Transitory and trend shocks to productivity.

Transcription

1 Aguiar and Gopinath (JPE 2007) Stochastic Growth Model Single-good, single-asset SOE model Transitory and trend shocks to productivity. Technology: Cobb-Douglas production function, capital, K t, and labor, L t, Y t = e zt K 1 α t (Γ t L t ) α, α (0, 1) represents labor s share of output.

2 Aguiar and Gopinath JPE 2007 Parameters z t and Γ t represent productivity processes. z t = ρ z z t 1 + ε z t (1) ρ z < 1, and ε z t N(0, σz 2 ). i.i.d draws. Γ t = e gt Γ t 1 = Γ 0 t s=1 e gs g t = (1 ρ g )µ g + ρ g g t 1 + ε g t, ρ g < 1 and ε g t N(0, σg 2 ) µ g :Productivity s long-run mean growth rate.

3 Aguiar and Gopinath JPE 2007 Period utility is Cobb-Douglas, (C γ t (1 L t ) 1 γ) 1 σ where 0 < γ < 1. u t = 1 σ Per-period resource constraint: C t +K t+1 = Y t +(1 δ)k t φ 2 One-period, risk-free bonds. ( ) 2 Kt+1 e µg K t B t +q t B t+1. K t (2)

4 (Closing the model) The price of debt is sensitive to the level of outstanding debt, [ ] 1 = 1 + r t = 1 + r + ψ e B t+1 Γt b 1, (3) q t r is the world interest rate b represents the steady-state level of normalized debt ψ > 0 governs the elasticity of the interest rate to changes in indebtedness.

5 Endogenous state variables: K, B Exogenous state: z, Γ Endogenous variables: C, K, B, L Representative agent s problem can be stated recursively: (C γ t (1 L t ) 1 γ) 1 σ V (K t, B t, z t, Γ t ) = max {C t,l t,k t+1,b t+1 } 1 σ + C t +K t+1 = Y t +(1 δ)k t φ 2 X t = K t+1 (1 δ)k t + φ 2 βe (V (K t+1, B t+1, z t+1, Γ t+1 )) ( ) 2 Kt+1 e µg K t B t +q t B t+1. K t ( ) 2 Kt+1 e µg K t K t (4)

6 Y, C, K, B are all non-stationary because of the stochastic trend Γ. Use hat to denote each variables de-trended counterpart: In normalized form: u t = Γ γ(σ 1) x t s.t. Ĉ t +e gt ˆK t = Ŷ t +(1 δ) ˆK t φ 2 x t Γ t 1. (Ĉ γ t (1 L t ) 1 γ) 1 σ 1 σ ( e gt ˆK t+1 ˆK t e µg ) 2 ˆK t ˆB t +e gt q t ˆB e gt ˆK t+1 = (1 δ) ˆK t + ˆX t φ 2 ( ) 2 ˆK t+1 e ˆK gt e µg ˆK t. t

7 Normalized Bellman: (Ĉ γ t (1 L) 1 γ) 1 σ V ( ˆK t, ˆB t, z, g) = max + (5) {Ĉ,L, ˆK t+1, ˆB t+1 } 1 σ ( ) βe gγ(1 σ) E t V ( ˆK t+1, ˆB t+1, z t+1, g t+1 (6) Given an initial capital stock, ˆK 0, and debt level, ˆB 0, the equilibrium of the economy is characterized by the first-order conditions of the problem (5), the technology, budget constraint, evolution of capital and the transversality conditions.

8 Consumption Euler equation: B t+1 Û c e gt q t + βe gtγ(1 σ) E t ( V B ( ˆK t+1, ˆB t+1, z t+1, g t+1 ) = 0 B t : V B ( ˆK t, ˆB t, z, g) = Û c Û c,t = βegt(γ(1 σ) 1) E t Û c,t+1 q t For well-behaved consumption of the linearized model in the steady state we require β(1 + r ) = e µg (1 γ(1 σ)).

