%slides_empirics.tex
%Slides for Chapter 1 of ``Open Economy Macroeconomics'' by Martín Uribe and Stephanie Schmitt-Grohé, Princeton University Press 2017. 
%© Martín Uribe and Stephanie Schmitt-Grohé.

\input{slides_preamble}


\begin{slide}
\begin{minipage}{4.5in}
\rightline{\includegraphics[width=4.5in]{USG_cover.eps}}

\centerline{\tiny  Princeton University Press,  2017}
\end{minipage}
\begin{minipage}{4.5in}
\centering 
{\small slides 

chapter 1   

business-cycle facts around the world
}



\end{minipage}
\end{slide}


\topleft{\textit{Open Economy Macroeconomics, Chapter 1} }
\topright{M.\ Uribe and S.\ Schmitt-Groh\'e}

\begin{slide}
To characterize business cycle facts we decompose a time series, $y_t$,  into a \\
$\bullet$ cyclical component, $y^c_t$, and a\\ 
$\bullet$ secular (or trend) component, $y^s_t$
\[
y_t = y^c_t + y^s_t
\]

  
There are various methods to extract the cyclical component:\\
$\bullet$ log-linear detrending\\
$\bullet$ log-quadratic detrending\\
$\bullet$ HP filtering\\
$\bullet$ Band pass filtering\\


\end{slide} 

\begin{slide}
{\bf Log-linear detrending}

Let 
\[
y_t\equiv \ln Y_t\]
denote the natural logarithm of a time series $Y_t$, where $t$ denotes time. Then write 
\[
y_t  = a + b t + \epsilon_t 
\]
where 
\[\mbox{cycle: } \quad y^c_t = \epsilon_t 
\]
\[\mbox{secular trend: } \quad y^s_t = a + b t 
\]
The parameters $a$ and $b$ can be estimated via ordinary least squares (OLS). 


{\small This is, for example, how King, Plosser, and Rebelo (JME, 1988) define the cyclical component of U.S. time series in their seminal real-business-cycle paper.  They further impose constant spending shares in the long run, that is, they impose that $b$ is the same for output, consumption, and investment. 


}
\end{slide} 


\begin{slide}
{\bf  Log-quadratic detrending}

\[
y_t = a + b t + c t^2  + \epsilon_t 
\]
\[\mbox{cycle: } \quad y^c_t = \epsilon_t 
\]
\[\mbox{secular trend: } \quad y^s_t = a + b t + c t^2 
\]

Again, 
the parameters $a$,  $b$, and $c$ can be estimated via OLS. 


{ This is, for example, how Mendoza (AER, 1991) defines the cyclical component of Canadian time series in his small-open-economy  real-business-cycle model. 


}


\end{slide} 


\begin{slide}
{\bf Log-quadratic detrending: Application to Argentine real GDP per capita, annual data, 1960-2011} 

\centerline{\includegraphics[height=4in]{empirics_argentina_quadratic_trend_gdp.eps}
\includegraphics[height=4in]{empirics_argentina_cycle_gdp.eps}}

%Produced with c:\data\uribe\book\empirics\wdi\Argentina_cycle.m
\end{slide} 


\begin{slide}
Result of quadratically detrending 
the Argentine output: \\
$\bullet$ 2.5 large cycles: (1) 1960-1990; (2) 1990-2002; (3) 2002-2011 (incomplete); \\
$\bullet$ in the 1960-1990 cycle output fell from 15 percent above trend at the peak in 1980 to 25 percent below trend at the trough in 1990, giving rise to what is now known as the lost decade\\
$\bullet$ The log-quadratic filter successfully identifies the contractions associated with\\
\indent -- the hyperinflation of  the late 1980s\\
\indent -- the demise of the Convertivility Plan and default  in   2001\\ 
$\bullet$ std($y^c_t) = $ 10.7 percent per year, implying a highly volatile business cycle \\
$\bullet$ corr($y^c_t,y^c_{t-1}) = 0.85$, implying  persistent cyclical fluctuations\\
\end{slide} 


\begin{slide}


{\small 


What are the characteristics of a typical business cycle? To answer this question, we compute business cycle facts for a large number of countries. 

We use  an annual data set from the World Development Indicators (WDI), starting in 1960 and ending in 2011. To be included in the sample, a country must have at least 30 consecutive observations of the following six  variables: 


- $ y_t$ (log of) real GDP per capita\\
- $ c_t$ (log of) real private consumption per capita\\
- $g_t$  (log of) real government  consumption per capita\\
- $i_t$  (log of) real investment per capita\\
- $x_t$  (log of) real exports per capita\\
- $m_t$  (log of) real imports per capita\\

There are 120 countries satisfying this requirement in the data set. 
Additionally, of those 120 countries, 94 have 30 consecutive years of data on the current account. Thus, statistics regarding the current account are based only on a sample of 94 countries.  


}
\end{slide} 
\begin{slide}
{\bf A Comment on the Consumption Data}
 

The WDI private consumption series includes  expenditures on nondurables, services, and durables. Typically,  business-cycle studies remove expenditures on  durables from the definition of consumption. 
The reason is that from an economic point of view,  expenditures on durable consumption goods, such as cars and washing machines,  represent  an investment in household physical capital. 
And this makes expenditures on durables  far more volatile than expenditures on nondurables and services. 


The next slides shows how much more volatile total  consumption is relative to that of  nondurables and services for the U.S. economy, a country for which we do have disaggregated consumption data. 

Unfortunately, most countries do not publish disaggregated consumption data. So, keep in mind that the volatility of consumption reported in the cross-country comparisons below is higher than what it would be were expenditures on durables excluded. 

