# I Auto Regressive Moving Average Model (ARMA) Ljung-Box Test

#### mertcan

Hi, according to ARMA model it is said that in order to check out white noise terms, Ljung-Box is applied involving sum of squared autocorrelations of errors with relevant lags. In short, sum of them is chi-square distribution but n-p-q degree of freedom when we have ARMA(p,0,q) model. My question : Is there a mathematical proof of why we subtract p+q from n?

By the way I know the proof of why we do the similar subtraction like above in multilinear regression. And mathematical proof of it is displayed in (https://stats.stackexchange.com/questions/24921/distribution-of-sum-of-squares-error-for-linear-regression/400261?noredirect=1#comment749409_400261). But even though there is a similarity I can not derive for ARMA model.

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#### BvU

Homework Helper
In short, sum of them is chi-square distribution but n-p-q degree of freedom when we have ARMA(p,0,q) model. My question : Is there a mathematical proof of why we subtract p+q from n?
No reponse so far, so I'll give it a try:
Degrees of freedom is number of points minus number of parameters in the model that are derived from the data. ARMA(p,0,q) has p+q parameters derived from the data. (I'd say p+q+1 because of the average, but apparently the 0 in there means the 0 is hypothesized).

Not really a proof, more an explanation....

#### mertcan

No reponse so far, so I'll give it a try:
Degrees of freedom is number of points minus number of parameters in the model that are derived from the data. ARMA(p,0,q) has p+q parameters derived from the data. (I'd say p+q+1 because of the average, but apparently the 0 in there means the 0 is hypothesized).

Not really a proof, more an explanation....
Thanks for return, according to definition : In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistics that are free to vary. Could you show me maybe using some mathematical demonstration how p+q+1 are not free to vary in ARMA? Could you help me imagine the case?

#### BvU

Homework Helper
Degrees of freedom is number of points minus number of parameters in the model that are derived from the data. ARMA(p,0,q) has p+q parameters derived from the data.
I don't feel like a genuine expert on ARMA (had to look it up for this thread). But I can try to answer
Could you help me imagine the case?
Simple rule that works for me: take a simple example !
e.g. ARMA(1,0) = AR(1) for two data points has no degrees of freedom: $c$ and $\phi_1$ are fully determined.

In general math: The $n+1$ coefficients for a polynomial of order $n$ through $n+1$ points can be calculated $\ \Rightarrow\$ no degrees of freedom

See also here: if you need to establish the quality of the model, the noise level comes in as well.

#### mertcan

I don't feel like a genuine expert on ARMA (had to look it up for this thread). But I can try to answer

Simple rule that works for me: take a simple example !
e.g. ARMA(1,0) = AR(1) for two data points has no degrees of freedom: cc and ϕ1ϕ1 are fully determined.

In general math: The n+1n+1 coefficients for a polynomial of order nn through n+1n+1 points can be calculated ⇒ ⇒ no degrees of freedom

See also here: if you need to establish the quality of the model, the noise level comes in as well.
Thanks for return: What do you say about the following:

Let's say we have ARMA(4,0,0) process and

$$yt=ϕ1∗yt−1+ϕ2∗yt−2+ϕ3∗yt−3+ϕ4∗yt−4+error_t$$​

As you can see EXPECTATION OF $$y_1=\phi_1*y_{0}+\phi_2*y_{-1}+\phi_3*y_{-2}+\phi_4*y_{-3}$$
EXPECTATION OF $$y_2=\phi_1*y_{1}+\phi_2*y_{0}+\phi_3*y_{-1}+\phi_4*y_{-2}$$
EXPECTATION OF $$y_3=\phi_1*y_{2}+\phi_2*y_{1}+\phi_3*y_{0}+\phi_4*y_{-1}$$
EXPECTATION OF $$y_4=\phi_1*y_{3}+\phi_2*y_{2}+\phi_3*y_{1}+\phi_4*y_{0}$$

