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What is the difference between (1) and (2)? Why does $\theta$ (theta) have a negative sign in the ARMA model?

$$ \begin{align} \hat y_t &= \mu + \varphi_{1} y_{t - 1} + \cdots + \varphi_{p} y_{t - p} - \theta_{1} e_{t - 1} - \cdots - \theta_{q} e_{t - q} &(1) \\ X_t &= c + \varepsilon_t + \sum_{i = 1}^p \varphi_i X_{t - i} + \sum_{i = 1}^q \theta_i \varepsilon_{t - i} \qquad &(2) \end{align} $$

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  • $\begingroup$ did you have any source where you had taken those? $\endgroup$ – rapaio Nov 9 '16 at 8:57
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Different authors define it differently. There's nothing magical about either choice, they're each convenient in a different context.

The most obvious reason why some like to set the coefficients to be negative comes when you write the models in terms of polynomials in the backshift or lag operator (which I'll call $B$ here, but which is as often called $L$).

[However, first note that what your equation (1) calls $\mu$ should conventionally be $\phi_0$. Usually in these models $\mu$ is reserved for an estimate of the mean, in which case there should be a $\mu$ term subtracted from each of the $y$ terms on the right hand side ($(y_{t-i}-\mu)$ in place of each $y_{t-i}$. We could either proceed as if you had $\phi_0$ instead of $\mu$ or each $y$ term had a $-\mu$ associated with it. The second is neater for what I want to illustrate but for consistency with equation (2) I'll do the first. Note further than your equation (1) is a model for $hat{y}$ rather than for $y$. The model for $y_t$ should have an $\epsilon_t$ in it. Even then the equation is still wrong because it treats the population parameters and their estimates as the same thing. I will reserve $\phi$ and $\theta$ for the population parameters and would use $\hat{\phi}$ and $\hat{\theta}$ for their estimates.]

So properly rewritten your equation (1) would take the form:

$y_t=\phi_0+\phi_1 y_{t-1}+\phi_2 y_{t-2}\, ...\,+\phi_p y_{t-p}+\epsilon_t-\theta_1\epsilon_{t-1}-\theta_2\epsilon_{t-2}\, ...\,-\theta_q\epsilon_{t-q}$

$y_t-\phi_1 y_{t-1}-\phi_2 y_{t-2}\,...\,-\phi_1 y_{t-1}=\phi_0+\epsilon_t-\theta_1\epsilon_{t-1}-\theta_2\epsilon_{t-2}\, ...\,-\theta_q\epsilon_{t-q}$

$(1-\phi_1B-\phi_2B^{\,2}\,...\,-\phi_pB^{\,p})y_t=\phi_0+(1-\theta_1B-\theta_2B^{\,2}\,...\,-\theta_qB^{\,q})\epsilon_t\,,$ or

$\phi(B)y_t=\phi_0+\theta(B)\epsilon_t$

where $\phi(B) = (1-\phi_1B-\phi_2B^{\,2}\,...\,-\phi_pB^{\,p})$ and $\theta(B)=(1-\theta_1B-\theta_2B^{\,2}\,...\,-\theta_qB^{\,q})\,$.

The symmetry of the two polynomials in $B$ (where both sets of parameters appear with negative signs in front of the parameters) is a good reason to choose that parameterization.

For the model in equation (2) the positive sign is chosen so that when it's written with all the parameters on the right hand side they all have positive signs. Whichever choice is made there's a benefit and a cost (it's simpler in one form more complex in another), and which is nicest to use depends on what you'll be doing with it.

It's best to simply get used to the fact that there are two different common ways to write an ARMA model.

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