9 Solow Residual sr t = z t + αln(γ t ) Beveridge-Nelson Decomposition sr t = τ t + s t τ t = lim j sr t+j ( α τ t = αµ g + τ t ρ g is a random walk (with drift) and ( αρg s t = z t 1 ρ g is a stationary series. ) ε g t ) (g t µ g )

10 Solow Residual The random walk component of the Solow residual can be expressed as, σ τ 2 σ sr 2 = [( 2 1+ρ z ) α 2 σ 2 g (1 ρ g ) 2 σ 2 z + α2 σ 2 g (1 ρ 2 g ) ]. (7) Cochrane (1988) that lim K 1 Var(sr t sr t K ) = σ τ 2. (8) K The relative variance σ2 τ can be approximated empirically by σ SR 2 fixing K and calculating the sample variances of (sr t sr t K ) and sr. Given the (short) length of the time series, most estimations are inconclusive.

11 Structural Estimation Using GMM: Estimate θ = (σ g, σ z ), m(θ) = [σ(y), σ(c), cov(nx/y, y)]. Using the volatility of filtered consumption and income E[m 1 (θ) 2 y 2 ] = 0 E[m 2 (θ) 2 c 2 ] = 0 Using the volatility of filtered income and covariance of filtered income and (trade balance as a ratio of GDP) E[m 1 (θ) 2 y 2 ] = 0 E[m 3 (θ) 2 nx y y] = 0

12 Table 3: Benchmark Parameter Values Time preference rate β 0.98 Consumption Exponent (utility) γ 0.36 Steady-state debt to GDP b 10% Coefficient on interest rate premium ψ Labor Exponent (Production) α 0.68 Risk Aversion σ 2 Depreciation Rate δ 0.05 Capital Adjustment Cost φ 4.0 Notes: Benchmark parameters used in all specifications. Capital adjustment cost parameter is set at 4, except for specification (IV) of Table 4, where it is estimated.

13 Table 4: Estimated Parameters Mexico Canada Parameter: (I) (II) (III) (IV) (I) (II) (III) (IV) σ g (0.37) (0.56) (0.52) (0.29) (0.18) (0.32) (0.61) (0.37) σ z (0.27) (0.65) (0.22) (0.34) (0.09) (0.06) (0.08) (0.14) ρ g (0.10) (0.05) (0.54) (0.36) ρ z (0.09) (0.02) µ g (0.15) (0.13) φ (0.39) (0.45) Random Walk Component (0.07) (0.06) (0.05) (0.11) (0.07) (0.13) (0.29) (0.24) Moments Used σ y, σ c σ y, Cov(nx,y) σ y, σ c, Cov(c,y) All σ y, σ c σ y, Cov(nx,y) σ y, σ c, Cov(c,y) All Notes: GMM estimates with standard errors in parentheses. Moments Used refer to which empirical moments were matched during estimation. All refers to the following eleven moments: the standard deviations of income, consumption, investment, net exports, first-differenced (unfiltered) income; the covariances of income with lagged income, consumption, investment, and net exports; the autcovariance of first-differenced (unfiltered) income; and the mean of first-differenced (unfiltered) income. The Random Walk Component is calculated as in equation (14). Standard deviations are reported in percentage terms. All parameters not estimated in each specification were fixed at the benchmark values reported in Table 3. When not estimated, we set ρ g =0.01 and ρ z =0.95.