\end{slide}

\begin{slide}
\centerline {\bf Relative Volatilty of Disaggregated Consumption, $\boldsymbol{\sigma_c / \sigma_y}$}
\centerline{(annual U.S. data, 1965-2011, data source, bls.gov)} 
\centerline{\begin{tabular}{lcccc}
\hline
\hline
Measure of $C_t$&Avg.& log-linear& log-quadratic & HP \\
&Share&detrending &detrending& filter \\
\hline
Total Consumption&1&  1.02&  1.01&  0.88
\\
Nondurables and Services& 0.87&  0.87&  0.84&  0.64
\\
Durables&0.13&  2.47&  2.53&  2.95\\
 \hline
\hline
\end{tabular}}
%produced with read_data.m in directory: z:\uribe\book\empirics\US_C

$\bullet$ all standard deviations are scaled by $\sigma_y$

$\bullet$  nondurable and services consumption  is less volatile than output. 

$\bullet$ whereas durable consumption is much more volatile than output.

$\bullet$ the standard deviation of total consumption is nearly 20 percent higher than that of nondurables and services (even though durable consumption represents only 13 percent of total consumption expenditure).

\end{slide}





\begin{slide}

We first present business-cycle facts based on 
quadratically detrended versions of  $y_t$, $c_t$, $g_t$, $i_t$, $x_t$, and $m_t$. 

Trade Balance, $TB_t\equiv X_t-M_t$
can take negative values, so its log doesn't exist. Instead, we detrend it by   first 
scaling it  by the trend component of output to obtain  
$tb_t \equiv  \frac{X_t - M_t}{\exp(y^s_t)}$ and then removing a quadratic trend. In this way, the deviations from trend are measured in percent of trend output.  
Same with the Current Account,  denoted $CA_t$; thus, 
$ca_t \equiv  \frac{CA_t}{\exp(y^s_t)}$. 

For each country, we compute standard deviations, contemporaneous correlations with output, and serial correlations of all  variables. 
We then compute a population-weighted average of each statistic. 

\end{slide}




\begin{slide}

\centering 

{\bf \large Business Cycles Around the World: 

Ten Facts}


\end{slide}




\begin{slide}


\begin{center}
{\large  \bf High Global Volatility}

\begin{tabular}{ccc}
\hline \hline
Business-Cycle\\
Statistic&World&U.S.\\
\hline
$\sigma_y$&6.2\%&2.9\%\\
\hline \hline
\end{tabular}
\end{center}

\vspace{1in}


{\bf Fact 1:} The cross-country average standard deviation of output is twice as large as its U.S.\ counterpart. 




\end{slide}


\begin{slide}

\centerline{ \large \bf Excess Consumption Volatility}

\begin{center}

\begin{tabular}{ccc}
\hline \hline
Business-Cycle&World&\\
Statistic&Average \\\hline
$\frac{\sigma_c}{\sigma_y}$&1.05\\
\hline \hline
\end{tabular}

\end{center}

\vspace{1in}

{\bf Fact 2:} On average, across countries, private consumption (including durables) is more volatile than output. 


\end{slide}

 

\begin{slide}
\begin{center}
{\large \bf Global Ranking of Volatilities}


\begin{tabular}{ccc}
\hline \hline
Business-Cycle&World\\
Statistic&Average\\\hline
$\frac{\sigma_m}{\sigma_y}$&3.23\\\\
$\frac{\sigma_i}{\sigma_y}$&3.14\\\\
$\frac{\sigma_x}{\sigma_y}$&3.07\\\\
$\frac{\sigma_g}{\sigma_y}$&2.26\\\\
$\frac{\sigma_c}{\sigma_y}$&1.05\\
\hline \hline
\end{tabular}

\end{center}

{\bf Fact 3:} The ranking of cross-country average standard deviations from top to bottom is imports, investment, exports, government spending, consumption, and output. 


\end{slide}


\begin{slide}
\centerline{ \large \bf Cyclicality }

\begin{center}

\begin{tabular}{cc}
\hline \hline
Business-Cycle Statistic&World Average \\\hline
corr$(c,y)$&0.69\\
corr$(i,y)$&0.66\\
corr$(x,y)$&0.19\\
corr$(m,y)$&0.24\\
\hline
corr$(tb,y)$&-0.18\\
corr$(ca,y)$&-0.28\\
\hline 
corr$(g/y,y)$&-0.02\\
\hline \hline
\end{tabular}
\end{center}

{\bf Fact 4:}\\ Consumption, investment, exports, and imports are procyclical. 
\\
{\bf Fact 5:} \\The trade balance and the current account are countercyclical. 
\\
{\bf Fact 6:} \\The share of government consumption in output  is acyclical. 

\end{slide}

\begin{slide}
\begin{center}
{\large \bf Persistence}


\begin{tabular}{ccc}
\hline \hline
Business-Cycle&World\\
Statistic&Average\\\hline
corr$(y_t,y_{t-1})$&0.71\\
corr$(c_t,c_{t-1})$&0.66\\
corr$(g_t,g_{t-1})$&0.76\\
corr$(i_t,i_{t-1})$&0.56\\
corr$(x_t,x_{t-1})$&0.68\\
corr$(m_t,m_{t-1})$&0.61\\
\hline \hline
\end{tabular}

\end{center}
{\bf Fact 7:} All components of demand ($c, g, i, x$) and supply ($y,m$) are positively serially correlated. 


