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%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Income Processes} % 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{Fall 2024} % 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{Announcements}
  \begin{itemize}
\item Be sure you  have done the following:
  \begin{itemize}
  \item Setting up programming languages.
  \item Set up \LaTeX.
  \item Look into using SSH and the campus cluster.
  \item I will email IT this afternoon to make sure everyone has access to the cluster.
  \end{itemize}
\item Possible change to syllabus: add topic on market power?
  \end{itemize}
\end{frame}

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

\section{Basic Income Processes} % 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{Recap from last time}
  \begin{itemize}
  \item In conducting our quantitative exercises, we want an accurate model of income and wages for calibration \& evaluation.
  \item Basic approach:
    \begin{enumerate}
    \item Condition on observables in data and estimate residual earnings.
    \item Use model to understand the contributions to residual earnings.
    \end{enumerate}
  \item Some considerations:
    \begin{enumerate}
    \item Make your data look as much like the model as possible.
    \item Does your model have an hours choice, i.e., an {\it intensive margin}? Then you want to match hourly wages.
    \end{enumerate}
  \item Good sources (where much of this comes from):
    \begin{itemize}
    \item Chris Tonetti's Write-up on income processes
    \item Gianluca Violante's slides on income processes
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Residual Earnings}
  \begin{itemize}
  \item Estimate the following:
    \begin{align}
      Y_{i,j,t} &= w_{t}\exp(f(X_{i,j,t})+\epsilon_{i,j,t})h_{i,j,t}
    \end{align}
  \item where
    \begin{align*}
      Y_{i,j,t}&: \text{Total income of ind. i at age j, at time t}\\
      w_{t}&: \text{Hourly wages}\\
      X_{i,j,t}&: \text{Predictable component}\\
      \epsilon_{i,j,t}&: \text{Stochastic component}\\
      h_{i,j,t}&: \text{Hours worked}
    \end{align*}
  \item Note: if your model has an {\it intensive margin}, i.e., choice of hours, estimate
    \begin{align}
      y_{i,j,t} &= \frac{Y_{i,j,t}}{h_{i,j,t}} = w_{t}\exp(f(X_{i,j,t})+\epsilon_{i,j,t})
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Residual Earnings}
  \begin{itemize}
  \item Estimate the following:
    \begin{align}
      ln(Y_{i,j,t}) &= ln(w_{t}) + ln(h_{i,j,t}) + f(X_{i,j,t}) + \epsilon_{i,j,t}
    \end{align}
  \item $X_{i,j,t}$ includes covariates (heterogeneity) that is observable and not central to the research question.
  \item Some examples:
    \begin{itemize}
    \item Race, gender (usually just use males), marital status
    \item Year, state, age
    \end{itemize}
  \item You might try to explain the effect of a covariate, i.e., race on labor earnings.
  \item Usually, write down a model to understand the residual.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Statistical Model of Income}
  \begin{itemize}
  \item We can think about residual earnings as being partially transitory:
    \begin{itemize}
    \item Temporary layoff
    \item Temporary hours cut
    \end{itemize}
  \item And persistent:
    \begin{itemize}
