Christiano, Motto, and Rostagno (2014) (Risk shocks)

Model

The model in this paper is a mashup of the DSGE framework of Cristiano, Eichenbaum, and Evans and the entrepreneur and financial friction component of Bernanke, Gertler, and Gilchrist. To this framework they add an additional feature they term a “risk shock”, which is stochastic volatility in the distribution of entrepreneur productivity.

The model has too many pieces for me to describe in detail, so I will attempt to focus on the parts that are most important for understanding the key takeaway of the paper.

Firms:

• Representative, competitive final goods producer that aggregates intermediate outputs using a CES technology. The CES parameter in this technology is stochastic.
• Each intermediate good is produced by a monopolist with Cobb Douglass technology in capital and labor. Monopolists face a Calvo-style pricing friction. They also face stochastic fixed costs that vary over time but are common for all intermediate good producers.

Labor markets:

• A representative labor contractor aggregates a continuum of differentiated labor services using a CES technology with a constant parameter. They rent labor services to intermediate goods producers at a wage W
• For each type of differentiated labor, there is a monopoly union that determines the wage rate for that labor type, subject to Calvo style fractions.

Households:

• Large number of identical and competitive households (i.e. a representative household)
• Each household contains every type of differentiated labor and has a large number of entrepreneurs (will talk about these separately)
• Households own physical capital, and rent it to entrepreneurs, who in turn rent it to intermediate firms each period. Household’s face quadratic adjustments costs to capital
• The representative household has log preferences over consumption and a linear disutility from aggregate labor supplied by the household
• Households have access to a one period bond as well as a ten year bond (parameterization is quarterly, so ten period bond is 40 periods.)
• Finally households own firms, so they receive profits from all firms as a lump sum adjustment in their budget constraint.

Entrepreneurs:

• Within each household there is a large number of entrepreneurs.
• Entrepreneurs enter each period with a net worth. They combine this with a loan from a mutual fund to purchase capital from households in a competitive market
• Entrepreneurs then rent out omega times this value to firms, where omega is log-normally distributed with mean 1 and a stochastic variance. This is the key feature of their model. They call this stochastic volatility in entrepreneur efficiency the risk shock.
• If omega is below some cutoff, then entrepreneurs don’t rent their capital and declare bankruptcy. The mutual fund pays a monitoring cost and then claims all the entrepreneur’s assets

Other features:

• There are 12 aggregate shocks in the model
• Each of these is modeled as an AR(1) in log-deviations from the steady state value
• The innovation in this AR(1) process is the sum of 9 terms: one un-anticipated shock and the sum of news shocks from the previous 8 periods.

Note: I’ve skipped over main details needed to formally close the model

Data and estimation

The authors approximate a solution and do Bayesean estimation using Dynare.

The use quarterly from 1985 to 2010 and include 8 “standard” macro variables:

1. GDP
2. consumption
3. investment
4. inflation
5. the real wage
6. the relative price of investment goods
7. hours worked,
8. the federal funds rate.

In addition, they also use four financial variables:

1. Credit to non financial firms (flow of funds)
2. Difference between 10 year US government bond and the federal funds rate
3. Dow Jones Wilshire 5000 index (proxy for aggregate value of entrepreneur net worth)
4. Difference between interest rate on BAA-rated corporate bonds and the 10 year US government bond

Results

The main result from their empirical analysis is that when

• data for financial variables is included
• prices and wages are subject to Calvo-style frictions

the risk shock is the most significant shock driving business cycles. Why these two things? First let’s understand the mechanism by which risk shocks generate business cycles. Consider a rise in the risk shock, then

• Probability of a low omega increases
• banks raise interest rates to to cover resulting costs
• Entrepreneurs then borrow less, so credit falls and entrepreneurs purchase less capital from households
• This lowers investment, causing output and consumption, and the price of capital to fall
• Lower price of capital causes the net worth of entrepreneurs to fall, which magnifies the entire effect

Now, with that in mind, if financial data were left out, then the link between lower entrepreneur credit and investment in capital may not be fully captured by the data.

Furthermore if prices were fully flexible (no Calvo-frictions), then the decline in investment (in other words a lower demand for current goods) would immediately drive prices down and cause consumption to increase. The business cycle data show that consumption and investment move together (in contrast to what I just said would happen in a flexible price version of this model), which would cause the estimation to downplay the importance of the risk shock.

This is the main contribution of the paper. They have some discussions on why the risk shock has a larger impact on business cycles than the other shocks in the model as well as a talk about robustness; but we won’t talk about that now.