14 Table 5a: Moments for Emerging Market (Mexico) Data (I) (II) (III) σ (y) (0.35) (0.35) (0.31) (0.27) σ ( y) (0.25) (0.26) (0.22) (0.17) σ ( c) σ ( y) σ ( I ) σ ( y) σ ( nx) σ ( y) (0.08) (0.08) (0.03) (0.05) (0.29) (0.10) (0.04) (0.33) (0.09) (0.04) (0.02) (0.10) ρ (y) (0.07) (0.01) (0.002) (0.02) ρ ( y) (0.09) (0.02) (0.01) (0.08) ρ ( y, nx) (0.08) (0.10) (0.04) (0.06) ρ ( y, c) (0.02) (0.02) (0.01) (0.03) ρ ( y, I) (0.03) (0.003) (0.002) (0.03) Notes: Theoretical moments are calculated from the model using the parameters reported in Tables 3 and 4. Columns (I), (II), and (III), use the estimated parameters from Table 4, Columns (I), (II), and (IV), respectively. Standard errors reported in parentheses are calculated from the parameter standard errors reported in Table 4 using the Delta method. Data moments are estimated with GMM. Note that estimation of the autocorrelation of the growth rate of income reduces the sample size by two quarters. We drop the first two quarters for the other moments to maintain a constant number of observations per moment. This truncation implies that the empirical moments above may differ slightly from those reported in Table 2.

15 Table 5b: Moments for Developed Market (Canada) Data (I) (II) (III) σ (y) (0.20) (0.20) (0.20) (0.11) σ ( y) (0.09) (0.14) (0.15) (0.09) σ ( c) σ ( y) σ ( I ) σ ( y) σ ( nx) σ ( y) (0.05) (0.05) (0.05) (0.07) (0.25) (0.05) (0.05) (0.23) (0.09) (0.03) (0.03) (0.11) ρ (y) (0.04) (0.002) (0.002) (0.01) ρ ( y) (0.10) (0.01) (0.01) (0.04) ρ ( y, nx) (0.18) (0.11) (0.08) (0.19) ρ ( y, c) (0.05) (0.01) (0.004) (0.07) ρ ( y, I) (0.09) (0.004) (0.002) (0.06) Notes: Theoretical moments are calculated from the model using the parameters reported in Tables 3 and 4. Columns (I), (II), and (III), use the estimated parameters from Table 4, Columns (I), (II), and (IV), respectively. Standard errors reported in parentheses are calculated from the parameter standard errors reported in Table 4 using the Delta method. Data moments are estimated with GMM. Note that estimation of the autocorrelation of the growth rate of income reduces the sample size by two quarters. We drop the first two quarters for the other moments to maintain a constant number of observations per moment. This truncation implies that the empirical moments above may differ slightly from those reported in Table 2.

16 Table A1: Data Sources Quarters Source Emerging Markets Argentina IFS Brazil NP Ecuador IFS Israel IFS Korea OECD Malaysia IFS Mexico OECD Peru IFS Philippines IFS Slovak Republic OECD South Africa IFS Thailand IFS Turkey OECD Developed Markets Australia OECD Austria OECD Belgium OECD Canada OECD Denmark OECD Finland OECD Netherlands OECD New Zealand OECD Norway OECD Portugal NP Spain OECD Sweden IFS Switzerland OECD See Appendix for discussion of data sources. NP stands for Neumeyer and Perri (2005). NP s Brazil data are from Instituto Brasileiro de Geografia e Estatística, Novo Sistema de Contas Nacionais (IBGE/SCN novo). NP s Portugal data are from OECD.

17 Figure 2: Random Walk Component of Solow Residual 3 Relative Variance of Random Walk Component Mexico Canada K Notes: This figure plots σ 2 τ/σ 2 sr, where σ 2 τ is estimated as T/(K(T-K)(T-K+1))Σ t=k (y t -y t-k -Kµ) 2, where y t is the log Solow residual at time t and µ is the sample average of the growth rate of the Solow residual. Each point corresponds to the choice of K depicted on the horizontal axis. σ 2 sr is the value of σ 2 τ when K=1 (i.e., the variance of the firstdifference of log Solow residuals). The solid line depicts Mexico and the dashed line depicts Canada.

