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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Why is this linear combination of random variables from a white noise stationary?

I'm looking at the following definition of a causal AR(p) (autoregressive) model: An AR(P) model $\phi(B)x_t= \epsilon_t$ is said to be causal if it has a stationary solution $$x_t=\epsilon_t +\sum^{\...
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Getting different AIC / BIC values for AR(2) estimation via AutoReg(2) vs ARIMA(2,0,0) through python statsmodels

I am trying to fit an AR(2) model to a data series claims_df['initial claims'] via statsmodels.tsa.ar_model.AutoReg and ...
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Understanding Auto-Regression in details

I was analysing the equation of auto-regression model, e.g. AR(1), so that: Yhat = Beta0 + Beta1Yt-1 Lets say I have a 36 observation series and I want to forecast 3 periods ahead and Beta1 is 1.1 (...
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Expectation of the ratio of sum (XY) and sum(X)

I want to know (mathematically) how the following expression changes as $M$ increases but still have no clue after thinking about it for a while. Any suggestions or comments will be much appreciated. $...
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How do I prove the general formulation of LRV to verify that LRV for AR(1) is in fact (σ²)/(1-α)²

I understand the equations separately but don’t know how they are connected! Please give me an explanation / hint Given the AR(1) structure, the long run variance (LRV) of Xt is known to be (σ²)/(1-α)...
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Using the Yule-Walker equation to calibrate an autoregressive model with the method of moments

Consider the following discrete autoregressive $\epsilon_t$, where $\epsilon_t \in (\pm 1 ) \forall \ t \geq 1$. We think of $\epsilon_t$ as the child of a previous sign at time $t-l$, where $l$ is ...
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Stationarity condition for specific Autoregressive AR(2) Process: $y_{t} = y_{t-1} + Cy_{t-2} + z_{t}$

The Autoregressive (AR) process $y_{t} = y_{t-1} + Cy_{t-2} + z_{t}$ is stationary provided that $-1 < C < 0$. We know that a general AR(2) process i.e., $y_{t} = a_1 y_{t-1} + a_2 y_{t-2} + z_{...
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Kalman forecast of AR(1)

I'm trying to work out the details of the proof of the following statement: Suppose $\xi_t = \rho \xi_{t-1} + \epsilon_t$ is an AR(1) process. Using Kalman filter, one can prove that $\mathbb{E}_t\{\...
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Unbiasedness and consistency of OLS in an AR(1) model with AR(1) residuals [duplicate]

consider equation 1 : , Now let , where the error component is iid with mean 0 and constant variance, and Is the OLS estimator of the coefficients in equation 1 unbiased and consistent under this ...
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Absolute Summability of Coefficients of MA(inf) representation of stationary AR(p)

Not self-study. This is more of a maths question: Let $\Phi(B)X_t = e_t$ be a (weakly) stationary process. Let $$\Phi(B) = \prod_{i=1}^p (1-\alpha_iB)$$ So from stationarity we will have $|\alpha_i|&...
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What are pros and cons of AR and other time series models with overlapping datapoints

Anyone have any links or resources on pros/cons of building a timeseries model with overlapping data points? Generally, a lot of text build models on daily returns, but let's say the daily variable ...
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Overlapping data in time series model?

I have some time series data that I am trying to model - basically historical elasticity reads with daily sales data. I really am just interested in finding the 'mean' value - not necessarily predict ...
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Generating Autoregressive x(m) Terms in R [closed]

Attempting to create code that generates x1...xm values, time series plot, and autocorrelogram given certain coefficients and white noise. Code built so far: ...
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Forward/Backward Iteration and Stationary/Stability

Suppose I have an AR(1) process of the form: $$y_t = \phi y_{t-1} + \epsilon_t$$ where $\epsilon_t$ is a white noise process with mean zero and variance $\sigma^2$. If $|\phi| < 1 $ , the model is ...
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invertible moving average (MA) and its inverted form

Since we can write any invertible MA(q) time series in the inverted form as $z_t \approx \sum_{j=1}^{p}\pi_j z_{t-j} + a_t$, does this mean that we can fit an AR(p) model (with p being high enough) ...
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Long-term variance of AR(p)

