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\title[Consumption Risk]{Business Cycle Basics}
\author[]{Emi Nakamura and Jon Steinsson}
\institute[]{UC Berkeley}
\date[]{August 2025}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}


\begin{frame}{Business Cycle Basics}
	\begin{itemize}
		\item Trend vs. cycle decompositions
		\item Business cycle asymmetries: unemployment
		\item Is output trend or difference stationary?
		\item Welfare gains from stabilization policy
		\begin{itemize}
			\item Lucas' Macroeconomic Priorities
			\item Connection to asset pricing and equity premium puzzle
		\end{itemize}
		\item Great Moderation
	\end{itemize}
\end{frame}



\begin{frame}{Burns and Mitchell}
	\begin{figure}
		\centering
		\includegraphics[width =1.3   \textheight]{FiguresTables/BurnsMitchellquote.png}
	\end{figure}

	\footnotesize (Peak to trough analysis)
\end{frame}

\begin{frame}{How to Define Cycles?}
	\begin{itemize}
		\item Peak-to-trough analysis
		\item Linear detrending
		\item First Differences (aka Growth Rate)
		\item Flexible detrending (e.g., HP Filter, Bandpass)
	\end{itemize}
\end{frame}


\begin{frame}{Stock and Watson (1999): Level of GDP}
	\begin{figure}
		\centering
		\includegraphics[width =1\textwidth]{FiguresTables/stockwatson1999_levelofgdp.png}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}{Stock and Watson (1999): Linearly Detrended}
	\begin{figure}
		\centering
		\includegraphics[width =1\textwidth]{FiguresTables/stockwatson1999_linearlydetrended.png}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}{Stock and Watson (1999): Growth Rate}
	\begin{figure}
		\centering
		\includegraphics[width =1\textwidth]{FiguresTables/stockwatson1999_growthrate.png}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}{Band Pass Filter Definition}
	\begin{figure}
		\centering
		\includegraphics[width =0.7\textwidth]{FiguresTables/stockwatson1999_bandpass1.png}
	\end{figure}
	\vspace{-4mm}
	Also: Hodrick-Prescott (1981) Filter
\end{frame}

\begin{frame}{Band Pass Filter}
	\begin{figure}
		\centering
		\includegraphics[width =0.7\textwidth]{FiguresTables/stockwatson1999_bandpass2.png}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}
	\frametitle{Business Cycles: Standard View}
	\centering
	\makebox[\linewidth]{%
		\centering
		\includegraphics[height=7cm,width=12cm]{FiguresTables/businesscycle.png}
	}
\end{frame}


\begin{frame}
	\frametitle{Friedman 1964}
	\centering{
	\begin{quote}
		``Consider an elastic string stretched taut
		between two points on the underside of a rigid
		horizontal board and glued lightly to the board.
		Let the string be plucked at a number of points
		chosen more or less at random with a force
		that varies at random, and then held down at
		the lowest point reached. The result will be
		to produce a succession of apparent cycles in
		the string whose amplitudes depend on the
		force used in plucking the string.''
	\end{quote}}

\vspace{20pt}
\emph{\footnotesize{``Output is
				viewed as bumping along the ceiling of
				maximum feasible output except that every
				now and then it is plucked down by a cyclical
				contraction.''}}
	
\end{frame}	


\begin{frame}
	\frametitle{The Last Trough-to-Peak-to-Trough in the US}
	\centering
	\makebox[\linewidth]{%
		\centering
		\includegraphics[height=7cm,width=12cm]{FiguresTables/LastCycle2.pdf}
	}
\end{frame}



\begin{frame}
	\frametitle{Through the Lens of the Plucking Model}
	\centering
	\makebox[\linewidth]{%
		\scalebox{-1}[1]{\includegraphics[height=7cm,width=12cm,angle=180]{FiguresTables/WoodenBoard.pdf}}
	}
\end{frame}

\begin{frame}
	\frametitle{Friedman 1964: Implication}
	\begin{itemize}
		\item Plucking Model view: Contractions are drops below potential
		\item Plucking Property: One implication stressed by Friedman
		\begin{itemize}
			\item Size of a contraction predicts the size of the following expansion
			\item Size of an expansion doesn't predict the size of the following contraction
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}
	\frametitle{The Plucking Property}
	\centering
	\makebox[\linewidth]{%
		\centering
		\includegraphics[height=5cm, width=11cm]{FiguresTables/PluckingProperty.pdf}
	}
	
