# 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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### 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 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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### (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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### 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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