\end{slide}

\begin{slide}
\begin{center}
{\bf \large The U.S.\, Business Cycle as 

A Point of Comparison}
\end{center}
\end{slide}


\begin{slide}
\begin{minipage}{3in}
Comparing Business Cycles around the World to the U.S. Business Cycle
\end{minipage}\,
\begin{minipage}{6in}

{\tiny \begin{tabular}{lcc}
\hline\hline
Statistic&United&All\\
% &Poor&Emerging&Rich\\
&States&Countries
\\%&Countries&Countries&Countries\\
\hline
\multicolumn{3}{l}{\underline{Standard  Deviations}}\\
$\sigma_y$&\fbox{2.94}&  \fbox{6.22}\\
$\sigma_c/\sigma_y$&\fbox{1.02}&  \fbox{1.05}
\\%&  1.12&  0.98&  0.87\\
$\sigma_g/\sigma_y$&1.93&  2.26
\\%&  2.46&  2.00&  1.73\\
$\sigma_i/\sigma_y$&  3.52&3.14
\\%&  3.24&  2.79&  3.20\\
$\sigma_x/\sigma_y$&3.49&  3.07
\\%&  3.08&  2.82&  3.36\\
$\sigma_m/\sigma_y$& 3.24& 3.23
\\%&  3.30&  2.72&  3.64\\
$\sigma_{tb/y}$&0.94&  2.34
\\%&  2.12&  3.80&  1.25\\
$\sigma_{ca/y}$&1.11&  2.16
\\%&  2.06&  3.08&  1.39\\
\multicolumn{3}{l}{\underline{Correlations with $y$}}\\
$y$&   1.00&1.00
\\%&   1.00&   1.00&   1.00\\
$c$&   0.90&0.69
\\%&   0.66&   0.75&   0.76\\
$g/y$&  \fbox{-0.32}&\fbox{-0.02}
\\%&   0.08&  -0.08&  -0.39\\
$i$& 0.80&  0.66
\\%&   0.60&   0.77&   0.77\\
$x$& -0.11&  0.19
\\%&   0.14&   0.35&   0.17\\
$m$& 0.31&  0.24
\\%&   0.14&   0.50&   0.34\\
$tb/y$&-0.51&  -0.15
\\%&  -0.11&  -0.21&  -0.26\\
$tb$&-0.54&  -0.18
\\%&  -0.14&  -0.24&  -0.25\\
$ca/y$&-0.62&  -0.28
\\%&  -0.28&  -0.24&  -0.30\\
$ca$&-0.64&  -0.28
\\%&  -0.28&  -0.26&  -0.31\\
\multicolumn{3}{l}{\underline{Serial Correlations}}\\
$y$&0.75&  0.71
\\%&  0.65&  0.87&  0.76\\
$c$&0.82&  0.66
\\%&  0.62&  0.74&  0.75\\
$g$&0.91&  0.76
\\%&  0.71&  0.80&  0.89\\
$i$&0.67&  0.56
\\%&  0.49&  0.72&  0.67\\
$x$& 0.75& 0.68
\\%&  0.65&  0.74&  0.74\\
$m$&0.63&  0.65
\\%&  0.61&  0.74&  0.69\\
$tb/y$& 0.79& 0.61
\\%&  0.59&  0.62&  0.69\\
$ca/y$& 0.79& 0.57
\\%&  0.55&  0.52&  0.71\\
\multicolumn{3}{l}{\underline{Means}}\\
$tb/y$&-1.5&  -1.3
\\
$(x+m)/y$&\fbox{18.9}&  \fbox{36.5}
\\
\hline\hline
\end{tabular}
}
\end{minipage}

\end{slide}


\begin{slide}
{\bf  Some  Observations} 

{\small 
{\bf US1:} The world is a much more volatile place than the U.S.:
 The cross-country average volatility of output is twice as large as its U.S.~counterpart. Important question in Macro: Why? Is the U.S. lucky, that is, is it hit by  smaller  shocks?; or does the U.S.\, have better policy?


{\bf US2:} Besides that, business cycle facts for the  United States are not that different from  what is observed around the world. 


{\bf US3:}  Consumption is as volatile as output both in the U.S.\ and  around the world ($\sigma_c/\sigma_y \approx 1$). You might find this fact surprising in the case of the United States, as you might be used to the result that consumption is less volatile than output. The reason is that the WDI's consumption series used here includes expenditures on durables.

{\bf US4:}  The share of government spending is more countercyclical in the United States than around the world (-0.32 versus -0.02), that is, the U.S. appears to employ more strongly countercyclical fiscal policy than the rest of the world. 

{\bf US5:} One other difference: the U.S. is less open, $(x+m)/y=18.9$\% compared to 36.5\%. 

}

\end{slide}




\begin{slide}
To sum up, the two most striking differences between U.S. business cycles and business cycles around the world are: 

 $\bullet$ Business cycles are half a volatile in the U.S.


$\bullet$ Fiscal policy is more strongly countercyclical in the U.S. 

\end{slide}

\begin{slide}
\begin{center}
{\bf \large Business Cycles 
in 

Rich, Emerging, and Poor Countries}
\end{center}
\end{slide}



\begin{slide}
{\bf \large Question: Are business cycles different in rich, poor, and middle-income (or emerging)  countries? }

To answer this question, the first thing 
to do is to determine which countries are rich, poor, or middle income. This, in turn, requires  coming up with a measure of   income  per capita that is comparable across countries.  
We use  the  geometric average of  PPP-converted GDP per capita
in U.S.~dollars of 2005  over the period 1990-2009.    

