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A Modern Approach

Jeffrey M. Wooldridge

Chapter 11

Further Issues in Using OLS with Time Series Data - all with Video Answers

Educators

Chapter Questions

04:38

Problem 1

Let $\left\{x_{t}: t=1,2, \ldots\right\}$ be a covariance stationary process and define $\gamma_{h}=\operatorname{Cov}\left(x_{t}, x_{t+h}\right)$ for $h \geq 0$ [Therefore, $\left.\gamma_{0}=\operatorname{Var}\left(x_{t}\right) .\right]$ Show that $\operatorname{Corr}\left(x_{t}, x_{t+h}\right)=\gamma_{h} / \gamma_{0}$.

Chris Trentman
Chris Trentman
Numerade Educator
05:18

Problem 2

Let $\left\{e_{t}: t=-1,0,1, \ldots\right\}$ be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by $$x_{t}=e_{t}-(1 / 2) e_{t-1}+(1 / 2) e_{t-2}, t=1,2, \ldots$$ i. Find $\mathrm{E}\left(x_{t}\right)$ and $\operatorname{Var}\left(x_{t}\right) .$ Do either of these depend on $t ?$
ii. Show that $\operatorname{Corr}\left(x_{t}, x_{t+1}\right)=-1 / 2$ and $\operatorname{Corr}\left(x_{t}, x_{t+2}\right)=1 / 3 .$ (Hint: It is easiest to use the formula
in Problem $1 .$ )
iii. What is $\operatorname{Corr}\left(x_{t}, x_{t+h}\right)$ for $h>2 ?$
iv. Is $\left\{x_{t}\right\}$ an asymptotically uncorrelated process?

Heather Duong
Heather Duong
Numerade Educator
03:41

Problem 3

Suppose that a time series process $\left\{y_{t}\right\}$ is generated by $y_{t}=z+e_{t},$ for all $t=1,2, \ldots .$ where $\left\{e_{t}\right\}$ is an i.i.d. sequence with mean zero and variance $\sigma_{e}^{2}$. The random variable $z$ does not change over time; it has mean zero and variance $\sigma_{z}^{2} .$ Assume that each $e_{t}$ is uncorrelated with $z$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:31

Problem 4

Let $\left\{y_{t}: t=1,2, \ldots\right\}$ follow a random walk, as in $(11.20),$ with $y_{0}=0 .$ Show that $\operatorname{Corr}\left(y_{t}, y_{t+h}\right)=\sqrt{t /(t+h)}$ for $t \geq 1, h>0$.

Ameer Said
Ameer Said
Numerade Educator
06:30

Problem 5

For the U.S. economy, let gprice denote the monthly growth in the overall price level and let gwage be the monthly growth in hourly wages. [These are both obtained as differences of logarithms:
gprice $=\Delta \log (\text {price}) \text { and gwage }=\Delta \log (\text {wage}) .]$ Using the monthly data in WAGEPRC, we estimate the following distributed lag model: i. Sketch the estimated lag distribution. At what lag is the effect of gwage on gprice largest? Which lag has the smallest coefficient? ii. For which lags are the $t$ statistics less than two? iii. What is the estimated long-run propensity? Is it much different than one? Explain what the LRP tells us in this example. iv. What regression would you run to obtain the standard error of the LRP directly? v. How would you test the joint significance of six more lags of $g$ wage? What would be the $d f$ s in the $F$ distribution? (Be careful here; you lose six more observations.)

