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%	TITLE PAGE
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\title[]{Macro II} % The short title appears at the bottom of every slide, the full
% title is only on the title page

\author{Professor Griffy} % Your name
\institute[University at Albany, SUNY] % Your institution as it will appear on the bottom of
% every slide, may be shorthand to save space
{
UAlbany  \ % Your institution for the title page
}
\date{Spring 2025} % Date, can be changed to a custom date

\begin{document}

\begin{frame}
  \titlepage % Print the title page as the first slide
\end{frame}



%----------------------------------------------------------------------------------------
%	PRESENTATION SLIDES
%----------------------------------------------------------------------------------------

%------------------------------------------------
\section{Introduction} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
%------------------------------------------------

\begin{frame}
\frametitle{Introduction}
\begin{itemize}
\item Hand back midterms.
\item A few people forgot Blackwell's Sufficient Conditions.
\item Everyone demonstrated they understood the material.
\item Scheduling:
  \begin{itemize}
  \item No class next Thursday (4/10)
  \item I'm hosting a macro conference here that weekend (4/11-4/12).
  \item I know first year is busy, but please come if you are interested!
  \end{itemize}
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Motivation I}
\begin{itemize}
\item Baseball season is starting.
\visible<2->{\item Adrian (Prof. Masters) is a big Yankees fan.}
\visible<3->{\item Anyone who sees him and says ``Go Yankees! Ben said you're a huge fan!'' gets extra credit.}
\visible<4->{\item If he says non-sense like ``I'm a Red Sox fan,'' (more) extra credit if you say something like ``oh, well that's okay too, I guess.''}
\visible<5->{\item Extra, extra bonus points if he's wearing his Red Sox hat.}
\end{itemize}
\end{frame}

%------------------------------------------------



\begin{frame}
\frametitle{Motivation II}
\begin{itemize}
\item Goal of RBC literature: quantitatively understand the macroeconomy.
\item Interpret previous phenomena; make predictions about future phenomena.
\item To do this, we need a credible way of parametrizing the model:
\begin{enumerate}
\item Historically, macroeconomists have used calibration: just-identified method of moments.
\item Recently, economists have begun to employ alternative techniques like maximum likelihood.
\item With knowledge of these more advanced techniques, we might be able to explore more important issues, like identification.
\end{enumerate}
\item Here: provide background for maximum likelihood estimation and calibration and compare the results.
\end{itemize}
\end{frame}

%------------------------------------------------

\section{Standard RBC Model} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk

%------------------------------------------------

\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\item Reintroduce Hansen's model:
\begin{itemize}
\item Standard RBC: all fluctuations of hours worked on the {\it intensive} margin, i.e. average number of hours worked.
\item Data: little fluctuation in average hours worked; lots of fluctuation in whether or not people are working ({\it extensive} margin).
\item Standard RBC: missed badly on labor fluctuations (Frisch Elasticity, i.e. response of labor to change in wage too low).
\item Solution: Modify model to have extensive margin with high Frisch Elasticity.
\item Now: households pick the {\it probability} of working, but have to work a set number of hours.
\item This is a {\it nonconvexity} in that it forces individuals to work either 0 or h hours.
\end{itemize}
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Hansen (1985)}
\begin{itemize}
\item Neoclassical growth model with labor-leisure lottery.
\item A social planner maximize the following:
\begin{equation}
E(\sum_{t = 0}^{\infty}\beta^{t}[ln(C_{t}) - \gamma H_{t}]
\end{equation}
\item Subject to the following constraints:
\begin{equation}
Y_{t} = A_{t}K_{t}^{\theta}(\eta^{t}H_{t})^{1 - \theta}
\end{equation}
\begin{equation}
ln(A_{t}) = (1 - \rho)ln(A) + \rho ln(A_{t - 1}) + \epsilon_{t}, \hspace{3 mm} \epsilon_{t}\sim N(0,\sigma_{\epsilon}^{2})
\end{equation}
\item The goods market clears and capital evolves in a predetermined fashion.
\item Here, we assume that per capita labor productivity grows at rate $\eta$.
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Equilibrium}
\begin{itemize}
\item First step: detrend appropriate variables by per capita growth to get stationarity: i.e. $y_{t} = Y_{t}/\eta^{t}$.
\item The system of equations that characterize the equilibrium are:
\begin{equation}\label{eq:AggOutput}
y_{t} = a_{t}k_{t}^{\theta}h_{t}^{1 - \theta}
\end{equation}
\begin{equation}\label{eq:ProductivityEvolution}
ln(a_{t}) = (1 - \rho)ln(A) + \rho ln(a_{t - 1}) + \epsilon_{t}
\end{equation}
\begin{equation}\label{eq:GoodsMarket}
y_{t} = c_{t} + i_{t}
\end{equation}
\begin{equation}\label{eq:CapitalEvoluation}
\eta k_{t + 1} = (1 - \delta)k_{t} + i_{t}
\end{equation}
\item Combine FOC[c] and FOC[h]:
\begin{equation}\label{eq:LaborLeisure}
\gamma c_{t}h_{t} = (1 - \theta)y_{t}
\end{equation}
\item Euler Equation:
\begin{equation}\label{eq:Euler}
\frac{\eta}{c_{t}} = \beta E_{t}[\frac{1}{c_{t + 1}}(\theta(\frac{y_{t + 1}}{k_{t + 1}}) + 1 - \delta)]
\end{equation}
\end{itemize}
\end{frame}

