Macroeconomics - Data & Theory

Transcription

1 Macroeconomics - Data & Theory Wouter J. Den Haan University of Amsterdam October 24, 2009

2 How to isolate the business cycle component? Easy way is to use the Hodrick-Prescott (HP) lter min fx τ,t g T t=1 T 1 t=2 (x t x τ,t ) 2 + λ Filtered series is then de ned as T 1 n[(x τ,t+1 x τ,t ) (x τ,t x τ,t 1 )] 2o t=2 x hp,t = x t When λ = 1, 600, then the lter turns out to be similar to a frequency domain lter that takes out all frequencies associated with cycles that have a period that exceeds 32 quarters. x τ,t

3 How to isolate the business cycle component in Matlab? Suppose "data" is a T n matrix with observations for n variables HP-trend is given by datatrend = hp lter(data,1600); Business cycle component is given by databc = data-datatrend;

4 Key stylized facts (1951Q1-2005Q4) x t (all in real terms) σ x Cor(x t 1,y t ) Cor(x t,y t ) Cor(x t+1,y t ) ln(gdp)=y 1.57% ln(non-dur. cons.) 1.09% ln(dur. cons.) 4.86% ln(priv. invest.) 4.89% ln(priv. n-r. invest.) 4.83% ln(priv. r. invest.) 9.48% ln(wage rate) 0.87% ln(gdp/person) 1.31% ln(gdp/hour) 1.04% ln(tfp) Q1-2002Q2 0.93% unemploy. rate 0.72% employment rate 0.58% participation rate 0.24% inventory/gdp 0.41% investment/gdp 0.46%

5 In words Output more volatile than TFP This measure of TFP not adjusted for capacity utilization, thus... GDP/hour less volatile than GDP/person, why? Residential investment very volatile Unemployment rate = #unemployed/labor force Employment rate = #employed/population (>16yr) 6= 1-unemp rate One small component is key for business cycle uctuations Wages more cyclical if corrected for composition e ects high autocorrelation coe cients

6 Correlation of GDP and TFP has declined Filtered GDP and TFP GDP TFP

7 Correlation GDP & TFP from x value till 02Q

8 Same for GDP & GDP/Hours Filtered GDP and GDP/Hour GDP GDP/Hour

9 Same for GDP & GDP/Hours

10 Standard RBC model with leisure i max fct+j,k t+1+j,h t+j g E h j=0 βj u(c t+j, 1 h t+j )ji t j=0 s.t. c t+j + k t+1+j w t+j h t+j + r k t+j k t+j + (1 k t+1+j 0 k t predetermined δ)k t+j

11 First-order conditions Households u(c t,1 h t ) u(c t,1 h t ) c t c t w t = u(c t,l t ) l t l t = 1 h h t = E t β u(c t+1,1 h t ) c (1 δ t+1 + r k t+1 i

12 Firm problem & rst-order conditions max fht,k t g θ t kt αh1 t w t h t r k t k t w t = (1 α)θ t kt αh t α r k t = αθ tkt α 1 h 1 t α α

13 What if wages are acyclical? Suppose u c "=) u l " Suppose u c #=) u l # u(c t, l t ) c t w t = u(c t, l t ) l t Thus if consumption is procyclical then hours are countercyclical Thus if consumption is procyclical then hours are countercyclical

14 Special case with analytical solution Log utility: u(c t, l t ) = ln(c t ) + B ln(l t ) Complete depreciation: δ = 1 First-order conditions: " 1 = E t β 1 αθ t+1 c t c t+1 w t c t = B 1 h 1 kt+1 h t+1 α 1 #

15 Solution to rst-order conditions Similar to model in notes without leisure c t = (1 αβ)y t = (1 αβ)k α t h 1 α t =) c t y t = (1 αβ) Plug this into rst-order condition for labor/leisure (1 α)y t /h t (1 αβ)y t = h t = h = B 1 h t =) 1 α B(1 αβ) + (1 α) Thus the savings rate and hours decision are constant

16 Failures of the special case Savings rate and hours decision are constant Wage rate is too volatile and too procyclical ln w t = ln(1 α) + ln y t ln h Consumption is too volatile and too procyclical ln c t = ln(1 αβ) + ln y t Investment is not volatile enough ln i t = αβ + ln y t Observations are related! More volatile investment requires procyclical savings rate

17 What about propagation? Data: TFP has less persistence than output For example, ρ( θ t, θ t 1 ) = 0.11 and ρ( y t, y t 1 ) = (1951Q1-2002Q2) Does the model endogenously generate more persistence? For capital? For output? ln k t+1 = constant + ln θ t + α ln k t ln y t = constant + ln θ t + α ln k t = constant + ln θ t + α ln y t 1 Propagation for capital survives for realistic depreciation, but result for output does not.

18 Shocks versus the model Policy rule in DSGE model: z t+1 = a 0 + A 1 z t + A 2 shocks t How much can the contemporaneous values of the shocks explain by themselves? That is, how bad is this SGE model: z t+1 = ã 0 + Ã 2 shocks t

19 Output & current productivity shock 0.08 Actual & fitted output actual fitted: no lags

20 Hours & current productivity shock 1.44 Actual & fitted hours actual fitted: no lags

21 Consumption & current productivity shock 0.14 Actual & fitted consumption actual fitted: no lags

22 Investment & current productivity shock 0.06 Actual & fitted investment actual fitted: no lags

23 Capital & current productivity shock 2.24 Actual & fitted capital actual fitted: no lags

24 Adding 20 lagged values of the shock 2.25 Actual & fitted capital actual fitted: no lags 2.1 fitted: 20 lags

25 Adding 40 lagged values of shock 2.25 Actual & fitted capital actual fitted: no lags fitted: 20 lags fitted: 40 lags