# Tag Info

### Confused about Autoregressive AR(1) process

You have two problems--and one of them is interesting. Without a noise term, the series is no longer stationary. Its value is increasing asymptotically, but definitely, toward $1:$ ARIMA applies ...
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### How does ACF & PACF identify the order of MA and AR terms?

The quotes are from the link in the OP: Identification of an AR model is often best done with the PACF. For an AR model, the theoretical PACF “shuts off” past the order of the model. The phrase “...
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### Why do we care if an MA process is invertible?

Invertibility is not really a big deal because almost any Gaussian, non-invertible MA$(q)$ model can be changed to an invertible MA$(q)$ model representing the same process by changing the parameter ...
• 11.2k
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### Why is OLS estimator of AR(1) coefficient biased?

As essentially discussed in the comments, unbiasedness is a finite sample property, and if it held it would be expressed as $$E (\hat \beta ) = \beta$$ (where the expected value is the first moment ...
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### If an auto-regressive time series model is non-linear, does it still require stationarity?

If the purpose of your model is prediction and forecasting, then the short answer is YES, but the stationarity doesn't need to be on levels. I'll explain. If you boil down forecasting to its most ...
• 61.5k
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### What is the difference between deterministic and stochastic model?

The video is talking about deterministic vs. stochastic trends, not models. The highlight is very important. Both your models are stochastic, however, in the model 1 the trend is deterministic. The ...
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### What is the difference between deterministic and stochastic model?

As Aksakal mentioned in his answer, the video Ken T linked describes properties of trends, not of models directly, presumably as part of teaching about the related topic of trend- and difference-...
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### Why is the dickey fuller test different from a simple t-test

You are right that the test statistic is just a standard t-statistic. It, however, follows a different null distribution, i.e., using critical values from the t or normal distribution would lead to ...
• 33.6k
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### How to write variance covariance matrix of AR(1) process in R

The covariance between an observation at time $t_i$ and time $t_j$ is $$\frac{\sigma^2}{1-\phi^2}\phi^{|t_i-t_j|}$$ If the delta's are time gaps between the time ...
• 11.2k
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### Where is $|\theta|<1$ used in recursive method derivation of invertibility of MA(1)?

It's not a silly question --- this is a common misconception in the recursive method, and I've seen many people make the same mistake. The problem here is that your "and so on" glosses over ...
• 126k
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### Why are exponential smoothing models not considered auto-regressive?

For an autoregressive model, non-linear or linear, the number of lags must be finite. An ETS(A,N,N) model can be written as an AR($\infty$) model, but not as an autoregressive model with finite lags. ...
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### Exponential decay of ACF of AR(p) process

$f(h) =\phi^h$ is an exponential function.
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### Under what circumstances is an MA process or AR process appropriate?

I can provide what I think is a compelling answer to the first part of the question ("whence MA?") but am presently pondering an equally compelling answer to the second part of the question ("whence ...
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### Why is OLS estimator of AR(1) coefficient biased?

@Alecos nicely explains why a correct plim and unbiasedbess are not the same. As for the underlying reason why the estimator is not unbiased, recall that unbiasedness of an estimator requires that all ...
• 33.6k
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### Random Forest regression for time series prediction

How is this different from utilizing 'later' data in the time series as testing? The approach you quote is called "rolling origin" forecasting: the origin from which we forecast out is "rolled ...
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### Simulate AR(1) process in R with specified nonzero mean and AR coefficient

Notice that this is the same model as $$X_t - 10 = .5(X_{t-1} - 10) + Z_t .$$ $10$ is the mean, while $5$ was the intercept. This means we can add ten to the mean zero series. In other words, if ...
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### Generate AR(1) process with different $y_0$ values in R

arima.sim does not allow such fine control. But if you look at the function definition, you will notice that it is just applying ...
• 57k
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### Finding variance of AR process

$$\text{Var}(y_t)=\text{Var}(\phi y_{t-1}) + \text{Var}(\varepsilon_{t}).$$ As we know, $E(\varepsilon_{t}^2)=\sigma^2$. Then we have: $$\text{Var}(y_t)=\text{Var}(\phi y_{t-1}) + \sigma^2.$$ Now ...
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### How does ACF & PACF identify the order of MA and AR terms?

Robert Nau from Duke's Fuqua School of Business gives a detailed and somewhat intuitive explanation of how ACF and PACF plots can be used to choose AR and MA orders here and here. I give a brief ...
Accepted

### Difference between MA and AR

A key difference which I failed to appreciate: the MA model predictions of $x_t$ include $\epsilon_{t-1}$ in its computation whereas the AR model only predicts based on $x_{t-1}$ without (explicit) ...
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### Why this OLS fitting will converge to (0,-0.5)?

This is a law of large numbers in action. Let $\rho$ be the parameter (the lag-1 correlation) and let $\varepsilon_i$ be a sequence of iid standard Normal variables, so that for $i=1, 2, \ldots,$ the ...
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### Expectation of the ratio of sum (XY) and sum(X)

I will assume $a=0$ and $b=1$ in the following. Here is a simulation experiment to look at the variability of the expectation in $M$: ...
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### AR Forecasting one day ahead

+1 to Richard's answer. A few additions: First off, as Richard notes, you are using the full training set to estimate the autoregression coefficients, which is what ...
• 125k
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### A question reagarding stationarity of an AR(2) model

This is a consequence of the (general) fact that (auto)correlation matrices like $$\begin{pmatrix} 1&\rho_1&\rho_2\\ \rho_1&1&\rho_1\\ \rho_2&\rho_1&1\\ \end{pmatrix}$$ are ...
• 33.6k
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### How many lags should I include in time series prediction?

Looking at individual autocorrelations may help in simple cases, but this way you could miss lags that are important only jointly but not individually. Alternatively, you may try the following: ...
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Expanding on two good answers. Write down the OLS estimator: $$\hat\beta =\beta + \frac{\sum_{t=2}^Ty_{t-1}\varepsilon_t}{\sum_{t=2}^Ty_{t-1}^2}$$ For unbiasedness we need E\left[\frac{\sum_{t=2}^...