Loosely speaking, PPP-converted GDP in a given country is the value of all goods and services produced in that country evaluated at U.S.\ prices.  
By evaluating production of  goods
and services
 in different countries at the same prices,  
PPP conversion makes cross-country comparisons more sensible. 


{\tiny A good source for PPP-converted GDP numbers is the World Bank's {\it International Comparison Program (ICP)}.


}

\end{slide}

\begin{slide}

{\bf  PPP-Converted GDP --- An example}

{\small Suppose that in a given year  country X produces: 
 3 hair cuts and  1 ton of grain.

Suppose the unit prices of these items  in country X are, 1 and 200 dollars, respectively.

Hence the nonconverted measure of GDP in country X is 203 dollars.

Suppose that in  the United States a hair cut  costs 20 dollars and 1 ton of grain 200 dollars.
 
Then, the PPP-converted measure of GDP in country X is 260 dollars.  

Country X is 28\% (260/203-1) richer when GDP is measured at PPP prices than when it is measured at domestic prices. The reason is that nontradable services are more expensive in the U.S.\ than in  country X. 

This hypothetical result is indeed typical for poor and emerging countries, where labor-intensive services are far cheaper than in the U.S. 


}
\end{slide}

\begin{slide}
{\bf The Country Distribution of GDP  Per Capita}

The next figure displays the distribution of PPP-converted  GDP per capita across countries. The horizontal axis measures the average PPP-converted GDP per capita
in U.S. dollars of 2005 over the period 1990 to 2009. The vertical axis measures the number of countries with GDP less than or equal to the associated level on the horizontal axis. 

The median GDP per capita is \$6,615, and the mean GDP per capita is \$11,254. Eighty countries (or 2/3) have per capita incomes below the mean. 

That income is unevenly distributed across countries is also reflected in the fact that  the plotted line is so steep at low levels of output.

{\tiny Q: How would the figure look if income was evenly distributed across countries?}

\end{slide}

\begin{slide}
\begin{center}
{\bf Country Distribution of GDP Per Capita} 
%\label{fig:empirics_mgdp_ppp_frequency}
\includegraphics[height=6in]{empirics_mgdp_ppp_frequency.eps}
%Produced with z:\uribe\book\empirics\wdi\mgdp_ppp_frequency.m
\end{center}
\end{slide}



\begin{slide}
Divide our sample of 120 countries into three groups: Poor, emerging, and rich countries. 
These groups  are defined as all countries with average PPP converted GDP per capita
 within the ranges:

less than  \$3,000,\\
\$3,000 to \$25,000,\\
more  than \$25,000,

 respectively. 

This results in 40 poor (1/3), 58 emerging (1/2), and 22 (1/6) rich countries.  


\end{slide}

\begin{slide}
{\small
{\bf  Poor Countries} 
\\
--Benin, Bhutan, Burkina Faso, Burundi, Central African Republic,
Comoros, Gambia, Guyana, Honduras, Lesotho, Malawi, Mali, Mauritania,
Mongolia, Niger, Papua New Guinea, Rwanda, Senegal, Sierra Leone, Togo, Zambia,
Zimbabwe.
\\
--Cameroon, Congo, C\^ote d'Ivoire, Ghana, Kenya, Madagascar, Mozambique, Nepal, Sri Lanka, Sudan, Uganda.
\\
--Bangladesh, China, Ethiopia, India, Indonesia, Pakistan, Philippines.

{\bf  Emerging Countries}
\\
--Albania, Antigua and Barbuda, Bahrain, Barbados, Bolivia,
Botswana, Bulgaria, Chile, Costa Rica, Cuba, Cyprus, Dominica, Dominican
Republic, Ecuador, El Salvador, Fiji, Gabon, Greece, Grenada, Guatemala, Hungary,
Israel, Jordan, Malta, Mauritius, Namibia, New Zealand, Panama, Paraguay,
Portugal, Puerto Rico, Seychelles, St.\ Kitts and Nevis, St.\ Lucia, St. Vincent and the
Grenadines, Suriname, Swaziland, Tonga, Trinidad and Tobago, Tunisia, Uruguay.
\\
--Algeria, Argentina, Colombia, Iran, Malaysia,
Morocco, Peru, South Africa, South Korea, Spain, Syria, Thailand, Turkey,
Venezuela.
\\
--Brazil, Egypt, Mexico.

{\bf  Rich Countries}
\\
--Austria, Belgium, Denmark, Finland, Hong Kong, Iceland, Ireland,
Luxembourg, Macao, Netherlands, Norway, Singapore, Sweden, Switzerland.
\\
--Australia, Canada, France, Italy, United Kingdom.
\\
--Germany, Japan, United States.

}
\end{slide}

\begin{slide}
Choice of classification thresholds is somewhat arbitrary. 

World Bank uses GNI per capita with thresholds set ad hoc in 1989 and adjusted for inflation since then to classify countries as low-, middle-, or high income. 


Standard and Poor's  (S\&P) classifies countries into groups with  developed, emerging, and frontier markets. Its classification system is not rule based, instead it considers a country's market and regulatory structure, the trading environment, and the operational efficiency. All countries that have developed markets according to  S\&P  as of 2011 fall into the group of rich countries in our sample, with the exception of Israel, Portugal, Spain, and Greece, which by our classification are emerging economies. Overall there is a high degree of overlap between the S\&P classification of countries with emerging and frontier markets and our emerging and poor country classification. A limitation of the S\&P   classification system is that it covers only 77 countries. 
 





