
\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.00.0.2606}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{Created=Thursday, April 03, 2008 21:49:12}
%TCIDATA{LastRevised=Thursday, April 03, 2008 23:28:32}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Blank - Standard LaTeX Article">}
%TCIDATA{CSTFile=40 LaTeX article.cst}

\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
%\input{tcilatex}

\begin{document}


\bigskip EC 327 PROBLEM SET 3

Prof. Baum, Mr. Dmitriev

11.7

(i) (5 marks) We plug the first equation into the second to get
$$y_t-y_{t-1}=\lambda(\gamma_0+\gamma_1x_t+e_t-y_{t-1})+a_t$$
and rearranging,
$$y_t=\lambda\gamma_0+(1-\lambda)y_{t-1}+\lambda\gamma_1x_t+a_t+\lambda e_t\equiv
\beta_0+\beta_1y_{t-1}+\beta_2x_t+u_t$$
where $\beta_0\equiv\lambda\gamma_0$, $\beta_1\equiv(1-\lambda),\beta_2\equiv\lambda\gamma_1$, and
$u_t\equiv a_t+\lambda e_t$.

(ii) (10 marks) An OLS regression of $y_t$ on $y_{t-1}$ and $x_t$ produces consistent, asymptotically normal estimators of the
$\beta_j$. Under $E(e_t|x_t,y_{t-1},x_{t-1},...)=E(a_t|x_t,y_{t-1},x_{t-1},...)=0$ it follows that $E(u_t|x_t,y_{t-1},x_{t-1},...)=0$, which means that the model is dynamically complete [see equation (11.37)]. Therefore, the errors are serially uncorrelated. If the homoskedasticity assumption $Var(u_t|x_t,y_{t-1})=\sigma^2$
holds, then the usual standard errors, $t$ statistics and $F$ statistics are asymptotically valid.

(iii) (5 marks) Because $\beta_1=(1-\lambda)$, if $\widehat{\beta}_1=0.7$, then $\widehat{\lambda}=0.3$. Further,
$\hat{\beta}_2=\hat{\lambda}\hat{\gamma}_1$, or $\hat{\gamma}_1=\hat{\beta}_2/\hat{\lambda}=\frac{0.2}{0.3}\approx0.67$.

11.8

(i) (10 marks) Sequential exogeneity does not rule out correlation between, say, $u_{t-1}$ and $x_{ij}$ for any regressors $j=1,2,...,k$. The differencing generally induces correlation between the differenced errors and the differenced regressors. To see why, consider a sing explanatory variable, $x_t$. Then $\delta u_t=u_t-u_{t-1}$ and $ \delta x_t=x_t-x_{t-1}$. Under sequential exogeneity, $u_t$ is uncorrelated with $x_t$ and $x_{t-1}$, and $u_{t-1}$
is uncorrelated with $x_{t-1}$. But $u_{t-1}$ can be correlated with $x_t$, which means that $\delta u_t$ and
$\delta x_t$  are generally correlated. In fact, under sequential exogeneity, it is always true that $Cov(\delta x_t,\delta u_t)=-Cov(x_t,u_{t-1})$.

(ii) (10 marks) Strict exogeneity of the regressors in the original equation is sufficient for OLS on the first-differenced equation to be consistent. Remember, stric exogeneity implies that the regressors in any time period are uncorrelated with the errors in any time period. Of course, we could make the weaker assumption: for any $t$, $u_t$ is uncorrelated with $x_{t-1,j},x_{tj}$, and $x_{t+1,j}$ for all $j=1,2,...,k$. The strengthening beyond sequential exogeneity is the assumption that $u_t$ is uncorrelated with all of next period's outcomes on all regressors. In practice, this is probably similar to just assuming strict exogeneity.

(iii) (10 marks) If we assume sequential exogeneity in a static model, the condition can be written as
$$E(y_t|z_t,z_{t-1},z_{t-2},...)=E(y_t|z_t)$$
which means that, once we control for the current (contemporaneous) values of all explanatory variables, no lags matter. Although some relationships in economics are purely static, many admit distributed lag dynamics. Therefore, if one wants to capture lagged effects, it is a good idea to explore distributed lag models - whether or not we think there might be feedback from $u_t$ to future values of the explanatory variables.

18.6

(i) (5 marks) This is given by the estimated intercept, 1.54. Remember, this is the
percentage growth at an annualized rate. It is statistically different from
zero since t = 1.54/.56 = 2.75.

(ii) (5 marks) 1.54 + .031(10) = 1.85. As an aside, you could obtain the standard
error of this estimate by running the regression.

pcipt on pcipt-1, pcipt-2, pcipt-3, (pcspt-1 -- 10),

and obtaining the standard error on the intercept.

(iii) (5 marks) Growth in the S\&P 500 index has a statistically significant effect on
industrial production growth -- in the Granger causality sense -- because
the t statistic on pcspt-1 is about 2.38. The economic effect is reasonably
large.

