Calibration, Estimation, and Effects of Technology Shocks - PowerPoint PPT Presentation

Calibration, Estimation, and Effects of Technology Shocks Jose-Victor Rios Rull Frank Schorfheide Penn Grad Students University of Pennsylvania June 8, 2007

A Question A Model Empirical Analysis A Question • What fraction of the variation in output and hours worked is due to technology shocks? • This is a long-standing question in business cycle research, see, for instance, Kydland and Prescott (1982) and Fisher (2006). • We’ll focus on hours worked. Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Households • There is a continuum of households solving the following problem � ∞ � �� � ln C t − B H 1+1 /ν t β t max I E 0 (1) 1 + 1 /ν t =0 C t + P k t X t = W t H t + R k t P k s.t. (2) t K t K t +1 = (1 − δ ) K t + X t (3) • C t is consumption, H t is hours worked, X t is investment (physical units), P k t is the price of the unit of the investment good (using the consumption good as numeraire), W t is the wage, and R k t the rental rate of capital. Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Households • Labor supply � 1 � ν W t H t = B t C t • Euler Equation � P k �� � t +1 / C t +1 (1 − δ ) + R k 1 = β I E t t +1 P t / C t Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Firms • Firms rent capital and labor services from Households and produce consumption and investment goods. • Technology: C t + X t = A t K α t H 1 − α t V t • Profits: Π t = C t + P k t X t − W t H t − R k t P k t K t • For the firms to be willing to produce both consumption and investment goods it has to be the case that P k t = 1 / V t . • The optimal choice of capital and labor implies W t = (1 − α ) A t K α t H − α R k t P k t = α A t K α − 1 H 1 − α , t t t Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis NIPA and Exogenous Processes • NIPA: investment is measured as I t = X t P k t . Hence, Y t = C t + I t = A t K α t H 1 − α . t • Neutral technological shocks A t = exp { γ a + � a t } A t − 1 , � a t = ρ a � a t − 1 + σ a ǫ a , t • Investment-specific technology shocks V t = exp { γ v + � v t } V t − 1 , � v t = ρ v � v t − 1 + σ v ǫ v , t • To estimate the model, we make the preference shock time-varying ln( B t / B ) = ρ b ln( B t − 1 / B ) + σ b ǫ b , t . Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Equilibrium Conditions • Endogenous variables: Y t , C t , I t , K t +1 , W t , R k t , H t . • The endogenous variables have to satisfy the following set of (nonlinear) rational expectations equations � 1 � ν W t H t = B t C t � �� � C t V t (1 − δ ) + R k 1 = β I E t t +1 C t +1 V t +1 = (1 − δ ) K t + I t V t K t +1 A t K α t H 1 − α = Y t t = C t + I t Y t = (1 − α ) Y t / H t W t R k α Y t / ( P k = t K t ) t Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Detrending • Along a balanced growth path the following variables are stationary Y t C t I t K t +1 , W t R k , , , , t , H t . Q t Q t Q t Q t V t Q t • where 1 α 1 − α 1 − α Q t = A V . t t • We denote a detrended version of X t by ˆ X t . Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Equilibrium Conditions • The (detrended) endogenous variables have to satisfy the following set of (nonlinear) rational expectations equations � � ν ˆ 1 W t H t = ˆ B t C t � � ˆ � � C t Q t V t (1 − δ ) + R k 1 = β I E t t +1 ˆ C t +1 Q t +1 V t +1 Q t − 1 V t − 1 ˆ (1 − δ ) ˆ + ˆ K t +1 = K t I t Q t V t � Q t − 1 V t − 1 � α ˆ ˆ Y t = ˆ ˆ C t + ˆ K α H 1 − α Y t = , I t t t Q t V t ˆ Y t Q t V t ˆ (1 − α ) ˆ R k = Y t / H t , t = α W t ˆ Q t − 1 V t − 1 K t Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Equilibrium Conditions • Recall that � � 1 α 1 − α ( γ a + � 1 − α ( γ v + � Q t / Q t − 1 = exp a t ) + v t ) , exp { γ v + � v t } V t / V t − 1 = • Define q t = Q t / Q t − 1 and v t = V t / V t − 1 . Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Equilibrium Conditions � � ν ˆ 1 W t 1 K t +1 = (1 − δ ) ˆ ˆ + ˆ H t = , K t I t ˆ B t q t v t C t � � ˆ � � C t (1 − δ ) + R k 1 = β I E t t +1 ˆ C t +1 q t +1 v t +1 � 1 � α ˆ K α ˆ H 1 − α Y t = ˆ ˆ C t + ˆ = , Y t I t t t q t v t ˆ Y t ˆ (1 − α ) ˆ R k W t = Y t / H t , t = α q t v t ˆ K t � � 1 α q t = exp 1 − α ( γ a + � a t ) + 1 − α ( γ v + � v t ) v t = exp { γ v + � v t } Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Solving the Model • We can now calculate a steady state (in terms of the detrended variables), log-linearize the equilibrium conditions around the steady state, and apply a solution technique to solve the system of linear rational expectations difference equations. • We show subsequently the relevant steady state ratios and log-linearized equations. Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Steady States e ( γ a + γ v ) / (1 − α ) R ∗ − 1 + δ = β α e ( γ a + γ v ) / (1 − α ) K ∗ = Y ∗ R ∗ � � I ∗ K ∗ 1 − (1 − δ ) e − ( γ a + γ v ) / (1 − α ) = Y ∗ Y ∗ I ∗ 1 − (1 − δ ) e − ( γ a + γ v ) / (1 − α ) = K ∗ Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Log-linearizations � ν ( � W t − � C t − � H t = B t ) � � + I ∗ � � K ∗ � (1 − δ ) e − ( γ a + γ v ) / (1 − α ) K t +1 = K t − � q t − � v t I t � � R ∗ C t − � � 1 − δ + R ∗ � 0 = I E t C t +1 − ( � q t +1 + � v t +1 ) + R t +1 � α � K t + (1 − α ) � = H t − α [ � q t + � v t ] Y t � � 1 − I ∗ C t + I ∗ � � Y ∗ � Y t = I t Y ∗ � Y t − � � W t = H t � Y t − � � R k K t + � q t + � = v t t Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

