2.3 Real business cycle fluctuations
2.3.3 Matching time series
Simulation results have shown that the RBC model solves the Shimer puzzle but does not solve the equity premium puzzle in simulations. This section studies whether the model can match the time series of key macroeconomic variables. This section matches the unemployment time series to estimate a process for TFP from the model’s policy function; using this TFP series as an
7See Hairault et al. (2010) and Den Haan et al. (2020).
input, I simulate the model and compare key macroeconomic variables to their empirical counterparts.
In applications like the canonical New-Keynesian model, researchers apply a Kalman filter to estimate parameters of a linear model and match time series. However, this model is solved globally, which renders the Kalman filter inapplicable.8 Schorfheide et al. (2018) demonstrate how to use a non-linear filter to estimate a higher order approximation of the standard Bansal-Yaron model. Compared to the models outlined in this paper, the standard
Bansal-Yaron model is much simpler and can be solved with perturbation.
A global solution is too time intensive and thus a non-linear filter is not an economical choice.
Still, it is possible to match unemployment and a second time series with the parametrized model: I feed a starting point for employment and stock prices in April 1929 into the model. The monthly unemployment series allows to estimate a time series for productivity, zt, such that the model’s policy function maps into the unemployment time series. Then, the output time series (linearly interpolated to monthly frequency) allows us to estimate theiid growth innovationsa,t. The orthogonality ofa,t to unemployment is the key to estimate the two series separately: oncea,t is revealed it contains no information about future changes of productivity.
The investment decision of the forward-looking firm is not affected; so investment, and therefore unemployment, must be completely driven by zt. See Appendix 2.C for details. By construction, the model-generated output and unemployment series should match the empirical time series.
At times, the policy function does not allow us to reach a high employment rate, leading to minor discrepancies between model-generated and empirical series.
Figures 2.2 - 2.4 compare matched time series of the RBC model against data. The model is parametrized to post-war data, but the longer time frame used here allows us to check how a endogenous disaster model performs in a true economic disaster - the Great Depression. In the title of each panel,ρ denotes the correlation coefficient of the annualized model-generated and
8Petrosky-Nadeau and Zhang (2017) and Fern´andez-Villaverde and Levintal (2018) show that the perturbation solution is a bad approximation of the global policy function in a rare disaster model.
empirical time series.
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -15
-10 -5 0 5 10 15 20
rateinpercent
GDP Growth (annual);=0.97223
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -15
-10 -5 0 5 10 15 20
rateinpercent
Consumption Growth (annual);=0.41826
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -0.15
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
pct.deviationsfromHP--lteredtrend
Total Factor Productivity (quarterly)
Model estimation Fernald ; =0.79154 Fernald (utility adjusted) ; =0.28915
Figure 2.2:Matched time series of the RBC model: output, consumption and productivity. Grey bands denote NBER recessions. ρdenotes the correlation coefficient of annualized simulated and empirical data. HP-filter smoothing parameters:λ= 129,600 for monthly andλ= 100,000 for quarterly data.
Consistent with simulation results, the RBC model matches post-war consumption, but fails to match the consumption series before the war.
Unsurprisingly, the model fails to reproduce the low-consumption, high-output, and low-unemployment war years 1941-1945. The last panel of Figure 2.2 shows that my estimated TFP series is more volatile than estimates by Fernald (2014). The panel also illustrates that, to match the pre-1947 unemployment series, we need to assume large fluctuations of TFP.
Figure 2.3 shows the matched labour market series. Model-generated separations are slightly too smooth, but job-finding rates are matched well.
Figure 2.3 reveals the appeal of endogenous separations when firms face a non-negativity constraint on vacancies. Endogenous separations allow us to match wide fluctuations of the unemployment rate, as exemplified during depressions or the war years. The minimum of employment growth
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 0
5 10 15 20 25 30
rateinpercent
Unemployment (monthly),;=0.99945
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 1
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
rateinpercent
Separation rate (quarterly),;=0.80442
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 0
5 10 15 20 25 30 35 40 45 50
rateinpercent
Job-nding (quarterly),;=0.88992
RBC model data
Figure 2.3:Matched time series of the RBC model: labour market. Grey bands denote NBER recessions.ρdenotes the correlation coefficient of annualized simulated and empirical data.
is min [loglt+1−loglt] =−πeu,t. If separations are fixed, employment cannot fall as quickly as it does during the Great Depression. In the opposite direction, employment can, at most, grow at rate
max [loglt+1−loglt] = 1−lt
lt πue,t−πeu,t.
For a low unemployment rate, the term1−ll is very small. Under fixed separa-tions, employment can only grow rapidly if the job-finding rate is excessive.
