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A Modern Approach

Jeffrey M. Wooldridge

Chapter 18

Advanced Time Series Topics - all with Video Answers

Educators

Chapter Questions

09:56

Problem 1

Consider equation (18.15) with $k=2 .$ Using the IV approach to estimating the $\gamma_{h}$ and $\rho,$ what would you use as instruments for $y_{t-1} ?$

Benjamin Angeles
Benjamin Angeles
Numerade Educator
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Problem 2

An interesting economic model that leads to an econometric model with a lagged dependent variable relates $y_{t}$ to the expected value of $x_{t},$ say, $x_{t}^{*},$ where the expectation is based on all observed information at time $t-1:$
$$y_{t}=\alpha_{0}+\alpha_{1} x_{t}^{*}+u_{t}$$
A natural assumption on $\left\{u_{t}\right\}$ is that $\mathrm{E}\left(u_{t} | I_{t-1}\right)=0,$ where $I_{t-1}$ denotes all information on $y$ and $x$ observed at time $t-1 ;$ this means that $\mathrm{E}\left(y_{t} | I_{t-1}\right)=\alpha_{0}+\alpha_{1} x_{t}^{*} .$ To complete this model, we need an assumption about how the expectation $x_{i}^{*}$ is formed. We saw a simple example of adaptive expectations in Section $11-2$, where $x_{i}=x_{t-1}$. A more complicated adaptive expectations scheme is
$$x_{i}^{*}-x_{t-1}^{*}=\lambda\left(x_{t-1}-x_{t-1}^{*}\right)$$
where $0<\lambda<1 .$ This equation implies that the change in expectations reacts to whether last period's realized value was above or below its expectation. The assumption $0<\lambda<1$ implies that the change in expectations is a fraction of last period's error.
i. Show that the two equations imply that
$$y_{t}=\lambda \alpha_{0}+(1-\lambda) y_{t-1}+\lambda \alpha_{1} x_{t-1}+u_{t}-(1-\lambda) u_{t-1}$$
[Hint: Lag equation (18.68) one period, multiply it by ( $1-\lambda$ ), and subtract this from ( 18.68 ). Then, use (18.69).]
ii. Under $\mathrm{E}\left(u_{t} | I_{t-1}\right)=0,\left\{u_{t}\right\}$ is serially uncorrelated. What does this imply about the new errors,
$$
v_{t}=u_{t}-(1-\lambda) u_{t-1} ?
$$
iii. If we write the equation from part (i) as
$$y_{t}=\beta_{0}+\beta_{1} y_{t-1}+\beta_{2} x_{t-1}+v_{t}$$
how would you consistently estimate the $\beta_{j}$ ?
iv. Given consistent estimators of the $\beta_{j}$, how would you consistently estimate $\lambda$ and $\alpha_{1}$ ?

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
02:41

Problem 3

Suppose that $\left\{y_{t}\right\}$ and $\left\{z_{t}\right\}$ are $\mathrm{I}(1)$ series, but $y_{t}-\beta z_{t}$ is $\mathrm{I}(0)$ for some $\beta \neq 0 .$ Show that for any $\delta \neq \beta$ $y_{t}-\delta z_{t}$ must be I(1).

Tanishq Gupta
Tanishq Gupta
Numerade Educator
04:14

Problem 4

Consider the error correction model in equation $(18.37) .$ Show that if you add another lag of the error correction term, $y_{t-2}-\beta x_{t-2},$ the equation suffers from perfect collinearity. (Hint: Show that $\left.y_{t-2}-\beta x_{t-2} \text { is a perfect linear function of } y_{t-1}-\beta x_{t-1}, \Delta x_{t-1}, \text { and } \Delta y_{t-1} .\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
04:07

Problem 5

Suppose the process $\left\{\left(x_{t}, y_{t}\right): t=0,1,2, \ldots\right\}$ satisfies the equations
$$y_{t}=\beta x_{t}+u_{t}$$
and
$$\Delta x_{t}=\gamma \Delta x_{t-1}+v_{t}$$
where $\mathrm{E}\left(u_{t} | I_{t-1}\right)=\mathrm{E}\left(v_{t} | I_{t-1}\right)=0, I_{t-1}$ contains information on $x$ and $y$ dated at time $t-1$ and earlier, $\beta \neq 0,$ and $|\gamma|<1\left[\text { so that } x_{t}, \text { and therefore } y_{t}, \text { is } I(1)\right] .$ Show that these two equations imply an error correction model of the form
$$\Delta y_{t}=\gamma_{1} \Delta x_{t-1}+\delta\left(y_{t-1}-\beta x_{t-1}\right)+e_{t}$$
where $\gamma_{1}=\beta \gamma, \delta=-1,$ and $e_{t}=u_{t}+\beta v_{t}$. (Hint: First subtract $y_{t-1}$ from both sides of the first equation. Then, add and subtract $\beta x_{t-1}$ from the right-hand side and rearrange. Finally, use the second equation to get the error correction model that contains $\Delta x_{t-1}$.)