By the way we do not know $$y_{0},y_{-1},y_{-2},y_{-3}$$ so may be we can write
$$y_1=error_1$$
$$y_2=\phi_1*y_{1}+error_2$$
$$y_3=\phi_1*y_{2}+\phi_2*y_{1}+error_3$$
$$y_4=\phi_1*y_{3}+\phi_2*y_{2}+\phi_3*y_{1}+error_4$$
$$error_1=\phi_1*y_{0}+\phi_2*y_{-1}+\phi_3*y_{-2}+\phi_4*y_{-3}$$
$$error_2=\phi_2*y_{0}+\phi_3*y_{-1}+\phi_4*y_{-2}$$
$$error_3=\phi_3*y_{0}+\phi_4*y_{-1}$$
$$error_4=\phi_4*y_{0}$$
In short we have 8 equations and 8 unknowns as $$error_1, error_2, error_3, error_4, y_{0}, y_{-1}, y_{-2}, y_{-3}$$
So error terms 1 to 4 can not be random because we solved them in linear system they are not free as in $$y_5,y_6.......$$ so we have n-4 degree of freedom.What do you say?

#### FactChecker

Gold Member
2018 Award
In short we have 8 equations and 8 unknowns as $$error_1, error_2, error_3, error_4, y_{0}, y_{-1}, y_{-2}, y_{-3}$$
So error terms 1 to 4 can not be random because we solved them in linear system they are not free as in $$y_5,y_6.......$$ so we have n-4 degree of freedom.What do you say?
That looks good to me. I think that your example is a higher-dimensional example of the very simple one that @BvU gave. I like the simple example to make the point and I like your higher-dimensional example to show the generalization.

#### mertcan

@FactChecker and @BvU could you help me for the case ARMA (0,0,q)?
Actually I converted MA to infinite AR process but can not get proper result. I think derivation is different than AR processes?

#### mertcan

I think I carved out it but I also wonder your return for cross check...?

#### mertcan

That looks good to me. I think that your example is a higher-dimensional example of the very simple one that @BvU gave. I like the simple example to make the point and I like your higher-dimensional example to show the generalization.
I don't feel like a genuine expert on ARMA (had to look it up for this thread). But I can try to answer

Simple rule that works for me: take a simple example !
e.g. ARMA(1,0) = AR(1) for two data points has no degrees of freedom: cc and ϕ1ϕ1 are fully determined.

In general math: The n+1n+1 coefficients for a polynomial of order nn through n+1n+1 points can be calculated ⇒ ⇒ no degrees of freedom

See also here: if you need to establish the quality of the model, the noise level comes in as well.
I also tired to set some equations for ARMA(3,0,2) model. As you know we lose p+q=5 degree of freedom which means we set constraints 5 error terms. So could you check my proof if it?

$$y1=error_1$$
$$y2=ϕ1∗y1+error_2$$
$$y3=ϕ1∗y2+ϕ2∗y1+θ1∗error2+θ2∗error_1+error_3$$
$$error1=ϕ1∗y0+ϕ2∗y−1+ϕ3∗y−2+θ1∗error_0+θ2∗error_−1$$
$$error2=ϕ2∗y0+ϕ3∗y−1+θ2∗error_0$$​

$$error_3=\phi_3*y_0$$

$$y4=ϕ1∗y3+ϕ2∗y2+ϕ3∗y1+θ1∗error_3+θ2∗error_2+error_4$$
$$y5=ϕ1∗y4+ϕ2∗y3+ϕ3∗y2+θ1∗error_4+θ2∗error_3+error_5$$​

if we set error_4 and error_5 are zero then we have 8 equations and 8 unknowns

$$error_1,error_2,error_3,error_0,error_−1,error_−2,y0,y−1,y−2error_1,error_2,error_3,error_0,error_−1,error_−2,y0,y−1,y−2$$​

. In short

$$error_1,error_2,error_3,error_0,error_−1,error_4,error_5error_1,error_2,error_3,error_0,error_−1,error_4,error_5$$​

has been set without considering their randomnesses. What do you think about that?

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