This problem can be observed in the war years. The employment rate is already at 97% in late 1942 and rises to 99% in 1945. In an exogenous separation model, positive employment growth at a 97% employment rate demands an excessively high job-finding rate. Additionally, under exoge-nous separations, the burden of creating fluctuations in unemployment lies completely on the job-finding rate, which must fluctuate too strongly.
Turning to Figure 2.4, dividends in the first panel are almost constant,
whereas empirical dividends fluctuate strongly pro-cyclically. Model div-idends are the residual of output after wages and investment costs. In a boom, firms generate higher profits but households increase savings. In a recession, the reverse holds. These net transfers from firms to households are very stable compared to empirical dividends, which are true profit shares.
Model predicted wages are less volatile than their empirical counterpart, but the two series display a high degree of correlation.
Turning to asset prices, the model does not match the time series of the risk-free rate. Expected monthly consumption growth (and the value function) determines the risk-free rate entirely; a link that is clearly too strong. Adding New-Keynesian frictions and a Taylor rule would probably improve the model’s fit here. The model simulation neither matches the volatility of stock prices nor the mean, volatility or direction of equity returns. The RBC model has problems turning modest fluctuations of productivity, which are sufficient to match the unemployment rate and output growth, into strong fluctuations of equity returns and prices.
Finally, Figure 2.5 depicts the return predictability in the post-war U.S.
economy. Empirically, a below-average price-to-earnings ratio is a good predictor of higher future returns. To erase the effect of cyclical volatility, the figure plots the ratio of price to earnings over the past ten years (PE10) and the average future return in the following ten years. In the panel titles,ρ denotes the correlation of PE10 and future returns in the data and the model respectively. Empirically, the correlation coefficient is -0.65; the model-generated coefficient is about null, i.e. the model model-generated PE10 does not predict future equity returns. Quantitatively, the model does not reproduce returns or price-to-earnings ratios as volatile as in the data (see y-axes).
Concerning levels, the price-to-earnings ratio is much larger in the model than in the data, while the stock returns in the model are low compared to data averages. The former follows from the definition of model dividends as net transfers from firms to households. The latter follows from the model’s difficulties solving the equity premium puzzle. Interestingly, the model underestimates returns in the 2010s. Recall that I match output growth and the unemployment rate. Output growth has been modest compared to returns. In the 2010s, the unemployment rate fell to record lows, but at 3-4 percent, has almost no room to decrease further. Hence, the model fails to
match large returns in 2010s. A labour-leisure decision could account for employment growth beyond the unemployment rate and improve data fit.
In summary, the RBC model is effective in matching post-war output, consumption and labour market data. Endogenous separations improve data fit, especially in volatile periods like the Great depression. However, the model’s ability to match asset pricing data is miserable. Petrosky-Nadeau et al. (2018) claim that a very similar model solves the equity premium puzzle via endogenously generated disasters. The next section examines their model.
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -100
-50 0 50 100 150 200
unit deviations from linear trend x100
Dividend (monthly),;=0.032549
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -10
-5 0 5 10 15
pct.deviationsfromHP--lteredtrend
Wages (quarterly),;=0.80725
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -15
-10 -5 0 5 10 15 20
rateinpercent
Risk-free rate (annualized);=-0.1737
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -40
-20 0 20 40 60 80
rateinpercent
Stock return (annual);=0.053332
RBC model data
1927 1937 1947 1957 1967 1977 1987 1997 2007 2017 -100
-80 -60 -40 -20 0 20 40 60
pct.deviationsfromHP--lteredtrend
Stock price (monthly);=0.44557
RBC model data
Figure 2.4:Matched time series of the RBC model: asset prices. Grey bands denote NBER recessions.ρdenotes the correlation coefficient of annualized simulated and empirical data. HP-filter smoothing parameters:λ= 129,600 for monthly andλ= 100,000 for quarterly data.
1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 2004 -20
-10 0 10 20
future 10y return average log PE10 (right scale)
-1 -0.5 0 0.5 1
Return predictability: Data;=-0.65611
1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 2004 -5
0 5
-0.2 0 0.2
Return predictability: Model generated;=-0.0013432
Figure 2.5:Matched time series of the RBC model: return predictability. The price earnings ratio (PE10) is the ratio of the current price to average earnings over the past ten years (also called Cyclically Adjusted PE Ratio (CAPE) or Shiller PE Ratio). I compute log PE10 as logD+1P to avoid explosive PE10 for close to zero dividends. Future 10 year returns are the geometrical average of annual returns over the next ten years. The figure shows absolute deviations from mean. Here,ρdenotes the correlation coefficient of PE10 and CAPE in both panels.