Hot answers tagged

4

You can't make your process stationary. It is simply not stationary, because it has a unit root. However, there is a related process which has no unit root, and is stationary. Your process can be written like this, with the backshift operator $B$: $$\left(1-\frac{7}{6}B-\frac{1}{6}B^2+\frac{1}{3}B^3\right)r_t = 0.05 + a_t$$ As you computed, the ...


3

It looks exactly as I would expect this kind of a model should forecast the mean. This is an AR(1) model (ignoring MA terms here). It's of the form: $$x_t=c+\phi_1 x_{t-1}+\varepsilon_t$$ After a few steps its forecast becomes a straight line at $\bar\mu=\frac c {1-\phi_1}$ level. Even if you include MA terms the long run forecast is the same, only variance ...


2

I would go for auto.arima() in the forecast package for R, or the more recent ARIMA() in the follow-up package fable. This is not the most recent implementation (the relevant paper is Hyndman & Khandakar, 2008), but: It has been the workhorse for academic and practical forecasting for the last decade. Any bugs or systematic weaknesses must have been ...


1

Remark: Mind, that we can solve this only is $(X_t)$ is casual. A sufficient condition for that is $|\varphi|<1$. Using the representation with a lag operator $B$ \begin{equation} (1-\varphi B^{12})X_t = (1 + \theta B)Z_t. \end{equation} We can evaluate the casual form of $(X_t)$, which is \begin{equation} X_t = (1+\theta B)\sum_{i = 0}^{\infty}\varphi^...


1

Stock prices are the worst kind of a toy problem to train yourself on when studying time series. The simple reason: they're essentially unpredictable. At least the kind of the series you are using in the time frame that you chose. Just think logically: if the price series were predictable why wouldn't everyone predict them and make money? See what Fama got ...


1

I'll be hand-wavy but your outputs are fine for an ARIMA. The main difference between your two graphs of fitted and out of sample forecasts are that you are feeding actual past values in the top graph but your forecast uses past forecast values to forecast. Simple ARIMA orders will never have the bips and bops that your actual data has in it's forecast in ...


1

Brief answers to some of your questions: The ARMA model is estimated in one go (not split between MA and AR) e.g. using its state space representation and Kalman filtering. $\epsilon_i$ are not always Gaussian. Their distribution depends on the assumption on standardized innovations in the GARCH conditional variance equation. $\mu$ is the constant and it ...


1

Low p-values in an ADF test mean you reject the null hypothesis of presence of a unit root. The ADF test does not tell whether the series are autoregressions or not. Instead, the ADF tests for presence of a unit root. In sum, your findings are not contradictory.


1

I don't have a fully satisfactory answer, but it may be helpful. If you really want to understand the details, you may want to step through the source code of forecast.Arima() (note the capital A). Your arima() call fits an ARIMA(0,1,0)(0,0,1)[12] model. The formula for this model is $$ (1-B)y_t = (1-\Theta B^{12})\epsilon_t, $$ where $B$ indicates the ...


1

The transfer function (I/O relationship) can be written as (assuming the form $ \phi(z)X_t = \theta(z)Z_t $) $$\begin{align}T(z)&=\frac{\theta(z)}{\phi(z)}=\frac{1+0.4z-0.6z^2}{1+1.9z+0.9z^2}=\frac{(1-0.6z)(1+z)}{(1+0.9z)(1+z)}\\&=\frac{1-0.6z}{1+0.9z}=\frac{5/3}{1+0.9z}-2/3\\&=5/3(1-0.9z+(-0.9z)^2...)-2/3, & \text{s.t.}\ \ \ |0.9z|<1\\&...


1

You cannot say anything about overfitting from (log-)likelihood, AIC or BIC alone. You could say something by comparing the model's in-sample fit with its out-of-sample prediction accuracy. If the in-sample residuals are subtantially smaller than the out-of-sample forecasting errors, the model suffers from overfitting; otherwise, it does not. Time series ...


1

Coefficients can be greater than 1 as long as roots are outside the unit circle. For the case of AR(1) the single AR coefficient less than one (in magnitude) is equivalent to root outside the unit circle. But not for AR(2), etc.


1

Your ACF is (more or less) exponentially decaying. Your PACF has a clear peak at a lag of 1, and much less clear ones at lags 2 and 34. I recommend Forecasting: Principles and Practice (2nd ed.) by Athanasopoulos & Hyndman. Based on the section on ACF/PACF reading (scroll down), it looks like your data may be best described by an ARIMA(1,1,0) process. ...


1

Simultaneous estimation will surely produce different output than stepwise estimation. The conditional mean model is estimated assuming a GARCH-type conditional variance in the simultaneous case but a constant variance in the stepwise case, yielding different optima. The conditional variance model is estimated in line with the first optimum in the ...


Only top voted, non community-wiki answers of a minimum length are eligible