    \item College degree
    \item Innate ability
    \item Job-specific skills
    \end{itemize}
  \item A statistical model (i.e., one with no explicit structural interpretation) will estimate the size of these two components over the life-cycle.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Statistical Model of Income}
  \begin{itemize}
  \item If you are not micro-founding the income process (i.e., writing down a search model), need alternate model of income.
  \item Basic approach: specify two equation transitory and persistent model of income.
  \item Let $\tilde{Y}_{i,j}$ be residual log earnings of individual i at age j (remember, we controlled for time in the first stage).
    \begin{align}
      \tilde{Y}_{i,j} &= z_{i,j} + \psi_{i,j}\\
      z_{i,j} &= \rho z_{i,j-1} + \nu_{i,j}\\
      \psi_{i,a}&\sim iid (0, \sigma_{\psi})\\
      \nu_{i,a}&\sim iid (0, \sigma_{\nu})\\
      z_{i,0}&= 0
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Identification}
  \begin{itemize}
  \item Even in macro, we need to worry about identification.
  \item What parameters do we need to estimate?
  \item $\rho, \sigma_{\psi}, \sigma_{\nu}$.
    \begin{align}
      \tilde{Y}_{i,j} &= z_{i,j} + \psi_{i,j}\\
      z_{i,j} &= \rho z_{i,j-1} + \nu_{i,j}\\
      \rightarrow Var(\tilde{Y}_{i,0}) &= Var(z_{i,0}) + Var(\psi_{i,0}) + \cancel{Cov(z_{i,0},\psi_{i,0})}\\
      \rightarrow Var(\tilde{Y}_{i,0}) &= \sigma_{\psi}^{2}\\
      Var(\tilde{Y}_{i,j}) &= Var(z_{i,j}) + \sigma_{\psi}^{2}\\
      Var(z_{i,j}) &= \rho^{2}Var(z_{i,j-1}) + \sigma_{\nu}^{2}\\
      Cov(\tilde{Y}_{i,j},\tilde{Y}_{i,j-n}) &= Cov(z_{i,j},z_{i,j-n})\\
      Cov(z_{i,j},z_{i,j-n}) &= \rho^{n}Var(z_{i,j-n})
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Identification II}
  \begin{itemize}
  \item First, the peristence:
    \begin{align}
      \frac{Cov(z_{i,j},z_{i,j-2})}{Cov(z_{i,j-1},z_{i,j-2})} = \frac{\rho^{2}Var(z_{i,j-2})}{\rho Var(z_{i,j-2})} = \rho
    \end{align}
  \item Initial distribution is simple: $Var(\tilde{Y}_{i,0}) = \sigma_{\psi}^{2}$
  \item The persistent shock:
    \begin{align}
      Var(z_{i,j}) &= \rho^{2}Var(z_{i,j-1}) + \sigma_{\nu}^{2}\\
      Var(\tilde{Y}_{i,j}) - \sigma_{\psi}^{2} &= \rho^{2}Var(z_{i,j-1})  + \sigma_{\nu}^{2}\\
      Var(\tilde{Y}_{i,j}) - \rho Cov(\tilde{Y}_{i,j},\tilde{Y}_{i,j-1}) - \sigma_{\psi}^{2} &= \sigma_{\nu}^{2}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Statistical Model of Income}
  \begin{itemize}
  \item We've now identified an earnings process and can estimate it from the data.
    \begin{align}
      \tilde{Y}_{i,j} &= z_{i,j} + \psi_{i,j}\\
      z_{i,j} &= \rho z_{i,j-1} + \nu_{i,j}\\
      \psi_{i,a}&\sim iid (0, \sigma_{\psi})\\
      \nu_{i,a}&\sim iid (0, \sigma_{\nu})\\
      z_{i,0}&= 0
    \end{align}
  \item What can we use it for?
    \begin{enumerate}
    \item Calibration targets for a structural model.
    \item External validation of structural model.
    \item Better understanding of data.
    \end{enumerate}
  \item This also tells us roughly what fraction of income shocks are insurable (transitory) and what are not (permanent).
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Statistical Model of Income}
\centering\includegraphics[width=0.8\textwidth]{HSV2012_IncomeShocks.png}
\end{frame}

