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Mathematics
The AR§ model
A. can be written as ${Y}_{t}={\beta }_{0}+{\beta }_{1}{Y}_{t-1}+{\epsilon }_{t-p}$Y_(t)=beta_(0)+beta_(1)Y_(t-1)+epsi_(t-p)
B. can be represented as follows: ${Y}_{t}={\beta }_{0}+{\beta }_{1}{X}_{t}+{\beta }_{p}{Y}_{t-p}+{\epsilon }_{t}$Y_(t)=beta_(0)+beta_(1)X_(t)+beta_(p)Y_(t-p)+epsi_(t)
C. is defined as ${Y}_{t}={\beta }_{0}+{\beta }_{p}{Y}_{t-p}+{\epsilon }_{t}$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:
${Y}_{t}={\beta }_{0}+{\beta }_{1}{Y}_{t-1}+{\beta }_{2}{Y}_{t-2}+...+{\beta }_{p}{Y}_{t-p}+{\epsilon }_{t}$Y_(t)=beta_(0)+beta_(1)Y_(t-1)+beta_(2)Y_(t-2)+...+beta_(p)Y_(t-p)+epsi_(t)
where ${\beta }_{0},{\beta }_{1},...,{\beta }_{p}$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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