According to the Yule-Walker equations, the long-term variance of an AR(p) process is $$\frac{\sigma^2}{1-\phi_1\rho_1 \ldots - \phi_p\rho_p}$$ Where the AR(p) is defined as $$Y_{t}=\delta + \phi_1 y_{...
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Autocovariance of Explosive AR(1) model with $|\phi|>1$ expressed as a stationary process

I am working through the book called Time Series Analysis and Its Applications by Shumway and Stoffer. I am stuck deriving an equation given in example 3.4 in the book (page 80 for the fourth edition),...
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Get Yule Walker estimates from autocorrelations

I'm looking to be able to estimate parameters of an AR(2) time series using the sample autocovariances. I have: ...
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Handling many short times series with exogenous variables simultaneously

I am trying to find a solution to a problem, having a dataset with multiple short time series and exogenous variables. Read this, this, and this. And many other resources. Still cannot find a clear ...
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How to turn many variables into one for the PACF equation

I have been trying to calculate the PACF manually, but I encountered some issues with the following equation: $PACF = \frac{Covariance ([Y_{t}|Y_{t-1}, Y_{t-2},...,Y_{t-k+1}],[Y_{t-k}|Y_{t-1}, Y_{t-2}...
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I don't seem to interpret my program generated AR(1) model correctly

For a high school maths paper I have been attempting to model the monthly unemployment rate in the USA since January 1948. I chose to create an AR/Auto-regression model to forecast future unemployment ...
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How do I calculate $\sigma^2$ within the $\varepsilon = N(0, \sigma^2)$ notation

The error term for an AR(1) model is assumed to be $\varepsilon = N(0, \sigma^2)$, a mean of 0. An example of a source that mentions this: The AR(1) model can be written in intercept form, $$z_t = \...
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How do I find my coefficient for AR(1)

I've been working on trying to create an AR model for the unemployment rate in the USA. I started with de-trending etc to make my data stationary, then tested with a dickey fuller test to make sure ...
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Thresholded AR(1) Forecast

I want to find the forecast of an threshold AR(1) process: I am confused on how to do it. Do I take the expected value of the process making $x_{t-1}$ become $x_n$, and $w_t$ becomes $w_{n+1}$, and ...
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What would be a continuous-time version of a VAR process?

It is often said that a AR(1) process can be viewed as a discretized version of the continuous-time Ornstein-Uhlenbeck process. Can we really claim this to be valid considering that the Ornstein-...
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Autocovariance for an explosive AR(1) process

I am struggling to understand how the below result of the autocovariance of an explosive AR(1) process is derived, taken from Time Series Analysis and its Applications (R. H. Shumway & D. S. ...
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Conditional expectation of $X_t$ in a time series, given that other draws were below $c$

I'm interested in the moments of a given draw, $X_t$, of a time series conditional on the knowledge that all other draws within some window before and after $t$ were below a fixed threshold, $c$. For ...
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Why, empirically, does strict stationarity only hold *asymptotically* for an AR(1) process?

A Gaussian AR(1) process with autocorrelation $|\phi|<1$ is strictly stationary, meaning that: $$F_{X}(x_{t_1+\tau} ,\ldots, x_{t_n+\tau}) = F_{X}(x_{t_1},\ldots, x_{t_n}) \quad \text{for all } \...
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Distribution of Variance of draws from Multivariate Normal Distribution

Let the vector $\boldsymbol x$ be a draw of $n$ values from a multivariate normal distribution with zero mean. $$ \boldsymbol x \sim \mathcal{N}(\boldsymbol 0, \Sigma) $$ It may be assumed that $\...
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How to check assumptions of an AR(1) process

My question is, if a model allows the errors to follow an AR(1) process, how to check for the model adequacy? Can we use ACF and PACF plots?
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How to detect an AR(1) process of residuals from a correlogram?

I am estimating a dynamic factor model which allows the errors to follow an AR(1) process. Thus, an approximate dynamic factor model. So for residual diagnostics I plotted the correlogram of residuals....
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What is the structure of a GAM fit with `mgcv::bam()` with the `rho` parameter set?

If I fit the model mgcv::bam(y~s(x), rho=0.8), where y and x are ordered by time, my understanding is that the model can be described in math notation as: $ y_t = \...
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Are there circumstances where the Yule-Walker is preferable to OLS?