%------------------------------------------------
\section{Calibration}
%------------------------------------------------


\begin{frame}
\frametitle{Formally}
\begin{itemize}
\item Calibration is mathematically equivalent to just-identified GMM.
\item Select a set of moments that we believe have a "high signal-to-noise" ratio.
\item Generally, choose parameter so that steady-state variables match well-known quantities.
\begin{equation}
\Omega(\{X_{t}^{M}\}_{t = 1}^{T}) = \Omega(\{X_{t}\}_{t = 1}^{T})
\end{equation}
\item Informally, use other implied moments to consider the "fit" of these parameters.
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Selecting Moments for Hansen's Model}
\begin{itemize}
\item We will start by considering a relationship between wages and output:
\begin{equation}
w_{t} = \frac{\partial y_{t}}{\partial h_{t}} =(1 - \theta)a_{t}(\frac{k_{t}}{h_{t}})^{\theta}
\end{equation}
\begin{equation}
\Rightarrow \frac{w_{t}h_{t}}{y_{t}} = (1 - \theta)
\end{equation}
\item That is, our theory implies that the ratio of real wages to output should equal $1 - \theta$, or the share of income paid to workers.
\begin{equation}
ln(Y_{t + 1}) - ln(Y_{t}) \approx (1 - \theta)ln(\eta)
\end{equation}
\item If we assume that the capital stock is approximately constant quarter to quarter, then this might be a reasonable approximation, given that A and H have little trend.
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Selecting Moments for Hansen's Model - Cont.}
\begin{itemize}
\item \textcolor{blue}{Cooley} (1995) suggests that the steady-state capital-output ratio is 3.32 yearly:
\item Then $\beta^{4}$ solves equation \autoref{eq:Euler}:
\begin{equation}
3.32 = \frac{\theta}{\frac{4\eta}{\beta} - 1 + 4\delta}
\end{equation}
\item We also take $\delta = 0.012$ from Cooley.
\item Hours have been observed to be roughly trendless, thus we can find $\gamma$ from the following:
\begin{equation}
 h^{*} = (\frac{1 - \theta}{\gamma})[1 - (\frac{\theta(\eta - 1 + \delta)}{\frac{\eta}{\beta} - 1 + \delta})]^{-1}
\end{equation}
\item From the following, we can estimate TFP and its associated parameters, $\rho$ and $\sigma_{\epsilon}$:
\begin{equation}
\Delta ln(Y_{t}) - [ (1 - \theta)[\Delta ln(H_{t}) + ln(\eta)] \approx \Delta ln(A_{t})
\end{equation}
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Readying the Data}
\begin{itemize}
\item We must match theoretical moments to the correct empirical moments:
\begin{enumerate}
\item Our model doesn't include government or international trade, so these need to be removed from GDP.
\item Use personal consumption and private investment.
\item We have no prices, so each variable needs to be in real terms.
\item Each of the variables is defined to be per-capita, so we need to divide by population.
\end{enumerate}
\item Further preparations are needed:
\begin{enumerate}
\item Series decomposed into trend and cycle using Hodrick-Prescott Filter.
\item Solving for $\theta$ requires further detrending: divide per-capita variable by $\eta^{t}$.
\end{enumerate}
\item Most of the data taken from BEA.
\item Real wages per capita are taken from FRED, and not explicitly in the model.
\item \textcolor{red}{Really lean into the model as the ``true'' model of the world}
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Calibration Results}
\begin{table}[ht]
\caption{Calibration Estimates} % title of Table
\centering % used for centering table
\begin{tabular}{c c c c c c c} % centered columns (4 columns)
\hline\hline %inserts double horizontal lines
\multicolumn{2}{c}{Preferences}&\multicolumn{5}{c}{Technology} \\ [0.5ex] % inserts table
%heading
 % inserts single horizontal line
$\beta$  &  $\gamma$  & $\theta$  & $\eta$  & $\delta$  & $\rho$  & $\sigma_{\epsilon}$  \\ % inserting body of the table
\hline
0.9903 & 0.0076 & 0.3739  & 1.0061 & 0.0120  & 0.9972  & 0.0129 \\

