Choose most appropriate model from Minitab output of AR(1) model and AR(2) stationary series

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

Minitab was used to fit both an AR(1) model and an AR(2) to a stationary series. The following table summarises the results obtained:

	AR(1) Model		AR(2) Model
Parameter	Estimate	p-Value	Estimate	p-Value
$\alpha_1$	{est11}	{p11}	{est12}	{p12}
$\alpha_2$	$\ldots$	$\ldots$	{est22}	{p22}
$\mu$	{me1}	$\ldots$	{me2}	$\ldots$

Give any introductory information the student needs.

			Name	Type	Generated Value

est12	number	0.325
est11	number	0.34
flse122	string	strong evidence against
p22	number	0.045
an	string	and the one before that
me1	integer	100
me2	integer	80
ine	string	\<
flse	string	AR(1)
flse121	string	strong evidence against
q1	number	0.05
q3	number	0.01
q2	number	0.05
correct11	string	strong evidence against
flse111	string	no evidence against
flse221	string	slight evidence against
flse222	string	slight evidence against
flse321	string	no evidence against
flse322	string	no evidence against
tol	number	0.01
correct	string	AR(2)
est22	number	0.34
that	integer	365
forc	number	254.90
correctr22	string	reject this hypothesis
thisno	string	two
forc1	number	174.80
p11	number	0.005
p12	number	0.04
me	integer	80
thismany	integer	320
flse311	string	moderate evidence against
thatmany	integer	365
flser22	string	retain this hypothesis
correct21	string	moderate evidence against
flse211	string	slight evidence against
correct22	string	moderate evidence against
p	number	0.325
s	string	s
u	integer	2
t	integer	2
w	integer	2
v	integer	1
y	number	0.34

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(0.255..0.455#0.005)

JME function reference

Description

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Generated value: `number`

0.325

→ Used by:

forc
p

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Write down the full estimated AR(1) and AR(2) models:

$\text{AR(1)}:\;\;Y_t=a_1+b_1(Y_{t-1}-c_1)+\epsilon_t$

Find $a_1=\;$ Gap 0. $b_1=\;$ Gap 1. $c_1=\;$ Gap 2.

$\text{AR(2)}:\;\;Y_t=a_2+b_2(Y_{t-1}-c_2)+d_2(Y_{t-2}-e_2)+\epsilon_t$

Find $a_2=\;$ Gap 3., $b_2=\;$ Gap 4., $c_2=\;$ Gap 5., $d_2=\;$ Gap 6., $e_2=\;$ Gap 7.

Advice

a) Using the data given we have the models are given by:

AR(1):

$Y_t=a_1+b_1(Y_{t-1}-c_1)+\epsilon_t$

where $a_1=\var{me1},\;\;b_1=\alpha_1=\var{est11},\;\;c_1=\var{me1}$

AR(2):

$Y_t=a_2+b_2(Y_{t-1}-c_2)+d_2(Y_{t-2}-e_2)+\epsilon_t$

where $a_2=\var{me2},\;\;b_2=\alpha_1=\var{est12},\;\;c_2=\var{me2},\;\;d_2=\alpha_2=\var{est22},\;\;e_2=\var{me2}$

Using the above diagram for making decisions on retaining or rejecting the hypotheses on the coefficients we see that:

For the AR(2) fit:

$\var{est12}$ has a p-value of $\var{p12}$ and so there is {Correct21} the hypothesis that $\text{H}_0:\;\alpha_1=0$ . So we reject this hypothesis.

$\var{est22}$ has a p-value of $\var{p22}$ and so there is {Correct22} the hypothesis that $\text{H}_0:\;\alpha_2=0$ . So we {Correctr22}.

For the AR(1) fit:

$\alpha_1=\var{est11}$ has a p-value of $\var{p11} \lt \var{q3}$ and so there is {Correct11} the hypothesis that $\text{H}_0:\;\alpha_1=0$ .

So we reject this hypothesis.

We see that since in either case we reject the hypothesis that $\alpha_1=0$ , but that for the AR(2) model we {Correctr22} that $\alpha_2=0$ , so we choose the {Correct} model as the most appropriate one.

Using the {Correct} model we have the forecast for the next period:

Forecast = $\simplify[all,!collectNumbers]{{me}+{p}({thismany}-{me})+{y}({thatmany}-{me})}=\var{forc}$ to 2 decimal places.

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Choose most appropriate model from Minitab output of AR(1) model and AR(2) stationary series

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Taxonomy: mathcentre

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Contributors

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `number`

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Gap-fill

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Error

Choose most appropriate model from Minitab output of AR(1) model and AR(2) stationary series

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`