\end{slide}


\begin{slide}

\begin{center}
{\bf \Large Three important differences emerge in business cycels acrosss rich, poor, and emerging countries: }

{\large \bf Excess Volatility of\\ Poor and Emerging Countries}

\begin{tabular}{cccc}
\hline \hline
Business-Cycle\\
Statistic&Poor&Emerging&Rich\\
\hline
$\sigma_y$&6.1\%&8.7\%&3.3\%\\
\hline \hline
\end{tabular}
\end{center}

\bigskip

{\bf Fact 8:} Business Cycles in poor and emerging countries are about {\bf twice} as volatile as business cycles in rich countries. 


and ...

\end{slide}

\begin{slide}

\begin{center}

{\large \bf Less Consumption Smoothing in Poor and Emerging than in Rich Countries}

\begin{tabular}{cccc}
\hline \hline
Business-Cycle\\
Statistic&Poor&Emerging&Rich\\
\hline
$\sigma_c/\sigma_y$&1.12&0.98&0.87\\
\hline \hline
\end{tabular}
\end{center}

\bigskip

{\bf Fact 9:} 
The relative consumption volatility is higher in poor and emerging countries than in rich countries. 


and ...

\end{slide}




\begin{slide}

\begin{center}

{\large \bf The Countercyclicality of Government Spending Increases With Income}

\begin{tabular}{cccc}
\hline \hline
Business-Cycle\\
Statistic&Poor&Emerging&Rich\\
\hline
corr$(g/y,y)$&0.08&-0.08&-0.39\\
\hline \hline
\end{tabular}
\end{center}

\bigskip

{\bf Fact 10:} 
The share of government consumption 
is countercyclical in rich countries, but acyclical in emerging and poor countries. 
\end{slide}




\begin{slide}
To recap: Three important differences between poor and emerging and rich country business cycles: 

$\bullet$ Poor and emerging countries are twice as volative as rich countries (Fact 8). 

$\bullet$ Poor and emerging countries have a higher relative consumption volatility than rich countries (Fact 9). 

$\bullet$ Only in rich countries is the share of government consumption countercyclical (Fact 10). 

\end{slide}

\begin{slide}
\begin{center}
{\bf \large Country Size and Business Cycles}
\end{center}
\end{slide}



\begin{slide}





Countries are sorted into three size categories: small, medium, and large. These three 
categories are defined, respectively,   as all countries with population in 2009 of  less than 20 million, between 20 and 80 million, and more than 80 million. 

Small: 77 countries; Medium: 30 countries; Large: 13 countries. 

\end{slide}



\begin{slide}
\begin{center}
{\large \bf Volatility of Output\\ Controlling for Country Size and Income}


\begin{tabular}{lccc}
\hline \hline
&\multicolumn{3}{c}{$\sigma_y$}\\
&Poor&Emerging&Rich\\
\hline
All&6.1\%&8.7\%&3.3\%\\
Small&8.2\%&9.5\%&4.3\%\\
Medium&9.5\%&9.0\%&3.1\%\\
Large&5.6\%&7.9\%&3.3\%\\
\hline \hline
\end{tabular}
\end{center}

Fact 8 is robust to controlling for country size, poor and emerging economies are at least twice as volatile as rich economies. 

\end{slide}


\begin{slide}
\begin{center}
{\large \bf Relative Volatility of Consumption, $\sigma_c/\sigma_y$, \\ Controlling for Country Size and Income}




\begin{tabular}{lccc}
\hline \hline
&\multicolumn{3}{c}{$\sigma_c/\sigma_y$}\\
&Poor&Emerging&Rich\\
\hline
All&1.1&0.98&0.87\\
Small&1.4&0.97&0.92\\
Medium&1.1&0.93&0.93\\
Large&1.1&1.1&0.84\\
\hline \hline
\end{tabular}
\end{center}

Fact 9 is robust to controlling for country size, in poor and emerging countries consumption is excessively volatile. 

\end{slide}

\begin{slide}
{\bf \large Are the business cycle facts we document robust to alternative detrending methods?} 

Consider next: \\
(a) Hodrick-Prescott (HP) filtered data
\\
(b) First-differenced data

\end{slide}


\begin{slide}
\begin{center}
{\bf \large HP-Filtered  Business Cycles}
\end{center}
\end{slide}



\begin{slide}
{\bf \large (a) The Hodrick and Prescott (1997) Filter}

Given a time series $y_t$ , for $t=1, 2,\dots T$, pick $y^c_t$ and $y^s_t$ to 
\[
%\label{eq:HP_loss_function}
\min_{\{y^c_t,y^s_t\}_{t=1}^T}\left\{
 \sum_{t=1}^T
(y^c_t)^2 + \lambda  \sum_{t=2}^{T-1} 
\left[
(y^s_{t+1}-y^s_t)-(y^s_t-y^s_{t-1})
\right]^2
\right\}
\]
\[\mbox{subject to} \quad y^s_t + y_t^c = y_t \]

where $\lambda$ is a parameter. 

When $\lambda\rightarrow \infty$, changes in the growth rate of $y^s_t$ become infinitely costly, and the HP trend component converges to a log-linear trend. 

When $\lambda\rightarrow0$, the cycle disappears ($y^c=0$), and the secular trend is the time series itself ($y^s_t=y_t$).
\end{slide}

\begin{slide}
The optimality conditions to this minimization problem can be written as
\[
Y = (I + \lambda A ) Y^s
\]
where $Y$ is the vector of observations of $y_t$ , $Y^s$ is the vector of the secular components $y^s_t$, and the matrix $A$ is a matrix of integers. 