\hline %inserts single line
\end{tabular}
\label{table:nonlin} % is used to refer this table in the text
\end{table}
\begin{table}[ht]
\caption{Steady-States} % title of Table
\centering % used for centering table
\begin{tabular}{c c c c c c} % centered columns (4 columns)
\hline\hline %inserts double horizontal lines
%heading
 % inserts single horizontal line
$y^{*}$  &  $c^{*}$  & $i^{*}$  & $h^{*}$ & $k^{*}$ & $a^{*}$ \\ % inserting body of the table
\hline
8,834 & 6,694 & 2,140  & 108.61 & 118,320  & 17.8309 \\

\hline %inserts single line
\end{tabular}
\label{table:nonlin} % is used to refer this table in the text
\end{table}
\end{frame}


%------------------------------------------------

\section{Maximum Likelihood Estimation} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
%------------------------------------------------

\begin{frame}
\frametitle{Maximum Likelihood}
\begin{itemize}
\item An alternative approach to estimation is maximum likelihood via the Kalman Filter.
\item With equation (40), we can now write the system in state-space form:
\begin{equation}
f_{t} = \Pi_{1} s_{t} + \eta_{t}
\end{equation}
\begin{equation}
s_{t + 1} = \Pi_{2}s_{t} + \epsilon_{t}
\end{equation}
\item We typically include $\eta_{t}$ as measurement errors for the observed variables to avoid stochastic singularity.
\item Having written the model like this, we can apply the Kalman Filter for different parameter values to find the likelihood maximizing parameter vector.
\end{itemize}
\end{frame}

%------------------------------------------------


\begin{frame}
\frametitle{Log-Linearizing the System}
\begin{itemize}
\item We can now write the system as:
\begin{equation}
\Psi_{1}\zeta_{t} = \Psi_{2}\xi_{t} + \Psi_{3}\tilde{a}_{t}
\end{equation}
\begin{equation}
\Psi_{4}E_{t}(\xi_{t + 1}) = \Psi_{5}\xi_{t} + \Psi_{6}\zeta_{t} + \Psi_{7}\tilde{a}_{t}
\end{equation}
\item $\zeta_{t}$ are static predetermined and nonpredetermined variables, $[\tilde{y}_{t}, \tilde{h}_{t}, \tilde{i}_{t}]'$.
\item $\xi_{t}$ are dynamic predetermined and nonpredetermined variables, $[\tilde{k}_{t}, \tilde{c}_{t}]'$.
\item $\tilde{a}_{t}$ is the technology process.
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Matrices}
\begin{equation*}
\kappa = \eta/\beta - 1 + \delta
\end{equation*}

\begin{equation*}
\lambda = \eta - 1 + \delta
\end{equation*}
\begin{center}
\includegraphics[width=5in]{matrices.PNG}
\end{center}

\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Comparing Results}
\begin{table}[ht]
\caption{MLE Results Fixing $\beta$ and $\delta$} % title of Table
\centering % used for centering table
\begin{tabular}{c c c c c c c} % centered columns (4 columns)
\hline\hline %inserts double horizontal lines
\multicolumn{2}{c}{Preferences}&\multicolumn{5}{c}{Technology} \\ [0.5ex] % inserts table
%heading
 % inserts single horizontal line
$\beta$  &  $\gamma$  & $\theta$  & $\eta$  & $\delta$  & $\rho$  & $\sigma_{\epsilon}$  \\ % inserting body of the table
\hline
0.99 & 0.0045 & 0.2292  & 1.0051 & 0.0250  & 0.9987  & 0.0052  \\

\hline %inserts single line
\end{tabular}
\label{table:nonlin} % is used to refer this table in the text
\end{table}
\begin{table}
\caption{Calibration Estimates} % title of Table
\centering % used for centering table
\begin{tabular}{c c c c c c c} % centered columns (4 columns)
\hline\hline %inserts double horizontal lines
\multicolumn{2}{c}{Preferences}&\multicolumn{5}{c}{Technology} \\ [0.5ex] % inserts table
%heading
 % inserts single horizontal line
$\beta$  &  $\gamma$  & $\theta$  & $\eta$  & $\delta$  & $\rho$  & $\sigma_{\epsilon}$  \\ % inserting body of the table
\hline
0.9903 & 0.0076 & 0.3739  & 1.0061 & 0.0120  & 0.9972  & 0.0129 \\

