minimum distance equations I want to solve. The objective is to estimate the parameters of the random variables in the following DGP: $$ x_t= \phi_t(\...
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1answer
148 views

EM for MAP variance of AR(1)

this question is about MAP inference in an AR(1) model (exercise 1.6 from West, M., Time Series: Modeling, Computation and Inference). It's not a homework assignment. Assume $n$ observations were ...
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1answer
480 views

Model selection and estimation for pseudo out-of-sample forecasting

I have quarterly data on inflation from 1990 Quartal 1 to 2016 Quartal 3. If I want to perform the pseudo out-of-sample forecasting one quarter ahead with an autoregressive function, do I have to ...
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247 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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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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569 views

Longitudinal data analysis where meaning and metric of response variable varies over time

Determining what factors predict change over time is a topic of investigation in many fields and there are a variety of readily implemented methods for analysing repeated measures in the same metric. ...
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316 views

Determining parameters in AR model for non-stationary time series

I am currently trying to fit an AR model to some financial data. The time series $Y_t$ in levels is clearly non-stationary; however it appears the first differences $dY_t$ are stationary (and this is ...
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453 views

How to fit log-linear poisson autoregressive mixed model?

I have time-series count data $N_{i,j}$ (population sizes in site $i$ and year $j$) and I want to correlate year-to-year changes with the environmental conditions $x_{i,j}$. For this, I want to fit ...
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410 views

Fit Negbin glm model with autoregressive correlation structure

I am attempting to estimate the effect of various variables on the time-series of counts of reported cattle stillbirths. We investigate the effect of day-of-week, month, holidays etc…and also the ...
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4answers
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Time Series for each customer

Is it possible to create Time Series Analysis for each customer? Say if have 100 customers and I wanted to predict how much amount they are going to spend next. I have done the Time Series for the ...
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1answer
5k views

How to write variance covariance matrix of AR(1) process in R

I'm trying to write autocovariance matrix of AR(1) process in R and I'm having difficulty. The autocovariance matrix that I'm using in my project takes the form as shown in the picture: I also ...
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2answers
243 views

Differencing a time series

I am looking to find the ACF of a time series, but after it is differenced. $y_t = a_1y_{t-1} + \epsilon_t , \mid a_1 \mid < 1$ I understand that to find the ACF this process needs to be ...
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2answers
947 views

Stationarity of AR(1) process, stable filter

This section of the Wikipedia article about the Autoregressive Model reads: An AR(1) process is given by: $$X_t = c + \varphi X_{t-1}+\varepsilon_t$$ where $\varepsilon_t$ is a white noise process ...
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863 views

How to show the inconsistency of the OLS estimator for unit-root AR(1) processes by simulation?

From what I understand, OLS gives consistent estimates for stationary AR(1) time series but not for unit-root ones. I am trying to illustrate this phenomenon with a small simulation in R but the OLS ...