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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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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26 views

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
30 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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42 views
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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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45 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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16 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
28 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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32 views

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 ...
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19 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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27 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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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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39 views

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 ...
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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 ...
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51 views

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$\Delta x_{t-2}$ ...
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2answers
49 views

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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43 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 ...
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1answer
28 views

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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29 views

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.} \: ...
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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 ...
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89 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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Expectation of the distant future values conditional on the current information today

I am trying to understand a structural econometric model, in particular the one presented in "Stock J. H. and Wise,D. A., 1990. Pension, the option value of work, and retirement, Econometrica, 58 (5), ...
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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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Multi level growth model - modelling the covariance structure

I am performing a multi level growth model in R with rank as the DV, seasons since draft as the time-based predictor and a random effect for position (intercept and slope). I am attempting to model ...
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1answer
43 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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Individual versus group-wise significance in ARDL context

In an ARDL model approach, what is one supposed to do if the F-bound test shows insignificance while some variables have significant long run and short run coefficients (the error correction term is ...
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26 views

How to estimate a two-level, driven, AR(1) process

I would like to estimate the following model: $$ y_t = (1-a)\,y_{t-1} + a\,\mathbf{b}^T\mathbf{x}_t + \epsilon_t \\ z_t = (1-c)\,z_{t-1} + c\,y_{t} $$ where $z_t$ and $\mathbf{x}_t$ are observed, ...
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62 views

Vector autoregression: many variables (10), short sample (100)

Suppose there are ten observation sites along the road. A, B, C, D, E, F, G, H, I, J. We obtain data at each site once in a day, in this order. That is, first go to the site A at 9:00a.m., and when ...
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Sum of the AR coefficients and First Order Autocorrelation Coefficient

I'm working with quarterly inflation, usually a AR(4) and I want to obtain different measures of persistence, that are: 1. the sum of the AR coefficients Σα 2. First Order Correlation Coefficient, ...
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Combining continuous spatial and discrete time series methods for spatial prediction

Here's something I've been pondering. Wondering if anyone can shed come light on it/recommend some references/tell me why it makes no sense, please. In my field (predicting crime risk by location), ...
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Order of AR and MA

How to determine the order of AR and MA Process for the time series data of length 8000?? Also let me know how to find coefficients of AR and MA as the references provides only for a0 and a1...
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2answers
131 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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124 views

How to calculate interim and long-run multipliers in ARDL models with >1 lag?

I have calculated an ARDL(24,36) model with 1 independent variable. The data is monthly, hence the inclusion of so many lags. I am trying to calculate the interim multiplier (the cumulative effect at ...
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26 views

How can I test for seasonality when the trend is not supposed to be monotonic but sinusoidal?

My knowledge of time-series analysis is limited. So far I have only assessed whether there was a seasonality in my time series data with the assumption of monotonic trend. To test that I would have ...
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How estimate Autoregressive model of order 2 on dataframe with missing values and multiple columns in R

I have a dataframe with 4000 companies, each company present as a column in the dataframe. The complexity of my dataframe is that companies belong to a stock exchange and all companies that had been ...
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Reproducing an AR Model

I recently found this paper https://static.googleusercontent.com/media/www.google.com/en//googleblogs/pdfs/google_predicting_the_present.pdf where the authors predict economic trends with Google ...
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Mitigate autocorrelation in time series with AR(2) process

I have a dataframe with 4000 companies and I have calculated a liquidity measure of each of the company in the dataframe. Liquidity is highly persistent. And my analysis shows that in these indiviual ...
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2answers
190 views

Time series with autoregressive distributed lags: Forecasting for future

I have daily data from last 2 years. I want to do ARIMAX and the regressor component being autoregressive distributed lag of the same variable. Since it has impact, along with dummy variables to ...
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19 views

How to write this ARIMA model mathematically? [duplicate]

I am trying to analyze a time series: I see a strong seasonal pattern, so from every value, I subtract the value from the same month the previous year (12 periods prior). Also, I am using 1 AR term ...
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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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1answer
187 views

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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How to interpret linear filter formula

I am taking my first course in time-series analysis, and I recently encountered the so-called linear filter for the first time. I thought I could just skip this section. Apparantly though, this ...
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heteroscedastic time series in SAS autoreg - white noise matter?

I normally work with categorical outcomes, so a lot of this is new to me. Attempting to model monthly interrupted time-series in proc autoreg. There were 11 intervention changes of varying potency ...