18 Figure 3: Impulse Responses Ratio of Net Exports to GDP 0.5% 0.4% g shock z shock 0.3% 0.2% 0.1% 0.0% -0.1% % -0.3% -0.4% -0.5% -0.6% Ratio of Consumption to GDP 0.4% 0.2% g shock z shock 0.0% % -0.4% -0.6% -0.8% Ratio of Investment to GDP 1.2% 1.0% 0.8% g shock z shock 0.6% 0.4% 0.2% 0.0% -0.2% % -0.6% Notes: Impulse response of net exports, consumption, and investment relative to income in response to a one percent shock to ε g ( g shock, solid line) and a one percent shock to ε z ( z shock, dashed line). The values plotted are deviations from steady state. The parameters ρ g and ρ z are set to 0.01 and 0.95, respectively. All other parameters are as reported in Table 3.

19 Figure 4: Sensitivity of Moments to the Relative Volatility of Trend Shocks Relative Volatility of Net Exports, Consumption, and Investment σ_{inv}/σ_{y} σ_{c}/σ_{y} σ_{nx}/σ_{y} σ_{g}/σ_{z} Correlations ρ(y,y(-1)) ρ( y, y(-1)) ρ(nx,y) ρ(c,y) ρ(inv,y) σ_{g}/σ_{z} Notes: The top panel plots the standard deviation of filtered investment, consumption, and net exports relative to the standard deviation of filtered income as a function of alternative σ g /σ z. The bottom panel plots the autocorrelation of filtered income; the autocorrelation of unfiltered income growth; and the contemporaneous correlations of filtered net exports, consumption, and investment with filtered income; as functions of alternative σ g /σ z.

20 Figure 5: Autocovariance Function of Solow Residual: Data and Model Mexico 7.E-04 6.E-04 Autocovariance 5.E-04 4.E-04 3.E-04 2.E-04 1.E-04 Data Model -1.5*SE +1.5*SE 0.E+00-1.E E-04 Lag Canada 2.E-04 Autocovariance 1.E-04 5.E-05 0.E+00 Data Model -1.5*SE +1.5*SE E-05-1.E-04 Notes: This figure plots the autocovariance function of filtered log Solow residuals from the data (solid line) and the model (dashed line) for Mexico (top panel) and Canada (bottom panel). The model is generated from parameters reported in Column (IV) of Table 4. The dashed lines represent 1.5 standard error bounds. Lag

21 Figure 6: Sudden Stop Mexico Tequila Crisis ( ) Data NX/Y Model NX/Y Log Deviation from 1991Q Q1 1992Q1 1993Q1 1994Q1 1995Q1 1996Q1 1997Q Notes: Dashed line represents the empirical rartio of net exports to GDP in Mexico. The solid represents the ratio predicted by the model using the observed Solow residuals and the parameters reported in Column (IV) of Table 4. Both series are log deviations from 1991Q1.

22 Financial Frictions, Financial Accelerator, Real Economy Neumeyer and Perri (JME, 2005) Firms pay part of the factors of production before production takes place, creating a need for working capital. Preferences that generate a labor supply that is independent of consumption (GHH)

23 Neumeyer and Perri (2005) The need for working capital to finance the wage bill makes the demand for labor sensitive to the interest rate. Since firms have to borrow to pay for inputs, increases in the interest rate make their effective labor cost higher and reduce their labor demand for any given real wage. The impact of this fall in labor demand on equilibrium employment will depend on the nature of the labor supply. GHH preferences: labor supply is independent of shocks to interest rates. Declines in labor demand induce a fall in equilibrium employment At business cycle frequencies the capital stock is relatively stable, declines in equilibrium employment translate into output declines.

24 Neumeyer and Perri (2005) Data: Measures of expected real interest rate For emerging markets use dollar denominated bonds and subtract expected U.S. inflation. Government bonds Evidence that interest rates on private corporate and government bonds highly correlated.

25 Figure 4. Timeline Period t (t-1) + t - t + (t+1) - R(s t ) and A(s t ) are revealed. Labor and capital are hired. Firms issue bonds at rate R(s t-1 ). Final good is produced. Bonds issued in (t-1) + and t - mature. Households buy/issue bonds at rate R(s t ). Consumption and investment take place.