\end{frame}

\begin{frame}
	\frametitle{The Plucking Property}
	\centering
	\resizebox{\textwidth}{!}{% <------ Don't forget this %
		\begin{tabular}{lcc} \hline\hline
			& $\beta$ & $R^2$ \\ \hline 
			Subsequent expansion on contraction & 1.09    & 0.58 \\ [-0.4ex]
			& (0.33)  & \\ 
			Subsequent contraction on expansion & -0.38   & 0.22 \\ [-0.4ex]
			& (0.27)  & \\ \hline
		\end{tabular}%
	}
\end{frame}

\begin{frame}
	\frametitle{Alternative: Forest Fires}
	Jackson and Tebaldi (2017)
	\begin{itemize}
		\item Forest fires: the longer the expansion, the more ``underbrush'' there are to burn during the contraction		
		\item Recessions: Perhaps longer expansions beget larger contractions
		\item Aggregate data not supportive: Neither size nor duration of expansion predictssize of subsequent contraction 
		\item Goes (slightly) the other way
		\begin{itemize}
			\item Longest
			post-WW2 expansions (1961-1968, 1982-1989, 1992-2000) were followed by among the mildest recessions
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}
	\frametitle{Unemployment: Up the Elevator, Down the Stairs}
	\centering
	\makebox[\linewidth]{%
		\centering
		\includegraphics[height=7cm,width=12cm]{FiguresTables/Unemployment.png}
	}
\end{frame}

\begin{frame}
	\frametitle{Contractions are Faster than Expansions}
	\begin{itemize}
		\item Neft\c{c}i (1984)'s \emph{``sudden jumps and slower drops''}
	\end{itemize}
	\bigskip
	\centering
	\resizebox{0.8\textwidth}{!}{% <------ Don't forget this %
		\begin{tabular}{lcc} \hline\hline
			& pp/year   \\ \hline
			Speed of expansions \hspace{0.5in}  & 0.88  \\ 
			Speed of contractions  & 1.89 \\ [1ex]
			P-value for equal speed  & 0.002  \\ \hline
		\end{tabular}%
	}
\end{frame}

\begin{frame}
	\frametitle{Positive skewness of the unemployment rate}
	\begin{itemize}
		\item Sichel (1993)'s ``deepness'':\emph{``when troughs are deeper than peaks are tall''}
		\makebox[\linewidth]{%
			\centering
			\includegraphics[height=4cm, width=10cm]{FiguresTables/Histogramskewness.pdf}
		}
		\item Skewness of the unemployment rate = 0.63
		\item Reject [H0: No Skewness] at p-value = 0.071
	\end{itemize}
\end{frame}


\begin{frame}{Covid Recovery: Why Was This Time So Fundamentally Different?}
Not well understood. Some ideas: 
	\begin{itemize}
		\item Shock was (and expected to be) less persistent 
		\begin{itemize}
			\item Labor hoarding, temporary layoffs
		\end{itemize}
		\item Shock involved ``rationing'' which led households to build up huge savings
		\begin{itemize}
			\item Pent up demand for vacation travel, restaurants etc. 	
		\end{itemize}		
		\item Huge government stabilization policies (Fed, Treasury)
	\end{itemize}
\end{frame}



\begin{frame}{Is output trend or difference stationary?}
	Intellectual history:
\begin{itemize}
	\item Big debate in macro in 1980's \\ (Nelson and Plosser, 1982; Watson, 1986; \\ Clark, 1987; Campbell and Mankiw, 1987)
	\item Important implications for business cycle theory \linebreak (Are recessions permanent?)
\end{itemize}
Papers:
\begin{itemize}
		\item Cochrane (1988): US data on GNP, Variance ratios
		\item Cogley (1990)	
	\end{itemize}