Yule-Walker and OLS seem to both provide comparable estimates in AR models for large N, but Yule-Walker's estimates "are not so good" for smaller N (Chatfield, 2019, p. 84). Under what ...
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Autoregressive Timeseries (AR-1) and interpretation of autocorrelation plots

My understanding is as follows: ACF-Plot: AR-k model shows highest correlation at lag k PACF-Plot: PACF to only describe the direct relationship between an observation and its lag, so that there would ...
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Estimate AR(1) parameters when DGP is AR(2) and the aim is a recursive forecast

I am exploring different options for making recursive forecasts for time series in the ML realm. I have found that it is easier to understand the basic concepts if I first use simpler models from the ...
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2 votes
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Using the method of moments or GMM to estimate the parameters of a specific problem

Given $(X_t)_{t \in \mathbb{Z}}$ an AR(1) process: $$X_t = c+ \phi X_{t-1} + \epsilon_t, \quad \epsilon_t\sim WN(0,\sigma^2)$$ We can show that $E(X_t) = \frac{c}{1- \phi}$ and $E(X_t^2) = \frac{\...
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(Sums of dependent random variables) This problems develops a central limit theorem for a sum of dependent random variables

Let $X_1, X_2,...$ be i.i.d. r.v.s with zero mean and unit variance. Define $Z_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n X_jX_{j+1}$. (a) Show $Var(Z_n) = 1$ (b) Show $Z_n \to \mathcal{N}(0,1)$. Hint: First ...
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Understanding Dickey-Fuller Test vs. t-test

I am working to understand why it is that in an AR(1) regression, the regression coefficient is not asymptotically t-distributed. Specifically, I'm trying to understand which assumptions about a ...
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Differencing data with missing values?

I have a non-stationary dataset that I would like to model using a VAR model. I need to difference it to make it stationary, however my dataset contains a lot of NaN's at random points, so using ...
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Spearman rank correlation of AR1 process or bivariate normal

In a first-order autoregressive process (AR1), a time-series is generated which correlates with itself whereby datapoints close in time are more correlated than datapoints further away from each other....
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The prob. limit of the OLS estimator of AR(1) process with AR(1) errors

Given the model: \begin{aligned} Y_t &= \delta Y_{t-1}+u_t, \\ u_t &= \rho u_{t-1}+\epsilon_t, \end{aligned} where $\epsilon_t\sim i.i.d. (0,\sigma^2)$, $|\delta|,|\rho|<1$. Then how to ...
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Can HAR models also be applied on non-volatility data?

Currently, I am trying to forecast several cash flows of accounts receivable and payable of a company. I want to apply the HAR model due to the simple structure of the model; it incorporates the short-...
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Does a constant term in a nonstationary AR model always imply a trend?

Given an $AR(k)$ model of the form $$y_t = \alpha_1y_{t-1}+...+\alpha_ky_{t-k} + \mu + \varepsilon_t$$ with $\alpha$ satisfying $(1-\alpha_1 z - ... -\alpha_k z^k ) = 0$ for $z=1$, does a nonzero $\mu$...
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Can there exist a unit root series that’s Granger-caused, or better predicted with a model other than the AR process we tested using ADF?

If a series has a unit root, then it is a function of random white noise. Therefore, it follows a random walk process. Is it then possible for: Some other series to Granger-cause the unit root series?...
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Yule walker equation [duplicate]

y_t * y_(t - tau) = phi * y_(t - 1) * y_(t - tau) + epsilon_t * y_(t - tau) → gamma_tau = phi * gamma_(tau - 1) Hello. I am just really unclear how you get the second equation from the first. I read ...
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yule walker equation [closed]

Can you please explain me how by taking the expectations of both sides one arrives at the final equation?
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Calculate AIC of an AR process knowing the coefs and n?

In a paper I'm reading, the authors determine the order of an AR process via AIC. Fine. But they do so from the AR coefficients and length of the process and not from the time series itself. The ...
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logLik of an AR1 in R package stats

...
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Why is AR(1) process with $|\phi| > 1$ not stationary? [duplicate]

I saw the top answer on the following post: Stationarity of AR(1) model. It says that an AR(1) process $X_{t} = \phi X_{t-1} + \epsilon_{t}$ where $\epsilon_{t} \sim WN(0, \sigma^{2})$ has a ...
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Sample Period for Model Selection?

I am currently trying to perform pseudo-out-of-sample forecasting for monthly exchange rates with a 10-year rolling window. Before that, I select an autoregressive model using Box Jenkins ...
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