C18.2

(i)(5 marks)  We run the regression

ginvpct= --.786 -- .956 log(invpct-1) + .0068 t

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.170) \ \ \ \ \ \ \ \ \ \ \ \ \ (.198) \ \
\ \ \ \ \ \ \ \ \ (.0021)

+ .532 ginvpct-1 + .290 ginvpct-2

\ \ \ \ \ \ \ \ (.162) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.165)

n = 39, R2 = .437,

where ginvpct = log(invpct) -- log(invpct-1). The t statistic for the
augmented Dickey-Fuller unit root test is --.956/.198 --4.82, which is well
below --3.96, the 1\% critical value obtained from Table 18.3. Therefore, we
strongly reject a unit root in log(invpct). (Incidentally, remember that the
t statistics on the intercept and time trend in this estimated equation to
not have approximate t distributions, although those on ginvpct-1 and
ginvpct-2 do under the usual null hypothesis that the parameter is zero.)

(ii)(5 marks)  When we apply the regression to log(pricet) we obtain

tgprice = --.040 -- .222 log(pricet-1) +.00097 t

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.019) \ \ \ \ \ \ \ \ \ (.092) \ \ \ \ \
\ \ (.00049)

+ .328 gpricet-1 + .130 gpricet-2

\ \ \ \ \ \ \ \ (.155) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.149)

n = 39, R2 = .200,

Now the Dickey-Fuller t statistic is about --2.41, which is above --3.12,
the 10\% critical value from Table 18.3. [The estimated root is 1 -- .222 =
.778, which is much larger than for log(invpct).] We cannot reject the unit
root null at a sufficiently small significance level.

(iii) (5 marks) Given the very strong evidence that log(invpct) does not contain a
unit root, while log(pricet) may very well, it makes no sense to discuss
cointegration between the two. If we take any nontrivial linear combination
of an I(0) process (which may have a trend) and an I(1) process, the result
will be an I(1) process (possibly with drift).

\bigskip

\bigskip

\bigskip

\bigskip

\bigskip

C18.5

(i) (5 marks) The estimated equation is

= .078 + 1.027 hy36thyt-1 - 1.021$change$hy3t - .085 $change$hy3t-1

\ \ (.028) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0.016) \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ (0.038) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.037) \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \

- .104 $change$hy3t-2

\ \ \ \ \ \ \  (.037)

n = 121, R2 = .982, se = .123.

The t statistic for H0: $B$ = 1 is (1.027 -- 1)/.016 = 1.69. We do not
reject H0: $B$ = 1 at the 5\% level against a two-sided alternative,
although we would reject at the 10\% level.

(ii) (5 marks) The estimated error correction model is

hy6t= .070 + 1.259 $change$hy3t-1 - .816 (hy6t-1 -- hy3t-2)

\ \ \ \ \ \ \ \ \ \ (.049) \ \ \ \ \ \ \ \ (.278) \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ (.256)

+ .283 $change$hy3t-2 + .127 (hy6t-2 -- hy3t-3)

\ \ \ \ \ \ (.272) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.256)

n = 121, R2 = .795.

Neither of the added terms is individually significant. The F test for their
joint significance gives F = 1.35, p-value = .264. Therefore, we would omit
these terms and stick with the error correction model estimated in (18.39).

C18.11

(i) (5 marks) For lsp500, the ADF statistic without a trend is t = -.79; with a trend,
the t statistic is -2.20. This are both well above their respective 10\%
critical values. In addition, the estimated roots are quite close to one.
For lip, the ADF statistic without a trend is -1.37 without a trend and
-2.52 with a trend. Again, these are not close to rejecting even at the 10\%
levels, and the estimated roots are very close to one.

(ii) (5 marks) The simple regression of lsp500 on lip gives

lip500lsp= -2.402 + 1.694

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (.095) (.024)

n = 558, R2 = .903

The t statistic for lip is over 70, and the R-squared is over .90. These are
hallmarks of spurious regressions.

(iii) (5 marks) Using the residuals obtained in part (ii), the ADF statistic (with two
lagged changes) is -1.57, and the estimated root is over .99. There is no
evidence of cointegration. (The 10\% critical value is -3.04.) tu

(iv) (5 marks) After adding a linear time trend to the regression from part (ii), the
ADF statistic applied to the residuals is -1.88, and the estimated root is
again about .99. Even with a time trend there is no evidence of
cointegration.

(v) (5 marks) It appears that lsp500 and lip do not move together in the sense of
cointegration, even if we allow them to have unrestricted linear time
trends. The analysis does not point to a long-run equilibrium relationship.

C18.13 (i) (5 marks) The DF statistic is about -3.31, which is to the left of the
2.5\% critical value (-3.12), and so, using this test, we can reject a unit
root at the 2.5\% level. (The estimated root is about .81.)

(ii) (5 marks) When two lagged changes are added to the regression in part (i), the t
statistic becomes -1.50, and the root is larger (about .915). Now, there is
little evidence against a unit root.

(iii) (5 marks) If we add a time trend to the regression in part (ii), the ADF
statistic becomes -3.67, and the estimated root is about .57. The 2.5\%
critical value is -3.66, and so we are back to fairly convincingly rejecting
a unit root.

(iv) (5 marks) The best characterization seems to be an I(0) process about a linear
trend. In fact, a stable AR(3) about a linear trend is suggested by the
regression in part (iii).

(v) (5 marks) For prcfatt, the ADF statistic without a trend is -4.74 (estimated root
= .62) and with a time trend the statistic is -5.29 (estimated root = .54).
Here, the evidence is strongly in favor of an I(0) process whether or not we
include a trend.



\end{document}