Solve for $Y^s$: 
\[
Y^s = (I+ \lambda A)^{-1} Y
\]
With $Y^s$ in hand, $Y^c$, is
\[
Y^c = Y-Y^s
\]

$\Rightarrow$ HP filter is a linear filter. 
\end{slide}

\begin{slide}
{\bf The Size of $\lambda$ Matters}

Annual data:  $\lambda = 100$. But Ravn and Uhlig (2001) suggest $\lambda = 6.25$



\begin{center}
{\bf Example: Argentina}

\begin{tabular}{lcc}
\hline
\hline
&\multicolumn{2}{c}{$\lambda$}\\
& 100 & 6.25\\
\hline  
$\sigma_y$& 5.7\%&3.6\%\\
\hline
\hline
\end{tabular}
\end{center}

%produced by running argentina_cycle.m in directory
%z:\uribe\book\empirics\wdi
%Note: for lambda=6.25 one must change manually lambda in bc_analysis.m 
%and set HP =1 and set dohp=1
\end{slide}
 
\begin{slide}

\centerline{\bf \Large HP Filtered Trend  of Argentine Output} 

\bigskip

\centerline{ ($\lambda=100$) \hspace{3in} ($\lambda =  6.25$)}

\centerline{{\includegraphics[width=5in]{empirics_argentina_cycle_hp.eps}  }
{\includegraphics[width=5in]{empirics_argentina_cycle_hp6p25.eps}  }}
%Produced with c:\data\uribe\teaching\econ366\wdi\argentina_cycle.m having previously set HP=1 in bc_analysis
%Produced with c:\data\uribe\teaching\econ366\wdi\argentina_cycle.m having previously set HP=1 and la=6.25  in bc_analysis

HP-6.25 attributes bulk of the 1989 crisis and of the  2001 crisis to trend. But both were cyclical rather than secular for both were followed by rapid recovery. Thus, we will use $\lambda = 100$ for remainder of section. 

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
\centering

{\large \bf 
Cyclical Component of Argentine Output: }

{ HP 100 Filter  Versus Quadratic Trend} 

\centerline{\includegraphics[width=5in]
{empirics_hp_vs_quad.eps}  }
%Produced with c:\data\uribe\teaching\econ366\wdi\hp_vs_quad.m 
\end{slide}







\begin{slide}

\begin{center}

{\large \bf High Global Volatility}

\begin{tabular}{ccc}
\hline \hline
Detrending&\multicolumn{2}{c}{$\sigma_y$} \\
\cline{2-3}
Method&World Average&USA\\
\hline
QT &6.2\%&2.9\%\\
HP &3.8\%&2.0\%\\
\hline \hline
\end{tabular}
%produced with: z:\uribe\book\empirics\wdi\tablehp.m 

\end{center}

{\bf Fact 1}: The cross-country average volatility of output is twice as large as its U.S.~counterpart.

\begin{itemize}
\item  { Fact 1} continues to hold. 
\item Under HP filtering volatility falls by about 2/3.  
\end{itemize}
\end{slide}





 \begin{slide}

\begin{center}

{\large \bf Countercyclicality of the Trade-Balance-to-Output Ratio}

\begin{tabular}{lcc}
\hline \hline
corr$(tby,y)$&QT&HP\\\hline
All&-0.15& -0.18\\
Poor &-0.11& -0.08\\
Emerging&-0.21& -0.34\\
Rich &-0.26& -0.37\\
\hline \hline
\end{tabular}

\end{center}

{\bf Fact 5:} On average across countries the share of the trade balance in output is negatively correlated with output. 


$\bullet$ Fact 5 continues to hold under HP filtering

\end{slide}

\begin{slide}

\begin{center}

{\large \bf Excess Volatility of\\ Poor and Emerging Countries}

\begin{tabular}{cccc}
\hline \hline
Detrending&\multicolumn{3}{c}{$\sigma_y$}\\\cline{2-4}
Method&Poor&Emerging&Rich\\
\hline
QT&6.1\%&8.7\%&3.3\%\\
HP&4.1\%&4.0\%&2.0\%\\
\hline \hline
\end{tabular}
\end{center}

\bigskip

{\bf Fact 8:} Business Cycles in rich countries are about half as volatile as business cycles in emerging or poor countries. 

$\bullet$ Fact 8 continues to hold under HP filtering


\end{slide}

\begin{slide}

\begin{center}

{\large \bf Less Consumption Smoothing In Poor and Emerging Countries}

\begin{tabular}{cccc}
\hline \hline
Detrending&\multicolumn{3}{c}{$\sigma_c/\sigma_y$}\\
\cline{2-4}
Method&Poor&Emerging&Rich\\
\hline
QT&1.12&0.98&0.87\\
HP & 1.09&1.23&0.87\\
\hline \hline
\end{tabular}
\end{center}

\bigskip

{\bf Fact 9:} The relative consumption volatility is higher in poor and emerging countries than in rich countries. 

$\bullet$  Fact 9 continues to hold under HP Filtering

\end{slide}

\begin{slide}

\begin{center}

{\large \bf The countercyclicality of Government Spending Increases With Income}

\begin{tabular}{cccc}
\hline \hline
Detrending&\multicolumn{3}{c}{corr$(g/y,y)$}\\
\cline{2-4}
Method&Poor&Emerging&Rich\\
\hline
QT&0.08&-0.08&-0.39\\
HP&0.02&-0.06&-0.56\\
\hline \hline
\end{tabular}
\end{center}

\bigskip

{\bf Fact 10:} 
The share of government consumption 
is countercyclical in rich countries, but acyclical in emerging and poor countries. 

$\bullet$ Fact 10 continues to hold under HP filtering
\end{slide}


\begin{slide}
{\bf \large Summary of Comparison}

\begin{itemize}
\item QT and HP result in largely the same business cycle facts!