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

\section{Restricted vs. Heterogeneous Income Profiles} % 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{Income Profile Heterogeneity}
  \begin{itemize}
  \item What have we done thus far? Specified an income process with persistent and transitory components.
  \item But, we've assumed that it fluctuates around some trend, in which there is no heterogeneity.
  \item Is this a reasonable assumption?
  \item Guvenen (2009) suggests that it is not.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Income Profile Heterogeneity}
  \begin{itemize}
  \item (Mostly following Guvenen's notation)
  \item Model of log labor income of individual $i$ at age/experience $j$:
    \begin{align}
      y_{j}^{i} &= \beta^{i}\times j + z_{j}^{i}\\
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \eta_{j}^{i}
    \end{align}
  \item There is a deterministic trend, $\beta^{i}$, which is potentially heterogeneous between individuals $i$.
  \item There is a persistent shock $z_{j}^{i}$.
  \item (Full specification in subsequent discussion)
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Income Profile Heterogeneity}
    \begin{align}
      y_{j}^{i} &= \beta^{i}\times j + z_{j}^{i}\\
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \eta_{j}^{i}
    \end{align}
  \begin{itemize}
  \item Two views of $\beta^{i}$:
    \begin{itemize}
    \item Restricted income profiles (RIP): $\sigma_{\beta} = 0$, i.e., profiles fluctuate around the same trend.
    \item i.e., $y_{j}^{i} &= \beta\times j + z_{j}^{i}$
    \item Heterogeneous income profiles (HIP): $\sigma_{\beta} > 0$, i.e., profiles have different individual trends.
    \end{itemize}
  \item Why important?
    \begin{itemize}
    \item As before, these profiles are often used as calibration targets.
    \item Difference isn't an innocuous assumption.
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Interpretation}
    \begin{align}
      y_{j}^{i} &= \beta^{i}\times j + z_{j}^{i}\\
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \eta_{j}^{i}
    \end{align}
  \begin{itemize}
  \item What does it mean for individuals to have heterogeneous income growth rates?
  \item What are some empirical implications for restricted growth rates?
  \item How could we use a model to think about this income process?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Guvenen (2009) Statistical Model of Earnings}
  \begin{itemize}
  \item Estimate the following:
    \begin{align}
      y_{j,t}^{i} &= g(\theta_{t}^{0},X_{j,t}^{i}) + f(\alpha^{i},\beta^{i},X_{j,t}^{i}) + z_{j,t}^{i} + \phi_{t}\epsilon_{j,t}^{i}
    \end{align}
    \begin{align*}
      g(\theta_{t}^{0},X_{j,t}^{i}) &: \text{Predictable common growth rate \& intercep}t\\
      f(\alpha^{i},\beta^{i},X_{j,t}^{i}) &: \text{Individual specific growth rate \& intercept ($\alpha$)}\\
      \phi_{t}\epsilon_{j,t}^{i} &: \text{Transitory shocks/Measurement error, iid}
    \end{align*}
  \item Profile heterogeneity:
    \begin{align}
      f(\alpha^{i},\beta^{i},X_{j,t}^{i}) &= \alpha^{i} + \beta^{i}\times j
    \end{align}
  \item Permanent shocks:
    \begin{align}
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \pi_{t}\eta_{j}^{i},\hspace{3mm} z_{0,t}^{i} = 0, \eta\sim iid
    \end{align}
  \item Heterogeneity uncorrelated with shocks.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Identification (ex-ante)}
  \begin{itemize}
  \item As before, identification using second-order moments.
  \item We can denote the income residual $\tilde{y}_{j,t}^{i}$, i.e., the residual after estimating the polynomial $g$.
    \begin{align}
      var(\tilde{y}_{j,t}^{i}) &= [\sigma_{\alpha}^{2} + 2\sigma_{\alpha\beta}j + \sigma_{\beta}^{2}j] + var(z_{j,t}^{i}) + \phi_{t}^{2}\sigma_{\epsilon}^{2}\\
      cov(\tilde{y}_{j,t}^{i},\tilde{y}_{j+n,t+n}^{i}) &= [\sigma_{\alpha}^{2} + \sigma_{\alpha\beta}(2j + n) + \sigma_{\beta}^{2}j(j + n)] + \rho^{n}var(z_{j,t}^{i})
    \end{align}
  \item Note: n years/experience ahead.
  \item Remaining parameters identified from the persistent shock:
    \begin{align}
      var(z_{1,t}^{i}) &= \pi_{t}^{2}\sigma_{\eta}^{2}\\
      var(z_{j,1}^{i}) &= \pi_{1}^{2}\sigma_{\eta}^{2}\sum_{j=0}^{h-1}\rho^{2j}\\
      var(z_{j,t}^{i}) &= \rho^{2}var(z_{j-1,t-1}^{i}) + \pi_{t}^{2}\sigma_{\eta}^{2}, t > 1, j > 1
    \end{align}
  \item j: experience, t: year.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Thoughts on Identification}
  \begin{itemize}
  \item What is the inherent problem with identifying parameters from persistent component $z_{j,t}^{i}$?
  \item It is unobserved!
  \item Another name for two equation income process (and other specifications):
    \begin{itemize}
    \item ``Measurement equation'': $y_{j}^{i} = \beta^{i}\times j + z_{j}^{i}$
    \item ``Transition equation'': $z_{j}^{i} = \rho z_{j - 1}^{i} + \eta_{j}^{i}$
    \end{itemize}
  \item Learn transition equation from measurement equation observations.
  \item i.e., place structural restrictions on transition equation and back-out values.
  \item As we saw in restricted process: identification hinges on relating everything to measurement equation.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Intuitive Identification}
  \begin{itemize}
  \item Heterogeneity will bias up the estimate of shock persistence.
  \item Good example of classical measurement error.
  \end{itemize}
\centering\includegraphics[width=0.75\textwidth]{Guvenen2009_Identification1.png}
\end{frame}

