@@ -63,7 +63,7 @@ A **Kesten process** is a stochastic process of the form
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x_{t+1 } = a_{t+1 } x_t + \eta _{t+1 }
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\quad \text {with } x_0 \text { given}
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- where :math: `\{ a_t\} _{t \geq 1 }` and :math: `\{\eta _t\} _{t \geq 1 }` are iid
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+ where :math: `\{ a_t\} _{t \geq 1 }` and :math: `\{\eta _t\} _{t \geq 1 }` are IID
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sequences.
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We will focus on the nonnegative scalar case, where :math: `x_t` takes values
@@ -73,9 +73,9 @@ In particular, we will assume that
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* the initial condition :math: `x_0 ` is nonnegative,
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- * :math: `\{ a_t\} _{t \geq 1 }` is a nonnegative iid stochastic process and
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+ * :math: `\{ a_t\} _{t \geq 1 }` is a nonnegative IID stochastic process and
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- * :math: `\{\eta _t\} _{t \geq 1 }` is another nonnegative iid stochastic process, independent of the first.
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+ * :math: `\{\eta _t\} _{t \geq 1 }` is another nonnegative IID stochastic process, independent of the first.
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@@ -122,7 +122,7 @@ The GARCH(1, 1) volatility process takes the form
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\sigma _{t+1 }^2 = \alpha _0 + \sigma _t^2 (\alpha _1 \xi _{t+1 }^2 + \beta )
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- where :math: `\{\xi _t\}` is iid with :math: `\mathbb E \xi _t^2 = 1 ` and all parameters are positive.
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+ where :math: `\{\xi _t\}` is IID with :math: `\mathbb E \xi _t^2 = 1 ` and all parameters are positive.
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Returns on a given asset are then modeled as
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@@ -131,7 +131,7 @@ Returns on a given asset are then modeled as
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r_t = \sigma _t \zeta _{t+1 }
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- where :math: `\{\zeta _t\}` is again iid and independent of :math: `\{\xi _t\}`.
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+ where :math: `\{\zeta _t\}` is again IID and independent of :math: `\{\xi _t\}`.
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Notice that the volatility sequence :math: `\{\sigma _t\}`, which drives the dynamics, is a Kesten process.
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@@ -152,7 +152,7 @@ Wealth then evolves according to
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where :math: `\{ R_t\}` is the gross rate of return on assets.
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- If :math: `\{ R_t\}` and :math: `\{ y_t\}` are both iid , then :eq: `wealth_dynam `
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+ If :math: `\{ R_t\}` and :math: `\{ y_t\}` are both IID , then :eq: `wealth_dynam `
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is a Kesten process.
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@@ -386,7 +386,7 @@ We can express this idea by stating that a suitably defined measure
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\frac {s_{t+1 }}{s_t} = a_{t+1 }
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- for some positive iid sequence :math: `\{ a_t\}`.
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+ for some positive IID sequence :math: `\{ a_t\}`.
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One implication of Gibrat's law is that the growth rate of individual firms
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does not depend on their size.
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s_{t+1 } = a_{t+1 } s_t + b_{t+1 }
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- where :math: `\{ a_t\}` and :math: `\{ b_t\}` are both iid and independent of each
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+ where :math: `\{ a_t\}` and :math: `\{ b_t\}` are both IID and independent of each
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other.
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In the exercises you are asked to show that :eq: `firm_dynam ` is more
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