\item The main difference is that HP filtering implies that the volatility of output and aggregate demand is 2/3 that implied by log-quadratic detrending. 
\end{itemize}


\end{slide}

\begin{slide}


{\bf \large (b) First-differenced data} 

\[
\Delta y_t \equiv \ln Y_t - \ln Y_{t-1}
\]

\end{slide}

\begin{slide}
\begin{minipage}{6in}
{\tiny 
\begin{tabular}{lccccc}
\hline\hline
Statistic&All &Poor&Emerging&Rich\\
&Countries&Countries&Countries&Countries\\
\hline
\multicolumn{5}{l}{\underline{Standard  Deviations}}\\
$\sigma_{\Delta y}$&  4.39&  4.94&  {\color{red}{4.08}}&{\color{blue}{2.38}}\\
$\sigma_{\Delta c}/\sigma_{\Delta y}$&  1.14&  1.14& {\color{red}{1.34}}& {\color{blue}{0.85}}\\
$\sigma_{\Delta g}/\sigma_{\Delta y}$&  2.14&  2.28&  2.39&  1.17\\
$\sigma_{\Delta i}/\sigma_{\Delta y}$&  3.81&  3.80&  4.06&  3.49\\
$\sigma_{\Delta x}/\sigma_{\Delta y}$&  3.37&  3.22&  3.98&  3.22\\
$\sigma_{\Delta m}/\sigma_{\Delta y}$&  3.60&  3.50&  3.84&  3.76\\
$\sigma_{tb/y}$&  2.34&  2.12&  3.80&  1.25\\
$\sigma_{ca/y}$&  2.16&  2.06&  3.08&  1.39\\
\multicolumn{5}{l}{\underline{Correlations with $\Delta y$}}\\
$\Delta y$&   1.00&   1.00&   1.00&   1.00\\
$\Delta c$&   0.60&   0.54&   0.64&   0.79\\
$g/y$&  -0.10&  -0.02&  -0.18&  -0.32\\
$\Delta i$&   0.64&   0.59&   0.66&   0.83\\
$\Delta x$&   0.21&   0.18&   0.15&   0.42\\
$\Delta m$&   0.33&   0.26&   0.40&   0.57\\
$tb/y$&  -0.10&  -0.08&  -0.20&  -0.07\\
$ca/y$&  -0.07&  -0.06&  -0.12&  -0.07\\
\multicolumn{5}{l}{\underline{Serial Correlations}}\\
$\Delta y$&  0.29&   0.28&   0.29&   0.32\\
$\Delta c$&  0.02&  -0.03&   0.02&   0.27\\
$\Delta g$&  0.18&   0.14&   0.11&   0.48\\
$\Delta i$&  0.01&  -0.01&   0.03&   0.08\\
$\Delta x$&  0.07&   0.08&  -0.00&   0.10\\
$\Delta m$&  0.04&   0.08&  -0.02&  -0.04\\
$tb/y$&  0.61&   0.59&   0.62&   0.69\\
$ca/y$&  0.57&   0.55&   0.52&   0.71\\
\hline\hline
\end{tabular}
}
\end{minipage}
\begin{minipage}{3in}
{\bf First-differenced Business Cycles}

{\small Note. The variables  $\Delta y$, $\Delta c$, $\Delta g$, $\Delta i$, $\Delta x$,   and $\Delta m$  denote, respectively the log differences of output, consumption, government consumption, investment, exports, and imports. The  variables $g/y$,  $tb/y$, and $ca/y$ are quadratically detrended in levels.  All variables are expressed in percent.}
%produced with: c:\data\uribe\teaching\econ366\wdi\table4.m
\end{minipage}


\end{slide}


\begin{slide}
{\bf \large Summary of Comparison}

\begin{itemize}
\item QT,  HP, and first-differencing  result in largely the same business cycle facts!

\item In particular, Facts 8 and 9 continue to hold: Emerging countries are about twice as volatile as rich countries, and emerging countries have a larger relative consumption volatility. 
\end{itemize}


\end{slide}


\begin{slide}
{\bf Business Cycle Facts with Quarterly Data}

Main problem of quarterly data: not many long time series.

Include countries that have quarterly data on $y$, $c$, $g$ $i$, $x$, and $m$ for at least 30 years.  This requirement reduces  the number of countries from 120 to {\bf 28}! 

poor countries: 0 (out of 40)\\
emerging countries: 11 (out of 58)\\
rich countries: 17 (out of 22)

The sample period is 1980Q1 to 2012Q4.  
The data is online in file \verb7usg_data_quarterly.xls7 on the book's Web site.  We collected this data  from  national statistical agencies, the OECD, IFS, and Eurostat. (It would be great if in the future some international organization, like the IMF, would provide a quarterly data base for more countries, especially, poor and emerging ones that goes back to the 1980s or even further!)