\end{frame}


\begin{frame}{Suppose Output is a Random Walk?}
		\[ \log Y_{t+1} = \mu + \log Y_{t} + \epsilon_{t+1} \]
	\begin{itemize}
		\item What does this imply about $\partial \log Y_{t+j} / \partial \epsilon_{t+1}$ as $j \rightarrow \infty$? \pause 
		\begin{itemize}
		\item  $\partial \log Y_{t+j} / \partial \epsilon_{t+1} = 1$ for all $j$
		\end{itemize}
		\item What does it imply about $\mbox{var}_{t}(\log Y_{t+j})$ as $j \rightarrow \infty$? \pause
		\begin{itemize}
			\item Goes to infinity!!
		\end{itemize} 
		\item But does US GDP look like a random walk with drift?
	\end{itemize}
\end{frame}


\begin{frame}{US GDP}
	\begin{figure}
		\centering
		\includegraphics[width =1.2   \textheight]{FiguresTables/Cochrane1988figure2.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}{Suppose Output is Trend Stationary?}
	\begin{itemize}
		\item Trend stationary model:
		\[\log Y_{t+1} = bt + \sum_{j=0}^{\infty} a_{j} \epsilon_{t-j} \]
		($ \sum_{j=0}^{\infty} a_{j} \epsilon_{t-j}$ stationary: $a_{j}$ approaches zero for large $j$) 
		\item How do long run forecasts change in response to  $\epsilon_{t}$? \pause
		\begin{itemize}
			\item Invariant to shocks!
		\end{itemize}
		\item $\mbox{var}_{t}(\log Y_{t+j}) \rightarrow  \sum_{j=0}^{\infty} a_{j}^{2} \sigma$ as $j \rightarrow \infty$

	\end{itemize}
\end{frame}

\begin{frame}{Variance Ratios for Consumption Growth}
	\begin{itemize}
		\item Cochrane (1988) advocated using variance ratios:
		\[\mbox{VR}_{i,k} = \frac{1}{k} \frac{\mbox{var}(Y_{i,t} - Y_{i,t-k})}{\mbox{var}( Y_{i,t} - Y_{i,t-1})} \] \pause
		
		\item Random walk: $VR_{i,k} = 1$ for all $k$
		\item Trend stationary: $VR_{i,k} \rightarrow 0$ as $k \rightarrow \infty$
		\item Positively autocorrelated growth:  $VR_{i,k} > 1$ for large $k$
	\end{itemize}
\end{frame}

\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[width =1.2   \textheight]{FiguresTables/Cochrane1988figure1.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}{Cogley (1990)}
	\begin{itemize}
		\item Extends Cochrane's estimation approach to \linebreak 9 OECD countries for 1871-1985
		\item Critiques small sample properties of Cochrane's \linebreak asymptotic standard errors
	\end{itemize}
\end{frame}

\begin{frame}
	\begin{figure}{Log Per Capita Consumption in France}
		\centering
		\includegraphics[ width=\textwidth]{FiguresTables/FranceConsumption.png}
	\end{figure}
	\vspace{-4mm}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[width =1   \textheight]{FiguresTables/Cogley1990table2.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[width =1.3   \textheight]{FiguresTables/Cogley1990table4.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}{Welfare Gains from Stabilization Policy}
	\begin{itemize}
		\item Robert Lucas' thought experiment
		\item Macroeconomic Priorities (Lucas, 2003)
		\item Famously led Lucas to stop working on business cycles and start working on growth
	\end{itemize}
\end{frame}