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

\begin{frame}
  \frametitle{Sample Selection}
  \begin{itemize}
  \item Data: Panel Study of Income Dynamics (PSID), 1968-1993.
  \item Satisfy the following criteria for at least 20 years:
    \begin{enumerate}
    \item Male head of household;
    \item Between 20 and 64 years old;
    \item Positive labor earnings and hours;
    \item Hours each year between 520 and 5110;
    \item Wage floor and wage cap;
    \item Not in the Survey of Economic Opportunity (SEO) sample.
    \end{enumerate}
  \item This yields 1270 individuals.
  \item Thoughts on this selection?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Estimated Parameters}
\centering\includegraphics[width=\textwidth]{Guvenen2009_EstimatedParameters.png}
\begin{itemize}
\item What are some interesting features of the estimated parameters?
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Restricted vs. Heterogeneous Persistence}
\centering\includegraphics[width=0.9\textwidth]{Guvenen2009_PersistentShock.png}
\begin{itemize}
\item A one time shock to persistent component.
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Decomposition}
\centering\includegraphics[width=\textwidth]{Guvenen2009_Decomposition.png}
\begin{itemize}
\item How much income inequality is persistent shocks, i.e., AR(1), and how much is profile heterogeneity?
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Covariance Structure: College}
\centering\includegraphics[width=0.9\textwidth]{Guvenen2009_Covariance1.png}
\begin{itemize}
\item Each line is cov. between $ln(\tilde{y})$ at exp./age h (left-most point) and h + n.
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Covariance Structure: High School}
\centering\includegraphics[width=\textwidth]{Guvenen2009_Covariance2.png}
\begin{itemize}
\item Each line is $Cov(ln(\tilde{y}_{h}),ln(\tilde{y}_{h+n}))$.
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Covariance Structure: College}
\centering\includegraphics[width=0.5\textwidth]{Guvenen2009_Covariance1.png}%
\includegraphics[width=0.5\textwidth]{Guvenen2009_Covariance3.png}
\begin{itemize}
\item College appears to have both an AR(1) component and an HIP component.
\item AR(1) decreases in importance as HIP increases.
\item How could this be interpreted?
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Covariance Structure: High School}
\centering\includegraphics[width=0.55\textwidth]{Guvenen2009_Covariance2.png}%
\includegraphics[width=0.5\textwidth]{Guvenen2009_Covariance4.png}
\begin{itemize}
\item High school appears to be better approximated by restricted profiles and an AR(1) shock.
\item How could we interpret this?
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Income Processes and Macro Models}
  \begin{itemize}
  \item How do we relate these income processes to calibrating macro models?
  \item Straightforward in a model with competitive labor market.
  \item As a reminder:
    \begin{align}
      y_{j}^{i} &= \beta^{i}\times j + z_{j}^{i}\\
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \eta_{j}^{i}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Simple Model (Mostly Storesletten, Telmer, Yaron (2004))}
  \begin{itemize}
  \item As a reminder
    \begin{align}
      y_{j}^{i} &= \beta^{i}\times j + z_{j}^{i} + \psi_{j}^{i}\\
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \eta_{j}^{i}
    \end{align}
  \item An agent solves the following:
    \begin{align}
      V_{t}(h_{t},a_{t},\mu) &= \max_{c_{t},a_{t + 1}} u(c_{t}) + \beta E[V_{t +_1}(h_{t + 1},a_{t + 1},\mu)]\\
      \text{s.t.   } c_{t} + a_{t + 1} &\leq (1 + r_{t})a_{t} + w_{t}h_{t}\\
      ln(h_{t + 1}) &= ln(h_{t}) + \mu + z_{t + 1} + \epsilon_{t+1}\\
      z_{t + 1} &= \rho z_{t} + \nu_{t + 1},\hspace{3mm} z_{0} = 0\\
      \epsilon&\sim\;iid\; N(0,\sigma_{\epsilon}),\hspace{3mm}\nu\sim\;iid\; N(0,\sigma_{\nu})
    \end{align}
  \item Rental rates determined competitively (exogenously, etc.):
    \begin{align}
      w_{t} &= F_{H}(K,H)\\
      r_{t} &= F_{K}(K,H)\\
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Mapping between our income process and the macro model}
  \begin{itemize}
  \item As a reminder
    \begin{align}
      y_{j}^{i} &= \beta^{i}\times j + z_{j}^{i} + \psi_{j}^{i}\\
      z_{j}^{i} &= \rho z_{j - 1}^{i} + \eta_{j}^{i}
    \end{align}
  \item Model specified income process
    \begin{align}
      ln(h_{t + 1}) &= ln(h_{t}) + \mu + z_{t + 1} + \epsilon_{t+1}\\
      z_{t + 1} &= \rho z_{t} + \nu_{t + 1},\hspace{3mm} z_{0} = 0\\
      \epsilon&\sim\;iid\; N(0,\sigma_{\epsilon}),\hspace{3mm}\nu\sim\;iid\; N(0,\sigma_{\nu})
    \end{align}
  \item Assume a cohort-wide rental rate and $ln(h_{t})$ is proportional to log-income.
  \item Then, $\mu\sim N(\mu_{\beta}, \sigma_{\beta})$;
  \item $\epsilon\sim N(0, \sigma_{\psi})$;
  \item $\nu\sim N(0, \sigma_{\eta})$, and $\rho$ is identical.
  \item Estimate $ln(h_{t})$ from the age-0 earnings distribution.
  \end{itemize}
\end{frame}

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

\section{Conclusion}
% ------------------------------------------------


\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Review panel data methods.
  \item Overview of other empirical regularities in the labor market.
  \item To do between now and Tuesday:
    \begin{enumerate}
  \item Install the appropriate programming languages.
  \item Start playing around with one of the panel data sources discussed in first slides (PSID, NLSY, SIPP). Or CPS (non-panel).
    \end{enumerate}
  \item I'll post helpful code next week, but I want you to check them out first.
  \end{itemize}
\end{frame}


\end{document}