Heather Duong
Heather Duong
Numerade Educator
04:07

Problem 6

Let $h y 6_{t}$ denote the three-month holding yield (in percent) from buying a six-month T-bill at time $(t$- 1) and selling it at time $t$ (three months hence) as a three-month T-bill. Let $h y_{t-1}$ be the three-month holding yield from buying a three-month T-bill at time $(t-1) .$ At time $(t-1), h y 3_{t-1}$ is known, whereas $h y 6_{t}$ is unknown because $p s_{t}$ (the price of three-month T-bills) is unknown at time $(t-1) .$ The expectations hypothesis (EH) says that these two different three-month investments should be the same, on average. Mathematically, we can write this as a conditional expectation: $$\mathrm{E}\left(h y 6_{t} | I_{t-1}\right)=h y 3_{t-1}$$ and testing $\mathrm{H}_{0}: \beta_{1}=1 .$ (We can also test $\mathrm{H}_{0}: \beta_{0}=0,$ but we often allow for a term premium for buying assets with different maturities, so that $\beta_{0} \neq 0 .$ ) i. Estimating the previous equation by OLS using the data in INTQRT (spaced every three months) gives $$\begin{aligned}
\widehat{h y 6_{t}} &=-.058+1.104 \mathrm{hy}_{t-1} \\
&\quad\quad(.070)(.039) \\
n &=123, R^{2}=.866
\end{aligned}$$
Do you reject $\mathrm{H}_{0}: \beta_{1}=1$ against $\mathrm{H}_{0}: \beta_{1} \neq 1$ at the $1 \%$ significance level? Does the estimate seem practically different from one? ii. Another implication of the EH is that no other variables dated as $t-1$ or earlier should help explain $h y 6_{t},$ once $h y g_{t-1}$ has been controlled for. Including one lag of the spread between six-month and three-month T-bill rates gives $$\begin{aligned}
\widehat{h y 6}_{t}=&-.123+1.053 \mathrm{hy} 3_{t-1}+.480\left(\mathrm{r} 6_{t-1}-r 3_{t-1}\right) \\
&(.067)\quad\space\space(.039) \\
n &=123, R^{2}=.885
\end{aligned}$$Now is the coefficient on $h y 3_{t-1}$ statistically different from one? Is the lagged spread term significant? According to this equation, if, at time $t-1, r 6$ is above $r 3,$ should you invest in six-month or three-month T-bills? iii. The sample correlation between $h y$ s $_{t}$ and $h y s_{t-1}$ is. $914 .$ Why might this raise some concerns with the previous analysis?
iv. How would you test for seasonality in the equation estimated in part (ii)?

Heather Duong
Heather Duong
Numerade Educator
04:07

Problem 7

A partial adjustment model is $$\begin{aligned}
y_{t}^{*} &=\gamma_{0}+\gamma_{1} x_{t}+e_{t} \\
y_{t}-y_{t-1} &=\lambda\left(y_{t}^{*}-y_{t-1}\right)+a_{t}
\end{aligned}$$ where $y_{i}$ is the desired or optimal level of $y$ and $y_{t}$ is the actual (observed) level. For example, $y_{i}$ is the desired growth in firm inventories, and $x_{t}$ is growth in firm sales. The parameter $\gamma_{1}$ measures
the effect of $x_{t}$ on $y_{i}$. The second equation describes how the actual $y$ adjusts depending on the relationship between the desired $y$ in time $t$ and the actual $y$ in time $t-1 .$ The parameter $\lambda$ measures the speed of adjustment and satisfies $0<\lambda<1$ i. Plug the first equation for $y_{i}$ into the second equation and show that we can write $$y_{t}=\beta_{0}+\beta_{1} y_{t-1}+\beta_{2} x_{t}+u_{t}$$ In particular, find the $\beta_{j}$ in terms of the $\gamma_{j}$ and $\lambda$ and find $u_{t}$ in terms of $e_{t}$ and $a_{t}$. Therefore, the partial adjustment model leads to a model with a lagged dependent variable and a contemporaneous $x$ ii. If $\mathrm{E}\left(e_{t} | x_{t}, y_{t-1}, x_{t-1}, \ldots\right)=\mathrm{E}\left(a_{t} | x_{t}, y_{t-1}, x_{t-1}, \ldots\right)=0$ and all series are weakly dependent, how
would you estimate the $\beta_{j} ?$
iii. If $\hat{\beta}_{1}=.7$ and $\hat{\beta}_{2}=.2,$ what are the estimates of $\gamma_{1}$ and $\lambda ?$

Heather Duong
Heather Duong
Numerade Educator
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Problem 8

Suppose that the equation $$y_{t}=\alpha+\delta t+\beta_{1} x_{t 1}+\cdots+\beta_{k} x_{t k}+u_{t}$$ satisfies the sequential exogeneity assumption in equation (11.40)
i. Suppose you difference the equation to obtain $$\Delta y_{t}=\delta+\beta_{1} \Delta x_{t 1}+\cdots+\beta_{k} \Delta x_{t k}+\Delta u_{t}$$ Why does applying OLS on the differenced equation not generally result in consistent estimators of the $\beta_{j} ?$ ii. What assumption on the explanatory variables in the original equation would ensure that OLS on the differences consistently estimates the $\beta_{j} ?$ iii. Let $z_{t 1}, \ldots, z_{t k}$ be a set of explanatory variables dated contemporaneously with $y_{t}$. If we specify the static regression model $y_{t}=\beta_{0}+\beta_{1} z_{t 1}+\cdots+\beta_{k} z_{t k}+u_{t},$ describe what we need to assume for $x_{t}=z_{t}$ to be sequentially exogenous. Do you think the assumptions are likely to hold in economic applications?

Shu Naito
Shu Naito
Numerade Educator