\begin{frame}{Welfare Gains from Stabilization Policy}
	\begin{itemize}
		\item Imagine that we could magically eliminate all consumption variation around trend
		\item Consumer's stochastic consumption stream:
		\[ c_{t} = Ae^{\mu t}e^{-(1/2)\sigma^{2}}\epsilon_{t} \]
		\item $\log(\epsilon_{t}) \sim \mbox{N}(0,\sigma^{2})$ implies useful fact:
		\[E(e^{-(1/2)\sigma^{2}}\epsilon_{t}) = 1\]
		\item Consumer's utility function
		\[ E\left\{ \sum_{t=0}^{\infty} \beta^{t} \frac{c_{t}^{1-\gamma}}{1-\gamma} \right\} \]
		\item $\beta$ is subjective discount factor, $\gamma$ coefficient of risk aversion
	\end{itemize}
\end{frame}

\begin{frame}{Welfare Gains from Stabilization Policy}
	\[ E\left\{ \sum_{t=0}^{\infty} \beta^{t} \frac{((1+\lambda)c_{t})^{1-\gamma}}{1-\gamma} \right\} =  \sum_{t=0}^{\infty} \beta^{t} \frac{(Ae^{\mu t})^{1-\gamma}}{1-\gamma} \]
	\vspace{0.2in}
	\[ \lambda \simeq \frac{1}{2} \gamma \sigma^{2} \]
	\begin{itemize}
		\item For 1947-2001, the standard deviation of the log of U.S. real, per capita consumption about a linear trend: 0.032.
		\item Estimates of $\gamma$ from macro and public finance: 1-4
	\end{itemize}
	\[ \lambda = \frac{1}{2} (0.032)^{2} = 0.0005 \]
\end{frame}

\begin{frame}{Welfare Gains from Stabilization Policy}
	\begin{itemize}
		\item Conclusion: Welfare gains from stabilization policy are trivial.
		\item Is this convincing? \pause
			\begin{itemize}
				\item Trend vs. difference stationary consumption
				\item Disasters, long-run risks
				\item Uninsurable risks
				\item Inequality and heterogeneity 
		\end{itemize} 
		Deeply connected to ``equity premium puzzle''
	\end{itemize}
\end{frame}

\begin{frame}{Digression: Consumption Based Asset Pricing}
	\begin{itemize}
		\item Consumption based asset pricing starts from the \linebreak Consumption Euler equation:
		\[U'(C_{t}) = \beta E_{t} [ (1+R_{i,t+1}) U'(C_{t+1})] \]
		\item A little manipulation yields:
		\[ 1 = E_{t} \left[ (1+R_{i,t+1}) \frac{\beta U'(C_{t+1})}{U'(C_{t})} \right] \]
		\[ 1 = E_{t} [(1+R_{i,t+1}) M_{t+1}] \]
		\item where $M_{t+1}$ is call the stochastic discount factor
	\end{itemize}
\end{frame}

\begin{frame}{Power Utility}
	\begin{itemize}
		\item Now suppose that
		\[ U(C_{t}) = \frac{C_{t}^{1-\gamma}-1}{1-\gamma}\]
		\item This implies:
		\[ U'(C_{t}) = C_{t}^{-\gamma} \]
		\[M_{t+1} = \frac{\beta U'(C_{t+1})}{U'(C_{t})} = \beta \left( \frac{C_{t+1}}{C_{t}} \right)^{-\gamma}\] 
	\end{itemize}
\end{frame}

\begin{frame}{Euler Equation with Power Utility}
	\[ 1 = E_{t} [(1+R_{i,t+1}) M_{t+1}] \]
	\[ 1 = E_{t} [(1+R_{i,t+1}) ]  \beta \left( \frac{C_{t+1}}{C_{t}} \right)^{-\gamma} \]
	Taking logs and assuming log-normality:
	\[0 = E_{t} r_{i,t+1} + \log \beta - \gamma E_{t} \Delta C_{t+1} + \frac{1}{2} [\sigma_{i}^{2} + \gamma^{2} \sigma_{c}^{2} - 2 \gamma \sigma_{ic}]\]
	And for risk-free rate:
	\[0 = r_{f,t+1}  + \log \beta - \gamma E_{t} \Delta C_{t+1} + \frac{1}{2}\gamma^{2} \sigma_{c}^{2} \]
	Combining these last two yields:
	\[ E_{t} r_{i,t+1} - r_{f,t+1} + \frac{1}{2} \sigma_{i}^{2} = \gamma \sigma_{ic} \]
\end{frame}

