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Mathematics
The AR§ model
A. can be written as Y t = β 0 + β 1 Y t 1 + ε t p Y_(t)=beta_(0)+beta_(1)Y_(t-1)+epsi_(t-p)
B. can be represented as follows: Y t = β 0 + β 1 X t + β p Y t p + ε t Y_(t)=beta_(0)+beta_(1)X_(t)+beta_(p)Y_(t-p)+epsi_(t)
C. is defined as Y t = β 0 + β p Y t p + ε t Y_(t)=beta_(0)+beta_(p)Y_(t-p)+epsi_(t)
D. represents Y t Y_(t) as a linear function of p p of its lagged values.
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Answer: D. represents Y t Y_(t) as a linear function of p p of its lagged values.
Explanation:
The AR§ model stands for autoregressive model of lag p. In this model, the current value of Y t Y_(t) depends on its past p values, along with an error term. It can be expressed mathematically as:
Y t = β 0 + β 1 Y t 1 + β 2 Y t 2 + . . . + β p Y t p + ε t Y_(t)=beta_(0)+beta_(1)Y_(t-1)+beta_(2)Y_(t-2)+...+beta_(p)Y_(t-p)+epsi_(t)
where β 0 , β 1 , . . . , β p beta_(0),beta_(1),...,beta_(p) are the parameters to be estimated and are known as autoregression coefficients.
Option A is incorrect because it has only one lag term Y t 1 Y_(t-1) in the equation, whereas the AR§ model involves p lag terms.
Option B is incorrect because it has included an additional independent variable X t X_(t), which is not part of the AR§ model.
Option C is incorrect because it has no lagged term in the equation, whereas the AR§ model involves p lag terms.
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