26 Neumeyer and Perri (2005) To transfer w(s t )l(s t ) to workers that earn w(s t ) goods per unit of time, firms need to set aside a fraction θ of the wage bill at t and a fraction (1 θ) at t +. The worker receives w(s t )l(s t ) at t +. An alternative model of the need for working capital would have been to assume some form of limited participation, that is, that a fraction of the workers are excluded from asset markets between t and t + but need resources to consume. Firms have to borrow θw(s t )l(s t ) (working capital) between t and t + at rate R(s t 1 ) The market for the services of capital is frictionless

27 Neumeyer and Perri (2005) Firms and Technology y(s t ) = A(s t )k(s t 1 ) α ((1 + γ) t l(s t )) 1 α γ: deterministic growth rate of labor-augmenting technological change Firm s problem y(s t ) w(s t )l(s t ) k(s t 1 )r(s t ) [R(s t 1 ) 1]θw(s t )l(s t ) FOC wrt l(s t ) w(s t ) [ 1 + θr(s t 1 1) ] = y l (s t )

28 Neumeyer and Perri (2005) Household s problem max β t π(s t )U(c(s t ), l(s t )) t=0 s t c(s t ) + x(s t ) + b(s t ) + κb(s t ) = w(s t )l(s t ) + r(s t )k(s t 1 ) + R(s t 1 )b(s t 1 ) κ(.) is a convex function. x(s t ) = k(s t ) (1 δ)k(s t 1 ) + Φ(k(s t 1 ), k(s t )) Country s net foreign asset position in period t is b(s t 1 ) θw(s t )l(s t )

29 Neumeyer and Perri (2005) Interest Rates Loans to the domestic economy are risky assets because there can be default on payments to foreigners. Real interest rates change as the perceived default risk changes Even if the default risk stays constant, interest rates can change because the preference of international investors for risky assets change over time. R(s t ) = R (s t )D(s t ) R (s t ): international rate for risky assets (which is not specific to any emerging economy) D(s t ): country spread over R (s t ) paid by borrowers in a particular economy

30 Neumeyer and Perri (2005) One asset: all agents (domestic or foreign, borrower or lender) face the same rate of interest R. Foreigners lend positive amounts to the domestic economy all along the equilibrium path. What determines D(s t )? Case 1: Exogenous to local productivity shocks Case 2: D(s t ) = ηe(a(s t+1 ))

31 Neumeyer and Perri (2005) D(s t ) = R(s t )/R (s t ) R(s t ) : 3-month real yield on Argentine dollar denominated sovereign bonds R (s t ): Redemption real yield on an index of non-investmentgrade U.S. domestic bonds. Figure 5 Estimate two independent first order autoregressive processes for R (s t ) and D(s t )

32 Neumeyer and Perri (2005) Calibration: u(c, l) = 1 [ c ψ(1 + γ) t l µ] 1 σ 1 σ ( ) κb(s t ) = y(s t b(s ) ) y(s t ) b Φ = φ ( k(s t ) k(s t 1 )(1 + γ) 2 k(st 1 ) k(s t 1 ) 2

33 Neumeyer and Perri (2005) Labor markets u l u c = w(s t ) = Log linearize with GHH preferences y l (s t ) [1 + θr(s t 1 1)] ˆl t+1 = 1 ˆR 1 ɛ s 1 t + α ˆk 1 ɛ d ɛ s 1 t ɛ d