\begin{frame}{Equity Premium}
	\[ E_{t} r_{i,t+1} - r_{f,t+1} + \frac{1}{2} \sigma_{i}^{2} = \gamma \sigma_{ic} \]
	\begin{itemize}
		\item Equity premium is a function of covariance between consumption growth and return on equity
		\item What is the intuition for this?
	\end{itemize}
\end{frame}

\begin{frame}{Measurement}
	\[ E_{t} r_{i,t+1} - r_{f,t+1} + \frac{1}{2} \sigma_{i}^{2} = \gamma \sigma_{ic} \]
	Suppose equity returns and consumption growth are perfectly correlated:
	\[ E_{t} r_{i,t+1} - r_{f,t+1} + \frac{1}{2} \sigma_{i}^{2} = \gamma \sigma_{i} \sigma_{c} \]
\end{frame}

\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[width =1.3   \textheight]{FiguresTables/Table5_campbell.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}

\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[width =1.3   \textheight]{FiguresTables/barroursua2008.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}


\begin{frame}{Equity Premium}
	\begin{itemize}
		\item Volatility of consumption seems to be relatively modest
		\item World seems to be a relatively safe place
		\item People must be very risk averse to not want to \linebreak bid up prices of stocks 
	\end{itemize}
\end{frame}


\begin{frame}{Introspection about $\gamma$}
	\begin{itemize}
		\item Consider the following gamble: I flip a coin and ...
		\begin{itemize}
			\item If it comes up heads, I multiply your lifetime income by 1 million
			\item If it comes up tails, I reduce your lifetime income by 10\%
		\end{itemize}
		\onslide+<2->
		\item If you accept, your CRRA is less than 10
		\onslide+<3->
		\item What about 20\% reduction? If yes, CRRA < 5
		\onslide+<4->
		\item What about 30\% reduction? If yes, CRRA < 3
		\onslide+<5->
		\item What about 50\% reduction? If yes, CRRA < 2
	\end{itemize}
\end{frame}

\begin{frame}{Great Moderation}
Empirical observation:
\begin{itemize}
	\item Rolling window quarterly standard deviation of output \\ shows sharp drop in 1984!
	\item[] Note: This is the ``classic'' graph but perhaps it is misleading \\ (more on this later...)
\end{itemize}	
Early papers: 
	\begin{itemize}
		\item McConnell and Perez-Quiros (2000)
		\item Blanchard and Simon (2001)
		\item Stock and Watson (2002))
	\end{itemize}

\end{frame}	

{
	\setbeamercolor{background canvas}{bg=}
	\includepdf[pages={2-2}]{FiguresTables/QuarterlyRealGDPSA.pdf}
} 



\begin{frame}{Early work: McConnell and Perez-Quiros (2000)}
			\begin{itemize}
				\item Sharp decline in volatility in 1984Q1
				\item Possible hypotheses: 
				\begin{itemize}
					\item Shift toward service sector
					\item Better monetary policy
				\end{itemize}
				\item  But largest declines in durable goods: production but not sales \\ (i.e. role of inventories)
				\begin{itemize}
					\item Conclude that inventory management was important
				\end{itemize}
			\end{itemize}
	\end{frame}
	
\begin{frame}{McConnell and Perez-Quiros (2000)}
	\begin{figure}
		\centering
		\includegraphics[width =1   \textheight]{FiguresTables/Mcconnellandperezquirosfig1.png}
	\end{figure}
	