A Question A Model Empirical Analysis Log-linearizations 1 α � = 1 − α � a t + 1 − α � q t v t � = ρ a � a t − 1 + ǫ a , t a t � = ρ v � v t − 1 + ǫ v , t v t � ρ b � B t = B t − 1 + ǫ b , t Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

Data A Question Calibration A Model DSGE Model Estimation Empirical Analysis VAR Analysis Deterministic Trends Answering the Question: Three Approaches • A Calibration • Bayesian estimation of the DSGE model • A structural VAR, loosely based on the model Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

Data A Question Calibration A Model DSGE Model Estimation Empirical Analysis VAR Analysis Deterministic Trends Data (To be Updated) • Sample Range: 1955:Q1 to 2004:Q4. • Relative Price of Investment: (1 / P k t ); Source: Fisher’s (2006) interpolation of Violante’s series (Equipment and Structures). • Labor Share: computed as in Cooley and Prescott (1995) • Population (for conversion into per capita terms): total civilian noninstitutional (thousands, NSA); Source: DRI-Global Insight. • Hours: Aggregate Hours Index (ID PRS85006033); Source: Bureau of Labor Statistics. Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks

Data A Question Calibration A Model DSGE Model Estimation Empirical Analysis VAR Analysis Deterministic Trends Data (To be Updated) • According to our model Y t = C t + I t = C t + X t P k t where output, investment (and consumption) are measured in terms of consumption goods. • In the data, we start from nominal output, consumption, and investment. Roughly: GDP nom = C nom + I nom + G nom + NetEX nom • We have to take a stand on what to do with G nom and NX nom . How about: treating NX nom as investment, splitting G nom (attributing government expenditures on investment goods to investment and the remainder to consumption). What should we do with consumer durables? Rios-Rull, Schorfheide, Grad Students Calibration, Estimation, and Effects of Technology Shocks