\hline %inserts single line
\end{tabular}
\label{table:nonlin} % is used to refer this table in the text
\end{table}

\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Rios-Rull et al. (2012)}
\begin{itemize}
\item Attempt to compare calibrated and Bayesian results.
\item Estimate Hansen's model with investment shocks and different labor supply elasticities.
\item Three different calibration approaches to identifying elasticity:
\begin{enumerate}
\item Use long-run hours worked: elasticity around 2.
\item Use lotteries (equivalent to what we have done here): elasticity of $\infty$.
\item Use estimates from microeconomic studies: between 0.2 - 0.76.
\end{enumerate}
\item The models result in around the same results if identifying assumption 3 is used.
\item They conclude that identification is more important than estimation technique.
\end{itemize}
\end{frame}

%------------------------------------------------

\section{Criticisms and Extensions of General RBC Model}

\begin{frame}\frametitle{Criticism 1: no independent evidence for technology shocks}

\begin{itemize}
\item Hard to identify specific shocks

\item Alexopoulos (AER 2011). Publication of new technology books seem to
signal changes in TFP. But this explains only a fraction of Solow residual

\item Negative shocks -- Are they technological regress?

\item Oil price shocks act like technology shocks in some ways, but are best
modeled separately\pagebreak
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Criticism 2: Solow residual is correlated with demand shocks}

% \begin{itemize}
% \item Labor hoarding

\begin{itemize}
\item Measured labor hours don't account for intensity of effort

\item During recessions, reduce effort rather than hours

\item During expansions, increase effort, rather than hours

\item Can reflect matching and training costs, implicit insurance
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item Suppose output is given by
\begin{equation*}
Y_{t}=K_{t}^{\alpha }\left( A_{t}L_{t}U_{t}\right) ^{1-\alpha }
\end{equation*}%
where $U$ denotes utilization (effort)

\item The true Solow Residual is
\begin{equation*}
SR_{t}=\left( 1-\alpha \right) \left( \ln A_{t}+\ln U_{t}\right) =\ln
Y_{t}-\alpha \ln K_{t}-\left( 1-\alpha \right) \ln L_{t}
\end{equation*}%
implying that changes in utilization affect the Solow Residual

\item Increases in utilization are mistaken for increases in productivity

\item Addresses Criticism 2: Demand shocks increase utilization and thus
increase utilization-unadjusted Solow residual

\item Keynesian AD-AS model with sticky wages, demand driven fluctuations
and no labor hoarding implies $corr(y,y/\ell )<0.$ Data shows $corr(y,y/\ell
)\approx 1/2.$ Adding labor hoarding can generate a positive
correlation
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Propagation Mechanisms: Economic dynamics that extend and transform
the effects of an exogenous shock}

\begin{itemize}
\item Intertemporal substitution of labor: higher productivity today induces
more work today

\item Capital accumulation

\item Problem 1: Without indivisible labor, small IES$_{L}$ implies small
labor propagation

\item Problem 2: Capital accumulation generates little propagation

\item Even with indivisible labor, the dynamics of output are the dynamics
of technology
\end{itemize}
\end{frame}

\begin{frame}\frametitle{RBC models that use measured Solow residuals cannot produce (Cogley
and Nason 1995)}

\begin{itemize}
\item Positive autocorrelation in output growth (not output)

\item Note: Solow residuals follow AR(1) with $\rho \leq 1$

\item A 'hump-shaped" impulse response function for transitory
shocks
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Generating persistence: need to slow down the economy's response to
the initial shock}

\begin{itemize}
\item Labor search (Merz, 1995; Andolfatto, 1996); Employers and workers
need time to make matches

\item Finance constraints (Carlstrom and Fuerst, 1997; Bernanke, Gertler and
Gilchrist, 1999): Over time, higher productivity allows firms to borrow more

\item Factor hoarding (Burnside, Eichenbaum and Rebelo, 1996): Firms first
increase effort and capital utilization, then increase hours and capital
\end{itemize}
\end{frame}

\section{Conclusion}

\begin{frame}
\frametitle{Conclusion}
\begin{itemize}
\item Baseball extra credit!
\item Calibration uses model implied restrictions and estimates on data.
\item Maximum likelihood (and related techniques) use all variation in the data.
\item MLE and calibration provide estimates that are relatively similar in this context.
\item Others have shown similar results for more complex models (Rios et al., 2012).
\item Next: Most likely, Heterogeneous agents
\item No class next Thursday (come to conference that weekend instead!)
\end{itemize}
\end{frame}


\end{document}