\end{slide}


\begin{slide}
 
\centerline{\small \bf Business Cycles in Emerging and Rich Countries, Quarterly Data, 1980Q1-2012Q4}

\bigskip

\centerline{\tiny
\begin{tabular}{l*{9}{c}}
\hline\hline
&\multicolumn{3}{c}{Log-Quadratic Time Trend}&\multicolumn{3}{c}{HP Filter}&\multicolumn{3}{c}{First Differences}\\
\cline{2-10}
Statistic&All&Emerging&Rich&All&Emerging&Rich&All&Emerging&Rich\\
%Statistic& Countries& Countries& Countries& Countries& Countries& Countries& Countries& Countries& Countries\\
\hline
\multicolumn{5}{l}{\underline{Standard  Deviations}}\\
$\sigma_y$&  3.26&  \fbox{4.27}&  \fbox{2.74}&  1.80&  \fbox{2.60}&  \fbox{1.38}&  1.12&  \fbox{1.70}&  \fbox{0.81}\\
$\sigma_c/\sigma_y$&  0.99&  \fbox{1.23}&  \fbox{0.87}&  1.01&  \fbox{1.32}&  \fbox{0.85}&  1.18&  \fbox{1.48}&  \fbox{1.03}\\
$\sigma_g/\sigma_y$&  1.46&  2.07&  1.15&  1.30&  2.02&  0.93&  2.07&  3.33&  1.41\\
$\sigma_i/\sigma_y$&  3.44&  3.67&  3.31&  3.73&  3.88&  3.65&  4.32&  4.95&  3.99\\
$\sigma_x/\sigma_y$&  3.77&  3.97&  3.67&  4.01&  3.80&  4.11&  4.38&  4.65&  4.25\\
$\sigma_m/\sigma_y$&  3.52&  3.55&  3.51&  4.44&  3.65&  4.84&  4.60&  4.26&  4.77\\
$\sigma_{tb/y}$&  1.80&  2.93&  1.21&  1.09&  1.95&  0.64&  1.80&  2.93&  1.21\\
\multicolumn{5}{l}{\underline{Correlations with $y$}}\\
$y$&   1.00&   1.00&   1.00&   1.00&   1.00&   1.00&   1.00&   1.00&   1.00\\
$c$&   0.83&   0.72&   0.88&   0.78&   0.78&   0.78&   0.61&   0.62&   0.61\\
$g/y$&  -0.43&  \fbox{-0.11}& \fbox{ -0.59}&  -0.58& \fbox{-0.22}&  \fbox{-0.78}&  -0.16&  -0.17&  -0.15\\
$i$&   0.86&   0.82&   0.88&   0.84&   0.77&   0.87&   0.65&   0.57&   0.70\\
$x$&   0.17&  -0.00&   0.26&   0.43&  -0.05&   0.67&   0.33&   0.04&   0.48\\
$m$&   0.60&   0.48&   0.66&   0.68&   0.52&   0.76&   0.44&   0.37&   0.47\\
$tb/y$&  {-0.44}& \fbox{ -0.52}& \fbox{ -0.41}&{ -0.39}& \fbox{ -0.56}& \fbox{ -0.31}&  -0.02&  -0.11&   0.02\\
$tb$&  -0.44&  -0.51&  -0.40&  -0.39&  -0.56&  -0.31&     &     &     \\
\hline
\hline
\end{tabular}
}

{\tiny Notes: Moments are averaged across countries using population weights. 
Rich Countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Hong Kong, Italy, Japan, Netherlands, Norway, Sweden, Switzerland, United Kingdom,  United States. Emerging Countries: Argentina,  Israel, South Korea, Mexico, New Zealand, Peru, Portugal, South Africa, Spain, Turkey,  and Uruguay.
%produced with: c:\data\uribe\book\empirics\quarterly\table_q.m 

} 
\end{slide}

\begin{slide}
Observations on the table

In quarterly data, as in annual data, 

$\bullet$  Business cycles in emerging  countries  are about twice as volatile as business cycles in rich countries. (Fact 8) 

$\bullet$  There is less consumption smoothing in emerging countries than  in rich countries. (Fact 9) 

$\bullet$  The countercyclicality of government spending increases with income. (Fact 10)  

$\bullet$ The trade balance is negatively correlated with output. (Fact 5)
\end{slide}

\begin{slide}

\centerline{\bf  \small Summary of Chapter 1:}



\centerline{\bf \small  10 Business-Cyle Facts Around the World} 

{\tiny 

{\bf Fact 1: [High Global Volatility]}
The cross-country average standard deviation of  output 
is about twice as large as its U.S.~counterpart. 
\\
{\bf Fact 2: [Excess Consumption Volatility]} On average across countries, private  consumption including durables  is  more  volatile than output.
\\
{\bf Fact 3: [Global Ranking of Volatilities]}
The ranking of cross-country average standard deviations from top to bottom is imports, investment, exports, government spending, consumption, and output. 
\\
{\bf Fact 4: [Procyclicality of the Components of Aggregate Demand] }
On average across countries, consumption, investment, exports, and imports are  positively correlated with output. 
\\
{\bf Fact 5: [Countercyclicality of the Trade Balance and the Current Account]}  On average across countries, the trade balance,  trade-balance-to-output ratio,  current account, and  current-account-to-output ratio are  negatively correlated with output. 
\\
{\bf Fact 6:  [Acyclicality of the Share of Government Consumption in GDP]} On average across countries, the share of government consumption in output  is roughly uncorrelated with output. 
\\
{\bf Fact 7: [Persistence]}
The components of aggregate supply (output and imports) and aggregate demand (consumption, government spending, investment, and exports) are all positively serially correlated.
\\
{\bf \color{red} Fact 8: [Excess Volatility of Poor and Emerging Countries]}
Business cycles in emerging or poor countries  are about twice as volatile as business cycles in rich countries. 
\\
{\bf \color{red} Fact 9: [Excess Consumption Volatility  in Poor and Emerging Countries]}
The relative consumption volatility is higher in poor and emerging countries than in rich countries. 
\\
{\bf \color{red} Fact 10: 
[The Countercyclicality of Government Spending Increases with Income]}
The share of government consumption is countercyclical in rich countries, but acyclical in emerging and poor countries. 



}
\end{slide}


\end{document}