34 Real wage L s Re wa L d (R 0 ) L d (R 1 ) l 1 l 0 Employment

35 Neumeyer and Perri (2005) Linearized first order condition for bonds ĉ t+1 ĉ = 1 ( 1 w ) ˆR t + w(ˆl t+1 ˆl t ) σ ν

36 Real wage L s (c(r 0 )) L s (c(r 1 )) L d (R 1 ) L d (R 0 ) l 0 l 1 Employment

37 Table 2. Baseline parameter values Shocks Name Process Parameter Values Productivity  s t = ρ A  s t 1 + ε A s t ρ A =0.95 σ(ε A )=Varies Intenational rate ˆR s t = ρ 1 ˆR s t 1 + ε R s t ρ 1 =0.81 σ(ε R )=0.63% Country risk (independent) ˆD s t = ρ 2 ˆD s t 1 + ε D s t ρ 2 =0.78 σ(ε D )=2.59% Country risk (induced) ˆD s t = ηe t ( s t+1 )+ε I s t η =1.04 σ(ε I )=1.7% Preference parameters Value Name Symbol GHH Cobb Douglas Discount factor β Utility curvature σ 5 5 Labor curvature v Labor weight ψ Consumption share µ technology parameters Name Symbol Value Technological progress growth γ 0.62% Capital exponent (production) α 0.38 Depreciation rate δ 4.4% % labor income paid in advance θ 1 Bond holding cost κ 10 5 Capital adjustment costs φ Varies See the notes in table 3 for the value of the parameter in different experiments

38 Table 3. Simulated and Actual Argentine Business Cycles %Standard Dev. of x % Standard Dev. %Standard Dev. of GDP GDP R NX TC INV HRS Argentine Data (0.36) (0.52) (0.11) (0.03) (0.13) (0.08) No country risk a) R shocks b) R and A shocks Independent country risk c) R and D shocks d) R, D and A shocks Induced country risk e) R and A shocks Correlation of GDP with R NX TC INV HRS Argentine Data (0.08) (0.02) (0.01) (0.01) (0.11) No country risk a) R shocks b) R and A shocks Independent country risk c) R and D shocks d) R, D and A shocks Induced country risk e) R and A shocks Correlation of R with NX TC INV HRS Argentine data (0.06) (0.07) (0.09) (0.12) No country risk a) R shocks b) R and A shocks Independent country risk c) R and D shocks d) R, D and A shocks Induced country risk e) R and A shocks Notes: See the notes in table 1A for a definition of the data series and for a description of how data statistics are computed. Model series are treated exactly as the data series. Statistics computed on the model series are averages across 500 simulations, each simulation of same length as the data sample. The capital adjustment cost parameter is φ is set to 8 (models a and b), 25.5 (models c and d), and 40 (model e). The standard deviation of productivity shocks σ(ε A ) is set to 1.98% (model b), 1.75% (model d), and 1.47% (model e).

39 Table 4. Sensitivity analysis Preferences GHH (ν =1.2) GHH (Baseline) GHH (v =4) CD (σ =5) CD (σ = 50) σ(y) σ(y DAT A ) Corr(Y,R) σ(y) σ(y DAT A ) Corr(Y,R) σ(y) σ(y DAT A ) Corr(Y,R) σ(y) σ(y DAT A ) Corr(Y,R) σ(y) σ(y DAT A ) Corr(Y,R) θ = % 55% % % % θ =1/ % 34% % % % 0.20 θ = % 21% % % % 0.69 Notes: The capital adjustment cost parameter φ is set to 25.5 in all experiments. All remaining parameters are set to their baseline values.

40 3 Figure 7. Impulse Responses to a Shock in International Interest Rates 2 Percent deviation from steady state Savings R NX GDP Employment Consumption -3 Investment Years after shock Note: Impulse responses are computed using baseline parameter values and a capital adjustment costs parameter (φ) equal to 25.1.

41 0.5 Figure 8. Correlation between GDP(t) and R(t+J) Model without Country Risk Model with Independent Country Risk Model with Induced Country Risk Data 0 Correlation J (Lead/Lag of R) Note: The dashed lines are two standard error bands around the cross-correlations in the data. Shocks to international interest rates and to TFP are present in all three models.

42 Figure 9. Business Cycles in Argentina 0.1 Data Model (shocks to international rates and independent country risk) Output Cycles in Argentina Deviations from trend of GDP Quarters