Heather Duong
Heather Duong
Numerade Educator
08:17

Problem 6

Using the monthly data in VOLAT, the following model was estimated:
$$
\begin{aligned}
\widehat{p c i p} &=1.54+.344 p c i p_{-1}+.074 p c i p_{-2}+.073 p c i p_{-3}+.031 p c s p_{-1} \\
&(.56)(.042) \\
n &=554, R^{2}=.174, \overline{R^{2}}=.168
\end{aligned}
$$
where $p c i p$ is the percentage change in monthly industrial production, at an annualized rate, and $p c s p$ is the percentage change in the Standard \& Poor's 500 Index, also at an annualized rate.
i. If the past three months of pcip are zero and $p c s p_{-1}=0,$ what is the predicted growth in industrial production for this month? Is it statistically different from zero?
ii. If the past three months of pcip are zero but $p c s p_{-1}=10,$ what is the predicted growth in industrial production?
iii. What do you conclude about the effects of the stock market on real economic activity?

Heather Duong
Heather Duong
Numerade Educator
01:34

Problem 7

Let $g M_{t}$ be the annual growth in the money supply and let unem, be the unemployment rate. Assuming that unem_ follows a stable AR(1) process, explain in detail how you would test whether $g M$ Granger causes unem.

EA
Erwin Antoni
Numerade Educator
04:07

Problem 8

Suppose that $y_{t}$ follows the model
$$
\begin{aligned}
y_{t} &=\alpha+\delta_{1} z_{t-1}+u_{t} \\
u_{t} &=\rho u_{t-1}+e_{t} \\
\mathrm{E}\left(e_{t} | I_{t-1}\right) &=0
\end{aligned}
$$
where $I_{t-1}$ contains $y$ and $z$ dated at $t-1$ and earlier.
i. Show that $\mathrm{E}\left(y_{t+1} | I_{t}\right)=(1-\rho) \alpha+\rho y_{t}+\delta_{1} z_{t}-\rho \delta_{1} z_{t-1}$. (Hint: Write $u_{t-1}=y_{t-1}-\alpha-\delta_{1} z_{t-2}$ and
plug this into the second equation; then, plug the result into the first equation and take the conditional expectation.
ii. Suppose that you use $n$ observations to estimate $\alpha, \delta_{1},$ and $\rho .$ Write the equation for forecasting $y_{n+1}$
iii. Explain why the model with one lag of $z$ and $\mathrm{AR}(1)$ serial correlation is a special case of the model
$$y_{t}=\alpha_{0}+\rho y_{t-1}+\gamma_{1} z_{t-1}+\gamma_{2} z_{t-2}+e_{t}$$
iv. What does part (iii) suggest about using models with $\mathrm{AR}(1)$ serial correlation for forecasting?

Heather Duong
Heather Duong
Numerade Educator
03:18

Problem 9

Let $\left\{y_{t}\right\}$ be an $\mathrm{I}(1)$ sequence. Suppose that $\hat{g}_{n}$ is the one-step-ahead forecast of $\Delta y_{n+1}$ and let $\hat{f}_{n}=\hat{g}_{n}+y_{n}$ be the one-step-ahead forecast of $y_{n+1} .$ Explain why the forecast errors for forecasting $\Delta y_{n+1}$ and $y_{n+1}$ are identical.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
07:34

Problem 10

Consider the geometric distributed model in equation (18.8), written in estimating equation form as in equation (18.11):
$$y_{t}=\alpha_{0}+\gamma z_{t}+\rho y_{t-1}+v_{t}$$
where $v_{t}=u_{t}-\rho u_{t-1}$
i. Suppose that you are only willing to assume the sequential exogeneity assumption in (18.6). Why is $z_{t}$ generally correlated with $v_{t} ?$
ii. Explain why estimating (18.11) by IV, using instruments $\left(z_{t}, z_{t-1}\right),$ is generally inconsistent under $(18.6) .$ Using the IV estimator, can you test whether $z_{t}$ and $v_{t}$ are correlated?
iii. Evaluate the following proposal when only (18.6) holds: Estimate (18.11) by IV using instruments $\left(z_{t-1}, z_{t-2}\right)$
iv. Explain what you gain by estimating (18.11) by 2 SLS using instruments $\left(z_{t}, z_{t-1}, z_{t-2}\right)$
v. In equation $(18.16),$ the estimating equation for a rational distributed lag model, how would you estimate the parameters under (18.6) only? Might there be some practical problems with your approach?

Heather Duong
Heather Duong
Numerade Educator