\end{frame}
	
	
\begin{frame}{ Blanchard and Simon (2001)}
		\begin{itemize}
			\item Longer-term trend in volatility over the twentieth century
			\item Confounding factors: 
			\begin{itemize}
				\item World Wars
				\item Much more volatile data before modern national accounts (Romer (1986))
			\end{itemize}
			\item Nevertheless, it appears there has been a ``long and large decline'' in macro volatility over the post-WWII period
			\begin{itemize}
				\item Exact timing varies across countries
			\end{itemize}
			\end{itemize}
	\end{frame}
	
	\begin{frame}
		\begin{figure}
			\includegraphics[scale =0.75]{FiguresTables/Figure5StochVolUSUKCanada.pdf}
		\end{figure}
	Figure from Nakamura et al. (2013)
	\end{frame}

	
	\begin{frame}{Stock and Watson (2002)}
Comprehensive view of the evidence:
		\begin{itemize}
			\item Broad decline in volatility across sectors
			\item Associated with declines in conditional variances, not means (based on ARIMA type model)
			\item Break rather than trend
			\begin{itemize}
				\item Formally test for break date (as in Bai, Lumsdaine, Stock, 1998)
			\end{itemize}
			\item Break date around 1984
			\item Much less support for ``inventories'' view when focusing on annual data 
		\end{itemize}
	\end{frame}
	
	\begin{frame}
		\begin{figure}
			\centering
			\includegraphics[width =1.1   \textheight]{FiguresTables/stockandwatson2002_table7.pdf}
		\end{figure}
		\vspace{-4mm}
	\end{frame}
	

\begin{frame}{Why Did the Great Moderation Occur?}
	\vspace{-10pt}
Stock and Watson consider several hypotheses
		\begin{itemize}
			\item Inventories (McConnell and Perez-Quiros, 2000)
			\begin{itemize}
				\item But little evidence in annual (as opposed to quarterly) data
			\end{itemize}
			\item Sectoral Shift to less volatile service sector 
			\begin{itemize}
				\item  Not large enough (but ignores feedback)
			\end{itemize}
			\item Weaker propagation of shocks
			\begin{itemize}
				\item Inconsistent with VAR evidence (see next slide)
			\end{itemize}
				\item Changes in residential housing market
				\begin{itemize}
					\item Big drop in volatility and changes in institutions, but occurred gradually! 
					\item Might be a candidate
				\end{itemize}
		\end{itemize}
	 What is left? Smaller shocks!
	\end{frame}
	
\begin{frame}{Stock and Watson: Shocks not Propagation!}
	\begin{figure}
		\centering
		\includegraphics[width =1.1   \textheight]{FiguresTables/stockandwatson2002_table5.pdf}
	\end{figure}
	\vspace{-4mm}
\end{frame}

	

\begin{frame}{Smaller Shocks}
\begin{itemize}
	\item Identified productivity and commodity price shocks  (20-30\%)
	\item Changes in Monetary Policy (10-25\%)
	\begin{itemize}
			\item Change in Taylor rule
			\item Estimate structural model, allowing Taylor rule to change
				\item Lucas critique?
			\end{itemize}
			\item Unknown sources of ``Good Luck'' (40-60\%)
			\item Has our good luck ended?
		\end{itemize}
	\end{frame}


	\begin{frame}{Jaimovich and Siu (2007)}
		\begin{itemize}
			\item Argue for demographic explanation for Great Moderation
			\item Key insight: Population is aging in many countries
						\item ...And volatility of hours worked much higher for certain age groups
						\begin{itemize}
							\item Consider age groups 15-19, 20-24, 25-29,...,60-64 
							\item U-shaped pattern in cyclical volatility by age
							\end{itemize}
		\end{itemize}
	\end{frame}
	
	\begin{frame}
		\begin{figure}
			\centering
			\includegraphics[width =1.2   \textheight]{FiguresTables/jaimovichandsiu_fig2.pdf}
		\end{figure}
		\vspace{-4mm}
	\end{frame}

\end{document}






















